Paper Title: The Hausdorff dimension of planar elliptic measures via quasiconformal mappings.


	
	Planar elliptic measures via quasiconformal mappings]The Hausdorff dimension of planar elliptic measures via quasiconformal mappings
	
	Ignasi Guillén-Mola]Ignasi Guillén-Mola
	
	Ignasi Guillén-Mola,
		Departament de Matemàtiques, Universitat Autònoma de Barcelona.
	
	ignasi.guillen@uab.cat
	
	
	
	Supported by the PIF UAB 2020/21 from the Universitat Autònoma de Barcelona, and by Generalitat de Catalunya’s Agency for Management of University and Research Grants (AGAUR) (2021 FI_B 00637 and 2023 FI-3 00151). Also partially supported by the Spanish State Research Agency (AEI) project PID2021-125021NAI00 (Spain) and by MICINN (Spain) under the grant PID2020-114167GB-I00. The author thanks CERCA Programme/Generalitat de Catalunya for institutional support.


        

        28A12, 30C62, 31A15, 35J25. Secondary: 28A25, 28A78, 31B05, 35J08.
	
	
	§ ABSTRACT 
		In this paper we study the dimension of planar elliptic measures via the application of quasiconformal mappings. In fact, in our case studies, we find a quasiconformal mapping that relates the elliptic measure in a domain to the harmonic measure in its image domain, and we deduce bounds for the Hausdorff dimension of the elliptic measure by the known results on the harmonic side.
	
	
	[
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    August 5, 2024
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§ INTRODUCTION AND MAIN RESULTS

	
	Let Ω⊂^n+1, n≥ 1, be a domain (open and connected set), and let A ∈^(n+1)× (n+1) (Ω) be a uniformly elliptic matrix. That is, a real and measurable matrix A(·)=(a_ij (·))_1≤ i,j ≤ n+1 such that there exists λ≥ 1, the so-called ellipticity constant, with
	
    λ^-1|ξ|^2≤⟨ A(x) ξ, ξ⟩  and  ⟨ A(x) ξ, η⟩≤λ |ξ||η|,

	for all ξ,η∈^n+1 and a.e. x∈Ω. The uniform ellipticity condition implies that the matrix has bounded coefficients.
 
    We consider the second-order operator in divergence form L_A u  - (A∇ u), understood in the weak sense. We say that a function u∈ W^1,2_ (Ω) is L_A-harmonic in Ω if
	
    ∫_Ω A(y) ∇ u(y) ∇φ (y)   dy = 0   for all φ∈ C_c^∞ (Ω) .

	In this case, we write L_A u=0 in Ω for short.
	
	The elliptic (L_A-harmonic) measure in Ω⊂^n+1 (bounded if n≥ 2) with pole p∈Ω is the unique Radon probability measure ω_Ω,A^p, given by the Riesz representation theorem, such that
	
    u_f^L_A(p) = ∫_∂Ω f(ξ)   dω_Ω,A^p (ξ)   for every f∈ C(∂Ω),

	where u_f^L_A is the L_A-harmonic extension of f in Ω via Perron's method. If the set Ω, the matrix A, or the pole p∈Ω is clear from the context, we will omit them in ω_Ω, A^p. For a detailed construction of elliptic/harmonic measures, see <cit.>.
	
	We are interested in studying the Hausdorff dimension of elliptic measures arising from uniformly elliptic matrices, defined as
	
    _ω_Ω, A^p inf{_ F : ω_Ω, A^p (F^c) = 0}.

        With abuse of terminology, given a measure μ we will say that ω_Ω,A^p has σ-finite μ if there is a set F⊂∂Ω satisfying ω_Ω,A^p(F)=1 and with σ-finite μ. That is, F is of the form F=⋃_i≥ 1 F_i with μ(F_i)<∞.
 
	First, let us recall the definition of absolutely continuous measures. Given two measures μ and ν on X, we say that μ is absolutely continuous with respect to ν if
	
    ν (S) = 0 μ (S) =0   for all  S ⊂ X .

	We write μ≪ν in this case.
	
	Note that by the Harnack inequality of positive L_A-harmonic functions, given any two poles x,y ∈Ω we have
	
    ω_Ω, A^x ≪ω_Ω, A^y ≪ω_Ω, A^x.

	Therefore, we can omit the pole in the study of the dimension of the elliptic measure.
	
	Given ρ≥ 1 and C>0, we define
	
    φ_ρ,C (r) = r^2/ρ+1exp( C 2ρ/ρ+1√(log r^-ρlogloglog r^-ρ)).

	When ρ=1, this is precisely the gauge function that appears in <cit.>. We remark that ^φ_ρ,C≪^α for any α <2/ρ+1 because φ_ρ,C (t)/t^α→ 0 as t → 0. The precise definition of the h-Hausdorff measure ^h obtained via the Carathéodory's construction can be found in definition hausdorff measure. The measure ^1 will also be referred to as length.
	
	In the plane, the most precise results concerning the dimension of the harmonic measure ω = ω_Id are attributed to Makarov <cit.> and Wolff <cit.>. Makarov showed that there is a constant C>0 such that for any simply connected domain Ω, its harmonic measure ω_Ω satisfies ω_Ω≪^φ_1,C, with φ_1,C defined in makarov gauge function. Recall that we noted that ^φ_1,C≪^α for any α <1, which implies that _ω_Ω≥ 1. Actually, Makarov also proved in <cit.> that _ω = 1 for simply connected domains. On the other hand, the upper bound _ω_Ω≤ 1 for any planar domain Ω is due to Jones and Wolff in <cit.>. This was later improved by Wolff in <cit.>, who showed that ω_Ω has σ-finite length. For more geometric properties of the harmonic measure, see the introduction of <cit.> and the references therein.
	
	For future reference, we write here the precise statements mentioned above.
	
	
		There exists a universal constant C>0 such that ω_Ω≪^φ_1,C for every simply connected domain Ω⊂^2.
	
	
	
		For any domain Ω⊂^2, ω_Ω has σ-finite length.
	

        The behavior of the Hausdorff dimension of elliptic measures is different. For every ε>0, using quasiconformal mappings in <cit.>, Sweezy constructed a domain Ω⊂^2 and a uniformly elliptic matrix A such that _ω_Ω,A≥ 2-ε. Through a different technique, David and Mayboroda in <cit.> defined a continuous scalar function a≈ 1 on the complementary of the 1-dimensional four corner Cantor set C⊂^2 such that ω_^2∖ C, a Id is comparable to ^1|_C. Operators of the form L_aId=- a∇ that satisfy the above comparability are the so-called “good elliptic operators” in the terminology used in <cit.>. By the same strategy, for 1<d<log 4 /log 3, Perstneva in <cit.> constructed a “good elliptic operator” L_a_d Id on the (unbounded component of the) complementary of a d-dimensional Koch-type snowflake S_d ⊂^2 satisfying that ω_^2∖ S_d,a_d Id is comparable to ^d|_S_d.
        
        When adding regularity to the matrix, one could expect that elliptic measures behave as the harmonic measure. Recently, in collaboration with Prats and Tolsa in <cit.>, for Lipschitz matrices we proved the analog of Wolff's <Ref> for planar Reifenberg flat domains with small constant. More precisely, given a uniformly elliptic matrix A with Lipschitz coefficients, and a Reifenberg flat domain Ω⊂^2 with small constant depending on the ellipticity of A, ω_Ω,A has σ-finite length.
	
        In the same spirit as the previous theorems, our main results in <Ref> are the following.

	
		Let Ω⊂^2 be a domain, and A∈^2× 2 (^2) be a uniformly elliptic (with ellipticity constant λ≥ 1) and symmetric matrix with A=1. Then ω_Ω, A has σ-finite ^2λ/λ+1.
	
        Let φ_·, · be as in makarov gauge function. Then there exists C>0, the same constant as in Makarov's <Ref>, such that for any finitely connected domain Ω, ω_Ω, A≪^φ_λ,C.
	

        
            Assume, in addition to the hypotheses of <Ref>, that A∈ C^α_ (^2), 0<α < 1. Then ω_Ω, A has σ-finite length. If in addition the domain is finitely connected, then there is C>0, the same constant as in Makarov's <Ref>, such that ω_Ω, A≪^φ_1,C.
        
	
	For the proof of <Ref>, see proof of non-regularity case main thm,proof of holder-regularity case main thm respectively. We remark that the constant C>0 is the same as in Makarov's <Ref>, but the constant inside the function φ_ρ,C is C 2ρ/(ρ+1), i.e., it also depends on ρ.
	
	Note that the previous results are coherent with the results in the harmonic case of Makarov's <Ref> and Wolff's <Ref>. If A=Id, we can apply <Ref> using that the coefficients are Hölder, or we can just also use that the ellipticity constant is λ = 1 in <Ref> and hence 2/(λ+1)=2λ /(λ+1)=1. The consistency of using <Ref> with λ=1 is due to the sharpness of the distortion of the Hausdorff dimension and measures under quasiconformal mappings, see dimension distortion theorem, <Ref> and <Ref>.
 
    In the case of symmetric with determinant 1 matrices, the scheme of the proof only needs the uniform ellipticity condition on the matrix, and the key point is that the quasiconformal mapping which relates the elliptic measure of a domain with the harmonic measure in the image domain, is uniquely determined by the matrix. In particular, the more regular the coefficients of the matrix are, the more regular the quasiconformal mapping is. Because of this reason, from the proof of <Ref> we obtain the Hölder case in <Ref> and the following cases:

    
        Under the assumptions of <Ref>: 
        
            
  * If A(z) = A((z)) (or A(z) = A((z))), A∈ W_^1,p (^2) with p>2λ^2/λ^2+1 or A∈ DMO_p (see <Ref>) with p>1, then ω_Ω, A has σ-finite length.
            
  * If A∈_log (see <Ref>), then ω_Ω, A has σ-finite ^t/log1/t.
            
  * If A∈ (see <Ref>), then _ω_Ω,A≤ 1.
        
        If in addition the domain is finitely connected, the following hold:
        
            
  * If A(z) = A((z)) (or A(z) = A((z))), or A∈ DMO_1, then ω_Ω, A≪^φ_1,C.
            
  * If A∈_log, then ω_Ω, A≪^(φ_1,C) ∘ (tlog1/t).
            
  * If A∈, then _ω_Ω,A=1.
        
        Here C>0 is the same constant as in Makarov's <Ref>.
    

    We remark that <Ref> hold as long as the determinant is constant. The proofs of all cases can be found on proof of non-regularity case main thm (Hölder case), proof of main thm only one variable dependence (A(z)=A((z)) case), proof of vmo matrix everywhere ( case), proof of main thm with sobolev (Sobolev case), proof of dini mean oscillation elliptic measure (DMO case) and proof of square mean oscillation elliptic measure (_log case). Also, as noted in <Ref>, for 1≤ p < ∞ there holds
    
    Hölder⊂ Dini⊂ DMO_p ⊂ DMO_1 ⊂_log⊂,

    where Dini consists of functions satisfying the pointwise Dini condition <ref>.
	
	In the same way, as a corollary of <cit.> and by the same approach of <Ref>, using the fact that the quasiconformal mapping and its inverse are C^1 when the matrix is DMO_p with p>λ, we derive a dimension drop in the following situation. The details are left to the reader. 
	
	
		Let A∈^2× 2 (^2) be a uniformly elliptic (with ellipticity constant λ≥ 1) and symmetric matrix with constant A. Let E⊂^2 be a compact subset contained in a C^1 curve which is (s,C_0)-AD regular for some s>0, that is,
		
    C_0^-1 r^s ≤^s (E ∩ B(x,r)) ≤ C_0 r^s   for all  x∈ E  and  0<r≤(E).

		Let Ω = ^2 ∖ E, and assume also that A∈ DMO_p with p>λ. We have that:
		
			
  * If s∈[1/2,1), then _ω_Ω, A < s.
			
  * There exists ε = ε_C_0 >0 small enough such that if s∈[1/2 - ε,1), then _ω_Ω, A < s.
		
	
	
	In <Ref>, we obtain a result analogous to <Ref> for general uniformly elliptic matrices in CDC domains. Roughly speaking, a domain satisfies the CDC condition if its complementary is uniformly “fat” with respect to the scale; see <Ref>.
 
        
		Let Ω⊂^2 be a CDC domain, A ∈^2× 2(Ω) be a uniformly elliptic matrix with ellipticity constant λ≥ 1, and
        
    K_λ = λ   if  A  is symmetric,
    λ + √(λ^2 - 1)   otherwise.

        Then ω_Ω,A has σ-finite ^2K_λ /(K_λ +1).
		
		Let φ_·,· be as in makarov gauge function. Then there is C>0, the same constant as in Makarov's <Ref>, such that for any finitely connected domain Ω,[In particular it is CDC.] its elliptic measure satisfies ω_Ω, A≪^φ_K_λ, C.
	
	
	Contrary to what we had in the symmetric and determinant 1 case, for general uniformly elliptic matrices we cannot find a quasiconformal mapping such that the elliptic measure is the push-forward of the harmonic measure in some other domain. This will not allow us to relate, for any domain, its elliptic measure with the harmonic measure in some other domain. However, if the domain is CDC, we find a quasiconformal mapping that sends the elliptic Green function to the harmonic Green function in the image domain, which allows us to obtain the comparability between the elliptic measure and the harmonic measure in the image domain. Using this, the proof follows the same scheme as in the previous case in <Ref>.
	
    We remark that, in the above situation, the quasiconformal mapping depends not only on the matrix but also on the Green function. Therefore, we cannot easily deduce better regularity of the quasiconformal mapping even if the matrix has additional regularity. However, if the matrix is continuous up to the boundary, then via a localization argument we deduce the following result.
	
    
        Let Ω⊂^2 be a CDC domain and A∈^2× 2 (Ω) be a uniformly elliptic matrix with continuous coefficients up to ∂Ω, that is, A∈ C^0 (Ω). Then _ω_Ω, A≤ 1. If Ω is finitely connected, then _ω_Ω, A = 1.
    

    Recall that in <cit.>, Prats, Tolsa, and the author obtained the σ-finite length of ω_Ω,A under the assumptions that A is Lipschitz and Ω⊂^2 is Reifenberg flat. In contrast, the result above shows that the upper bound ω_Ω,A≤ 1 holds under much weaker assumptions: Ω is CDC and A is continuous up to the boundary.

    We note that <Ref> does not contradict the result of Perstneva in <cit.>. In fact, the scalar function a_d ≈ 1 that she defined is continuous in the CDC domain Ω, the (unbounded component of the) complementary of a d-dimensional Koch-type snowflake, while in <Ref> we require the matrix to be continuous up to the boundary.

    

    Acknowledgments.
        The author would like to thank Martí Prats and Xavier Tolsa for their guidance and advice.

    

§ PRELIMINARIES


    

 §.§ Notation


    
	
  * We use c,C≥ 1 to denote constants that may depend only on the dimension and the constants appearing in the hypotheses of the results, and whose values may change at each occurrence.
	
	
  * We write a≲ b if there exists a constant C≥ 1 such that a≤ Cb, and a≈ b if C^-1 b ≤ a ≤ C b.
	
	
  * If we want to stress the dependence of the constant on a parameter η, we write a≲_η b or a≈_η b meaning that C=C(η)=C_η.
	
	
  * The ambient space is ^2. However, some auxiliary results will be stated in general, i.e., in ^n+1 for n≥ 1.
	
	
  * The diameter of a set E⊂^n+1 is denoted by E sup_x,y∈ E |x-y|.
	
	
  * We denote by B_r (x) or B(x,r) the open ball with center x and radius r, i.e., B_r (x)=B(x,r)={y∈^n+1 : |y-x|<r}. We denote B_r  B_r (0).

        
  * Given a ball B, we denote by r_B or r(B) its radius, and by c_B or c(B) its center.

        
  * We denote by Q_r(x) or Q(x,r) the open cube with center x and side length 2s, i.e., Q_r(x) = Q(x,r) = {y∈^n+1 : |y_i-x_i|<r for all 1≤ i ≤ n+1}.

        
  * Given a cube Q, we denote by ℓ(Q) its side length, and by c_Q or c(Q) its center. That is, Q=Q(c_Q, ℓ(Q)/2).
	
	
  * We say that a matrix A is Hölder continuous with exponent α∈ (0,1] in a set U, or briefly C^0,α (U), if its coefficients are Hölder continuous with exponent α. That is, there exists a constant C_α >0 (called the Hölder seminorm) such that
	
    |a_i j(x)-a_i j(y)| ≤ C_α |x-y|^α  for all  x, y ∈ U  and  1 ≤ i, j ≤ n+1 .

	For shortness we write C^α instead of C^0,α if α∈ (0,1), and when α = 1 we say “Lipschitz continuous”. In this case we write C_L instead of C_1, i.e.,
	
    |a_i j(x)-a_i j(y)| ≤ C_L |x-y|   for all  x, y ∈ U  and  1 ≤ i, j ≤ n+1 .

	
	
  * We say that a function f is κ-Lipschitz in U if |f(x)-f(y)| ≤κ |x-y| for all x,y∈ U.
	
	
  * We denote the characteristic function of a set E by _E.
	
	
  * Given t>0 and a set E⊂^n+1, we write U_t (E) {x∈^n+1 :  (x,E) < t } for the t-neighborhood E.

        
  * Given a space of function X, we say that a matrix A=(a_ij)_i,j=1^n+1 belongs to X, A∈ X for short, if a_ij∈ X for all 1≤ i,j≤ n+1.


        
    

 §.§ Reduction to bounded Wiener regular domains

	
	We present the lemmas to see that for the study of the dimension of planar elliptic measures, we can reduce to bounded Wiener regular domains. For the proof of the results in <Ref> for bounded Wiener regular domains, refer to <Ref>.

        
            Let Ω⊂^n+1, n≥ 1, be a domain and A be a real and uniformly elliptic matrix. We say that ξ∈∂Ω is a Wiener regular point for L_A if for every f ∈ C(∂Ω), the L_A-harmonic extension u_f^L_A of f in Ω via Perron's method satisfies lim_Ω∋ x →ξ u_f^L_A(x) = f(ξ). We say that Ω is Wiener regular for L_A if every point ξ∈∂Ω is a Wiener regular point for L_A.
        
        

        
            In fact, the characterization of Wiener regular points does not depend on the operator L_A, see <cit.>. Therefore, we say that a point ξ∈Ω is Wiener regular if and only if it is Wiener regular for some (and hence for all) L_A.
        
        
        In particular, using the Wiener regularity characterization for the Laplace operator, we directly note that finitely connected domains Ω⊂^2 are Wiener regular, and in order to study the size of Wiener regular points (for any L_A) we can use the classical potential theory for the Laplacian operator. More precisely, from this and the same proof of <cit.>, we obtain the following.

        
            Let A be a real and uniformly elliptic matrix. To prove that for any domain Ω⊂^2, either _ω_Ω,A≤β (with 0<β≤ 2) or ω_Ω,A has σ-finite (gauge) ^h, it suffices to prove it for Wiener regular domains.
            
                Indeed, in the proof of <cit.> one can replace with no modification harmonic measure by elliptic measure, the set G (given by assuming the result for Wiener regular domains) is constructed so that ω_Ω,A (G)=1 and either _ G≤β or ^h (G) is σ-finite respectively, and the set F satisfies ^γ (F)=0 for all 0<γ≤ 2, see also <cit.>. In particular, the set G∪ F has full elliptic measure ω_Ω,A and either _ (G∪ F)≤β or ^h (F∪ G) is σ-finite respectively, as claimed.
            
        
	
	Before presenting the lemmas for reducing to bounded domains, we first introduce the capacity.
	
	
		Let K be a compact subset of an open set Ω. Its capacity is
		
    (K,Ω) = inf{∫_Ω |∇ u|^2   dx : u∈ C^∞_c (Ω)  and  u≥ 1  on  K  }.

	
	
	Let Ω⊂^n+1, with n ≥ 1, be a (bounded if n≥ 2) Wiener regular domain and B be a ball centered at ∂Ω. By the maximum principle <cit.> we have 
	
    ω^z_Ω∩ B,A (E) ≤ω^z_Ω,A (E)   for any  E⊂∂Ω∩ B  and  z∈Ω∩ B .

	The converse inequality may fail. However, the following weaker relation for (bounded if n≥ 2) Wiener regular domains holds, see <cit.>. If (B ∩∂Ω, 4B)≠ 0, for a Borel set E⊂∂Ω∩ B and z_E ∈∂ 1.5 B such that ω^z_E_Ω,A (E) = max_z∈∂ 1.5 Bω^z_Ω,A (E), then
    
    ω^z_E_Ω,A (E) ≲ (2B, 4B)/ (B ∩∂Ω, 4B)ω^z_E_Ω∩ 4B,A (E),

    where the constant involved depends on the ellipticity constant of the matrix A and the dimension. We remark that (B ∩∂Ω, 4B)=0 would imply ω_D,A (B∩∂Ω)=0 for any domain D with B∩∂Ω⊂∂ D, see <cit.>.
	
	The following two lemmas allow us to reduce to the case of bounded domains.
	
	
		Let Ω⊂^2 be a (possibly unbounded) Wiener regular domain and A be a real uniformly elliptic matrix. Let {B_i}_i be a disjoint collection of balls centered at ∂Ω with ω_Ω,A(∂Ω∖⋃_i B_i )=0, and let F_i ⊂∂ (Ω∩ 4B_i) with ω_Ω∩ 4 B_i ,A (F_i)=1. Then F∂Ω∩⋃_i F_i satisfies ω_Ω,A (F)=1. The same also holds for bounded Wiener regular domains Ω⊂^n+1 when n≥ 2.
	
	
	
		Let Ω⊂^2 be a finitely connected domain, A be a real and uniformly elliptic matrix and F⊂∂Ω with ω_Ω,A (F)=1. For any small enough ball B centered at ∂Ω, each connected component of Ω∩ B is simply connected, and ω_Ω∩ B,A (∂Ω∖ F) = 0. Equivalently, (ω_Ω∩ B,A)_|∂Ω≪ω_Ω,A. In particular ω_Ω,A≥ω_Ω∩ B,A.
		
			Since Ω is finitely connected, each connected component of Ω∩ B is simply connected. By easy inequality localized elliptic measure we have ω_Ω∩ B,A (∂Ω∖ F) ≤ω_Ω,A (∂Ω∖ F) = 0.
		
	
	
	
		By <Ref> and <Ref> it suffices to prove our results for bounded Wiener regular domains. By translation, we can also assume Ω⊂ B_1/4 (0) if necessary.
	
	
	
		Assuming Ω⊂ B_1/4 (0), it may seem natural to assume that A=Id in B_1(0)^c by replacing A by the uniformly elliptic (with the same ellipticity constant) matrix
		
    Aφ A + (1-φ) Id,

		for some function φ such that 0≤φ≤ 1, φ = 1 in B_1/2 (0) and φ=0 in B_1(0)^c. If φ is smooth, then A often has the same regularity as A. Note that A is symmetric wherever A is symmetric. However, A may no longer have determinant 1 even when A has determinant 1. If necessary, if we wanted to preserve also the determinant 1 property, we could replace the matrix A by the uniformly elliptic (also with the same ellipticity constant) matrix A defined in <Ref>.
  
        Instead of modifying the matrix, in most of the cases we will modify the Beltrami coefficient, see the proof of the non-regularity case of <Ref> in proof of non-regularity case main thm, the proof of the  case of <Ref> in proof of vmo matrix everywhere, and the proof of <Ref> in proof of main thm cdc domains, for example.
	
	
	We also want to note that from easy inequality localized elliptic measure and hard inequality localized elliptic measure, the dimension of elliptic measures only depends on how the matrix is around the boundary of the domain. We prove this in the following result.
	
	
		Let Ω⊂^n+1, n≥ 1, be a (possibly unbounded if n=1) Wiener regular domain, ε>0 and let A,B be two real uniformly elliptic matrices with A=B in U_ε (∂Ω){x : (x,∂Ω) < ε}. Then ω_Ω, A (F) = 1 if and only if ω_Ω, B (F)=1. That is, ω_Ω, A≪ω_Ω, B≪ω_Ω, A.
		
			Let ω_·ω_·, A and ω_·ω_·, B. We prove ω_Ω (F) =1 ⇒ω_Ω(F)=1. The converse direction follows by symmetry.
			
			Let F⊂∂Ω with ω_Ω (F)=1. Consider the pairwise disjoint countable subfamily {B_i/5}_i ⊂{B_ε/100 (ξ)/5}_ξ∈∂Ω such that ∂Ω⊂⋃_i B_i, by a 5r-covering theorem. Since all the balls have the same radius and {B_i /5}_i is pairwise disjoint, it also holds that {B_i}_i has finite overlapping with constant depending only on the dimension.
   
            We have
		
    ω_Ω (F^c) ≤∑_i ω_Ω (F^c ∩ B_i).

		We claim that each term on the right-hand side is zero, and in particular ω_Ω (F)=1. Indeed, for each i fix a pole p_i ∈Ω∩ 4B_i. Since both matrices A and B coincide in Ω∩ 4B_i we have ω^p_i_Ω∩ 4B_i (·) = ω^p_i_Ω∩ 4B_i (·), and using easy inequality localized elliptic measure and ω^z_i_Ω (F)=1 we get
			
    ω^p_i_Ω∩ 4B_i (F^c ∩ B_i)
                    =ω^p_i_Ω∩ 4B_i (F^c ∩ B_i)
    			≤ω^p_i_Ω (F^c ∩ B_i)
    			≤ω^p_i_Ω (F^c) = 0.

        Arguing as in the proof of <cit.>, where hard inequality localized elliptic measure is used, ω^p_i_Ω∩ 4B_i (F^c ∩ B_i)=0 implies ω_Ω (F^c ∩ B_i) =0, as we claimed.
		
	
	
	

 §.§ Small perturbation of the constant case

	
	In this subsection, we study the dimension of elliptic measures arising from a uniformly elliptic matrix whose coefficients are close in the L^∞ norm to a uniformly elliptic constant matrix. The main result of this section and its proof will be used in the proof of the Dini mean oscillation case in <Ref> and <Ref>.
	
	For λ≥ 1 we denote
	
    _(λ)    = sup{_ω_Ω, A : Ω⊂^n+1 and  A  has ellipticity constant λ},
    _^sc (λ)    = inf{_ω_Ω, A : Ω⊂^2 is simply connected and  A  has ellipticity constant λ}.

	Using this notation, in the harmonic case, i.e., λ =1, in the plane we have _ (1)≤ 1 and _^sc(1)≥ 1 by Wolff's <Ref> and Makarov's <Ref> respectively.
	
	In the result below, we estimate the dimension of the elliptic measure in terms of how close the matrix A is to a uniformly elliptic constant matrix.
	
	
		Let Ω⊂^n+1 be a domain and C be a constant uniformly elliptic matrix with ellipticity constant λ≥ 1. Let A be a (not necessarily constant) matrix and
		
    εmax_i,j∈{1,…,n+1}a_ij (·)- c_ij_L^∞ (Ω).

		If ε is small enough (depending on λ and n) then A is uniformly elliptic in Ω and _ω_Ω,A≤_ ((1-(n+1)ελ)^-1). If Ω⊂^2 is simply connected, then _ω_Ω,A≥_^sc ((1-2ελ)^-1).
	
	
	Note that when the matrix A is constant, the result agrees with the dimension of the harmonic measure. Before the proof, we need the following lemma.
	
	
		Let Ω⊂^n+1 be a domain and C be a uniformly elliptic and constant matrix. Then, in the weak form, for any u∈ W^1,2_ (Ω) we have
		
    C∇ u =  C_s ∇ u,

		where C_s denotes the symmetric part of the matrix C, that is C_s = (C+C^T)/2.
		
			For u ∈ C^∞ (Ω), and as C is constant, we have
			
    C∇ u = ∑_i,j c_ij∂_ij u = ∑_ijc_ij+c_ji/2∂_ij u =  C_s ∇ u .

			
			Let us check the weak form of this identity for u∈ W^1,2_ (Ω). Given φ∈ C_c^∞(Ω), let U be an open set such that φ⊂ U ⊂U⊂Ω, and we have u ∈ W^1,2 (U). As C^∞(U) is dense in W^1,2 (U), take {u_k}_k≥ 1⊂ C^∞ (U) so that lim_k→∞u_k - u_W^1,2(U) =0. As C∇ u_k =  C_s ∇ u_k in the weak form for each k≥ 1, we get
			
    ∫_Ω C∇ u ∇φ 
    				=   ∫_Ω C∇ u_k ∇φ + ∫_Ω C∇ (u-u_k) ∇φ
    
    				=   ∫_Ω C_s ∇ u_k ∇φ + ∫_Ω C∇ (u-u_k) ∇φ
    
    				=   ∫_Ω C_s ∇ u ∇φ + ∫_Ω C_s ∇ (u_k-u) ∇φ + ∫_Ω C∇ (u-u_k) ∇φ.

			The lemma follows since ∫_Ω |∇ (u-u_k)| |∇φ| ≲∇ (u-u_k)_L^2 (U)→ 0 as k →∞.
		
	
	
	Using the previous lemma, we prove <Ref>.
	
		If ε is small enough depending on λ and n then A is uniformly elliptic in Ω.
		
		By <Ref>, in the weak form we have
		
    A∇ =  C∇  +  (A-C)∇ 
    			=  C_s∇  +  (A-C)∇
    			=  (C_s+ A-C)∇.

		Note that the matrix C_s+ A-C is uniformly elliptic as it has bounded coefficients and ⟨ B ξ, ξ⟩ = ⟨ B_s ξ, ξ⟩ for any (not necessarily constant) matrix B. Then, taking the linear change of variables given by S^T, where S is the square root of C_s (that is, S=√(C_s), the matrix satisfying S^T S=C_s) and
		
    M (z)     (S^T)^-1( C_s + A(S^T z)- C ) S^-1
    
                =   Id + (S^T)^-1( A(S^T z)- C ) S^-1    for z∈ (S^T)^-1 (Ω),

		by <Ref> below we have
		
    ω_Ω, A^p (E) 
    			= ω_(S^T)^-1Ω, M^(S^T)^-1 p ((S^T)^-1 E),   for any  p ∈Ω and E⊂∂Ω.

		
		We claim that the ellipticity constant of M in (S^T)^-1Ω is at most (1-(n+1)ελ)^-1. Indeed, for ξ,η∈^n+1 and x∈Ω, by the Cauchy-Schwartz inequality and the well-known fact |Bv|≤ (n+1)δ |v| if |b_ij|≤δ for all 1≤ i,j ≤ n+1, we have
		
    ⟨{(S^T)^-1( A(x)- C ) S^-1}ξ,η⟩≤ |( A(x)- C ) S^-1ξ||S^-1η|
                ≤ (n+1)ε |S^-1ξ||S^-1η|.

            Note that for any vector v there holds |Sv|^2 = ⟨ Sv,Sv⟩ = ⟨ S^T S v,v⟩ = ⟨ C_s v, v ⟩≥λ^-1 |v|^2. All in all,
            
    ⟨{(S^T)^-1( A(x)- C ) S^-1}ξ,η⟩≤ (n+1)ελ |ξ||η|,

		which implies
		
    ⟨ Mξ, η⟩≤ (1+(n+1)ελ)|ξ||η| and ⟨ Mξ,ξ⟩≥ (1-(n+1)ελ)|ξ|^2.

		The claim follows as 1/(1-(n+1)ελ)≥ 1+(n+1)ελ.
		
		Since the linear mapping given by S^T is bi-Lipschitz and the ellipticity constant of M is at most (1-(n+1)ελ)^-1, the lemma follows from relation elliptic measure after square root change of variables and the definition of _ and _^sc.
	
	
	Note that if the constant matrix C is symmetric, i.e., C=C_s, then matrix after change of constant sym change of variables reads as
	
    M (z)  (S^T)^-1 A(S^T z) S^-1 for z∈ (S^T)^-1Ω.

	
	As we mentioned in the introduction of this section, we will apply <Ref> and adapt its proof in some proofs below. More precisely, we will compare the matrix A with the frozen coefficient matrix C=A(ξ), for some point ξ∈∂Ω.
	
	
		Taking C=A(ξ) (for some point ξ∈∂Ω) when the matrix A is symmetric with constant determinant, we have that the matrix M in matrix after change of constant sym change of variables - sym case is symmetric and has determinant 1.
	
	
	

§ QUASICONFORMAL MAPPINGS

	
	In this section, we provide the definition and main properties of quasiconformal mappings. This is based on the monograph <cit.>.
	
	
		An orientation-preserving homeomorphism ϕ : Ω→Ω^' between planar domains is called K-quasiconformal, 1≤ K <∞, if f∈ W^1,2_ (Ω) and the directional derivatives satisfy
		
    max_α∈[0,2π] |∂_α f| ≤ K min_α∈[0,2π] |∂_α f|  a.e. in Ω .

	
	
	
		In particular, a mapping is 1-quasiconformal if and only if it is conformal.
	
	
	The condition in quasiconformal condition 1 is equivalent to
	
    |∂̅f| ≤ k |∂ f|   a.e. in Ω,  for  k K-1/K+1,

	where ∂∂_z = 1/2 (∂_x - i∂_y) and ∂̅∂_z̅ = 1/2 (∂_x + i∂_y) are the Wirtinger derivatives.
	
	A fundamental property of onto K-quasiconformal mappings f:Ω→Ω^' is that its inverse function f^-1 : Ω^'→Ω is also K-quasiconformal, see <cit.>. Also, global K-quasiconformal mappings f:→ are quasisymmetric, that is, there exists an increasing homeomorphism η_K : [0,∞) → [0,∞), only depending on K, such that for each triple z_0, z_1, z_2 ∈ we have
	
    |f(z_0)-f(z_1)|/|f(z_0)-f(z_2)|≤η_K (|z_0 - z_1|/|z_0 - z_2|) ,

	see <cit.>. In this case, we say that f is η_K-quasisymmetric. For a local equivalence between quasiconformal and quasisymmetric mappings, see <cit.>
	
	Another important property is that quasiconformal mappings are locally Hölder continuous, see <cit.>.
	
	
		Every K-quasiconformal mapping f:Ω→Ω^' is locally 1/K-Hölder continuous. More precisely, if B⊂ 2B ⊂Ω, then
		
    |f(z)-f(w)| ≤ C_K  (f (B)) |z-w|^1/K/ (B)^1/K,    z,w∈ B,

		where the constant C_K depends only on K.
	
	
	Quasiconformal mappings are related to solutions of the Beltrami equation, see <cit.>. Given a homeomorphic W^1,2_ (Ω)-mapping f:Ω→Ω^', write μ (z) = ∂̅f(z) / ∂ f(z) when ∂ f(z)≠0 and μ(z)=0 otherwise. With this, we have the equivalence
	
    f  is  K-quasiconformal∂̅f = μ∂ f  a.e. in Ω,

	where μ, called the Beltrami coefficient of f, is a bounded measurable function satisfying μ_∞≤ (K-1)/(K+1) = k < 1.
	
	
		With this relation, given μ as above, we say that a function f ∈ W^1,2_ (Ω) is μ-quasiconformal if f solves the Beltrami equation ∂̅f = μ∂ f a.e. in Ω.
	
	
	The measurable Riemann mapping theorem ensures the existence (and uniqueness up to normalization) of solutions to a Beltrami equation, see <cit.>.
	
	
		Let |μ| ≤ k < 1 for almost every z∈. Then there is a solution f:→ to the Beltrami equation
		
    ∂̅f = μ∂ f  a.e. in ,

		which is a K-quasiconformal homeomorphism normalized by the three conditions f(0)=0, f(1)=1 and f(∞)=∞. Furthermore, the normalized solution f is unique.
	
	
	For a compactly supported Beltrami coefficient μ, we call principal solution to the function f∈ W^1,2_(Ω) with ∂̅f=μ∂ f normalized by the condition f(z)=z+(1/z) near infinity.
	
	

 §.§ Quasiconformal mappings and A-harmonic functions

	
	Given a uniformly elliptic matrix A∈^2× 2 and a global quasiconformal mapping f of , we define the uniformly elliptic matrix A_f as
	
    A_f (x)  ( Df(x)) Df(x)^-1 A(f(x)) (Df(x)^-1)^T.

	We remark that the matrix A_f is uniformly elliptic by the quasiconformal condition of f, see <cit.>, and in particular, planar uniformly elliptic operators are invariant under quasiconformal homeomorphisms. Moreover, these matrices satisfy
	
    (A_f)_f^-1 = A = (A_f^-1)_f.

	
	With the notation in matrix pushforward, we observe the behavior of L_A-harmonic functions and elliptic measures under quasiconformal changes of variables in the following two lemmas.
	
	
		Let Ω⊂^2 be an open set, A∈^2× 2 be a uniformly elliptic matrix, and f be a quasiconformal homeomorphism of . Then u is L_A-harmonic in Ω if and only if u∘ f is L_A_f-harmonic in f^-1(Ω).
	
	
	
		Let Ω⊂^2 be a domain, A∈^2× 2 be a uniformly elliptic matrix and f be a quasiconformal homeomorphism of . For any x∈Ω and E⊂∂Ω, we have
		
    ω_Ω,A^x (E)=ω_f^-1(Ω),A_f^f^-1(x) (f^-1(E)).

	
	
	For a detailed proof for bi-Lipschitz mappings see <cit.>. For a more general version of these results see <cit.> respectively.
	
	

 §.§ Distortion under quasiconformal mappings

	
	First, we recall Carathéodory's construction (see <cit.>) to define (gauge) Hausdorff measures. Let ℬ denote the family of all closed balls contained in , and let h:ℬ→ [0,∞) be any function defined in ℬ with h(B(x,r))→ 0 as r→ 0, for all x∈. Given 0<δ≤∞ and a set E⊂^n+1 define
	
    _δ^h (E) inf{∑_i h (B_i) : E ⊂⋃_i B_i,  with B_i∈ℬ and  r(B_i) ≤δ} .

	Define the h-Hausdorff measure ^h as
	
    ^h (E) lim_δ→ 0^h_δ (E) =sup_δ >0^h_δ (E) ,

	as ^h_δ (E) increases as δ decreases.
	The δ = ∞ case _∞^h is called h-Hausdorff content. When h(B(x,r))=r^α we simply write ^α (resp. ^α_∞) to denote the α-Hausdorff measure (resp. content).
	
	It is clear that _∞^h (E) ≤^h (E). Despite the converse inequality does not hold, _∞^h (E)=0 indeed implies ^h (E)=0. Although _∞^h is not measure, we write ^h ≪_∞^h ≪^h.
	
	Now we turn our attention to the distortion of the Hausdorff dimension and measure under K-quasiconformal mappings. Astala in <cit.> (see also <cit.>) proved the area distortion theorem, that is, there is a constant C_K depending only on K≥ 1, such that for any K-quasiconformal mapping f:→, for any disk B⊂ and for any subset E⊂ B, we have
	
    1/C_K(|E|/|B|)^K ≤|f(E)|/|f(B)|≤ C_K (|E|/|B|)^1/K .

	As a corollary, see <cit.> and also <cit.>, the area distortion theorem governs the distortion of Hausdorff dimension
	
    1/K( 1/_ (E) - 1/2)
    		≤1/_ (f(E)) - 1/2≤ K ( 1/_ (E) - 1/2) .

	
	In the preprint <cit.> (see also <cit.> for the published version), Tolsa generalized the previous two results by obtaining the control of the Hausdorff measure.
	
	
		Let 0<t<2 and denote t^' = 2Kt/2+(K-1)t. Let f  :→ be K-quasiconformal. For any ball B and any compact set E⊂ B, we have
		
    ^t^' (f(E))/ (f(B))^t^'≤ C_K ( ^t (E)/ (B)^t)^t^'/Kt ,

		where the constant C_K depends only on K. In particular, if ^t (E) is finite (resp. σ-finite), then also ^t^' (f(E)) is finite (resp. σ-finite).
	
	
	The case t=t^' = 2 is precisely the area distortion theorem in area distortion theorem. The σ-finiteness in the particular case t^'=1 had been solved in <cit.>. The same control in distortion of hausdorff measure with Hausdorff content _∞ instead of Hausdorff measures was proved in <cit.>.
	
	

  §.§.§ Distortion of gauge Hausdorff measures

	
	In this section, we will start by presenting a technical lemma used in the proof in <cit.> of <Ref> above, and later we will deduce the precise distortion of the gauge Hausdorff measure ^φ under principal quasiconformal mappings, where φ is the Makarov gauge function in makarov gauge function
	
	Let us introduce the notation used in <Ref> below, see <cit.>. Let ε : ℬ→ [0,∞) be any function defined on ℬ, the family of all closed balls in . We set ε (x,r) ε (B(x,r)). Given an arbitrary bounded set S⊂, let ℬ (S) denote the family of balls with minimal diameter containing S, and set ε (S) inf_B ∈ℬ (S)ε (B).
	
	As in <cit.>, the families 𝒢_1 and 𝒢_2 are defined as:
	
		
  * We say that a function ε : ℬ→ [0,∞) belongs to 𝒢_1 if there is a constant C_1 such that
		
    C_1^-1ε (x,r) ≤ε (y,s) ≤ C_1 ε (x,r)

		whenever |x-y|≤ 2r and r/2 ≤ s ≤ 2r.
		
		
  * For each fixed 0<t<2, a function ε = ε (x,r) belongs to the family 𝒢_2 = 𝒢_2 (t) if there is a constant C_2 such that
		
    ∑_k≥ 0 2^-k (2-t)ε (x,2^k r) ≤ C_2 ε (x,r) .

	
	
	
		Let 0<t<2 and t^' = 2Kt/2+(K-1)t. Let ε∈𝒢_1 and set h(x,r)=r^t^'ε (x,r). Suppose that for any principal K-quasiconformal mapping ψ :→ the function (ε∘ψ)^d belongs to 𝒢_2 for any t^'/Kt≤ d ≤ 1. Let E ⊂ B(0,1/2) be compact and ϕ : → a principal K-quasiconformal mapping, conformal on ∖𝔻. Set
		
    ε(x,r) = ε (ϕ (B(x,r)))^Kt/t^',   h (x,r) = r^t ε(x,r).

		Then we have
		
    _∞^h (ϕ (E)) ≤ C_K _∞^h (E)^t^'/Kt .

	
	
	Here we state the distortion of the Makarov gauge function in <Ref> under quasiconformal mappings. The proof will be a routine computation to check the conditions of the previous lemma for this particular case. For this reason, its proof is located at the end of the subsection.
	
	
		Let ϕ : → be a principal K-quasiconformal mapping and conformal on ∖𝔻, and E⊂ B(0,1/2) be compact. For s>e set[Note that 𝕃og (s) is not defined when 0<s≤ e, and 𝕃og (s)≥ 0 whenever s≥ e^e.]
		
    𝕃og (s) log s logloglog s,

		and for s_0 > e^e and s>0, set
        
    𝕃og_s_0 (s) 𝕃og (s_0),   if  0<s≤ s_0,
    𝕃og (s),   if  s>s_0.

        Then there exists an small enough 0<r_0 <e^-e and a constant C_K,ϕ such that if for r>0 we define
		
    φ (r) = r exp(C√(𝕃og_r_0 (r^-1)))

		and
		
    φ(r) = r^2/K+1exp( C 2K/K+1√(𝕃og_r_0(C^-1_K,ϕ r^-K) )),

		then 
		
    _∞^φ (ϕ (E)) ≤ C_K_∞^φ (E)^K+1/2K.

	
	
	Since for sets with zero Hausdorff measure and content we only need to care about the gauge function for small values, and ^h ≪_∞^h ≪^h, we obtain the following immediate corollary. 
	
	
		Let ϕ : → be a principal K-quasiconformal mapping and conformal on ∖𝔻, and φ_·,· as in makarov gauge function. Then ^φ_K,C (E) = 0 implies ^φ_1,C (ϕ (E)) = 0 for all E⊂ B(0,1/2).
		
			For any constant T>0, the function
			
    Q_T(r)log(T1/r)logloglog(T1/r)/log1/rlogloglog1/r
    			=(log T/log1/r+1)
    			(
    			log(
    				log(
    					log T/log1/r
    					+1
    					)
    				/loglog1/r
    				+1
    				)
    			/logloglog1/r
    			+1
    			).

			satisfies
			
    lim_r→ 0(√(Q_T(r))-1)√(log1/rlogloglog1/r) = 0.

			In particular, this implies that for any constant S>0, if r>0 is small enough then
			
    φ_K,C (Sr)    = S^2/K+1φ_K,C (r) exp(C 2K/K+1(√(Q_S^-K(r^K))-1)√(log r^-Klogloglog r^-K))
       ≤ 1.1 S^2/K+1φ_K,C (r),

			and the result follows from this and <Ref>.
		
	
	
	
		Take t^' = 1, t=2/K+1, ε (x,r) φ (r) / r and h(x,r)  r^t^'ε (x,r) in <Ref>. Note that in this case the functions ε and h do not depend on x. So, we write ε (r) and h(r) instead of ε (x,r) and h(x,r) respectively. Recall that when r>0 is small enough, we have that
		
    ε (r) = φ (r)/r =  exp( C √(log r^-1logloglog r^-1))

		is decreasing with lim_r→ 0ε (r) = +∞.
		
		Let us first see how <Ref> implies this corollary, and later we will check its hypothesis. By the choice of t, t^', ε(r) and h(r) at the beginning of this proof, setting
		
    ε(x,r) ε (ϕ (B(x,r)))^2K/K+1,
    		  h(x,r)  r^2/K+1ε(x,r),

		by <Ref> we obtain
		
    _∞^φ (ϕ (E)) ≤ C_K _∞^h (E).

		Now we want to remove the dependence of the quasiconformal mapping ϕ on the gauge function h. Recall ε (ϕ (B(x,r))) = inf_B ∈ℬ (ϕ(B(x,r)))ε (r_B) where ℬ (ϕ(B(x,r))) denotes the family of balls with minimal diameter containing ϕ(B(x,r)). Since ε does not depend on the point x, we get 
		
    ε (ϕ (B(x,r))) = inf_B ∈ℬ (ϕ(B(x,r)))ε (r_B)= ε( (ϕ(B(x,r)))/2) .

		From the local Hölder continuity of K-quasiconformal mappings in <Ref>, there exists C_K,ϕ with C_K,ϕ^-1 r^K ≤ (ϕ(B(x,r)))/2 ≤ C_K,ϕ r^1/K, and since ε is decreasing we have
		
    ε (ϕ (B(x,r))) ≤ε (C_K,ϕ^-1 r^K) .

		This implies 
		
    h(x,r) 
    		= r^2/K+1ε (ϕ (B(x,r)))^2K/K+1≤ r^2/K+1ε (C_K,ϕ^-1 r^K)^2K/K+1 
    		=φ(r),

		which together with aplicacion of distortion gauge measures concludes the lemma.
		
		It remain to check the assumptions in <Ref>, i.e., ε∈𝒢_1, and (ε∘ψ)^d ∈𝒢_2 for any principal K-quasiconformal mapping ψ :→ and any t^'/Kt≤ d ≤ 1.
		
		Since the function ε does not depend on the point, in order to check that ε∈𝒢_1, it suffices to see that there exists a constant C_1 such that C_1^-1ε (r) ≤ε (s) ≤ C_1 ε (r) whenever r/2 ≤ s ≤ 2r. This follows if ε (r) ≤ 2 ε (2r) for small enough r. Indeed, from this and that the function ε is decreasing then 1/2ε (r) ≤ε (2r)≤ε (s) ≤ε (r/2) ≤ 2 ε (r). Let us see ε (r) ≤ 2 ε (2r) if r is small enough. Using the function Q from the proof of <Ref> we have
		
    Q_2 (2r)log1/rlogloglog1/r/log1/2rlogloglog1/2r
    		= 
    		(log 2/log1/2r+1)
    		(
    		log(
    			log(
    				log 2/log1/2r
    				+1
    				)
    			/loglog1/2r
    			+1
    			)
    		/logloglog1/2r
    		+1
    		),

		and
		
    ε(r)/ε(2r) 
    		= φ(r)/r/φ(2r)/2r
    		= exp( C ( √(Q_2(2r))-1 ) √(log1/2rlogloglog1/2r)).

		As lim_r→ 0{( √(Q_2(2r))-1 ) √(log1/2rlogloglog1/2r)} = 0, recall from the proof of <Ref>, we obtain
		
    lim_r→ 0ε(r)/ε(2r) 
    		= exp( C lim_r→ 0{( √(Q_2(2r))-1 ) √(log1/2rlogloglog1/2r)}) = 1 .

		This implies that there exists r_0 >0 small such that ε (r) ≤ 2 ε (2r) for every 0<r<r_0, as claimed.
		
		Let us see now that (ε∘ψ)^d ∈𝒢_2 for any principal K-quasiconformal mapping ψ :→ and any t^'/Kt≤ d ≤ 1, that is, there exists a constant C_2 >0 such that
		
    ∑_k≥ 0 2^-k (2-t)(( ε∘ψ )(x,2^k r))^d ≤ C_2 ((ε∘ψ) (x,r))^d .

