On BMO and Carleson measures on Riemannian manifolds
Denis Brazke Armin Schikorra Yannick Sire


Let ℳ be a Riemannian n-manifold with a metric such that the manifold is Ahlfors-regular. We also assume either non-negative Ricci curvature, or that the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. We characterize BMO-functions u: ℳ→ by a Carleson measure condition of their σ-harmonic extension U: ℳ× (0,∞) →. We make crucial use of a T(b) theorem proved by Hofmann, Mitrea, Mitrea, and Morris.

As an application we show that the famous theorem of Coifman–Lions–Meyer–Semmes holds in this class of manifolds: Jacobians of W^1,n-maps from ℳ to ^n can be estimated against BMO-functions, which now follows from the arguments for commutators  recently proposed by Lenzmann and the second-named author using only harmonic extensions, integration by parts, and trace space characterizations.





Open and closed magnetic configurations of twisted flux tubes
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    December 30, 2023
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§ INTRODUCTION

It is a classical result that in Euclidean space there is a relation between BMO-functions u: ^n→ and Carleson measures in ^n+1_+. Precisely, the following statement can be found, e.g., in <cit.>.

Let u ∈ C_c^∞(^n) and denote by U(x,t): ^n+1_+ → the harmonic extension, i.e. the unique solution to the following equation:

    _x,t U ≡ (∂_tt + _x) U = 0      
    
    U = u    
    lim_|(x,t)| →∞ U(x,t) = 0

Then the following two BMO-seminorms are equivalent: The integral one

    [u]_BMO = sup_B ⊂^n |B|^-1∫_B|u-(u)_B|,

and the Carleson-measure version

    [u]_B̃M̃Õ := sup_B ⊂^n |B|^-1∫_T(B) t|∇_(x,t) U(x,t)|^2 dx dt ^1/2.

Here, (u)_B = |B|^-1∫_B u and T(B) is the tent in ^n × (0,∞) over the ball B, namely if B = B(x_0,r) then T(B) = {(x,t): |x-x_0| < r-t}.


While relations between Carleson measures and certain extensions of functions have been extended to spaces of homogeneous type (see e.g. <cit.>),  these extensions are usually of a potential type (with conditions on kernel decay). The main drawback is that these extensions in general  do not satisfy an equation such as (<ref>). On the other hand, 
in applications such as proving sharp commutator estimates, see <cit.>, it is beneficial (and maybe even crucial) to have the extension satisfying certain PDEs such as (<ref>) (or more generally satisfying a Dirichlet-to-Neumann principle <cit.>). 

The aim of the present work is to prove an equivalence result like in the previous theorem involving a natural PDE-extension to the half-space in a rather general geometric framework.  Let (ℳ,g) be an n-dimensional smooth Riemannian manifold which is also Ahlfors regular, meaning that the measure of a ball of radius r is (uniformly) comparable to r^n. By _ℳ we denote the Laplace-Beltrami operator on ℳ. We equip  with the Carnot-Carathéodory metric d. Without further assumptions on the manifold it seems implausible that, e.g., harmonic extensions satisfy a statement such as <Ref>. In Theorem <ref> we introduce such assumptions on the manifold and its heat kernel.

Besides the classical harmonic extension we also take σ-harmonic extensions into account. To define them we follow the semigroup representation (cf. <cit.> for the Euclidean analogue but as stated in the latter it extends to much more general contexts).

Let us clarify the differential operators that we use. Denote by d  the exterior derivative and ⋆ the Hodge operator. We define the Laplace-Beltrami operator (or Laplace-de-Rham operator, which are the same for us since they act only on functions/0-forms) by Δ = Δ_ := ⋆ d⋆d, and the gradient of a smooth function f by ∇ f = ∇_ f= d f, (or depending on the context ∇ f = (d f)^♯). With this setup, we have (with standard abuse of notation) |∇ f|^2 = g(∇ f,∇ f) and

    ⟨∇ f,∇ h ⟩_L^2 = ∫_⋆d f ∧d h

		
    i.e. ∇ f_L^2^2 = ∫_ g(∇ f,∇ f)  dx = ∫_ |∇ f|^2  dx.


The σ-harmonic extension is defined as follows.

 Let ℳ be as above and 0 < σ < 1.
For u ∈ C_c^∞(ℳ) the σ-harmonic extension U: ℳ× [0,∞) → is the solution to                                                                                                                                                          

    _ℳ U  + 1-2σ/t∂_t U + ∂_tt U = 0      
    
     U(x,0) = u(x)    
    lim_|(x,t)| →∞ U(x,t) = 0.

This solution is formally given by

    U(x,t) = 1/4^σΓ(σ) t^2σ∫_0^∞ e^s_ℳ u(x)  e^-t^2/4s ds/s^1+σ,

and explicitly one has

    U(x,t) =   1/4^σΓ(σ)∫_0^∞∫_ℳ p(x,y,s) u(y) dy  t^2σ e^-t^2/4s ds/s^1+σ,	
    
     			=    1/Γ(σ)∫_0^∞∫_ p(x,y,t^24s)   u(y)  dy   e^-s  s^σ - 1 ds,

where p(x,y,s) is the heat kernel for ℳ, i.e. 

    (∂_t - _x)p(x,y,s) = 0      
    
                p(x,y,0) = δ_x,y                                                                                                                            .

   

The previous definition is not explicitly stated in <cit.> but it is easy to check that the semi-group approach automatically carries over to such a geometric setting under very weak assumptions on the manifold, see Section <ref>. For more information and properties about the heat kernel, see <cit.>.
 	
We define the following semi-norms: Let U(x,t) be the σ-harmonic extension of u to ℳ× (0,∞). Denote the usual BMO-norm as

    [u]_BMO(ℳ) := sup_B ⊂ℳ |B|^-1∫_B|u-(u)_B|,

where the supremum is taken over balls B. Furthermore, we define a notion of BMO in terms of the σ-harmonic extension and Carleson measures, namely

    [u]_B̃M̃Õ(ℳ)≡ [u]_B̃M̃Õ_σ(ℳ):= sup_B ⊂ℳ |B|^-1∫_T(B) t|∇_x,t U(x,t)|^2 dx dt ^1/2.

