Paper Title: On almost everywhere convergence of planar Bochner-Riesz mean.





§ ABSTRACT 
We demonstrate that the almost everywhere convergence of the planar Bochner-Riesz means for L^p functions in the optimal range when 5/3≤ p≤ 2.  
This is achieved by establishing a sharp L^5/3 estimate for a maximal operator closely associated with the Bochner-Riesz multiplier operator. The estimate depends on a novel refined L^2 estimate, which may be of independent interest.



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    August 5, 2024
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§ INTRODUCTION





For any Schwartz function f, the n-dimensional Bochner-Riesz mean T^_tf is defined as

    T_t^ f(x):=(2π)^-n∫_ℝ^n(1-|ξ|^2/t^2)^_+f(ξ)e^ix·ξdξ.

It can be extended to a multiplier operator with the Fourier multiplier (1-|·|^2/t^2)^_+. Analogous to the Gauss mean (where the multiplier is replaced by e^-|·|^2/t) or the Able mean (where the multiplier is replaced by e^-|·|/t), the Bochner-Riesz mean is a summation method that was introduced to study the radial convergence of Fourier transform. 
The issue of almost everywhere convergence concerning the Bochner-Riesz mean stands out as one of the most intriguing and significant problems in modern analysis. It can be precisely formulated as follows: 





For any L^p function f, what is the optimal regularity  required for the Bochner-Riesz mean T_t^ f to converge to f almost everywhere?


In the higher range p≥2, the almost-everywhere convergence problem is completely solved in <cit.>. However, the lower range p<2 poses significantly greater challenges. 
It is worth noting that, when p<2, Stein's maximal principle suggests that the following two claims are equivalent:

    
  * lim_t→∞T_t^ f(x) 
    =f(x) for a.e. x∈ℝ^n and  every L^p function f.
    
  * The maximal Bochner-Riesz operator T^_∗ is bounded in L^p.

Here the maximal operator T^_∗ is defined as

    T^_∗ f(x):=sup_t>0|T_t^ f(x)|.



In his investigation into weak-type L^p-estimates for maximal operators, Tao put forth an insightful conjecture concerning the maximal operator T^λ_∗. This conjecture, outlined in his cited work <cit.>, offers a compelling speculation on the operational dynamics and possible constraints of T^λ_∗.
Allow us to articulate this conjecture herein.



For any 1≤ p<2, the maximal Bochner-Riesz operator T^_∗ extends to a bounded operator in L^p(^n) when >max{0, 2n-1/2p-n/2}, that is, 

    T^_∗ f_L^p(ℝ^n)≤ C f_L^p(ℝ^n) .



The conjecture has remained remarkably intriguing and challenging, with very limited progress made over the years.
Its motivation lies in the pointwise convergence of circular Fourier partial sums, rendering the scenarios most compelling and challenging as  approaches zero.
Prior to our work, there had been no positive conclusion for  near zero. Additionally, for higher dimensional cases when n≥ 4, no affirmative (non-trivial) result has been reached even when  is away from zero.
In the planar case, we are positioned to offer a partial resolution of this conjecture, marking a notable improvement over previous research, particularly as it's the first instance of attaining sharp and non-trivial conclusions for p<2 and λ near zero. 
Our approach significantly diverges from prior studies.
Primarily, it revolves around a quantitative version of localized L^2 estimates
concentrating on some unit balls. 
These refined L^2 estimates not only deepen our understanding of the conjecture but also may pave the way for future investigations and improvements in this field. Moreover, they extend beyond addressing the maximal Bochner-Riesz operators, such as the reverse square function inequality in the plane, suggesting broader applications for the quantitative local L^2 inequalities. It is reasonable to anticipate further applications stemming from these insights.  We now state our main result as follows. 



Conjecture <ref> is true when n=2 and 5/3≤ p≤ 2.

 




As a direct implication  of Theorem <ref>,  within the range 5/3≤ p≤2, the Bochner-Riesz mean T^_t f converges to f for any f∈ L^p() if >0.
However, the assertion does not hold when λ=0,  as demonstrated by Fefferman's renowned example <cit.>. Theorem <ref> encompassed all prior findings in the plane from <cit.>, and <cit.> which collectively affirm the pointwise convergence conclusion for 
(1/p, λ) within the quadrilateral region C depicted in Figure <ref>.
Theorem <ref> extends its coverage to a significantly larger area,  specifically, the triangular region defined by three points: (1/2,0), (3/5, 0) and (1, 1/2).  Region B remains open, while Region A is invalidated by Tao's example, where  f(x_1,x_2)=a(R^-1/2x_1)a(x_2)e^ix_1 is set in (<ref>) for a standard bump function a. 



Our main theorem, Theorem <ref>, is established through L^p estimates of a specific maximal operator, which bears relevance to the classical Bochner-Riesz multiplier. Before proceeding, let's introduce the definition of this maximal operator.
For t∈ [1,2] and any positive real number R, define S_t, R by

    S_t, Rf(ξ) = a(R(t-|ξ|))  f(ξ)

for any ξ∈ℝ^2 and any Schwartz function f, where the function a represents a standard bump function on [-1,1].  
The multiplier a(R(t-|ξ|)) here is a bump function supported on the annulus 
A_t, R:={ξ∈ℝ^2: t-R^-1≤ |ξ| ≤ t+R^-1}.  We define the corresponding maximal operator
S^*_R by,  

    S^*_R f(x) =sup_t∈ [1,2]| S_t, Rf(x)| ,

for any x∈ℝ^2. 



By means of particular simplifications detailed in Section <ref>, establishing Theorem <ref> can be accomplished by focusing solely on the following L^p-estimates of the maximal operator, which formulates our central theorem in this article. 


For any >0 and any 5/3≤ p≤ 2,  there is a constant C_ such that 

    S^*_R f_L^p(ℝ^2)≤ C_ R^f_L^p(ℝ^2) ,

for any Schwartz function f and any R≥ 1. 


In the planar case, Tao's conjecture can be comprehensively addressed upon solving the conjecture stated below.

The L^p-estimate (<ref>) holds for p=3/2.  




One of the main challenges posed by Conjecture <ref> arises from the absence of orthogonality in the radial direction. Research by <cit.> demonstrates that substituting the L^p_xL^∞_t norm in (<ref>) with a larger L^p_xL^p'_t norm (1/p+1/p'=1) leads to the breakdown of Conjecture <ref> within the entire isosceles right triangular area illustrated in Figure <ref>. This essentially means that an interpolation between the established outcomes at the endpoints L^1_xL^∞_t and L^2_xL^2_t offers an almost sharp bound for the L^p_xL^p'_t norm. To extend the results for the maximal problem beyond this interpolation line, it becomes imperative to establish a certain level of radial orthogonality. As we'll explain shortly, this is largely achieved implicitly through the consideration of specific weighted estimates.







Surprisingly, Tao's example f(x_1,x_2)=a(R^-1/2x_1)a(x_2)e^ix_1, among some reductions, also serves as a sharp example of this L^p_xL^p'_t problem. In Section <ref>, we will briefly explain how our approach in this paper can, to some extent, rule out Tao's example for Conjecture <ref>.






By employing certain reductions (refer to Section <ref>), demonstrating Theorem <ref> can be narrowed down to primarily examining 
the L^p estimates for the following model operator:


    ∑_jS_j f_F_j ,

where F_j's are disjoint sets in an R-ball B_R in the plane,
each of them is a collection of unit balls, _F_j represents the indicator
function of F_j, 
and S_j is a multiplier operator given by S_jf(ξ)=a(R(t_j-|ξ|)) f(ξ) for some t_j∈[1,2]. Notice that the operator S_j depends on both j and R, yet we abstain from explicitly including R in the notation S_j.


More precisely, it suffices to establish that 

    ∑_jS_j f_F_j_p ≲_ R^f_p,

for one endpoint with p=5/3 and any ϵ>0, as the L^p estimates (<ref>) can then be derived through interpolation with the other known endpoint at p=2.
This observation leads us to explore the weighted L^p-norm S_jf_L^p(_F_j) for any fixed j. Given that each S_j behaves similarly to the spherical multiplier operator S:=S_1, R, we are interested in analyzing the L^p-norm restricted to a subset F⊂ B_R, that is,    


    Sf_F_p.





Decoupling inequalities have been utilized as a method in previous studies aiming towards (<ref>). In <cit.>, (<ref>) is bounded by a decoupling norm, albeit without incorporating the information from the weight _F. In a more recent paper <cit.>, the authors proposed to bound this weighted L^p-norm using a decoupling norm with a saving term expressed as

    (|F|/R^n)^

for some >0. Such decoupling inequalities have contributed to advancements regarding Conjecture <ref> in dimensions two and three. However, for the maximal Bochner-Riesz problem, the decoupling norm might not be the most suitable choice. One reason is that decoupling inequalities are powerful tools primarily aimed at analyzing constructive interference, which is often the main challenge in L^p-problems when p>2. However, in Conjecture <ref>, we are required to consider L^p functions in the lower range of p, where the constructive interference might not be the main obstacle. 



We naturally anticipate the gain (<ref>) for the weighted L^2-norm of Sf. Unfortunately, the Knapp example f(x)=e^itx_na(R^-1/2x̅)a(R^-1x_n) with (x̅, x_n)∈ℝ^n-1×ℝ demonstrates that we cannot generally expect a weighted inequality in the form

    Sf_F_L^2(ℝ^n)≲(|F|/R^n)^Sf_2.

However, if we replace the operator S with its multilinear analog Mul_n (S),  the Bennett-Carbery-Tao multilinear restriction theorem <cit.> indicates that

    Mul_n(S)f_F_L^2(ℝ^n)⪅(|F|/R^n)^1/2nSf_2.

The multilinear weighted L^2 estimate (<ref>) is sharp, by testing F=B_r for any 1≤ r ≤ R. Nevertheless, these are essentially the only sharp examples.


This observation suggests that if S could be substituted freely with its multilinear analog, then (<ref>) would prove quite useful for the lower range p<2. To see why, 
we first assume 

    (# j)|F_j|∼ R^2,

which can be achieved by pigeonholing. Henceforth, on the one hand, we have from the weighted L^2 estimate (<ref>) and the assumption (<ref>), 

    ∑_jS_jf_F_j_2^2≲∑_jBil(S_j)f_F_j_2^2 ⪅(|F_j|/R^2)^1/2∑_jS_jf_2^2≲1/(# j)^1/2f_2^2.

On the other hand, invoke the sharp L^4/3 estimate for the Bochner-Riesz operator to get

    ∑_jS_jf_F_j_4/3^4/3≲∑_jf_4/3^4/3≲(#  j)f_4/3^4/3.

An interpolation between the above two estimates gives

    ∑_jS_jf_F_j_p^p⪅f_p^p

for p=16/9, which proves Conjecture <ref> when n=2 and 16/9≤ p≤ 2.