		Recall that ℬ (S) denotes the family of balls with minimal diameter containing the set S, (ε∘ψ) (x,s) = ε (ψ (B(x,s))) = inf_B ∈ℬ (ψ(B(x,s)))ε (r_B), and that the function ε does not depend on the point. Since ψ is an homeomorphism, we have ψ (B(x,2^k r)) ⊂ψ(B(x,2^k+1 r)) for any k≥ 0. In particular, a minimizer ball for ψ(B(x,2^k+1 r)) contains the set ψ(B(x,2^k r)), and therefore the minimizer ball of ψ(B(x,2^k r)) has smaller diameter than the minimizer ball of ψ(B(x,2^k+1 r)). This and the fact that the function ε is decreasing imply
		
    (ε∘ψ) (x,2^k+1 r) ≤ (ε∘ψ) (x,2^k r)

		for any k≥ 0. Consequently, (ε∘ψ) (x,2^k r) ≤ (ε∘ψ) (x, r), and the 𝒢_2 condition is satisfied with C_2 ∑_k≥ 0 2^-k (2-t) < ∞ since
		
    ∑_k≥ 0 2^-k (2-t)(( ε∘ψ )(x,2^k r))^d 
    		≤((ε∘ψ) (x,r))^d ∑_k≥ 0 2^-k (2-t) .

	
	
	

 §.§ The A-harmonic conjugate in simply connected domains

	
	This section is a brief summary of <cit.>. Let Ω⊂^2 be a bounded simply connected domain. We remark that the simply connected condition on Ω used in this section is just an auxiliary assumption to be able to define the auxiliary L_A-harmonic conjugate function below.
	
	Consider u a solution to
	
    A(z) ∇ u(z) =0,    u∈ W^1,2_ (Ω),

	and consider the vector fields ∇ u and A(z) ∇ u. The vector field ∇ u has zero curl since ∇ u = ∂_y ∂_x u - ∂_x ∂_y u =0, and A(z) ∇ u is divergence-free as a solution to the previous equation.
	
	The Hodge star operator * is defined as
	
    * [  0 -1;  1  0 ] : ^2 →^2,

	which is simply the counterclockwise rotation by 90 degrees, and so * * = -Id. The Hodge star operator * transforms curl-free fields into divergence-free fields and vice versa. Therefore
	
    (*∇ u) = 0,  and  (*A(z)∇ u)=0 .

	
	By the Poincaré lemma[It is in this step where we need to work with simply connected domains Ω⊂^2.], see <cit.> for the precise statement, we have that *A(z)∇ u is a gradient field. That is, there exists a real-valued function v∈ W^1,2_ (Ω), called the L_A-harmonic conjugate of u, such that 
	
    ∇ v = *A(z) ∇ u .

	
	Define now
	
    A^*  *^t A^-1 * = -* A^-1 *,

	which is also uniformly elliptic and
	
    A^* (z) ∇ v = A^* (z) (*A(z) ∇ u) = (-* A^-1 *) (*A(z) ∇ u) = *∇ u .

	Recall that ∇ u = 0, and therefore (*∇ u) = 0, i.e., the function v satisfies
	
    A^* (z) ∇ v =0,    v∈ W^1,2_ (Ω).

	In the converse direction, the existence of v∈ W^1,2_ (Ω) such that ∇ v = *A∇ u implies A∇ u = 0, because (*A∇ u) = ∇ v = 0, the Hodge star operator * sends zero-curl to zero-divergence and **=-Id. As before, A∇ u = 0 implies A^* ∇ v = 0.
	
	
		Wherever the matrix is symmetric with determinant 1 then A^*=A, and hence both u and its L_A-harmonic conjugate function v are L_A-harmonic. Indeed, in this particular case, the inverse matrix is
		
    A^-1 = [  a_22 -a_12; -a_12  a_11 ] ,

		and hence
		
    A^* = -*  [  a_22 -a_12; -a_12  a_11 ] * = A .

	
	
	The following theorem describes the strong connection between the Beltrami equation and PDEs.
	
	
		Let Ω⊂^2 be a simply connected domain, and let u∈ W^1,1_ (Ω) be a solution 
		
    A(z) ∇ u = 0.

		If v∈ W^1,1 (Ω) is a solution to ∇ v = *A ∇ u, then the function f=u+iv satisfies the generalized Beltrami equation
		
    ∂̅f = μ_A(z) ∂ f + ν_A(z) ∂ f,

		where the coefficients μ_A,ν_A are given by
		
    μ_A =   1/ (Id + A)( a_22-a_11-i(a_12+a_21) ), 
    ν_A =   1/ (Id + A)( 1- A + i(a_12-a_21) ) .

		
		Conversely, if f∈ W^1,1_ (Ω) is a mapping satisfying the generalized Beltrami equation generalized Beltrami equation, then u= f and v =  f satisfy ∇ v = *A∇ u with A given by solving the complex equations in relation coefficients beltrami and pde.
	
	
	Notice also from relation coefficients beltrami and pde that when the matrix is symmetric and with A=1, then ν_A=0 and generalized Beltrami equation reads as the classical Beltrami equation
	
    ∂̅f = μ_A(z) ∂ f  with μ_A =  a_22 - a_11 - 2ia_12/2+a_11 + a_22.

	In this case, we have the following relations:
	
		Let A = (a_ij)_i,j=1^2 be a uniformly elliptic matrix. Wherever A is symmetric with determinant 1, then the Beltrami coefficients in relation coefficients beltrami and pde are
		
    μ_A =  a_22 - a_11 - 2i a_12/2+a_11 + a_22 ,   ν_A = 0,

		and given a domain Ω⊂^2 and f∈ W^1,1_ (Ω), the following are equivalent:
		
			
  * ∂̅f = μ_A (z) ∂ f,
			
  * ∇ ( f) = *A∇ ( f),
			
  * ∂̅(f^-1) = - μ_A (f^-1 (z)) ∂ (f^-1),
			
  * ∇( (f^-1) ) = *B∇( (f^-1) ),
		
		where
		
    B [     1/a_11  a_12/a_11; -a_12/a_11     1/a_11 ]∘ f^-1 and 
    			B^*
    			=[     1/a_22 -a_12/a_22;  a_12/a_22     1/a_22 ]∘ f^-1.

		are uniformly elliptic. In this case, we also have,
		
    A∇ ( f) = 0,         A∇ ( f) = 0,
        B ∇( (f^-1) ) = 0,        B^* ∇( (f^-1) ) = 0 .

		
			During the proof we write μ=μ_A. From <Ref> we have ∂̅f = μ∂ f if and only if ∇ ( f) = *A∇ ( f), with the precise Beltrami coefficient
			
    μ =  a_22 - a_11 - 2ia_12/2+a_11 + a_22 ,

			which gives thm qc pde cond1⇔thm qc pde cond2.
			
			By <cit.>, ∂̅f = μ (z) ∂ f if and only if
			
    ∂̅(f^-1) = - μ (f^-1 (z)) ∂ (f^-1),

			and so thm qc pde cond1⇔thm qc pde cond3. Hence, from <Ref> there exists[Actually, <Ref> gives us how to find the matrix B from A. Directly from beltrami equation inverse and relation coefficients beltrami and pde we deduce that the matrix B is antisymmetric with a common diagonal term.] a matrix B (depending on A, that is, on μ) such that (<ref>) is equivalent to having ∇( (f^-1) ) = *B∇( (f^-1) ). The matrix B defined in matrix of inverse function beltrami satisfies (<ref>) with ν = -μ∘ f^-1, i.e.,
			
    - a_22 - a_11 - 2ia_12/2+a_11 + a_22 =-μ 
    			= 1- (B ∘ f)  + 2 i b_12/(Id + B ∘ f) ,

			and hence we have thm qc pde cond3⇔thm qc pde cond4. Moreover, a routine computation B^* = -∗ B^-1∗ gives the second matrix in matrix of inverse function beltrami.
			
			It remains to see the last part. Since ∇( (f^-1) ) = *B∇( (f^-1) ) by thm qc pde cond4, we have that 
			
    B ∇( (f^-1) )=0,    B^* ∇( (f^-1) ) = 0,

			as we did in the paragraph before <Ref>. The same holds for the real and imaginary parts of f with the matrix A and A^* respectively, but A^* = A by <Ref>.
		
	
	
	Notice that even though the matrix A is symmetric with determinant 1, the new matrix B in matrix of inverse function beltrami is no longer symmetric with determinant 1. However, we still have that (f^-1) (resp. (f^-1)) is L_B-harmonic (resp. L_B^⋆-harmonic).
	
	The following result gives the precise distortion of a quasiconformal mapping solving the generalized Beltrami equation generalized Beltrami equation.
	
	
		Let A∈^2× 2 be a uniformly elliptic matrix with ellipticity constant λ≥ 1, μ_A and ν_A as in relation coefficients beltrami and pde, and K_λ as in K-lambda ellipticity condition beltrami coefficient. Then
		
    |μ_A| + |ν_A|_∞≤K_λ -1/K_λ + 1<1.

		
		
			See <cit.> for general matrices and <cit.> for the symmetric case.
		
	
	
	
		Note that in both cases K_λ→ 1^+ as λ→ 1^+. In particular K_λ -1/K_λ + 1→ 0^+ as λ→ 1^+.
	
	
	
		Let A be a uniformly elliptic and symmetric matrix with determinant 1. Recall from <Ref> we noted that the modification A= φ A + (1-φ)Id is symmetric but it has no longer determinant 1 in general.  The right way to interpolate the matrices A and Id to preserve also that the modified matrix has determinant 1 is by replacing A by the uniformly elliptic (with the same ellipticity constant as A), symmetric and with determinant 1 matrix
		
    A=[     a_11(1+φ)^2+a_22(φ-1)^2+2(1-φ^2)                              4φ a_12;                              4φ a_12 a_11(φ -1)^2+(1+φ)(a_22(1+φ)+2(1-φ)) ]/a_11(1-φ^2)+a_22(1-φ^2)+2(1+φ^2),

		which agrees with A when φ=1 and with Id when φ=0. This is precisely the matrix such that μ_A=φμ_A and ν_A = φν_A=0, by the relation between uniformly elliptic matrices and Beltrami coefficients in <Ref>. As μ_A_∞≤μ_A_∞, in particular we have that A has the same ellipticity constant as A.
	
	
	

 §.§ Harmonic factorization in general domains

	
	In this section, we show that there exists a quasiconformal homeomorphism factorizing L_A-harmonic functions into harmonic functions in its image domains.

        Let Ω⊂^2 be an open set, A be a uniformly elliptic matrix, and μ_A and ν_A as in relation coefficients beltrami and pde. Given an L_A-harmonic function u∈ W^1,2_ (Ω) and any simply connected domain D⊂Ω, let v ∈ W^1,2_ (D) be the L_A-harmonic conjugate function of the L_A-harmonic function u∈ W^1,2_ (D). By <Ref>, the function f = u+iv ∈ W^1,2_ (D) satisfies the Beltrami equation
		
    ∂̅f = μ_A ∂ f + ν_A ∂ f a.e. in  D.

		In particular, the function f satisfies the classical Beltrami equation
        
    ∂̅f = μ∂ f  a.e. in D, with μμ_A + ν_A ∂ f/∂ f   where ∂ f/∂ f  is defined,
    
    				0    otherwise.

        Note that the Beltrami coefficient μ seems to depend on ∂ f / ∂ f, but in fact it only depends on (the derivative of) u. Indeed, by the relation ∇ v = *A(z) ∇ u, see (<ref>), we can write in ∂ f the terms ∂_x v and ∂_y v using ∂_x u and ∂_y u. More precisely,
        
    ∂ f = (1+ a_11)∂_x u + a_12∂_y u 
    				- i(a_21∂_x u+(1+a_22)∂_y u
    				),

        and in particular, μ in classical beltrami equation f depends on μ_A, ν_A and the derivatives of the function u, but not on the auxiliar L_A-harmonic conjugate function v in D. By classical beltrami equation f and partial f only depends on u we can redefine μ in whole Ω as
        
    μ_A,uμ_A + ν_A (1+ a_11)∂_x u + a_12∂_y u 
    				+ i(a_21∂_x u+(1+a_22)∂_y u
    				)/
    				(1+ a_11)∂_x u + a_12∂_y u 
    				- i(a_21∂_x u+(1+a_22)∂_y u
    				) a.e. in Ω.

	
	The main result of this section is the following.
	
	
		Let Ω⊂^2 be an open set, A be a uniformly elliptic matrix with ellipticity constant λ≥ 1, and μ_A and ν_A as in relation coefficients beltrami and pde. For any L_A-harmonic function u∈ W^1,2_ (Ω), if ϕ∈ W^1,2_(Ω) be a homeomorphic solution of the Beltrami equation
        
    ∂̅ϕ = μ_A,u∂ϕ a.e. in Ω,  where μ_A,u as in coefficients classical beltrami equation in terms of u and A,

	then u∘ϕ^-1: ϕ (Ω) → is harmonic.
	
	
	The existence of ϕ is given by the Riemann mapping theorem, see <Ref>. Moreover, ϕ is K_λ-quasiconformal with K_λ as in K-lambda ellipticity condition beltrami coefficient, since μ_A,u_∞≤|μ_A|+|ν_A|_∞≤K_λ -1/K_λ +1 <1 by <Ref>.
	
	
		We claim that it suffices to prove the lemma for simply connected domains. Indeed, for a general open set Ω, for each point x∈Ω take an open ball B with x∈ B ⊂Ω, and in particular u∈ W_^1,2 (B) and u is L_A-harmonic in B. Assuming the lemma for simply connected domains, we have that u∘ϕ_u,A^-1: ϕ_u,A (B) → is harmonic. Since it is true for any x ∈Ω, we obtain that u ∘ϕ_u,A^-1 is harmonic in ϕ_u,A (Ω).
		
		Let us see the proof assuming also that Ω is simply connected. Let v ∈ W^1,2_ (Ω) be the L_A-harmonic conjugate function of the L_A-harmonic function u∈ W^1,2_ (Ω). By <Ref>, the function f = u+iv ∈ W^1,2_ (Ω) satisfies the Beltrami equation ∂̅f = μ_A ∂ f + ν_A ∂ f a.e. in Ω, and in particular, also safisfies the classical Beltrami equation in classical beltrami equation f. By the discussion before the statement of this lemma, we have μ=μ_A,u in Ω and therefore both f=u+iv ∈ W^1,2_ (Ω) and ϕ∈ W^1,2_ (Ω) solve classical beltrami equation in terms of u and A in Ω. By Stoilow factorization (see <cit.>), the function f ∘ϕ^-1 :  ϕ (Ω) → is holomorphic. Taking the real part on both sides we obtain
		
    u(ϕ^-1 (z))=  ( f ) (ϕ^-1 (z))  =  ( f ∘ϕ^-1)  ( z ),

		and so u∘ϕ^-1 : ϕ (Ω) → is harmonic.
	
	
	

  §.§.§ Symmetric matrices with determinant 1

	
	Let us move to the case where the matrix A is symmetric with A =1 in Ω. Recall that we simply have K_λ = λ in Ω since the matrix is symmetric there, see K-lambda ellipticity condition beltrami coefficient. Note that in this case, by beltrami coefficients for sym and det 1 matrices in <Ref> we have that, for any L_A-harmonic function u∈ W^1,2_ (Ω), the Beltrami coefficient μ_A,u in coefficients classical beltrami equation in terms of u and A equals μ_A in Ω. In particular, we directly obtain the following result.
	
	
		Let Ω⊂^2 be an open set and A be a uniformly elliptic matrix with ellipticity constant λ≥ 1. Assume also that A is symmetric with A=1 in Ω. Let ϕ∈ W^1,2_ (Ω) be a homeomorphic solution of the Beltrami equation
		
    ∂̅ϕ = μ_A ∂ϕ a.e. in Ω, where μ_A =  a_22 - a_11 - 2ia_12/2+a_11 + a_22.

		For any L_A-harmonic function u∈ W_^1,2 (Ω), then u∘ϕ^-1: ϕ (Ω) → is harmonic.
	
	
	As the function ϕ above factorizes every L_A-harmonic function in Ω into a harmonic function in ϕ (Ω), the corollary suggests, and in fact it holds, that
	
    A_ϕ^-1 = Id  a.e. in ϕ (Ω);

	recall the definition of A_ϕ^-1 in matrix pushforward. That is, the factorization from L_A-harmonic functions to harmonic functions is not only given by the Stoilow factorization (in the proof of <Ref>) but by the previous pointwise relation given by just applying the change of variables ϕ in the weak form of L_A.
	
	Let us see the proof of change of variables instead of stoilow factorization. Recall that the equality above is equivalent to Id_ϕ=A in a.e. Ω, which reads as
	
    1/ Dϕ Dϕ· A · Dϕ^T=Id  a.e. in Ω.

	Writing ϕ=u+iv, μ_1=a_22-a_11, μ_2=-2a_12 and μ_d=2+a_11+a_22 so that μ_A = μ_1+iμ_2/μ_d, the pointwise equality ∂̅ϕ=μ_A∂ϕ a.e. in Ω is equivalent to the 2× 2 system
	
    [     μ_2 μ_d+μ_1; μ_d+μ_1    -μ_2 ][ ∂_y u; ∂_y v ]
    	=
    	[    μ_d-μ_1        μ_2;        μ_2 -(μ_d-μ_1) ][ ∂_x u; ∂_x v ],

	implying that we can write ∂_y u, ∂_y v in terms of ∂_x u, ∂_x v as
	
    [ ∂_y u; ∂_y v ]
    	=
    	[     μ_2 μ_d+μ_1; μ_d+μ_1    -μ_2 ]^-1[    μ_d-μ_1        μ_2;        μ_2 -(μ_d-μ_1) ][ ∂_x u; ∂_x v ].

	Using this relation and taking into account that A=1, the claim follows.
	
	

§ SYMMETRIC MATRICES WITH DETERMINANT 1

	
	Let Ω⊂^2 be a domain and A ∈^2× 2 (Ω) be a symmetric and uniformly elliptic (with ellipticity constant λ≥ 1) matrix with A = 1. Note that during all <Ref> we will use K_λ = λ since the matrix is symmetric, see K-lambda ellipticity condition beltrami coefficient.
	
	

 §.§ Push-forward of the harmonic measure and proof of Theorem <ref>

	
	Given a pole p∈Ω, the harmonic factorization of L_A-harmonic functions given by change of variables instead of stoilow factorization leads us to obtain in the following lemma that the elliptic measure ω^p_Ω, A is precisely the push-forward of the harmonic measure ω^ϕ (p)_ϕ (Ω), Id under the map ϕ^-1 given by <Ref>. The following result is a consequence of <Ref> and <Ref>.
	
	
		Under the assumptions in <Ref>, then
		
    ω^p_Ω, A (ϕ^-1 (E)) =  ω^ϕ (p)_ϕ (Ω), Id (E)   for any  p∈Ω and E⊂∂ (ϕ (Ω)).

	
	
	By means of the distortion results of Hausdorff dimension and measure under quasiconformal mappings in <Ref> we obtain the non-regularity case in <Ref>.
	
	
		Recall we can assume that Ω⊂ B_1/4 (0) is bounded Wiener regular by <Ref>, and we can replace finitely connected by simply connected by <Ref>.
		
		Let ϕ_A ∈ W^1,2_ () be a principal solution of the Beltrami equation
		
    ∂̅ϕ_A = μ_A _B_1 (0)∂ϕ_A  a.e. in ,  where μ_A  as in beltrami coefficients for sym and det 1 matrices,

		given by the measurable Riemann mapping theorem, see <Ref>. In particular, ϕ_A is a λ-quasiconformal homeomorphism, conformal in ∖𝔻, and ϕ_A^-1 is also a λ-quasiconformal homeomorphism.
		
		By Wolff's <Ref> there is a set F'⊂∂ (ϕ_A (Ω)) such that ω^ϕ_A (p)_ϕ_A (Ω), Id (F') = 1 and with σ-finite length. Therefore, the set F = ϕ_A^-1 (F') ⊂∂Ω satisfies ω^p_Ω, A (F) = 1 by the relation relation elliptic measures between elliptic and harmonic measures, and by the distortion of Hausdorff measures under quasiconformal mappings in <Ref>, the set F has σ-finite ^2λ/λ+1.
		
		Now consider the case where Ω is simply connected. Recall that the λ-quasiconformal homeomorphism ϕ_A is conformal in ∖𝔻. By the relation relation elliptic measures between harmonic and elliptic measures, Makarov's <Ref> and <Ref> respectively, there exists C>0 such that
		
    ω_Ω, A^p (·) = ω^ϕ(p)_ϕ_A (Ω), Id (ϕ(·)) ≪^φ_1,C (ϕ (·)) ≪^φ_λ,C (·).

	
	
	In the following sections, we will assume that the matrix A, and therefore also the Beltrami coefficient μ_A, enjoys more regularity. By regularity theory, we will have that the quasiconformal mapping ϕ_A given in <Ref> has better estimates than the Hölder continuity in <Ref> or the distortion of Hausdorff dimension (resp. measure) in dimension distortion theorem (resp. <Ref> or <Ref>). The proof of the results in the following sections will have the same structure as the proof above, though we will use that the change of variables ϕ_A is not only λ-quasiconformal, but it is more regular.
	
	

 §.§ Mean oscillation under quasiconformal mappings

	
	In this subsection, we define the space BMO functions and collect some properties to control the regularity of the coefficients arising in the following sections.
	
	
		Let 1≤ p <∞, f∈ L^p_ (Ω) and a set S ⊂Ω. Denote the p-mean oscillation of f in S the quantity
		
    [f]_S, p(_S | f(z) - m_S f |^p   dz)^1/p,

		where m_S f _S f(y)  dy 1/|S|∫_S f(y)   dy.
	
	
	Bounded functions satisfy [f]_S, p^p ≲f_L^∞ (S)^p, where the constant involved depends on 1≤ p <∞. If p=1, the constant is 2 by triangle inequality. In this lemma, we collect some fundamental bounds of the p-mean oscillation. We omit its proof.
	
		Let 1≤ p < ∞. The following holds:
		
			
  *  [f+g]_S,p^p ≲ [f]_S,p^p + [g]_S,p^p.
			
			
  *  [fg]_S,p^p ≲g_L^∞ (S)^p [f]_S,p^p + f_L^∞ (S)^p [g]_S,p^p.
			
			
  *  [1/f]_S,p^p ≲ 1/f _L^∞ (S)^2p [f]_S,p^p.
			
			
  *  [f/g]_S,p^p 
			≲1/g_L^∞ (S)^p [f]_S,p^p + f_L^∞ (S)^p   1/g _L^∞ (S)^2p [g]_S,p^p.
		
	
	
	Let us now recall the definition of bounded mean oscillation functions.
	
	
		A function f∈ L_^1 (Ω) is of bounded mean oscillation in Ω, f∈ (Ω), if
		
    f_Ω,*sup_B ⊂Ω ball_B |f(z)-m_B f |   dz  <∞.

		If Ω = ^n+1 we write f_*.
	
	
	With the notation in <Ref>, f_Ω,∗ = sup_B⊂Ω ball [f]_B,1. Recall that f∈ implies |f|∈. Note also that it is equivalent considering cubes Q instead of balls B in the (^n+1) norm.
	
	Given a function f∈ (^n+1), the John-Niremberg inequality, see <cit.> for example, states that there exist dimensional constants C_1 and C_2 such that for every cube Q ⊂^n+1 and every λ > 0,
	
    | { z∈ Q : |f(z) - m_Q f| > λ}| ≤ C_1 e^-C_2 λ / f_* |Q| .

	Consequently, see <cit.> for a proof, for all 1<p<∞,
	
    f_* ≈_p f_*,psup_B ⊂Ω ball( _B |f(z)-m_B f |^p   dz )^1/p .

	
	A close look at the proof of the John-Niremberg inequality in <cit.> reveals that, in fact, for every cube Q⊂^n+1 and every λ > 0, if
	
    f_Q,*^cubessup_R⊂ Q  cube_R |f(z)-m_R f |   dz

	then
	
    | { z∈ Q : |f(z) - m_Q f| > λ}| ≤ C_1 e^-C_2 λ / f_Q,*^cubes |Q|.

	Following the steps in the proof of <cit.>, given a cube Q⊂^n+1, we have
	
    sup_R⊂ Q  cube_R |f(z)-m_R f |   dz ≈_p sup_R⊂ Q  cube( _R |f(z)-m_R f |^p   dz )^1/p .

	
	Functions of bounded mean oscillations are invariant under quasiconformal mappings. This is a result of Reimann in <cit.>, which, in fact, also proved that if a W^1,1_-homeomorphism f:→ preserves the  space, then f is K-quasiconformal, with K=K(f_*). In the following lemma, we control the mean oscillation under global quasiconformal mappings. As the statement of the lemma and its proof are slightly different from the one in <cit.> and <cit.>, for the sake of completeness we include the details.
	
	
		Let K≥ 1, ϕ∈ W_^1,2 () be a K-quasiconformal homeomorphism of , and B⊂^2 be an open ball. Given an open ball B ⊂B (with center c_B and radius r_B), consider
		
    B_-  B ( ϕ^-1 (c_B) , max_|ξ-c_B|=r_B |ϕ^-1 (ξ)-ϕ^-1 (c_B)| ) ,

		Then r_B_-≤ C_K,B,ϕ· r_B^1/K, and for any p∈ (K,∞),
		
    _B |f (ϕ^-1 (x)) - m_B (f ∘ϕ^-1)|   dx
    			≲_p,K( _ B_- |f (x)- m_ B_- f |^p   dx )^1/p .

		In particular f∘ϕ^-1_* ≲_K f_*.
		
			The notation “-” in B_- is used to specify that the ball B_- “lives” in the image of ϕ^-1.
			
			We control the radius of B_-. From the definition of B_- and the Hölder continuity with exponent 1/K of quasiconformal mappings in <Ref> applied to ϕ^-1, we conclude
			
    r_B_- = max_|ξ-c_B|=r_B |ϕ^-1 (ξ)-ϕ^-1 (c_B)| 
    				≤ C_K,B,ϕmax_|ξ-c_B|=r_B |ξ- c_B |^1/K =  C_K,B,ϕ· r_B^1/K .

			
			It remains to see distortion of mean oscillation under qc. We can change from m_B (f ∘ϕ^-1) to m_ B_- f because
			
    _B |f (ϕ^-1 (x)) - m_B (f ∘ϕ^-1)|   dx
    					   ≤_B _B |f (ϕ^-1 (x)) - f (ϕ^-1 (y))|    dy   dx 
       ≤ 2 _B |f (ϕ^-1 (x)) - m_ B_- f |   dx .

			Let q be the Hölder conjugate of p, i.e., 1/p+1/q=1. In particular p∈ (K,∞) if and only if q∈(1,K/(K-1)). By the change of variables x=ϕ (y), ϕ^-1 (B) ⊂  B_- and Hölder's inequality, we bound the last term in (<ref>) by
			
    _B |f (ϕ^-1 (x)) - m_ B_- f |   dx 
    				   = 1/|B|∫_ϕ^-1 (B) |f (y)- m_ B_- f | | Dϕ (y)|   dy 
       ≤1/|B|( ∫_ B_- |f (x)- m_ B_- f |^p   dx )^1/p( ∫_ B_-|  Dϕ (x) |^q   dx )^1/q .

			Since q∈ (1,K/(K-1)) ⊂ [0,K/(K-1)), the higher integrability of |  D ϕ|, see <cit.>, implies that the last term in (<ref>) is controled by
			
    ( ∫_ B_-|  Dϕ (x) |^q   dx )^1/q≲_p,K | B_-|^1/q|ϕ ( B_-)|/| B_-| .

			The inequality in higher integrability determinant implies that (<ref>) reduces to
			
    _B |f (ϕ^-1 (x)) - m_ B_- f |   dx 
    					   ≲_p,K| B_-|^1/p/|B|( _ B_- |f (x)- m_ B_- f |^p   dx )^1/p
    					| B_-|^1/q|ϕ ( B_-)|/| B_-|
       = |ϕ ( B_-)|/|B|( _ B_- |f (x)- m_ B_- f |^p   dx )^1/p,

			where we used 1/p + 1/q = 1 in the last inequality.
			
			Now we want to bound |ϕ ( B_-)|/|B|. We have the inclusion
			
    ϕ ( B_-) ⊂  B_+  B ( c_B , max_|ξ-ϕ^-1(c_B)|=r_B_- |ϕ (ξ)-c_B|).

			Here the notation B_+ is used to specify that the ball B_+ “lives” in the image of ϕ. By <cit.>, there exists a constant δ = δ (K) such that δ B_+ ⊂ϕ (B_-) ⊂ B_+. By definition δ B_+ ⊂ B, and hence
			
    |ϕ ( B_-)|/|B|≤|B_+|/|δ B_+|≈_K 1 .

			Inequality in distortion of mean oscillation under qc follows from <ref>, and comparable balls applied to vmo preimage. Therefore, <Ref> is proved. 
		
	
	
	

 §.§ Proof of Theorems <ref> and <ref> via quasiconformal regularity

	
	In this section, via quasiconformal theory, we will see that assuming regularity of the matrix, the global λ-quasiconformal homeomorphism ϕ_A in <Ref> gains more regularity.
	
	Let us present the sketch of this. If ϕ_A is a solution of ∂̅ϕ_A = μ_A ∂ϕ_A, then ϕ_A and ϕ_A^-1 are λ-quasiconformal, in particular ϕ_A, ϕ_A^-1∈ C^1/λ_, and moreover ϕ_A^-1 satisfies the Beltrami equation ∂̅(ϕ_A^-1) = -(μ_A ∘ϕ_A^-1) ∂ (ϕ_A^-1). From ϕ_A^-1∈ C^1/λ_, assuming regularity on the matrix A will not only provide regularity on μ_A, but also on -μ_A ∘ϕ_A^-1. As a consequence, we will obtain extra regularity on the functions ϕ_A and ϕ_A^-1. For example, if A∈ C^α, then μ_A ∈ C^α and -μ_A ∘ϕ_A^-1∈ C^α/λ_, whence we obtain that ϕ_A∈ C^1,α_ and ϕ_A^-1∈ C^1,α/λ_, see <cit.>.
	
	

  §.§.§ Hölder continuous coefficients


        In this subsection we prove <Ref>. The sketch is what is already explained above. For completeness, we provide the proof in full detail.
 
        Although we saw in <Ref> that the dimension of elliptic measures only depends on the nature of the matrix around the boundary, during the following proof we emphasize that the regularity of the matrix is only needed around the boundary. Recall U_t (∂Ω) denotes the t-neighborhood of the boundary, i.e.,
	
    U_t (∂Ω) {x∈^2 :  (x,∂Ω) < t} .

	
	
		Recall we can assume that Ω⊂ B_1/4(0) is bounded Wiener regular by <Ref>, and we can replace finitely connected by simply connected by <Ref>. Without loss of generality, we can also assume A=1.
		
		Assume that A∈ C^α (U_ε(∂Ω)) for ε >0. Let ϕ_A ∈ W^1,2_() be the normalized solution of the Beltrami equation
		
    ∂̅ϕ_A = μ_A ∂ϕ_A  a.e. in ,  where μ_A  as in beltrami coefficients for sym and det 1 matrices,

		given by the measurable Riemann mapping theorem, see <Ref>. In particular ϕ is a λ-quasiconformal homeomorphism. Since A is uniformly elliptic, in particular a_11 , a_22≈ 1 and hence 2+a_11 + a_22≥ 2, and therefore A∈ C^α (U_ε (∂Ω)) implies μ_A ∈ C^α (U_ε (∂Ω)).
		
		By the Hölder regularity of quasiconformal mappings in <Ref> we have that ϕ_A^-1∈ C^1/λ_, and hence -μ_A ∘ϕ_A^-1∈ C^α/λ_ in an open neighborhood of ∂ (ϕ_A (Ω)). Therefore, we get that ϕ_A ∈ C^1,α_ (resp. ϕ_A^-1∈ C^1,α/λ_) in an open neighborhood of ∂Ω (resp. ∂ (ϕ_A (Ω))), see <cit.>. In particular both ϕ_A and ϕ_A^-1 are Lipschitz in an open neighborhood of ∂Ω and ∂ (ϕ_A (Ω)) respectively.
		
		The rest of the proof follows by the same proof of <Ref> in proof of non-regularity case main thm using that ϕ_A is bi-Lipschitz in an open neighborhood of ∂Ω.
	
	
	

  §.§.§ Only one variable dependence coefficients

	
	In this subsection, we consider matrices with no regularity, but satisfying A(z) = A((z)) (or A(z) = A((z))). Even with no regularity on the coefficients (only uniformly elliptic), in this case, the quasiconformal change of variables is bi-Lipschitz.
	
	
		Let μ : → be an arbitrary measurable function with μ_∞≤ k < 1 that depends on x=(z) only and let
		
    f(x) = ∫_0^x 1+μ(t)/1-μ(t)  dt .

		Then ϕ(z) = f(x) + iy represents a unique quasiconformal mapping on  onto itself with complex dilatation μ and normalizations ϕ(0)=0, ϕ(i)=i and ϕ(∞)=∞. Moreover, ϕ is bi-Lipschitz.
	

        
            Recall we can assume that Ω is Wiener regular by <Ref>, and we can replace finitely connected by simply connected by <Ref>. Without loss of generality, we can also assume A=1.
			
			We assume without loss of generality that A(z) = A((z)). In particular, μ_A as in beltrami coefficients for sym and det 1 matrices satisfies μ_A(z) = μ_A((z)) and hence ϕ_A, given by <Ref>, is bi-Lipschitz. The proof follows as in the non-regularity case of <Ref> in proof of non-regularity case main thm using that ϕ_A is bi-Lipschitz.
        
	
	

  §.§.§ VMO coefficients

	
	Let us introduce the space of functions with vanishing mean oscillation.
	
	
		A functions f has vanishing mean oscillation, written f∈, if f∈ and
		
    lim_r→ 0sup_x∈^2_B_r (x) |f(z)-m_B_r(x) f |   dz =0.

	

        For the classical Beltrami equation with _c coefficient, its quasiconformal solution preserves the dimension of sets.

	
		Let μ∈_c be a Beltrami coefficient and ϕ∈ W^1,2_ () be a K-quasiconformal homeomorphism of  satisfying ∂̅ϕ = μ∂ϕ. Then both ϕ and ϕ^-1 are C^α_ for every α∈ (0,1). In particular
		
    _ (ϕ (S)) = _ (S)   for any set  S.

	

        This is well-known in the literature. For the sake of completeness, a proof of <Ref> can be found below, as the author did not find any proof in the literature.
	
        Using the result above, for  matrices we recover the Jones and Wolff theorem in <cit.> and the lower bound of the dimension of elliptic measure in simply connected domains.
	
		
			Recall we can assume that Ω⊂ B_1/4(0) is bounded Wiener regular by <Ref>, and we can replace finitely connected by simply connected by <Ref>. Without loss of generality, we can also assume A= 1.
			
			Let us fix a function φ∈ C^∞_c (^2) satisfying
			
    0≤φ≤ 1, φ_|B_1/2(0)=1, φ_|B_1(0)^c=0  and  |∇φ|≲ 1,

			and let ϕ_A ∈ W^1,2_() be the normalized solution of the Beltrami equation
			
    ∂̅ϕ_A = φμ_A ∂ϕ_A  a.e. in ,  where μ_A  as in beltrami coefficients for sym and det 1 matrices,

			given by the measurable Riemann mapping theorem, see <Ref>. Since A is uniformly elliptic, in particular a_11 , a_22≈ 1 and hence 2+a_11 + a_22≥ 2. Since A∈ (^2), by <ref> in <Ref> we have μ_A ∈ (^2) and so φμ_A∈_c (^2). Therefore, by <Ref>, the function ϕ_A preserves the dimension of sets. The theorem follows from this by the same proof of <Ref>, see proof of non-regularity case main thm.
		

	We now turn to the proof of <Ref>. First, in the following result we see the regularity of a quasiconformal mapping with _c coefficient.
	
	
		Let μ∈_c be such that μ_∞ = k < 1, and p ∈ (0,∞). Then for each h∈ L^p the equation
		
    ∂̅f - μ∂ f = h

		admits a solution f with ∇ f ∈ L^p which is unique up to an additive constant.
	
        This result also holds for the generalized Beltrami equation, see <cit.> and <cit.>.
        
	
	    Note that if ∂̅ϕ = μ∂ϕ with μ∈_c, then ϕ∈ L^p_ for all p∈ (1,∞). Thus, ϕ∈ C_^α for every α∈ (0,1), and as a consequence, _ (ϕ (S))≤_ (S) for any set S.
	
        
        Next we want to see the regularity of the Beltrami coefficient of ϕ^-1. Before that, as a consequence of the estimate distortion of mean oscillation under qc in <Ref>, we see that _c∩ L^∞ is invariant under quasiconformal mappings.

        
            _c∩ L^∞ is invariant under global homeomorphic quasiconformal mappings.
            
                Let f∈_c∩ L^∞ be non-identically zero, otherwise we are done. By <Ref>, for all ε>0 there is r_0 = r_0 (ε) such that if r<r_0 then
                
    sup_x∈^2_B_r(x) |f-m_B_r(x)f|<ε.

                Let ϕ∈ W^1,2_() be a K-quasiconformal homomorphism of , with K≥ 1, B be a ball such that (f∘ϕ^-1) ⊂B/2 and fix C C_K,2B,ϕ, the constant in <Ref>. We want to see that for all ε^'>0 there is r_0^' = r_0^' (ε^') ≤ r(B)/100 such that if r<r_0^' then
                
    sup_x∈^2_B_r(x) |f∘ϕ^-1-m_B_r(x)(f∘ϕ^-1)|<ε^'.

                Given x∈B and r<r_0^' (otherwise the integral above is zero), let B=B_r(x) and B_- as in <ref>. By <ref> (with p=K+1) we have
                
    _B |f∘ϕ^-1-m_B(f∘ϕ^-1)|
                       ≤ C_K (_B_- |f-m_B_-f|^K+1)^1/(K+1)
       ≤ C_K (2f_L^∞)^K/(K+1)(_B_- |f-m_B_-f|)^1/(K+1),

                where r(B_-)≤C r(B)^1/K. So, given ε^', take ε = (ε^' /(C_K (2f_L^∞)^K/(K+1)))^K+1 and r_0^' = min{(r_0 (ε)/C)^K,r(B)/100}. With this choice we have r(B_-) < C (r_0^')^1/K≤ r_0 (ε) and therefore
                
    C_K (2f_L^∞)^K/(K+1)(_B_- |f-m_B_-f|)^1/(K+1) < ε^',

                as claimed.
            
        
	
	In the following lemma we prove that if ϕ solves the Beltrami equation ∂̅ϕ=μ∂ϕ with μ∈_c, then ϕ^-1 not only solves the generalized Beltrami equation ∂̅(ϕ^-1)=-(μ∘ϕ^-1)∂(ϕ^-1) with μ∘ϕ^-1∈_c, but it also satisfies ∂̅(ϕ^-1)=μ∂ϕ with μ∈_c. In particular, we can also apply <Ref> to ϕ^-1, see <Ref>.
	
	
		Let μ∈_c be a Beltrami coefficient and ϕ∈ W^1,2_ () be a K-quasiconformal homeomorphism of  satisfying ∂̅ϕ = μ∂ϕ. Then ϕ^-1 satisfies
		
    ∂̅(ϕ^-1)
    			= - (μ∘ϕ^-1) ∂ (ϕ^-1) 
    			= - ((μ∂ϕ/∂ϕ) ∘ϕ^-1)  ∂ (ϕ^-1)

		with μ∘ϕ^-1, (μ∂ϕ / ∂ϕ) ∘ϕ^-1∈_c.
		
			The equalities in general to classical beltrami equation are proved in <Ref> and <cit.>.
			
			From <ref> in <Ref> and <Ref> it suffices to see ∂ϕ / ∂ϕ∈. The strategy is the same as in the proof of <cit.>. Due to the work of Hamilton in <cit.> we have λ = log∂ϕ∈. Writing ∂ϕ = e^λ, in particular we have
			
    ∂ϕ/∂ϕ 
    			= e^λ/e^λ 
    			= e^λ e^i λ/e^λ e^-i λ
    			= e^2i λ .

			The function λ belongs in  as λ∈. Since ∋ t↦ e^it is 1-Lipschitz, for any ball B we have
			
    _B | e^2iλ (x) - m_B e^2iλ|   dx 
    				   ≤_B _B | e^2iλ (x)-e^2iλ (y)|   dy   dx 
       ≤ 2_B _B | λ (x)-λ (y) |   dy   dx 
    				≤  4_B | λ (x) - m_B λ|   dx,

			implying ∂ϕ / ∂ϕ = e^2iλ∈ as λ∈.
		
	

        We are now ready to prove <Ref>.

        
            It follows from <Ref>, <Ref> and <Ref>.
        
	
	

  §.§.§ Sobolev coefficients

	
	In <Ref>, we had that the quasiconformal mapping ϕ_A was bi-Lipschitz, which allowed us to prove the σ-finite length property of the elliptic measure. In this section we assume A∈ W_^1,p with p>2λ^2/λ^2+1. Although this Sobolev regularity on A, and so also for the Beltrami coefficient μ_A, does not provide a bi-Lipschitz quasiconformal solution in general, it still preserves sets with σ-finite length, see <Ref> below. In particular, this will give the Wolff's <Ref> in this situation.
	
	
		Let K≥ 1 and 2K^2/K^2+1 < p ≤ 2. Let μ∈ W^1,p_c (^2) be a compactly supported Beltrami coefficient with μ_∞≤K-1/K+1, and ϕ any μ-quasiconformal mapping. For every compact set E,
		
    ^1 (E)  is σ-finite^1 (ϕ(E))  is σ-finite.

	
	
	For a non-trivial example of non-bi-Lipschitz μ-quasiconformal mapping with μ∈ W^1,2, see the function in <cit.>. We remark that there are also bi-Lipschitz μ-quasiconformal mappings with μ∉W^1,2. In fact, for L≥ 1, L-bi-Lipschitz mappings are μ-quasiconformal with μ_∞≤ (L^2-1)/(L^2+1), and in general μ∉W^1,2. Also, μ(z)=1/2χ_𝔻 (z) ∉W^1,2 provides a bi-Lipschitz μ-quasiconformal mapping. The latter is a particular case in <cit.>.
	
	The result above is precisely what we need to obtain the σ-finite length property when we invoke Wolff's <Ref>.
	
		
			Recall we can assume that Ω⊂ B_1/4(0) is bounded Wiener regular by <Ref>. Without loss of generality, we can also assume A = 1. When p>2 we have the Sobolev embedding into Hölder continuous functions W^1,p⊂ C^1-2/p by Morrey's inequality, and hence the result follows from <Ref>. So, assume p∈( 2λ^2/λ^2+1 , 2].
			
			Let φ be as in definition of the support function and ϕ_A ∈ W^1,2_() be the normalized solution of the Beltrami equation
			
    ∂̅ϕ_A = φμ_A ∂ϕ_A  a.e. in ,  where μ_A  as in beltrami coefficients for sym and det 1 matrices,

			given by the measurable Riemann mapping theorem, see <Ref>. Since A is uniformly elliptic, in particular a_11 , a_22≈ 1 and hence 2+a_11 + a_22≥ 2. Since A∈ W_^1,p (^2), we have that μ_A ∈ W_^1,p (^2) and therefore φμ_A∈ W_c^1,p(^2). The theorem follows by the same proof of <Ref> in proof of non-regularity case main thm, using Wolff's <Ref> and that the function ϕ_A preserve sets of σ-finite length, see <Ref>. That is, if F⊂∂(ϕ_A (Ω)) is the set given by Wolff's theorem, then ϕ_A^-1 (F)⊂∂Ω has full elliptic measure and σ-finite length.
		
	
	

 §.§ Proof of Theorem <ref> via PDE regularity

	
	Similarly as we did in the previous <Ref>, but using a PDE approach instead of applying quasiconformal theory, in this section we deduce better regularity of the quasiconformal change of variables assuming regularity on the matrix A, which allows proving the remaining cases of <Ref>.

    Let us sketch this when the matrix has Hölder coefficients. As in the proof of <Ref> in proof of holder-regularity case main thm, we can assume that A∈ C^α (U_ε (∂Ω)) for 0<α<1 and ε>0. Let ϕ_A ∈ W^1,2_() be the normalized solution of the Beltrami equation
    
    ∂̅ϕ_A = μ_A ∂ϕ_A  a.e. in ,  where μ_A  as in beltrami coefficients for sym and det 1 matrices,

    given by the measurable Riemann mapping theorem, see <Ref>. By <Ref>, uϕ_A, vϕ_A, Rϕ_A^-1 and Iϕ_A^-1 satisfy A ∇ u = 0, A ∇ v = 0, B ∇ u = 0 and B^* ∇ v = 0 respectively, where B and B^* are the uniformly elliptic matrices in <ref>. By the Hölder regularity of quasiconformal mappings we have that ϕ^-1_A ∈ C^1/λ_, and hence A∈ C^α (U_ε (∂Ω)) implies B, B^*∈ C^α/λ_ in an open neighborhood of ∂(ϕ_A (Ω)). All in all, u,v,I,R satisfy a PDE with Hölder coefficients and therefore ϕ_A is bi-Lipschitz in an open neighborhood of ∂Ω. <Ref> follows from the same proof of <Ref> in proof of non-regularity case main thm using that ϕ_A is bi-Lipschitz in an open neighborhood of ∂Ω.
	
	

  §.§.§ Dini mean oscillation

	
	We consider matrices with p-Dini mean oscillation.

        
		For 1≤ p <∞ and f∈ L_^p(^n+1), denote
		
    f_ MO,p (r) sup_x∈^n+1(_B(x,r)|f(z)-m_B(x,r) f|^p   dz)^1/p,

		where MO stands for “mean oscillation”. We say that f is of p-Dini mean oscillation, f∈ DMO_p, if f_ MO,p satisfies the Dini condition
		
    ∫_0^1 f_ MO,p (t)  dt/t < ∞ .

	

        
            From Hölder's inequality, we have the inclusion DMO_q ⊂ DMO_p if 1≤ p < q <∞. The Dini condition of the mean oscillation directly implies DMO_p ⊂ for every 1≤ p <∞. Moreover, every DMO_p (1≤ p <∞) contains the (pointwise) Dini space, that is, functions f satisfying the Dini condition
            
    ∫_0^1 τ (t)  dt/t < ∞,   where τ (t) = sup_|x-y|≤ t |f(x)-f(y)|.

            All in all,
            
    Hölder⊂ Dini⊂ DMO_q ⊂ DMO_p ⊂,   for any 1≤ p < q <∞.

        
	
	Under the 1-Dini mean oscillation regularity on the matrix coefficients, L_A-harmonic functions are C^1. More precisely, we have the following result.
	
	
		Suppose the coefficients of A(x)=(a_ij(x))_i,j=1^n+1 satisfy
		
    max_1≤ i,j≤ n+1∫_0^1 η_a_ij (t)  dt/t < ∞,   where η_a_ij (r) sup_x∈ B_3 (0)_B(x,r)|a_ij(z)-m_B(x,r) a_ij|   dz ,

		and let u ∈ W^1,2 (B_4 (0)) be a L_A-solution in B_4 (0). Then, we have u ∈ C^1 (B_1 (0)).
	
	
	In the following lemma, we see how the 1-Dini mean oscillation regularity changes under K-quasiconformal mappings.
	
	
		Let K≥ 1 and let ϕ∈ W^1,2_ () be a K-quasiconformal homeomorphism of . For any function f and any K<p<∞, there exists a constant C_K,ϕ such that
		
    ∫_0^1 η_(f∘ϕ^-1) (t) dt/t≲_p,K∫_0^C_K,ϕ f_ MO,p (t) dt/t.

		
			Fix B=B_100 (0). Recall that ϕ^-1 is η_K-quasisymmetric as it is global K-quasiconformal, see quasisymmetric condition. Take T=T_K≥ 2 big enough so that η_K (1/T) ≤ 1/2, and denote r_m = T^-m. Let x∈ B_3 (0). First note that for r_m+1≤ r ≤ r_m, and changing from m_B(x,r) (f ∘ϕ^-1) to m_B(x,r_m) (f ∘ϕ^-1) as we did in vmo to other constant, we have
			
    η_(f∘ϕ^-1) (x,r) _B(x,r)|(f∘ϕ^-1 )(z)-m_B(x,r) (f ∘ϕ^-1)|   dz ≲η_(f∘ϕ^-1) (x,r_m) .

			Thus, ∫_0^1 η_(f∘ϕ^-1) (x,t) dt/t≲∑_m=0^∞η_(f∘ϕ^-1) (x,r_m).
			
			Denote now r_m (x) max_|ξ-x|=r_m |ϕ^-1 (ξ) - ϕ^-1 (x)|, recall the ball defined in ball in the image vmo. We simply write r_m as x∈ B_3 (0) is fixed in this proof and the constants below do not depend on x. By distortion of mean oscillation under qc in <Ref> we have
			
    η_(f∘ϕ^-1) (x,r_m) ≲_p,K(_B(ϕ^-1(x),r_m)|f(z)-m_B(ϕ^-1(x),r_m (x)) f|^p )^1/pη_f (ϕ^-1 (x),r_m,p).