Again, T(B) is the tent in ℳ× (0,∞) over the ball B, namely if B = B(x_0,r) then T(B) = {(x,t): d(x,x_0) < r-t}.


Our main result is the following:

Let  be a complete path-connected and Ahlfors regular manifold without boundary, such that the Ricci curvature is bounded from below. 



If moreover the heatkernel of ℳ satisfies 

    sup_x,y ∈ℳ |∇ p(x,y,t)|  t^-n+1/2,

then for any 0 < σ < 1 the semi-norms of BMO defined above are equivalent, i.e  for any u ∈ C_c^∞(ℳ) we have 

    [u]_BMO(ℳ) [u]_B̃M̃Õ(ℳ).



In case of non-negative Ricci curvature of the manifold, we can drop the assumption of the gradient bound and have the following:


Let  be a complete path-connected and Ahlfors regular manifold without boundary, such that the Ricci curvature is non-negative. Then for any 0 < σ < 1 the semi-norms of BMO defined above are equivalent, i.e  for any u ∈ C_c^∞() we have 

    [u]_BMO()≈ [u]_B̃M̃Õ().



Theorem <ref> and Theorem <ref> can be very useful for example when estimating commutators via harmonic extensions, as recently proposed in <cit.>, who gave new proofs for a large class of commutator estimates (in Euclidean space). Their argument is based on integration by parts and trace space characterizations for σ-harmonic extensions. Since in this paper we obtained the latter characterization for BMO, one can follow the ideas in <cit.> almost verbatim for manifolds. For example, the following estimate on Jacobians was obtained for ℳ=  ^n in the celebrated work <cit.>, which lead to several breakthroughs in regularity theory e.g. of harmonic maps. We can extend it to manifolds.

Let ℳ be an n-manifold as in Theorem <ref> or Theorem <ref>, n ≥ 2.
For f ∈ W^1,n(ℳ,^n) and any φ∈ C_c^∞(ℳ) we have

    ∫_ℳdf^1∧…∧df^n φ≤ C(ℳ)  ∇ f_L^n()^n [φ]_BMO.



The remainder of this paper will be as follows: In Section <ref> we introduce the notion of an admissible manifold, which is more general than the one in Theorem <ref> or Theorem <ref>, but more complicated to check. In Section <ref> we use the T(b)-theorem from <cit.> to obtain square function estimates. In Section <ref> we prove Theorem <ref> and Theorem <ref>. In Section <ref> we prove the Jacobian estimate, Theorem <ref>. Computations concerning the σ-harmonic extensions are moved to the appendix, Section <ref>.

Acknowledgments 
The authors would like to warmly thank Laurent Saloff-Coste for highly valuable discussions on the topic of this paper and Fabrice Baudoin and Ryan Alvarado for providing crucial references. Part of this work was carried out while D.B. was visiting the University of Pittsburgh, he likes to thank the Math Department for its hospitality.

Y.S. acknowledges funding from Simons Foundation. A.S. acknowledges funding from the Simons foundation and the Daimler and Benz foundation.





§ ADMISSIBLE MANIFOLDS AND HEAT KERNEL ESTIMATES

The proofs of <Ref> and <Ref> are based on heat kernel estimates which allow to deduce a T(b) theorem, see Section <ref>. We exhibit a large class of Riemannian manifolds for which our theorem applies.  We first define a general setting in which our theory actually works. 


A manifold  is said to be admissible, if it is complete, path-connected, Ahlfors regular, without boundary and its heat kernel p(x,y,t) satisfies the following conditions:
For every σ > 0 there exists ν > 0 such that for all t > 0, x, y ∈ℳ:

    ∫_0^∞ p(x,y,t^2 s) 1+1/s e^-1/4s ds/s^1+σt^ν/d(x,y)^2+t^2^n+ν/2,


    ∫_0^∞ |∇_xp(x,y,t^2 s)|   e^-1/4s ds/ s^1+σt^ν-1/d(x,y)^2+t^2^n+ν/2,


    ∫_0^∞ |∇_y p(x,y,t^2 s)| 1+1/s e^-1/4s ds/s^1+σt^ν-1/d(x,y)^2+t^2^n+ν/2,


    ∫_0^∞ |∇_x ∇_y p(x,y,t^2 s)|   e^-1/4s ds/ s^1+σt^ν-2/d(x,y)^2+t^2^n+ν/2.



The following lemma is providing a more treatable class of admissible manifolds. 

Let  be a complete path-connected and Ahlfors regular manifold without boundary. 
If the heat kernel satisfies
	
    p(x,y,t) ≲ t^-n/2  e^-c d(x,y)^2/t,

			
    |∇ p(x,y,t)| ≲ t^-n+1/2  e^-c d(x,y)^2/t,

			
    |∇_ -2pt x∇_ -2pt y p(x,y,t)| ≲1/t^n/2 + 1 e^-cd(x,y)^2/t,

for some constant c > 0, then  is admissible with ν = 2σ.

 The geometry of the manifold is the same as in Definition <ref>, so we only have to check the conditions (<ref>) to (<ref>).

As for (<ref>):
Using the change of variables, we see that
		
    ∫_0^∞ p(x,y,t^2 s) 1+1/s e^-1/4s ds/s^1+σ    = t^2 + 2σ - 2∫_0^∞ p(x,y,s) 1+t^2/s e^-t^2/4s ds/s^1+σ
       ≲ t^2σ∫_0^∞1+t^2/s e^-c d(x,y)^2/s e^-ct^2/sds/s^n/2 + 1 + σ.

	In the last step we employed (<ref>). Set Λ := d(x,y)^2 + t^2, and making the change of variables s ⟼Λ s, we see
		
    …    = t^2 σ∫_0^∞1+t^2/Λ s e^-c/sds/s^n/2 + 1 + σ Λ^n/2 + σ≲t^2σ/Λ^n + 2σ/2∫_0^∞1+1/s e^-c/sds/s^n/2 + 1 + σ.

	The integral can be estimated by a multiple of Γ(n/2 + σ) + Γ(n/2 + σ + 1), so it is finite. This shows (<ref>). 