The first obstacle we encounter is how to transition from the linear operator to the bilinear one. Typically, the Bourgain-Guth broad-narrow method (see <cit.>) is a standard tool for this purpose. However, given the intricacy of our problem, we need to utilize more nuanced concepts of broadness, akin to those introduced by Guth in <cit.>.


We call an interval τ⊂^1 a ρ-cap if the Lebesgue measure of the cap τ is ∼ρ. We also define  

    C_τ:={ξ∈^2:ξ/|ξ|∈τ}

as the conic region determined by τ.


For any cap τ,  S_τ f denotes the smooth Fourier restriction of Sf in τ, that is, 
S_τ f = a_τSf. Here a_τ is a smooth bump function on the conic region C_τ. 



Suppose ={} is a collection of finitely overlapping (log R)^-1-caps. Let M be a large number. For any function f and any set E⊂ B_R, define the broad norm Sf_L^2__M(E) as 

    Sf_L^2__M(E):=max_'⊂,
    #'=Mmin_∈'{S_f_L^2(E)}.




The quantity Sf_L^2__M(E) is determined by the cap  such that S_f_L^2(E) is the M-th largest element among the family {S_f_L^2(E)}_∈. When E is a unit ball, the definition coincides with the one introduced in  <cit.>. Compared to the usual one Sf_L^2(E) or its bilinear analog Bil(S)f_L^2(E), the broad norm Sf_L^2__M(E) possesses a stronger directional non-concentration property.



To establish an analog of the inequality (<ref>), we find ourselves needing certain refined L^2 estimates associated with the broad norm, which are the cornerstone of our proof of Theorem <ref>.  To elucidate our new refined L^2 results, we first need to introduce some definitions. We denote the ρ-neighborhood of a set E by N_ρ(E) and the Lebesgue measure of E by |E|. Moreover, A≫ B signifies that the number A is much greater than the number B, and # S represents the number of elements in a (finite) set S. 



Let κ∈ [r^-1/2,1] for some given real number r≥ 1. A subset E of an r-ball (or cube) B_r in the plane is called  regular in B_r  if

    
  * For any r^1/2× r-tube T in ℝ^2,
    
    |N_r^1/2(E)∩ T|≲ |T|.

    
  * There exists a collection _ of r^1/2× r tubes  with #_≲|N_r^1/2(E)|/( r^3/2) so that E⊂⋃_T∈_T.





Given a large number R≫ K≫1 with K=R^2^-n, let r_1<⋯<r_n be scales such that r_n=R, r_k=r_k+1^1/2 for all k∈{1, ⋯, n-1}. 

We say a set E is regular with factors (_r_1,…,_r_n) if for every k∈{1, ⋯, n}, there exists 
 _r_k∈ [r_k^-1/2, 1] such that 

    
  * E is _r_k regular in an r_k-ball (or cube) B_r_k whenever B_r_k∩ E≠∅. 
    
  * |N_r_k-1(E)∩ Q_r_k|, for every dyadic r_k-cubes of Q_r_k intersecting with N_r_k-1(E),   
    is the same up to a constant multiple which may depend on k. Note that 
    it is equivalent to  
    
    |N_r_k-1(E)∩ Q_r_k|/|Q_r_k|∼|N_r_k-1(E)|/|N_r_k(E)|,

 since ∑_Q_r_k: Q_r_k∩ N_r_k-1(E)≠∅|N_r_k-1(E)∩ Q_r_k| = 
  |N_r_k-1(E)| and |N_r_k(E)|∼ |Q_r_k|#{Q_r_k: Q_r_k∩ N_r_k-1(E)≠∅}.







Let R≫ K≫ 1. Suppose F, a union of K-balls (or cubes) B's, is regular with factors (_r_1,…, _r_n) and M≥ 3n. Then for any function f∈ L^2,   we have

    ∑_B⊂ FSf_L^2__M(B)^2≲(log R)^5n∏_k=1^n_r_k^-1(|F|/R^2)Sf_2^2.

and

    Sf_L^2(F)^2≲∏_k=1^n_r_kSf_2^2.

In particular, the geometric mean of (<ref>) and (<ref>) yields that 

    ∑_B⊂ FSf_L^2__M(B)^2≲(log R)^5n(|F|/R^2)^1/2Sf_2^2.

Here ∑_B⊂ F denotes a summation over all K-balls (or cubes) contained within F. 








Although Proposition <ref> is stated with Sf being the smooth Fourier restriction of f on the R^-1-neighborhood of the unit circle, it is also valid when the unit circle is replaced by a C^2 curve with nonzero curvature.




Proposition <ref> enables us to derive the following refinement of (<ref>) in ℝ^2: for any L^2(ℝ^2) functions f_1, f_2, we have

    Bil(S)f_1_F_2Sf_2_F_2⪅(|F|/R^2)^1/2f_1_2f_2_2.

This improvement over (<ref>) is notable since it eliminates the requirement for a multilinear structure in the second term Sf_2_F_2 on the left side. Consequently, we can derive an L^5/3 estimate rather than an L^16/9 estimate. 
The method developed to prove Proposition <ref> may be of independent interest. 

Additionally, we anticipate further applications of the refined L^2 estimates and its analogs. For instance, at a cost of R^, we can reprove the reverse square function estimate in ℝ^2 by solely working in the L^2 space and using L^2 orthogonality. The transition from L^4 space to L^2 space is facilitated by Hölder's inequality and the examination of level sets. This approach circumvents the reliance on the algebraic property of the number 4, namely, 4=2× 2, which was pivotal in its original proof in <cit.>. Below, we will outline a sketch of its proof using refined L^2 estimates. It is our aspiration that this perspective will shed lights on L^p-problems in harmonic analysis.




    

By employing the standard broad-narrow argument, we identify a dyadic number ∈[R^-1/2, 1] along with the corresponding α-caps τ such that

    Sf_4^4⪅∑_τS_τ f_L^4_(B_R)^4.

Choosing K=(log R)^C, for each τ, via pigeonholing, we find a dyadic number λ and a set E_λ of unit balls such that for any unit ball
 B⊂ E_, we have S_τ f_L^4_(B)^4∼. Furthermore,

    S_τ f_L^4_(B_R)^4≲(log R)^CS_τ f_L^4_(E_)^4.

Dividing B_R into Rα× R-rectangles T with direction τ, we have

    S_τ f_L^4_(E_)^4=∑_TS_τ f_L^4_(E_∩ T)^4.

Inside each E_∩ T, by reverse Hölder's inequality, we get

    S_τ f_L^4_(E_∩ T)^4≲1/|E_∩ T|S_τ f_L^2_(E_∩ T)^4.

Applying a rescaled version of (<ref>) through parabolic rescaling,  at a cost of K^O(1)⪅1, we see that 

    S_τ f_L^2_(E_∩ T)^4⪅|E_∩ T|/|T|S_τ f_L^2(T)^4⪅|E_∩ T|/|T|(∫_T∑_θ⊂τ|S_θ f|^2)^2

by L^2-orthogonality. Utilizing Cauchy-Schwarz's inequality and combining the above two estimates yield

    S_τ f_L^4_(E_∩ T)^4⪅∫_T(∑_θ⊂τ|S_θ f|^2)^2.

Consequently, from (<ref>) and (<ref>), we obtain  

    Sf_4^4≲∑_τS_τ f_L^4_(B_R)^4⪅∑_τ∫(∑_θ⊂τ|S_θ f|^2)^2⪅∫(∑_θ|S_θ f|^2)^2,

which precisely corresponds to the reverse square function estimate.





Notations:  


    
  ∙ We denote A≲ B to signify that A≤ CB for some constant C, and A⪅ B to indicate that A≤ C_ R^ B for any >0. We define A∼ B if A≲ B and B≲ A. 
    
  ∙ In the paper, we consider a large number R≫1. Another large number, denoted as K, satisfies K∼exp(log R/loglog R)⪅1. The value of K remains fixed starting from equation (<ref>).
   
    
  ∙ θ (or θ') always represents an R^-1/2-cap.
    
  ∙B_r stands for a ball (or cube) in the plane with radius (or side length) r>0. 
    




Acknowledgment. The first author is supported by Simons collaboration grants. The second author thanks Shengwen Gan for some helpful conversation regarding the appendix.




§ CREATING LARGE REGULAR SUBSETS AND THE REFINED L^2 ESTIMATES



In this section, our attention is directed towards investigating the weighted L^2 estimate Sf_L^2(E), where E represents a subset of an R-ball (or cube), denoted as B_R. Our aim is to provide a demonstration for Proposition <ref>. The demonstration relies on the construction of sizable regular subsets based on a greedy algorithm.  






 §.§ Algorithms for discovering extensive regular subsets


Let us first state a lemma on finding significant regular subsets in a single scale.  We will use this lemma repeatedly to obtain large regular subsets in multi-scales in the next lemma. 



Suppose E⊂ B_r is a union of r^1/2-balls (or cubes). Then there exists a factor  and a union of r^1/2-balls (or cubes) E_⊂ E with |E_|≳ (log R)^-1|E| and such that E_ is  regular in B_r.



We present a greedy algorithm described as follows.  The algorithm is designed to finding a subset F of B_r consisting of r^1/2-balls and a union of r^1/2-balls E.  Once the algorithm stops, a subset E_κ of E is the large regular set as desired in the lemma. 



First, let us see how to make initial selections for F and E.    
We initiate F:= E.  Define 

    _F^* =max_ T  any r^1/2× r tube|F∩ T|/|T|.

It is easy to see that ^*_F ∈ [r^-1/2, 1].
Take κ_F ∈ [r^-1/2, 1] to be the maximal dyadic number κ∈ [r^-1/2, 1] 
with 2κ_F^*≥κ≥κ_F^*/4.  
We now choose an r^1/2× r-tube T that maximizes |F∩ T|.  
Then we see that 

    κ_F ∼  |F∩ T|/|T| .

For the chosen tube T, let _F(T) be the collection of r^1/2-balls (or cubes) such that Q⊂ F and Q∩ T≠∅. Define _T ⊂ F as

    _T = ⋃_Q⊂_F(T)Q.

Then |_T|/|T|∼κ_F since the chosen tube T maximizes |F∩ T|.
We now initiate E: = {_T}, which consists of some r^1/2-balls from the chosen tube T.   




Update F by removing the union of r^1/2-balls _T, i.e., set F := F ∖_T. Then repeat the 
previous process for the updated F to continue the updating procedure by setting E:=E∪_T, where T represents the chosen tube associate to the updated F. 
We terminate the algorithm until |F|≤ |E|/2.  
Since at each step we remove at least one r^1/2-ball from F, a termination will inevitably occur after a finite number of steps. 



When the algorithm stops, after grouping together those subsets of E generated by those F's with the same dyadic number κ_F=κ, 
we obtain a collection of disjoint sets {E_}_, where κ ranges over certain dyadic numbers in [r^-1/2, 1], to form a partition of the set E acquired at the end of the algorithm. Because |E|≥ |E|/2, we see that 

    ∑_κ| E_κ|≥  |E|/2 .