			
			By the quasisymmetry condition we have 1/η_K (T) ≤r_m / r_m-1≤η_K (1/T) ≤ 1/2, and hence 1 ≤ 2 ∫_r_m^r_m-1dt/t. As before, for every r_m ≤ r ≤r_m-1 we have η_f (ϕ^-1 (x), r_m,p) ≲η_f (ϕ^-1 (x),r,p). By <Ref> we have r_m ≤ C_K,B,ϕ r_m^1/K, and in particular r_-1≤ C_K,B,ϕ T^1/K. As a consequence of these observations respectively, we get 
			
    ∑_m=0^∞η_(f∘ϕ^-1) (x,r_m)
    					   ≲∑_m=0^∞η_f (ϕ^-1 (x),r_m, p) ≲∑_m=0^∞∫_r_m^r_m-1η_f (ϕ^-1 (x),r_m, p) dt/t
       ≲∑_m=0^∞∫_r_m^r_m-1η_f (ϕ^-1 (x),t, p) dt/t
    					= ∫_0^C_K,B,ϕ T^1/Kη_f (ϕ^-1 (x),t, p) dt/t.

			The last element is bounded by ∫_0^C_K,B,ϕ T^1/K f_ MO,p (t) dt/t. Taking the supremum over all the points x∈ B_3 (0) we obtain the lemma with the constant C_K,B,ϕ T^1/K.
		
	
	
	Using the previous lemma we deduce the C^1 regularity ϕ_A^-1 in the proof of the following result when A∈ DMO_p with p>λ.
 
		
			Recall we can assume that Ω⊂ B_1/4 (0) is bounded Wiener regular by <Ref>, and we can replace finitely connected by simply connected by <Ref>. Without loss of generality, we can also assume A =1.
			
			Let us see first the simply connected case. Assume A∈ DMO_1 and let ϕ_A ∈ W^1,2_() be the normalized solution of the Beltrami equation
			
    ∂̅ϕ_A = μ_A ∂ϕ_A  a.e. in ,  where μ_A  as in beltrami coefficients for sym and det 1 matrices,

			given by the measurable Riemann mapping theorem, see <Ref>. In particular uϕ_A and vϕ_A satisfy ∇ v = *A∇ u, and A ∇ u = 0, see <Ref>. Now, since A ∈ DMO_1 we have that |∇ u| is bounded in B_1/2 (0), see <Ref>, and by the relation ∇ v = *A∇ u we also have that |∇ v| is bounded in B_1/2 (0). In particular, u and v are Lipschitz and hence ϕ is Lipschitz in B_1/2 (0) ⊃∂Ω. This implies ω_Ω,A≪^φ_1,C by the same proof of <Ref> in proof of non-regularity case main thm.
			
			We now turn to the σ-finite length for general domains. Suppose first A∈ DMO_p with p > λ, and later we will see we can always reduce to this case. Let ϕ_A be as above. Recall from <Ref> that the functions R (ϕ^-1) and I (ϕ^-1) satisfy ∇ I = * B ∇ R, and B ∇ R = 0, where B is the uniformly elliptic matrix in matrix of inverse function beltrami. By <Ref>, <ref> in <Ref>, and A∈ DMO_p with p>λ we obtain that B∈ DMO_1. By <Ref> we have that R∈ C^1, i.e., |∇ R| is bounded around ∂ϕ_A(Ω), and by the relation ∇ I = * B ∇ R we also get that |∇ I| is bounded there. Hence, R= (ϕ^-1) and I= (ϕ^-1) are Lipschitz, and in particular ϕ^-1 is Lipschitz. The same proof of <Ref> in proof of non-regularity case main thm implies the σ-finite length.
			
			Let us see now the case A∈DMO_p with p>1 follows from the previous reduction. As A∈DMO_p ⊂DMO_1, by <cit.> we have that it agrees a.e. with a matrix with (uniformly) continuous coefficients, which by the L_A weak form we can assume that we are in this later case. In particular, given ε>0, for each ξ∈Ω there is δ_ξ,ε>0 such that
			
    z∈6B_δ_ξ,ε(ξ) |a_ij(z)-a_ij(ξ)|<ε for any i,j∈{1,2}.
 
			Let {B_i}_i⊂{B_s(ξ)}_ξ∈∂Ω,0<s≤δ_ξ,ε be a countably subfamily of disjoint balls such that ω^p (∂Ω∖⋃_i B_i)=0, by Vitali's covering theorem. The goal is to prove that for each index i there exists a set F_i⊂∂(Ω∩ 4B_i) satisfying ω_Ω∩ 4B_i,A(F_i)=1 and with σ-finite ^1(F_i), because by <Ref> we would have that the set F=⋃_i F_i satisfies ω_Ω,A (F)=1 and of course it has σ-finite length.
			
			Let us now fix an index i. Let ξ_i denote the center of the ball B_i, and r_i its radius. Let S=√(A(ξ_i)) be the matrix satisfying S^T S=A(ξ_i), and M as in matrix after change of constant sym change of variables, i.e.,
			
    M(z)=Id + (S^T)^-1(A(S^T)-A(ξ))S^-1matrix after change of constant sym change of variables - sym case= (S^T)^-1A(S^T)S^-1,

			which is symmetric and with determinant 1, see <Ref>. Then, by <Ref> we have
			
    ω^q_Ω∩ 4B_i, A (E) = ω_(S^T)^-1(Ω∩ 4B_i), M^(S^T)^-1q ((S^T)^-1E),

			for any q∈Ω∩ 4B_i and E∈∂ (Ω∩ 4B_i). Given φ∈ C^∞_c (6B_i) such that 0≤φ≤ 1, φ_|5B_i=1 and |∇φ |≲ 1/r_i, consider the function φ= φ∘ S^T. Using this function φ, we can directly replace the matrix M in the right-hand side above by simply the uniformly elliptic, symmetric and with determinant 1 matrix M such that φμ_M = μ_M, see the precise definition in <Ref>. That is,
			
    ω^q_Ω∩ 4B_i, A (E) = ω_(S^T)^-1(Ω∩ 4B_i), M^(S^T)^-1q ((S^T)^-1E)

			for any q∈Ω∩ 4B_i and E∈∂ (Ω∩ 4B_i).
			
			From the proof of <Ref>, M has ellipticity constant λ_i=(1-2ελ)^-1 in (S^T)^-1(6B_i). This implies that |μ_M|≤ (λ_i-1)/(λ_i+1) in (S^T)^-1(6B_i). In particular, |μ_M| = |φμ_M|≤ |_(S^T)^-1(6B_i)μ_M|≤ (λ_i-1)/(λ_i+1) everywhere, implying that M has ellipticity constant λ_i=(1-2ελ)^-1 everywhere. If ε>0 is small enough then 1≤λ_i=(1-2ελ)^-1 <p; recall that p>1 is the value so that A∈DMO_p.
			
			Let us see M∈DMO_p. As A∈DMO_p, we have that (S^T)^-1A (·)S^-1∈DMO_p, and as z↦ S^T z is a linear mapping we get (S^T)^-1A(S^T ·)S^-1∈DMO_p. Using now that φ= φ∘ S^T is smooth, bounded and with compact support, applying <Ref> we conclude that M∈DMO_p.
			
			As the ellipticity constant of M is 1≤λ_i <p and M∈DMO_p, we are in the case of the second paragraph in this proof, and hence there is a set F_i⊂∂(Ω∩ 4B_i) satisfying ω_(S^T)^-1(Ω∩ 4B_i), M(F)=1 and with σ-finite ^1(F_i). In particular ω_Ω∩ 4B_i, A (S^T(F))=1 and S^T (F) has σ-finite length, and the theorem follows.
		
	
	

  §.§.§ Mean oscillation with logarithm decay

	
	In this section, we consider matrices with the quantitative log decay of the mean oscillation in <cit.>.
	
		We say that f∈ L_^1 (^n+1) is of vanishing mean oscillation with logarithmic decay, f ∈_log for short, if
		
    sup_x∈^n+1
     s∈ (0,r][(1+|log s|)_Q_s(x)|f-m_Q_s(x) f| ]  0,

		recall Q_s(x) denotes the cube with center x and side length 2s.
	

        
            The logarithmic decay in <ref> implies _log⊂. On the other hand, for a function f∈ DMO_1, given ε >0 we can take s_0 =s_0(ε) such that ∫_0^s_0 f_ MO,1 (t)  dt/t < ε, and hence (1+|log t|)f_ MO,1 (t) < t (1+|log t|) ε / s_0 for all 0<t< s_0. Thus, we can take δ = δ (ε) small enough so that 0<t<δ implies (1+|log t|)f_ MO,1 (t) <ε. Interchanging balls B(x,t) by cubes Q_t(x) in the definition of f_ MO,1 (t) we obtain that f satisfies <ref>. That is, DMO_1 ⊂_log. All in all,
            
    DMO_1 ⊂_log⊂.

        
	
	The fundamental property local equivalence bmo norm of  functions implies that, for 1≤ p <∞, f∈ and r∈(0,1), we have
	
    sup_x∈^n+1
     s∈ (0,r][(1+|log s|) _Q_s(x) |f-m_Q_s(x) f| ]
            ≈_p sup_x∈^n+1
     s∈ (0,r][(1+|log s|) (_Q_s(x) |f-m_Q_s(x) f|^p)^1/p],

	and therefore f∈_log if and only if for some (and hence for all) 1≤ p <∞ the right-hand side term above goes to 0 as r→ 0. Indeed, for x∈^2 and s∈ (0,r], using local equivalence bmo norm we have
	
    (_Q_s(x) |f-m_Q_s(x) f|^p)^1/p≤sup_Q ⊂ Q_s(x)(_Q |f-m_Q f|^p)^1/p≈_p sup_Q ⊂ Q_s(x)_Q |f-m_Q f|.

	Since |log t| is decreasing in (0,1) and every Q⊂ Q_s(x) satisfies ℓ (Q) ≤ 2s, we obtain
	
    sup_Q⊂ Q_s(x) (1+|log s|) _Q |f-m_Q f| 
    			   ≤sup_Q⊂ Q_s(x)(1+|logℓ(Q)/2|) _Q |f-m_Q f| 
       ≤sup_x∈^2
     s∈ (0,r][(1+|log s|) _Q_s(x) |f-m_Q_s(x) f| ],

	and inequality ≳_p in eq:vmo log only matters one exponent follows from eq:vmo log only matters one exponent step 1 and eq:vmo log only matters one exponent step 2 by taking the supremum over x∈^2 and s∈(0,r]. The other inequality in eq:vmo log only matters one exponent is just Hölder's inequality.
	
	Assuming that the matrix has _log coefficients, the following result states that L_A-harmonic solutions have the gradient in .
	
	
		Let A ∈ L^∞ (Q_3(0)) ∩_log and u ∈ W^1,2 (Q_3 (0)) be a L_A-solution in Q_3 (0). Then ∇ u ∈ (Q_1 (0)).
	
	
	In the following lemma, we see that functions with derivatives in the space  satisfy the so-called log-Lip regularity.
	
	
		Let f : B_1(0) ⊂^n+1→ be a function with ∇ f ∈ L^1 (B_1 (0)) ∩ (B_1(0)), and let x,y∈ B_1/2 (0). If |x-y| is small enough, only depending on the L^1 and  norms, then
		
    |f(x)-f(y)| ≲∇ f_* |x-y| log1/|x-y|.

		
			Consider B B(x+y/2, |x-y|) ⊂ B_1 (0), and write r_B = |x-y|. Clearly,
			
    |f(x)-f(y)| ≤ |f(x)-m_B f|+|f(y)-m_B f|

			and by symmetry, it suffices to control one of them. By <cit.>,
			
    |f(x)-m_B f| ≲∫_B |∇ f (z)|/|x-z|  dz .

			Splitting the domain of integration B of the right-hand side integral in annulus, we obtain
			
    ∫_B |∇ f (z)|/|x-z|  dz 
    				   = ∑_k≥ 0∫_2^-kB ∖ 2^-(k+1)B|∇ f (z)|/|x-z|  dz 
    				≈2^k/r_B∑_k≥ 0∫_2^-kB ∖ 2^-(k+1)B |∇ f (z)|   dz 
       ≲ r_B ∑_k≥ 0 2^-k_2^-k B |∇ f (z)|   dz.

			
			Let us bound each element on the right-hand side in terms of k≥ 0. For each k≥0, let N_k ∈ the largest index such that 2^N_k r_2^-k B = 2^N_k 2^-k r_B ≤ 1/2, i.e., N_k ⌊log_2 2^k/2|x-y|⌋. With this choice,
			
    _2^-k B |∇ f (z)|   dz
    				= ∑_m=0^N_k -1(_2^m 2^-k B |∇ f (z)|  dz - _2^m+1 2^-k B |∇ f (z)|   dz) 
    				+_2^N_k B |∇ f (z)|   dz .

			Each element in the sum is controlled in absolute value by ≲∇ f_*. The term _2^N_k B |∇ f (z)|   dz is controlled by ≲∇ f_L^1 (B_1 (0)) as 2^N_k B ⊂ B_1 (0) and 2^N_kr_B ≈ 1 by the choice of N_k. In particular
			
    _2^-k B |∇ f (z)|   dz ≲ N_k ∇ f_* + ∇ f_L^1 (B_1 (0)) .

			If |x-y| is small enough depending on L^1 and  norm, more precisely, take small enough |x-y| to satisfy ∇ f_L^1 (B_1 (0))≤⌊log_2 1/2|x-y|⌋∇ f_*, then ∇ f_L^1 (B_1 (0))≤ N_k ∇ f_*, and therefore
			
    _2^-k B |∇ f (z)|   dz ≲ N_k ∇ f_* .

			
			Summing over k≥ 0,
			
    |f(x)-m_B f| ≲∫_B |∇ f (z)|/|x-z|  dz 
    			≲ r_B ∑_k≥ 0 2^-k_2^-k B |∇ f (z)|   dz 
    			≲ r_B ∇ f_* ∑_k≥ 0 2^-k N_k .

			Recall N_k =  ⌊log_2 2^k/2|x-y|⌋≈log2^k/|x-y|. Hence, the sum in the right-hand side is controlled by
			
    ∑_k≥ 0 2^-klog2^k/|x-y| 
    			= ∑_k≥ 0 2^-klog 2^k 
    			+ log1/|x-y|∑_k≥ 0 2^-k .

			As ∑_k≥ 0 2^-klog 2^k < ∞, for small enough |x-y| we have ∑_k≥ 0 2^-klog 2^k ≤log1/|x-y|, and hence ∑_k≥ 0 2^-klog2^k/|x-y|≲log1/|x-y|.
		
	
	
	Here we see the distortion of Hausdorff measures under log-Lip functions.
	
	
		If f : ^n+1→^n+1 satisfies |f(x)-f(y)| ≤ C_logLip |x-y| log1/|x-y|, g(t) = t/log1/t, h(t) = tlog1/t, and A ⊂^n+1, then
		
			
  * ^g (f(A)) ≤ 2C_logLip^1 (A), and
			
  * ^φ_ρ,C (f(A)) ≤ 2C_logLip^2/ρ+1^(φ_ρ,C) ∘ h (A).
		
		
			For small enough δ >0, the function t ↦ tlog1/t is increasing in t∈ (0,δ). Let η = C_logLipδlog1/δ>0, and note that η→ 0 as δ→ 0.
			
			For ε >0, let {E_i}_i such that A⊂⋃_i E_i, E_i < δ and ∑_i  E_i ≤^1_δ (A) + ε. Then f(A) ⊂⋃_i f(E_i) and (f(E_i)) ≤ C_logLip E_i log1/ E_i≤ C_logLipδlog1/δ = η. As g(t) = t / log1/t is increasing in (0,η) if η is small enough, we have
			
    g( f(E_i))    ≤ g(C_logLip E_i log1/ E_i) = C_logLip E_i log1/ E_i/log(1/C_logLip E_i log1/ E_i)
        = C_logLip E_i log1/ E_i/log1/C_logLip + log1/ E_i + loglog1/ E_i≤ 2C_logLip E_i ,

			where we used that E_i <δ is small enough. All in all,
			
    ^g_η (f(A)) ≤∑_i g( f(E_i)) ≤ 2C_logLip∑_i  E_i ≤ 2C_logLip (^1_δ (A) + ε) ≤ 2C_logLip (^1 (A) + ε) .

			Letting ε→ 0 and δ→ 0, i.e., η→ 0, we get ^g (f(A)) ≤ 2C_logLip^1 (A) as claimed.
			
			Similarly, for ε >0, let {E_i}_i such that A⊂⋃_i E_i, E_i < δ and ∑_i ((φ_ρ,C) ∘ h) ( E_i) ≤_δ^(φ_ρ,C) ∘ h (A) + ε. As before,
			
    ^φ_ρ,C_η (f(A))    ≤∑_i φ_ρ,C ( f(E_i)) ≤∑_i φ_ρ,C (C_logLip h( E_i)) 
       ≤ 2C_logLip^2/ρ+1∑_i ((φ_ρ,C) ∘ h) ( E_i)
    				≤ 2C_logLip^2/ρ+1 (^(φ_ρ,C) ∘ h (A) + ε),

			where we used φ_ρ,C (C_logLip h( E_i))≤ 2C_logLip^2/ρ+1φ_ρ,C(h( E_i)), see the proof of <Ref>. Letting ε, δ→ 0, and hence also η→ 0, we obtain ^φ_ρ,C (f(A))≤ 2C_logLip^(φ_ρ,C) ∘ h (A).
		
	
	
	
		_log,c is invariant under global homeomorphic quasiconformal mappings.
		
			Let f∈_log be non-identically zero (otherwise we are done) with compact support and ϕ∈ W^1,2_ () be a K-quasiconformal homeomormism of , with K≥ 1. Let B be a ball such that (f∘ϕ^-1)⊂B/2, r < r(B)/100 and fix C C_K,2B, ϕ, the constant in <Ref> so that r_B_-≤C r_B^1/K for every ball B⊂ 2B. For x∈B and s∈ (0,r] (otherwise the first integral below is zero), by <Ref> and local equivalence bmo norm[The last step using local equivalence bmo norm is unnecessary since the rest of the proof works with the exponent 2K by <Ref>. However, to simplify the notation, we remove the exponent 2K.] we have
			
    _Q_s(x)|f∘ϕ^-1-m_Q_s(x) (f∘ϕ^-1)|
    					   ≤sup_Q ⊂ Q_s(x)_Q|f∘ϕ^-1-m_Q (f∘ϕ^-1)| 
       ≲sup_Q ⊂ Q(ϕ(x),Cs^1/K)(_Q|f-m_Q f|^2K)^1/2K
       ≈sup_Q ⊂ Q(ϕ(x),Cs^1/K)_Q|f-m_Q f|.

			As we did in eq:vmo log only matters one exponent step 2, we obtain
			
    (1+|log (Cs^1/K)|) sup_Q ⊂ Q(ϕ(x),Cs^1/K)_Q|f-m_Q f|
    				≤sup_x∈^2
     s∈ (0,Cr^1/K][(1+|log s|) _Q_s(x)|f-m_Q_s(x) f| ].

			The lemma follows from log decay new matrix B step 1, log decay new matrix B step 2, the fact that (1+|log s|) ≤ 2K (1+|log (Cs^1/K)|) for small enough s, and finally taking r→ 0.
		
	
	
	Using the previous results, we conclude this section with the following result about the dimension of the elliptic measure for _log matrices.
 
		
			Recall we can assume that Ω⊂ B_1/4 (0) is bounded Wiener regular by <Ref>, and we can replace finitely connected by simply connected by <Ref>. In particular Ω⊂ Q_1/3 (0) (cube centered at 0 with side length 2/3). Without loss of generality, we can also assume A=1.
			
			Let φ(·) = φ(·/100), with φ as in definition of the support function, and let ϕ_A ∈ W^1,2_() be the normalized solution of the Beltrami equation
			
    ∂̅ϕ_A = φμ_A ∂ϕ_A  a.e. in ,  where μ_A  as in beltrami coefficients for sym and det 1 matrices,

			given by the measurable Riemann mapping theorem, see <Ref>. Let A be the uniformly elliptic, symmetric and with determinant 1 matrix such that φμ_A = μ_A, see <Ref>. Therefore, the Beltrami equation above reads as
                
    ∂̅ϕ_A = μ_A∂ϕ_A  a.e. in ,  with μ_A = μ_A  in  B_50 (0).

            In particular uϕ_A and vϕ_A satisfy A ∇ u = 0 and A ∇ v = 0 in B_50 (0), see <Ref>. Now, since A∈_log we have that ∇ u, ∇ v ∈, see <Ref>.
			
			Now we want to see the same property for ϕ_A^-1. By <Ref>, the functions R (ϕ_A^-1) and I (ϕ_A^-1) satisfy B∇ R=0 and B^* ∇ I=0 in ϕ_A(B_50(0)), where B and B^* are the uniformly elliptic matrices in matrix of inverse function beltrami. Since A∈_log, by <Ref> we have that μ_A ∈_log and hence μ_A∈_log,c. By the invariance of _log,c under quasiconformal mappings in <Ref>, we have that -μ_A∘ϕ_A^-1∈_log,c, the coefficient of the conjugated Beltrami equation in <ref> of <Ref>. Therefore, by <Ref> the uniformly elliptic matrices B and B^* in <ref> (in terms of A) belong to _log. By <Ref> and since B = B and B^* = B^* in ϕ_A(B_50(0)), we conclude ∇ R, ∇ I∈.
			
			Since both ∇ϕ_A, ∇ϕ_A^-1∈, by the log-Lip regularity in <Ref> and the distortion under these type of functions in <Ref>, continuing as in the end of the proof of <Ref> in proof of non-regularity case main thm we have that there is a set F⊂∂Ω satisfying ω_Ω, A (F)=1 and with σ-finite ^t/log1/t measure; and if the domain is simply connected, by Makarov's <Ref> there exists C>0 such that ω_Ω, A≪^(φ_1,C) ∘ (tlog1/t).
		
	
	

§ UNIFORMLY ELLIPTIC MATRICES IN CDC DOMAINS

	
	This part is dedicated to the proof of <Ref>. That is, we work with general uniformly elliptic matrices A∈^2× 2 (not necessarily symmetric nor constant determinant) in CDC domains. 

        

 §.§ Elliptic measure and Green's function in CDC domains

	
	
		A domain Ω⊂^n+1, n≥ 1, satisfies the capacity density condition, CDC for short, if there exist a constant c_0>0 and a radius r_0 >0 such that 
		
    (B(x_0,r)∩Ω^c, B(x_0,2r) ) ≥ c_0 r^n-1

		for all x_0∈∂Ω, x_0 ≠∞ and r≤ r_0. We write (c_0,r_0)-CDC in case we want to stress the parameters, and c_0-CDC if there is r_0>0 such that Ω is (c_0,r_0)-CDC.
	

        In particular, CDC domains are Wiener regular, see <cit.>.
        
        A key property in CDC domains is the following uniform control of the elliptic measure, the so-called Bourgain's lemma, see <cit.> for the harmonic case.

        
            Let Ω⊂^n+1, n≥ 1, be a bounded (c_0, r_0)-CDC domain, and let A be a uniformly elliptic matrix. Then there exists a constant τ∈ (0,1), depending only on n, c_0 and the ellipticity constant of A, such that for every E⊂∂Ω, x_0∈∂Ω and 0<r≤ r_0, the following holds:
            
                
  * if B(x_0, 2r)∩ E =∅, then ω_Ω,A^p (E)≤ 1-τ <1, and
                
  * if B(x_0, 2r)∩∂Ω⊂ E, then ω_Ω,A^p (E)≥τ >0,
            
            for any point p∈ B(x_0,r)∩Ω.
        

        Let Ω⊂^n+1, n≥ 1, be a bounded Wiener regular domain and A∈^(n+1)× (n+1) be a uniformly elliptic matrix. We denote by g_x^Ω,A (·) the Green function in Ω with pole at x∈Ω of the operator L_A = - A∇ constructed in <cit.> in the planar case and <cit.> in higher dimensions. For all x∈Ω, g_x^Ω,A satisfies[If Ω⊂^2 is bounded, then Y^1,2_0 (Ω) in <cit.> is precisely W^1,2_0 (Ω), see between Lemma 3.1 and Section 3.1 of <cit.>.]
    
    g_x^Ω,A∈ W^1,2 (Ω∖ B_r(x)) ∩ W^1,s_0 (Ω),  for all s∈ [1,(n+1)/n)  and  0<r<(x,∂Ω),

    and
    
    ∫_Ω  A (z) ∇ g_x^Ω,A (z) ∇φ (z)   dz = φ (x),   for all φ∈ C_c^∞ (Ω).

    In particular, g_x^Ω,A is L_A-harmonic in Ω∖{x}. By <ref> and a density argument, <ref> remains true for φ∈ W^1,s^'_c (Ω), with n+1 < s^' < ∞.

    We remark that g_x^Ω,A(y) ≥ 0 for each y∈Ω∖{x}, see <cit.> in higher dimensions and <cit.> for the planar case. For extra properties with proofs in the planar case, see <cit.> and the references therein.

    
        If the set Ω, the matrix A, or the pole x∈Ω is clear from the context, we will omit it in g_x^Ω,A. We will mainly skip the domain Ω in g_x^Ω,A, that is, g_x^A.
    
	
        Given a bounded Wiener regular domain Ω⊂^n+1, n≥ 1, a uniformly elliptic matrix A, and a ball B centered at ∂Ω, we claim
	
    ω^p_Ω,A (B) ≲ r_B^n-1max_x∈Ω∩ 2B g_p^A^T (x)   for all p∈Ω∖ 4B,

	with constant depending on the ellipticity of A. This follows from the identity
    
    ∫_∂Ωφ (ξ)   d ω^x_Ω,A (ξ) -φ(x) = -∫_Ω A^T (z)  ∇ g_x^T (z) ∇φ (z)   dz,   for a.e.  x∈Ω,

    where φ∈ C(Ω)∩ W^1,2(Ω), see <cit.> for a proof in the planar case. Indeed, taking φ∈ C_c^∞ (^2) such that φ=1 in B, φ=0 in (1.5B)^c and |∇φ| ≲ 1/ r_B in 1.5B ∖ B, by <ref>, the ellipticity condition, and Hölder's and Cacciopolli's inequalities, for a.e. p∈Ω∖ 4B we get
	
    ω^p_Ω,A (B) ≤∫_∂Ωφ(ξ)   dω^p_Ω,A (ξ) 
    	= -∫_Ω A^T ∇ g_p^A^T∇φ≲1/r_B∫_1.5B |∇ g_p^A^T|
    	≲ r_B^n (1/r_B^2_2B |g_p^A^T|^2 )^1/2.

	The last element is trivially bounded by the maximum value of r_B^n-1 g_p^A^T in Ω∩ 2B, and elliptic measure bounded by max of green function holds for all p∈Ω∖ 4B by continuity.
	
	For CDC domains, the following weaker reverse inequality holds.
	
	
		Let Ω⊂^n+1, n≥ 1, be a bounded (c_0,r_0)-CDC domain, A be a uniformly elliptic matrix and B be a ball centered at ∂Ω. If r_B≤ r_0 then
		
    r_B^n-1 g_p^A^T (x) ≲ω_Ω,A^p (4B)  for  x∈Ω∩ B  and  p∈Ω∖ 2B,

        with constant depending only on c_0 and the ellipticity of A.
	

        In higher dimensions, n≥ 2, for bounded Wiener regular domains there holds the refined inequality[In fact, one can replace 4B by aB with a>0 and the involved constant does not depend on a.]
        
    r_B^n-1 g_p^A^T (x) inf_z∈ 2Bω_Ω,A^z (4B) ≲ω_Ω,A^p (4B)  for  x∈Ω∩ B  and  p∈Ω∖ 2B,

        see <cit.> for a proof in the harmonic case, which also works in the elliptic case, see <cit.>. In particular, when the domain is CDC, by <Ref> we have inf_z∈ 2Bω_Ω,A^z (4B) ≳ 1, and therefore the finer formulation thinner green bounded by elliptic measure is equivalent to <Ref>.

        We remark that the proof of thinner green bounded by elliptic measure relies on the good behavior of the Green function (in fact, the fundamental solution) in higher dimensions, in contrast to what happens in the planar case. However, for planar CDC domains the Green function behaves well in a region around the pole.

        
            Let Ω⊂^2 be a bounded (c_0,r_0)-CDC domain, and A be a uniformly elliptic matrix. Let y ∈Ω be such that (y,∂Ω)<r_0. Then
            
    g_y^A (x) ≈ 1  for  x ∈∂ B(y,(y,∂Ω)/2),

            with constant depending only on c_0 and the ellipticity of A.
        

        We leave its proof for the end of this section. Granted this, we obtain <Ref>.

        
            Fix x∈Ω∩ B. Since r_B≤ r_0, by <Ref> we have g_x^A (·) ≲ 1 in ∂ B(x,(x,∂Ω)/2). On the other hand, by <Ref> there holds ω_Ω,A^· (4B)≳ 1 in 2B⊃∂ B(x,(x,∂Ω)/2). In particular, g_x^A (·)≲ω_Ω,A^· (4B) in ∂ B(x,(x,∂Ω)/2), and trivially also in ∂Ω. By the maximum principle, the same inequality holds in Ω∖ B(x,(x,∂Ω)/2), and the lemma follows by taking p∈Ω∖ 2B and g_p^A^T(x)=g_x^A (p).
        

        Although the good behavior in <Ref> is not known for general planar domains, in the harmonic case, in <cit.>, there is a planar version of thinner green bounded by elliptic measure with _B |g^Id_x (y) - m_B g^Id_x|  dy instead of g_p^Id(x). Moreover, in <cit.>, a pointwise estimate involving the logarithmic capacity is proved, which in particular gives <Ref> in the harmonic case, with different dilatations of the ball.

        We now turn to the proof of <Ref>. In the harmonic case, it is stated and proved in <cit.>[The CDC is with respect the logarithmic capacity, see <cit.>.]. Even though the same proof can be applied with minor technical changes to the elliptic case, for the sake of completeness all the details are provided here, as the author has found neither the statement nor any proof in the literature for the elliptic case. The rest of this subsection is dedicated to proving <Ref>.

        Although in <cit.> only the harmonic version of the following lemma is stated, exactly the same proof applies in this case.

        
            Let Ω∈^n+1, n≥ 1, be a domain and A be a uniformly elliptic matrix, and ξ∈∂Ω. Suppose that there exists δ∈ (0,1) such that ω_Ω∩ B(ξ,r),A^x (∂ B(ξ,r)∩Ω) ≤β < 1 for x∈∂ B(ξ, δ r)∩Ω and r∈ (0,R). Then there is α = α(β,δ) so that, for all r∈ (0,R),
            
    ω_Ω,A^x (B(ξ,r)^c) ≲(|x-ξ|/r)^α for  x∈Ω∩ B(ξ,r),

            with constant depending only on β and δ.
        

        We note that if Ω is (c_0,r_0)-CDC, then for any ball B centered at ∂Ω, the domain Ω∩ B⊂Ω is also (c_0,r_0)-CDC, just because the complementary increases. Therefore, for any ξ∈∂Ω and r ≤ 2r_0, by <Ref> we have ω_Ω∩ B(ξ,r),A^x (∂ B(x,r)∩Ω) ≤ 1-τβ for all x∈ B(ξ, r/2)∩Ω, and in particular, we can apply the previous lemma, with δ = 1/4 say. Consequently, for (c_0,r_0)-CDC domains Ω, <Ref> holds with R=2r_0 for all ξ∈∂Ω, where the constant now depends on c_0 and the ellipticity of A.
        
        In particular, given a (c_0,r_0)-CDC domain Ω and ξ∈∂Ω, since Ω∩ B(ξ,r_0) is (c_0,r_0)-CDC as well, for all r∈ (0,2r_0) there holds
        
    ω_Ω∩ B(ξ,r_0),A^x (B(ξ,r)^c)
            ≲(|x-ξ|/r)^α for  x∈Ω∩ B(ξ,r),

        where the constant depends only on c_0 and the ellipticity of A.

        We adapt the proof in <cit.> with the technical changes for the elliptic case.

        
            The ellipticity constant of A is denoted by λ≥ 1. To simplify the notation, we also write ω_·,A = ω_·, g_y = g_y^A and R(y,∂Ω)<r_0.

            For the lower bound, let g_y^R g_y^B_R(y),A and note that g_y^R≤ g_y in B_R(y) by the maximum principle, as g_y-g_y^R is L_A-harmonic in B_R(y) and positive in ∂ B_R (y). Finally, g_y^R≈_λ 1 on ∂ B(y,R/2) and the lower bound is proved. This follows by writing g_y^R in terms of the Green function g_0^B_1(0),A where A (·)= A(y+R·) has the same ellipticity constant, and the pointwise estimates of the fundamental solution, see <cit.> and the references therein. Note that the lower bound holds for general Wiener regular domains.
            
            We now turn to the upper bound. Let M_0 = sup_∂ B(y,R/2) g_y (·), and in particular, g_y (·) ≤ M_0 in Ω∖ B(y,R/2) by the maximum principle. Let y^∗∈∂Ω be such that |y-y^∗|=(y,∂Ω)=R. Thus, by the maximum principle we have
            
    g_y (z) ≤ M_0 ω_Ω∩ B(y^∗, R/2)^z (Ω∩∂ B(y^∗, R/2))  for z∈Ω∩ B(y^∗, R/2).

            By <Ref> there exists 0<ε_1=ε_1 (c_0,λ)<1/4 such that if z∈Ω∩ B(y^*,2ε_1 R), then
            
    ω_Ω∩ B(y^∗, R/2)^z (Ω∩∂ B(y^∗, R/2))
                ≤ω_Ω∩ B(y^∗, R/2)^z (B(y^*, R/4)^c)
                ≤ C ε_1^α≤1/2.

            Therefore, combining the latter estimates we get
            
    g_y (z)≤ M_0 /2,  for z∈Ω∩ B(y^*,2ε_1 R).


            We will compare the Green function g_y with the Green function g_y^𝐁 g_y^𝐁,A in the ball 𝐁 B(y, (1-ε_1)R). Let us cover ∂𝐁 with a finite number N=N(ε_1)=N(c_0,λ) of balls. Let y_1 be the point in the segment yy^∗ with |y_1-y^∗|=ε_1 R. Fix N⌈200π/ε_1⌉ and assume that we have already defined y_1,… y_j. Let y_j+1 be the point in ∂𝐁 with ∠ (y_j+1-y,y_j-y)=2π/N >0.[This condition is just to ensure that we always choose the point going in the anticlockwise direction. We could do the same procedure in the clockwise direction as well.] This procedure gives N points as y_N+1=y_1. Let r |y_2-y_1|, and for each 1≤ j≤ N, we define B_j B(y_j,r). As all the points are equidistributed in ∂𝐁, in particular y_j-1,y_j+1∈∂ B_j for all 1≤ j≤ N, with the correspondence y_0=y_N and y_N+1=y_1, and clearly ∂𝐁⊂⋃_j=1^N B_j. Moreover, r=|y_2-y_1|≤^1 (arc (y_1 y_2))=2π(1-ε_1)R/N≤2π R/N≤ε_1 R/100 by the choice of N. In particular, g_y (·)≤ M_0 / 2 in B_1.

            We claim that if g_y ≤ (1-α)M_0 in B_j for some α>0, then g_y≤ (1-τα)M_0 in B_j+1, with τ∈ (0,1) as in <Ref>. Indeed, since the annulus 𝒜 25B_j∖B_j satisfies 𝒜̅⊂ B(y,R)∖B(y,R/2), in particular g_y≤ M_0 in 𝒜̅. Since ω_𝒜^z (∂ 25B_j)=1 for z∈∂ 25 B_j and vanishes on ∂ B_j, we can write
            
    g_y (z) ≤α M_0 ω_𝒜^z (∂ 25B_j) + (1-α)M_0  for  z∈∂𝒜.

            By the maximum principle, the same holds for z∈𝒜̅. The annulus 𝒜 is a (c_0,9r)-CDC domain with universal constant c_0 and radius r_0=9r, see <cit.>. In particular, for z∈B_j+1⊂ 2B_j+1=B(y_j+1,2r), since 4B_j+1∩∂ 25B_j = ∅, by <Ref> we have ω_𝒜^z (∂ 25B_j)≤ 1-τ. That is, g_y ≤ (1-τα)M_0 in B_j+1, as claimed.
            
            Since g_y≤ M_0 /2 in B_1, from the previous paragraph we conclude that g_y ≤ (1-c)M_0 on ∂𝐁, with c=0.5τ^N, that is, c=c(c_0,λ)∈ (0,1). Then,
            
    g_y^𝐁 (z) = g_y(z) - ∫_∂𝐁 g_y(ξ)  dω_𝐁,A^z (ξ) ≥ g_y(z) - (1-c)M_0  for  z∈𝐁,

            and hence
            
    sup_z∈∂ B(y,R/2) g_y^𝐁 (z) ≥ M_0 - (1-c)M_0 ≥ cM_0.

            As in the proof of the lower bound, we have g_y^𝐁 (z) ≈_λ 1 on ∂ B(y,R/2). Therefore, M_0 ≤ C(c_0,λ) and the upper bound is also proved.
        
	
	

 §.§ Proof of Theorem <ref>

	
	Let us start by the harmonic factorization of the Green function for the operator L_A, which turns out to be the harmonic Green function in the image domain.
	
	
		Let Ω⊂^2 be a bounded Wiener regular domain, p∈Ω, A be a uniformly elliptic matrix with A=Id in an open neighborhood of p. If μ_A,g_p^A and ϕ are as in <Ref>, then g_p^Ω,A∘ϕ^-1 = g_ϕ (p)^ϕ(Ω),Id is the Green function in ϕ(Ω) with pole ϕ(p) for the Laplacian operator L_Id=-Δ.
	
	
	
		Note that the Green function g_p^A has a singularity at p∈Ω. However, the Beltrami coefficient μ_A,g_p^A should be understood as μ_A,g_p^A=μ_A in the open neighborhood of p where the matrix A=Id, as ν_A=0 in this case. In particular, g_p^A is L_A-harmonic in Ω∖{x: A(x)≠ Id}⊂Ω∖{p}.
	
	
	
		We check ∫_ϕ (Ω)∇ (g_p^A ∘ϕ^-1) ∇φ =φ (ϕ (p)) for any φ∈ C^∞_c (ϕ (Ω)). Let B⊂ 10B⊂Ω be a ball centered at p∈Ω with small enough radius such that A=Id in 2B, and let ψ∈ C^∞_c (^2) be so that ψ_|B/2=1 and ψ_|B^c=0. Then
		
    ∫_ϕ (Ω)∇ (g_p^A ∘ϕ^-1) ∇φ
    		=∫_ϕ (Ω)∇ (g_p^A ∘ϕ^-1) ∇ (φ·(ψ∘ϕ^-1))
    		+∫_ϕ (Ω)∇ (g_p^A ∘ϕ^-1) ∇ (φ·(1-ψ∘ϕ^-1)) .

		Since φ (1-ψ∘ϕ^-1)∈ W^1,2_c (ϕ (Ω)) with φ (1-ψ∘ϕ^-1)=0 in ϕ(B/2)∋ϕ(p), and g_p^A ∘ϕ^-1 is harmonic in ϕ(Ω∖ B/2) by <Ref>, we have that the second term on the right-hand side is zero, and therefore
		
    ∫_ϕ (Ω)∇ (g_p^A ∘ϕ^-1) ∇φ
    		=∫_ϕ (Ω)∇ (g_p^A ∘ϕ^-1) ∇ (φ·(ψ∘ϕ^-1)) .

		As the matrix A=Id in 2B and ϕ is conformal in 2B, we have that the matrix after the change of variables is A_ϕ^-1 = Id in ϕ(2B), recall its definition in matrix pushforward. In particular, since ∇ (φ· (ψ∘ϕ^-1)) ⊂ϕ(B), we can just write
		
    ∫_ϕ (Ω)∇ (g_p^A ∘ϕ^-1) ∇φ
    		=∫_ϕ (Ω)A_ϕ^-1∇ (g_p^A ∘ϕ^-1) ∇ (φ·(ψ∘ϕ^-1)).

            By <cit.> (applied to every Ω^'⊂Ω∖{p}) we have
            
    ∇ (g_p^A ∘ϕ^-1) (z) = (D ϕ^-1 (z))^T ∇ g_p^A (ϕ^-1 (z)) for a.e.  z∈ϕ(Ω),

            and applying the change of variables given by ϕ, see <cit.>, we have
		
    ∫_ϕ (Ω)A_ϕ^-1∇ (g_p^A ∘ϕ^-1) ∇ (φ·(ψ∘ϕ^-1))
    		=∫_Ω  A ∇ g_p^A ∇( (φ∘ϕ)·ψ).

            By the higher integrability of quasiconformal mappings in <cit.>, see also higher integrability determinant, there exists ε>0 such that (φ∘ϕ)·ψ∈ W^1,2+ε_c (Ω), see <cit.> for instance[Cover φ by a finite number of balls B_i with 2B_i⊂ϕ(Ω).], and as noted right below <ref>, the integral above equals φ(ϕ(p))ψ(p)=φ(ϕ(p)), as claimed.
		
		As we have that both g_p^A ∘ϕ^-1 and g_ϕ(p)^Id are the dirac delta in the weak sense, we obtain
		
    ∫_ϕ (Ω)∇ (g_ϕ(p)^Id - g_p^A ∘ϕ^-1) ∇φ =0  for any φ∈ C^∞_c (ϕ (Ω)) .

            Since g_p^A ∈ W^1,s (Ω) and g_ϕ(p)^Id∈ W^1,s (ϕ(Ω)) for all 1≤ s < 2 by <ref>, in particular g_p^A ∘ϕ^-1, g_ϕ(p)^Id∈ W^1,1 (ϕ (Ω)), see <cit.> for instance. The existence of weak derivatives in L^1 (ϕ(Ω)) implies that the left-hand side is -∫_ϕ(Ω)φΔ (g_ϕ(p)^Id - g_p^A ∘ϕ^-1), and therefore
            
    ∫_ϕ (Ω)φΔ (g_ϕ(p)^Id - g_p^A ∘ϕ^-1) =0  for any φ∈ C^∞_c (ϕ (Ω)).

		By Weyl's lemma, g_ϕ(p)^Id - g_p^A ∘ϕ^-1∈ C^∞ (ϕ(Ω)) and is harmonic in ϕ(Ω). Since both functions g_ϕ(p)^Id and g_p^A ∘ϕ^-1 vanish on ∂ (ϕ (Ω)), in particular g_ϕ(p)^Id - g_p^A ∘ϕ^-1 vanishes on ∂ (ϕ(Ω)). By the maximum principle for harmonic functions, the function g_ϕ(p)^Id - g_p^A ∘ϕ^-1 must be zero in ϕ(Ω), and therefore g_ϕ(p)^Id = g_p^A ∘ϕ^-1 in ϕ(Ω), as claimed.
	
	
	
	Given a function ϕ as in <Ref> for the matrix A^T, using the notation in matrix pushforward and <Ref>, recall that Id_ϕ denotes the matrix such that
	
    u  is harmonic in ϕ(Ω)  u∘ϕ is  L_Id_ϕ-harmonic in Ω .

        With this definition, in the previous lemma we saw that g_p^A^T is the Green function in Ω for both operators L_A^T and L_Id_ϕ. Consequently, we obtain the following relation between their elliptic measures. When applying elliptic measure bounded by max of green function and <Ref> in the following proof, note that the matrix Id_ϕ is symmetric.
	
	
		Under the same hypothesis in <Ref>, assume also that Ω is (c_0,r_0)-CDC. If μ_A^T,g_p^A^T and ϕ are as in <Ref>, then for any ball B centered at ∂Ω with r_B≤ r_0 /2 and satisfying that p∈Ω∖ 4B, there holds
		
    ω_Ω,A^p (B) ≲ω_Ω,Id_ϕ^p (8B)  and ω_Ω,Id_ϕ^p (B) ≲ω_Ω,A^p (8B),

            where the constants depend only on c_0 and the ellipticity of A.
		
			Let ω = ω_Ω,A^p and ω= ω_Ω,Id_ϕ^p. By elliptic measure bounded by max of green function we have ω^p (B) ≲max_x∈Ω∩ 2B g_p^A^T (x). By <Ref>, g_p^A^T is also the Green function for the operator with matrix Id_ϕ, that is g_p^A^T = g_p^Id_ϕ in Ω. Applying <Ref>, we get g_p^Id(x)≲ω(8B) for any x∈Ω∩ 2B. All in all, ω (B) ≲ω(8B). We can do the same argument to obtain ω(B) ≲ω (8B).
		
	
	
	
		Under the same hypothesis in <Ref>, assume also that Ω is c_0-CDC. If μ_A^T,g_p^A^T and ϕ are as in <Ref>, then
		
    ω_Ω,A^p (E) ≈ω_Ω,Id_ϕ^p (E)  for any  E⊂∂Ω ,

            where the constant depends only on c_0 and the ellipcity of A. By <Ref>, this is equivalent to
		
    ω^p_Ω, A (E) ≈ω_ϕ(Ω), Id^ϕ(p) (ϕ(E))  for any  E⊂∂Ω .

		
			Let ω = ω_Ω,A^p and ω= ω_Ω,Id_ϕ^p. For every ε>0 let V_ε⊃ E be an open set such that ω(V_ε) ≤ω(E) + ε.
			
			Split E=E_𝖽∪ E_𝗇𝖽, where “𝖽” stands for doubling and “𝗇𝖽” for non-doubling, as
			
    E_𝖽   {x∈ E : ∃{r_k}_k≥ 1 with r_kk→∞⟶ 0  such that ω(8B_r_k(x))≤ 100ω(B_r_k(x))},  and
    
    				E_𝗇𝖽    E∖ E_𝖽 .

			With this choice we have ω (E_𝗇𝖽) =0, see <cit.>, and hence ω (E) = ω (E_𝖽).
			
			For each x∈ E_𝖽 take an small enough radius r_x>0 such that
			
				
  *  8B_r_x(x) ⊂ V_ε,
				
  *  ω (8B_r_x (x)) ≲ω (B_r_x (x)), since x∈ E_𝖽; and
				
  *  ω(B_r_x (x)) ≲ω(8 B_r_x (x)), by <Ref>,
			
			and let {8B_i}_i be the subfamily of {8B_r_x (x)}_x∈ E_𝖽 covering E_𝖽 and with finite overlapping, given by the Besicovitch's covering theorem. Therefore,
			
    ω (E) =ω (E_𝖽) ≤ω(⋃_i 8B_i) ≤∑_i ω (8B_i) 
    				doubling ball,relation two elliptic measure≲∑_i ω(8B_i) ≈ω( ⋃_i 8B_i ) doubling balls inside open neigh of E≤ω(V_ε) ≤ω(E) + ε.

			Letting ε→ 0 we conclude ω (E) ≲ω(E). The lemma follows since the exactly same argument holds if we interchange ω and ω.
		
	
	
	Let us see the proof of <Ref>.
	
	
		Recall we can assume that Ω⊂ B_1/4 (0) is bounded Wiener regular by <Ref> and we can replace finitely connected by simply connected by <Ref>. Furthermore, if ε>0 is small enough so that B_2ε(p)⊂Ω, by <Ref> we can replace A by _B_1(0)∖B_ε(p) A + (_B_1^c(0)∪B_ε(p))Id.
		
		Let ϕ_A^T,g_p^A^T∈ W_^1,2() be the normalized solution of the Beltrami equation
		
    ∂̅ϕ_A^T,g_p^A^T = μ_A^T,g_p^A^T∂ϕ_A^T,g_p^A^T a.e. in ,  where μ_A^T,g_p^A^T as in coefficients classical beltrami equation in terms of u and A,

		given by the measurable Riemann mapping theorem, see <Ref>. In particular ϕ_A^T,g_p^A^T is a K_λ-quasiconformal homeomorphism with K_λ as in K-lambda ellipticity condition beltrami coefficient by <Ref>, and conformal in ∖𝔻. We remark that the dependence of the function ϕ_A^T,g_p^A^T in terms of g_p^A^T and A^T is due to the fact that μ_A^T,g_p^A^T depends on (the derivatives of) g_p^A^T and the coefficients of A^T.
		
		The theorem follows from the exactly same proof of the non-regularity case of <Ref> in proof of non-regularity case main thm, using ω_Ω, A≪ω_ϕ(Ω), Id (ϕ (·)) ≪ω_Ω, A by <Ref> (although we do not have the equality in relation elliptic measures) and that ϕ_A^T,g_p^A^T is K_λ-quasiconformal instead of λ-quasiconformal.
	
	
	

 §.§ Continuous matrices. Proof of Theorem <ref>

	
	In this section, we prove <Ref> via a localization argument. More precisely, we use the continuity of the matrix to see that locally, it is as close as we want to a constant matrix. Once in this case, we use the following remark.
	
	
		A quick look of the proof of <Ref> reveals that the same result remains true for CDC domains if we replace _(λ) by
		
    _^CDC(λ)sup{_ω_Ω, A : Ω⊂^n+1 is CDC and  A  has ellipticity constant λ}.