As for (<ref>) and for (<ref>): It suffices to show (<ref>), since in our setting ∇_-2pt x p(x,y,t) = ∇_-2pt y p(x,y,t) by the symmetry of the heat kernel. Using the gradient estimate (<ref>) we deduce
		
    ∫_0^∞ |∇_-2pt y p(x,y,t^2 s)| 1+1/s e^-1/4s ds/s^1+σ    = t^2 + 2σ - 2∫_0^∞ |∇_-2pt y p(x,y,s)| 1+t^2/s e^-t^2/4s ds/s^1+σ
       ≲ t^2σ∫_0^∞1+t^2/s e^-c d(x,y)^2/s e^-ct^2/sds/s^n + 1/2 + 1 + σ.

	As in (<ref>), setting Λ := d(x,y)^2 + t^2, and making the change of variables s ⟼Λ s, we see
		
    …    = t^2 σ∫_0^∞1+t^2/Λ s e^-c/sds/s^n + 1/2 + 1 + σ Λ^n + 1/2 + σ≲t^2σ/Λ^n + 1 + 2σ/2∫_0^∞1+1/s e^-c/sds/s^n + 1/2 + 1 + σ.

	The integral can be estimated by a multiple of Γ(n + 1/2 + σ) + Γ(n + 1/2 + σ + 1), so it is finite. Moreover, since Λ≥ t^2, we get in the end
		
    ∫_0^∞ |∇_-2pt y p(x,y,t^2 s)| 1+1/s e^-1/4s ds/s^1+σ≲t^2σ - 1/Λ^n + 2σ/2.

	 This shows (<ref>), and hence also (<ref>). 

As for (<ref>): We proceed as in (<ref>) to (<ref>). Using the estimate (<ref>), we get
		
    ∫_0^∞ |∇_-2pt x∇_-2pt y p(x,y,t^2 s)|   e^-1/4s ds/s^1+σ    = t^2 + 2σ - 2∫_0^∞ |∇_-2pt x∇_-2pt y p(x,y,s)|   e^-t^2/4s ds/s^1+σ
       ≲ t^2σ∫_0^∞  e^-c d(x,y)^2/s e^-ct^2/sds/s^n + 2/2 + 1 + σ.

	As before, setting Λ := d(x,y)^2 + t^2, and making the change of variables s ⟼Λ s, we see
		
    …    = t^2 σ∫_0^∞ e^-c/sds/s^n + 2/2 + 1 + σ Λ^n + 2/2 + σ≲t^2σ/Λ^n + 2 + 2σ/2∫_0^∞  e^-c/sds/s^n + 2/2 + 1 + σ.

	The integral can be rewritten as a multiple of Γ(n + 2/2 + σ), so it is finite. Moreover, since Λ≥ t^2, we get in the end
		
    ∫_0^∞ |∇_-2pt x∇_-2pt y p(x,y,t^2 s)| e^-1/4s ds/s^1+σ≲t^2σ - 2/Λ^n + 2σ/2.

	This finishes the proof.




Let  be as in <Ref>. Then  is admissible.



The statement follows from known zero-order and first-order bounds on the heat kernel on  which we recall below.

By the curvature assumption we have (<ref>), see <cit.>, see also <cit.>.
				
By <cit.> the assumption on the heat kernel together with (<ref>) implies
			
    |∇ p(x,y,t)| ≲ t^-n+1/21+d^2(x,y)/t  e^-c d(x,y)^2/t.

Since (1+|x|) e^-|x|≤ C e^-1/2|x| this readily implies (<ref>). Recall that the heat kernel is symmetric, so the gradient estimate holds both for ∇_x and ∇_y.

For (<ref>) we use the semi-group property of the heat kernel, i.e.
			
    p(x,y,2t) = ∫_ p(x,z,t)   p(y,z,t)  dz.

		Keep in mind, that if a ≤ b + c, then a^2 ≤ 2(b^2 + c^2). 
		So using Hölder's inequality, we arrive at
			
    e^d(x,y)^2/At  |∇_ -2pt x∇_ -2pt y p(x,y,2t)|	   ≤∫_ e^2d(x,z)^2/At  |∇_ -2pt x p(x,z,t)|   e^2d(y,z)^2/At  |∇_ -2pt y p(y,z,t)|  dz	
       ≤( ∫_ e^4d(x,z)^2/At  |∇_ -2pt x p(x,z,t)|^2  dz )^1/2( ∫_ e^4d(y,z)^2/At  |∇_ -2pt y p(y,z,t)|^2  dz )^1/2
       ≲( ∫_ e^4d(x,z)^2/At 1/t^n + 1  e^-cd(x,z)^2/t dz )^1/2
         ×( ∫_ e^4d(y,z)^2/At 1/t^n + 1  e^-cd(y,z)^2/t dz )^1/2
        = 1/t^n + 1∫_ e^4d(x,z)^2/At  e^-cd(x,z)^2/t dz.

		We now choose A so big, such that 4/A < c in the equation above, which means we look for an estimate of the form
			
    ∫_ e^-cd(x,z)^2/t dz ≲ t^n/2,

		where c > 0. Let B_0 = B(x,√(t)) and let B_k = B(x, 2^k √(t)) ∖ B(x,2^k - 1√(t)). Then we have = ∪ B_k. Furthermore, it holds
			
    ∫_B_0 e^-cd(x,z)^2/t dz ≲ |B_0| ≲ t^n/2

		by the Ahlfors regularity. Moreover, it holds
			
    ∫_B_k e^-cd(x,z)^2/t dz ≲ 2^nk  e^-c   2^k - 1  t^n/2

		again by the Ahlfors regularity. Together, we have
			
    ∫_ e^-cd(x,z)^2/t dz ≤∑_k = 0^∞∫_B_k e^-cd(x,z)^2/t dz ≲ t^n/2∑_k = 0^∞ 2^nk  e^-c   2^k - 1≲ t^n/2.

		This estimate together with (<ref>) gives (<ref>) as desired.



Let  be as inTheorem <ref>. Then  is admissible.