Additionally,  each E_ obeys E_=_T∈__T and |_T|/|T|∼ for any tube T∈𝕋_. Here 𝕋_κ is a collection of r^1/2× r-tubes.   Moreover, since E_ is contained in an F with κ_F=κ, we have that for any r^1/2× r-tube T, 

    |N_r^1/2(E_)∩ T|≲ |F∩ T|≲ |T|.



By pigeonholing, there exists a  such that |E_|≳(log R)^-1|E|. This yields #_≲|E_|/( r^3/2) since E_=_T∈__T is 
a disjoint union and |_T|∼|T| for any T∈_. Finally, because _T⊂ 2T, we can enlarge the tube set _ so that the estimate #_≲|E_|/( r^3/2) remains valid, and simultaneously, E_κ⊂⋃_T∈_T. Therefore, 
by Definition <ref>, we see that E_κ is κ-regular in B_r with 
|E_κ| ≳ (log R)^-1|E| as desired. 





Let R≫ K≫1 with K=R^2^-n. Let r_1<⋯<r_n be scales such that r_n=R, r_k=r_k+1^1/2 for all k∈{1, ⋯, n-1}.  Suppose that  E⊂ B_R is a union of K-balls (or cubes). Then there exists a collection of factors  (_r_1,…,_r_n) with each _r_k∈ [r_k^-1/2, 1] and a union of K-balls (or cubes) E'⊂ E with |E'|≳ (log R)^-3n|E| such that E' is regular with factors (_r_1,…,_r_n).


We create the set E' inductively from the smallest scale r_1=K^2 to the largest one r_n=R.
Let E_0=E and r_0=K. The algorithm starts at scale r_1, where Lemma <ref>
starts to kick in.  Let k∈{1,⋯, n}. At beginning of the k-th step of the algorithm, there is a union of r_k-1-balls E_k-1 such that E_k-1 is regular with factors (_r_1,…,_r_k-1) and |E_k-1|≳(log R)^-3(k-1)|E|. 


If k=n then the algorithm terminates.  Otherwise, we use dyadic cubes Q of dimensions r_k× r_k to partition the plane, and then consider each dyadic r_k-cube Q that E_k-1∩ Q≠∅. Apply Lemma <ref> to the pair (E_k-1∩ Q, Q) so that there exists a factor _Q and a union of r_k-1-balls (or cubes) E__Q⊂ E_k-1∩ Q such that 

    
  * |E__Q|≳(log R)^-1|E_k-1∩ Q|
    
  * E__Q is _Q regular in Q.

By dyadic pigeonholing, there exists a collection of dyadic r_k-cube  such that 

    
  * _Q are the same for all Q∈.
    
  * |N_r_k-1(E__Q)∩ Q| are the same for all Q∈ up to a constant multiple.
    (Hence (<ref>) holds).
    
  * |E_k-1∩⋃_Q∈Q|≳(log R)^-2|E_k-1|.

We define

    E_k:
        
        = ⋃_Q∈ E__Q

Then it is easy to see that |E_k|≳(log R)^-3|E_k-1|, which yields 
|E_k|≳(log R)^-3k|E|, through the inductive hypothesis. 
E_k is _B-regular in B_r_k whenever B_r_k∩ E_k≠∅. 
Hence E_k is regular with factors (_r_1,…,_r_k) and |E_k|≳(log R)^-3k|E|. 



When the algorithm stops, we obtain set E_n with |E_n|≳(log R)^-3n|E| such that E_n is regular with factors (_r_1,…,_r_n). Take E'=E_n and then we finish the proof. 







 §.§ Refined L^2 estimates for a κ-regular set


We now provide a quantitative characterization of the local L^2 norm of Sf_F where F is 
a -regular set. The demonstration hinges on a combinatorial approach, employing multiple applications of the pigeonhole principle. 


For a ball (or cube) B, we define the weight function w_B to be

    w_B(x) = 1/( 1+  dist(x, B)/r(B))^L ,

where r(B) is the radius of the ball B, dist(x, B) represents the distance between x and B, and 
L is sufficiently large constant. 
The weighted norm f_L^2(w(B)) is expressed as  

    f_L^2(w(B)) = ( ∫_ℝ^2|f|^2 w_B dx)^1/2 .

 Similarly the weighted broad norm Sf_L^2_ br_M(E) is defined by substituting the corresponding 
 L^2(E)-norm with the weighted norm L^2(w(E)) when E is a ball or a cube. 
 The weight function w_B is specifically crafted to address contributions from Schwartz tails. Essentially, it can be regarded as the indicator function _B, since the primary contribution arises from B.












If F, a union of r^1/2-balls (or cubes) {B}, is _r regular in B_r. Then for any function 
f∈ L^2, we have

    ∑_B⊂ FSf_L^2__M(B)^2≲(log R)^5(|F|/_r |B_r|)Sf_L^2__M-2(w(B_r))^2

and

    Sf_L^2(F)^2≲_rSf_L^2(w(B_r))^2.




Denote by d(τ) the diameter (length) of a cap τ and by 𝕋_r the collection of r^1/2× r tubes intersecting B_r.
By Definition <ref> and  L^2-orthogonality, we have

    Sf_L^2(F)^2   ≲∑_Q⊂ F∑_τ:d(τ)=r^-1/2∑_T∈_τ∫_Q |S_τ f|^2_T
       ≲_r∑_τ,d(τ)=r^-1/2∑_T∈_τ∫ |S_τ f|^2_T≤_rSf_L^2(w(B_r))^2,

which gives (<ref>). Here Q ranges over all dyadic r^1/2-cubes in F. 

 
Next, let us prove (<ref>). Since F is  regular, there is a collection __r of r^1/2× r-tubes  such that #__r≲|N_r^1/2(F)|/( r^3/2)∼|F|/( r^3/2) and 

    ∑_B⊂ FSf_L^2__M(B)^2≲∑_T∈__r∑_B⊂ F∩ TSf_L^2__M(B)^2.

By dyadic pigeonholing, there exists a __r'⊂__r, such that

    
  * ∑_B⊂ F∩ TSf_L^2__M(B)^2 are the same up to a constant multiple, for all T∈__r'.
    
  * We have
    
    ∑_T∈__r'∑_B⊂ F∩ TSf_L^2__M(B)^2≳(log R)^-1∑_T∈__r∑_B⊂ F∩ TSf_L^2__M(B)^2.




Fix an r^1/2× r tube T∈__r. By dyadic pigeonholing again, there exists a collection of r^1/2-balls (T) such that

     
  ∙ Sf_L^2__M(B)^2 are the same up to a constant multiple, for all B∈(T).
     
  ∙  We have
    
    ∑_B∈(T)Sf_L^2__M(B)^2≳(log R)^-1∑_B⊂ F∩ TSf_L^2__M(B)^2.


Denote by τ_T⊂^1 the directional cap associated to T, so |τ_T|∼ r^-1/2. Recall the definition of the broad norm in Definition <ref>. For each B⊂ T, let (B) be the collection of (log R)^-1-caps such that #(B)=M and for any ∈(B),

    Sf_L^2__M(B)^2≲S_ f_L^2(B)^2.

For each (log R)^-1-cap  and each r^-1/2-cap τ, define a collection of r^1/2× r-tubes 
_, τ as

    _,τ := {T: τ_T=τ⊂} ,

and
_,τ(B) as

    _,τ(B):={T⊂ B_r: T∩ 2B≠∅, T∈_,τ}.

Then by L^2-orthogonality, 

    S_ f_L^2(B)^2   ≲∑_τ⊂∑_T∈_,τ(B)S_τ f_T_L^2(2B)^2
       ≲ r^-1/2∑_τ⊂∑_T∈_,τ(B)S_τ f_T_2^2.




Denote by _T(B)⊂(B) the collection of (log R)^-1-caps σ obeying (θ_T,)≳ (log R)^-1. Then #_T(B)≥ M-2. Let us consider the pairs (,B)∈×(T). Note that

    ∑_B∈(T)∑_∈_{∈_T(B)}≥ (M-2)#(T).

Since #≲log R, we have

    (<ref>)=∑_∈∑_B∈(T)_{∈_T(B)}≲ (log R)#(T).

Therefore, since for any , #{B∈(T):∈_T(B)}≤#(T), there exists a set (T)⊂ such that 

    
  ∙ #(T)≥ M-2.
    
  ∙ For any ∈(T), 
    
    #{B∈(T):∈_T(B)}≳(log R)^-1#(T).


 Since Sf_L^2__M(B)^2 are the same up to a constant multiple for all B∈(T), for any ∈(T), from (<ref>), it follows that 

    ∑_B∈(T):∈_T(B)S_ f_L^2(B)^2≳(log R)^-2∑_B∈ F∩ TSf_L^2__M(B)^2.

Since (θ(T), )≳(log R)^-1,  we see that any tube T'∈𝕋_σ, τ
is transversal to the tube T. Thus, there are at most O(log R) many r^1/2-balls (or cubes) B belonging to 
ℬ(T)∩ T'.  Henceforth,  from (<ref>),  we get

    ∑_B∈(T):∈_T(B)S_ f_L^2(B)^2   ≲ r^-1/2∑_τ⊂∑_T'∈_,τ∑_B∈(T)∩ T'S_τ f_T'_L^2(B_r)^2
       ≲ (log R)r^-1/2S_ f_L^2(w(B_r))^2,

which yields that for any ∈(T),

    ∑_B∈ F∩ TSf_L^2__M(B)^2≲(log R)^3r^-1/2S_ f_L^2(w(B_r))^2.




Finally, consider the pairs (,T)∈×__r'. Since #(T)≥ M-2 for any T∈__r', similarly, by pigeonholing, there exists a set (B_r)⊂ so that 

    
  ∙ #(B_r)≥ M-2.
    
  ∙ For any ∈(B_r), 
    
    #{T∈__r':∈(T)}≳(log R)^-1#__r'.


 Since  ∑_B⊂ F∩ TSf_1_L^2__M(B)^2 are the same up to a constant multiple for all T∈__r', by (<ref>) and (<ref>), we end up with, for any ∈(B_r), 

    ∑_T∈__r∑_B∈ F∩ TSf_L^2__M(B)^2   ≲(log R)^2∑_T∈__r'
    ∈(T)∑_B∈ F∩ TSf_L^2__M(B)^2
       ≲(log R)^5r^-1/2(#__r)S_ f_L^2(w(B_r))^2.

Using  (<ref>) and #__r≲|N_r^1/2(E)|/( r^3/2),  we can finally conclude from the above estimate that for any ∈(B_r), 

    ∑_B⊂ FSf_L^2__M(B)^2≲(log R)^5(|F|/ r^2)S_ f_L^2(w(B_r))^2,

which yields that

    ∑_B⊂ FSf_L^2__M(B)^2≲(log R)^5(|F|/ |B_r|)S_ f_L^2__M-2(w(B_r))^2.