        That is, given a CDC domain Ω⊂^n+1 and a constant uniformly elliptic matrix C with ellipticity constant λ≥ 1, if A is a (not necessarily constant) matrix with small enough
		
    εmax_i,j∈{1,…,n+1}a_ij (·)- c_ij_L^∞ (Ω)

        depending on λ and n, then A is uniformly elliptic in Ω and _ω_Ω,A≤_^CDC ((1-(n+1)ελ)^-1).
	
	
	
		First, we prove the upper bound for CDC domains, and then the lower bound for finitely connected domains.
		
		“Upper bound”. It suffices to see that for every τ >0 there exists F_τ⊂∂Ω with _ F_τ≤ 1+τ and ω_Ω,A (F_τ)=1. Indeed, F⋂_n≥ 1 F_1/n satisfies _ F ≤ 1 and ω_Ω,A (F)=1.
		
		Since A has continuous coefficients in Ω, for each ξ∈∂Ω and every ε > 0 there is δ_ξ, ε >0 such that
		
    z∈4B_δ_ξ,ε (ξ) ∩Ω |a_ij (z) - a_ij (ξ)| < ε for any  i,j∈{1,2}.

		
		Fix ε >0. Let {B_i}_i ⊂{B_s (ξ)}_ξ∈∂Ω, 0<s≤δ_ξ, ε be a countable subfamily of disjoint balls such that ω^p (∂Ω∖⋃_i B_i) = 0, by Vitali's covering theorem.
		
		We claim that it suffices to see that for each index i there exists a set F_i ⊂∂ (Ω∩ 4B_i) with ω_Ω∩ 4B_i,A (F_i) = 1 and _ F_i ≤ 1+τ provided ε>0 is small enough. Indeed, this implies that the set F_τ∂Ω∩⋃_i F_i satisfies _ F_τ≤ 1+τ by definition, and ω_Ω,A (F_τ)=1 by <Ref>.
		
		Applying <Ref> for each index i, by continuity of the matrix C0 and as Ω∩ 4B_i is CDC we have that there exists a set F_i⊂∂ (Ω∩ 4B_i) with ω_Ω∩ 4B_i,A (F_i) = 1 and _ F_i ≤_^CDC ((1-2ελ)^-1). By <Ref> and taking ε>0 small enough respectively we have
		
    _^CDC ((1-2ελ)^-1) ≤2K_(1-2ελ)^-1/K_(1-2ελ)^-1+1≤ 1+τ,

		and the upper bound is proved.
		
		“Lower bound”. We turn now to the proof of the lower bound for finitely connected domains. Let F⊂∂Ω with ω_Ω,A(F)=1 and fix τ >0. Given ε >0, for each ξ∈∂Ω let δ_ξ, ε>0 be small enough to satisfy continuity of the matrix C0 and that Ω∩ B(ξ,δ_ξ,ε) is simply connected, see <Ref>. Let {B_i}_i ⊂{B_s (ξ)}_ξ∈∂Ω, 0<s≤δ_ξ, ε be a countable subfamily with finite overlapping by Besicovitch covering theorem. For each ball B_i, by <Ref> we have ω_Ω∩ B_i (∂Ω∖ F) =0, which implies 
		
    ω_Ω∩ B_i ((∂ B_i ∩Ω)∪(F∩ B_i)) = 1.

		From this, _∂ B=1, continuity of the matrix C0 and <Ref> we have
		
    _ F ≥_ (F∩ B_i) ≥_^sc ((1-2ελ)^-1).

		By <Ref> and taking ε>0 small enough respectively we have
		
    _^sc ((1-2ελ)^-1) ≥2/K_(1-2ελ)^-1+1≥ 1-τ.

		As this can be done for every τ>0 we have that _ F≥ 1 and the lower bound of _ω_Ω,A for finitely connected is proved.
	
	
	
	
	
	
	alpha
	
	 
	
	


 Paper Title: Comments on the regularity of harmonic maps between singular spaces.












Comments on the regularity of harmonic maps between singular spaces
    Luca Gennaioli [luca.gennaioli@sissa.itluca.gennaioli@sissa.it] Nicola Gigli [nicola.gigli@sissa.itnicola.gigli@sissa.it] Hui-Chun Zhang
[zhanghc3@mail.sysu.edu.cnzhanghc3@mail.sysu.edu.cn] Xi-Ping Zhu
[stszxp@mail.sysu.edu.cnzstszxp@mail.sysu.edu.cn]
    August 5, 2024
===============================================================================================================================================================================================================================================================





§ ABSTRACT 
In this work we are going to establish Hölder continuity of harmonic maps from an open set Ω in an RCD(K,N) space valued into a CAT(κ) space, with the constraint that the image of Ω via the map is contained in a sufficiently small ball in the target. Building on top of this regularity and assuming a local Lipschitz regularity of the map, we establish a weak version of the Bochner-Eells-Sampson inequality in such a non-smooth setting. Finally we study the boundary regularity of such maps.








§ INTRODUCTION

In the last 50 years the study of harmonic maps has been blooming and gained a lot of interest from the mathematical community. One of the main questions is the one of existence of such mappings and parallel to that there is the issue of their regularity.

When u:Ω⊆ M^n→ N^k is an harmonic map between Riemannian manifolds (M^n,g_M) and (N^k,g_N) the picture nowadays is quite clear: the existence of such mappings has been established via the study of parabolic problems by Hamilton (see <cit.> for a discussion on the topic) and then by looking at the problem in a variational way. The latter approach can be tailored to the non-smooth setting as well, indeed in the recent <cit.> the author has been able to prove the existence of harmonic maps between an RCD space and a CAT(κ) space if the image of u is contained in a sufficiently small ball. 

Back to the case of an harmonic map between smooth Riemannian manifolds, the Bochner-Eells-Sampson formula states that 

    Δ(| u|_ HS^2/2)=|∇ u|_ HS^2+ Ric_g_M(∇ u,∇ u)-∑_i,j≤ n⟨ u_*ℛ^N(e_i,e_j)e_i,e_j⟩,

where Ric_g_M is the Ricci tensor of the source space, u_*ℛ^N is the pullback of the curvature tensor of the target space via the map u and e_α_α=1^n is an orthonormal frame for the tangent bundle TM. If we assume that Ric_g_M≥ -K (lower bound on the Ricci tensor) and R_N≤κ (upper bound on the sectional curvatures), the previous identity can be turned into the following inequality

    Δ(| u|_ HS^2/2)≥ |∇ u|_ HS^2+ K| u|_ HS^2-κ | u|_ HS^4.

From this inequality, at least if κ=0 it is possible to quickly deduce that harmonic maps are locally Lipschitz, as in this case we have

    Δ(| u|_ HS^2/2)≥ K| u|_ HS^2

and thus a De Giorgi-Nash-Moser argument shows that the function f:=| u|_ HS^2 is locally bounded. The case κ>0, say κ=1, is more delicate and is known to require the additional assumption that the range of u is contained in a ball B_r(p)⊂ N^k of radius r<π2 (otherwise there are known counterexamples to regularity <cit.>). On top of this,  the term | u|^4 is a priori not in L^1, making  it hard to extract information from (<ref>). To overcome these difficulties,  Serbinowski argued as follows: the function f(x):= d_N(u(x),p) satisfies -Δcos(f)≥ | u|^2_ HScos(f) (as a consequence of the fact that u is harmonic and of the curvature assumption on N), and quite trivially we have || u|_ HS|^2≤ |∇ u|_ HS^2. These consideration and little algebraic manipulation show that (<ref>) implies

    | u|_ HScos(f) div(cos^2(f)∇(| u|_ HScos(f)))≥ K| u|^2_ HS,

and since cos(f) is far from zero, a Moser iteration argument can be called into play to prove that | u|_ HScos(f), and thus | u|_ HS, is locally bounded, as desired.

This type of reasoning allows to conjecture that, in the non-smooth setting, one should impose a lower bound on the Ricci curvature of the source and an upper bound on the sectional curvature on the target to get that an harmonic maps is (locally) Lipschitz.

Many contributions in this direction have appeared in the recent years: for an account of the story we refer to the extensive introductions in <cit.>, <cit.>, <cit.> and <cit.>. Here  we just recall that in  <cit.> the authors proved the Lipschitz regularity of harmonic maps between Alexandrov spaces and a weak Bochner-Eells-Sampson inequality. Building on this, in the more recent <cit.> and <cit.> the authors where able to establish such regularity when the source space is an RCD(K,N) space, namely a space with a synthetic notion of Ricci curvature bounded below by K and dimension bounded above by N, and the target is a CAT(0) space, namely a space with a synthetic notion of sectional curvature bounded above by 0. 



Very roughly said, the basic argument to get a sort of (<ref>) and local Lipschitz regularity of harmonic maps is to build two families (g_t),(h_t) of functions  (via a kind of Hopf-Lax formula for metric-valued maps) converging to | u|^2 in L^1 as t↓0 satisfying

    12Δ g_t≥ K h_t    ∀ t>0.

Quite clearly, from this it is possible to pass to the limit and obtain that

    Δ(| u|^2/2)≥ K| u|^2.

Notice that in this the quantity | u| is the operator norm of u, not its Hilbert-Schmidt norm as in (<ref>), thus  (<ref>) is not the same as (<ref>), but the effect is the same: a Moser iteration argument shows that | u| must be locally bounded and thus that u is locally Lipschitz.

When dealing with the case κ>0 this strategy encounters a problem, as the approximation procedure does not work well in conjunction with Serbinowski's technique: shortly said, at the approximated level the right hand side of (<ref>) still contains a term that does not go to 0 in L^1 as t↓0.


Because of these difficulties, we do not achieve Lipschitz regularity of harmonic maps in the mote general setting, our main results are rather:


  1) the proof of Hölder continuity, see Theorem <ref>. Here we follow the strategy in <cit.>.

  2) the higher integrability of the energy density, see Theorem <ref>, by using a Caccioppoli inequality and the Gehring lemma in <cit.>.

  3) Under the a priori assumption that the harmonic function is Lipschitz, possibly with a sub-optimal control on the Lipschitz constant, we prove a version of inequality (<ref>), see Theorem <ref>. To achieve this we suitable combine ideas from <cit.>, <cit.> and <cit.>. Once we have this,  following the arguments in <cit.> one can obtain a sharp estimate on the Lipschitz constant and, as a consequence, a Liouville-type of result,  Theorem <ref> and Corollary <ref> for the precise statements.

  4) the boundary regularity, see Theorem <ref>.  


    N.G. acknowledges the support of the European Union - NextGenerationEU, in the framework of the PRIN Project 'Contemporary perspectives on geometry and gravity' (code 2022JJ8KER – CUP G53D23001810006). The views and opinions expressed are solely those of the authors and do not necessarily reflect those of the European Union, nor can the European Union be held responsible for them. H.C.Z. is partially supported by NSFC 12025109, and X.P.Z. is partially supported by NSFC 12271530. 









 













§ PRELIMINARIES



 §.§ The source: RCD(K,N) spaces

We say that (,_,) is a metric measure space if (,_) is a complete and separable metric space and  is a Radon measure which is finite on balls. For a function f:→ we set 


    f(x):=lim sup_y→ x|f(y)-f(x)|/_(x,y)        if x is not isolated
     
        0        if x is isolated

and we call it local Lipschitz constant of f, while with f we denote the classical Lipschitz constant of f.

In order to develop Sobolev calculus on metric measure space, following <cit.>, we introduce the Cheeger energy Ch:L^2(_)→[0,∞] as 

    Ch(f):=inf{lim inf_k→∞1/2∫_^2(f_k)_:  (f_k)_k∈_ bs(), f_k→ f  in L^2(_)}.

It can be proved that if Ch(f)<∞ there exists a function, which we call |∇ f|, such that |∇ f|∈ L^2(_) and 

    Ch(f)=1/2∫_|∇ f|^2_.

If that is the case we say that f∈ W^1,2(). The latter set is actually a vector space with its natural operation and, if endowed with the norm f_W^1,2=f_L^2+2 Ch(f), it is also a Banach space. In order to introduce a well-behaved notion of Laplacian of a function we shall now speak about infinitesimal Hilbertianity.
We say that a metric measure space is infinitesimally Hilbertian, following <cit.>, if Ch is a quadratic form. In this case via polarization it is possible to give a meaning to the object 

    ∫_⟨∇φ,∇ f⟩_

by setting 

    ∫_⟨∇φ,∇ f⟩_:=  Ch(f+φ)-  Ch(f)-  Ch(φ)


    We say that f:→ in W^1,2() is such that f∈ D(Δ)⊂ L^2(_) if there exists g∈ L^2(_) such that 
    
    -∫_⟨∇φ,∇ f⟩_=∫_ gφ_

    for all φ∈ W^1,2(). We shall set Δ f:=g.


    We say that f:→ in W^1,2_ loc() has measure-valued Laplacian in Ω if there exists a Radon measure μ∈ℳ(Ω) such that 
    
    -∫_⟨∇φ,∇ f⟩_=∫_φμ

    for all φ∈_c(Ω), the latter being the space of Lipschitz functions with compact support inside Ω.


    With a little bit of abuse of notation we shall call Δ f=μ the measure-valued Laplacian as well. We will do this since if μ≪_ with density in L^2_ loc, then μ=Δ f_. Notice also that we are using the term Radon measures to denote what are more properly called Radon functionals (see <cit.>).




We are now ready to introduce the class of spaces which we will use as source space for the definition of our harmonic map u. 
We can introduce RCD(K,N) spaces building on the tools we have just presented. Following an Eulerian approach it is possible to characterize them via the Bochner inequality (see <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>). For a more detailed discussion on such notions and for the interplay with optimal transport we refer to the recent <cit.> and <cit.>.

We say that a metric measure space (,_,_) is an RCD(K,N) space if the following conditions are met:

    
  * There exists c_1,c_2≥0 such that for some x∈ we have
    
    (B_r(x))≤ C_1 e^c_2 r^2.

    
  * W^1,2() is a Hilbert space. 
 
    
  * If f∈ W^1,2() is such that | f|≤ 1 -a.e., then f has a 1-Lipschitz representative. 

    
  * For every f∈ D(Δ) with Δ f∈ W^1,2() and g∈ L^∞()∩ D(Δ) the following Bochner inequality holds 
    
    ∫_| f|^2/2Δ g≥∫_ g(K| f|^2+(Δ f)^2/N + ⟨∇ f,∇Δ f⟩).



The final object we shall introduce is the heat semigroup h_t:L^2(_)→ L^2(_): it can be introduced as the gradient flow of the Cheeger energy. Therefore we shall call (h_tf)_t≥ 0 such a gradient flow starting from f∈ L^2(_). For an account of its properties the reader can consult <cit.>. If the space  is an RCD(K,N) space then it is possible to consider the EVI_K gradient flow of the entropy functional on the space of probability measures. If we denote with h_tδ_x the gradient flow of the entropy starting from a Dirac mass centered at x we have h_tδ_x≪_ and we shall call p_t(x,y):= h_tδ_x/_(y). It can be proved that h_tf:=∫_ p_t(x,·)f(x)_ and that p_t is Hölder continuous and satisfies the following Gaussian estimates 

    c/_(B_√(t)(x))e^-_^2(x,y)/3t-C_1t≤ p_t(x,y)≤C/_(B_√(t)(x))Ce^-_^2(x,y)/5t+C_2t,

for all x,y∈, t>0 and for some c,C,C_1,C_2>0.
There is also a gradient bound thanks to the Li-Yau inequality but for the sake of exposition we shall limit ourselves to this presentation: the interested reader can consult <cit.>, <cit.>, <cit.> and <cit.> for more information on Gaussian estimates.



Since we are interested in giving a meaning to "Δ f≥η" we shall rigorously introduce such a notion:


    Let (,_,) be a metric measure space and Ω⊂ an open and bounded set. Let η:Ω→ be continuous and bounded.  We say that a function f∈ W^1,2_ loc(Ω) is such that Δ f≤η in the weak sense if for all φ∈_c^+(Ω) (being _c^+(Ω) the subset of _c(Ω) made of nonnegative functions) we have 
    
    -∫_∇ f·∇φ_≤∫_φη_.


To introduce another (weaker) notion of Laplacian bounds we need to introduce the following space

    Test_c^∞():={φ∈ D(Δ)∩ L^∞: |∇φ|∈ L^∞ , Δφ∈ L^∞∩ W^1,2}.

We write Test_c^∞(Ω) if suppφ⊂⊂Ω.



    Let (,_,_) be a metric measure space and Ω⊂ an open and bounded set. Let η:Ω→ be continuous and bounded.  We say that a function f∈ L^1_ loc(Ω) is such that Δ f≤η in the very weak sense if for all nonnegative φ∈ Test_c^∞(Ω) we have 
    
    ∫_fΔφ_≤∫_φη_.




    As it was pointed out in <cit.>, if  is an RCD(K,N) space and f∈ W^1,2_ loc(Ω) has a very weak Laplacian bound, then such a bound holds also in the weak sense due to the density of Test_c^∞(Ω) into _c(Ω).



Let (,_,_) be an infinitesimally Hilbertian metric measure space and Ω⊂ be an open and bounded set. Let f:Ω→ be a bounded and lower semicontinuous function and let η∈ C_b(Ω). We say that Δ f≤η in the heat flow sense if 

    lim sup_t→ 0h_tf(x)-f(x)/t≤η(x)

    for all x∈Ω, where f:→ is the global extension of f which is set to zero outiside of Ω.

Finally we recall the classical Laplacian comparison for the distance function from a point, which in this non-smooth setting has been obtained in <cit.>.
[Laplacian comparison]

Let (,_,_) be an RCD(K,N) space for some K∈, N∈ and fix x_0∈. Then the map x→^2_(x_0,x)=^2_,x_0(x) has measure-valued Laplacian and 

    Δ^2_,x_0/2≤ C(N,K,_,x_0(·))_

    in the weak sense. Moreover the same holds for the map x→_,x_0(x), on ∖{x_0}, namely
    
    Δ_,x_0 |∖{x_0}≤C(N,K,_,x_0(·))-1/_,x_0(·)_.



 The following simple but useful lemma concerns the properties of the Laplacian operator in product spaces (see also <cit.>)


    Let (,_,_) and (,_,_) be infinitesimally Hilbertian metric measure spaces. Then if ψ_1:→ and ψ_2:→ belong to L^∞(_)∩ W^1,2()∩ D(Δ_) and L^∞(_)∩ W^1,2()∩ D(Δ_) respectively, we have ψ_1ψ_2∈ W^1,2(×)∩ D(Δ_×) and
    
    Δ_×(ψ_1ψ_2)=ψ_1Δ_ψ_2+ψ_2Δ_ψ_1.



The fact that ψ_1ψ_2∈ W^1,2(×) is elementary and follows from Hölder inequality and the chain rule for differentials.

For the second part  notice that h_t^×=h_t^ h_t^=h_t^ h_t^  so that we have
    
    h_t^×(ψ_1ψ_2)-ψ_1ψ_2/t=h_t^ψ_1h_t^ψ_2-ψ_1ψ_2/t=h_t^ψ_1h_t^ψ_2-ψ_2/t+ψ_2h_t^ψ_1-ψ_1/t.

    We will show the convergence only for the first piece of the l.h.s., the second one being analogous. We have
    
    h_t^ψ_1h_t^ψ_2-ψ_2/t-ψ_1Δ_ψ_2_L^2(_⊗_)^2≤ C_1h_tψ_1-ψ_1_L^2(_)^2+C_2h_t^ψ_2-ψ_2/t-Δ_ψ_2_L^2(_)^2.

    The right hand side of the previous expression clearly goes to zero due to the assumptions and this concludes the proof.

With the previous lemma at hand and (<ref>) we state <cit.>, giving an alternative proof.

    Let (,_,_) be an RCD(K,N) space and consider the map (x,y)∈×→_^2(x,y), then 
    
    Δ_×_^2(·,·)≤ 2C(N,K,(·,·))_

    in the weak sense.


    Let us show that the bound holds in the very weak sense and then conclude using that _^2(·,·)∈ W^1,2_ loc(×). Using the product of nonnegative test functions ψ_1ψ_2 and applying Lemma <ref> and Theorem <ref> we get 
    
    ∫_×_^2Δ_×ψ_1ψ_2_⊗_=∫_ψ_1(x)∫__,x^2Δψ_2__(x) 
     
    +∫_ψ_2(y)∫__,y^2Δψ_1__(y) 
    ≤2∫_×ψ_1(x)ψ_2(y)C(N,K,_,x(y))_⊗_(x,y),

    where we also exploited the simmetry of the distance function. This proves the bound in the very weak sense and therefore also in the weak sense.





 §.§ The target: CAT(κ) spaces


For what concerns the target space, for our harmonic map we will consider a complete CAT(κ) space, namely a metric space with sectional curvature bounded above by κ. Let M_κ be the model space, namely the 2-dimensional connected, simply-connected and complete Riemannian manifold with constant sectional curvature equal to κ. Let us further denote by _κ the geodesic distance on such a space and with D_κ= diam( M_κ) its diameter, i.e.

    D_κ=
        π/√(κ)        if κ>0 
    
           +∞    if κ≤ 0.

We also set R_κ:= D_κ/2.
We have the following:

   Let (,_) be a complete metric space. We say that (,_) is a CAT(κ) space if it is geodesic and for any triple of points a,b,c∈ such that _(a,b)+_(b,c)+_(a,c)<2 D_κ and any intermediate point d between b and c there exist comparison points a,b,c,d∈ M_κ such that _(a,b)=_κ(a,b), _(b,c)=_κ(b,c), _(a,c)=_κ(a,c) and
   
    _(a,d)≤_κ(a,d).



We now have a key technical Lemma holding in general CAT(κ) spaces which is <cit.>: we shall discuss only the case κ=1 for the sake of exposition.


    Let (,) be a CAT(1) space. Take any ordered sequence of points {P,Q,R,S}⊂ with _(P,Q)+_(Q,R)+_(R,S)+_(S,P)≤ 2π and let Q_m be the mid-point of the geodesic joining Q and R (which in this case is unique). Then for any α∈[0,1] and β>0 we get
    
    1-α/2(4sin^2(_QR/2)-4sin^2(_PS/2))+2αsin(_QR/2)(2sin(_QR/2)-2sin(_PS/2)) 
    ≤[1-1-α/2(1-1/β)]4sin^2(_PQ/2)+2cos(_QR/2)(cos(_PQ_m)-cos(_QQ_m)) 
      
            + [1-1-α/2(1-β)]4sin^2(_RS/2)+2cos(_QR/2)(cos(_SQ_m)-cos(_RQ_m)).





 §.§ Sobolev spaces with metric targets and Harmonic maps


Following <cit.> (after the seminal work <cit.>) we shall now introduce the Korevaar-Schoen energy and its main properties, being the main tool we need to speak about harmonic functions.


Let u∈ L^2(Ω,) with Ω⊆ open set. We call the 2-energy density of u at scale r inside Ω the quantity 𝐤𝐬_2,r[u,Ω]:→_+, defined as 

    𝐤𝐬_2,r[u](x):=
        (_B_r(x)_^2(u(x),u(y))/r^2(x))^1/2      if B_r(x)⊂ U
    
        0    otherwise.

Moreover we introduce the total energy of u in Ω as 
    E_2[u,Ω]:=lim inf_r→ 0∫_Ω𝐤𝐬_2,r[u,Ω]^2(x)(x).

We can now define Sobolev spaces as follows

    We say that a function u∈ L^2(Ω,) is in KS^1,2(Ω,) if E_2[u]<+∞. We say that u is harmonic in Ω if u=_v∈ KS^1,2(Ω,) E_2[v,Ω].

Existence of minimizers for E_2[·,Ω] has been established in the recent <cit.> (see Theorem 1.2 therein) under the condition that the image of such maps is contained in a sufficiently small ball of the target space.


We shall assume the reader to be familiar with these concepts as we are going to recall only part of <cit.>, stating it for RCD spaces instead of the more general class of strongly rectifiable metric measure spaces.

    Let (,_) be an RCD(K,N) space and (,_) a complete metric space. Then for every u∈ KS^1,2(,) there exists a function e_p[u]∈ L^2(), called p-energy density of u, such that 
    
    𝐤𝐬_2,r[u]→ e_2[u]  - a.e.  and in L^2  as r→ 0.

    In particular the lim inf in (<ref>) is actually a limit.

We shall now present a representation formula of the energy density e_2[u] in terms of the Hilbert-Schmidt norm of the differential | u|_ HS: we will not discuss the meaning of the object u, referring to <cit.> for the details. What follows is <cit.>.

    Let (,_,_) be an RCD(K,N) space and Ω⊂ an open set. Let (,_) be a CAT(κ) space and u∈ KS^1,2(Ω,), then for its energy density we have the following representation formula
    
    e_2[u]=(d+2)^-1/2| u|_ HS.



    Note that in <cit.> the theorem is stated for  which is a strongly rectifiable space and  which is a CAT(0) space. On one hand the proof for the case of CAT(κ) target is the same of the one for CAT(0) spaces, exploting the universal infinitesimal Hilbertianity of such spaces (see <cit.>), on the other hand we shall avoid speaking about strongly rectifiable metric measure spaces since our main results are only stated for RCD(K,N) spaces.


Finally we have the following definition:

Let (,_) be a complete and geodesic metric space. We say that a function E:→ is λ-convex if for all x,y∈ and for all geodesics γ connecting x=γ_0 and y=γ_1 we have

    E(γ_t)≤ tE(γ_1)+(1-t)E(γ_0)-λ/2t(1-t)_^2(γ_0,γ_1).

    




§ MAIN RESULTS



 §.§ Hölder regularity of harmonic maps

In this section we will prove Hölder regularity of our harmonic map with values in a sufficiently small ball of a CAT(κ) space. Note that without this assumption there may be a "big" set of discontinuity (singular set), for examples and a detailed discussion one can consult <cit.>. Since we can always renormalize the target space in such a way that it becomes a CAT(1) space, to ease the notation and the computations we shall assume (,_) to be a CAT(1) space here and in the rest of the work. 

In the following we shall prove the convexity of three functions, namely 1-cos(_,o), _,o and ^2_,o. The proof of the λ convexity of the squared distance is contained <cit.> and the convexity of the distance _,o is well-known but we shall prove them here anyway because they are natural consequences of the convexity of 1-cos(_,o).


    Let (,_) be a CAT(1) space and consider B_ρ(o)⊂ with ρ<π/2. Then  the distance function _,o=_(o,·) is convex on B_ρ(o), ^2_,o is λ-convex and the function cos(_(o,·)) is λ^'-concave, with 
    
    λ = 2cosρ,    λ^'=cosρ.

    Finally _(·,·) restricted to B_ρ/2(o) is jointly convex.


We show that the distance from the north pole on 𝕊^2 is convex on the upper hemisphere. Consider three points N, p, q∈𝕊^2. Denote with _N(y):=_𝕊^2 (N,y) the distance from the north pole for every y∈𝕊^2 and let γ be the geodesic connecting p and q. By the cosine law for the sphere we can consider the triangle whose vertex are p, q and N and write

    cos(f(t))=cos(t_𝕊^2(p,q))cos(_N(p))+sin(t(p,q))sin(_N(p))sin(θ),

    where f(t)=_N(γ(t)) and θ is the angle between γ^'(0) and η^'(1) (η being the geodesic connecting the north pole and the point p). Note that we also used the fact that _𝕊^2(p,γ(t))=t_𝕊^2(p,q). Now differentiate twice the previous identity to get 
    
    (cos(f(t)))^''=-^2_𝕊^2(p,q)cos(f(t))≤ -_𝕊^2^2(γ_1,γ_0)cosρ,

whence cos(f(t)) is a λ^'-concave function with λ^'=cos(ρ).  Now write f=arccoscos(f) and let us call g(t):=cos(f(t)): we have 

    ^2/ t^2f=(g^')^2g-g^''(1-g^2)/(1-g^2)^3/2≥ 0,

meaning that f is a convex function (we have used that Im(g)⊆(0,1] and g^''≤ 0)- this is fully justified if g≠ 1, i.e. f≠ 0, otherwise the argument is justified by slightly moving the north pole N combined with the stability properties of convexity.

For what concerns the squared distance f^2 just use the product rule for the derivative to get 

    ^2/ t^2f^2= 2 |f^'|^2+2ff^''≥ 2ff^''.

Now plug (<ref>) into the previous expression to get 

    ^2/ t^2f^2≥ 2f[(g^')^2g-g^''(1-g^2)/(1-g^2)^3/2]≥ -2fg^''(1-g^2)/(1-g^2)^3/2≥ 2^2_(p,q)cosρf/sin f≥ 2^2_𝕊^2(p,q)cosρ,

which is the λ convexity with λ=2cosρ.

Now consider three points x, y∈ B_ρ(o)⊆ and let p,q,N be three comparison points of x,y,o in 𝕊^2: by the CAT(1) condition we have _(γ(t),o)≤_𝕊^2(γ(t),N) (with γ geodesic joining p and q and with γ geodesic joining x and y and), meaning that

    cos(_(γ(t),o))≥cos(_𝕊^2(γ(t),N)).

 The definition of comparison points together with the previous observation allows to write

    cos(_(γ(t),o))≥ t cos(_(q,o))+(1-t) cos(_(p,o))+t(1-t)/2_^2(p,q)cosρ,

which is the sought λ^'-concavity with λ^'=cosρ. Analogous arguments apply for _(o,·) and _^2(o,·).

For the final part of the proof fix x∈ B_ρ/2(o) and notice that for all y∈ B_ρ/2(o) we must have _(x,y)< ρ by triangle inequality. Therefore we can use the fact that B_ρ(x) is convex and conclude.




    Being the distance function nonnegative it is apparent that also ^2_N/2 is convex. 

 

Letting R_k:=π/√(k) we have the following (see <cit.>)


Let (,_,p) be a pointed complete  CAT(κ) space and let >0, then the function ^2_o:B_(R_k-)/2(o)→[0,∞) is λ-convex, namely for all x,y∈B_(R_k-)/2(o) and γ geodesic between the two points we have

    ^2_o(γ_t)≤ t ^2_o(y)+(1-t) ^2_o(x)-λ t(1-t) ^2(x,y)  ∀ t∈[0,1],

    where λ = (R_k-)tan(/2).



We recall now some lemmas of gradient flow theory on locally CAT(κ)-spaces which will be useful to prove some Laplacian bounds. Let us start with the following, which is part of <cit.>, to which we also refer for the relevant definitions:

    Let  be a locally CAT(κ)-space, E: →∪{+∞} a λ-convex and lower semicontinuous functional. Then, the following hold:

    
  * Existence

For every y∈ D(E) there exists a gradient flow trajectory for E starting from y.

  * Uniqueness and λ-contraction

For any two gradient flow trajectories (y_t),(z_t) we
have

    _(y_t,z_t)≤ e^-λ(t-s)_(y_s,z_s)  ∀ t≥ s≥ 0.


    



Then we have the following a priori estimates for the gradient flow trajectory which is <cit.>, following the ideas contained in <cit.>:

Let  be locally CAT(κ) and E: →∪{+∞} be a
λ-convex and lower semicontinuous functional, λ∈. Let y,z∈ and consider the
gradient flow trajectories (y_t),(z_t) associated with E.
Then, for any t≥ s>0, it holds

    ^2_(y_t,z_s)≤   e^-2λ s(^2_(y,z)+2(t-s)(E(z)-E(y)) 
       +2|∂^-E|^2(y)∫_0^t-sθ_λ(r) r-λ∫_0^t-s^2_(y_r,z) r).

where θ_λ(t):=∫_0^te^-2λ r r.

With the previous two lemmas at hand we can prove the analogue of <cit.> for CAT(κ) spaces. Below we shall denote with _ bs() the space of Lipschitz functions with bounded support and with _ bs^+() the subset of _ bs() made of nonnegative functions.


    Let (,,) be an RCD(K,N) space,  a locally CAT(κ)-space and  Ω⊂ open and bounded. Also, let f∈() be λ-convex, λ∈, and u∈ KS^1,2(Ω,). For g∈_ bs()^+ define the (equivalence class of the) variation map u_t(x):= GF^f_tg(x)(u(x)) ∀ t > 0, x∈Ω. Then, u_t∈ KS^1,2(Ω,) for every t>0 and there is a constant C>0 depending
on f,g such that

    | u_t|^2_ HS≤ e^-2λ t g(| u|^2_ HS-2t⟨ g, (f∘ u)⟩ + Ct^2) - q.o.  in Ω

holds for every t∈[0,1]. In particular

    lim sup_t→ 0E^ KS(u_t)-E^ KS(u)/t≤ -1/d+2∫_Ω(λ g| u|^2_ HS+⟨ g, (f∘ u)⟩).



The fact that u_t∈ L^2(Ω,) easily follows from the following inequalities and the fact that the support of g is bounded:

    _^2(u_t(x),o)≤2_^2(u_t(x),u(x))+2_^2(u(x),o) 
    ≤ 2_^2(u(x),o)+2te^2|λ|t^2(f)g(x),

where for the second inequality we applied the a priori estimates (<ref>) and exploited the fact that |∂^-f|(y)≤(f) for all y∈.
Now thanks to (<ref>) we have (w.l.o.g. assume g(y)≥ g(x))

    ^2_(u_t(x),u_t(y))≤ e^2|λ||g(x)-g(y)|^2_(u(x), GF^f_t|g(y)-g(x)|(u(y))).

Now we can use the sharp dissipation rate of the gradient flow (see <cit.>) to establish the Lipschitzianity of the map t→ GF^f_t(u(x)) and get

    ^2_(u(x), GF^f_t|g(y)-g(x)|(u(y)))   ≤ 2^2_(u(x), GF^f_t|g(y)-g(x)|(u(x))) 
       +2^2_( GF^f_t|g(y)-g(x)|(u(x)), GF^f_t|g(y)-g(x)|(u(y))) 
       ≤ C_1 t^2|g(x)-g(y)|^2+2e^2|λ||g(y)-g(x)|^2_(u(y),u(x)) 
       ≤ C_1 t^2 ^2(x,y)+ C_2^2_(u(y),u(x)).

Dividing by r^2:=^2(x,y) and (B_r(x)) and integrating over B_r(x)⊆Ω we get

    ks^2_2,r[u_t,Ω](x)≤ C_1 t^2 + C_2  ks^2_2,r[u,Ω](x).

The fact that (Ω)<+∞ allows to conclude u_t∈ KS^2(Ω,). 

For what concerns estimate (<ref>) the proof is verbatim the one in <cit.>.

Finally for the last point we just need to subtract from both sides of (<ref>) the quantity | u|^2_ HS and then integrate over Ω and divide by 2t(d+2). Taking the lim sup as t→ 0^+ and exploiting a dominated convergence argument allows to conclude with (<ref>).

The following is a generalization to CAT(κ) spaces of well-known inequalities holding for functions in CAT(0) spaces. We begin with the following:


   Let (,,) be an RCD(K,N) space and (,_) be a locally CAT(κ) space. Let Ω⊂ be open and bounded and let u:Ω→ be an harmonic map and f:→ be a Lipschitz and λ-convex map, then f∘ u∈ W^1,2(Ω) and 
   
    Δ(f∘ u)≥λ| du|^2_ HS

   in the weak sense. In particular Δ(f∘ u) is a signed Radon measure.


   The fact that f∘ u∈ W^1,2(Ω) is well-known (see <cit.>). To prove (<ref>) first observe that being u harmonic implies 
   
    lim sup_t→ 0E^KS(u_t)-E^KS(u)/t≥ 0,

   so that (<ref>) gives
   
    λ∫_Ωg| u|^2_ HS≤ - ∫_Ω⟨ g,(f∘ u)⟩=∫_ΩΔ(f∘ u)g

   for all g∈_bs^+(), whence (<ref>) follows.


    Let (,_,) be an RCD(K,N) space and (,_) a CAT(1) space. Let u:Ω⊂→ be an harmonic mapping such that u(Ω)⊂ B_ρ(o) for some ρ<π/2, then consider the function f_o:→ [0,1] given by f_o(x):=cos(_(u(x),o)). We have f_o∈ W^1,2(Ω) and 
    
    Δ f_o ≤ -cosρ | u|^2_ HS

    in the weak sense in Ω.


    This is indeed a consequence of Proposition <ref> in combination with Proposition <ref>. Indeed one just needs to apply those results with the space (B_ρ(o),_), which is a CAT(1) space.

We now have the following result which holds in a more general setting than the present one (see <cit.>) but we shall present it in the setting of RCD spaces to avoid further technicalities.
[Elliptic Harnack inequality]
    Let (,,) be an RCD(K,N) space and u:→ be a weakly subharmonic function in B_4r(x_0), i.e. u∈ W^1,2(B_4r(x_0)) and
    
    Δ u≥ 0

    in the weak sense in B_4r(x_0). Then the following estimate holds
    
    sup_z∈ B_r/2(x_0)max{u,0}(z)≤ C(K^-r^2,N)(1/(B_r(x_0))∫_B_r(x_0)u^2)^1/2,

    where C is equibounded as r→ 0^+.


    As a consequence of (<ref>) we get that any weakly subharmonic function is locally bounded from above.


We shall now introduce the following notation: for a function v:→ we set 

    v_R:=_B_R(x_0)v,

where x_0∈ is a point which will be clear from the context. We further set 

    v_+,R:=sup_x∈ B_R(x_0)max{v,0}(x)

The following is a combination of <cit.> and <cit.>:


    Let u:→ be as in the previous Theorem and nonnegative, then there exists δ_0>0 independent of R such that 
    
    sup_B_R(x_0) u≤ (1-δ_0)u_+,4R + δ_0u_R.

   Moreover if ∈(0,1/4) there exists m∈ (independent of u and ) such that 
   
    u_+,^mR≤^2u_+,R+(1-^2)u_R^'

   where R^' (possibly depending on  and u) is such that  ^mR≤ R^'≤ R/4.



We proceed recalling another useful lemma which again extends to the context of CAT(κ) spaces without modifications:

    Let (,,) be an RCD(K,N) space and let (,_,o) be a pointed complete metric space, then for every u∈ KS^1,2(,) there exists C=C( diam(Ω),K,N)≥ 1 such that for every r>0 and p∈Ω for which B_rC(p)⊆Ω we have 
    
    _B_r(p)∫_B_r(p)^2_(u(x),u(y))(x)(y)≤ Cr^2∫_ B_rC(p)e_2^2[u].



  The proof can be found in <cit.>. 

The next Lemma is basically <cit.> adapted to CAT(κ) setting.



    Let (,,) be an RCD(K,N) space, Ω∈ an open set, and (,_) be a CAT(1) space. Let u:Ω→ be an harmonic map with values in B_ρ(o) with ρ<π/2 and let B_4R(x_0)⊂⊂Ω, then 
    
    R^2_B_R(x_0)| u|^2_ HS≤ C(v_+,4R-v_+,R),

    where v(x)=^2_(u(x),o) and C=C( diam(Ω),K,N).


    To begin with let us consider a mollified version of the Green function (whose existence can be proved for instance via Lax-Milgram theorem) which solves in the weak sense the following
     
    -Δ G_p=χ_B_R(p)/(B_R(p))     on B_2R(p) 
    
                
                G_p=0     on B^c_2R(p).
 
      We have (we shall omit the point p center of the ball)
      
    ∫_B_2R⟨φ, G_p⟩ = _B_Rφ

      for all φ∈_bs() with suppφ⊂⊂ B_2R(p). Now following <cit.> we define for convenience a rescaled version of G, namely we set
      
    G_p,R:=(B_R(p))/R^2G_p,

      which satisfies
      
    ∫_B_2R⟨φ, G_p,R⟩ = 1/R^2∫_B_Rφ

      and the following estimates (again we refer to <cit.>, which deals with more general metric spaces which  include the class of RCD(K,N) spaces)
      
    0<C_1≤ G_p,R   on B_R, 
    
              0≤ G_p,R≤ C_2   on B_2R,

      where C_1,C_2 only depend on K,N and diam(Ω).
      Now we can define z:=v-v_+,4R and write, exploiting (<ref>) for f(·) equal to  _^2(·,o) with λ=2cosρ by Proposition <ref>,
      
    λ∫_B_2R| u|^2_ HSG_p,R^2≤∫_B_2R(Δ z) G^2_p,R=-2∫_B_2R⟨ z, G_p,R⟩ G_p,R.

      Now we can use the Leibniz rule for the differential (G_p,Rv)=G_p,R z+z G_p,R and write
      
    λ∫_B_2R| u|^2_ HSG_p,R^2≤-2∫_B_2R⟨ G_p,R,(G_p,Rz)⟩+2∫_B_2R⟨ G_p,R, G_p,R⟩ z.

      Being z≤ 0 we can neglect the second term and obtain
      
    λ∫_B_2R| u|^2_ HSG_p,R^2   ≤-2∫_B_2R⟨ G_p,R,(G_p,Rz)⟩=-1/R^2∫_B_RG_p,Rz
       ≤ -C_1(B_R)/R^2(v_R-v_+,4R)=C_1(B_R)/R^2(v_+,4R-v_R)

      where we used the definition of the mollified Green function. Finally, applying Corollary <ref>, we get the thesis.

We are now in position to prove the desired Hölder continuity of harmonic maps.


    Let u:Ω⊆→ be an harmonic map such that Im(u)⊆ B_ρ(o) with ρ<π/2 and with (,,) which is an RCD(K,N) space and  which is a  CAT(1) space. Then u is locally Hölder continuous in Ω.


    The proof closely follows <cit.>. Let us fix x_0∈Ω in such a way that B_4R(x_0)⊂⊂Ω. Let us define the mean of u on a ball centered at x_0 with radius r, denoted by u_r, as one of the minimums of
    
    ∋ q↦_B_r(x_0)_(u(x),q)(x).

Finally set v_p(x):=_^2(u(x),p) where p∈ will be chosen later and w(x):=^2_(u(x),u_R/4). We want to exploit the result in Corollary <ref>: let us therefore fix ≤ 1/10 so that ^mR≤ R^'≤ R/4 and estimate as follows 

    w_R^'^m=1/(B_R^'(x_0))∫_B_R^'(x_0))^2_(u(x),u_R/4)(x)≤C/(B_R/4(x_0)∫_B_R/4(x_0)^2_(u(x),u_R/4)(x)

    where C is independent of R, exploiting the (uniformly) doubling property of the measure  on Ω.
    Now applying Poincaré inequality to the previous expression we get 

    C/(B_R/4(x_0))∫_B_R/4(x_0)^2_(u(x),u_R/4)(x)≤ C_1R^2/(B_R(x_0))∫_B_R/(4λ)(x_0)| u|_ HS^2,

for some λ∈ (0,1). Now we shall apply Lemma <ref> and the doubling inequality again to obtain 

    w^m_R^'≤ C(v_p,+,R/λ-v_p,+,R/4λ).


Choose now p∈ conv(u(B_^mR(x_0)) so that we have 

    sup_x∈ B_^mR(x_0)^2_(u(x),p)≤ 2 sup_x∈ B_^mR(x_0)^2_(u(x),u_R/4)+2^2_(u_R/4,p)≤ 4 sup_x∈ B_^mR(x_0)^2_(u(x),u_R/4)

and at the same time 

    sup_x∈ B_R(x_0)^2_(u(x),u_R/4)≤ 4 sup_x∈ B_R(x_0)^2_(u(x),p)

Combining estimate (<ref>) and the result of Corollary <ref> we get 

    sup_x∈ B_^mR(x_0)^2_(u(x),u_R/4)
         ≤ 4^2sup_B_R(x_0)^2_(u(x),u_R/4) + C(v_p,+,R/λ-v_p,+,R/4λ) 
    ≤ 16^2sup_x∈ B_R(x_0)^2_(u(x),p)+C(v_p,+,R/λ-v_p,+,^mR),

where in the last line we also used that ^m≤ (1/8)^m≤ 1/4≤ 1/4λ. In the end we obtain

    sup_x∈ B_^mR(x_0)^2_(u(x),p)≤ 64^2sup_x∈ B_R(x_0)^2_(u(x),p)+C(v_p,+,R/λ-v_p,+,^mR).


Setting ω(r):=sup_x∈ B_r(x_0)^2_(u(x),p) we can rewrite the previous inequality as 

    (1+C)ω(^mR)≤ 64^2ω(R)+Cω(R/λ)≤ (64/100+C)ω(R/λ),

which means 

    ω(^mR)≤ cω(R/λ),

where  and λ are fixed and c<1. By an iteration of the latter estimate (holding for every R≤ R_0 for which B_R_0(x_0)⊂⊂Ω) we get

    ω(r)/r^α≤ Cω(R_0)/R_0^α,

where α∈ (0,1), C>0 and r≤ R_0. Choosing p=u_r we get 

    √(ω(r))≤ osc(u,B_r(x_0)) ≤ 2  √(ω(r))

and this proves the (local) Hölder continuity of u.




 §.§ Higher integrability  of energy densities


Let Ω⊂ X be an open bounded set in an RCD(K,N) space with X∖Ω≠∅ , K∈ℝ and N∈[1,∞).  Let (Y,_) be a CAT(1κ) space with κ>0 and suppose that u∈ KS(Ω;Y) is an harmonic map with values in a ball B_ρ(o)⊂ Y with ρ∈(0,π/2√(κ)). We shall always fix a Hölder continuous representative of u.

Let us recall some notations in <cit.>.

    For any q>1, a nonnegative -measurable function w on Ω belongs to the weak q-Reverse Hölder class RH^weak_q if there exists a constant C_q such that 
    
    (_B w^q)^1/q⩽  C_q ∫_2Bw
 
for all ball B:=B_r(y) with 2B:=B_2r(y)⊂Ω.


We need the following Gehring lemma, see <cit.> and <cit.>.

    If 1<q<∞ and w∈ RH^weak_q, then there exists ε>0 such that w∈ RH^weak_q+ε.    
Now we will prove the higher integrability of energy density.

    
    Let Ω, Y and u be as above. Then there exists an ε=ε(N,K, diam(Ω),ρ)>0 such that | u|_ HS∈ W^1,2+ϵ_ loc(Ω) and  
    
    (_B | u|^2+ε_ HS)^2/2+ε⩽ C_ε_B| u|_ HS^2

    for any ball B with 2B⊂Ω, where the constant C_ϵ>0 depends only on ϵ.
    
    

    Fix any ball B with 2B⊂Ω, then by Lemma <ref>, we have 
    
    Δ (f_o-a)⩽ -cosρ | u|_ HS^2,  ∀ a∈ℝ,

where f_o(x)=cos(_(u(x),o)). Let ϕ:Ω→ [0,1] be a cut-off function with ϕ=1 on B, ϕ=0 out of 3/2B, and 
    |∇ϕ|⩽ C_1r^-1,   |Δϕ|⩽ C_2r^-2,

where the constants C_1,C_2 depend only on K,N and diam(Ω). Then we get

    ∫_B| u|^2_ HS⩽∫_3/2B| u|^2_ HSϕ⩽C_3/r^2∫_3/2 B |f_o-a|,

for all a∈ℝ, where C_3=C_2/cosρ. It is well-known that a weak (1,2)-Poincaré inequality holds on RCD(K,N) spaces and since the weak (1,s)-Poincaré inequality is an open ended condition (see <cit.>), there exists  a number s_0∈(1,2) such that the weak (1,s_0)-Poincaré inequality holds on  RCD(K,N) spaces. Therefore, we have 

    inf_a∈ℝ_3/2 B|f_o-a|⩽ C_K,N, diam(Ω),s_0· r(_2B |∇ f_o|^s_0)^1/s_0.

Combining the above two inequalities, we conclude that 

    (_B| u|^2_ HS)^1/2⩽ C_4(_2B|∇ f_o |^s_0)^1/s_0⩽ C_4 (_2B| u |_ HS^s_0)^1/s_0,

where we have used |∇ f_o|⩽ |sin_Y(o,u)|· |∇_Y(o,u)|⩽   | u|_ HS. Now, applying <ref> to | u|^s_0_ HS, we obtain
| u|_ HS is in W^2+ε_ loc(Ω), and moreover 

    (_B| u|^2+ε_ HS)^1/(2+ε)⩽    C_ε(_2B| u |_ HS^s_0)^1/s_0⩽ C_ε(_2B| u |_ HS^2)^1/2,

since s_0<2 and Hölder inequality.









 §.§ Auxiliary results


In this section we shall work under the following assumptions:
    
        
  * (,_,) is an RCD(K,N) space with essential dimension d∈. 
        
  * Ω⊂ is an open bounded set with X∖Ω≠∅, which is equivalent to  (∖Ω)>0. 
        
  * (,_) is a CAT(1) space: the results obtained for general CAT(κ) spaces will be obtained by a rescaling of the distance function. 
        
  * u∈ KS^1,2(Ω;) is harmonic with values in a ball B_ρ(o)⊂ with ρ<π/2. Finally we shall fix Borel representatives of u (the Hölder continuous one) and of e_2[u] (and of | u|_ HS).
        
  * Let Ω^'⊂⊂Ω be open and consider r>0 and x̂∈Ω^' such that B_4r(x̂)⊂Ω^' and u_C^α(Ω^')r^α<π/10. Finally call B=B_r(x̂), 2B:=B_2r(x̂) and B^'=B_3r/2(x̂). 