Using <cit.>, we obtain the gradient estimate (<ref>). The claim then follows as in the proof of Corollary <ref>.



The previous results, thanks to <cit.>, extend straightforwardly to Lie groups with polynomial volume (notice that these are spaces of homogeneous type in the sense of Coifman and Weiss <cit.>). 




§ T(B)-THEOREM AND SQUARE FUNCTION ESTIMATES


We use the following important version of the local T(b)-theorem on manifolds, which is proven in much greater generality in <cit.>. It allows us to pass from local estimates in small balls of ℳ (i.e. essentially the Euclidean space) to global estimates.















	
Let  be an admissible manifold and let T be an operator, acting on functions f:→ via

    Tf(x,t) := ∫_ℳκ(x,y,t)   f(y)  dy,

where κ : ×× (0,∞) ⟶ is integrable and satisfies 

    ∫_κ(x,y,t)  dy = 0   for all  x ∈,  t > 0,


    |κ(x,y,t)| t^ν/d(x,y)^2+t^2^n+ν/2  for all  x,y ∈,  t > 0


    |κ(x,y_1,t)-κ(x,y_2,t)|/d(y_1,y_2)t^ν - 1/d(x,y_1)^2+t^2^n+ν/2  for all  d(y_1,y_2) ≤1/2 (d(x,y_1)^2 + t^2)^1/2.


Then,

    ∫_ℳ∫_0^∞ |Tf(x,t)|^2 dt/tdx^1/2∫_ℳ |f(x)|^2 dx^1/2.



All conditions in <cit.> are satisfied, once we confirm 1.,2.,3.: We choose a smooth decomposition of unity b_Q ∈ C_c^∞() each supported within a coordinate patch of  and constantly one in a small Whitney cube Q. Then it suffices to show that for some α∈ (0,1] the following holds for any η∈ C_c^∞(B(x_0,r)):

    ∫_B(x_0,r)∫_0^r |Tη(x,t)|^2 dt/t dx  r^n+2α [η]^2_C^α.

But observe that because of ∫_ℳκ(x,y,t)  dy = 0 by assumption (and κ is integrable by the assumptions as well)

    |Tη(x,t)| =    |∫_ℳκ(x,y,t)   (η(y)-η(x)) dy  |
        [η]_C^α ∫_ℳt^ν/d(x,y)^2+t^2^n+ν/2  d(x,y)^α dy
    
     =    [η]_C^α  t^-n+α∫_ℳd(x,y)/t^α/d(x,y)/t^2+1^n+ν/2 dy

which holds for any α∈ [0,1].
Now,

    ∫_ℳd(x,y)/t^α/d(x,y)/t^2+1^n+ν/2 dy	
     =   ∑_k=1^∞∫_B(x,2^k t) \ B(x,2^k-1t)d(x,y)/t^α/d(x,y)/t^2+1^n+ν/2 dy
       +∫_B(x,t)d(x,y)/t^α/d(x,y)/t^2+1^n+ν/2 dy.

Observe that 

    ∫_B(x,t)d(x,y)/t^α/d(x,y)/t^2+1^n+ν/2 dy ∫_B(x,t) 1 dy ≲ t^n

and

    ∫_B(x,2^k t) \ B(x,2^k-1t)d(x,y)/t^α/d(x,y)/t^2+1^n+ν/2 dy 
         (2^k t)^n  2^α k/2^2k+1^n+ν/2≲ t^n 2^k(α -ν).

We conclude that for α < ν,

    |Tη(x,t)|  
      [η]_C^α  t^α.

This implies (<ref>).


The main point is that an admissible manifold allows for the T(b)-theorem to be applied to the σ-harmonic extension.


As a corollary we obtain a result which is essentially a square function estimate.

Let  be admissible and 0 < σ < 1. Let U be the σ-harmonic extension of u ∈ C_c^∞(), then

    ∫_ℳ× (0,∞) t  |∂_t U(x,t)|^2  dx dt u_L^2(ℳ)^2,


    ∫_ℳ× (0,∞) t  |∇_ -2pt x U(x,t)|^2  dx dt u_L^2(ℳ)^2.



Let p(x,y,s) be the heat kernel for ℳ. Then

    U(x,t) :=   1/4^σΓ(σ)∫_0^∞∫_ℳ p(x,y,s)   u(y)  dy   t^2σ e^-t^2/4s ds/s^1+σ
    
     =   1/4^σΓ(σ)∫_0^∞∫_ℳ p(x,y,t^2 s)   u(y)  dy   e^-1/4s ds/s^1+σ.

Regarding (<ref>), we have

    t  ∂_t U(x,t) = 1/4^σΓ(σ)∫_0^∞∫_ℳ p(x,y,s)   u(y)   t  ∂_t t^2σ e^-t^2/4s dy ds/s^1+σ.

Thus, for

    κ(x,y,t) 	    := 1/4^σΓ(σ)∫_0^∞ p(x,y,s)  t  ∂_t t^2σ e^-t^2/4s ds/s^1+σ
        = 2  t^2σ/4^σΓ(σ)∫_0^∞ p(x,y,s)  σ - t^2/4s  e^-t^2/4s ds/s^1+σ,


and

    Tu(x,t) := ∫_ℳκ(x,y,t)   u(y) dy,

we have
	
    ∫_ℳ× (0,∞) t   |∂_t U(x,t)|^2  dx dt = ∫_ℳ∫_0^∞ |Tu(x,t)|^2 dt/t dx.

Since for u constant we know that U(x,t) is also constant (see Appendix <ref>), one has t  ∂_t U ≡ 0. We conclude that

    ∫_ℳκ(x,y,t) dy = 0   for all  x ∈ℳ,  t > 0.


It remains to establish the estimates (<ref>) and (<ref>), then the claim follows from <Ref>.

We estimate

    |κ(x,y,t)| ≲ t^2σ∫_0^∞ p(x,y,s)  |σ - t^2/4s |   e^-t^2/4s ds/s^1+σ.