Lemma <ref> now follows from (<ref>) and (<ref>).








 §.§ Proof of Proposition <ref>



We will use Lemma <ref> at each scale, from the smallest one r_1 to the biggest one r_n. We can assume the weighted norm associated to w(B) in (<ref>) and 
(<ref>) to be the norm associated to B, because the weighted function w_B can be viewed essentially as _B.  

Denote N_r_k(F) by F_k for k∈{1, ⋯, n}. Since F is regular with factors (_r_1,…, _r_n), for each k we have

    |F_k-1∩ Q_r_k|/|Q_r_k|∼|F_k-1|/|F_k|.

for every dyadic r_k-cube Q_r_k with F_k-1∩ Q_r_k≠∅.


At the k-th scale, apply (<ref>) for each dyadic r_k-cube Q_r_k and the set F_k-1∩ Q_r_k to get 

    ∑_Q_r_k-1⊂ F_k-1∩ Q_r_kSf_L^2__M-2k+2(Q_r_k-1)^2
           ≲(log R)^5(|F_k-1∩ Q_r_k|/_r_k |Q_r_k|)Sf_L^2__M-2k(Q_r_k)^2
       ≲(log R)^5(|F_k-1|/_r_k |F_k|)Sf_L^2__M-2k(Q_r_k)^2.

Sum up all dyadic cubes Q_r_k's in F_k so that

    ∑_Q_r_k-1⊂ F_k-1Sf_L^2__M-2k+2(Q_r_k-1)^2
        ≲(log R)^5(|F_k-1|/_r_k |F_k|)∑_Q_r_k⊂ F_kSf_L^2__M-2k(Q_r_k)^2,

for every k∈{1,⋯, n}. 
 Multiplying these inequalities together results in obtaining (<ref>) as desired.


Similarly, apply (<ref>) for each Q_r_k to get

    Sf_L^2(F_k-1∩ Q_r_k)^2≲_r_kSf_L^2(Q_r_k)^2.

Sum up all Q_r_k in N_r_k(F) so that

    Sf_L^2(F_k-1)^2≲_r_kSf_L^2(F_k)^2,

for every k∈{1, ⋯, n}.
Consequently, (<ref>) follows by taking product of these inequalities. Therefore, we complete the proof. 














§ EXPLORING THE MAXIMAL OPERATOR T_*^: INITIAL INSIGHTS



We are interested in the maximal operator T_*^λ defined as in (<ref>). It is clearly connected with the Fourier multiplier 

    m_t^(ξ):= (2π)^-n(1-|ξ|^2/t^2)^_+ ,

for ξ∈ℝ^n. Going forward, we concentrate on the planar case with 
n=2,  while noting that many conclusions in this section remain applicable to higher-dimensional Euclidean spaces. 






 §.§ Reduction and linearization for the maximal operator


We perform a standard frequency decomposition to the Fourier multiplier m_t^ by localizing the frequency space via a smooth partitioning of unity.  In fact, 
let η_0 be a bump function of a small ball, say,  B_1/2, and η_k be a bump function of the annulus {ξ∈^2:1-2^-k≤ |ξ|≤ 1-2^-k-1} for k∈ℕ. 
Then ∑_k=0^∞η_k =1 on the unit ball. We now define 

    m_k, t^λ (ξ)= m_t^(ξ) η_k(ξ/t) ,

for any ξ∈ℝ^2. It is clear that m_t^λ=∑_k=0^∞ m^_t. Let
K_k, t^λ to denote the inverse Fourier transform of m_k, t^, i.e., 

    K_k, t^λ = m_k, t^ .

By the triangle inequality, to prove Theorem <ref>, it suffices to prove that for each k≥ 0, 

    sup_t>0|K_k,t^∗ f|_p≲_ 2^k 2^-kf_p

for any >0 and 5/3≤ p ≤ 2. To further localize the variable t, we need the following proposition proved by Tao <cit.>. A heuristic proof is presented in <cit.>, Section 4. For a detailed exposition, see also the appendix of <cit.>. 



Fix any 1≤ p≤ 2, k∈, and >0. Suppose that

    sup_t∈[1/2,1]|K_k,t^∗ f|_L^p(B_2^k)^p≲_ 2^k2^-kf_p^p.

for any >0 and any f∈ L^p. Then (<ref>) is true. 


For a fixed k, let R=2^k.  Observe that 

    K_k, t^∼  2^-k a( R(t-|ξ|)) ,

for some bump function on the annulus {ξ∈^2:t-R^-1≤ |ξ|≤ t+R^-1}. Consequently,  from the definition of S_R^* in (<ref>) and Proposition <ref>, we deduce that (<ref>) is equivalent to 

    S^∗_R f_L^p(B_R)⪅f_p,

for 5/3≤ p≤ 2, which precisely corresponds to (<ref>) stated in Theorem <ref>. 




Next, we are going to linearize the maximal function S_R^∗ f. Let {t_j}_j=1^[R] be a collection of R^-1-separated points in [1,2]. For 1≤ t_j≤ 2, recall that A_t_j, R is defined in Section <ref> as
the annulus with radius t_j and thickness R^-1 given as follows: 

    A_t_j, R ={ξ∈^2:t_j-R^-1≤ |ξ|≤ t_j+R^-1}.

As in Section <ref>, S_j (=S_t_j, R) is defined as 

    S_j f(ξ) = a( R(t_j-|ξ|) )  f(ξ) .
 





For any 1≤ p≤ 2, there exists a collection of disjoint sets {F_j}_j=1^[R], where each F_j⊂ B_R is a union of unit balls, such that

    S_R^∗ f_p⪅∑_jS_jf_F_j_p.



Recall that S_R^*f=sup_t∈ [1, 2]|S_t, Rf|.  
Consider S_t, Rf(x) as a function of (x,t) in ℝ^2×ℝ.  
Then its Fourier transform on ℝ^2×ℝ equals to 
e^-iτ|ξ| f(ξ) a(τ/R)_B_2(ξ), a function of (ξ,τ) that is essentially supported in the region B_2× [-R,R].  Thus, as a function of (x,t), |S_t, Rf| is essentially constant in B× I, where  B⊂ B_R is any unit ball and I⊂[1,2] is any R^-1 interval. Henceforth, 

    ∫sup_t∈ I|S_t, Rf|^p_B∼sup_t∈ I∫|S_t, Rf|^p_B⪅∫ |S_j f|^p_B ,

whenever t_j∈ I. Moreover, 

    ∫sup_t∈ [1,2]|S_t, Rf|^p_B∼sup_I⊂[1,2]∫sup_t∈ I|S_t, Rf|^p_B.


For each unit ball B, define j_B as the minimal choice of j∈{1, ⋯, [R]} such that 

    t_j∈ I  and sup_t∈ I∫|S_t, Rf|^p_B∼∫sup_t∈ [1,2]|S_t, Rf|^p_B.

Now define F_j as a union of unit balls B with j_B=j. This gives

    ∫sup_t∈ [1,2]|S_t, Rf|^p⪅∫∑_j|S_jf|^p_F_j ,

as desired.










 §.§ An L^2 estimate localized to a single ball


We aim to present a local L^2-estimate associated with a single ball. Before stating it, let us introduce some notations.  







We use Γ to denote the circle {ξ∈ℝ^2: |ξ|=1} and N:→^1 to represent its Gauss map. 
Recall that τ stands for a cap defined as in Definition <ref>. 
Let φ_τ be a bump function of the interval 

    {ξ_1∈ℝ: N(ξ_1, ξ_2)∈τ   for   (ξ_1, ξ_2)∈Γ} .

We define S_τ, j by 

    S_τ,jf(ξ) = S_j f(ξ)φ_τ(ξ_1) ,

for ξ=(ξ_1, ξ_2). The function φ_τ helps localizing smoothly the operator S_jf 
on the cap τ in the frequency space.  Thus, the Fourier transform of S_τ, jf is supported around 
a 1/R-neighborhood of the cap τ.  
The operator S_τ, j may vary from line to line depending on the change of the bump functions a and φ_τ, but such variation is harmless to our argument. 
For a ball (or cube) B in the plane, let ψ_B be a bump function on 2B so that 
ψ_B(x)∼ 1  for x∈ B and |ψ_B^(k)|≲_k (d(B))^-k for any k∈ℕ,
where d(B) stands for the diameter of B. 



We need to figure out what happens when the function f is localized on a single ball (or cube). 
This scenario can be addressed by the following lemmas,  
the first of which is essentially the local L^2 estimate proved in <cit.>.



Let B be an Rα-ball with 0≤α≤ 1. 
For any positive integer j and f_j∈ L^2,   

    ∑_jS_j(f_j)ψ_B_2^2⪅∑_j|f_j|_2^2.



 
We will utilize the TT^*-method to prove Lemma <ref>. 
Via the method of stationary phase, the kernel of the operator S_j, denoted by K_j, exhibits the following asymptotic behavior (see for instance <cit.> Chapter IX)

    K_j (x)≈φ(R^-1x)R^-1|x|^-1/2(e^it_j|x|∑_j=0^∞a_j|x|^-j+e^-it_j|x|∑_j=0^∞ b_j|x|^-j),

where φ is a smooth function, a_j and b_j are constants depending on j.  The principal contribution arises from 
the first term when j=0. Henceforth, we only need to prove that 

    ∑_jS'_j(f_j)ψ_B_2^2⪅∑_j|f_j|_2^2,

where S'_j f= K'_j*f with 

    K'_j(x) = φ(R^-1x)R^-1|x|^-1/2 e^it_j|x| .




For a dyadic number 1≤ r≲ R, let η_r(x) be a bump function of the dyadic annulus {x: |x|∼ r}, and let η_0 be a bump function of the unit ball so that 

    K'_j=η_0 K'_j+∑_rη_r K'_j.

For each r, define 

    K_j,r(x):=η_r(x) K_j'(x)  ,

which is clearly ∼ R^-1r^-1/2η_r(x)e^it_j|x|. Moreover, we define S_j,rf:=K_j,r∗ f. 





Since K_j,r is supported in  x:|x|∼ r, by a standard localization argument (see, for example, <cit.> Lemma 2.2), it suffices to prove that for any r-ball B_r,

    ∑_jS_j,r(f_j_B_r)ψ_B_2^2≲∑_j|f_j|_B_r_2^2.


Without loss of generality,  we can confine the 
 variable x in (<ref>) to the conic direction  {x:0≤ x/|x|≤π/4}.
 For any y_1∈[-R,R], we define g_j(·)=|f_j(y_1,·)|_B_r(y_1,·). Hence, by freezing the first variable y_1 and Cauchy-Schwarz, it suffices to prove that

    ∑_jS_τ, j(g_j)ψ_B_2^2≲ r^-1∑_jg_j_2^2.