        
    

Let us first define F:→ as the following

    F(t):=2sin(t/2)+4sin^2(t/2)

and observe that F is such that F^', F^''≥ 0 on [0,π/2]. With a little abuse of notation let us also set 

    F(z,w):=2sin(_(z,w)/2)+4sin^2(_(z,w)/2)

for any z,w∈.
    


    Consider the map W(x,y):=_(u(x),u(y)) and let x_1,x_2∈Ω. Then W∈ W^1,2_ loc(Ω×Ω) and 
    
    Δ_×W≥ 0     on  B_R_1(x_1)× B_R_2(x_2)

in the weak sense, as long as l:= diam(u(B_R_1(x_1))∪u(B_R_2(x_2)))<ρ.


The Sobolev regularity is trivial. 
We shall therefore prove that (<ref>) holds in the very weak sense and exploit Remark <ref> to conclude. 

    To this aim let ψ,ϕ∈ Test_c^∞(Ω) be nonnegative and such that suppψ⊂⊂ B_R_1(x_1) and suppϕ⊂⊂ B_R_2(x_2): using Lemma <ref> and Fubini we have
    
    ∫_×WΔ_×(ψϕ)(_⊗_)   =∫_ϕ(y)∫__(u(x),u(y))Δ_ψ(x)_(x)_(y) 
       +∫_ψ(x)∫__(u(x),u(y))Δ_ϕ(y)_(y)_(x).

    Now thanks to the assumptions for every y∈ B_R_2(x_2) the map x→_(u(x),u(y)) has nonnegative Laplacian in the very weak sense, since z→_(z,u(y)) is convex on B_l(u(y)). The same holds for the map y→_(u(x),u(y)) for every x∈ B_R_1(x_1), meaning that 
    
    ∫_×WΔ_×(ψϕ)(_⊗_)≥ 0

    and concluding the proof.




    Let Z(x,y):=F(u(x),u(y)), then Z∈ W^1,2_ loc(Ω×Ω)
    
    Δ_× Z≥ 0     in B_R_1(x_1)× B_R_2(x_2)

    in the weak sense as long as l:= diam(u(B_R_1(x_1))∪u(B_R_2(x_2)))<ρ.


  Again we shall gliss on the Sobolev regularity since it is apparent. To obtain the Laplacian bound let us first set G(t):=2sin(t/2)+4sin^2(t/2). The latter function is such that G^'(t),G^''(t)≥ 0 for all t∈[0,π/2]. Let us call again W(x,y):=_(u(x),u(y)): again we shall test the inequality first in the very weak sense.  Let again ψ,ϕ∈ Test_c^∞(Ω) be nonnegative and such that suppψ⊂⊂ B_R_1(x_1) and suppϕ⊂⊂ B_R_2(x_2), then
   
    ∫_×ZΔ_×(ψϕ)(_⊗_)   =∫_ϕ(y)∫_ Z(x,y)Δ_ψ(x)_(x)_(y) 
       +∫_ψ(x)∫_ Z(x,y)Δ_ϕ(y)_(y)_(x).

  Arguing as in Lemma <ref> we conclude observing that for all y∈ B_R_2(x_2) the map x↦ Z(x,y) has nonnegative Laplacian. Indeed by the chain rule we have 
  
    Δ_XW(·,y)=F^''(t)|∇_(u(·),u(y))|^2+F^'(t)Δ__(u(·),u(y))≥ 0.

  The same holds for the other variable and we thus conclude.












    
 We introduce the following quantities, in order to produce an Hopf-Lax formula for the function u,

    f(x,y)= 
        
           -F(u(x),u(y))      if x,y∈ B^'
    
           -6      otherwise.

Notice that f is lower semiconinuous since F is bounded between 0 and 6. We call f_t the p-Hopf-Lax semigroup applied to the function f, namely we set

    f_t(x):=inf_y∈[_^p(x,y)/pt^p-1+f(x,y)],

where we avoid to include p in the definition of f_t to lighten the notation.
Notice that 0≥ f_t(x)≥ -6 for every x∈. Moreover the infimum in (<ref>) is actually a minimum (this follows by Weierstrass theorem exploiting the semicontinuity of the function we are minimizing). We also have a quantitative estimate for where to find a minimum, indeed denoting with y_t,x a minimizer for f_t(x), choosing x as a competitor, we get

    f_t(x)≤_^p(x,y_t,x)/pt^p-1+f(x,y_t,x)≤ 0.

This means _(x,y_t,x)≤ (6pt)^p-1/p so that there exists t_*=t_*(p)>0 such that we have

    f_t(x):=inf_y∈ B_12√(t)(x)[_^p(x,y)/pt^p-1-F(u(x),u(y))]    ∀ x∈ B

for t∈ (0,t_*).

Now set 

    S_t(x):={y∈: f_t(x)=_^p(x,y)/pt^p-1-F(u(x),u(y))}

and observe that the latter set is non-empty if t<t_*. Finally set 

    L_t(x):=min_y∈ S_t(x)_(x,y)     and     D_t(x):=L_t^p(x)/pt^p-1-f_t(x).

We now present a slight modification of <cit.> since we still don't know that the map u is Lipschitz continuous but we have Hölder regularity instead: if the map is assumed to be Lipschitz the proof works in the same way replacing α with 1.


    With the above notation and assumptions, we have
    
        
  * f_t is Hölder continuous on B. 
        
  * L_t and D_t are lower semicontinuous. 
        
  * There exists a constant C=C(p,u_C^α,k)>0 such that 
        
    L_t≤ C t^β,     D_t≤Ct^β^',     -f_t≤Ct^β^'   on  B,

    
where β=(p-1)/(p-α), β'=αβ, and the constant C  depends on p, the Hölder norm of u, u_C^α.  


     The proof of (1) is immediate since the infimum of equi-Hölder functions is Hölder.

    The proof of (2) is contained in <cit.>.

    For the proof of (3) consider y_t(x)∈ S_t(x) such that L_t(x)=_(x,y_t). We get, using sinθ≤θ for θ>0 and that _(u(x),u(y_t))≤π,
    
    D_t(x)=L_t^p(x)/pt^p-1-f_t(x)=F(x,y_t)≤_(u(x),u(y_t))+_^2(u(x),u(y_t)) 
    ≤ (1+π)_(u(x),u(y_t))≤(1+π)u_C^αL_t^α(x)=C L_t^α(x).

    At the same time, being f_t≤ 0 we have 
    
    L_t^p(x)/pt^p-1≤ D_t(x)≤ C L_t^α(x)

    so that we get L_t≤ C t^β, with β=(p-1)/(p-α). For D_t we have instead
    
    D_t(x)≤ C L_t^α(x)≤Ct^β^'

    with β^'=αβ. Finally for f_t we have, since -f_t≤ D_t,
    
    -f_t≤Ct^β^'.




To establish the key variational inequality we shall exploit the following simple but useful lemma


    With the above assumptions we have 
    
    Δ f(·,y)≤ 0   on B

    in the weak sense, for all y∈ B.


    Thanks to the assumptions it is sufficient to compute Δ F(u(·),u(y)) in the weak sense. By the chain rule we get 
    
    Δ F(u(·),u(y))=F^''|∇_(u(·),u(y))|^2+F^'Δ_(u(·),u(y)),

    whence the claim follows by the nonnegativity of the factors on the right hand side (recall that the Laplacian of x↦_(u(x),u(y)) is nonnegative thanks to Proposition <ref>).




We now have a lemma on the heat flow Laplacian of the Hopf-Lax semigroup (the idea is from <cit.>, see also <cit.> and <cit.>)

    Let f:→ be a bounded Borel function. Assume that for some x,y∈ we have
    
    Q_t^p f(x)=f(y)+^p_(x,y)/pt^p-1.

    Then
    
    Δ Q_t^pf(x)≤Δ f(y)-K^p_(x,y)/t^p-1.

    holds in the heat flow sense.


    First of all let π_s∈𝒫(×) be an optimal transport plan between h_sδ_x∈𝒫() and h_sδ_y∈𝒫() for the cost _^p.
    Moreover we have the following estimate, which is the Wasserstein contractivity of the heat flow (holding in general RCD(K,∞) spaces, see <cit.>),
    
    W^p_p(h_sδ_x,δ_y)≤ e^-pKs_^p(x,y).

    We can now estimate as follows
    
    h_sQ_tf(x)   =∫_Q_tf(z) h_sδ_x(z)=∫_×Q_tf(z)π_s(z,z^') 
       ≤∫_×[f(z^')+^p(z,z^')/pt^p-1]π_s(z,z^') 
    (by optimality of π_s)           =∫_ f(z^') h_sδ_y(z)+1/pt^p-1W^p_p(h_sδ_x,h_sδ_y) 
       =h_sf(y)+1/pt^p-1W^p_p(h_sδ_x,h_sδ_y).

   Finally applying (<ref>) to the previous inequality we get (note that the following would hold for any w∈ in place of y)
   
    h_sQ_tf(x)≤ h_sf(y)+e^-pKs/pt^p-1_^p(x,y).

   Subtracting (<ref>) from (<ref>), dividing by s>0 and taking the lim sup as s→ 0 finally gives (<ref>).



We now proceed with a refinement of (<ref>), following <cit.>, which will be crucial for obtaining an elliptic inequality involving the function f_t.



Let u:Ω→ be an harmonic map with Ω⊂ open set, (,_) which is a CAT(1) space and Im(u)⊆ B_ρ(o) with o∈, ρ<π/2. Let further f_o(x):=cos(_(u(x),o)), then we have f_o∈ W^1,2(Ω) and 

    Δ f_o≤ -f_o | u|_ HS^2=-f_o(n+2) e_2^2[u]     in Ω

    in the weak sense.




 Let us first set R(x):=_(u(x),o), denote with x→ G_t^u(x),o the map which associates to each x∈Ω the point at time t lying in the geodesic (recall that geodesics are unique in our case) connecting o and u(x). Finally set u_η:=G_η^u,o where η∈ W^1,2(Ω)∩ C_c(Ω) is such that 0≤η≤ 1: then by  <cit.> we have

    e_2^2[u_η t]≤sin^2[(1-η t)R]/sin^2R(e_2^2[u]-e_2^2[R])+e_2^2[(1-η t)R]

 -a.e. in Ω, where t is a positive parameter that we will eventually send to zero. Now we shall use the duplication formula for the sinus to get 

    | u_η t|_ HS^2≤[cos^2(tη R)+sin^2(tη R)cos^2R/sin^2R-cos Rsin(2tη R)/sin R](| u|^2_ HS-| R|^2_ HS)+| R-tη R|^2_ HS.

Note that we have simultaneously used that | u|^2_ HS=(n+2)e_2^2[u] (recall that if f:→ then | f|=| f|_ HS).
We proceed integrating over Ω, we divide by t and exploit the fact that E_2(u_tη)-E_2(u)≥ 0 (as u is harmonic) together with the asymptotics of the involved functions to get 

    0≤∫_Ω[-η Rcos R/sin R| u|^2_ HS+η Rcos R/sin R| R|^2-⟨ R,(η R)⟩].

We can now use the following identity

    ⟨∇(ηR/sin R),∇cos R⟩ = η Rcos R/sin R| R|^2-⟨ R,(η R)⟩

to get 

    0≤∫_Ω-η Rcos R/sin R| u|^2_ HS+⟨∇(ηR/sin R),∇cos R⟩.

    Note that now we can choose the magnitude of η to be whatever we want since the inequality doesn't change if we divide everything by a positive constant. Now pick φ∈_c(Ω) nonnegative and set η:=φ R/sin R: it is clear that η∈ W^1,2(Ω)∩ C_c(Ω) because it is the product of a bounded W^1,2(Ω) and continuous function and a Lipschitz function with compact support. Finally this means that for all φ∈_c(Ω) nonnegative we have
    
    ∫_Ωφcos R| u|^2_ HS≤∫_Ω⟨∇φ,∇cos R⟩.

    The latter is the conclusion.











Let f∈ W^1,2(×) be such that for all y∈Ω_2 Δ_f^y≥ 0 in the very weak sense in Ω_1⊂ and ∀ x∈Ω_1 Δ_ f^x≥ 0 in the very weak sense in Ω_2⊂ with Ω_1,Ω_2 open and bounded, then 

    Δ_×f≥ 0     in Ω_1×Ω_2

in the very weak sense.


    By standard arguments we have that if φ∈_c(Ω_1×Ω_2) then there exists φ_n = ψ^n_1ψ^n_2 where ψ^n_1:→ and ψ^n_2:→ are nonnegative and in Test_c^2  such that φ_n→φ in W^1,2(Ω_1×Ω_2). Therefore to prove (<ref>) it is enough to test the inequality against functions of the latter form. Now thanks to the tensorization of Sobolev space and to Lemma <ref> we have 
    
    ∫_×∇_×f·∇_×(ψ_1ψ_2)(_⊗_)   =∫_×∇_f^y·∇_ψ_1ψ_2+ψ_1∇_ f^x·∇_ψ_2(_⊗_) 
       =-∫_×fΔ_ψ_1 ψ_2+ψ_1Δ_ψ_2(_⊗_) 
       =-∫_ψ_2∫_Δ_fψ_1__-∫_ψ_1∫_Δ_fψ_2__,

    and the latter terms are clearly nonpositive due to the assumptions. 




    Let π_t=h_tδ_x⊗ h_tδ_y∈𝒫(×), which is an admissible transport plan between P_tδ_x and P_tδ_y and consider the usual setting with u:→ harmonic function with values in a sufficiently small ball, then we have 
    
    lim sup_t→ 01/t(-∫_×_(u(z),u(w))π_t(z,w)+_(u(x),u(y)))≤ 0.



    The idea is to use the tensorization of Sobolev spaces: indeed for any p∈Ω, since the distance function _ is convex on B_(R_k-)/2(o), we have 
    
    Δ(_,p∘ u)≥ 0.

    Since the values of u are contained in the aforementioned ball we have 
    
    Δ(_,u(z)∘ u)≥ 0   for ∀ z∈.

    Now by Lemma <ref> we get
    
    Δ_×(_∘(u,u))≥ 0.

   Now computing the derivative (we are allowed since we can extend u to a continuous and bounded function over all ) we get 
   
    lim sup_t→ 01/t(-∫_×_(u(z),u(w))π_t(z,w)+_(u(x),u(y))) 
    ≤-lim inf_t→ 0∫_×_(u(z),u(w))∂_tH_×((z,x),(w,y),t)⊗.



We can now exploit the fact that the heat kernel solves the heat equation we can substitute the time derivative with the Laplacian on × and integrate by parts to get 

    lim sup_t→ 01/t(-∫_×_(u(z),u(w))π_t(z,w)+_(u(x),u(y))) 
    ≤-lim inf_t→ 0∫_X×Δ_×_∘(u,u)H_×⊗≤ 0.


    



Finally define some parametric functions depending on the distance of the target space _ and deduce some Laplacian bounds on them that we shall exploit later in the proof of the "good" distributional bound.


    Let u:Ω⊂→ be an harmonic map with Im(u)⊂ B_ρ(o) and ρ<π/2. Consider for any z∈Ω and y∈  the function
    
    w_a,b,y,z(x)=a_^2(u(x),u(z))+bcos(_(u(x),y)).

    For -a.e. x_0∈Ω we have 
    
    Δ w_a,b,o,x_0(x_0)   ≤(2a-bcos(_(u(x_0),o)))(n+2)e_2^2[u](x_0) 
       =(2a-bcos(_(u(x_0),o)))| u|^2_ HS(x_0)

    in the heat flow sense.


        First of all we shall notice that <cit.> holds also in this setting with the same proof since by Lemma <ref> we have the (Hölder) continuity of u. Therefore we have 
        
    h_t(_^2(u(·),u(x_o))(x_0)=2| u|^2_ HS(x_0)t+o(t)     as  t→ 0^+.

        for -a.e. x_0.
        Secondly by the results contained in <cit.> and Lemma <ref> we have 
        
    lim sup_t→ 0h_tcos(_(u(·),o))(x)-cos(_(u(x),o))/t≤ -cos(_(u(x),o))| u|^2_ HS.

        Combining (<ref>) with (<ref>) we finally get the thesis.



 §.§ A variant of the Bochner-Eells-Sampson inequality

The authors in <cit.> are able to prove the Lipschitz continuity of harmonic maps between Alexandrov spaces exploiting the properties of the Hopf-Lax semigroup. Moreover in <cit.>, given the Lipschitz continuity of the harmonic map proved in <cit.>, they are able to prove a weak version of the Bochner-Eells-Sampson inequality for maps from a Riemannian domain to a CAT(1) space. Here we shall exploit the ideas contained in <cit.> and fuel them with the ideas of <cit.> (see also <cit.> for the non-smooth counterpart, as in our case) to obtain a variational inequality (the "good" distributional bound) which in the limit will be the desired inequality.




 For any t∈ (0,t_*), we have the following properties:
 

  * f_t∈ W^1,2_ loc(B_R)∩ C(B_R);

  * L_t and D_t are lower semi-continuous on B_R;

  * there exist constants C_1, C_2>0 depending only p and A such that 


    L_t(x)/t   ≤ C_2·[ lipu(x,C_1t^p-1/p)]^1/p-1;
    D_t(x)/t   ≤ C_2·[ lipu(x,C_1t^p-1/p)]^p/p-1;
    -f_t(x)/t   ≤ C_2·[ lipu(x,C_1t^p-1/p)]^p/p-1.




The first two items were proved in <cit.>.

For item (3), let y_t∈ S_t(x) be such that L_t(x)=_(x,y_t). Firstly, from f_t≤0 and F(x,y)≤ A(1+A), we get that 

    L_t^p(x)/pt^p-1=F(x,y_t)+f_t(x)≤ A(A+1)   ⟹   L_t(x)≤ C_1t^p-1/p,

where C_1:=[pA(A+1)]^1/p.
By the definition of lipu(x,r), we have that 

    D_t(x)=F(x,y_t)≤ (1+A)_(u(x),u(y_t))≤ (1+A) · lipu(x,C_1t^p-1/p)· L_t(x).

Substituting   this into  

    L_t^p(x)/pt^p-1=D_t(x)+f_t(x)≤ D_t(x),

we conclude that 

    L_t (x)/ t    ≤[p (1+A) · lipu(x,C_1t^p-1/p)]^1/p-1;
    D_t (x)/ t    ≤  (1+A) · lipu(x,C_1t^p-1/p)·L_t(x)/t≤ C_p,A·[ lipu(x,C_1t^p-1/p)]^p/p-1.


 
At last, by using -f_t(x)=D_t(x)-L^p_t(x)/pt^p-1≤ D_t(x), we obtain the desired upper bound of -f_t(x)/t. The proof is now complete.




We now recall <cit.>.


    There exists T>0 such that, given a Borel set E⊂ B^' such that (B^'∖ E)=0, we have: for all 0<t<T there exists z_t∈ B such that for -a.e. x∈ E∩ B_4r/3(x)=:E∩ B^'' and every n∈ the function
    
    y↦ g_t(x,y,z_t):=_^p(x,y)/pt^t-1+f(x,y)+^2_(y,z_t)/2n

    admits a minimizer T_t(x) and such minimizer belongs to the set E∩ B^''.


    The difference with respect to <cit.> lies in the different definition of f, however since the proof follows with minor modifications we decided to omit it. Note moreover that from the proof in <cit.> we can infer that Sobolev regularity is not necessary for the function f. It would be sufficient to ask for f to be continuous and with a Laplacian bound Δ f≤ L in the weak sense.

We further define

    f_t,n(x):=inf_y∈[_^p(x,y)/pt^p-1+f(x,y)+^2_(y,z_t)/2n].

We now have the following distributional bound for the function f_t,n.

    Possibly choosing a smaller t_* the following holds. Let f_t,n be defined as in (<ref>) and p⩾ 2: we have 
    
    Δ f_t,n≤ C(K,N,p, diam(Ω))(1/t+1/n)     on B

    in the weak sense, for all t<t_*, for all n∈.


Fix y∈ B: by Theorem <ref> and p⩾ 2, we have 

    Δ_,y^p= p(p-1)_,y^p-2|∇_,y|^2·+p_,y^p-1Δ_,y⩽ C(K,N,p, diam(Ω))·.

Combining this with
 Lemma <ref> and <cit.> we infer the result.

To obtain the "good" distributional bound we need the following lemma for the function F to be able to let the heat flow and the Hopf-Lax semigroup combine in an efficient way.

    Consider 4 points P,Q,R,S inside u(B_r(x)) in such a way that P:=u(x), Q:=u(x), R:=u(y), S:=u(y). Let us further set  l_0:=2sin_(Q,R)/2, l_1:=2cos_(Q,R)/2, α:=1/(1+2l_0) and finally let β>0. We have
    
    F(u(x),u(y))-F(u(x),u(y))≤[w_a_1,b,Q_m,x(x)-w_a_1,b,Q_m,x(x)]+[w_a_2,b,Q_m,y(y)-w_a_2,b,Q_m,y(y)]/α l_0,

   where Q_m is the middle point of the geodesic joining Q and R,  
  
    a_1:=1-1-α/2(1-1/β),   b:=l_1,   a_2:=1-1-α/2(1-β)

  and the function w is defined in Lemma <ref>.


    We can  apply (<ref>) to get  
    
    α l_0(F(Q,R)-F(P,S)))   =α l_0(4sin^2_QR/2-4sin^2_PS/2)+α l_0(2sin_QR/2-2sin_PS/2) 
       ≤[1-1-α/2(1-1/β)]4sin^2_PQ/2+l_1(cos_PQ_m-cos_QQ_m) 
       +[1-1-α/2(1-β)]4sin^2_RS/2+l_1(cos_SQ_m-cos_RQ_m) 
       ≤[w_a_1,b,Q_m,x(x)-w_a_1,b,Q_m,x(x)]+[w_a_2,b,Q_m,y(y)-w_a_2,b,Q_m,y(y)],

 
  which concludes the proof.

The second tool we need is an improvement of the earlier distributional bound: this is the aim of the following proposition.

    We have
    
    Δ f_t_≤ -KL_t^p/t^p-1+(1+o_t(1))D_t| u|^2_ HS   on  B

    in the weak sense, for all t<t^* and p⩾2.


    First of all let us recall the definition of f_t,n 
    
    f_t,n(x):=inf_y∈[^p_(x,y)/pt^p-1+f(x,y)+^2_(y,z_t)/2n].

    Thanks to the Lemma <ref> we can find z_t in such a way that a minimizer of g_t(x,y,z_t), i.e. a point T_t(x) for which g_t(x,T_t(x),z_t)=f_t,n(x), lies inside E∩ B^'' for -a.e. x∈ E∩ B^'' and we can choose E to be the set of regular points of the space intersected with the set of Lebesgue points of | u|_ HS (which is clearly of full measure). Now let us fix x∈ E∩ B^'' and call y the "good" minimiser of f_t,n(x). Clearly for such points we have 
    
    f_t,n(x)=f(x,y)+_^p(x,y)/pt^p-1+_^2(y,z_t)/2n=-F(u(x),u(y))+_^p(x,y)/pt^p-1+_^2(y,z_t)/2n.

    Now fix any other two points x,y∈Ω. Setting  P:=u(x), Q:=u(x), R:=u(y), S:=u(y). Using the inequality (<ref>) of the key techcnical lemma (and its notation) we get

    f_t,n(x)   =inf_y∈[^p_(x,y)/pt^p-1+f(x,y)+^2_(y,z_t)/2n] 
       =-F(u(x),u(y))+inf_y∈[^p_(x,y)/pt^p-1+F(u(x),u(y))-F(u(x),u(y))+^2_(y,z_t)/2n] 
       ≤_(<ref>)-F(u(x),u(y))+w_a_1,b,Q_m,x(x)-w_a_1,b,Q_m,x(x)/α l_0
           +Q_t[w_a_2,b,Q_m,y(·)-w_a_2,b,Q_m,y(y)/α l_0+_^2(·,z_t)/2n](x),

with equality if x=x.
We now proceed to obtain a bound on the Laplacian of f_t,n in the heat flow sense at the point x, therefore we shall estimate

    lim sup_s→ 0^+h_s(f_t,n)(x)-f_t,n(x)/s=Δ f_t,n(x).

Exploiting the previous inequalities and the monotonicity of the heat flow (h_tf≤ h_t g if f≤ g) we get 

    Δ f_t,n(x)≤Δ w_a_1,b,Q_m,x(x)/α l_0+Δ Q_t[w_a_2,b,Q_m,y(·)/α l_0+_^2(·,z_t)/2n](x).

Moreover thanks to the properties of the Hopf-Lax semigroup (namely (<ref>)) we get 

    Δ Q_t[w_a_2,b,Q_m,y(·)/α l_0+_^2(·,z_t)/2n](x)≤Δ w_a_2,b,Q_m,y(y)/α l_0+1/nΔ_^2(·,z_t)(y)-KL_t^p(x)/t^p-1.

Now we can apply Lemma <ref> and the Laplacian comparison to obtain 

    Δ f_t,n(x)   ≤C(K,N,r)/n-KL_t^p(x)/t^p-1+2a_1-bcos(_(u(x),Q_m))/α l_0| u|^2_ HS(x) 
       +2a_2-bcos(_(u(y),Q_m))/α l_0| u|^2_ HS(y).

 Since cos(_(Q_m),y)=cos(_(Q_m),x)=l_1/2 and 1-l_1^2/4=l_0^2/4 we can choose β such that 
  a_2=l_1^2/4, so that 2a_2-bcos(_(u(y),Q_m))=0. This is achieved with  
  
    β = 1-l_0(1+2l_0)/4.

  Via standard computations we get 
  
    2a_1-bcos(_(Q_m),x)/α l_0=2l_0(1+2l_0)(1/4+1/4-l_0(1+2l_0)).

  Therefore we get 


    Δ f_t,n(x)   ≤C(K,N,r)/n-KL_t^p(x)/t^p-1+2l_0(1+2l_0)(1/4+1/4-l_0(1+2l_0))| u|^2_ HS(x) 
       ≤C(K,N,r)/n-KL_t^p(x)/t^p-1+(1+o_t(1))D_t(x)| u|^2_ HS(x).

where we also used that D_t(x)=l_0+l_0^2 and that u is Hölder continuous to estimate the remainder in o_t(1)  (observe also that x̂ does not depend on n∈).
Combining the latter with Lemma <ref> and <cit.> (recalling that u is continuous on Ω) we end up with

    Δ f_t,n≤C(K,N,r)/n-KL_t^p(·)/t^p-1+(1+o_t(1))D_t(·)| u|^2_ HS(·)   on  B

in the weak sense, for all n∈ and for all t<t_* .

Now since f_t,n converges to f_t uniformly as n→∞, thanks to the regularity of f_t and the stability of the Laplacian bounds we infer (<ref>).






  



We now recall <cit.>:


    Let q be such that 1/q+1/p=1. For all x∈ B we have 
    
    lim inf_t→ 0f_t(x)/t≥-1/q^qu(x).

    Moreover, in adding   to assume that u is locally Lipscithz continuous,  for -a.e. x∈ B (namely any point in B where u is metrically differentiable) we have
    
    lim_t→ 0^+f_t(x)/t=- lip^qu(x)/q

    and
    
    lim_t→ 0^+L_t(x)/t= lip^q/pu(x),    lim_t→ 0^+D_t(x)/t= lip^qu(x).



    The proof follows as in <cit.> combined with   <cit.>.



[A variant of the BES inequality]
Let u be as above and assume that it is locally Lipschitz in Ω, then the inequality  

    Δ(^2u/2)≥|∇ u|^2-K^2(u)- e_2^2[u]^2u

holds in the very weak sense in Ω.


     By the chain rule it is easy to infer that (<ref>) is equivalent to 
    
    Δ u≥-K u- e_2^2[u] u.

We shall now verify that there exists a neighborhood B_R(x) with B_2R(x)⊂Ω such that (u)∈ W^1,2(B_R(x)) and (<ref>) holds in the sense of distributions in B_R(x).

Due to the continuity of u there exists R>0 such that u(B_2R(x))⊂ B_π/4(u(x)), so that diam(u(B_2R(x)))<π/2 and R<r/2. By (<ref>) and (<ref>) we have Δ f_t/t≤ C( u) on B_2R for all t∈(0,t_*). Combining the elliptic inequality (<ref>) with Lemma <ref> and a Caccioppoli inequality we get f_t/t∈ W^1,2(B_3R/2(x)) with f_t/t_W^1,2(B_3R/2(x))≤ C and C depending only on the Lipschitz norm of u in Ω^'. Therefore, exploiting Lemma <ref>, up to a subsequence we have that -f_t/t converges weakly in W^1,2 to ^q(u)/q and we get

    Δ(^qu/q)≥ K^q u- e_2^2[u]·^q u

in B_3R/2(x) in the weak sense.
Exploiting the Lipschitz continuity of u  we get 

    Δ(^qu/q)≥ K( u)^q-( u)^q+2≥ -C

where the constant is uniform in q. Now again by Caccioppoli inequality we get ^q u/q_W^1,2(B_R)≤ C as q→ 1. This means that ^q(u)/q converges to (u) in W^1,2(B_R(x)) and we can pass to the limit in (<ref>) and get (<ref>), whence we also deduce (<ref>). 

Finally we shall mention that the theorems in <cit.> hold also in the present setting: we refer to <cit.> for the proofs which work mutatis mutandis in our context.

    Let u be as above but with values in B_ρ(o)⊂, where (,_) is a CAT(κ) space and ρ<π/2√(κ). Then letting R>0 be such that B_2R(x_0)⊂Ω we have
    
    sup_x∈ B_R/2(x_0) lip(u)(x)≤C_N,√(K)R,π/(2√(κ)-ρ)/R,

    where the constant C only depends on the parameters listed at its subscript.

As a consequence we obtain a Liouville type theorem for harmonic maps, which follows by estimate (<ref>).

    Let (,_,_X) be an RCD(0,N) space and (,_) be a CAT(κ) space. Consider an harmonic map u:→ such that u()⊂ B_ρ(o) for some o∈ and ρ<π/(2√(κ)) with sublinear growth, i.e.
    
    lim inf_R→∞sup_y∈ B_R(x_0)_(u(y),o)/R=0

    for some o∈. Suppose that u is locally Lipschtiz continuous. Then u must be a constant map.



 

 §.§ Boundary regularity for harmonic maps

 
In this section, we continue to assume that  Ω⊂ X is an open bounded set in an RCD(K,N) space with X∖Ω≠∅ , K∈ℝ and N∈[1,∞).  Moreover we let (Y,_) be a CAT(κ) space with κ>0. 

To study the boundary regularity of harmonic maps, we shall also impose some regularity conditions on the boundary of Ω.

    Let Ω⊂ X be a  domain. We say that Ω satisfies an exterior density condition if there exist two numbers λ∈(0,1) and R_0>0 such that  

    (Ω∖ B_r(x))⩾λ·(B_r(x))   ∀ x∈∂Ω,  ∀ r∈(0,R_0).

Additionally we say that Ω satisfies a uniform exterior sphere condition if there exists a number R_0>0 such that for each x_0∈∂Ω there exists a ball
 B_R_0(y_0) satisfying

    Ω∩ B_R_0(y_0)=∅   and   x_0∈∂ B_R_0(y_0).



    It is easy to see that if the space satisfies a volume doubling condition (which is the case of RCD(K,N) spaces, thanks to Bishop-Gromov inequality), then the exterior density condition is implies by the exterior sphere condition.

 
The main result of this section is the following.


  
Let Ω and Y be as above. Suppose that Ω⊂ X satisfies a uniform exterior sphere condition with constant R_0 and let w∈ Lip(Ω,Y). Let u∈ KS^1,2(Ω,Y) be an harmonic map with boundary data w such that Im(u)⊂ B_π/4-ρ(o) for some o∈ Y and ρ>0.  Then for any ϵ∈(0,1) it holds 

    _(u(x),w(x_0))⩽ C_ϵ   L_w   _^1-ϵ(x,x_0)

 for all  x_0∈∂Ω and x∈Ω with _X(x,x_0)<R_ϵ, where both R_ϵ and C_ϵ depend only on ϵ, N,K and diam(Ω), and  
  
    L_w:= sup_x,y∈Ω_(w(x),w(y))/_X(x,y).
 
  In particular, u is continuous at x_0 and u(x_0)=w(x_0).
 

To prove this result, we need the following two lemmas. 

 
 
 
 
 
 

 

 
  

  

Let Ω⊂ X be a bounded domain satisfying a uniformly exterior condition with constant R_0. Suppose that f∈   W^1,2(Ω) is a harmonic function on Ω with boundary data g∈ Lip(Ω). Suppose g(z_0)=0 for some z_0∈Ω. Then for any ϵ∈(0,1), there exists a number R_ϵ∈ (0,min{1,R_0/2}) (depending only on ϵ, N, K and diam(Ω)) such that for any ball B_r(x_0) with  x_0∈∂Ω  and r∈ (0,R_ϵ) it holds 

    sup_B_r(x_0)∩Ω |f(x)-f(x_0)|⩽ C_ϵL· r^1-ϵ,

  where the constant C_ϵ>0 depending only on ϵ, N,K, and the constant L is a Lipschitz constant of g.  


 
This  is Theorem 4.3 in <cit.>.   
  
 
 

 
    
 
Let Ω, Y be as above.  Suppose that u:Ω→ Y is an harmonic map. Then  for any 
P∈ Y such that Im(u)∈ B_π/2-ρ(P)  it holds
  
    Δ_(u(x),P)⩾ 0

  in the sense of distributions.  
  

Since the function _(P,·) is convex in B_π/2(P)⊂ Y, the assertion follows directly from  Proposition <ref>.






We are now in the position to prove Theorem <ref>, whose proof is a modification of the one in <cit.>.  
 

 
 Fix any a point x_0∈∂Ω, and set  P=w(x_0). Then, by the triangle inequality and the fact that Im(u)⊂ B_π/4-ρ(o), we have _(P,u(x))⩽π/2-2ρ for any x∈Ω. Moreover by Lemma <ref>, we observe that _(P,u(x)) is sub-harmonic on Ω. 
 
 We can now solve the Dirichlet problem 
 
    Δ f(x) =0   on Ω     and     f(x)-_(w(x_0),w(x))∈ W^1,2_0(Ω).

 Notice that, by the triangle inequality, the function g_x_0(x) :=_(w(x_0),w(x)) is Lipschitz continuous on Ω with a Lipschitz constant 
 
    L_g_x_0⩽  L_w   and   g_x_0(x_0)=0.

 According to Lemma  <ref>, we have 

    sup_B_r(x_0)∩Ω |f(x)-f(x_0)|⩽ C_ϵ  L_w  r^1-ϵ,

   for any ball B_r(x_0) with  x_0∈∂Ω  and r∈ (0,R'_ϵ). 
   
 At last, since _(u(x),w(x_0))-f(x) is sub-harmonic on Ω, and     
    [_(u(x),w(x_0))-f(x)]^+∈ W^1,2_0(Ω),

the maximum principle yields
   
    _(u(x),w(x_0))⩽ f(x),   a.e. in   Ω.
 
  Noticing that u∈ C(Ω) (by Theorem <ref>) and f∈ C(Ω), we get  
   
    _(u(x),w(x_0))⩽ f(x),  ∀ x∈Ω.
  
The combination of the latter with (<ref>) implies the desired result, concluding the proof.
   

 






    





    




§ TRYING TO PROVE LIPSCHITZ REGULARITY

Consider the function F(t)=2sint/2+4sin^2t/2. Via easy computations we have that the function F(t)/t is decreasing for t∈(0,π/2), whence exploiting also triangle inequality we have for any x,y,z∈Ω

    F(u(x),u(z))≤ F(u(x),u(y))+F(u(y),u(z)).

From now on we shall write F(x,y) in place of F(u(x),u(y)).
Now we shall try to modify a bit the perturbation argument in <cit.> in order to make it work also in our setting.
In particular we shall modify our function f_t as follows:

    f_t(x,λ):=inf_y∈Ω[e^-2Kλ_^2(x,y)/2t-e^kλF(x,y)],

where k is the upper bound of the sectional curvature in the target space, while K is the lower bound on the Ricci curvature in the domain. Then we define the following quantities

    S_t(x,λ):={y∈ B_2R: f_t(x,λ)=e^-2Kλ_^2(x,y)/2t-e^kλF(x,y)}, 
    
        L_t(x,λ)=min_y∈ S_t(x,λ)_(x,y),     D_t(x,λ):=L_t^p(x)/pt^p-1-f_t(x,λ).

The first step is to try and prove some modified versions of lemmas contained in <cit.>. By renormalizing our target space we can reduce to the case where k=1, so we shall work under this assumption. We shall start with the following, which is <cit.> (when possible we keep their notation as well)

    Consider any Ω^''⊂⊂Ω^' and set 
    
    C_*:=(2e· osc_Ω^'u+2)e^2K,     t_0:= dist^2(Ω^'',∂Ω^')/4C_*.

    Then for every t∈(0,t_0) we have 
    
        
  * For all λ∈[0,1] and x∈Ω^'' the set S_t(x,λ) is closed and non-empty. Moreover
        
    f_t(x,λ)=min_y∈ B_√(C_*t)(x)[e^-2Kλ_^2(x,y)/2t-e^λF(x,y)].
 

        
  * For all λ∈[0,1] and Ω^''⊂⊂Ω^' we have
        
    ∫_Ω^''|∇ f_t(x,λ)|^2(x)≤2/t^2 diam^2(Ω^') diam(Ω^'')+18E_2^u(Ω^'').
 

        
  * For all x∈Ω^'' the map λ→ f_t(x,λ) is Lipschitz in [0,1] and
        
    f_t(x,λ)-f_t(x,μ)≤|λ-μ|C(K)· C_* t.
 

        
  * (x,λ)→ f_t(x,λ) is in C(Ω^''×[0,1])× W^1,2(Ω^''×[0,1]) with respect to the measure ⊗ℒ^1.
    


   1. First of all it is easy to see that 0≥ f_t(x,λ)≥ -e^λ osc_Ω^'u. Due to the nonpositivity of f_t, for every minimizing sequence {y_j}_j∈ we have eventually
    
    _^2(x,y_j)≤ 2(1+e^λ osc_Ω^'u)e^2Kλt≤ C_*t,

    hence (<ref>) follows.

    2. Let z∈Ω^' be a minimizer for f_t(y,λ): we have
    
    f_t(x,λ)-f_t(y,λ)   ≤e^-2Kλ/2t(_^2(x,z)-_^2(y,z))+e^λ(F(y,z)-F(x,z)) 
       ≤e^-2Kλ/t diam(Ω^')_(x,y)+e^λ F(x,y),

    where we exploited (<ref>).
    Exchanging the roles of y and x and using the simmetry of the involved quantities we get 
    
    |f_t(x,λ)-f_t(y,λ)|≤1/t diam(Ω^')_(x,y)+e· F(x,y).

    Taking the square on both sides of previous inequality and applying Young inequality leads to
    
    |f_t(x,λ)-f_t(y,λ)|^2≤2/t^2 diam^2(Ω^')_^2(x,y)+18· F^2(x,y).

    Now divide by the opportune quantities to get the Korevaar-Schoen energy so that we end up with
    
    e_2[f_t(·,λ)](x)≤2/t^2 diam^2(Ω^')+18lim_→ 0^+_B_r(x)F^2(x,y)/^2(y).

Dobbiamo dimostrare che la roba a destra del limite coincide con e_2[u] ma sono abbastanza confident che si possa fare. (<ref>) follows after an integration over Ω^''.

3. Let now λ,μ∈[0,1] and fix x∈Ω^''. Pick z minimizing f_t(x,μ) and compute

    f_t(x,λ)-f_t(x,μ)≤(e^-2Kλ-e^-2Kμ)/2t_^2(x,z)≤ |λ-μ|C(K)· C_* t,

where point 1. and where C=C(K) is a constant depending only on K.

4. The proof follows the one of <cit.> without relevant modifications.


The following is instead <cit.> and we shall omit the proof since it follows without relevant modifications:

    Fix any domain Ω^''⊂⊂Ω^'. For each t∈(0,1), we have:
    
 
  * The function (x,λ)→ L_t,λ(x) is lower semi-continuous in Ω^''×[0,1]. 
 

  * For each λ∈[0,1],

    L_t,λ_L^∞(Ω^'')≤√(C_*t),

where the constant is the same appearing in the previous Lemma.


The following is a modification of <cit.>.

    Let Ω^''⊂⊂Ω^' and t∈(0,t_0). Then we have
    
    lim inf_μ→ 0f_t(x,λ+μ)-f_t(x,λ)/μ≥ preghiamo




    




     

§ APPENDIX







§ STRATEGY FOR THE LIPSCHITZ REGULARITY

First we want to prove a technical lemma holding in CAT(1) spaces. Let u_0,u_1∈𝐊𝐒^1,2(Ω,) with Im(u_i)⊂ B_ρ(o) for some ρ<π/2. Let moreover d(x):=_(u_0(x),u_1(x)) and R^v(x):=_(v(x),0) for any v:Ω→. Let further η∈_c(Ω) be such that Im(η)⊆ [0,1/2]. Set
 
    u_η(x):=(1-η(x)sin d(x)/d(x))u_0(x)+η(x)sin d(x)/d(x)u_1(x)   and
    
            u_η(x):=η(x)sin d(x)/d(x)u_0(x)+(1-η(x)sin d(x)/d(x))u_1(x).

 Now define the following weights for convex combinations
 
    φ = η(1-cos d)tan R^u_η/R^u_η   and  ψ:=η(1-cos d)tan R^u_1-η/R^u_1-η.

    Finally set 
    
    û_η(x)   :=(1-φ(x))u_η(x)+φ(x)o,
    û_1-η(x)   :=(1-φ(x))u_η(x)+φ(x)o 
    
             F_η(x)   :=√(2(1-cos d(x))/cos R^u_ηcos R^u_1-η).

    We have the following:


    Let u_0,u_1∈𝐊𝐒^1,2(Ω,) with Im(u_i)⊂ B_ρ(o) for some ρ<π/2. Then the following inequality holds
    
    |û_η|_ HS^2+|û_1-η|_ HS^2- | u_0|_ HS^2-| u_1|_ HS^2≤ -cos R^u_ηcos R^u_1-η⟨∇ F^2_η,∇η⟩ + 
    
            -2ηcos R^u_ηcos R^u_1-η| F_η|^2_ HS +   Quad(η,η)

    for -a.e. x∈Ω.


    From <cit.> we have 
    
    |û_η|^2_ HS≤sin^2[(1-φ)R^u_η]/sin^2R(| u_η|^2_ HS-| R^u_η|^2_ HS)+|[(1-φ)R^u_η]|^2_ HS

    and similarly
    
    |û_1-η|^2_ HS≤sin^2[(1-φ)R^u_1-η]/sin^2R(| u_1-η|^2_ HS-| R^u_1-η|^2_ HS)+|[(1-φ)R^u_1-η]|^2_ HS.






alpha


 Paper Title: Asymptotics in all regimes for the Schrödinger equation with time-independent coefficients.



§ ABSTRACT 
	Using the recent analysis of the output of the low-energy resolvent of Schrödinger operators on asymptotically conic manifolds (including Euclidean space) when the potential is short-range, we produce detailed asymptotic expansions for the solutions of the initial-value problem for the Schrödinger equation (assuming Schwartz initial data). Asymptotics are calculated in all joint large-radii large-time regimes, these corresponding to the boundary hypersurfaces of a particular compactification of spacetime.


Parameters Estimation in Source-Sink Space Population Evolutionary Models
     Erin AshleyPhD Candidate, Department of Aerospace Engineering, Iowa State University, IA 50011, USA. email: eashley1@iastate.edu, 
 Carla Simon Sanz Undergraduate Research Assistant, Department of Aerospace Engineering, Iowa State University, IA 50011, USA. email: carlass@iastate.edu,
 Simone ServadioAssistant Professor, Department of Aerospace Engineering, Iowa State University, IA 50011, USA. email: servadio@iastate.edu
 and Giovanni Lavezzi Postdoctoral Associate, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, MA 02139, USA. email: glavezzi@mit.edu

    Received XXX; accepted YYY
=======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================







§ INTRODUCTION


In recent work <cit.><cit.>, wave propagation on stationary, asymptotically flat spacetimes has been analyzed using Vasy's spectral methods <cit.>, which involve low-energy resolvent estimates for Schrödinger operators (i.e. for the “time-independent” Schrödinger equation, in physicists' preferred terminology) and are closely related to earlier work of Guillarmou–Hassell–Sikora <cit.>, with numerous precursors in the wider literature.
These tools apply to Schrödinger operators with short range potentials, which in <cit.> means decaying cubically or faster. The quadratic case is similar, requiring only minor modifications. Long range potentials (e.g. any Coulomb-like potential, decaying like ∼ r^-1) require serious modifications which we do not discuss here.

We focus on the 3-dimensional case of most physical interest, for which the spectral methods are most developed.
Let (X,g) denote an asymptotically conic manifold <cit.> (see below), and let ρ denote a boundary-defining-function (bdf) on X.
Let (X) = ⋂_k∈ρ^k C^∞(X) denote the Fréchet space of Schwartz functions on X.


The reader is invited to consider the case of exact Euclidean space. Then,

    X=^3 = ^3∪∞^2,

which is the compactification of ^3 constructed by adding on the 2-sphere ∞^2 at infinity, and g is the exact Euclidean metric. The results below are novel even in this case, where the analysis sharpens the seminal results of Jensen–Kato <cit.> by providing large-argument asymptotics in addition to long-time asymptotics.
Dispersive estimates for the Schrödinger equation (including scattering off obstacles, on manifolds, or with nonlinearities) have been a popular theme for a while now — see e.g. <cit.><cit.><cit.><cit.><cit.><cit.> for a few examples; <cit.> is a survey with many references — and global dispersive estimates probe the behavior of solutions in all large-t regimes.
Nevertheless, as far as we know the complete description of the asymptotics below is new even in the Euclidean case.

In the Euclidean case, a convenient choice of bdf is ρ = 1/⟨ r ⟩, where r is the Euclidean radial coordinate and ⟨ r ⟩ = (1+r^2)^1/2 is the Japanese bracket. Also, (X) = (^3) is just the usual set of Schwartz functions on ^3.


Because of our familiarity with the Euclidean case, even when working with general X it is usually easier to use the coordinate r(x) = ρ(x)^-1∈ C^∞(X^∘) in place of the bdf ρ(x). For example, g is given to leading order (with respect to some collar neighborhood of the boundary) by d r^2 + r^-2 g_∂ X for g_∂ X a Riemannian metric on X. This is the metric of an exact cone.
However, it should be kept in mind that, in the exact Euclidean case, r(x) = ⟨ r ⟩, where r is the Euclidean radial coordinate. This slightly overloaded notation should not cause confusion. Whenever we use `r' below, we mean ρ(x)^-1, not the Euclidean radial coordinate.



In this paper, we apply Hintz–Vasy's low-energy toolkit to the Schrödinger initial value problem

    -i ∂_t u = Δ_g u +  i  A·∇_g u + (V+2^-1 i∇_g· A) u, 
    
     u(0,x) = f(x),
     IVP

posed on the manifold _t× X^∘, where f∈(X),

	
  * Δ_g is the positive-semidefinite Laplace–Beltrami operator,
	
  * ∇_g : C^∞(X^∘) →(X^∘) is the gradient operator which is antisymmetric with respect to the L^2(X, g)-inner product,
	 A·∇_g u(x) = g(A(x),∇_g u(x)), and ∇_g· is the corresponding divergence operator, and
	
  * A ∈ r^-2(X;), and V∈ r^-3 C^∞(X;).

The vector field A plays the role of the vector-potential generated by a short-range magnetic field, e.g. that generated by a magnetic dipole, and V plays the role of the voltage generated by a short-range electric field, such as that generated by an electric quadropole. We exclude more slowly decaying fields.

Note that L^2(X,g) is not just L^2(X), where X is considered as a compact manifold-with-boundary, because g does not extend smoothly across the boundary of X. However, in the exact Euclidean case, L^2(X,g) = L^2(^3). To emphasize the distinction, L^2(^3)≅ L^2(^3)≠ L^2(^3), because e.g. the former contains all constant functions while the latter does not. Here, the isomorphism “≅” is that induced by the diffeomorphism ^3 ≅^3.



The coefficients of the PDE are static, i.e. constant in the time coordinate t.  Moreover, the differential operator

    P= Δ_g +  i A·∇_g + 2^-1 i∇_g· A + V ∈Diff^2(X^∘)

is formally symmetric with respect to the L^2(X, g)-inner product, ⟨ϕ,ψ⟩_L^2(X,g) = ∫_X ϕ^* ψVol_g. This enables the application of spectral-theoretic tools which otherwise might not apply or would require some work to justify.
We also make the following assumptions, as in Hintz's work:

	
  * (No zero energy resonance or bound state.) The operator P has trivial nullspace acting on ^1(X)=r^-1^0(X) ⊂ r^-1 L^∞ <cit.>.
	
  * The high energy estimates stated in <cit.> apply.