Since  was assumed to be admissible, we can use (<ref>) (after the transformation s ⟼ t^2 s) to conclude (<ref>).
In order to show (<ref>), we use the mean value theorem  to rewrite
	
    |κ(x,y_1,t) - κ(x,y_2,t)|/d(y_1,y_2)    = |∇_-2pt yκ (x,y,t)|	
       ≲ t^2σ∫_0^∞ |∇_-2pt y p(x,y,s)|  |σ - t^2/4s |   e^-t^2/4s ds/s^1+σ.

Again, since  was assumed to be admissible, we can use (<ref>) to deduce (<ref>).

In order to derive the second estimate (<ref>), we argue similarly for a slightly different kernel κ. By the representation formula for U, we can rewrite
	
    t  ∇_-2pt x U(x,t) = ∫_∇_-2pt x p(x,y,s)   u(y)  dy   t^2σ + 1  e^-t^2/4sds/s^1 + σ,

so we define the operator	
	
    Tu(x,t) = ∫_κ(x,y,t)   u(y)  dy

with the kernel

    κ(x,y,t) 	 :=     t^2σ + 1∫_0^∞∇_-2pt x p(x,y,s)   e^-t^2/4sds/s^1 + σ
    
    						 =     t ∫_0^∞∇_-2pt x p(x,y,t^2 s)   e^-1/4sds/s^1 + σ.

Then again it holds ∫κ = 0 and
	
    ∫_ℳ× (0,∞) t   |∇_-2pt x U(x,t)|^2   dx  dt = ∫_ℳ∫_0^∞ |Tu(x,t)|^2 dt/t dx.

So it suffices to show the estimates (<ref>) and (<ref>). This follows analogously to the first case by the admissibility of  and the mean value theorem.































Let  be admissible and 0 < σ < 1. Let U denote the σ-harmonic extension of u, then 

    Tu(x,t) := t  ∇_-2pt (x,t) U(x,t)

satisfies the conditions of <Ref>. In particular, the following estimate holds for all functions u ∈ C_c^∞():

    ∫_∫_(0,∞) t   |∇_-2pt(x,t)U(x,t)|^2  dt  dx ≲u_L^2()^2.



 Follows immediately from Proposition 3.2.





§ BMO AND CARLESON MEASURES: PROOF OF THE MAIN THEOREM

The proofs of <Ref> and <Ref> consist in proving two directions. The easier one is <Ref>, the more difficult one is  <Ref>. 
 Let  be admissible and 0 < σ < 1. Let U be the σ-harmonic extension of u ∈ C_c^∞(), then

    sup_B |B|^-1∫_T(B) |∇_(x,t) U(x,t)|^2  t dx dt  [u]_BMO(ℳ)^2.



We extend the argument from <cit.>.

Fix any ball B ⊂ℳ, and denote by B^∗ the ball with twice the radius.
We decompose

    ∇_(x,t) U = ∇_(x,t) U_1 + ∇_(x,t) U_2 + ∇_(x,t) U_3,

where U_i is the σ-harmonic extension of u_i, respectively, given as

    u_1 := χ_B^∗ (u-(u)_B^∗),


    u_2 := (1-χ_B^∗) (u-(u)_B^∗),


    u_3 := (u)_B^∗.

Observe that U_3 is constant and thus ∇ U_3 = 0.
Moreover,

    |B|^-1∫_T(B) |∇_(x,t) U_1(x,t)|^2  t dx dt  ≤ |B|^-1∫_ℳ× (0,∞) |∇_(x,t) U_1(x,t)|^2  t dx dt.

In view of <Ref>,

    |B|^-1∫_T(B) |∇_(x,t) U_1(x,t)|^2  t dx dt  |B|^-1∫_B^∗ |u-(u)_B^∗|^2

Since ℳ is supposed to be Ahlfors-regular,  |B|  |B^∗| with uniform constants, one has by John-Nirenberg inequality <cit.>, see also <cit.>,

    |B^∗ |^-1∫_B^∗ |u-(u)_B^∗|^2 [u]_BMO^2.

This implies,

    |B|^-1∫_T(B) |∇_(x,t) U_1(x,t)|^2  t dx dt   [u]_BMO^2.

It remains to estimate U_2.
As in the proof of <Ref>,

    |B|^-1∫_T(B) |∇_(x,t) U_2(x,t)|^2  t dx dt  = |B|^-1∫_T(B) |∫_ℳκ(x,y,t) u_2(y) dy |^2 dx dt/t.

In view of (<ref>), for some given ν > 0,

    |B|^-1∫_T(B) |∇_(x,t) U_2(x,t)|^2   t dx dt    |B|^-1∫_T(B)∫_ℳt^ν/d(x,y)^2 + t^2^n+ν/2 |u_2(y)| dy dx dt/t

We denote with B_k, k ∈, the ball concentric around B but with radius 2^k times the radius of B.

Since u_2 ⊂ℳ\ B^∗, we find that for any x ∈ B

    ∫_ℳt^ν/d(x,y)^2 + t^2^n+ν/2 |u_2(y)| dy ∑_k=0^∞t^ν/(2^kr)^2 + t^2^n+ν/2∫_B_k+1\ B_k|u(y)-(u)_B|  dy.

By triangle inequality and a telescoping sum,

    ∫_B_k+1\ B_k|u(y)-(u)_B|  dy ≤∫_B_k+1|u(y)-(u)_B_k+1|  dy + |B_k+1| ∑_i=0^k|(u)_B_i - (u)_B_i+1|.

Using the doubling property of the measure and the definition of BMO,
we have 

    ∫_B_k+1\ B_k|u(y)-(u)_B| dy  |B_k| k+1  [u]_BMO.

Consequently,

    ∫_ℳt^ν/d_ℳ(x,y)^2 + t^2^n+ν/2 |u_2(y)|  dy ≲∑_k=0^∞t^ν/(2^kr)^2 + t^2^n+ν/2  |B_k|k+1  [u]_BMO.

This implies

    |B|^-1∫_T(B) |∇_(x,t) U_2(x,t)|^2  t dx dt  A(r)  [u]_BMO,

where

    A(r) = ∫_0^r ∑_k=0^∞t^ν/(2^kr)^2 + t^2^n+ν/2  (2^kr)^n k+1  dt/t.