Apply (<ref>) so that the left-hand side of (<ref>)) equals to 

    ∑_j_1,j_2∫_^2g_j_1(z_1)g_j_2(z_2)K_τ,j_1,j_2(z_1,z_2)dz_1dz_2 ,

where K_τ,j_1,j_2(z_1,z_2) is defined by

    K_τ,j_1,j_2(z_1,z_2):= 1/rR^2∫ e^i(t_j_1|x-z_1|-t_j_2|x-z_2|)ψ_B^2(x)a_τ,j_1,j_2(x,z_1,z_2)dx .


Since ∇_x(t_j_1|x-z_1|-t_j_2|x-z_2|)≳ r^-1|z_1-z_2| due to some elementary geometry,  from the integration by parts, we end up with that

    |K_τ,j_1,j_2(z_1,z_2)|   ≲_N 1/rR^2min{|B|,r^2}(1+r^-1|B|^1/2|z_1-z_2|)^-N
       =min{^2/r,r/R^2}(1+(R/r)|z_1-z_2|)^-N.

This yields ∫|K_τ,j_1,j_2(z_1,z_2)|dz_1, ∫ |K_τ,j_1,j_2(z_1,z_2)|dz_2≲min{/R,1/ R^3}, which concludes (<ref>) by Schur's test since r≤ R. This establishes (<ref>) and consequently (<ref>), as desired. 





Let ∈[R^-1/2,1]. Suppose that B is an R-ball and {τ} is a collection of finitely overlapping -caps. Additionally, assume that {E_τ,j}_τ,j is a collection of disjoint sets
in the plane. Then

    ∑_τ∑_jS_τ, j(f_ B)_E_τ,j_2^2⪅f_B_2^2.




 By duality, it suffices to show that, for any f∈ L^2, 

    ∑_τ∑_jS_τ, j(f_E_τ,j)_L^2(B)^2⪅∑_τ∑_jf_E_τ,j_2^2.

Since ≥ R^-1/2, by L^2 orthogonality, 

    ∑_τ∑_jS_τ, j(f_E_τ,j)_L^2(B)^2≲∑_τ∑_jS_τ, j(f_E_τ,j)_L^2(B)^2.

Hence, to prove (<ref>), we only need to prove for a single τ,

    ∑_jS_τ, j(f_E_j)_L^2(B)^2⪅∑_jf_E_j_2^2,

where {E_j}_j is a collection of disjoint sets. It is more convenient to replace _B by a smooth cut-off ψ_B, 
so we will demonstrate the following slightly stronger inequality: 

    ∑_jS_τ, j(f_j)ψ_B_2^2⪅∑_jf_j_2^2,

where f_j∈ L^2 and  the supports of f_j's are disjoint.   



Observe that 

    S_τ,jf=S_j( S_τ,j f),

where S_τ, jf can be represented as K_τ, j*f such that 
the Fourier transform of the kernel K_τ, j is a smooth bump function supported in an 2α-ball containing the cap 
τ. Thus, the kernel K_τ, j obeys the decaying estimate,

    | K_τ,j(x)|≲_N ^2(1+ |x|)^-N,

which yields that sup_τ, j| K_τ, j| ∈ L^1.



Plugging f_j= S_τ,j (f_E_j) in (<ref>), we have

    ∑_jS_τ, j(f_E_j)ψ_B_2^2⪅∑_j| S_τ,j (f_E_j)|_2^2.

Thus, to prove (<ref>), we only need to show

    ∑_j| S_τ,j (f_j)|_2^2≲∑_jf_j_2^2 ,

for f_j's with disjoint supports.  
In fact, because sup_τ,j| K_τ,j| is integrable via (<ref>), we obatin 

    ∑_j| S_τ,j (f_j)|_2^2   ≲(sup_τ,j| K_τ,j|)∗∑_j|f_j|_2^2
       ≲∑_j|f_j|_2^2=∑_jf_j_2^2.

In the last equality, we use the fact that the supports of f_j's are disjoint. This gives (<ref>), and therefore
we complete the proof of Lemma <ref>. 





Another approach to prove (<ref>) is as follows: We first derive the explicit expression of the kernel of S_τ,j using the method of stationary phase. Then, we conclude (<ref>) via a similar idea in <cit.>. However, the explicit expression of the kernel of S_τ,j is not straightforward since it depends on the bump function 
φ_τ. We anticipate that proving (<ref>) by using this method would require more effort.









 §.§ L^4 orthogonality


In this subsection, we examine the orthogonality in L^4 concerning functions whose Fourier transforms are distributed around a neighborhood of the circle {ξ∈ℝ^2: |ξ|=1}. Recall that {t_1, t_2, ⋯, t_[R]} is a collection of 1/R-separated real numbers in [1, 2].  For given j∈ [1, R]∩ℤ and a cap τ,  let f_τ, j be a function whose Fourier transform is supported in  
C_τ∩ A_t_j, R, where C_τ is the conic region of the cap τ defined as in  (<ref>)
and A_t_j, R is the annulus given by (<ref>).  We use 𝒥_τ to denote a subset of 
[1, R]∩ℤ which may depend on τ. For any interval I in [1/2, 2] and any cap τ, we define

    _τ, I={j∈_τ:t_j∈ I} .







Let < be two numbers in [R^-1/2,1], and τ_0,τ_0' be two -caps with 
dist(τ_0, τ'_0) ⪅β.
Given an R-ball B, let _B,_B' be two collections of finitely overlapping -caps with #_B, #_B'≲μ_1 such that τ⊂τ_0, τ'⊂τ_0' for any τ∈_B, τ'∈_B'. Let ℐ= {I} be a finitely overlapping ^-1-intervals in [1/2,2].  Suppose also that there is a number μ_2 such that for each τ∈_B∪_B', #{I∈ℐ:_τ,I≠∅}∼μ_2. Then


    ∫_B|∑_τ∈_B∑_j∈_τf_τ,j|^2|∑_τ'∈_B'∑_j'∈_τ'f_τ',j'|^2
    ⪅    μ_1μ_2∑_τ∈_B∑_τ'∈_B'∑_I,I'∈ℐ∫_2B|∑_j∈_τ,If_τ,j|^2|∑_j'∈_τ',I'f_τ',j'|^2.




Let us define F_τ,I as

    F_τ,I:=∑_j∈_τ,If_τ,j.

Henceforth, it suffices to show that 

    ∫_B|∑_τ∈_B∑_IF_τ,I|^2|∑_τ'∈_B'∑_I'F_τ',I'|^2⪅μ_1μ_2∑_τ,τ'∑_I,I'∫_2B|F_τ,I|^2|F_τ',I'|^2.

Observe that  F_τ, I has its Fourier support in C_τ∩⋃_j∈ I A_t_j, R,  which is a subset 
of an ^-1×-rectangle, denoted by R_τ,I.




Let ψ_B be a Schwartz function such that its Fourier transform is essentially supported on an (R)^-1-ball centered at the origin and 1≤ψ_B(x) for any x∈ B. Then the left hand side of 
(<ref>) obeys 

    L.H.S.  (<ref>)   ≲∫|∑_τ∈_B∑_IF_τ,Iψ_B|^2|∑_τ'∈_B'∑_I'F_τ',I'ψ_B|^2
    ≲∑_τ_1,τ_2,τ_1',τ_2'∑_I_1,I_2,I_1',I_2'∫    ( F_τ_1, I_1∗ψ_B) ∗( F_τ_1', I_1'∗ψ_B)·( F_τ_2, I_2∗ψ_B)∗( F_τ_2', I_2'∗ψ_B),

where the sum is taken over all (τ_1, I_1), (τ_1', I_1'), (τ_2, I_2), (τ_2', I_2') such that

    (R_τ_1,I_1+R_τ_1',I_1')∩ (R_τ_2,I_2+R_τ_2',I_2')≠∅.

Thus, by Cauchy-Schwarz's inequality,

    (<ref>)≲   ∑_τ_1,I_1,τ_1', I_1'( ∫|( F_τ_1, I_1∗ψ_B) ∗( F_τ_1', I_1'∗ψ_B)|^2 )^1/2
       ∑_τ_2,I_2,τ_2', I_2'( ∫|( F_τ_2, I_2∗ψ_B) ∗( F_τ_2', I_2'∗ψ_B)|^2)^1/2,

where the sum of (τ_2,I_2,τ_2', I_2') is taken under (<ref>). 
Because each τ and τ' lie in a O(R^β)-conic region for any >0,
those rectangles R_τ_1, I_1, R_τ_1', I'_1,R_τ_2, I_2, R_τ_2', I'_2 can be essentially viewed as rectangles pointing to one direction. Hence, we see that  for a fixed quadruple (τ_1,τ_1',I_1,I_1'), there are ⪅μ_1μ_2 quadruples (τ_2,τ_2',I_2,I_2') such that (<ref>) holds.  Therefore, we end up with

    L.H.S.(<ref>)   ⪅μ_1μ_2∑_τ,I,τ', I'∫|( F_τ, I∗ψ_B) ∗( F_τ', I'∗ψ_B)|^2 
       ⪅μ_1μ_2∑_τ,τ'∑_I,I'∫_2B|F_τ,I|^2|F_τ',I'|^2 ,

as desired.


The foundation of the other lemma relies on a straightforward geometric insight as stated in <cit.>. An analogous outcome can also be discovered in Section 10 of <cit.>.



Given ∈[R^-1/2,1], let τ, τ' be two -caps and j, j'∈ [1, R]∩ℤ. Then

    ∫|f_τ,j|^2 | f_τ',j'|^2 dx ≲∑_θ⊂τ∑_θ'⊂τ'∫|f_θ,j|^2 | f_θ',j'|^2 dx.

Here θ and θ' are R^-1/2-caps forming partitions for τ and τ', respectively.



We represent 

    f_τ, j=∑_θf_θ, j     and    f_τ', j'=∑_θ'f_θ', j' .

By the Plancherel theorem, we have 

    ∫|f_τ,j|^2 | f_τ',j'|^2dx =  
     ∑_θ_1, θ_1', θ_2, θ_2'∫f_θ_1, j*f_θ_1', j'f_θ_2, j*
      f_θ'_2, j' ,

 where θ_1, θ_2⊂τ and θ'_1, θ_2'⊂τ' are R^-1/2-caps.
For k∈{1, 2}, the function f_θ_k, j*f_θ_k', j' is supported in 

    (f_θ_k, j) + (f_θ_k', j') .

Note that f_θ, j has its Fourier transform supported in C_τ∩ A_t_j, R, which is essentially    a rectangle of dimensions R^-1× R^-1/2. 
The key geometric observation is that the sets  (f_θ_k, j) + (f_θ_k', j') are finitely overlapped for all θ_k⊂τ and all θ_k'⊂τ'.
Consequently, combining this geometric observation with (<ref>), we derive that

    ∫|f_τ,j|^2 | f_τ',j'|^2dx≲∫sup_θ, θ'|f_θ, j*f_θ', j'|^2 ,

which is bounded above by 

    ∑_θ, θ'∫|f_θ, j*f_θ', j'|^2
    =  ∑_θ, θ'∫  |f_θ,j|^2 | f_θ',j'|^2 dx ,
 
as desired.  