There are many function spaces that can replace ^1(X) in (I). The point is just to rule out the presence of nonzero solutions u to Pu=0 decaying like 1/r.
The high energy estimates apply whenever the metric g is non-trapping or exhibits only normally hyperbolic trapping. In particular, (II) holds in the exact Euclidean case or in any sufficiently small perturbation thereof.


	The solution of the initial-value problem <ref> is of exponential-polyhomogeneous type on the compactification M↩ (0,∞)_t× X given by the iterated blowup
	
    M=[[ [0,∞]_t× X; {∞}×∂ X ]; β^-1({∞}×∂ X)∩clβ^-1( {∞}× X)  ],

	where β: [ [0,∞]_t× X; {∞}×∂ X ] → [0,∞]_t× X is the blowdown map.
	


We will explain the construction of M in more detail below. For now, see <Ref>, which describes M near β^-1([0,∞] ×∂ X) via an atlas (ignoring the angular degrees of freedom associated with ∂ X).
Also, see the discussion below for the definition of the notion of exponential-polyhomogeneity appearing in the theorem. It is a term used to state that asymptotic expansions hold without specifying the forms of those asymptotic expansions. More precise theorems (in particular, <Ref>) appear later.


As far as we are aware, <Ref> should hold if d≠ 2,  not just d=3, and for any A∈ r^-1(X) and V∈ r^-2 C^∞(X), regardless of whether or not there is a zero energy resonance or bound state.
Our techniques are quite general, but the spectral side has yet to be developed sufficiently for our techniques to allow us to deduce such a result using the tools below. For example, we cite <cit.> as a black box, but these works forbid the appearance of a zero energy resonance or bound state. Thus, one avenue for future work is extending the results here to more general settings. It should be remarked that <cit.> handles d>3 as well as d=3, so it actually would have been possible to treat d>3 in this work.





Finally, it is worth pointing out recent work <cit.> that develops microlocal tools which can be used, among other things, to produce asymptotic expansions for the Schrödinger equation with time-dependent coefficients that settle down to the constant-coefficient case. Their methods cannot handle coefficients that settle down to something non-constant. Conversely, our methods cannot handle any time-dependence in the coefficients whatsoever. So, there is no overlap between our results and those of Hassell et. al., besides the constant-coefficient case that is included in both. Moreover, the microlocal tools required are very different.

In the future, it will hopefully be possible to combine the analysis here with that in <cit.> to handle time-dependent coefficients which settle down to something stationary but with spatial variance. This would be analogous to Hintz's recent work <cit.> combining <cit.><cit.>.



 §.§ More precise form of the main theorem





For a function v ∈ C^∞(M^∘) to be of exponential-polyhomogeneous type on the manifold-with-corners (mwc) M means that v can be written as a finite sum v = ∑_n=1^N e^i θ_n v_n
for θ_n,v_n ∈ C^∞(M^∘) polyhomogeneous functions on M.
Polyhomogeneity is a widely used generalization of smoothness due to Melrose <cit.> which allows bounded powers of logarithms to appear in Taylor series. Such “generalized Taylor series” are called polyhomogeneous expansions.
The specific combinations of logarithms and powers allowed are specified by an index set _f⊂× at each boundary hypersurface f⊂ M of M. (An index set consists of a pair of numbers that are used to characterize the singularities of a distribution or a solution to a PDE.) Schematically, a polyhomogeneous function v:M^∘→ with index set _f at f admits the polyhomogeneous expansion

    v ∼∑_(j,k)∈_f v_f;j,kϱ_f^j log^k(ϱ_f)

at f,
where ϱ_f∈ C^∞(M) is a boundary-defining-function of f and the v_f;j,k∈ C^∞(f^∘) are polyhomogeneous functions on f, which itself is a mwc of one lower dimension.
Polyhomogeneity at corners guarantees the existence of joint asymptotic expansions there. This is equivalent to saying that the polyhomogeneous expansions at adjacent boundary hypersurfaces are compatible. Concretely, this means that (v_f;j,k)_f∩F;J,K = (v_F;J,K)_f∩F;j,k
for any adjacent boundary hypersurfaces f,F and (j,k)∈_f, (J,K)∈_F.
So, the notion of exponential-polyhomogeneous type is a formalization of the notion of admitting a full atlas of (term-by-term differentiable, in all directions) asymptotic expansions in terms of elementary functions.
If M is compact, then, in some sense, exponential-polyhomogeneity means that the provided atlas of asymptotic expansions is complete. (The opposite extreme is when M has no boundary, in which case exponential-polyhomogeneity just means smoothness and therefore says nothing about asymptotics.)

We refer to the cited works <cit.><cit.><cit.><cit.> for further discussion of polyhomogeneity and the function spaces capturing it, as well as for the related notion of conormality. Our notational conventions mostly follow <cit.> and are explained as needed. In particular, “^(,α)” is used to refer to partially polyhomogeneous behavior with index set  and a conormal error of order α∈, and “^” means purely polyhomogeneous behavior.
One abbreviation used throughout is that, for j∈ and k∈, “(j,k)” means the index set {(j+n,κ)∈× : n∈, κ≤ k }. Also, `∞' means empty index set, i.e. Schwartz behavior. For instance, ^∞(X) = (X), and ^(j,0)(X) = r^-j C^∞(X).


So, <Ref> states that solutions of the Schrödinger equation (with Schwartz initial data) are governed by four asymptotic regimes (five if we include the Cauchy hypersurface Σ = {t=0}⊂ M) one regime for each of the four boundary hypersurfaces nf,dilF,parF,kf⊂ M
of the mwc M appearing in the theorem.
A more precise version of the theorem – which, when combined with <Ref>, which studies the sum over eigenfunctions in <ref>, yields <Ref> – is:

	Let ϕ_1,⋯,ϕ_N ∈(X) denote the (automatically Schwartz) square-integrable eigenfunctions of P, so that Pϕ_n = -E_n ϕ_n for some E_n>0. Let f∈(X).
	If u ∈ C^∞([0,∞)_t× X^∘) solves the initial-value problem <ref>, and if χ∈ C_c^∞() satisfies 0∉supp(1-χ), then
	
    u(t,x) = exp[ -i (1-χ(t))/4 t ρ(x)^2]   u_phg(t,x)/(t+iϵ)^3/2 + ∑_n=1^N e^-i E_n tϕ_n(x) ⟨ϕ_n,f ⟩_L^2(X,g)

	for u_phg = u_phg[χ] polyhomogeneous on M, with
	
    u_phg∈^(0,0)∪ (1/2,0)∪, (0,0)∪,(0,0),∞,(0,0)(M)

	for some index sets ⊂ (2^-1^≥ 2)×, ⊂^≥ 1×, where the index sets are specified at kf, parF, dilF, nf, and Σ, respectively. So, the index set at kf is (0,0)∪ (1/2,0)∪, the index set at parF is (0,0)∪, the index set at nf is empty, and the remaining index sets are (0,0).
	Moreover, the leading order asymptotic at kf∪parF has the following form: for some polyhomogeneous w∈^1(X) and v ∈^1-(X),
	
    u_phg(t,x) - χ(r/t) w(x)-χ(r^2/t) v(x) ∈^(1/2,0)∪,,(0,0) ∪,∞,(0,0)(M).

	The restriction w|_∂ X(θ) does not depend on θ, and w|_∂ X=Λ(f) for some linear functional Λ:(X)→.  In fact, w(x) = -√(π i) r P^-1 f, where P^-1 f ∈^1(X) is the unique solution to Pw = f in ^0+(X).
	The functional Λ is given by
	
    Λ(f) = ⟨ f, u^(0)⟩_L^2(X,g),

	where u^(0)∈ C^∞(X) is the unique smooth solution of Pu^(0)=0 that is identically 1 at ∂ X.
	Letting
	
    L =-i d/dσ M_exp(- i σ r) P M_exp(i σ r) |_σ =0∈Diff^1(X^∘),

	in which M_∙: w ↦∙ w denotes a multiplication operator, the function v is given by v(x) = √(π i) P^-1 (-r  + L P^-1 )f.
	



	Under the stated assumptions, it is the case <cit.> that we have a well-defined one-sided inverse P^-1: ^2+α(X)→^α-(X)
	for any α∈ (0,1), where ^α-(X) = ⋂_ϵ>0^α-ϵ(X). So, P^-1 f ∈^1-(X). A standard argument lets us upgrade this to P^-1 f ∈^(1,0),2- (X); see <cit.>, specifically the main theorem in that section.
	Because L, which is given by
	
    L= -2  r^-1(r ∂_r+1)  r^-2Diff^1_b(X),

	(cf. <cit.>, our L being related to L(σ) there by L=-iL'(σ)|_σ=0)
	satisfies L ρ∈ρ^3 C^∞(X), it is the case that LP^-1 f ∈^3(X) and therefore that P^-1 L P^-1 f ∈^1-(X). The same theorem in <cit.> shows that P^-1 L P^-1 f is polyhomogeneous. This justifies the description of the profiles v,w in <Ref>. We refer to <cit.> for the details regarding the large r asymptotics of w.




For the reader uncomfortable with the notion of polyhomogeneity, the following L^∞-based corollary follows immediately from the theorem:

	For any K∈,
	
    u(t,x) =exp[ -i (1-χ(t))/4 t ρ(x)^2] χ(r/t) w(x) + χ(r^2/t) v(x)/(t+iϵ)^3/2 + ∑_n=1^N e^-i E_n tϕ_n(x) ⟨ϕ_n,f ⟩_L^2(X,g)
     +
    	O (1/⟨ t ⟩^3/2⟨r/t⟩^-K⟨t/r⟩^-1/2).

	The big-O term in <ref>, which is bounded above by O(r^1/2/t^2), is suppressed relative to the other terms as t→∞ in t≫ r. That is, for any c,ϵ>0 the big-O term is o(t^-3/2) if r=o(t).
	Moreover, for any ε>0,
	
    u(t,x) =exp[ -i (1-χ(t))/4 t ρ(x)^2] Λ(f)/(t+iϵ)^3/2  + O ( ⟨rt/t+r^2⟩^-4+ε).

	The big-O term in <ref> is suppressed relative to the other terms for t ∼ r^2; if cr^2 < t < C r^2 for some 0<c<C, then the big-O term is O(t^-2+ε/2)=O(r^-4+ε).


So, the solution consists of a sum over bound states and a dispersive remainder, whose large-t asymptotics have been given explicitly within r/t ≪ 1.


	One minor improvement of <Ref> is that the index sets , can be related to the polynomial decay rate of the coefficients of P-Δ_g_0 for g_0 the exactly conic metric on which g is modeled. The larger the degree, the smaller these index sets can be taken. This improvement follows from <Ref> and the corresponding fact in the main theorem of <cit.>.


Using Duhamel's principle, it is straightforward to show that: 

	The conclusions of the theorems above apply also to the inhomogeneous version of <ref> with forcing in (_t; (X)) = (_t× X), except without the explicit formulas for the terms w,v in the asymptotic profile.
	 


	One can still compute a formula for w,v in this case, it is just slightly more complicated. 


The proof below of the theorems above is essentially constructive, in the sense that it yields an algorithm for computing the asymptotic expansions of u_phg in all possible regimes, not just kf∪parF, and not just to leading order. The algorithm can be extracted from the proof. It produces expansions on M in terms of the coefficients of the expansions in <cit.>. Insofar as these coefficients are explicit, so too are the asymptotics on M.


	Since <cit.> is as-of-yet unpublished, it is worth mentioning that if <cit.> is used instead of <cit.>, then one gets <Ref> except with only conormal estimates of the remainder in <ref>. In particular, the L^∞-based corollary above can still be deduced.
	Moreover, this applies even if V,A merely satisfy symbolic estimates (so, do not necessarily extend smoothly to ∂ X), except in this case w,v are only going to be partially polyhomogeneous.


[Initial data with finite decay]
	In this paper, we only consider Schwartz initial data, but the same tools allow one to study the case where the initial data is polyhomogeneous on X with some finite decay rate at ∂ X. 


Finally, it is worth pointing out recent work <cit.> that develops microlocal tools which can be used, among other things, to produce asymptotic expansions for the Schrödinger equation with time-dependent coefficients that settle down to the constant-coefficient case. Their methods cannot handle coefficients that settle down to something non-constant. Conversely, our methods cannot handle any time-dependence in the coefficients whatsoever. So, there is no overlap between our results and those of Hassell et. al., besides the constant-coefficient case that is included in both. Moreover, the microlocal tools required are very different.

In the future, it will hopefully be possible to combine the analysis here with that in <cit.> to handle time-dependent coefficients which settle down to something stationary but with spatial variance. This would be analogous to Hintz's recent work <cit.> combining <cit.><cit.>.




 §.§ Geometric setup and spacetime compactification




Concretely, that (X,g) be an asymptotically conic manifold means that X is a smooth manifold-with-boundary and g is a Riemannian metric on X^∘ satisfying the following: for some ρ̅>0 and embedding ι: [0,ρ̅]_ρ×∂ X → X satisfying ι(0,-) = id_∂ X (that is, a collar neighborhood of the boundary), and for some Riemannian metric g_∂ X on ∂ X, the pullback ι^* g has the form

    ι^* g - ρ^-4dρ^2- ρ^-2 g_∂ X∈ρ C^∞(Sym^sc T^* ([0,ρ̅)_ρ×∂ X) ),

where ^sc T^* X is the vector bundle over X whose smooth sections are given by C^∞(X)ρ^-2dρ, C^∞(X)ρ^-1ω for ω∈Ω^1(∂ X) and linear combinations thereof. That is, g differs from the exactly conic metric ρ^-4dρ^2 + ρ^-2 g_∂ X by suitably decaying terms. In the exact Euclidean case, g_∂ X is the standard metric (or any scalar multiple thereof) on the 2-sphere at infinity.


The first component of ι^-1 serves as a boundary-defining-function (bdf). That is, there exists a bdf ρ∈ C^∞(X; [0,∞) ) such that ρ(ι(ϱ,θ) ) = ϱ for all ϱ∈ [0,ρ̅]. That this is a bdf means that ρ^-1({0}) = ∂ X and that dρ is nonvanishing on ∂ X (and also that it is nonnegative). Going forwards, we will identify [0,ρ̅]_ρ×∂ X with its image under ι.
We will use the notation Ẋ[R] = [0, R^-1 ]_ρ×∂ X_θ,
and this can be considered as a subset of X as long as R>ρ̅^-1. The subscripts here signal preferred variable names used to parametrize each factor, and similar notation is used throughout below.




We now describe the construction of M in a bit more detail.
As a starting point, let C denote the “cylinder” C = [0,∞]_t×  X. Consider the mwc

    M/ parF=[C; {∞}×∂ X]  = C^∘∪Σ∪nf∪dilF_0 ∪kf_0

resulting from performing a polar blowup of the corner {∞}×∂ X⊂ C of C. Here, Σ = {t=0}, and the remaining three boundary hypersurfaces nf,dilF_0,kf_0 are the lift of [0,∞]_t×∂ X, the front face of the blowup, and the lift of {∞}× X, respectively. Then, M can be constructed in terms of M/ parF as M = [M/ parF; dilF_0 ∩kf_0 ],
which is the result of performing a polar blowup of the corner dilF_0 ∩kf_0 of [C; {∞}×∂ X].
This construction of M is depicted in <Ref>. The notation M/parF indicates that this mwc is, as a topological space, the quotient resulting from collapsing parF.







There are a number of other compactifications of ^+_t× X^∘ via mwcs used in this paper.
In the next section, we use

    C_1 = [0,∞]_t/r^2× X.

Topologically, this is just diffeomorphic to C (i.e. they are both cylinders), but it differs as a compactification of _t× X.
We refer to the boundary hypersurfaces {t/r^2=∞}, {r=∞}, and {t=0} of this mwc as kf, parF_1, Σ_1 respectively. As the notation suggests, a small neighborhood of kf in C_1 is identifiable with a neighborhood of kf in M, and the interiors of parF_1,Σ_1 are identifiable with their counterparts in parF,Σ, respectively. An alternative construction of M involves blowing up the lower corner of C_1 and then blowing up the new lower corner of the resultant mwc, which we call M/nf.


Another compactification of note is M/dilF, which is the result of blowing down dilF in M.
In <ref>, we address the question as to whether <Ref> holds on M/f for f∈{nf,dilF}. The upshot (which holds also for parF, but requires a different argument) is that <Ref> fails on both. (A similar analysis also applies to <Ref>, but we do not present it.)

In the Euclidean case, we can also define M/kf using the coordinates 1/t∈ [0,∞), x_j/t^1/2∈ near the blown down locus, but if bound states are present then it can be immediately concluded from <Ref> that exponential-polyhomogeneity does not hold on this compactification.
So, M is minimal among mwcs on which solutions of the Schrödinger equation with Schwartz initial data have the desired form.









 §.§ Sketch of proof



Consider the differential operator P = Δ_g + iA·∇ + 2^-1 i ∇_g· A + V.
The hypotheses are such that P: C_c^∞(X^∘) → L^2(X,g)
defines an essentially self-adjoint operator. The closure is the map H^2(X,g)→ L^2(X,g) given by restricting P: '(X)→'(X), defined in the sense of distributions, to the L^2-based Sobolev space H^2(X,g).
The spectrum σ(P) = σ_pp(P)∪σ_ac(P) of P consists of finitely many negative eigenvalues and a continuous spectrum on the whole nonnegative real axis, so  σ_pp(P) = {-E_1,⋯,-E_N}
for some 0<E_N ≤⋯≤ E_1, and σ_ac(P) = [0,∞).
Note the absence of embedded eigenvalues (recall that we are assuming the nonexistence of a bound state at zero energy)  or of singular continuous spectrum. The absence of embedded eigenvalues for E>0 follows from <cit.> as in <cit.>. That E_N ≠ 0 is the assumption that no bound state exists at zero energy.
Here N∈, with N=0 corresponding to the absence of pure-point spectrum.

Let Π:Borel()→(L^2(X,g)) denote the spectral measure of P.
So, for each Borel set S⊂, Π(S) is a projection operator on L^2(X).


Via the functional calculus, there exists a 1-parameter subgroup U:t↦ U(t) ∈U(L^2(X,g)) such that the solution u(t,x) : _t× X_x→ to <ref> is given by u(t,x) = (U(t) f)(x),
and U(t) is given by

    U(t) = ∫_-∞^∞ e^i E tΠ(E) =e^i Pt,

this integral being well-defined e.g. when applied to an element of (X). One fact that falls out of this formalism (and the ellipticity of P+1) is that, for f∈(X), then U(t)f ∈ C^∞(_t; (X)). This already gives <Ref> in any neighborhood of Σ disjoint from all of the other boundary hypersurfaces of M besides nf. 

For each n∈{0,…,N}, let ϕ_n be an L^2(X,g)-normalized bound state with Pϕ_n = E_n ϕ_n, such that ϕ_0,⋯,ϕ_N are orthogonal. Via a standard elliptic estimate – e.g. ellipticity in Melrose's Diff_sc(X) <cit.> – we have ϕ_n ∈(X) for each n. Let

    Π_pp(E) = ∑_n=1^N δ(E-E_n) ϕ_n  ⟨ϕ_n , - ⟩

be the spectral projection onto the pure-point spectrum.
Here, ⟨-,- ⟩ : L^2(X,g)× L^2(X,g)→ denotes the L^2(X,g)-inner product anti-linear in the first slot and linear in the second.

Stone's theorem says that the spectral projection onto the continuous spectrum, Π_ac(E) = Π(E) - Π_pp(E), which is supported on [0,∞)_E, has Radon–Nikodym derivative given by

    Π_ac(E) = 1/2π i (R(E+i0) - R(E-i0)),

where, for each E∉σ(P), R(E): L^2(X,g) → H^2(X,g)
denotes the resolvent R(E) = (P-E)^-1, and where, for each E>0, R(E± i0) = lim_ϵ→ 0^+ R(E± i ϵ), these limits existing in the strong operator topology.


Combining <ref> and <ref>,

    U(t) =  ∑_n=0^N e^-iE_n tϕ_n  ⟨ϕ_n ,- ⟩ +  1/2π i∫_0^∞ e^i E t  (R(E+i0) - R(E-i0))  E,

the integral being absolutely convergent in the strong sense. So, for any f ∈(X),

    (U(t)f)(x) =  ∑_n=0^N e^-iE_n tϕ_n(x)  ⟨ϕ_n , f ⟩  +  1/2π i∫_0^∞ e^i E t  (R(E+i0) - R(E-i0)) f(x)  E,

where, for each x∈ X^∘, the integral is absolutely convergent.[One conclusion of the analysis in <ref> is that the integrand in <ref> is, for each fixed t and x, continuous in E and rapidly decaying as E→∞. This is one way to see that the integral is absolutely convergent. With a bit more work, this can be turned into a proof that <ref> gives the solution to the Cauchy problem. ] 

The terms in the first sum in <ref> are easily analyzed — see <Ref>. The crux of our problem is to analyze the oscillatory integral I(t,x) = I_+(t,x) - I_-(t,x), where

    I_±(t,x) = ∫_0^∞ e^i E t  R(E± i0) f(x)  E = 2 ∫_0^∞ e^i σ^2 t R(σ^2 ± i 0) f(x) σσ .

(These integrals turn out to be conditionally convergent if t≠ 0, but not absolutely convergent because R(E± i0) f(x) is only O(1/E) as E→∞. We will quickly pass from these oscillatory integrals to ones whose integrands are Schwartz as E→∞, so this subtlety is not important.)


The key input, coming from <cit.><cit.><cit.>[Unfortunately, the order employed here for listing index sets in the superscript of the spaces ^∙ is the opposite of that used in the references. Our justification for the convention here was that it is natural to list the boundary hypersurfaces from left-to-right as they are depicted in <Ref>. This also has the advantage that the least important face ∞f is last.] is a detailed analysis of the output e^∓ i σ r R(σ^2 ± i0) e^± i σ rg(x) : ^+_σ× X_x→ of the “conjugated (limiting) resolvent” e^∓ i σ r R(σ^2 ± i0) e^± i σ r  for g∈(X) on the mwc

    X^sp_res = [ [0,∞]_σ×  X ; {∞}×∂ X] ↩^+_σ× X.

Label its faces zf,tf,bf,∞f, as in <Ref>.[The mwc X^sp_res differs from the mwc X^+_res in <cit.><cit.> because the former contains the face ∞f, whereas X^+_res is non-compact.]  The analysis that we need is summarized in <ref>. So, the integrands of I_± are well-understood. 


The main piece of analysis developed within this particular paper is the production of asymptotic expansions for oscillatory integrals of the same form as I_±:

	Let ,, denote index sets and α,β,γ∈∪{∞}, and suppose that min{α, j: (j,k)∈}>-2.
	Let ϕ∈^(,α),(,β),(,γ),∞(X_res^sp).
	Then, letting
	
    I_± [ϕ](t,x) =2∫_0^∞ e^i σ^2 t ± i σ r ϕ(σ,x) σσ : _t × X_x→,

	we have I_+[ϕ] ∈^(/2+1,α/2+1),(+2,β+2),(0,0)(C_1), and for any χ∈ C_c^∞() such that 0∉supp(1-χ), a decomposition of I_-[ϕ] of the form I_-[ϕ] =	exp( -i(1 -χ(t))/4 t ρ^2) I_osc[ϕ] + I_-,phg[ϕ]
	for some functions
	
    I_-,phg[ϕ] ∈^(/2+1,α/2+1),(+2,β+2),(0,0)(C_1)

	and I_osc[ϕ] ∈^(/2+1,α+1),(+2,β+2), (+1/2,γ+1/2),∞,(0,0)(M).
	Here, the index sets on C_1 are specified in the order at kf, parF_1, and Σ_1, respectively.
	
	


	The reason for the marked difference between the asymptotics of I_- and I_+ is that we are restricting attention to t≥ 0. If one wants to understand the asymptotics for t≤ 0, the roles of I_± are switched.


	Note that we are assuming in <Ref> that ϕ is Schwartz at ∞f. The resolvent output R(σ^2 ± i0) f is certainly not rapidly decaying, let alone Schwartz, as σ→∞, unless f=0. (If w=R(σ^2 ± i0) f were Schwartz as σ→∞, then f=(P -σ^2)w would be too, but since f is independent of σ this is only possible if f=0.) However, it turns out that the part of R(σ^2 ± i0) f that is not Schwartz at ∞f does not depend on the sign ±, which means that its contributions to I_-,I_+ exactly cancel when forming the difference I = I_+ - I_-. It therefore does not contribute to the asymptotic analysis of the Cauchy problem. The precise argument can be found in the next section.
	
	In <cit.>, Hintz instead studied the inhomogeneous problem with zero initial data, for which R(σ^2 ± i0)f is replaced by R(σ^2 ± i0) f(E,x) for f(E,x)∈(_E× X). Then, high energy estimates easily imply that R(σ^2 ± i0) f(E,x) is Schwartz at ∞f. Hintz then relates the solution of the Cauchy problem to the solution of the inhomogeneous problem.
	The analogous argument also works here. We preferred to study the Cauchy problem directly because it aids getting explicit formulas for the asymptotic profile.
	


<Ref> yields <Ref>. This deduction is contained in <ref>. Besides some straightforward but tedious computations, one additional piece of argument is required: we must explain why the non-oscillatory contribution I_phg is Schwartz at the lower corner of C_1. Unfortunately, <Ref> cannot be improved in this regard, as a numerical example in <ref> illustrates. Instead, we will prove Schwartzness at the lower corner of C_1 in the specific context of the proof of <Ref> using a direct argument. An alternative microlocal proof, using the PDE, is provided in <ref>. So, the exponential-polyhomogeneity of the resolvent output on X_res^sp – the conclusions of <cit.><cit.> – is not by itself completely sufficient to yield <Ref>.


	When <ref> is studied with initial data which is only polyhomogeneous with some finite rate of decay, u cannot be Schwartz at nf, because this would imply that the initial data is Schwartz. Our expectation in this case is that I_phg ends up not being Schwartz at the lower corner of C_1, but the rest of the asymptotic analysis is analogous, including the description of the resolvent output (except we get a new polyhomogeneous term at bf that does not depend on the sign ± in the limiting resolvent, so does not affect the asymptotic analysis of the Cauchy problem). So, there is a good reason why <Ref> cannot have in its conclusion that I=I_+-I_- is Schwartz at nf, and why, to prove this Schwartzness in the context of <Ref>, we need to use the PDE and the Schwartzness of the initial data. 



We will prove <Ref> over the course of three sections, <ref>, <ref>,  <ref>. The idea is to write

    I_±[ϕ] = I_±[χ_lowϕ]+I_±[χ_tf∩bfϕ]+I_±[χ_highϕ]

for χ_low, χ_tf∩bf ,χ_high∈  C^∞(X_res^sp) a partition of unity on X_res^sp such that the support of χ_low is disjoint from bf∪∞f, hence supported at low energies and rσ, the support of χ_tf∩bf is disjoint from zf∪∞f, and the support of χ_high  is disjoint from zf∪tf, hence supported at high energy.
The low energy contribution I_±[χ_lowϕ] is analyzed in <ref>, the high energy contribution I_±[χ_highϕ] is analyzed in <ref>, and the final contribution I_±[χ_tf∩bfϕ] is analyzed in <ref>. Each of the functions φ = χ_∙ϕ is polyhomogeneous on a mwc with only one corner, and this allows the analysis to be reduced to the study of elementary oscillatory integrals, each of which will be proven to be of exponential-polyhomogeneous type on M. The mwc M has four corners, there are three χ_∙ to consider, and there are two choices of sign ±, so spread throughout this paper are 4× 3× 2=24 basic lemmas which establish the exponential-polyhomogeneous form of one of the I_±[χ_∙ϕ] near one of the corners of M. (Some of these are stated together as one lemma, and some are spread over multiple lemmas. The “24” number should therefore not be taken literally.) Our main task is to prove these basic lemmas. Each is straightforward, but altogether their proof requires a fair amount of work.



The deduction of <Ref> from the work in <ref>, <ref>,  <ref> is contained in <ref>.







§ PROOF OF <REF>, <REF> FROM <REF>




The following theorem regarding the low-energy behavior of the resolvent output is contained in the conjunction of <cit.><cit.>:

	There exist some index sets _0⊆^≥ 3× and _0⊆^≥ 2×, with (2,1)∈_0, such that, for any f(x)∈ C^∞([0,∞)_σ;(X)), there exist u_0 ∈^(1,0)∪_0(X) and u_1 ∈^(1,1)∪_0(X) (which are allowed to depend on the sign ± if f depends on σ, but which are independent of the sign if f is independent of σ) such that
	
    e^∓ i σ r R( σ^2 ± i0) e^± i σ rf(σ,x)  =  u_0(x) ± i σ u_1(x)+ ϕ_±(σ,x) = φ_±(σ,x)

	for
	
    ϕ_±(σ,x)    ∈^(2,1)∪_0,_0,(1,1)∪_0_loc(X_res^sp\∞f ) 
    φ_±(σ,x)    ∈^(0,0)∪ (2,1)∪_0,(1,0)∪_0,(1,0)_loc(X_res^sp\∞f ).

	Here, the index sets are specified at the faces zf,tf,bf respectively.
	In fact, u_0(x) = P^-1 f(0,x) and u_1 = - P^-1 L P^-1 f(0,x) ∓ i P^-1 f'(0,x), where f'(σ,x) = ∂_σ f(σ,x) and L is as above.


	The properties of φ_± come from the main theorem of <cit.>. The conormality of ϕ_± comes from <cit.>, which says ϕ_±∈_loc^2-,2-,1-(X_res^sp\∞f). Polyhomogeneity, with the stated index sets, then comes from the fact that everything else in <ref> besides ϕ_± is already known to be polyhomogeneous.


Strengthening this slightly:

	Let f∈(_E× X) or f∈(X). For any ϵ>0, there exist
	
    ϕ_even   ∈^(2,1)∪_0,_0,∞(X_res^sp\∞f), 
    φ_even   ∈^(0,0)∪ (2,1)∪_0,(1,0)∪_0,(1,0)_loc(X_res^sp\∞f)

	such that
	
    R(σ^2 ± i0)f(E,x)    = e^± i σ r- ϵσ^2 (u_0(x) ± i σ u_1(x) ) + ϕ_even(σ,x)+ e^± i σ rϕ_±(σ,x) 
       = e^± i σ rφ_± + φ_even

	for some ϕ_±∈^(2,0)∪_0,_0,(1,1)∪_0,∞(X_res^sp ) and φ_±∈^(0,0)∪ (2,1)∪_0,(1,0)∪_0,(1,0),∞_loc(X_res^sp)
	(differing from the φ_±,ϕ_± in <ref>), where the final `∞' means Schwartz behavior as σ→∞.
	Moreover, 
	
    u_0(x)    = P^-1f̃(0,x) = P^-1f(0,x) 
    
    	u_1(x)    = - P^-1 L P^-1f̃(0,x) ∓ i P^-1f̃'(0,x) =  - P^-1 L P^-1 f(x) - P^-1 (r f(0,x)),

	where f̃ = e^∓ i σ r f(E,x).
	

Note that u_0,u_1,ϕ_even,φ_even do not end up depending on the sign ±. (This is the evenness to which the subscript “even” refers.)

	In order to apply <ref> to analyze the situation near σ=0 (and really just away from ∞f), we rewrite:
	
    e^∓ i σ r  R(σ^2 ± i0)f(E,x) = e^∓ i σ r  R(σ^2 ± i0)  e^± i σ rf̃(σ,x)

	for f̃(σ,x) ∈ C^∞([0,∞)_σ; (X)) given by f̃(σ,x) = e^∓ i σ r f(E,x).
	We can then apply <ref> with f̃ in place of f. This immediately gives the portion of the proposition involving φ, except for the behavior at high energy.


	Because of the essential self-adjointness of P, the σ→ 0^+ expansion of R(σ^2 ± i 0) f(E,x) at zf^∘ must have the form
	
    R(σ^2 ± i 0) f(E,x) ∼∑_(j,k)∈ (0,0) ∪ (2,1) ∪_0 (± i σ )^j log^k(± iσ ) ϕ_j,k(x)

	for some ϕ_j,k∈ C^∞(zf^∘), where the key point is that ϕ_j,k does not depend on the choice of sign. See <cit.>. The (j,k)=(2,1) term has the form - σ^2 (log(σ) ± i π/2 ) ϕ_2,1(x). Let ψ∈ C^∞(X^sp_res) be a cutoff Schwartz at bf∪∞f and identically =1 near zf. Then, we can take ϕ_even = -σ^2 log(σ) ψϕ_2,1(X) near σ=0. The properties of ϕ_± at low energy then follow.

	The formulas for u_0,u_1 follow from the corresponding formulas for the terms in <ref>. This completes the analysis of the low-energy behavior.

	If f∈(_E× X), the usual high-energy bounds <cit.> imply that
	
    e^∓ i σ r  R(σ^2 ± i0)f(E,x)∈^(1,0),∞_loc( (0,∞]_σ× X_x),

	where the `∞' denotes Schwartz behavior as σ→∞. So, <ref> also holds away from σ=0 if we take
	
    e^± i σ rϕ_± = R(σ^2 ± i0) f(E,x) -  e^± i σ r- ϵσ^2 (u_0(x) ± i σ u_1(x) ) - ϕ_even(σ,x)

	as a global definition, and the thusly defined ϕ_± lies in the desired function spaces. Likewise, if we define  
	
    φ_± = e^∓ i σ r R(σ^2 ± i 0)f(E,x),

	then the high energy estimates show that φ_± lie in the desired function spaces. 
	So, in this case, we can take φ_even = 0. To summarize, because f is Schwartz as E→∞, even the weak high-energy bounds in <cit.> suffice to show that the resolvent output has Schwartz behavior at high energy.
	
	If f∈(X), then, viewing f as a function on X×_σ, its large-σ semiclassical wavefront set lies in the zero section of the semiclassical cotangent bundle, where P-σ^2 is elliptic, which means (via the semiclassical parametrix) that there exists some v ∈⟨σ⟩^-2 L^∞(_σ;(X)) such that (P-σ^2)v - f ∈(_E× X). Then, 
	
    R(σ^2 ± i0)f = v + R(σ^2 ± i0) g
 
	for g= f-(P-σ^2)v  ∈(_E× X), as follows via the characterization of the output of the limiting resolvent in terms of the function spaces it lies in. We already know (by the previous paragraph) that R(σ^2 ± i0) g has the desired form, and v does not depend on ±, so it can be stuck in ϕ_even,φ_even. 


So, in the expansion of the spectral projection (R(σ^2 +i0) - R(σ^2-i0))f(E,x), all of the terms with even j up to the first logarithmic term cancel, and we even get some cancellation involving the logarithmic part of that term:

    (R(σ^2 +i0) - R(σ^2-i0))f(E,x) ∼ 2 i σϕ_1,0(x) - π i σ^2ϕ_2,1(x) + ∑_(j,k)∈_0 σ^j (logσ)^k ϕ̃_j,k(x)

for some ϕ̃_j,k∈ C^∞(zf^∘).

We are now in a position to prove <Ref>. So, let u be as in <ref>. This is then given by the right-hand side of <ref>.
The integral I in <ref> is given by I(t,x) = I_+[φ_+] - I_-[φ_-],
where

    φ_±∈^(0,0)∪ (2,1)∪_0,(1,0)∪_0,(1,0),∞(X_res^sp)

are as in <ref>. Note that φ_even does not contribute.


Each I_±[φ_±]  has the form described by our main lemma, <Ref>, and it turns out that the combination I_phg=I_+[φ]-I_-,phg[φ] must be Schwartz at dilF∪nf. We provide a proof using microlocal tools in <ref>, but the simplest way to see this is that each term in the Taylor series of I_phg at Σ={t=0} differs from that of the solution u by a Schwartz function (coming from I_osc, which is stated in <Ref> to be Schwartz at nf, and from the sum over bound states in <ref>, in which each bound state is Schwartz at nf). On the other hand, the initial data is Schwartz, so <ref> implies that each term in the Taylor series of u at Σ is Schwartz. So, each term in the expansion of I_phg at Σ is Schwartz. Since I_phg is polyhomogeneous already on C_1, this implies Schwartzness at dilF∪nf when viewed as a function on M.
So, in fact

    e^ir^2 (1-χ(t))/4 t I(t,x) ∈^(1,0)∪ (3/2,0)∪(2,1)∪ (1+_0/2),(3,0)∪(_0+2),(3/2,0),∞,(0,0)(M) 
    ⊆ (t+iϵ)^-3/2^(-1/2,0)∪ (0,0)∪ (1/2,1)∪,(0,0)∪,(0,0),∞,(0,0)(M)

where = (1,0) ∪(-1/2+ _0/2) ⊂ (2^-1^≥ 2)× and = _0-1 ⊂^≥ 1×.


So, combining <ref> and <ref>, we get the desired <ref> for

    u_phg∈^(-1/2,0)∪ (0,0)∪(1/2,1)∪, (0,0)∪,(0,0),∞,(0,0)(M)

defined by u_phg = (t+iϵ)^3/2 e^ir^2 (1-χ(t))/4 t I(t,x).


The rest of <Ref> requires a slightly more refined analysis of u_phg. To this end, write

    I(t,x) = 2 ∫_-∞^∞e^i σ^2 t + i σ r - ϵσ^2 (u_0(x) + iσ u_1(x)) σσ  + I_+[ϕ_+] - I_-[ϕ_-],

where
ϕ_±∈^(2,0)∪_0,_0,(1,1) ∪_0,∞(X_res^sp) are as in <ref>.
Note that the ϕ_even term in <ref> does not contribute.

The first term in <ref> is explicitly computable:

    exp( i r^2 / 4 (t+i ϵ )) ∫_-∞^∞e^i σ^2 t + i σ r - ϵσ^2 (u_0(x) + i σ u_1(x)) σσ = -√(π i)[ r u_0(x)/2 (t+iϵ)^3/2
      + i(r^2 + 2 i t - 2 ϵ )u_1(x)/4 (t+iϵ)^5/2]  ∈ (t+iϵ)^-3/2^(0,0),(0,0)∪ (1,1)∪_0,(0,1) ∪,(-1,1)∪(-1),(0,0)(M)  .



Applying <Ref> to I[ϕ] = I_+[ϕ_+] - I_-[ϕ_-], the conclusion is that I[ϕ] =	exp( -i(1 -χ(t))/4 t ρ^2) I_osc[ϕ] + I_phg[ϕ]
for

    I_osc[ϕ]∈^(2,0) ∪ (5/2,0)∪ (1+_0/2),_0+2,(3/2,1) ∪ (_0+1/2),∞,(0,0)(M) 
    
    = (t+iϵ)^-3/2^(1/2,0)∪,,(0,1)∪,∞,(0,0)(M)

and some I_phg[ϕ] ∈^,,(0,0)(C_1), which must be Schwartz at the low corner.  Combining <ref>, <ref>, and <ref>, the result is that

    u_phg∈^(0,0)∪ (1/2,0)∪, (0,0)∪,(0,1)∪,(-1,1)∪ (-1),(0,0)(M).

Combining this with <ref>, we get

    u_phg∈^(0,0)∪ (1/2,0)∪, (0,0)∪,(0,0),∞,(0,0)(M),

which was what was claimed in <Ref>.

We need to verify that u_phg has the claimed behavior at parF∪kf.
The analysis above showed that

    u_phg = -√(π i)exp( i (1-χ(t)) r^2/4t - ir^2/4(t+iϵ)) [ ru_0(x)   + i(r^2 + 2 i t - 2 ϵ )u_1(x)/2 (t+iϵ)] 
     + ^(1/2,0)∪,,(0,1)∪,(-1,1)∪(-1),(0,0)(M) .

This can be simplified using that (I)

    r^2 (t+iϵ)^-1 u_1(x) ∈^(1,0),(-1,1)∪ (-1),(0,0)(C)⊆^,(1,1)∪ (+1),(0,1)∪,(-1,1)∪ (-1),(0,0)(M),

(II) (1-χ(r/t)) ru_0(x) ∈^∞,∞,(0,0)∪,(0,0)∪,(0,0)(M) and

    (1-χ(r^2/t)) u_1(x) ∈^∞,(1,1)∪_0,(1,1)∪_0,(1,1)∪_0, (0,0)(M),

and (III) the exponential in <ref> fails to be smooth only at nf:

    exp( i (1-χ(t)) r^2/4t - ir^2/4(t+iϵ)) ∈ C^∞(M\nf).

Combining these observations with <ref> yields

    u_phg = -√(π i)  ( χ(r/t)ru_0 - χ(r^2/t)  u_1(x))+ ^(1/2,0)∪,,(0,1)∪,(-1,1)∪ (-1),(0,0)(M\nf),

where, since we are removing nf from consideration, the index set (-1,1)∪ (-1) does not signify anything.


Since χ(r/t)r u_0(x) ∈^(0,0),(0,0)∪, (0,0)∪,∞,∞(M) and χ(r^2/t) u_1(x) ∈^(0,0),(1,1) ∪,∞,∞,∞(M), combining <ref> with <ref> shows that the error term in <ref> lies in

    ^(1/2,0)∪,,(0,1) ∪,(-1,1) ∪ (-1),(0,0)(M\nf)   ∩(^(0,0)∪ (1/2,0)∪, (0,0) ∪,(0,0),∞,(0,0)(M) 
    ∪^(0,0),(0,0)∪, (0,0)∪,∞,∞(M)  ∪^(0,0),(1,1)∪,∞,∞,∞(M))
    
    =^(1/2,0) ∪, ,(0,0)∪,∞,(0,0)(M).

We have therefore improved <ref> to <ref>, which finishes the deduction of <Ref>. 


	Below, we will compute, for all ϕ_± as in <Ref>,
	full asymptotic expansions of I_phg[ϕ] at dilF∪nf.
	As already mentioned, it is not always the case, in this generality, that I_phg[ϕ] is Schwartz there — see <ref>.
	It may therefore seem a bit miraculous that, as stated above, Schwartzness does hold when ϕ_±(σ,x) = R(σ^2 ± i 0)f(x) for f∈(X). In principle, it should be possible to prove this fact by verifying that all of the terms in the expansions below vanish at dilF∪nf. Here (in this section and in <ref>), we have provided two alternative proofs.


The proof of <Ref> is completely analogous, except one has to add to f(x) a term g(E,x)∈(_E× X) that is the inverse Fourier transform of the forcing F(t,x)∈(_t× X). All the lemmas above were written in the level of generality needed to handle this.



§ LOW ENERGY CONTRIBUTION



We now analyze I_±[ϕ](t,x) = 2∫_0^∞ e^i σ^2 t ± i σ / ρ(x) ϕ(σ,x) σσ for ϕ polyhomogeneous on X_res^sp and supported away from bf∪∞f. This therefore constitutes the low energy contribution to our overall integrals.
We begin with a few geometric preliminaries.
Let λ = σ /ρ(x), so that the map ^+ × X^∘∋ (σ,x) ↦ (λ,x) ∈^+× X^∘ extends to a diffeomorphism

    ι:X_res^sp\ (bf∪∞f)  → [0,∞)_λ× X.

Let φ = ϕ∘ι^-1∈ C^∞((0,∞)_λ× X^∘_x). Then, ϕ being supported away from  bf∪∞f is equivalent to suppφ⋐ [0,∞)_λ× X. That is, φ(λ,-) vanishes identically if λ is sufficiently large. In terms of φ,

    2^-1 I_±[ϕ](t,x) = ∫_0^∞ e^i E t ± i σ /ρ(x) φ(σ / ρ(x),x)  σσ  = ρ(x)^2 ∫_0^∞ e^i λ^2 tρ(x)^2 ± i λφ(λ,x) λλ.


In order to express the spacetime asymptotics of I_±[ϕ], it is convenient to work with the compactification

    C_1 = [0,∞]_τ× X_x↩_t^+× X^∘_x

defined using τ = t ρ(x)^2. As mwcs, C_1≅ C, but these differ as compactifications of spacetime. The three boundary hypersurfaces of C_1 are _τ×∂ X, {∞}× X_x, and {0}× X_x. Note that

    M/ nf≅  [C_1 ; {0}×∂ X],

which is a precise way of saying that C_1 results from M by blowing down both nf and dilF. This blowdown identifies kf with {∞}_τ× X, parF\dilF with (0,∞]_τ×∂ X, Σ\nf with {0}× X^∘, and maps nf∪dilF to the corner {0}_τ×∂ X.


So, in order to specify asymptotics of I_±[ϕ] on M, it suffices to specify them on C_1.
The main proposition of this section, most of the details of the proof of which are relegated to <Ref>, below, reads:

	Suppose that φ(λ,x) ∈^(,α)_c ( [0,∞)_λ; ^(,β)(X_x)) for some index set ⊆{z∈:  z > -2}×, α∈^>-2∪{∞}, index set ⊂×, and β∈∪{∞}. Then,
	
    I_±[ϕ](t,x) ∈^(/2+1,α/2+1),(+2,β+2),(0,0)(C_1),

	where /2+1 is the index set at {∞}_τ× X, +2 is the index set at [0,∞]_τ×∂ X, and (0,0) is the index set at {0}× X.
	

See below for notational conventions regarding the Fourier transform.



	The proof shows that the expansion of I_±[ϕ] at [0,∞]_τ×∂ X, i.e. as r→∞, is just
	
    I_±[ϕ](t,x) ∼∑_(j,k)∈,  j ≤βρ(x)^j+2log^k ρ(x) _ξ→τ(e^± i ξ^1/2φ_j,k(ξ^1/2,θ) ) (τ),

	where φ_j,k(λ) ∈_c^(,α)([0,∞)_λ×∂ X_θ) are the coefficients in the polyhomogeneous expansion of φ(λ,x) at [0,∞)_λ×∂ X, i.e. as x→∂ X .


	Similarly, if we let φ^j,k(x) ∈^(,β)(X_x) denote the coefficients in the λ→ 0^+ expansion of φ(λ,x), then the τ→∞ expansion of I_±[ϕ] is
	
    I_±[ϕ](t,x) ∼ρ(x)^2 ∑_(j,k)∈,  j ≤α[ ∑_j_0=0^∞∑_K≥ k s.t. (j,K)∈(± i)^j_0/j_0! 2^Kφ^j-j_0,K(x)  c_j/2,K,k] (τ)^-j/2-1log^k (τ).

	If λ^j log^k(λ) φ^j,k(x) denotes the leading term in the λ→ 0^+ expansion of φ(λ,x), then the leading term in the τ→∞ expansion of I_±[ϕ] is given by I_±[ϕ] ∼ρ(x)^2 φ^j,k i^1+j/2 (-1)^k Γ(1+j/2) τ^-j/2-1log^k(τ).
	


	Rewriting the integral in terms of ξ = λ^2 σ^2 / ρ(x)^2, we have I_±(t,x)  = Ĩ_±(tρ(x)^2,x) for
	
    Ĩ_±(τ,x) = ρ(x)^2 ∫_0^∞ e^i ξτφ̃_±(ξ,x)  ξ = ρ(x)^2 _ξ→τ(Θ(ξ) φ̃_±(ξ,x))(τ),

	where φ̃_±(ξ,x) = e^± i ξ^1/2φ(ξ^1/2,x). Because φ(λ,x) ∈^(,α)_c ( [0,∞)_λ; ^(,β)(X_x)), we have
	
    φ̃_±(ξ,x) ∈^(/2,α/2)_c([0,∞)_ξ; ^(,β)(X_x)).

	So, via <Ref>, the Fourier transform on the right-hand side of <ref> lies in the function space ^(/2+1,α/2+1)(_τ; ^(,β)(X_x)).


	The form of the expansions given follow from <Ref>. Indeed, the large-τ expansions are stated as part of that proposition: letting φ̃_±;j,k∈^(,β)(X) denote the coefficients in the expansion of e^± i ξ^1/2φ(ξ^1/2,x) as ξ→ 0^+, then the expansion of ρ(x)^-2 I_±[ϕ] at {∞}_τ× X is given by
	
    φ(x)^-2 I_±[ϕ](t,x) ∼∑_(j,k)∈,  j ≤α |τ|^-j/2-1log^k |τ| [ ∑_(j,K) ∈, K≥ kφ̃_j/2,K(x) c_j/2,K,k] ,

	where the c_∙ = c_∙;+'s are given by <ref> below. Since φ̃_j/2,k(x) = ∑_j_0=0^∞ (± i)^j_0 (j_0! 2^k)^-1φ_j-j_0,k(x),
	<ref> follows.  Note that this sum is finite, since φ_j-j_0,k vanishes if j_0 is too large.

	Also, letting φ_j,k(λ) ∈_c^(,α)([0,∞)_λ×∂ X_θ) be the coefficients in the polyhomogeneous expansion of φ(λ,x) at [0,∞)_λ×∂ X, and defining φ_γ by
	
    φ(λ,x) = φ_γ(λ,x)+ ∑_(j,k)∈,  j ≤γρ(x)^j log^k ρ(x) φ_j,k(λ,θ),

	we have φ_γ(λ,x) ∈^(,α)_c ( [0,∞)_λ; ^γ(X_x)). <Ref> then says that the error in truncating the expansion in <ref> to order γ lies in ^(/2+1,α/2+1),γ+2,(0,0)(C_1),
	where the γ+2 is the order at [0,∞]_τ×∂ X. Since γ can be any real number ≤β, we conclude that <ref> holds.