By a substitution t ↦ rt we see that A(r) = A(1), and 

    A(1) ≤∫_0^1 t^ν-1 dt·∑_k=0^∞ 2^-ν k k+1 = C_ν < ∞.

This concludes the proof of <Ref>. 



Let  be admissible and 0 < σ < 1. Let U be the σ-harmonic extension of u ∈ C_c^∞(), then

    [u]_BMO(ℳ)^2 sup_B |B|^-1∫_T(B) |∇_(x,t) U(x,t)|^2  t dx dt



One technical ingredient in the proof of <Ref> is the following observation (cf. <cit.>).


Let Φ,U: ℳ× (0,∞) → be the σ-harmonic extension of φ and u, respectively. Then

    | ∫_ℳ u φ |   ∫_ℳ× (0,∞) t |∂_t Φ(x,t)|  |∂_t U(x,t)| dx dt
       + ∫_ℳ× (0,∞) t  |∇_x Φ(x,t)|  |∇_x U(x,t)| dx dt



By integration by parts and the decay as t →∞ (see <Ref>), we have for every x ∈ℳ:

    u(x) φ(x) = 1/2σ∫_0^∞ t^2σ∂_tt^1-2σ∂_t Φ(x,t) U(x,t) dt.

Since U is the σ-harmonic extension,

    ∂_tt^1-2σ∂_t U(x,t) = -t^1-2σ_x U(x,t)

and likewise for Φ











after integration by parts

    ∫_ u(x)  φ(x)  dx =     1/σ∫_× (0,∞) t  ∂_t U(x,t)  ∂_t Φ(x,t)  dx  dt	
        + 1/σ ∫_0^∞ t  ⟨∇_-2pt x U( ·  , t), ∇_-2pt xΦ( ·  , t)⟩_L^2() dt,

from which the claim follows immediately.


The following is proved in <cit.>. It is stated there in ^n, but the proof easily extends almost verbatim to Ahlfors regular spaces. 

Let ℳ be an Ahlfors-regular manifold and let F,G × (0,∞) ⟶ be measurable functions. Then,

    ∫_ℳ× (0,∞) t  |F(x,t)|   |G(x,t)|  dx dt 
       ∫_ℳ∫_d(x,y) < t|F(y,t)|^2 dy dt/t^n-1^1/2 dxsup_B |B|^-1∫_T(B) |G(x,t)|^2  t dx dt^1/2



















































Again, we essentially can follow Stein's book, namely <cit.>.

From <Ref> and <Ref> for F = ∇_ -2pt (x,t)Φ and G = ∇_ -2pt (x,t) U we obtain (using the notation (<ref>))

    | ∫_ℳ u φ  | ∫_ℳ∫_d(x,y) < t|∇Φ(y,t)|^2 dy dt/t^n-1^1/2 dx  [u]_B̃M̃Õ.

We can conclude once we show that

    ∫_ℳ∫_d(x,y) < t|∇Φ(y,t)|^2 dy dt/t^n-1^1/2 dxφ_ℋ^1,

where ℋ^1 is the Hardy space. Indeed, the claim then follows in view of the duality of Hardy spaces and BMO <cit.>.

To obtain (<ref>) we use an extrapolation result <cit.>, which essentially states that a suitable operator, if it is bounded from L^2 to L^2, can be extended to a bounded operator from the Hardy space into L^1. 
To apply this result, first observe that from Fubini we have

    x ↦∫_d(x,y) < t|∇Φ(y,t)|^2 dy dt/t^n-1^1/2_L^2()^2
    
      =   ∫_0^∞∫_∫_ t^1-nχ_d(x,y) < t |∇Φ(y,t)|^2 dx dy dt
       ∫_0^∞∫_  t  |∇Φ(y,t)|^2 dy dt.

From  <Ref> we conclude that the operator φ↦ Tφ := t ∇Φ as in (<ref>) satisfies the conditions imposed on the operator denoted by θ_E in <cit.>. That theorem implies (<ref>).







§ JACOBIAN ESTIMATE: PROOF OF THEOREM <REF>

By the trace space characterizations of BMO obtained in Theorem <ref> and <Ref> we can follow the strategy in <cit.> to prove the Coifman–Lions–Meyer–Semmes estimate on manifolds.


For some 1/2 < σ < 1, let F^ℓ be the σ-harmonic extension to × (0,∞) for f^ℓ for all ℓ = 1, …, n, and let Φ be the σ-harmonic extension of φ. Then by Stokes theorem we have
	
    |∫_df^1∧…∧df^n φ |=  |∫_× (0,∞)d_(x,t) F^1 ∧…∧d_(x,t) F^n ∧d_(x,t)Φ | .

We claim that 

    |∫_× (0,∞)d_(x,t) F^1 ∧…∧d_(x,t) F^n ∧d_(x,t)Φ | 
       ∫_× (0,∞) t  |∇_(x,t) F|^n-1  |∇_x ∇_(x,t) F|  |∇_(x,t)Φ|.

Assume we have (<ref>) (which will be proven below). Set

    M_f(x) := sup_d(x,y) < t |∇_(x,t) F(y,t)|.


Then, by Lemma <ref>, Hölder's inequality, and the BMO-characterization, Theorem <ref>

    ∫_ℳ× (0,∞) t |∇_(x,t) F|^n-1  |∇_x ∇_(x,t) F|  |∇_(x,t)Φ| 
       ∫_ℳ∫_d(x,y) < t|∇_(x,t) F|^n-1  |∇_x ∇_(x,t) F(y,t)|^2 dy dt/t^n-1^1/2 dx [φ]_BMO
       ∫_ℳ |M_f|^n-1∫_d(x,y) < t  |∇_x ∇_(x,t) F(y,t)|^2 dy dt/t^n-1^1/2 dx [φ]_BMO
       M_f_L^n()^n-1∫_d(·,y) < t  |∇_x ∇_(x,t) F(y,t)|^2 dy dt/t^n-1^1/2_L^n()  [φ]_BMO.

By Lemma <ref>, and the boundedness of the maximal function from L^n() to L^n() (which holds true on every space with doubling measure, <cit.>)

    M_f_L^n()∇ f_L^n().