§ ANALYZING THE MODEL OPERATOR: A THOROUGH INVESTIGATION



Lemma <ref> clarifies that our focus can be narrowed down to examining the model operator defined as

    ∑_jS_jf_F_j,

with S_j=S_t_j, R, {F_j} comprising disjoint subsets in the plane, and F_j⊂ B_R forming unions of unit balls. To validate Theorem <ref>, it suffices to demonstrate that


For p=5/3 and any f∈ L^p(^2), we have 

    ∑_jS_jf_F_j_p⪅f_p.



Theorem <ref> can be derived from Theorem <ref> via Proposition <ref>. Consequently, our focus now shifts to establishing Theorem <ref>.



We initiate the proof of Theorem <ref> by employing a broad-narrow reduction technique on our model operator (<ref>), reminiscent of the broad-narrow analysis method used in the work 
of Bourgain and Guth  in <cit.> and Guth in <cit.>.






 §.§ The broad-narrow analysis



For any ρ-cap τ, we define τ^* to be the 2(log R) ρ-cap that is concentric with τ.




Fix j. For each x∈ F_j, either one of the following statements is true.

    
  ∙ x is -broad for some dyadic number R^-1/2<≤ 1. This means that there exist an -cap τ and 
    at least 10loglog R many α-caps τ' contained in τ^* such that 
   
    
    |S_τ',jf(x)|⪆|S_τ,jf(x)|⪆ |S_jf(x)|.

    
    
  ∙ x is narrow. This means that there exists an R^-1/2-cap θ so that
    
    |S_θ,jf(x)|⪆|S_jf(x)|.




Without loss of generality, we can assume that log R is a dyadic number.
Let us  introduce ≲log R/loglog R many scales ρ_0,ρ_1 ⋯,ρ_n, where ρ_k=(log R)^-k for k∈{0, ⋯, n-1} and ρ_n≈ R^-1/2.
For each ρ_k, partition the unit circle into disjoint ρ_k-caps {_k}. 
We establish the lemma by running the following algorithm. 



First we define S__0,jf=S_j f. The algorithm starts from the first scale ρ_1, which we designate as the first step. In the k-th step, there is a function S__k-1,jf from the (k-1)-th step whose Fourier transform is  supported in A_t_j, R∩ C__k-1.  We partition the multiplier operator S__k-1,j using ρ_k-caps {_k} so that 

    S__k-1,jf=∑__k⊂_k-1S__k,jf.

If there exists a ρ_k-cap _k and at least 10loglog R many ρ_k-caps ω'_k's contained in _k^* such that 


    |S__k',jf(x)|⪆|S__k,jf(x)|≥ (10log R)^-1 |S__k-1,jf(x)|,

then we terminate the algorithm by setting  =ρ_k, τ'=_k', and τ=_k.




Otherwise, by the triangle inequality and pigeonholing, there exists a ρ_k-cap _k satisfying

    |S__k,jf(x)|≳ (loglog R)^-1|S__k-1,jf(x)|.

We then proceed with the algorithm to the (k+1)-step.



If the algorithm halts at some k<n, then (<ref>) satisfies the first part of the lemma, as k<n≲log R/loglog R and due to (<ref>),
confirming the α-broadness of x. 
Otherwise, for a similar rationale, there exists an R^-1/2-cap θ such that

    |S_θ, jf(x)|⪆|S_jf(x)|,

which fulfills the second part of the lemma and hence indicates the narrowness.   








 §.§ The preliminary reduction for the model operator


Using Lemma <ref>, we partition F_j as 

    F_j=(⋃_ E_, j)∪ E_R^-1/2, j,

where E_, j is a collection of the -broad points in F_j, and E_R^-1/2,j is a collection of the narrow points in F_j. By pigeonholing, there exists an ∈ [R^-1/2, 1] so that


    ∑_jS_jf_F_j_p^p⪅∑_jS_jf_ E_, j_p^p⪅∑_jS_τ(x), jf_ E_, j_p^p,

where τ(x) is an α-cap depending on the variable x. 
Hence there exists a collection of disjoint sets  {F_τ,j}_τ,j with F_τ,j⊂  E_, j so that

    ∑_jS_τ(x), jf_E_, j_p^p=∑_τ∑_jS_τ, jf_F_τ,j_p^p=∑_τ∑_jS_τ, jf_F_τ,j_p^p,

where τ ranges over all -caps. 
For each α-cap τ, we denote the normal vector of the cap τ at its center by e(τ).
Divide B_R into parallel tubes of dimensions Rα× R, oriented in the direction of e(τ). We denote the collection of these tubes as 𝕋_τ  
and let :=⋃_τ_τ where τ ranges over all -caps. 
For each T∈_τ, let T_j:=F_τ,j∩ T.  It is important to notice that {T_j}_T, j are disjoint. 
Since S_τ, jf is supported in a unit ball in the frequency space, T_j can be chosen as a collection of unit balls in the physical space For dyadic numbers ∈[1, R^2] and ν∈[1,R], let _,ν⊂ denote
the collection of tubes T in  satisfying 
#{j: T_j≠∅}∼ν and |T_j|∼λ for each j∈{j: T_j≠∅}.  
Since {T_j}_j are disjoint,  it is easy to see that 

    ν≤ R^2 .





 
 

By pigeonholing, there exist ,ν and a set _T defined for each T∈_,ν so that |_T|∼ν and

    ∑_τ∑_jS_τ, jf_F_τ,j_p^p⪅∑_τ∑_T∈_τ∩_,ν∑_j∈_TS_τ, jf_T_j_p^p.

We use _τ, λ, ν to denote _τ∩_,ν.  This finishes our broad-narrow reduction. Henceforth,  Theorem <ref> boils down to the estimate

    ∑_τ∑_T∈_τ, λ, ν∑_j∈_TS_τ, jf_T_j_p^p⪅f_p^p

for p=5/3. 



Let _λ, ν= ⋃_τ𝕋_τ, λ, ν. We observe that 
the collection _λ, ν of R× R tubes  obeys certain non-concentration properties. To elucidate this, we utilize the language associated with Wolff's axiom.




Let ρ_1≤ρ_2 be two positive numbers. We say  , a collection of, ρ_1×ρ_2 tubes obeys Wolff's axiom with factor M, if any ρ×ρ_2-tube, where ρ_1≤ρ≤ρ_2, contains ≤ M (ρ/ρ_1) tubes in .






_λ, ν is a collection of R× R tubes that obeys Wolff's axiom with a factor ≲ R^2/(ν).    


Assume r∈[R, R]. Let T̅ be an arbitrary r× R tube, and define  (T̅):={T∈: T⊂T̅}. It suffices to prove that #(T̅)≲R^2/νr/R. 

Notice that 

    T̅⊃⋃_T∈(T̅)⋃_j∈_TT_j

since T_j's are disjoint when j∈_T.   Thus, we see that 

    rR∼|T̅|≥∑_T∈(T̅)∑_j∈_T|T_j|.

Since |T_j|∼, #(T)∼ν, we obtain 

    rR≳#(T̅)ν,

which is exactly what we want.







 §.§ Refinement of the reduction process for the model operator


We now 
make a further reduction on the left-hand side of (<ref>). By dyadic pigeonholings, 
for each -cap τ, 
there exist a set _τ, , ν'⊂_τ, , ν and a union of unit balls T_j for each T∈_τ' and each j∈_T such that

    
  * T_j⊂ T_j, and |T_j| are the same up to a constant multiple for all T∈⋃_τ_τ' and each j∈_T.
    
  * |S_τ, jf(x)_T_j'(x)| are the same up to a constant multiple for all x∈T_j.
    
  * We have
    
    ∑_τ∑_T∈_τ, , ν'∑_j∈_TS_τ, jf_T_j'_p^p⪆∑_τ∑_T∈_τ, , ν∑_j∈_TS_τ, jf_T_j_p^p.



To simplify notation, we continue to denote _τ, , ν' by _τ, , ν, and represent T_j as T_j. Therefore, we have |T_j| ≲λ for all T ∈_τ, =⋃_τ_τ, , ν and j ∈𝒥_T.





Now, we shift our focus to a particular T_j and S_τ, jf_T_j_p^p. Through a linear transformation, let's assume, without loss of generality, that τ represents the cap {e∈𝕊^1: dist(e,e_2)≤α}, where e_2=(0,1) denotes the vertical unit vector. 
Under this assumption,  the Fourier transform of S_τ, jf is essentially contained in the set 

    {(ξ_1,ξ_2):|ξ_2-(1-ξ_1^2)^1/2|≤ R^-1 and |ξ_1|≲}.

Define  to be the parabolic rescaling (depending on T)

    (x_1,x_2)=((log R) x_1, (log R)^2 x_2).

Then, for some C^∞ function ϕ with |ϕ”|∼1, the Fourier transform of S_τ, jf∘ is contained in the set

    {(ξ_1,ξ_2):|ξ_2-ϕ(ξ_1)|≲ (log R)^-2R^-1 and |ξ_1|≲1}.


Fix K with K∼exp(log R/loglog R)⪅1 such that K^n=R(log R)^2 for some n≤loglog R. In particular, (log R)^O(n)⪅1. Let us define E_1:=(T_j). Then (T_j) is contained in an R(log R)^2-ball. Since |S_τ, jf(x)_T_j(x)| are the same up to a constant multiple, |(S_τ, jf_T_j)∘(x)| are the same up to a constant multiple for all x∈ E_1. By dyadic pigeonholing, there exists a set E_2⊂ E_1 such that

    
  * |E_2|⪆|E_1|
    
  * |B∩ E_2| are the same up to a constant multiple for all K-balls B⊂(T), if B∩ E_2≠∅.

Let E_3:=N_K(E_2). By Lemma <ref>, there exists a union of K-balls E_4⊂ E_3 with |E_4|⪆|E_3| such that E_4 is regular with some factors (_r_1,…,_r_n), where r_1=K, r_n=R^2. Denote by E_2'=E_2∩ E_4. Since |B∩ E_2| are the same up to a constant multiple for all K-balls B⊂(T) if B∩ E_2≠∅ and since |E_4|⪆|E_3|, we have |E_2'|⪆|E_2| and hence |E_2'|⪆|E_1|.

Let T_j':=^-1(E_2'). Since |E_2'|⪆|E_1|, 

    S_τ, jf_T_j'_p^p⪆S_τ, jf_T_j_p^p.

To ease notation, let us still use T_j to denote T_j'. 