 §.§ Fourier transforms of polyhomogeneous functions on the half-line


Our convention for the Fourier transform :'()→'() is

    ϕ(τ) = ∫_-∞^+∞ e^iξτϕ(ξ) ξ .

We will also write ϕ(τ) as _ξ→τ(  ϕ(ξ))(τ) when it is useful to name the dual variable, `ξ' in this case.

The main proposition of this subsection is:

	Suppose that = ⋂_n∈_n is an intersection of Banach spaces over , with _n+1⊆_n continuously.
	Fix α∈ (-1,∞) ∪{∞} and an index set ⊂{z∈:  z >-1}×, so that
	
    ^(,α)_c([0,∞);) ⊆ L^1(;).

	Then, if ϕ∈^(,α)_c([0,∞);),
	the Fourier transform ϕ satisfies ϕ(τ) ∈^(+1,α+1) (_τ;). Moreover, if ϕ_j,k∈ are the coefficients in the polyhomogeneous expansion
	
    ϕ(ξ) ∼∑_(j,k)∈ , j ≤αϕ_j,kξ^j log^k ξ

	of ϕ(ξ) as ξ→ 0^+, then
	
    ϕ(τ) ∼∑_(j,k)∈,  j ≤α[ ∑_K≥ k  s.t  (j,K)∈ϕ_j,K c_j,K,k;±] |τ|^-j-1log^k |τ|

	is the polyhomogeneous expansion of ϕ as τ→±∞, where the c_∙'s are given by <ref>.
	


Cf. <cit.>, which deals with polyhomogeneous functions on the compactified half-line.

Recall that the `c' subscript means that ϕ∈^(,α)_c([0,∞);) implies that there exists some ξ_0>0 such that ϕ (ξ)=0 for all ξ≥ξ_0.

For ϕ∈^(,α)_c[0,∞), we let ϕ(-ξ)=0 for ξ>0, this being implicit in <ref>.


	For simplicity, we prove the claim when =. The general case is completely analogous, as the assumptions on , as well as the compact support of ϕ, guarantee Bochner integrability throughout the argument.

	Let β∈ (-1,∞) satisfy β≤α. (If α<∞, then there is no reason not to take β=α.) We can write
	
    ϕ(λ) =   ϕ^(β)(ξ)+ ∑_(j,k)∈,   j ≤βϕ_j,kξ^j log^k ξ

	for ϕ_j,k∈ which do not depend on β, where ϕ^(β)∈^β_loc([0,∞)).
	Because  is an index set, the sum here is finite. (In the future, we will simply use that sums of this form are finite without stating so explicitly.)

	Let χ∈ C_c^∞() equal 1 identically on a neighborhood of {0}∪suppϕ. Then,
	
    ϕ(τ) = ∫_0^∞ e^i ξτχ(ξ)  ϕ^(β)(ξ) ξ+   ∑_(j,k)∈,   j ≤βϕ_j,k∫_0^∞ e^i ξτχ(ξ)  ξ^j log^k ξξ.

	Let E_β(τ) = ∫_0^∞ e^i ξτχ(ξ)  ϕ^(β)(ξ) ξ, and, for each (j,k)∈×, let
	
    I_j,k[χ](τ)  =  ∫_0^∞ e^i ξτχ(ξ)  ξ^j log^k ξξ,

	so that
	
    ϕ(τ) = E_β(τ) + ∑_(j,k)∈,  j≤βϕ_j,k I_j,k[χ](τ).

	It follows immediately from <cit.> that E_β∈^β+1(_τ).
	On the other hand, I_j,k[χ] can be written as
	
    I_j,k[χ] = χ(τ) *  _ξ→τ(Θ(ξ)ξ^j log^k ξ).

	By <Ref>,
	
    _ξ→τ(Θ(ξ)ξ^j log^k ξ) ∈'(_τ) ∩^(j+1,k)(_τ\{0}) ⊆'(_τ) ∩^+1(_τ\{0}).

	So, by <Ref>, I_j,k[χ](τ) ∈^+1(_τ).

	So, ϕ(τ) ∈^(+1,β+1) (_τ). Given the arbitrariness of β≤α, this implies the first clause of the proposition.

	The argument shows that the polyhomogeneous expansion of ϕ is given, at the level of formal series, by
	
    ϕ∼∑_(j,k)∈ϕ_j,k I_j,k[χ](τ).

	By <Ref>, the polyhomogeneous expansion of I_j,k[χ](τ) in this limit is the same as that of _ξ→τ(Θ(ξ)ξ^j log^kξ )(τ), which we compute in <Ref>.  Substituting this into <ref> yields  <ref>.




	For any j∈{z∈:  z>-1} and k∈, _ξ→τ(Θ(ξ)ξ^j log^k ξ ) is smooth away from the origin, and, for τ>0,
	
    _ξ→τ(Θ(ξ)ξ^j log^k ξ )(τ) = τ^-j-1∑_κ=0^k c_j,k,κlog^κτ

	for some c_j,k,κ∈. In fact, c_j,k,k = i^j+1 (-1)^k Γ(j+1),
	where Γ:\^≤ 0→ denotes Euler's gamma function and (± i)^z = exp(±π i z/2) for z∈.
	


	The proof shows that c_j,k,κ is given by
	
    c_j,k,κ = i^j+1 (-1)^κkκ∑_ϰ=0^k-κ( ±π i/2)^k-κ-ϰk-κϰd^ϰΓ(j+1) /d j^ϰ

	for all j∈{z∈:  z> -1}, k∈, and κ∈{0,…,k}.


	For τ≠ 0, the Fourier transform _ξ→τ(Θ(ξ)ξ^j log^k ξ ) is given by
	
    _ξ→τ(Θ(ξ)ξ^j log^k ξ ) = lim_ϵ→ 0^+∫_0^∞ e^i ξτ -ϵξ |τ| ξ^j log^k (ξ) ξ.

	Letting ξ = |τ| ξ, the  integral on the right-hand side can be written
	
    ∫_0^∞ e^i ξτ - ϵξ |τ|ξ^j log^k (ξ) ξ = |τ|^-j-1∑_κ=0^k (-1)^κkκlog^κ|τ|  ∫_0^∞ e^± i ξ - ϵξξ^j log^k-κ(ξ) ξ,

	where the ± is the sign of τ. The right-hand side has the same form as the right-hand side of <ref>, with an explicit formula for c_j,k,κ. All we need to do is compute the ϵ→0^+ limit of the integrals on the right-hand side.

	By Cauchy's integral theorem,
	
    ∫_0^∞ e^± i ξ - ϵξξ^j log^k (ξ) ξ   =  ∫_0^± i∞ e^± i z - ϵ z z^j log^k (z)  z 
       = ± i ∫_0^∞ e^- ξ∓ iϵξ (± iξ)^j log^k(± iξ) ξ,

	where we are using the principal branch of the logarithm in order to fix the phase of (± i ξ)^j = e^± jπ iξ^j and log(± iξ) = ±π i/2 + logξ. So, taking ϵ→ 0^+,
	
    lim_ϵ→ 0^+∫_0^∞ e^± i ξ - ϵξξ^j log^k (ξ) ξ  = ±  i ∫_0^∞ e^- ξ (± iξ)^j log^k (± iξ) ξ.

	The integral on the right-hand side can be written as
	
    d^k/d j^k∫_0^∞ e^- ξ (± iξ)^j ξ = d^k /d j^k( (± i)^j Γ(j+1)) = (± i)^j∑_κ=0^k kκ( ±π i/2)^k-κd^κΓ(j+1) /d j^κ.

	Chaining together these equalities yields the proposition.



Finally, to complete the proof of <Ref>:

	Suppose that χ∈() is identically 1 near the origin, and suppose that f∈'() ∩^_loc(_τ\{0}) = '() ∩^_loc([-∞,0)∪ (0,∞]_τ) for some index set ⊂×. Then,
	
    χ * f(τ) - (1- ψ(τ)) f(τ) ∈(_τ)

	for any ψ∈ C_c^∞(_τ) identically 1 near the origin.
	


	Write f(τ) = E(τ) + F(τ) for E∈() a compactly supported distribution and F∈^(_t). Then,
	
    χ * f(τ) - (1- ψ(τ)) f(τ) = (χ * F(τ) -  F(τ))+ χ * E(τ) - (1- ψ(τ)) E + ψ F(τ) .

	It is quick to see that the last three terms are Schwartz. Indeed, ψ F ∈ C_c^∞(), and necessarily singsupp E ⊆{0}, so (1-ψ) E ∈  C_c^∞() as well. The term χ * E(τ) being Schwartz is equivalent to χ^-1 E being Schwartz. Since E is compactly supported, ^-1 E is smooth, so χ^-1 E ∈ C_c^∞() is Schwartz.

	The remaining term on the right-hand side, χ * F -  F, requires a short argument. This term being Schwartz is equivalent to (1-χ(ξ)) ^-1 F(ξ) ∈(_ξ),
	which follows if ^-1 F is smooth except at the origin and Schwartz outside of some compact subset.
	We can write F(τ) = (1+τ^2)^j F_0(τ) for some F_0 ∈^1() and j∈. Because
	
    1+Δ_ξ = _τ→ξ^-1∘ M_1+τ^2∘_ξ→τ

	preserves the space of tempered distributions on the real line that are Schwartz except at the origin, it suffices to consider the case F∈^1(). Then, ∂_τ^k+ℓ  (τ^k F(τ))∈^1(_τ) ⊆ L^2(_τ) for all k,ℓ∈. Taking the inverse Fourier transform  yields
	
    ξ^k+ℓ∂_ξ^k ^-1 F(ξ)∈ L^2(_ξ)

	for all k,ℓ∈, which implies that ^-1 F is Schwartz except at the origin, as desired.





 §.§ Bound states



The contribution from bound states is a finite sum of functions v: _t× X_x→ of the form w(t,x) = e^-iEtφ(x) for some E∈ and Schwartz φ∈(X).


	If v(t,x) = e^-iEtφ(x) for some E>0 and Schwartz φ∈(X), then v is of exponential-polyhomogeneous type on M and Schwartz at nf∪dilF∪parF.
	


	On the cylinder C = [0,∞]_t× X, each such v is already of exponential-polyhomogeneous type, with
	
    v ∈ e^-i E t⋂_k∈ρ(x)^k C^∞(C) ⊆ e^- i E t⋂_k∈ϱ_nf^k ϱ_dilF^k ϱ_parF^k C^∞(M),

	where we used C^∞(C)⊂ C^∞(M), which follows from the construction of M via blowing up some corners of C.

	We can choose ϱ_kf = t^-1ρ(x)^-1(ρ(x)+1/t ρ(x))^-1, ϱ_parF = ρ(x)+1/t ρ(x), and ϱ_dilF = ρ(x)(ρ(x)+1/t ρ(x))^-1, as follows from the construction of M from M/parF, as described in the introduction.  So, t = ϱ_dilF^-1ϱ_parF^-2ϱ_kf^-1 is polyhomogeneous on M.





§ RADIATION-FIELD ANALOGUE AND THE HIGH ENERGY CONTRIBUTION



Fix an index set ⊂× and α∈∪{∞}.
Suppose that ϕ∈( _σ; _loc^(,α)(X)) vanishes identically in {σ< σ_0}⊂_σ× X for some σ_0>0. We denote the set of such functions as

    ( [σ_0,∞)_σ ; _loc^(,α)(X)).

In this section, we analyze

    I_±[ϕ] (t,x)=∫_0^∞ e^i σ^2 t ±  i σ r(x)ϕ(σ,x) σ,

where r(x) = ρ(x)^-1.
The argument is a straightforward application of the method of stationary phase, as we will see. The phase appearing in the oscillatory integral is θ_±(t,x;σ) = σ^2 t ±σ r, which has derivative ∂_σθ_±(t,x;σ) = 2σ t ± r.
Remembering that ρ>0 and σ,t≥ 0, ∂_σθ_+ is nonvanishing, while ∂_σθ_- vanishes at the “critical” frequency σ_crit = r/2t. Thus, following the oscillatory integral I_±[ϕ] along level sets of r/t ∈ C^∞(dilF^∘), an observer either sees rapid decay or else asymptotics in accordance with the stationary phase expansion.

For I_+[ϕ], the method of nonstationary phase also yields rapid decay at nf. It turns out that I_-[ϕ] also decays rapidly at nf.
The reason is that, in this asymptotic regime, r→∞ and t/r→ 0, which means that σ_crit→∞.
As ϕ(σ ,-) decays rapidly as σ→∞, the data in the stationary phase approximation decays rapidly as well. A less careful version of this reasoning (valid only in nf^∘) is that, in nf^∘, only r is a large parameter, so the relevant portion of the phase is θ_±,0 = ± r/2σ,
whose gradient ∂_σθ_±,0 is nonvanishing, so the method of nonstationary phase applies, regardless of the sign.



 §.§ Nonstationary case of sign


We first turn to the nonstationary case. Actually, in addition to discussing I_+[ϕ], we discuss the contribution I_-,non[ϕ(σ,-),ψ] to I_-[ϕ], where

    I_-,non[ϕ,ψ] (t,x)=∫_0^∞ e^i σ^2 t -  i σ r(x)[ 1 - ψ( σ - r/2t)] ϕ(σ,x) σ∈ C^∞(^+_t× X^∘_x),

where ψ∈ C_c^∞() is identically 1 in some neighborhood of the origin and, for convenience, suppψ⋐ (-σ_0,σ_0), where σ_0 is chosen such that ϕ(σ,-)=0 whenever σ≤σ_0.

In the next proposition, let ^∞,∞,(0,0)(C) denote the set of smooth functions on C = [0,∞]_t×  X that are Schwartz at ∂ C\{t=0}. Such functions are smooth on M and Schwartz at all faces except Σ = cl_M {t=0}.




	If ϕ∈ ( [σ_0,∞)_σ ; ^(,α)(X)), then I_+[ϕ],I_-,non[ϕ]∈^∞,∞,(0,0)(C).
	


	For any R>0, we have e^± i E^1/2 r(x)ϕ(E^1/2,x) ∈( [σ_0^2,∞) ; C^∞({r(x)<R} )). So, since the Fourier transform has the mapping property
	
    _E→ t: ( _E ; C^∞({r(x)<R} )) →( _t ; C^∞({r(x)<R} )),

	we deduce, since I_+[ϕ](t,x) = _E → t( e^i E^1/2 r(x)ϕ(E^1/2,x) )(t), that I_+[ϕ](t,x) ∈(_t;   C^∞ (X^∘_x)). Similarly,  if t is sufficiently large so that an R/2t-neighborhood of suppψ is still a subset of (-σ_0,σ_0), then if r(x)<R,
	
    I_-,non[ϕ](t,x) = _E → t( e^-i E^1/2 r(x)ϕ(E^1/2,x) )(t) .

	So, I_-,non[ϕ](t,x) ∈(_t;   C^∞ (X^∘_x)).
	So, in order to prove the proposition, it suffices to restrict attention to any neighborhood in C of [0,∞]_t×∂ X, at least one of which is identifiable with [0,∞]_t×Ẋ[R] for Ẋ[R]=(R,∞]_r ×∂ X_θ.

	Let ^∞,∞,(0,0)_loc(Ċ[R]) denote the set of smooth functions on Ċ[R] = [0,∞]_t×Ẋ[R]
	Schwartz at ∂Ċ[R]\{t=0}. It suffices to prove that
	
    I_+[ϕ] ∈^∞,∞,(0,0)_loc(Ċ[R]).



	Let L∈Diff_b(Ẋ), i.e. L is a differential operator in the C^∞(Ẋ)-algebra generated by vector fields tangent to ∂ X. For any j∈, we can write ∂_t^j L I_+[ϕ] = I_+[ϕ_j,L] for some
	
    ϕ_j,L∈(_E; _loc^(,α)(Ẋ))

	vanishing identically in {E<E_0=σ^2}.
	In order to prove that I_+[ϕ] ∈, it suffices to prove that
	
    I_+[ϕ_j,L]  ∈⟨ t +r ⟩^-K  L^∞_loc( [0,∞]_t ×Ẋ)

	for every K∈. Here, L^∞_loc( [0,∞]_t ×Ẋ) is the set of functions f(t,r,θ) on ^+_t×Ẋ_r,θ, such that, for each r_0>0, there exists some C_r_0>0 such that |f(t,r,θ)| ≤ C_f_0 whenever r≥ r_0.


	More generally, we show that the bound <ref> holds for I_+[ψ] whenever ψ(t,E,r,θ) ∈ C^∞(^+_t×_E ×Ẋ_r,θ)
	is vanishing identically on {E<E_0} and satisfies the following bounds: there exists some J∈ such that, for all k,K'∈, and for all Q ∈Diff_sc(Ẋ),
	
    ∂^k/∂ E^k Qψ(t,E,r,θ) ∈⟨ E ⟩^-K'⟨ t+r ⟩^J L^∞(^+_t×_E × (Ẋ_r,θ∩{r≥ r_0}) )

	for all r_0>0. For the ϕ_j,L above, this holds with J>0 sufficiently large such that _loc^(,α)(Ẋ) ⊂⟨ r ⟩^J L^∞_loc(Ẋ),
	but it will be useful to consider other values of J.


	Applying <ref> with k=0 and K'=2 yields
	
    |I_+[ψ](t,r,θ)| ≤∫_E_0^∞ |ψ(t,E,r,θ)|  E  ∈⟨ t+r  ⟩^J  L^∞_loc( [0,∞]_t ×Ẋ),

	so <ref> holds with K=-J. This is the base case of the inductive argument.


	Let K̂∈.
	Suppose we have shown that <ref>  holds for K = - J + K̃ for all K̃∈{0,…,K̂}, whenever ψ satisfies <ref>. The inductive step, which once handled completes the argument, is to show that the bound holds also for
	
    K = - J + K̂ + 1,

	i.e. that I_+[ψ] ∈⟨ t+r⟩^J - K̂-1 L^∞_loc([0,∞]_t×Ẋ). Writing
	
    I_+[ψ] = ∫_E_0^∞( ∂θ_+/∂ E)^-1( ∂/∂ Ee^i E t + i σ(E) r) ψ(t,E,-)  E

	and integrating by parts, the result is I_+[ψ] = I_+[ψ_1 ]+ I_+[ψ_2] for
	
    ψ_1 = -(∂θ_+/∂ E)^-1∂ψ/∂ E,     ψ_2= (∂θ_+/∂ E)^-2∂^2 θ_+/∂ E^2ψ .

	Note that ∂^2_E θ_+ = - 1/4 E^3/2. Thus, <Ref> implies that for each ν∈{1,2}, all k,K'∈, and for all Q ∈Diff_sc(Ẋ),
	
    ∂^k/∂ E^k Qψ_ν(t,E,r,θ) ∈⟨ E ⟩^-K'⟨ t+r ⟩^J-ν L^∞(^+_t×_E × (Ẋ_r,θ∩{r≥ r_0}) )

	for all r_0>0. In other words, each of ψ_1,ψ_2 also satisfies <ref>, except with a lower value of J. Thus, by the phrasing of the inductive hypothesis,
	
    I[ψ_ν] ∈⟨ t+r⟩^J - K̂-ν L^∞_loc([0,∞]_t×Ẋ) ⊆⟨ t+r⟩^J - K̂-1 L^∞_loc([0,∞]_t×Ẋ),

	as desired.



	For any j,k∈,  and for any r_0,σ_0>0, there exists a constant C = C(j,k,σ_0,r_0)>0 such that
	
    | ∂^j  ∂ ^k/∂σ ^j  ∂ r^k σ/2σ t+r| ≤C ⟨σ⟩/⟨ t+r ⟩^k+1

	holds for all t > 0, r≥ r_0, and σ≥σ_0.
	


	We have ∂_σ^j ∂_r^k (σ /(2σ t+r))  = (-1)^k k! ∂_σ^j (σ/(2σ t+r)^k+1),
	and
	
    ∂^j/∂σ^j σ/(2σ t+r)^k+1 = (k+j-1)! /k!j (-2t)^j-1/(2σ t+r)^k+j + (k+j)!/k!(-2t)^j σ/(2σ t +r)^k+j+1.

	The first term on the right-hand side, which is only nonzero if j≠ 0, satisfies the required estimate, as, for j≥ 1,
	
    0≤  t^j-1 (2σ t+r)^-k-j≤  (2 σ_0)^-j+1 (2σ_0 t + r)^-k-1≤ C_j,k⟨ t+r ⟩^-k-1

	for some C_j,k>0. The second term in <ref> is under control as well, as 0 ≤ t^j (2 σ t + r)^-k-j-1≤ (2σ_0)^-j (2σ_0 t+r)^-k-1≤ C_j,k⟨ t+r ⟩^-k-1,
	for a possibly different C_j,k. So, the bound <ref> follows.




 §.§ Stationary remainder

We now turn to the remaining contribution I_-,stat[ ϕ(σ,-),ψ ] to I_-[ϕ], where

    I_-,stat[ϕ,ψ](t,x) = ∫_0^∞ e^i σ^2 t -  i σ r(x)ψ( σ - r/2t) ϕ(σ,x) σ∈ C^∞(^+_t× X^∘_x).

The main proposition of this section says:

	Given ϕ∈([0,∞)_σ ; ^(,α)(X) ), ψ∈ C_c^∞() satisfying suppψ⋐ (-σ_0,σ_0), 0∉supp(1-ψ), and given  χ∈ C_c^∞() identically 1 near the origin,
	
    I_-,stat[ϕ,ψ] ∈ e^- i (1-χ(t)) r^2/4t^∞,∞, (+1/2,α+1/2),∞,∞(M),

	with I_-,stat[ϕ,ψ] vanishing identically in cl_M{r/2t ≤ϵ} for some ϵ>0.
	


	Since ϕ(σ,x)=0 for σ≤σ_0, and since we chose ψ such that suppψ⋐ (-σ_0,σ_0), and therefore suppψ⊂ (-σ_0+ϵ,σ_0-ϵ) for some ϵ>0, the integral I_-,stat[ϕ,ψ](t,x) is vanishing in {r/2t ≤ϵ}. Thus, we work on the sub-mwc
	
    M_ϵ,R=M ∩cl_M{r/2t > ϵ, r(x)> R}

	for R>0 sufficiently large such that we can identify X∩cl_X{r(x)>R} with Ẋ[R] = [0,R^-1)_ρ×∂ X_θ via a choice of boundary collar Ẋ[R] ↪ X.

	We can write M_ϵ,R = M_ϵ,R^∘∪ U_0 ∪ U, where, for any T>0 satisfying Tϵ > R and R_0>R satisfying R_0/2T > ϵ,
	
    U_0 ≅  [0,2T)_t×Ẋ[R_0],     
    		U ≅ (T,∞]_t×  ( 2ϵ,∞]_r/t×∂ X_θ.

	That is:
	
		
  * the map [0,2T)_t×Ẋ[R_0] ↪ M_ϵ,R, applying the boundary collar to the right factor, is a diffeomorphism onto U_0, and
		
  * the composition
		
    (T,∞)_t×  (2ϵ,∞)_r/t×∂ X_θ↪ (T,∞)_t×Ẋ[R]_r,θ↪ M^∘_ϵ,R,

		where the first map sends (t,r̂,θ)↦ (t,(tr̂,θ)) and the second map applies the boundary collar, extends to a diffeomorphism (T,∞]_t×  ( ϵ,∞]_r/t×∂ X_θ→ U.
	
	So, in order to conclude the proposition, it suffices to prove that I_-,stat[ϕ,ψ](t,r,θ)∈_loc ( [0,∞)_t×Ẋ) and
	
    I_-,stat[ϕ,ψ](t,r̂ t,θ)∈ e^-i r̂^2t/4_loc^(+1/2,α+1/2),∞( (T,∞]_t× (2ϵ,∞]_r̂×∂ X_θ ),

	where +1/2 is the index set at t=∞ and the ∞ denotes Schwartz behavior at r̂=∞.
	These claims are proven below. The first is in <Ref>, and the second is in <Ref>.



A modification of <ref>,

    I_-,stat[ϕ,ψ](t,r,θ) = ∫_0^∞ e^i σ^2 t -  i σ rψ( σ - r/2t) ϕ(σ,r,θ) σ∈ C^∞(^+_t×Ẋ^∘[R]_r,θ )

defines a function I_-,stat[ϕ,ψ]:_t×Ẋ[R] →  for any ϕ∈( _σ; _loc^(,α)(Ẋ)), for any index set  and any α∈∪{∞}, and for any ψ∈ C_c^∞().


	For any ϕ∈( _σ; _loc^(,α)(Ẋ)) and ψ∈ C_c^∞(), the function I_-,stat[ϕ,ψ] satisfies
	
    I_-,stat[ϕ,ψ] ∈_loc ( [0,∞)_t×Ẋ),

	i.e. is Schwartz at both boundary hypersurfaces {t=0} and [0,∞)_t×∂Ẋ.
	


	Using the rapid decay of ϕ(σ,-) as σ→∞ in some weighted L^∞-space ⟨ r ⟩^J L^∞_loc(Ẋ), we have, for all r_0>0 and r ≥ r_0,
	
    |I_-,stat[ϕ,ψ](t,r,θ) | ≤‖ψ‖_L^1sup_σ≥ r/2t - σ_0 |ϕ(σ,r,θ)| ≼⟨r/2t - σ_0 ⟩^-K⟨ r ⟩^J,

	which holds for some J≥ 0 and all K≥ 0, where the constant involved depends on r_0. Since ⟨ r/2t - σ_0 ⟩≽⟨ r ⟩⟨ t^-1⟩, we conclude that I_-,stat[ϕ,ψ] ∈⟨ r ⟩^-∞⟨ t^-1⟩^-∞ L^∞_loc ( [0,∞)_t×Ẋ).

	In order to control derivatives, we use the identities
	
    ∂_t I_-,stat[ϕ,ψ]    = i I_-,stat[σ^2 ϕ,ψ] + (r/2t^2) I_-,stat[ϕ,ψ'] 
    ∂_r I_-,stat[ϕ,ψ]    = -i I_-,stat[σϕ,ψ] - (1/2t) I_-,stat[ϕ,ψ'] + I_-,stat[∂_r ϕ,ψ].

	Applying these inductively, and applying the L^∞-bounds derived in the previous paragraph, it can be concluded that
	
    ∂_t^j ∂_r^k I_-,stat[ϕ,ψ] ∈⟨ r ⟩^-∞⟨ t^-1⟩^-∞  L^∞_loc ( [0,∞)_t×Ẋ)

	for all j,k∈. So, I_-,stat[ϕ,ψ] ∈_loc ( [0,∞)_t×Ẋ).



 For ϕ∈( _σ; _loc^(,α)(Ẋ)) and ψ as above,
	
    I_-,stat[ϕ,ψ](t,tr̂,θ) ∈ e^-i r^2/4t_loc^(+1/2,α+1/2),∞((0,∞]_t× (0,∞]_r̂×∂ X_θ ).

	The t→∞ expansion is given by
	
    I_-,stat[ϕ,ψ](t,tr̂,θ)    ∼ e^-i r^2/4t∑_j=0^∞Γ(j+1/2)/(2j)! (-i t)^j+1/2ϕ^(2j)(r̂/2,tr̂,θ), 
       ∼e^-i r^2/4t/√(-i t)∑_(j,k)∈[ ∑_j_0=0^∞r̂^-j_0Γ(j-j_0+1/2)/(2(j-j_0))! (-i)^j-j_0
                       ×∑_k_0=0^k log^k_0(r̂) k+k_0k_0ϕ^2(j-j_0)_j_0,k+k_0]  t^-jlog^k (t),

	where ϕ^(k)(σ,r,θ) = ∂_σ^k ϕ(σ,r,θ) for k∈, and ϕ^(k) = 0 if k<0, and where
	
    ϕ^(k)(σ,r,θ) ∼∑_(j,k)∈ϕ^(k)_j,k(σ,θ) r^-jlog^k(r)

	is the polyhomogeneous expansion of ϕ^(k) at r=∞, i.e. at bf. Here, each ϕ^(k)_j,k(σ,θ) is in ([σ_0, ∞) ; C^∞(∂ X_θ)).
	


	Let Ĩ_-,stat[ϕ,ψ](t,r,θ) = e^i r^2/4t I_-,stat[ϕ,ψ](t,r,θ). In terms of r̂=r/t, this can be written
	
    Ĩ_-,stat[ϕ,ψ](t,r̂t,θ) =  ∫_0^∞ e^it (σ-r̂/2)^2 ψ(σ-r̂/2) ϕ(σ,tr̂,θ)  σ.

	Our goal is to prove that this lies in _loc^(+1/2,α+1/2),∞ ((0,∞]_t× (0,∞]_r̂×∂ X_θ ).

		For each K∈, Taylor's theorem says that
	
    ϕ(σ,r,θ) = ∑_k=0^K 1/k!( σ - r̂/2)^k ϕ^(k)( r̂/2,r,θ)  + 1/K!∫_0^σ-r̂/2(σ - r̂/2-δ)^K ϕ^(K+1)( r̂/2+δ,r,θ)  δ,

	where the superscript on ϕ refers to differentiation in the first slot.


	Thus, Ĩ_-,stat[ϕ,ψ] = ∑_k=0^K ϕ^(k)(r̂/2,r,θ) I_-,stat,k[ψ] + I_-,stat,K,rem[ϕ,ψ ] for
	
    I_-,stat,k[ψ]    = 1/k!∫_-∞^∞  e^it(σ-r̂/2)^2( σ - r̂/2)^k  ψ( σ - r̂/2)  σ
       = 1/k!∫_-∞^∞  e^itδ^2δ^k  ψ(δ)  δ,
    		
    
    		I_-,stat,K,rem[ϕ,ψ]    = 1/K!∫_-∞^∞  e^itΔ^2ψ(Δ) [∫_0^Δ  (Δ-δ)^K ϕ^(K+1)( r̂/2+δ,r,θ) δ]Δ .

	The stationary phase approximation suffices to show that I_-,stat,k[ψ] ∈ t^-(k+1)/2 C^∞((0,∞]_t). In fact, since ψ=1 identically near the origin, the difference
	
    k! I_-,stat,k[ψ](t)-
    		
    			0     (k odd), 
    
    			(- it)^-(k+1)/2Γ((k+1)/2)     (otherwise),

	is, for large t, Schwartz.



	Since ϕ^(k)(r̂/2,r,θ)=ϕ^(k)(r̂/2,r̂ t,θ) lies in (_r̂; _loc^(,α)( (0,∞ ]_t ) ), it follows that
	
    ϕ^(k)(r̂/2,r,θ) I_-,stat,k[ψ] ∈(_r̂; t^-1/2_loc^(,α)( (0,∞ ]_t ) ).



	On the other hand, <Ref> shows that
	
    |I_-,stat,K,rem[ϕ,ψ](t,r̂t,θ)| ∈ t^-⌊ (K+1)/2 ⌋r̂^-∞_loc^0,0((0,∞]_t × (0,∞]_r̂×∂ X_θ ).

	Combining everything, I_-,stat[ϕ,ψ]∈(_r̂; _loc^(+1/2,min{α+1/2,⌊ (K+1) /2 ⌋}) (0,∞ ]_t ) ). Since K can be taken arbitrarily large, the result follows.




	For each J,K∈, ϕ∈(_σ ; _loc^0(Ẋ)), and ψ∈ C_c^∞(), consider the function _J,K[ϕ,ψ]:^+_t×Ẋ^∘_r,θ→ given by
	
    _J,K[ϕ,ψ] =∫_-∞^∞  e^itΔ^2Δ^J ψ(Δ) [∫_0^Δ  (Δ-δ)^K ϕ( r/2t+δ,r,θ) δ]Δ.

	Then,  _J,K[ϕ,ψ] ∈ t^-⌊ (J+K+1)/2 ⌋(t/r)^∞ L^∞_loc((0,∞]_t ×Ẋ_r,θ).

	In fact,
	_J,K[ϕ,ψ](t,r̂t,θ) ∈ t^-⌊ (J+K+1)/2 ⌋r̂^-∞_loc^0,0((0,∞]_t × [0,∞)_r̂×∂ X_θ ).
	




	We first prove the L^∞-bounds. For J+K=0, we have
	
    |_J,K[ϕ,ψ]| ≤ |suppψ | ‖ψ‖_L^1sup_δ∈suppψ|  ϕ( r/2t+δ,r,θ) | ∈  (t/r)^∞ L^∞_loc((0,∞]_t ×Ẋ_r,θ).

	To handle the J+K≥ 1 case, we integrate-by-parts, starting from
	
    2i t_J,K[ϕ,ψ] = ∫_-∞^∞( ∂/∂Δ e^itΔ^2) Δ^J-1ψ(Δ) [∫_0^Δ  (Δ-δ)^K ϕ( r̂/2+δ,r,θ) δ]Δ.

	Integrating-by-parts yields -2it _J,K[ϕ,ψ]=(J-1)_J-2,K[ϕ,ψ] + _J-1,K[ϕ,ψ'] + K _J-1,K-1 if K≠ 0 and
	
    -2it _J,0[ϕ,ψ]=(J-1)_J-2,0[ϕ,ψ] + _J-1,0[ϕ,ψ'] + _J-1[ϕ,ψ]

	otherwise,
	where the last of these functions is defined by <ref>.

	So, _J,K[ϕ,ψ] ∈ t^-⌊ (J+K+1)/2 ⌋(t/r)^∞ L^∞_loc((0,∞]_t ×Ẋ_r,θ) follows inductively.

	In order to control derivatives, we use L_J,K[ϕ,ψ]=_J,K[Lϕ,ψ], which holds for all L∈Diff(∂ X_θ), and the identities
	
    ∂_r̂_J,K[ϕ,ψ](t,r̂t,θ)    = 2^-1_J,K[∂_σϕ(σ,r,θ),ψ](t,r̂t,θ) + r̂^-1_J,K[r∂_rϕ(σ,r,θ),ψ](t,r̂t,θ) 
    ∂_t _J,K[ϕ,ψ](t,r̂t,θ)    = i_J+2,K[ϕ,ψ](t,r̂t,θ) + t^-1_J,K[r ∂_r ϕ(σ,r,θ),ψ](t,r̂t,θ) .

	Using these inductively, and using the L^∞-bounds already proven, the final clause of the lemma follows.




	For each J∈, ϕ∈(_σ ; _loc^0(Ẋ)), and ψ∈ C_c^∞(), consider the function _J[ϕ,ψ]:^+_t×Ẋ^∘_r,θ→ given by
	
    _J[ϕ,ψ](t,r,θ) =∫_-∞^∞  e^itΔ^2Δ^Jψ(Δ) ϕ( r/2t+Δ,r,θ) Δ.

	Then, for each K∈, we have _J[ϕ,ψ](t,r̂t,θ) ∈  t^-⌊ J/2 ⌋r̂^-K_loc^0,0((0,∞]_t × (0,∞]_r̂×∂ X_θ ).


	We first prove the L^∞-bounds.
	If J=0, then
	
    |_J[ϕ,ψ]| ≤‖ψ‖_L^2sup_Δ∈suppψ |ϕ( (r/2t) + Δ,r,θ) | ∈ (t/r)^∞ L^∞_loc((0,∞]_t ×Ẋ_r,θ) .

	If J≥ 1, then integration-by-parts yields
	
    -2 i t_J[ϕ,ψ] = (J-1)_J-2[ϕ,ψ]+ _J-1[ϕ,ψ'] +  _J-1[ϕ',ψ],

	where ϕ'(σ,r,θ) = ∂_σϕ (σ,r,θ). Applying this inductively allows the deduction of _J[ϕ,ψ] ∈ t^-⌊ J/2 ⌋ (t/r)^K L^∞_loc((0,∞]_t ×Ẋ_r,θ) from the J=0 case.

	To deduce conormality, we want to prove that the same L^∞-bounds apply to the quantity (t ∂ t)^j  ∂_r̂^k L _J[ϕ,ψ](t,r̂t,θ) for every j,k∈ and L∈Diff(∂ X_θ). Using the identities L _J[ϕ,ψ]= _J[Lϕ,ψ],
	
    ∂_r̂_J[ϕ',ψ](t,r̂t,θ)    = 2^-1_J[∂_σϕ(σ,r,θ) ,ψ](t,r̂t,θ) + r̂^-1_J[r ∂_r ϕ(σ,r,θ),ψ](t,r̂t,θ)], 
    ∂_t _J[ϕ',ψ](t,r̂t,θ)    = i _J+2[ϕ,ψ](t,r̂t,θ) - t^-1_J[r ∂_r ϕ(σ,r,θ),ψ](t,r̂t,θ),

	these bounds follow from those already proven.




§ REMAINING CONTRIBUTION




Finally, we examine I_±[ϕ](t,x) = 2∫_0^∞ e^i σ^2 t± i σ r(x)ϕ(σ,r,θ) σσ for ϕ polyhomogeneous on Ẋ^sp_res and supported near tf∩bf. Specifically, we consider the case ϕ(σ,r,θ) = φ(σ,σ r,θ) for

    φ(σ,λ,θ) ∈_c^(,α),(,β)([0,Σ)_σ× (Λ,∞]_λ×∂ X_θ)

for some Σ,Λ>0, index sets ,, and α,β∈. Here,  is the index set at σ = 0, i.e. at tf, and  is the index set at λ=∞, i.e. at bf.
In order to formulate the asymptotics of I_±, it is useful to work with the manifold Ṁ / nf, which is defined analogously to M/nf with Ẋ in place of X, and whose faces we label correspondingly.
Recall that Ċ_1 = [0,∞]_τ×Ẋ.
We summarize the results of this section in the following proposition:

	Given the setup above, I_+[ϕ]∈^∞,(+2,α+2),(0,0)_loc(Ċ_1), where the index sets are specified at {t=∞}, parF, and {t=0} respectively, and I_-[ϕ] = exp(-i r^2/4t )   Ĩ_-[ϕ] + I_-,phg[ϕ]
	for some
	
    Ĩ_-[ϕ] ∈^∞,(+2,α+2),(+1/2,β+1/2)_loc(Ṁ/ nf)

	and I_-,phg[ϕ]∈^∞,(+2,α+2),(0,0)_loc(Ċ_1), where the index sets on Ṁ/nf are specified at kf, parF, and dilF (and one has smoothness at Σ).
	Moreover, if χ∈ C_c^∞() satisfies suppχ⋐ (-1,1), then χ(2t ρ(x) Σ ) Ĩ_-[ϕ] is Schwartz.
	


	The statement for I_+[ϕ] comes immediately from <Ref> and <Ref>.

	The partial compactifications ^+_t×Ẋ↪ (Ṁ/ nf) \ ( dilF∪Σ) and ^+_t×Ẋ↪Ċ_1 \Σ are equivalent, in the sense that the identity map on the interior extends to a diffeomorphism (Ṁ/ nf) \ ( dilF∪Σ) ≅Ċ_1 \Σ.
	So, <Ref> tells us that
	
    I_-[ϕ]∈^∞,(+2,α+2)_loc((Ṁ/ nf) \ (dilF∪Σ)),

	where the index sets are specified at kf,parF, respectively.
	On the other hand, the partial compactifications ^+_t×Ẋ↪ (Ṁ / nf) \kf and ^+_t×Ẋ↪ [0,∞)_τ× (0,∞]_s×∂ X_θ given by (t,r,θ) ↦ (t/r^2,t/r,θ) are equivalent. So <Ref> tells us that
	
    Ĩ_-[ϕ] ∈_loc^(+2,α+2),(+1/2,β+1/2)((Ṁ/nf) \kf).

	This gives <ref>.

	The last clause of this proposition follows from the last clause of <Ref>.





 §.§ Control for very large time

The following establishes control of near kf:

	For φ(σ,λ,θ) ∈_c^(,α),(,β)([0,Σ)_σ× (Λ,∞]_λ×∂ X_θ) and ϕ(σ,r,θ) = φ(σ,σ r,θ), we have I_±[ϕ]( t,x) = ρ^2 I̅_±[φ](t ρ^2,x) for some
	
    I̅_±[φ](τ,x) ∈_loc^∞,(,α) (Ċ_1\Σ ),

	where Ċ_1\Σ = (0,∞]_τ×Ẋ, where  is the index set at (0,∞]_τ×∂ X_θ, i.e. as r→∞, and the ∞ denotes Schwartz behavior as τ→∞. Moreover, the expansion at (0,∞]_τ×∂ X_θ is given by
	
    I̅_±[φ](τ,x) ∼∑_(j,k)∈,  j ≤γ2/r^j∑_κ=0^k  (-1)^κkκlog^κ(r) ∫_Λ^∞ e^iλ^2 τ± i λφ_j,k(λ,θ) λ^1+jlog^k(λ) λ

	where φ(σ,λ,θ) ∼∑_(j,k)∈, j≤αφ_j,k(λ,θ)  σ^j logσ^k is the polyhomogeneous expansion of φ as σ→ 0^+, so that
	
    φ_j,k(λ,θ) ∈_c^(,β)((Λ,∞]_λ×∂ X_θ).

	The integrals on the right-hand side of <ref> are well-defined oscillatory integrals (though not necessarily absolutely convergent), e.g. via formal integration-by-parts.
	


	We have I_±[ϕ](t,x) = ρ^2 I̅_±[φ](tρ^2,x) for I̅_±[φ](τ,x) defined by
	
    I̅_±[φ](τ,x) = 2∫_Λ^∞ e^i λ^2 τ± i λφ(λ /r, λ,θ) λλ.

	Defining φ_γ(σ,λ,θ) = σ^-γ (φ(σ,λ,θ) - ∑_(j,k)∈, j≤γφ_j,k(λ,θ)  σ^j logσ^k)
	for γ∈ with γ≤α, we have
	
    φ_γ∈_c^0,(,β)([0,Σ)_σ× (Λ,∞]_λ×∂ X_θ).

	Let χ∈ C_c^∞() be identically 1 on [-1,1]. Then, we can write φ(σ,λ,θ) = χ(σΣ^-1) φ(σ,λ,θ) for all σ ,λ>0 and θ∈∂ X, so
	
    I̅_±[φ](τ,x) = ∑_(j,k)∈,  j ≤γ2/r^j∑_κ=0^k (-1)^κkκlog^κ(r)  I̅_±,j,k-κ[φ](τ,x) + r^-γI̅_±,γ[φ](τ,x)

	for
	
    I̅_±,j,k[φ](τ,x) = ∫_Λ^∞ e^iλ^2 τ± i λχ( λ/rΣ) φ_j,k(λ,θ) λ^1+jlog^k(λ) λ, 
    I̅_±,γ[φ](τ,x) = ∫_Λ^∞ e^iλ^2 τ± i λχ( λ/rΣ) φ_γ(λ/r,λ,θ) λ^1+γλ .

	By <Ref>, I̅_±,j,k(τ,x) ∈_loc^∞,(0,0)((0,∞]_τ×Ẋ_x)
	and I̅_±,γ(τ,x) ∈_loc^∞,0((0,∞]_τ×Ẋ_x), where the (0,0) denotes the index set at (0,∞]_τ×Ẋ_x and ∞ denotes Schwartz behavior as τ→∞.


	So, I̅_±[φ](τ,x) ∈_loc^∞,(,γ) (Ċ_1\Σ ), and since γ≤α was arbitrary we conclude <ref>.
	The explicit expansion follows from the argument above and the second half of <Ref>.



	For φ∈⋃_K∈_c^0,-K([0,Σ)_σ× (Λ,∞]_λ×∂ X_θ), for j∈ and k∈, and for χ∈ C_c^∞(), consider
	
    _±,j,k [φ,χ ](τ,r,θ) = ∫_Λ^∞ e^i λ^2 τ± i λχ( λ/rΣ) φ(λ/r,λ,θ) λ^j log^k( λ ) λ .

	Then, _±,j,k[φ,χ](τ,x) ∈_loc^∞,0((0,∞]_τ×Ẋ_x), where 0 is the order at (0,∞]_τ×∂ X, i.e. as r→∞, and the ∞ denotes Schwartz behavior as τ→∞.
	If φ(σ,λ,θ)=φ(λ,θ) does not depend on σ, and if 0∉supp(1-χ), then
	
    _±,j,k[φ,χ](τ,x) ∈_loc^∞,(0,0)((0,∞]_τ×Ẋ_x),

	and moreover
	
    _±,j,k[φ,χ](τ,x) - ∫_Λ^∞ e^i λ^2 τ± i λφ(λ,θ) λ^j log^k (λ ) λ∈_loc^∞,∞((0,∞]_τ×Ẋ_x).

	The second term on the right-hand side is a well-defined oscillatory integral, even though it may not be absolutely convergent.
	


	It suffices to consider the case  φ∈_c^0,0([0,Σ)_σ× (Λ,∞]_λ×∂ X_θ), as we can write _±,j,k[ φ,χ]=_±,j+K,k[ λ^-Kφ,χ].

	We first prove that _±,j,k[φ,χ](τ,x) ∈τ^-∞ L^∞_loc((0,∞]_τ×Ẋ_x). First of all, if j<-1, then _±,j,k[φ,χ](τ,r,θ) ∈   L^∞_loc((0,∞]_τ×Ẋ_x), as follows immediately from an L^∞-bound. Using
	
    2iτ_±,j,k [φ,χ ](τ,r,θ) =  ∫_Λ^∞[ ∂/∂λ e^i λ^2 τ]  e^± i λχ( λ/rΣ)φ(λ/r,λ,θ) λ^j-1log^k (λ)  λ,

	integrating-by-parts yields
	
    - 2 i τ_±,j,k [φ,χ ] = ± i _±,j-1,k[φ,χ] + r^-1Σ^-1_±,j-1,k[φ,χ']+ _±,j-2,k[σ∂_σφ(σ,λ,θ) ,χ] 
     + _±,j-2,k[λ∂_λφ(σ,λ,θ) ,χ]  + (j-1) _±,j-2,k[φ,χ] + k _±,j-2,k-1[φ,χ].

	Each term on the right-hand side has the same form as the original integral (possibly times an extra L^∞_loc factor), but with j with smaller real part. Since the left-hand side of <ref> has one extra factor of τ, this sets up an inductive argument to conclude O(τ^-∞) decay from the L^∞_loc estimate already proven in the j<-1 case.



	Now suppose that 0∉suppχ. Then, an L^∞-bound yields immediately that, if j<-1, then
	
    _±,j,k[φ,χ](τ,x) ∈   r^j+1+ϵ L^∞_loc((0,∞]_τ×Ẋ_x)

	for any ϵ>0. So, in this case the inductive argument above yields additionally rapid decay as r→∞, i.e. that _±,j,k[φ,χ](τ,r,θ) ∈τ^-∞ r^-∞ L^∞_loc((0,∞]_τ×Ẋ_x).

	We now prove two sets of estimates on derivatives of _±,j,k[φ,χ]. First of all, if n,m∈ and L∈Diff(∂ X_θ), then
	
    ∂_τ^n (r∂_r)^m  L _±,j,k [φ,χ](τ,r,θ) ∈τ^-∞  L^∞_loc((0,∞]_τ×Ẋ_x) .

	Secondly, if 0∉supp(1-χ) and if φ(σ,λ,θ) = φ(λ,θ) does not depend on σ, then, for m∈^+,
	
    ∂_τ^n ∂_r^m L _±,j,k [φ,χ](τ,r,θ) ∈τ^-∞ r^-∞ L^∞_loc((0,∞]_τ×Ẋ_x).

	These follow from applying repeatedly the identities
	
    L _±,j,k [φ,χ] = _±,j,k [Lφ,χ],     ∂_τ_±,j,k [φ,χ] = i_±,j+2,k [φ,χ]

	
    r∂_r_±,j,k [φ,χ] =  -_±,j+1,k[σ∂_σφ(σ,λ,θ),χ] - Σ^-1r^-1_±,j+1,k [φ,χ' ].

	For example, if φ(σ,λ,θ) = φ(λ,θ) does not depend on σ, then the first term on the right-hand side of <ref> is zero, so ∂_r _±,j,k [φ,χ] =   - Σ^-1r^-2_±,j+1,k [φ,χ' ], and if χ=1 identically near the origin, then 0∉suppχ', so we can apply the improved L^∞-estimates <ref> that apply to _±,j+1,k [φ,χ' ] in this case.

	These estimates say
	
    _±,j,k[φ,χ](τ,x) ∈_loc^∞,0((0,∞]_τ×Ẋ_x)

	and, under the stronger set of assumptions, 	_±,j,k[φ,χ](τ,x) ∈_loc^∞,(0,0)((0,∞]_τ×Ẋ_x). The fact that only the leading order term in the r→∞ expansion contributes in the latter case follow from the Schwartz estimates <ref> that apply when taking at least one derivative in r. To get <ref>, one only needs to verify that leading order term in the r→∞ expansion has the desired form, and this can be proven by integrating-by-parts until one is left working with absolutely convergent integrals. Then, the leading order term is computed by replacing χ with 1.