Moreover, similar as in (<ref>) together with the extrapolation result <cit.>, we obtain

    ∫_d(·,y) < t  |∇_x ∇_(x,t) F(y,t)|^2 dy dt/t^n-1^1/2_L^n()∇ f_L^n().

Here we also have used that ∇_x F is the σ-harmonic extension of ∇ f, which follows from the uniqueness of the σ-harmonic extension.

This concludes the proof of Theorem <ref>, up to proving (<ref>).



 
	Using one integration by parts in t-direction, together with the decay of the σ-harmonic extension for t →∞, we obtain
	
    |∫_× (0,∞)d_(x,t) F^1 ∧…∧d_(x,t) F^n ∧d_(x,t)Φ | 
    
    		=    |∫_× (0,∞) t^2σ ∂_t [ t^1 - 2σd_(x,t) F^1 ∧…∧d_(x,t) F^n ∧d_(x,t)Φ]  |.

By orthogonility of the coordinates in × (0,∞) we have 
    d_(x,t) F(x,t)= d_ F(x,t) + ∂_t F(x,t)  dt.

Since dt ∧dt = 0, we have two cases to estimate:

    |∫_× (0,∞) t^2σ ∂_t [ t^1 - 2σd_ℳ F^1∧…∧∂_t F^ℓ dt∧…∧d_ℳ F^n ∧d_ℳΦ]  |

for ℓ = 1,…,n and

    |∫_× (0,∞) t^2σ ∂_t [ t^1 - 2σd_ℳ F^1∧…∧d_ℳ F^n ∧∂_t Φ dt]  |.

Regarding (<ref>), 

    |∫_× (0,∞) t^2σ ∂_t [ t^1 - 2σd_ℳ F^1∧…∧∂_t F^ℓ dt∧…∧d_ℳ F^n ∧d_ℳΦ]  |
    ≤    |∫_× (0,∞) t d_ℳ F^1∧…∧∂_t F^ℓ dt∧…∧d_ℳ F^n ∧d_ℳ∂_tΦ |
       + ∫_× (0,∞) t^2σ |∇_(x,t) F|^n-1  |∂_t(t^1-2σ)∂_t F| |∇_x Φ|
       +∫_× (0,∞) t |∂_t ∇_x F| |∇_(x,t) F|^n-1 |∇_x	 Φ|.

The last term is already in the form of (<ref>). The second-to-last term is as well, if we use that ∂_t(t^1-2σ∂_t F) = t^1-2σ_ F.
For the first term we use an integration by parts (observe that  has no boundary)

    |∫_× (0,∞) t d_ℳ F^1∧…∧∂_t F^ℓ dt∧…∧d_ℳ F^n ∧d_ℳ∂_tΦ |
    
     =    |∫_× (0,∞) t d_ℳd_ℳ F^1∧…∧∂_t F^ℓ dt∧…∧d_ℳ F^n∂_tΦ |

Now it follows from Leibniz rule that this is of the form of (<ref>), so (<ref>) can be estimated as claimed.

Regarding the estimate for the term (<ref>) we argue similarly.

    |∫_× (0,∞) t^2σ ∂_t [ t^1 - 2σ d_ℳ F^1 ∧…∧d_ℳ F^n ∧∂_t Φ dt]  |
        |∫_× (0,∞) t^2σ  d_ℳ F^1 ∧…∧d_ℳ F^n ∧∂_t [ t^1 - 2σ∂_t Φ] dt  |
       +∫_× (0,∞) t |∇_(x,t)∇_x F|  |∇_x F|^n-1 |∂_t Φ|

The second term is already of the form (<ref>). For the first term we use again that ∂_t(t^1-2σ∂_t Φ) = t^1-2σ_Φ. An integration by parts along  (as we did for the estimate of (<ref>) above) results again in an estimate of the form (<ref>).
(<ref>) is now estimated, and we conclude that (<ref>) holds true.









§ COMPUTATIONS FOR THE SIGMA-HARMONIC EXTENSION

We recall our definition for the extension U, motivated by <cit.>. 
Given u ∈ C_c^∞(ℳ) and 0 < σ < 1, the σ-harmonic extension U: ℳ× [0,∞) → is formally given by

    U(x,t) := 1/4^σΓ(σ) t^2σ∫_0^∞ e^s_ℳ u(x)  e^-t^2/4s ds/s^1+σ.

More explicitly, one has

    U(x,t) :=   1/4^σΓ(σ)∫_0^∞∫_ℳ p(x,y,s)   u(y)  dy  t^2σ e^-t^2/4s ds/s^1+σ,	
    
     			=    1/Γ(σ)∫_0^∞∫_ p(x,y,t^24s)   u(y)  dy   e^-s  s^σ - 1 ds,

where p(x,y,s) is the heat kernel for ℳ. Furthermore, U is the smooth (in the interior) solution of

    _ℳ U  + 1-2σ/t∂ _t U + ∂_tt U = 0      
    
     U(x,0) = u(x)    
    lim_|(x,t)| →∞ U(x,t) = 0.

The most important property for us is that constants are extended by constants: our manifold being assumed to be Ahlfors regular, it is stochastically complete, i.e
	
    ∫_ p(x,y,s)  dy = 1	      for all  x ∈,  s > 0,

see <cit.>. Now using the representation formula, we deduce that for constant u ⟶ℝ
	
    U(x,t)	    = u/Γ(σ)∫_0^∞∫_ p(x,y,t^24s)  dy   e^-s s^σ - 1 ds = u.

One can also check, that the representation formula solves the PDE (<ref>). Using the second line in the representation formula and that p solves the heat equation (<ref>) 
 one sees that
	
    ∂_t U(x,t)	    = 1/Γ(σ)∫_0^∞∫_ (∂_t p)(x,y,t^24s)   u(y)  dy   e^-s [t/2  s^σ - 2]  ds	
    ∂_t^2 U(x,t)	    = 1/Γ(σ)∫_0^∞∫_ (∂_t p)(x,y,t^24s)   u(y)  dy   e^-s [-1/2  s^σ - 2 - s^σ - 1 - (1 - σ)s^σ - 2]  ds	
    Δ_x U(x,t) 	    =  1/Γ(σ)∫_0^∞∫_ (∂_t p)(x,y,t^24s)   u(y)  dy   e^-s [s^σ - 1]  ds.