Refer back to equations (<ref>) and (<ref>). Therefore, after another dyadic pigeonholing on T_j about the factors (_r_1,…,_r_n), demonstrating Theorem <ref> boils down to showing 


Let p=5/3 and let α∈ [R^-1/2, 1], ∈ [1, α R^2] and ν∈ [1, R] be dyadic numbers obeying (<ref>). Suppose _, ν=⋃_τ_τ, , ν, where 
τ ranges over all α-caps, 
is a collection of R× R tubes
such that 


   
  * for each T∈_, ν, there is a set _T with |_T|∼ν;
    
  * for each T∈_, ν and each j∈_T,  there is a subset T_j of T such that  
      T_j is a union of unit balls and |T_j|≲.    Moreover, for all T ∈_, ν and j∈_T, 
       T_j's are pairwise disjoint and  |T_j|'s are the same up to a constant multiple;
  
  * if α>R^-1/2, then any x∈ T_j is -broad whenever T∈_, ν and j∈_T;
    
  * |S_τ, jf(x)_T_j(x)| are the same up to a constant multiple for all x∈ T_j;
    
  * N_K((T_j)) is regular with some uniform factors (_r_1,…,_r_n) for each T∈_,ν and j∈_T, where r_1=K, r_n=R^2.

Then we have

    ∑_τ∑_T∈_τ, , ν∑_j∈_TS_τ, jf_T_j_p^p⪅f_p^p.










§  TECHNICAL ESTIMATES AND WRAPPING UP THE PROOF 




In this section, we provide a proof of Proposition <ref>.  We'll derive certain L^2-estimates and L^4/3-estimates, followed by employing interpolation to complete the proof.



Before stating the first L^2 estimate, we recall that 
the parabolic rescaling mapping ℒ defined in (<ref>) transforms the image of a R× R tube T∈, denoted as (T),
into a rectangle of dimensions R(αlog R)^2× Rα^2log R, which is contained in 
an R(log R)^2-ball. 





Let r≤ R(log R). For a given R× R tube T, we call an r× r(log R)^-1 tube Q an r-pseudo ball, if (Q)⊂(T) is an r(log R)-ball. 


Initially, we establish a basic yet useful lemma regarding the r-pseudo ball.


Let T∈𝕋_, ν and T be an R^1/2× R tube contained in T.
Then 

    #{Q:Q∩ T∩ T_j≠∅}≲_n (log R)R^1/2,

where Q denotes an R^1/2-pseudo ball and T_j is given as in Proposition <ref>.



Observe that ( T) becomes an R^1/2(log R)× R(log R)^2 tube. 
 Since N_K((T_j)) is regular with factors (_r_1,…,_r_n), we have

    #{(Q):(Q)∩( T)∩(T_j)≠∅}≲_n (log R)R^1/2,

where (Q), an R^1/2(log R)-ball, is the image set of an R^1/2-pseudo ball Q. 
Clearly, (<ref>) follows from (<ref>). 









 §.§  An application of the refined L^2 estimates for the regular set


The subsequent lemma emerges as a corollary of Proposition <ref>, presenting refined L^2-estimates for the regular set.


With the same assumption as Proposition <ref>, we have

    S_τ, jf_T_j_2^2⪅∏_k=1^n_r_k^-1(/R^2)S_τ^*, jf_2^2,

and

    S_τ, j(f_T_j)_2^2⪅∏_k=1^n_r_kf_T_j_2^2.

In addition, if Q⊂ T is an R^1/2-pseudo ball, then

    S_τ, j(f_T_j_Q)_2^2⪅ (R^-1/2^-1)∏_k=1^n-1_r_kf_T_j_Q_2^2.



Define E_5:= (T_j). By dyadic pigeonholing, there exists a set E_6⊂ E_5 such that

    
  * |E_6|⪆|E_5|=| (T_j)|.
    
  * |B∩ E_6| are the same up to a constant multiple for all dyadic unit cubes B⊂(T), if B∩ E_6≠∅.
     
  * S_τ, jf_T_j_2 ⪅ S_τ,jf_T_j_ℒ^-1(E_6) .

Let T_j':=^-1(E_6). Thus, we have

    S_τ, jf_T_j_2^2⪅S_τ, jf_T_j'_2^2,

and that for any dyadic unit cube B with B∩ E_6≠∅,

    |B∩ E_6|/|B|∼|E_6|/|N_1(E_6)|.


Let B be a dyadic unit cube with B∩ E_6≠∅. Since any x∈ T_j' is a broad point, there exists a set of finitely overlapping -caps {} with #{}≥ 10loglog R such that

    |S_, jf_T_j'∘ |⪆|S_τ, jf_T_j'∘|

in E_6∩ B for any . Since S_τ, jf_T_j'∘, S_, jf_T_j'∘ are essentially constant on a unit cube (or ball), we have any ,

    S_, jf∘_L^2(B)^2⪆S_τ, jf∘_L^2(B)^2∼|N_1(E_6)|/|E_6|S_τ, jf∘_L^2(E_6∩ B)^2.

 Thus, by taking M=10loglog R in the broad norm as given in Definition <ref>, 

    S_τ, jf∘_L^2(B)^2⪅S_τ^*, jf∘_L^2__M(B)^2.

Applying (<ref>) and (<ref>) and  summing up all dyadic  unit cubes B with B∩ E_6≠∅, we get 

    S_τ, jf_T_j'∘_2^2⪅|E_6|/|N_1(E_6)|∑_B,B∩ E_6≠∅S_τ^*, jf∘_L^2__M(B)^2.


Let F:=N_K(E_6), which is also a union of K-balls {B_K}. Then 

    ∑_B,B∩ E_6≠∅S_τ^*, jf∘_L^2__M(B)^2≲ K^2∑_B_K⊂ FS_τ^*, jf∘_L^2__M(B_K)^2.

Apply (<ref>) in Proposition <ref> (see also Remark <ref>) so that

    ∑_B_K⊂ FS_τ^*, jf∘_L^2__M(B_K)^2⪅∏_k=1^n_r_k^-1(|F|/R^2(log R)^4)S_τ^*, jf∘_2^2.

Note that |F|≲ K^2|N_1(E_6)|⪅|N_1(E_6)|. From (<ref>), (<ref>), (<ref>) and (<ref>), it follows that 

    S_τ, jf_T_j∘_2^2⪅S_τ, jf_T_j'∘_2^2⪅∏_k=1^n_r_k^-1(|E_6|/R^2(log R)^4)S_τ^*, jf∘_2^2.

Recall that |E_5|≥ |E_6|⪆|E_5|=| (T_j)|=(log R)^3|T_j| and |T_j|≲. Plug these to the above estimate and rescale to obtain (<ref>).


The proof for (<ref>) and (<ref>) essentially follows the same logic. Here, we'll solely focus on verifying (<ref>). Employing duality, it's enough to establish, for an L^2 function g:

    S_τ, jg_L^2(T_j∩ Q)^2⪅ (R^-1/2^-1)∏_k=1^n-1_r_kS_τ,jg_2^2.

Since ℒ(T_j) is regular with factors (κ_r_1,…,κ_r_n), it follows that (T_j∩ Q) is regular with factors (κ_r_1,…,κ_r_n-1, R^-1/2(αlog R)^-1). By applying (<ref>) from Proposition <ref>, we obtain (<ref>), as desired. 







We have the following L^2 estimate

    ∑_τ∑_T∈_τ, , ν∑_j∈_TS_τ, jf_T_j_2^2
        ⪅∏_k=1^n_r_k^-1(/R^2)f_2^2.





Note that {τ^*}_τ are log R-overlapping. 
(<ref>) follows from summing up (<ref>) for all possible j, T, τ and then employing Plancherel's theorem. 







 §.§ L^2 estimates via the local L^2 conclusion and Kakeya-type results




We need one more quantitative version of the L^2 estimates. To achieve that, let us make further reductions on the left-hand side of 
(<ref>).  





Partition B_R into R-cubes B's.  By pigeonholing, we can find a collection of R-cubes, denoted by ,  and a subset '_τ, , ν⊂_τ, , ν so that 

    
  * f_B_p, for all B∈,   are the same up to a constant multiple;
    
  * for all T∈'_τ, , ν,
    
    ∑_j∈_TS_τ, j(∑_B∈f_B)_T_j_p^p

    are the same up to a constant multiple;
    
  * we have
    
    ∑_τ∑_T∈_τ, , ν∑_j∈_TS_τ, jf_T_j_p^p⪅∑_τ∑_T∈'_τ, , ν∑_j∈_TS_τ, j(∑_B∈f_B)_T_j_p^p.


To simplify notation, we continue to use _τ, λ, ν to represent '_τ, λ, ν and _λ, ν to represent ⋃_τ'_τ, λ, ν.







Let _⊂ be the collection of R-cubes such that 2B intersects ∼ R× R-tubes in _λ, ν for each B∈_. By dyadic pigeonholings, we can find a dyadic number  and a collection of tubes _⊂_,ν so that 

    
  * |_|⪆|_, ν|.
    
  * For each T∈_, we have
    
    ∑_j∈_TS_τ, j(∑_B∈f_B)_T_j_p^p⪅∑_j∈_TS_τ, j(∑_B∈_f_B)_T_j_p^p.

    
  * For each T∈_, #{B∈_: B⊂ T} are the same up to a constant multiple.

Denote _τ,=_∩_τ, , ν. As a result, we have

    ∑_T∈_,ν∑_j∈_TS_τ, j(∑_B∈f_B)_T_j_p^p    ⪅∑_τ∑_T∈_τ,∑_j∈_TS_τ, j(∑_B∈_f_B)_T_j_p^p.




Our second proposition in this section is the following L^2 estimate.



For a fixed , we have 

    ∑_τ∑_T∈_τ,∑_j∈_TS_τ, j(∑_B∈_f_ B)_T_j_2^2⪅(R^2/ν)^-1f_2^2.



The proposition is built on Lemma <ref>,  a local L^2-estimate associated a single ball (or cube), and the following Kakeya-type lemma.



For fixed , we have 

    (#_)/(#_,ν)≲[(R^2)/(ν)]·(^2)^-1.



This follows from Kakeya estimates in ^2. Indeed, by the definition of _,

    ^2|B|(#_)≲∫(∑_T∈_, ν_T)^2≲∑_T_1∈_, ν∑_T_2∈_, ν|T_1∩ T_2|.

By Lemma <ref>, we see that 
 _,ν is a collection of R× R-tubes obeying Wolff's axiom with factor R^2/(ν).
Henceforth, we have 

    ∑_T_2∈_,ν|T_1∩ T_2|⪅ R^2/(ν)|T_1|.

 From (<ref>) and (<ref>), it follows that  

    ^2|B|(#_)⪅ R^2/(ν)|T|(#_,ν),

which proves the lemma since |B|∼|T|.



Note that the kernel of S_τ, j decays rapidly outside an R× R tube, centered at the origin, with the direction e(τ). By the triangle inequality, for a fixed R× R tube T∈⋃_τ_τ,, we have

    ∑_j∈_T∫|S_τ, j(∑_B∈_f_B)_T_j|^2≲#{B∈_:B⊂ T}∑_B⊂ T∑_j∈_T∫|S_τ, j(f_B)_T_j|^2.