 §.§ Asymptotics elsewhere, I_+

To complete our discussion of I_+, we prove:

		For φ(σ,λ,θ) ∈_c^(,α),(,β)([0,Σ)_σ× (Λ,∞]_λ×∂ X_θ) and ϕ(σ,r,θ) = φ(σ,σ r,θ), we have I_+[ϕ]( t,r,θ) = 2ρ^2 I̅_+[φ](t /r^2,r,θ) for some
		
    I̅_+[φ](τ,r,θ)∈_loc^(,α),(0,0)(Ċ_1 \kf) .

		


	We have I_+[ϕ]( t,r,θ) = 2ρ^2 I̅_+[φ](t /r^2,r,θ) for I̅_+[φ](τ,r,θ) defined by
	
    I̅_+[φ](τ,r,θ) = ∫_Λ^∞ e^i λ^2 τ + i λφ( λ/r,λ ,θ) λλ .

	Now let φ(σ,λ,θ) ∼∑_(j,k) ∈,  j ≤αφ_j,k(λ,θ) σ^j log^k σ denote the polyhomogeneous expansion of φ(σ,λ,θ) at σ=0, so
	
    φ_j,k∈_c^(,β) ((Λ,∞]_λ×∂ X_θ)

	and, letting φ_γ = σ^-γ (φ -  ∑_(j,k) ∈,  j ≤γφ_j,k(λ,θ) σ^j log^k σ), we have φ_γ∈_c^0,(,β)([0,Σ)_σ× (Λ,∞]_λ×∂ X_θ).
	Let χ∈ C_c^∞() be identically 1 on [-1,+1]. Then, φ(σ,λ,θ) = χ(σΣ^-1) φ(σ,λ,θ), so we can write
	
    I̅_+[φ](τ,r,θ) = ∑_(j,k)∈,  j ≤γ( 1/r)^j ∑_κ=0^k kκlog^κ(1/r)  I̅_+,j,k-κ(τ,r,θ) +  (1/r)^γI̅_+,γ(τ,x)

	for
	
    I̅_+,j,k[φ](τ,r,θ) = ∫_Λ^∞ e^iλ^2 τ+ i λχ( λ/r Σ) φ_j,k(λ,θ) λ^1+jlog^k(λ) λ, 
    I̅_+,γ[φ](τ,r,θ) = ∫_Λ^∞ e^iλ^2 τ+ i λχ( λ/rΣ) φ_γ( λ/r,λ ,θ) λ^1+γλ .

	We now appeal to <Ref> to conclude that 	I̅_+[φ](τ,r,θ)∈_loc^(,γ),(0,0)(Ċ_1 \kf). Since γ≤α was arbitrary, we can conclude <ref>.



	Using the explicit expansions in <Ref>, the proof of <Ref> shows that
	the expansion of I̅_+[φ](τ,r,θ) as r→∞ is given by
	
    I̅_+[φ]∼∑_(j,k)∈,  j ≤α(1/r)^jlog^k(1/r) ∑_K≥ k, (j,K) ∈Kk∫_Λ^∞
    	e^i λ^2 τ + iλφ_j,K(λ,θ) λ^1+jlog^K-k (λ) λ.



	Let Σ>0 and χ∈ C_c^∞([0,∞ )).
	Suppose that γ∈, k∈, and that ϕ∈^0,0,0,0_c([0,Σ)_σ× (Λ,∞]_λ× (0,∞]_r × [0,∞)_τ×∂ X_θ ). Consider the integral
	
    _+,γ,k[ϕ,χ](τ,r,θ) = ∫_Λ^∞ e^i λ^2 τ + iλϕ( λ/r ,λ,r, τ ,θ) χ( λ/r Σ) λ^γlog^k(λ) λ.

	Then, _+,γ,k[ϕ,χ](τ,r,θ) ∈^0,0_loc(Ċ_1\kf).
	If ϕ(σ,λ,r,τ,θ) = ϕ(λ,θ) does not depend on any of σ,τ,r, then
	
    _+,γ,k[ϕ,χ](τ,r,θ)
    		- ∫_Λ^∞ e^i λ^2 τ + iλϕ(λ,θ) λ^γlog^k(λ) λ∈^∞,(0,0)_loc(Ċ_1\kf),

	the integral on the left-hand side being a well-defined oscillatory integral. Here, the ∞ denotes Schwartz behavior as r→∞ for τ fixed.
	


	We first prove that
	
    _+,γ,k[ϕ,χ](τ,r,θ)∈ L^∞_loc([0,∞)_τ× (0,∞]_r×∂ X_θ ).

	If γ <-1, then ‖_+,γ,k[ϕ,χ] ‖_L^∞≤‖ϕ‖_L^∞‖χ‖_L^∞∫_Λ^∞λ^γ (logλ)^k λ < ∞. In order to prove the claim for γ≥ -1, we use an inductive argument.
	The key is the identity exp(i λ^2 τ + i λ )  = -i (2λτ+1)^-1∂_λexp(i λ^2 τ + i λ ), hence
	
    _+,γ,k[ϕ,χ](τ,r,θ) = ∫_Λ^∞[  -i/2λτ+1∂/∂λ e^i λ^2 τ + i λ] ϕ( λ/r, λ ,r, τ,θ) χ( λ/r Σ) λ^γlog^k (λ) λ.

	Integrate-by-parts, noting that no boundary terms arise:
	
    _+,γ,k[ϕ,χ] =_+,γ-1,k[φ,χ] +  _+,γ-1,k[ϕ_0,χ] +  _+,γ-1,k[ ψ,σχ'(σ)] 
    +  γ_+,γ-1,k[ ψ,χ] + k_+,γ-1,k-1[ ψ,χ] ,

	where
	
    φ(σ,λ,r,τ,θ)    = i/2λτ+1(  σ∂/∂σϕ(σ,r,τ,λ,θ) + λ∂/∂λϕ(σ,r,τ,λ,θ) )  
       ∈^0,0,0,0_c([0,Σ)_σ×  (Λ,∞]_λ× (0,∞]_r × [0,∞)_τ×∂ X_θ )
    		
    ϕ_0(σ,λ,r,τ,θ)    =  -2i λτϕ(σ,λ,r,τ,θ) /(2λτ+1)^2∈^0,0,0,0_c([0,Σ)_σ×  (Λ,∞]_λ×  (0,∞]_r × [0,∞)_τ×∂ X_θ ),

	and ψ = i (2λτ+1)^-1ϕ∈^0,0,0,0_c([0,Σ)_σ× (Λ,∞]_λ× (0,∞]_r   × [0,∞)_τ×∂ X_θ ).
	Since the right-hand side of <ref> involves only γ-1 in place of γ, this identity can be used repeatedly to reduce the to-be-proven claim <ref> to the γ<-1 case.

	A modification of this argument shows that if 0∉suppχ, then
	
    _+,γ,k[ϕ,χ](τ,r,θ)∈  (1/r)^∞ L^∞_loc([0,∞)_τ× (0,∞]_r×∂ X_θ ).

	Indeed, in this case, the integrand in <ref> is supported for λ∼ r. Thus, an initial L^∞-bound yields _+,γ,k[ϕ,χ](τ,r,θ)∈ r^γ + 1 L^∞_loc([0,∞)_τ× (0,∞]_r×∂ X_θ ). The inductive argument then shows that the same bound holds with γ+K in place of γ, for all K∈, which then implies <ref>.

	In order to deduce that  _+,γ,k[ϕ,χ] ∈^0,0_loc([0,∞)_τ× (0,∞]_r×∂ X_θ ), we want to show that
	
    (r∂_r)^j (τ∂_τ)^κ L _+,γ,k[ϕ,χ](τ,r,θ) ∈  L^∞_loc([0,∞)_τ× (0,∞]_r×∂ X_θ )

	for all j,κ∈ and L∈Diff(∂ X). As elsewhere, we use L _+,γ,k[ϕ,χ] = _+,γ,k[Lϕ,χ], and now we have τ∂_τ_+,γ,k[ϕ,χ] =  i τ_+,γ+2,k[ϕ,χ] +   _+,γ,k[ϑ,χ] for ϑ(σ,λ,r,τ,θ) = τ∂_τϕ(σ,λ,r,τ,θ) and r ∂_r _+,γ,k[ϕ,χ] = _+,γ,k[ϖ,χ] - _+,γ,k[ϕ,σχ'(σ) ]
	for ϖ(σ,λ,r,τ,θ) = (- σ∂_σ +r∂_r)ϕ(σ,λ,r,τ,θ). So, the desired bounds in <ref> follow inductively from the L^∞-bounds proven above.

	Suppose now that ϕ(σ,λ,r,τ,θ) = ϕ(λ,θ) does not depend on any of σ,τ,r. Then, ∂_τ_+,γ,k[ϕ,χ] =  i  _+,γ+2,k[ϕ,χ] and  ∂_r _+,γ,k[ϕ,χ] = - r^-1_+,γ,k[ϕ,σχ'(σ) ]. So, a similar inductive argument to the above shows that
	
    ∂_τ^κ L _+,γ,k[ϕ,χ]    ∈  L^∞_loc([0,∞)_τ× (0,∞]_s×∂ X_θ )  
    ∂_τ^κ∂_r^j+1L _+,γ,k[ϕ,χ]    ∈  r^-∞ L^∞_loc([0,∞)_τ× (0,∞]_r×∂ X_θ )

	for all j,κ∈ and L∈Diff(∂ X_θ). These estimates suffice to show that
	
    _+,γ,k[ϕ,χ](τ,r,θ)
    	∈^(0,0),(0,0)_loc([0,∞)_τ× (0,∞]_r×∂ X_θ )

	and that only the O(1) term in the r→∞ expansion is nontrivial. It can be checked, e.g. via the integration-by-parts argument above, that this leading term is that specified by <ref>.






 §.§ Asymptotics elsewhere, remaining case

Finally, to complete our discussion of I_-:

	For φ(σ,λ,θ) ∈_c^(,α),(,β)([0,Σ)_σ× (Λ,∞]_λ×∂ X_θ) and ϕ(σ,r,θ) = φ(σ,σ r,θ), we have
	
    I_-[ϕ](t,x) =  I_-,phg[φ](t/r^2,r,θ) +  I_-,osc[φ](t/r^2,t/r,θ)

	for
	
    I_-,phg[φ](τ, r,θ  ) ∈ r^-2_loc^(,α),(0,0)(Ċ_1\kf )

	and I_-,osc[φ](τ, s,θ  ) ∈ e^-i/4ττ^1/2 s^-2_loc^(,α), (,β) ([0,∞)_τ× (0,∞]_s×∂ X_θ ), where  is the index set as s→∞ and  is the index set as τ→ 0.
	Moreover, if χ∈ C_c^∞() satisfies suppχ⋐ (-1,1), then
	χ(2sΣ) I_-,osc[φ](τ,s,θ) ∈_loc^∞, ∞([0,∞)_τ× (0,∞]_s×∂ X_θ ).
	


	Let ψ∈ C_c^∞() satisfy suppψ⋐ (-1/2,1/2) and 0∉supp(1-ψ). Now define
	
    I_-,osc[φ,ψ](τ,s,θ)    =2τ^2/s^2∫_Λ^∞ e^iλ^2 τ - i λψ(2λτ - 1) φ( λτ/s , λ,θ)  λλ,  
    
    		I_-,phg[φ,ψ](τ,r,θ)    =2/r^2∫_Λ^∞ e^iλ^2 τ - i λ(1- ψ(2λτ -1))φ( λ/r , λ,θ) λλ   .

	Then <ref> holds. We just need to check that each of these integrals lies in the expected function spaces.
	We begin with I_-,phg. Let
	
    φ(σ,λ,θ) ∼∑_(j,k)∈ ,  j ≤αφ_j,k(λ,θ) σ^j log^k σ

	denote the σ→ 0^+ expansion of φ, so φ_j,k(λ,θ) ∈_c^(,β)((Λ,∞]_λ×∂ X_θ). Consider the function
	
    φ_γ(σ,λ,θ)  =
    	σ^-γ[ φ(σ,λ,θ) - ∑_(j,k)∈ ,  j ≤γφ_j,k(λ,θ) σ^j log^k σ] ∈_c^0,(,β)([0,Σ)_σ× (Λ,∞]_λ×∂ X_θ).

	Then, we have
	
    r^2/2 I_-,phg[φ,ψ]= ∑_(j,k) ∈,  j ≤γ∑_κ=0^k kκ( 1/r)^j  log^k-κ( 1/r)I_-,phg,1+j,κ[φ_j,k,ψ] + ( 1/r)^γ I_-,phg,1+γ,0[φ_γ,ψ],

	where the quantity I_-,phg,γ,κ is defined by <Ref>. That lemma then gives that each term on the right-hand side of <ref>, and therefore I_-,phg[φ,ψ] itself, has the required form, except with a conormal error of order γ. But since γ≤α was arbitrary, <ref> follows.

	Moving on to the other oscillatory integral, we introduce the coordinate δ = λτ-1/2. In terms of this coordinate,
	
    s^2 I_-,osc[φ,ψ](τ,s,θ) = e^-i/4τĨ_-,osc[φ,(δ+1)ψ(δ)](τ,s,θ)

	and
	
    Ĩ_-,osc[φ,ψ](τ,s,θ) =  ∫_-∞^∞ e^iδ^2/τψ(2δ) φ( 1/s(δ+1/2) , 1/τ(δ+1/2) ,θ)  δ.

	We can split this, for each γ∈,  as
	
    Ĩ_-,osc[φ,ψ]= ∑_(j,k) ∈,  j ≤γ∑_κ=0^k kκ (-1)^k-κ s^-jlog^k-κ (s)Ĩ_-,osc[φ_j,k,ψ_j,κ]  + s^-γĨ_-,osc[φ_γ,ψ_γ],

	where ψ_j,κ(2δ) = (δ+1/2)^j log^κ(δ+1/2)ψ(2δ) and ψ_γ(2δ) = (δ+1/2)^γψ(2δ). Now let
	
    φ_j,k(λ,θ) ∼∑_(j',k') ∈,  j' ≤βλ^-j'log^k'(λ) φ_j,k^j',k'(θ)

	denote the expansion of φ_j,k(λ,θ) as λ→∞, and let λ^-γφ_j,k^γ(λ,θ) denote the error from truncating the expansion to j' ≤γ. We use similar notation for φ_γ. Then, for each γ,γ'∈,
	
    Ĩ_-,osc[φ_j,k,ψ_j,κ] =  ∑_(j',k') ∈,  j' ≤γ∑_ϰ=0^k'k'ϰ (-1)^k'-ϰτ^j'log^k'-ϰ(τ) Ĩ_-,osc[ φ_j,k^j',k', ψ_j,κ^j',ϰ ] 
     + τ^γĨ_-,osc[φ_j,κ^γ, ψ_j,k^γ ],

	
    Ĩ_-,osc[φ_γ,ψ_γ] =  ∑_(j',k') ∈,  j' ≤γ'∑_κ=0^k'k'κ (-1)^k'-κτ^j'log^k'-κ(τ) Ĩ_-,osc[ φ_γ^j',k', ψ_γ^j',κ ] 
     + τ^γ'Ĩ_-,osc[φ_γ^γ', ψ_γ^γ' ],

	where ψ_j,κ^j',ϰ(2δ) = (δ+1/2)^j log^ϰ(δ+1/2) ψ_j,κ(2δ), and similarly for the other undefined terms.


	The result then follows from <Ref>. Indeed, we have that Ĩ_-,osc is a sum of four types of terms:
	
		
  * First, consider the “main” terms proportional to s^-jlog^k(s) τ^j'log^k'(τ) Ĩ_-,osc[φ_j,k^j',k', ψ_j,κ^j',ϰ ].

		Noting that Ĩ_-,osc[φ_j,k^j',k', ψ_j,κ^j',ϰ ] does not depend on s, the last clause of <Ref>
		yields
		
    Ĩ_-,osc[φ_j,k^j',k', ψ_j,κ^j',ϰ ] ∈τ^1/2 C^∞([0,∞)_τ; C^∞(∂ X_θ) ).

		
  * Now consider the terms s^-jlog^k(s) τ^γĨ_-,osc[φ_j,k^γ, ψ^γ_j,κ ]. Again noting that Ĩ_-,osc[φ_j,k^γ, ψ^γ_j,k ] does not depend on s, we have
		
    Ĩ_-,osc[φ_j,k^γ, ψ^γ_j,κ ] ∈τ^1/2^0_loc([0,∞)_τ×∂ X_θ).

		
  * Now consider the terms s^-γτ^j log^k(τ) Ĩ_-,osc[φ_γ^j,k, ψ_γ^j,k].

		By the last clause of <Ref>, Ĩ_-,osc[φ_γ^j,k, ψ_γ^j,k] ∈τ^1/2_loc^0,(0,0)([0,∞)_τ× (0,∞]_s ×∂ X_θ).
		
  * Finally, in τ^γ s^-γ'Ĩ_-,osc[φ_γ^γ',ψ_γ^γ'], we have Ĩ_-,osc[φ_γ^γ',ψ_γ^γ'] ∈τ^1/2_loc^0,0([0,∞)_τ× (0,∞]_s ×∂ X_θ)
		by <Ref>.
	

Putting this all together, Ĩ_-,osc[φ](τ, s,θ  ) ∈τ^1/2_loc^(,γ), (,γ') ([0,∞)_τ× (0,∞]_s×∂ X_θ ). Taking γ→α and γ'→β completes the proof that Ĩ_-,osc lies in the desired function spaces.

If χ∈ C_c^∞() satisfies suppχ⋐ (-1,1), then
χ(2sΣ) I_-,osc[φ](τ,s,θ) ∈_loc^∞, ∞([0,∞)_τ× (0,∞]_s×∂ X_θ ), as the corresponding clause of <Ref> shows that each of the terms Ĩ_-,osc appearing above is Schwartz when multiplied by χ(2sΣ).



	The proof shows that the r→∞ expansion of I_-,phg[φ,ψ] is given by
	
    I_-,phg[φ,ψ]∼2/r^2∑_(j,k)∈,  j ≤α(1/r)^jlog^k(1/r) ∑_K≥ k, (j,K) ∈Kk∫_Λ^∞
    	e^i λ^2 τ - iλ
    φ_j,K(λ,θ) λ^1+jlog^K-k (λ) λ,

	analogously to <ref>.
	The s→∞ expansion of I_-,osc[φ,ψ] is given by
	
    I_-,osc[φ,ψ] ∼e^-i/4τ/r^2∑_(j,k)∈,  j≤α s^-jlog^k(s) (-1)^k ∑_K≥ k, (j,K)∈KkĨ_-,osc[φ_j,K,(δ+1) ψ_j,K-k(δ) ]  ,

	where
	
    Ĩ_-,osc[φ_j,K, ψ_j,K-k ]    = ∫_-∞^∞ e^i δ^2/τ (δ+1/2)^1+jlog^K-k(δ+1/2) ψ(2δ) φ_j,K(1/τ(δ+1/2),θ) δ .

	We do not write the τ→ 0 expansion.



	Let ψ∈ C_c^∞() satisfy suppψ⋐ (-1/2,1/2) and 0∉supp(1-ψ). Fix Σ,Λ>0.
	Let ϕ∈^0,0,0,0_c([0,Σ)_σ× (Λ,∞]_λ× (0,∞]_r × [0,∞)_τ×∂ X_θ ), and consider, for γ∈ and κ∈,
	
    I_-,phg,γ,κ[ϕ,ψ](τ,r,θ) = ∫_Λ^∞ e^iλ^2 τ - i λ(1- ψ(2λτ -1))ϕ( λ/r , λ,r, τ,θ)  λ^γlog^κ(λ)λ.

	Then, we have I_-,phg,γ,κ[ϕ,ψ](τ, r,θ  ) ∈_loc^0,0([0,∞)_τ× (0,∞]_r×∂ X_θ ).
	If ϕ(σ,λ,r,τ,θ) = ϕ(λ,θ) depends only on λ,θ, then this can be improved to
	
    I_-,phg,γ,κ[ϕ,ψ](τ, r,θ  ) ∈_loc^(0,0),(0,0)([0,∞)_τ× (0,∞]_r×∂ X_θ ) =C^∞(Ċ_1\kf),

	and I_-,phg,γ,κ[ϕ,ψ](τ, s,θ  ) does not depend on r in this case.
	


	We first prove that I_-,phg,γ,κ[ϕ,ψ](τ, r,θ  ) ∈ L_loc^∞([0,∞)_τ× (0,∞]_r×∂ X_θ ). If γ< -1, this follows immediately from an L^∞-bound. Otherwise, we use an integration-by-parts argument as usual: exp(i λ^2 τ - i λ ) = -i (2λτ -1)^-1∂_λexp(i λ^2 τ - i λ ),
	so
	
    I_-,phg,γ,κ[ϕ,ψ] = ∫_Λ^∞[-i/2λτ-1∂/∂λ e^i λ^2 τ - i λ] (1- ψ(2λτ -1))ϕ( λ/r , λ,r, τ,θ)  λ^γlog(λ)^κλ.

	Note that the integrand is well-defined, since the factor 1-ψ(2λτ-1) vanishes when 2λτ-1 is sufficiently small. Integrating-by-parts, noting that no boundary terms arise,
	
    I_-,phg,γ,κ[ϕ,ψ] = I_-,phg,γ-1,κ[ϕ_0,ψ_0] - γ I_-,phg,γ-1,κ[ϕ_1,ψ_0] - κ I_-,phg,γ-1,κ-1[ϕ_1,ψ_0],

	where ψ_0 ∈ C_c^∞((-1/2,+1/2)) is identically 1 near the origin and satisfies  suppψ_0 ⋐ψ^-1({1}), and where
	
    ϕ_0(σ,λ,r,τ,θ) =  (1- ψ(2λτ -1)) [ -2i  λτ/(2λτ -1)^2   + i/2λτ-1 ( σ∂_σ   + λ∂_λ )] ϕ(σ,λ,r,τ,θ) 
    
    	- 2 i  λτ/2λτ - 1ψ'(2λτ - 1) ϕ(σ,λ,r,τ,θ),

	
    ϕ_1(σ,λ,r,τ,θ) =  (1- ψ(2λτ -1)) -iϕ(σ,λ,r,τ,θ)/2λτ-1.

	Since the three functions (1- ψ(2λτ -1)) / (2λτ - 1) , (1- ψ(2λτ -1))  λτ / (2λτ - 1)^2, ψ'(2λτ-1)λτ / (2λτ - 1) all lie in ^0,0,0,0_loc([0,Σ)_σ× (Λ,∞]_λ× (0,∞]_r × [0,∞)_τ×∂ X_θ ), we have
	
    ϕ_0(σ,λ,r,τ,θ),
    	ϕ_1(σ,λ,r,τ,θ) ∈^0,0,0,0_c([0,Σ)_σ× (Λ,∞]_λ× (0,∞]_r × [0,∞)_τ×∂ X_θ ).

	Thus, each term on the right-hand side of <ref> has the same form as the original oscillatory integral, except with γ-1 in place of γ and possibly κ-1 in place of κ, if κ>0.
	So, <ref> can be used inductively to conclude the desired L^∞-bound from the γ<-1 case.

	In order to prove conormality, we want to prove that
	
    (τ∂_τ)^j(r∂_r)^k L I_-,phg,γ,κ[φ](τ, r,θ  )  ∈ L_loc^∞([0,∞)_τ× (0,∞]_r×∂ X_θ )

	for all j,k∈ and L∈Diff(∂ X). The angular derivatives L are handled via differentiation under the integral sign as elsewhere, and for the other directions we use the following identities:
		
    τ∂_τ I_-,phg,γ,κ[ϕ,ψ] =  i τ I_-,phg,γ+2,κ[ϕ,ψ]  + I_-,phg,γ,κ[φ,ψ]
    		- I_-,phg,γ,κ[ϖ ,ψ_0]

		for φ(σ,λ,r,τ,θ) = τ∂_τϕ(σ,λ,r,τ,θ) and ϖ(σ,λ,r,τ,θ)=2λτψ'(λτ-1)ϕ(σ,λ,r,τ,θ), and
		
    r∂_r I_-,phg,γ,κ[ϕ,ψ](τ,r,θ) = I_-,phg,γ,κ[ς,ψ]

		for ς(σ,λ,r,τ,θ)=(- σ∂_σ + r∂_r) ϕ(σ,λ,r,τ,θ).
	Each of these has the same form as the original integral, so, a similar inductive argument to the above shows that the bounds <ref> follow from those already proven.

	Finally, suppose that  ϕ(σ,λ,r,τ,θ) = ϕ(λ,θ) depends only on λ,θ. Then, <ref> can be improved to
	
    ∂_τ I_-,phg,γ,κ[ϕ,ψ](τ,r,θ) =  i I_-,phg,γ+2,κ[ϕ,ψ](τ,r,θ) -  I_-,phg,γ+1,κ[ ,ψ_0](τ,r,θ)

	for =  (λτ)^-1ϖ, and now we simply have ∂_s I_-,phg,γ,κ[ϕ,ψ] = 0. Noting that each term on the right-hand side of <ref> has the same form as the original integral, the usual inductive argument yields polyhomogeneity.



	Let ψ∈ C_c^∞() satisfy suppψ⋐ (-1,1). Fix Σ,Λ>0, k∈.
	Let ϕ∈^0,0,0,0_c([0,Σ)_σ× (Λ,∞]_λ× (0,∞]_s × [0,∞)_τ×∂ X_θ ), and consider, for k∈,
	
    _k[ϕ,ψ](s,τ,θ) = ∫_-∞^+∞ e^i δ^2/τδ^kψ(2δ) ϕ( 1/s( δ+1/2)  , 1/τ( δ+1/2),s, τ,θ)   δ.

	Then,
	_k[ϕ,ψ](τ,s,θ) ∈τ^1/2+k_loc^0,0([0,∞)_τ× (0,∞]_s ×∂ X_θ). If χ∈ C_c^∞() satisfies suppχ⋐ (-1,1), then χ(2s Σ) _k[ϕ,ψ] is Schwartz.

	If ϕ(σ,λ,s,τ,θ) = ϕ(σ,s,θ) does not depend on λ,τ, then _k[ϕ,ψ](τ,θ) ∈τ^1/2+k_loc^0,(0,0)([0,∞)_τ×(0,∞]_s ×∂ X_θ), where the (0,0) is the index set as τ→ 0.
	


	
		
  * We begin by proving the weaker claim that
		
    _2k[ϕ,ψ] ∈τ^k L_loc^∞([0,∞)_τ× (0,∞]_s ×∂ X_θ).

		As with the other integrals analyzed elsewhere in this paper, this is proven using integration-by-parts: if k=0, then this bound is immediate, and otherwise, if k≥ 1, use
		
    _2k[ϕ,ψ] = -iτ/2∫_-∞^+∞[ ∂/∂δ  e^i δ^2/τ] δ^2k-1ψ(2δ) ϕ( 1/s( δ+1/2)  , 1/τ( δ+1/2),s, τ,θ)  δ.

		So,
		
    -2i/τ_2k[ϕ,ψ] = (2k-1)_2k-2[ϕ,ψ] +  _2k-2[ϕ,Δψ'(Δ)]  
     +
    		_2k-2[ σ∂_σϕ, (Δ+1)^-1Δψ(Δ)]  +
    		_2k-2[ λ∂_λϕ, (Δ+1)^-1Δψ(Δ)].

		Using this identity inductively, _2k[ϕ,ψ] ∈τ^k L_loc^∞([0,∞)_τ× (0,∞]_s ×∂ X_θ) follows from the k=0 case.
		
  * 

		The next goal is to improve this to the optimal
		
    _k[ϕ,ψ] ∈τ^1/2+k L_loc^∞([0,∞)_τ× (0,∞]_s ×∂ X_θ).

		In order to improve upon the previous bounds, we expand ϕ in Taylor series around δ=0:
		
    ϕ( 1/s( δ+1/2)  , 1/τ( δ+1/2),s, τ,θ) = ∑_j_1+j_2 ≤ Jδ^j_1+j_2/j_1! j_2! s^j_1τ^j_2ϕ^(j_1,j_2)( 1/2s  , 1/2τ,s, τ,θ) 
    
    		+  ∫_0^δ (δ-Δ)^J ∑_j_1+j_2=J+11/j_1! j_2! s^j_1τ^j_2ϕ^(j_1,j_2)( 1/s( Δ+1/2)  , 1/τ( Δ+1/2),s, τ,θ)Δ,

		where ϕ^(j_1,j_2)(σ,λ,s,τ,θ) = ∂_σ^j_1∂_λ^j_2ϕ (σ,λ,s,τ,θ).
		So, for any J∈,
		
    _k[ϕ,ψ] =_k,J+1[ϕ,ψ]+ ∑_j_1+j_2 ≤ J_k,j_1,j_2[ϕ,ψ],

		where
		
    _k,j_1,j_2[ϕ,ψ] = 1/j_1!j_2! s^j_1τ^j_2ϕ^(j_1,j_2)( 1/2s  , 1/2τ,s, τ,θ) ∫_-∞^+∞ e^i δ^2/τδ^k+j_1+j_2ψ(2δ)   δ

		and
		
    _k,J+1[ϕ,ψ] =  ∫_-∞^+∞ e^i δ^2/τδ^kψ(2δ) [∫_0^δ (δ-Δ)^J 
    ×∑_j_1+j_2=J+11/j_1! j_2! s^j_1τ^j_2ϕ^(j_1,j_2)( 1/s( Δ+1/2)  , 1/τ( Δ+1/2),s, τ,θ)Δ] δ.


		The method of stationary phase (which in this case amounts to Parseval–Plancherel), applied to the integral
		
    _j[ψ] = ∫_-∞^+∞ e^i δ^2/τδ^jψ(2δ)   δ,

		yields that _j[ψ] ∈τ^1/2+j C^∞[0,∞)_τ. So,
		
    _k,j_1,j_2[ϕ,ψ]  ∈τ^1/2+k+j_1+j_2^0,0_loc([0,∞)_τ× (0,∞]_s ×∂ X_θ).



		In order to estimate _k,J+1[ϕ,ψ], we use a similar integration-by-parts argument as before. We prove, via induction on k, that
		
    _k,J+1[ϕ,ψ] ∈τ^⌊ (k+J+1)/2 ⌋  L_loc^∞([0,∞)_τ× (0,∞]_s ×∂ X_θ),

		which is trivial in the k+J=0 case.
		Integrating-by-parts, we can write
		
    - 2i τ^-1_k,J+1[ϕ,ψ]    = (k-1)_k-2,J+1[ϕ,ψ] + 2_k-1,J+1[ϕ,ψ'] +J_k-1,J[ϕ,ψ]
    			if J≥ 1 and, otherwise, 
    			- 2i τ^-1_k,1[ϕ,ψ]    = (k-1)_k-2,1[ϕ,ψ] + 2_k-1,1[ϕ,ψ'] + _k-1 [φ,ψ̃]

		for ψ̃(2δ) = (δ+1/2)-J-1ψ(2δ) and
		
    φ(σ, λ,s,τ,θ) = ∑_j_1+j_2 ≤  1σ^j_1λ^j_2/j_1!j_2!∂^j_1/∂σ^j_1∂^j_2/∂λ^j_2ϕ (σ,λ,s,τ,θ),

		where _k-1 [φ,ψ] is defined by <ref>.

		Consider each of the terms on the right-hand sides of <ref>, <ref>.
		The method of nonstationary phase shows that
		
    _k-1,J+1[ϕ,ψ'] ∈τ^∞ L_loc^∞([0,∞)_τ× (0,∞]_s ×∂ X_θ).

		Moreover, by <ref>, we have _k-1 [φ,ψ̃] ∈τ^⌊ (k-1)/2 ⌋L_loc^∞([0,∞)_τ× (0,∞]_s ×∂ X_θ).
		So, if we know that
		
    _k-2,J+1[ϕ,ψ] , _k-1,J[ϕ,ψ] ∈τ^⌊ (k+J-1)/2 ⌋L_loc^∞([0,∞)_τ× (0,∞]_s ×∂ X_θ),

		then we can conclude that <ref> holds.

		As for <ref>, we can write _k-2,J+1[ϕ,ψ], _k-1,J[ϕ,ψ] as _k',J'+1 for k',J' satisfying k'+J'<J+K. This sets up an inductive algorithm to deduce the claim from the k+J=0 base case which is already known.

		Returning now to <ref>, this follows from <ref> combined with <ref> and <ref>, as long as J is sufficiently large.
		
  * 

		Having now proven the optimal L^∞-bounds <ref>, we upgrade this to conormality by estimating derivatives. We can do this using the usual argument: L_k,J+1[ϕ,ψ] = _k,J+1[Lϕ,ψ] for L∈Diff(∂ X_θ), s ∂_s _k,J+1[ϕ,ψ](τ,s,θ) =  _k,J+1[ (σ∂_σ + s ∂_s) φ (σ,λ,s,τ,θ),ψ̃],
		where φ,ψ̃ are as above, and
		
    τ∂_τ_k,J+1[ϕ,ψ](τ,s,θ) = - τ^-1i _k+2,J+1[ϕ,ψ](τ,s,θ) 
     +   _k,J+1[ (λ∂_λ + τ∂_τ) φ (σ,λ,s,τ,θ),ψ̃].

		So, via the usual inductive argument,  (s∂_s)^m (τ∂_τ)^n L_k,J+1[ϕ,ψ] ∈τ^⌊ (k+J+1)/2 ⌋  L_loc^∞([0,∞)_τ× (0,∞]_s ×∂ X_θ) for all m,n∈, which means that
		
    _k,J+1[ϕ,ψ] ∈τ^⌊ (k+J+1)/2 ⌋_loc^0,0([0,∞)_τ× (0,∞]_s ×∂ X_θ).

		
  * 

		Consider now χ(2sΣ) _k[ϕ,ψ], where χ is as in the statement of the proposition. Then, χ(2sΣ) _k,j_1,j_2[ϕ,ψ]=0
		identically, for all j_1,j_2∈ (due to the assumptions on the support of ϕ). So, the estimates above on _k,J+1, since J can be made arbitrarily large, give Schwartz behavior for χ(2sΣ) _k[ϕ,ψ] as τ→∞.
		
  * 

		Suppose now that  ϕ(σ,λ,s,τ,θ) = ϕ(σ,s,θ) does not depend on λ,τ. Then, _k[ϕ,ψ] = ∑_j=0^J _k,j,0[ϕ,ψ] + _k,J+1[ϕ,ψ], where
		
    _k,j,0[ϕ,ψ]    = 1/j! s^jϕ^(j)( 1/2s ,s,θ) ∫_-∞^+∞ e^i δ^2/τδ^k+jψ(2δ)   δ
    		= 1/j! s^jϕ^(j)( 1/2s ,s,θ) _k+j[ψ],
    		
    _k,J+1[ϕ,ψ]    =  ∫_-∞^+∞ e^i δ^2/τδ^kψ(2δ) [∫_0^δ(δ-Δ)^J /(J+1)!s^J+1ϕ^(J+1)( 1/s( Δ+1/2)  , s,θ)Δ] δ,

		where ϕ^(j) (σ,s,θ) = ∂_σ^j ϕ(σ,s,θ). Instead of <ref>, we now have
		
    _k,j,0[ϕ,ψ] ∈τ^1/2+k+j_loc^0,(0,0)([0,∞)_τ×(0,∞]_s ×∂ X_θ),

		since _j[ψ](τ) ∈τ^1/2+jC^∞([0,∞)_τ). On the other hand, we can improve <ref> to
		
    ∂_τ_k,J+1[ϕ,ψ] = τ^-2_k+2,J+1[ϕ,ψ](τ,s,θ).

		Note that the the growth of the τ^-2 factor is canceled out by the extra τ^2 decay of _k+2,J+1 versus _k,J+1. So, the usual inductive argument yields
		
    _k,J+1[ϕ,ψ] ∈τ^⌊ (k+J+1)/2 ⌋_loc^0,(0,0)([0,∞)_τ× (0,∞]_s ×∂ X_θ).

		Combining the estimates above and taking J to be sufficiently large, we conclude the final clause of the proposition.
	





§ PROOF OF MAIN LEMMA



We now turn to the proof of <Ref>, our “main lemma.”
Let ,,,α,β,γ,ϕ be as in the statement of that theorem.
Let χ_low, χ_tf∩bf, χ_high∈ C^∞(X_res^sp) be a partition of unity as in the introduction, so χ_low+ χ_tf∩bf+χ_high = 1 and

    suppχ_low∩ (bf∪∞f) = ∅, suppχ_tf∩bf∩ (zf∪∞f) = ∅,  suppχ_high∩ (zf∪tf) = ∅,

and we can choose that  suppϕ_low∩ ( suppχ_tf∩bf∪suppχ_high) = ∅. Moreover, χ_tf∩bf can be chosen to be supported over Ẋ[R] for some R, that is over a collar neighborhood of the boundary of X.



We split I_±[ϕ] as in <ref>.
We analyze each piece separately. First of all, according to <Ref>,

    I_±[χ_lowϕ ] ∈^(/2+1,α/2+1), (+2,β+2),(0,0)(C_1).

On the other hand, <Ref> says that I_+[χ_highϕ ] ∈^∞,∞,(0,0), and, together, <Ref> and  <Ref> say that

    I_-[χ_highϕ] ∈^∞,∞,(0,0)(C) + e^-i (1-χ(t))r^2/4t ^∞,∞,(+1/2,γ+1/2),∞,∞(M) 
     =  e^-i (1-χ(t))r^2/4t ^∞,∞,(+1/2,γ+1/2),∞,(0,0) (M).

Regarding I_+[χ_tf∩bfϕ], says <Ref> that I_+[χ_tf∩bfϕ] ∈^∞,(+2,β+2),(0,0)(C_1). Regarding I_-[χ_tf∩bfϕ], the same proposition says that I_-[χ_tf∩bfϕ] = exp(-i r^2/4t )   Ĩ_-[χ_tf∩bfϕ] + I_-,phg[χ_tf∩bfϕ]
for some

    Ĩ_-[χ_tf∩bfϕ] ∈^∞,(+2,β+2),(+1/2,γ+1/2),∞,∞(M)

and I_-,phg[χ_tf∩bfϕ]∈^∞,(+2,β+2),(0,0)(C_1).
So, summing up I_±[ϕ] = I_±[χ_lowϕ] + I_±[χ_tf∩bfϕ] + I_±[χ_highϕ], we conclude <Ref>.








§ MICROLOCAL SUPPLEMENT



The goal of this appendix is to sketch a microlocal proof of the following proposition:

	Suppose that f∈(X), and let φ_±(σ,x) = e^∓ iσ r R(σ^2 ± i0) f(x). Then, letting I_-[φ_-] = exp(-i(1-χ(t)) r^2/4t) I_osc[φ]+ I_-,phg[φ] denote the decomposition of I_-[φ_-] provided in <Ref>, it must be the case that the linear combination
	
    I_phg = I_+[φ_+] - I_-,phg [φ_-]
 is Schwartz at dilF.
	


First of all, note that because I_phg is already polyhomogeneous on C_1, it suffices to prove that I_phg is Schwartz in dilF^∘.
Let M_0 denote the manifold-with-boundary resulting from taking [_t× X ; {±∞}×∂ X ] subtracting the boundary hypersurfaces corresponding to {±∞}× X and blowing down the boundary hypersurfaces corresponding to _t×∂ X. Concretely, this mwc is identifiable with _t/r× [0,∞)_ρ×∂ X_θ near the boundary.
We utilize Melrose's sc-calculus on M_0. See <cit.> for an introduction.

To show that I_phg is Schwartz at dilF^∘ means to show that WF_sc(I_phg) = ∅, where WF_sc is Melrose's notion of sc-wavefront set.
Let ^sco^* M_0 denote the zero section of the sc-cotangent bundle over dilF^∘.
Because I_phg is conormal,

    WF_sc(I_phg) ⊆^sc o^*  M_0,

as follows e.g. via repeated applications of elliptic estimates in the sc-calculus. (This follows from b-vector fields being growing sc-vector fields.)


In order to study WF_sc(I_phg) further, we use the relation of I[φ] = I_+[φ_+]- I_-[φ_-] to u(t,x) = (U(t) f)(x) given by <ref>. First note that

    WF_sc(e^i E tφ(x)) = ∅

for any E∈ and φ∈(X). (If this looks strange, recall that ∂ M_0 does not contain any points where r≠∞.) So, WF_sc(u) = WF_sc(I[φ]).
Because I[φ] = exp(-i(1-χ(t)) r^2/4t) I_osc[φ_-]+ I_phg, we have

    WF_sc(I_phg) ⊆WF_sc(exp( -i(1 -χ(t))/4 t ρ^2)I_osc[φ_-] ) ∪WF_sc(I[φ]) 
     = WF_sc(exp( -i(1 -χ(t))/4 t ρ^2)I_osc[φ_-]) ∪WF_sc(u).

By the conormality of I_osc, we have WF_sc(exp( -i(1 -χ(t))/4 t ρ^2)I_osc) ⊆graph_∂ M_0(-2(r/t)  r + (r/t)^2  t)
the right-hand side being the graph over ∂ M_0 of the 1-form - (r^2/t) = - 2(r/t)  r + (r/t)^2  t, which is a smooth, nonvanishing section of ^sc T^* M_0. Because it is nonvanishing,

    WF_sc(exp( -i(1 -χ(t))/4 t ρ^2)I_osc) ∩^sc o^*  M_0 = ∅.

Combining this with <ref> and <ref>, we have WF_sc(I_phg) ⊆WF_sc(u) ∩^sc o^* M_0.
The upshot is that, in order to prove that WF_sc(I_phg)  = ∅, it suffices to prove that WF_sc(u) ∩^sc o^* M_0 = ∅.


In order to accomplish this, one can use a standard argument based on the splitting u = u_++u_-,  where u_±(t,x) = 1_± t>0 u(t,x).
As WF_sc(u) ⊆ ( WF_sc(u_-) ∪WF_sc(u_+)), it suffices to prove WF_sc(u_±) ∩^sc o^* M_0 = ∅ for each choice of sign.
To this end, note that u_± satisfy the PDE

    - i ∂_t u_± = P u_±∓ i δ(t) f(x)

in the sense of distributions.
As WF_sc(δ(t)f(x)) is contained at fiber infinity (as can be seen using the Fourier transform in a local coordinate patch), it is irrelevant as far as sc-wavefront set in the interior of the fibers is concerned.


We have P = Δ_g Diff_sc^1,-2(M_0), where the `-2' indicates two orders of decay. So, the principal symbol of P is

    p(τ,ξ) = τ + g^-1(ξ,ξ) ∈ C^∞(^scT^* M_0 ).

Associated to this function is the Hamiltonian vector field H_p = (∂_τ p)∂_t + (∂_ξ p)·∂_x = ∂_t + 2g^-1(ξ,-). Let 𝖧_p = ρ H_p, which restricts to a vector field on ^scT^*_∂ M_0 M_0. The Duistermaat-Hörmander theorem — see <cit.> for a precise statement in the context of the sc-calculus — then says that the portion of WF_sc(u_±) in ^scT^*_∂ M_0 M_0 consists of maximally extended integral curves of 𝖧_p. Those in the zero section ^sc o^* M_0 are of the form

    γ_θ_0 =( ^sc o^* M_0)∩{θ=θ_0},

as follows from the explicit formula for 𝖧_p.
Because u_± vanishes when ± t <0, it has no sc-wavefront set over the corresponding half of dilF^∘. So, γ_θ_0⊈WF_sc(u_±), which, by Duistermaat–Hörmander, implies WF_sc(u_±) ∩^sc o^* M_0 = ∅.



§ NECESSITY OF NF AND DILF



In this appendix, we summarize the role of nf,dilF and why it is not possible to blow down either while maintaining the exponential-polyhomogeneous form of solutions of the Schrödinger equation. We only sketch the argument, and,  for simplicity, we work in the X = 1 case.



	If θ_1,θ_2,a_0,a_1,a_2 are polyhomogeneous functions on a mwc M and p∈∂ M are such that θ_1,θ_2 are real-valued, a_0∈  C^∞(M;^×), a_1 extends continuously to a neighborhood of p, and this extension vanishes at p, and e^i θ_1 (a_0+a_1) = e^i θ_2 a_2,
 	then, near p, the difference θ_1 - θ_2 is, in a neighborhood of p, has no (j,k) for j<0 in its index set.
 	


	The function a_0 + a_1 is nonvanishing near p, so b=(a_0+a_1)^-1 is well-defined there, and b is polyhomogeneous, with a continuous extension to the boundary of M near p.

	Let b̃ be a globally polyhomogeneous function extending continuously to all of ∂ M and satisfying b̃= b near p. Then,

    e^iθ_1-i θ_2 = b̃ a_2,

	near p. Since a_2 is polyhomogeneous, and since a_0+a_1 is uniformly bounded near p (which implies that the same is true for a_2), it must be that a_2 extends continuously to a neighborhood of p. So, the right-hand side of <ref> extends continuously to a neighborhood of p.

	Moreover, the extension of a_2 to the boundary must be nonvanishing near p, and likewise for b̃, so b̃ a_2 is nonvanishing near p. This implies that the difference
	
    θ_1-θ_2 = - i log(b̃ a_2)

	is, near p, polyhomogeneous with all index sets in {z∈: z ≥ 0}×.




Consider now the Gaussian wavepacket

    G(t,x) = 1/√(1+2it)exp( -x^2/1+2it).

This solves the free Schrödinger equation in 1D with Schwartz initial data.


	The Gaussian wavepacket above is not of the form a e^i θ for θ real-valued and a,θ polyhomogeneous on M/nf or M/dilF.

Below, we will use the same names to refer to faces of the M/f as the corresponding faces in M. For example, though Σ,dilF are disjoint boundary hypersurfaces of M, the identically named boundary hypersurfaces of M/nf meet at the blown down locus.

	First, suppose, to the contrary, that G =e^i φ G_0 for some φ,G_0 polyhomogeneous on M/nf. Looking at the t→ 0^+ behavior of G in compact subsets worth of r, it must be the case that
	
		
  ⋆ the index set _Σ of φ at Σ can be taken to contain no terms (j,k) ∈× with j<0.
	
	Near the corner Σ∩dilF⊂ M/nf, we can use ρ=1/r and s= t/r as a coordinate system, where both of these are local boundary-defining functions, ρ of dilF and s of Σ. In terms of these coordinates,
	
    G = ρ^1/2/√(ρ+2i s )exp( - 1/ρ^2+2i s ρ).

	For s>0, this has the form ρ^1/2exp(i/(2 s ρ)) C^∞(^+_s× [0,∞)_ρ;^×) near ρ=0. Consequently, applying <Ref>,
	
    φ =  1/(2 s ρ) ^0-(^+_s× [0,∞)_ρ).

	But, for no index set _nf can φ∈^_nf,_Σ(([0,∞)_s× [0,∞)_ρ ) be consistent with <ref>, since this forces (-1,0)∈_Σ, in conflict with our earlier observation (⋆). So, the supposition that G has the stated form on M/nf is not tenable.

	

	Consider now M/dilF.
	Since (1+2i t)^-1/2 is polyhomogeneous on C, and therefore on M/dilF, in order to prove the desired result it suffices to prove that
	we do not have (1+2i t)^1/2 G = e^i φ a
	for φ real valued and a,φ polyhomogeneous on M/dilF.
	Near the corner nf∩parF on the blown down manifold M/dilF, we can use coordinates ϱ = 1/t^1/2 and τ = t/r^2, with ϱ a boundary-defining-function of parF and τ a boundary-defining-function of nf. In terms of these coordinates,
	
    (1+2i t)^1/2G = exp(- 1/ τ 	(ϱ^2+2i)).

	So, for ϱ>0, (1+2i t)^1/2G is Schwartz as τ→ 0^+. It follows that a is Schwartz at nf, however, restricting to ϱ=0, (1+2i t)^1/2G has magnitude 1, and therefore so does a. This contradicts the joint expandability of a at the corner, and therefore polyhomogeneity.





§ A NUMERICAL EXAMPLE





We end by considering the example I_±[ϕ] for ϕ(σ,x) = 2^-1 e^- σ^2 (1+σ^2 r^2)^-1. This is given by

    I_±[ϕ](t,x) = ∫_0^∞ e^i σ^2 t ± i σ r -σ^2σσ/1+σ^2 r^2.

As ϕ∈^(0,0),(0,0),(2,0)(X_res^sp), we know that I_+[ϕ] ∈^(1,0)∪ (1/2,0), (2,0),(0,0)  (C_1) and I_-[ϕ] = exp(-i (1-χ(t) r^2/4t) ) I_osc[ϕ] + I_-phg, for some

    I_osc[ϕ] ∈^(1,0) ∪ (1/2,0), (2,0), (5/2,0), ∞, (0,0)(M)

and I_-,phg∈^(1,0)∪ (1/2,0), (2,0),(0,0)  (C_1).  One point worth noting is that, despite ϕ being even, the difference I_+[ϕ] - I_-,phg[ϕ] does not decay rapidly at dilF, in contrast to what we observed in <ref> when ϕ is the output of the resolvent of a Schrödinger operator.
In other words, the integral

    I_+[ϕ] - I_-[ϕ] = ∫_-∞^∞ e^i σ^2 t + i σ r - σ^2σσ/1+σ^2 r^2

is not purely oscillatory at dilF, which can be seen clearly in the numerical plot <Ref>.