Comparing the square brackets shows that U indeed solves the PDE (<ref>).

Moreover, since the heat kernel is an approximation of the identity, we see that U has also the correct boundary data. Even more, as t →∞, we have U → 0. This follows from the admissibility of , i.e. given u ∈ C_c^∞
(), we can compute
	
    |U(x,t)|	   ≲ t^2σ∫_ℳ |u(y)| ∫_0^∞ p(x,y,s)   e^-t^2/4s ds/s^1+σ dy 
        = ∫_ℳ |u(y)| ∫_0^∞ p(x,y,t^2 s)   e^-1/4s ds/s^1+σ dy	
       ≲∫_ |u(y)|  t^ν/(d(x,y)^2 + t^2)^n + ν/2 dy	
       ≤1/t^n ∫_ |u(y)|  dy.








With an analogue computation, we see that
	
    |∂_t U(x,t)| ≲ t^-(n + 1) ∫_ |u(y)|  dy.

	

We also record the following estimate by maximal functions of the σ-harmonic extension.

Let  be as in Theorem <ref> or Theorem <ref>, and let U denote the σ-harmonic extension of u, for 1/2 < σ < 1. Then,

    sup_y,t:  d(x,y) < t |∇_(y,t) U(y,t)|   M|∇ u|(x)

where M denotes the Hardy-Littlewood maximal function	
	
    Mf(x) = sup_r > 01/|B(x,r)|∫_B(x,r) |f(y)|  dy.


The estimate in Lemma <ref> is false for σ≤1/2 even in Euclidean space. E.g. for σ = 1/2 we have

    |1 u(x)| = lim_t → 0^+ |∂_t U(x,t)| M|∇_x u|(x).

Indeed, the latter inequality has to be false because otherwise Lipschitz functions (i.e. functions with a finite right-hand side in the estimate above) have half-Laplacian bounded, which is false in general.

	Let p(x,y,s) be the heat kernel of , i.e. it solves (<ref>). Moreover, let t > 0, y ∈ and x ∈ B(y,t). We will estimate ∇_y U and ∂_t U seperately. Using the representation formula we see that 
		
    ∂_t U(y,t)	    = 1/Γ(σ)∫_0^∞∫_ (∂_t p)(y,z,t^24s)   u(z)  dz   e^-s [t/2  s^σ - 2]  ds	
        = 1/Γ(σ)∫_0^∞∫_Δ_z p(y,z,t^24s)   u(z)  dz   e^-s [t/2  s^σ - 2]  ds	
        = -1/Γ(σ)∫_0^∞∫_⟨∇_z p(y,z,t^24s), ∇ u(z)⟩ dz   e^-s [t/2  s^σ - 2]  ds.

	Taking the absolute value and using Fubini we arrive at
		
    |∂_t U(y,t)| ≲∫_ |∇ u(z)| ∫_0^∞ t   |∇_z p(y,z,t^24s)|   e^-s  s^σ - 2ds  dz.

	Using the gradient estimate (<ref>) of the heat kernel, we deduce
		
    ∫_0^∞ t   |∇_y p(y,z,t^24s)|   e^-s  s^σ - 2ds	   ≲1/t^n∫_0^∞ s^n + 1/2 + σ - 2  e^(-c d(y,z)^2/t^2 + 1)sds	
       ≲1/t^n 1/(c d(y,z)^2/t^2 + 1)^n/2 + σ - 1/2.

	Let B = B(x,2t) and B_k := B(x,2^k t) ∖ B(x,2^k - 1 t) for k ≥ 2. Then we can estimate
		
    |∂_t U(y,t)|	   ≲1/t^n∫_B   |∇ u(z)|  dz/(c d(y,z)^2/t^2 + 1)^n/2 + σ - 1 + ∑_k = 2^∞1/t^n∫_B_k|∇ u(z)|  dz/(c d(y,z)^2/t^2 + 1)^n/2 + σ - 1/2

	By the choice of the annuli, we can estimate for z ∈ B_k and k ≥ 2 
		
    d(y,z) ≥ |d(x,z) - d(x,y)| ≥ (2^k - 1 - 1) t.

	So it follows that
		
    sup_d(x,y) < t |∂_t U(y,t)|    ≲1/t^n∫_B |∇ u(z)|  dz + ∑_k = 2^∞1/2^2k( σ - 1/2) 1/(2^k t)^n∫_B_k |∇ u(z)|  dz
    ≲   M(|∇ u|)(x),


	since σ > 1/2 and  is Ahlfors regular.
	
The estimate of |∇_y U| is simpler (and works for any 0 < σ < 1). We simply observe that
		
    (∇_y U)(y,t) 	    = 1/Γ(σ)∫_0^∞∫_∇_y p(y,z,t^24s)   u(z)  dz   e^-s  s^σ - 1 ds	
        = 1/Γ(σ)∫_0^∞∫_∇_z p(y,z,t^24s)   u(z)  dz   e^-s  s^σ - 1 ds	
        = -1/Γ(σ)∫_0^∞∫_ p(y,z,t^24s)  ∇ u(z)  dz   e^-s  s^σ - 1 ds	
        = -1/Γ(σ)∫_∇ u(z) ∫_0^∞ p(y,z,t^24s)   e^-s  s^σ - 1 ds  dz.

	Taking the absolute value we arrive at
		
    |∇_y U(y,t)| 	   ≲∫_ |∇ u(z)| ∫_0^∞ p(y,z,t^24s)   e^-s  s^σ - 1 ds  dz.

	Using (<ref>) we see that
		
    ∫_0^∞ p(y,z,t^24s)   e^-s  s^σ - 1 ds ≲1/t^n 1/(d(y,z)^2/t^2 + 1)^n/2 + σ.

	In the same fashion as before we can obtain (for any 0 < σ < 1) the estimate
		
    sup_d(y,x) < t|∇_y U(y,t)| ≲ M(|∇ u|)(x).

This concludes the proof.


abbrv