Recall that #{B∈_:B⊂ T}, for all T∈𝕋_,  are the same up to a constant multiple. Hence

    #{B∈_:B⊂ T}≲(#_)/(#_),

and we have

    ∑_τ∑_T∈_τ,∑_j∈_TS_τ, j(∑_B∈_f_ B)_T_j_2^2 
    ≲    ∑_τ∑_T∈_τ,#{B∈_:B⊂ T}∑_B⊂ T∑_j∈_T∫|S_τ, j(f_B)_T_j|^2 
    ≲    (#_)/#_∑_B∈_∑_τ∑_T∈_τ,
    T⊃ B∑_j∈_T∫|S_τ, j(f_B)_T_j|^2.

Note that for fixed τ, there are only O(1) T∈_τ, such that B⊂ 2T. Invoke Lemma <ref> so that

    ∑_τ∑_T∈_τ,
    T⊃ B∑_j∈_T∫|S_τ, j(f_B)_T_j|^2⪅f_B_2^2.

Therefore, combining (<ref>) and (<ref>), we end up with

    ∑_τ∑_T∈_τ,∑_j∈_TS_τ, j(∑_B∈_f_ B)_T_j_2^2   ⪅(#_)/(#_)∑_B∈_f_B_2^2
       ⪅(#_)/(#_)f_2^2,

and the proposition follows from Lemma <ref> since #_⪆#_, ν. 








 §.§  L^4-estimates for the dual operator


The ultimate proposition in this section offers an L^4/3 estimate, which we will establish by examining its dual form in L^4.




Fix . We have the L^4/3 estimate 

    ∑_τ∑_T∈_τ,, ν∑_j∈_T∫|S_τ, j(∑_B∈_f_B)_T_j|^4/3⪅^1/3ν^2/3∏_k=1^n_k^2/3f_4/3^4/3.






Using the duality between L^4/3 and L^4, it suffices to prove that

    ∫∑_τ∑_T∈_τ,, ν∑_j∈_TS_τ, j(∑_B∈_f_B)f_T_j⪅^1/4ν^1/2∏_k=1^n_k^1/2f_4/3{f_T_j}_L^4(ℓ^4),

where {f_T_j}_L^4(ℓ^4) denotes 

    (∑_τ∑_T∈_τ, , ν∑_j∈𝒥_T∫| f_T_j|^4  )^1/4 ,

and f_T_j is a function supported on T_j.


Note that 

    ∫∑_τ∑_T∈_τ,, ν∑_j∈_TS_τ, j(∑_B∈_f_B)f_T_j
           =∫ f∑_τ∑_T∈_τ,, ν∑_j∈_T∑_B∈_S_τ,j(f_T_j)_B.

Hence by Hölder's inequality, it suffices to prove that 

    ∫|∑_τ∑_T∈_τ,, ν∑_j∈_T∑_B∈_S_τ,j(f_T_j)_B|^4⪅ν^2∏_k=1^n_k^2{f_T_j}_L^4(ℓ^4)^4.

We will investigate the integrand of the above estimate, that is, 

    |∑_τ∑_T∈_τ,, ν∑_j∈_T∑_B∈_S_τ,j(f_T_j)_B(x)|.


In parallel with the broad-to-narrow reduction outlined in Lemma <ref>, for every point x within B_R, one of the following statements holds concerning (<ref>):






A: x is _1-broad for some dyadic number R^-1/2<_1≤ 1. 
This means that there exist two caps τ_1 and τ_1' with (log R)α_1 ≳dist(τ_1,τ_1') ≳α_1, and |τ_1|,|τ_1'|∼α_1, such that one of the following conditions holds:

    
  * If _1≥, then for ∈{τ_1,τ_1'},
    
    (<ref>)⪅|∑_τ⊂∑_T∈_τ,, ν   ∑_j∈_T∑_B∈_S_τ,j(f_T_j)_B(x)|.

    
  * If _1<, then there exists an -cap τ such that τ_1,τ_1'⊂τ, and for ∈{τ_1,τ_1'}, we have
    
    (<ref>)⪅|∑_T∈_τ,, ν   ∑_j∈_T∑_B∈_S_,j(f_T_j)_B(x)|.


    
    
B: x is narrow. This means that there exists an R^-1/2-cap θ and an _1-cap τ such that θ⊂τ and

    (<ref>)⪅|∑_T∈_τ,, ν∑_j∈_T∑_B∈_S_θ,j(f_T_j)_B(x)|.

The proof of this dichotomy is almost identical to the one in Lemma <ref>. More precisely, 
this broad-narrow process can be carried out through an iteration commencing with the initial scale for α_1 approximately equal to 1. It concludes when reaching the scale of R^-1/2.  We omit its details here.  


Hence we see that there is a dyadic number α_1∈ [R^-1/2,1] such that 

    ∫|(<ref>)|^4⪅∫_E__1|∑_τ∑_T∈_τ,, ν∑_j∈_T∑_B∈_S_τ,j(f_T_j)_B|^4,

where  E_α_1 refers to the set of α_1-broad points if α_1 falls within the interval (R^-1/2, 1], and it signifies the set of narrow points if α_1=R^-1/2. 



Let's examine two distinct scenarios based on the magnitude of α_1:



  ∙ Case I, where α_1≥α;

  ∙ Case II, where α_1<α.




Case I,  where _1≥. Partition the interval [1/2,2] into _1^-1-intervals {I}. For each T∈_τ,, ν, decompose _T as a union of _T(I), where _T(I)={j∈_T:t_j∈ I}. By dyadic pigeonholing, there exists a dyadic number μ, a tube set _τ,, μ, ν, and a index set _T,μ for each T∈_τ,, μ, ν such that

    
  * #_T,μ(I)∼μ whenever _T,μ(I)={j∈_T, μ: t_j∈  I}≠∅.
    
  * We have
    
    ∫|(<ref>)|^4⪅∫_E__1|∑_τ∑_T∈_τ,, μ, ν∑_j∈_T,μ∑_B∈_S_τ,j(f_T_j)_B|^4.


Note that for each T∈_τ, , ν,  #_T∼ν. Thus we have

    #{I:_T, μ(I)≠∅}≲ν/μ.


We provide proof details exclusively for the scenario where α_1 > R^-1/2, as the narrow case can be addressed in a similar manner.
By the definition of _1-broad, we see that 

    ∫|(<ref>)|^4 ≲∑_τ_1∑_τ_1'
    (τ_1,τ_1')≈_1 ∑_B∈_𝔖_τ_1, τ_1', B,

where τ_1, τ'_1 are α_1-caps and 𝔖_τ_1, τ_1', B is defined as 

    ∫_B|∑_τ⊂τ_1∑_T∈_τ,, μ, ν∑_j∈_T, μS_τ,j(f_T_j)|^2|∑_τ'⊂τ_1'∑_T'∈_τ',, μ, ν∑_j'∈_T',μS_τ',j'(f_T_j'')|^2 .

Here τ, τ' are α-caps. 
Apply Lemma <ref> to (<ref>) so that 

    (<ref>)⪅(ν/μ)·
       ∑_τ⊂τ_1 
    τ'⊂τ_1'∑_T∈_τ,, μ, ν
    T'∈_τ',, μ, ν∑_I,I'∫_2B|∑_j∈_T, μ(I)S_τ,j(f_T_j)|^2|∑_j'∈_T', μ(I')S_τ',j'(f_T_j'')|^2.

Summing up all τ_1,τ_1' and B∈_, we obtain that 

    ∫|(<ref>)|^4
    ≲    (ν/μ) ∑_τ_1, τ_1'
    (τ_1,τ_1')≈_1∑_τ⊂τ_1
    τ'⊂τ_1'∑_T∈_τ,, μ, ν
     T'∈_τ',, μ, ν∑_I,I'∫|∑_j∈_T,μ(I)S_τ,j(f_T_j)|^2|∑_j'∈_T',μ(I')S_τ',j'(f_T_j')|^2.


Apply the Cauchy-Schwarz inequality to the index sets 𝒥_T, μ(I) and 𝒥_T',μ(I'),
then invoke Lemma <ref>. Consequently, we arrive at 

    ∫|(<ref>)|^4 ≲   νμ∑_τ_1, τ_1'
    (τ_1,τ_1')≈_1∑_τ⊂τ_1
    τ'⊂τ_1'∑_T∈_τ,, μ, ν
     T'∈_τ',, μ, ν∑_j∈_T
    j'∈_T'∑_θ⊂τ
    θ'⊂τ'∫|S_θ,j(f_T_j)|^2|S_θ',j'(f_T_j'')|^2.

For each τ_1, partition B_R into R_1× R-rectangles {T_τ_1}, pointing the direction e(τ_1).
Since for θ⊂τ_1∪τ_1', the kernel of S_θ,j decays rapidly outside an R_1× R-rectangle,  centered at the origin, 
with the direction e(τ_1). Thus we have

    ∫|(<ref>)|^4
         ≲νμ·
       ∑_τ_1, T_τ_1∑_τ_1'
    (τ_1,τ_1')≈_1∑_τ⊂τ_1
    τ'⊂τ_1'∑_T∈_τ,, μ, ν
     T'∈_τ',, μ, ν∑_j∈_T
    j'∈_T'∑_θ⊂τ
    θ'⊂τ'∫| S_θ, j(f_T_j_T_τ_1)|^2    |S_θ', j'(f_T_j''_T_τ_1)|^2.


Given that |S_θ, j(f_T_j_T_τ_1)| remains nearly constant within any R^1/2× R-tube oriented along e(θ), and considering the bilinear condition (τ_1,τ_1')⪆α_1, it becomes evident that 

    ∫ |S_θ, j(f_T_j_T_τ_1)|^2|S_θ', j'(f_T'_j'_T_τ_1)|^2⪅ R^-2_1^-1S_θ, j(f_T_j_T_τ_1)_2^2S_θ', j'(f_T_j''_T_τ_1)_2^2.

In the specific scenario where _1=R^-1/2, it's worth mentioning that (<ref>) remains valid even without relying on the bilinear structure. 



Note that #{τ_1', (τ_1,τ_1')≈_1}⪅1. We can thus sum up all τ⊂τ_1,θ,T,j and their minors in (<ref>) to get

    ∫|(<ref>)|^4⪅νμ R^-2_1^-1∑_τ_1∑_T_τ_1(∑_τ⊂τ_1∑_T∈_τ,, μ, ν∑_j∈_TS_τ, j(f_T_j_T_τ_1)_2^2)^2.

Invoke (<ref>) to obtain

    S_τ, j(f_T_j_T_τ_1)_2^2≲∏_k=1^n_r_kf_T_j_T_τ_1_2^2,

which yields

    ∫|(<ref>)|^4⪅νμ∏_k=1^n_r_k^2R^-2_1^-1∑_τ_1∑_T_τ_1(∑_τ⊂τ_1∑_T∈_τ,, μ, ν∑_j∈_Tf_T_j_T_τ_1_2^2)^2.

Because the supports of f_T_j's are disjoint, employing the Cauchy-Schwarz inequality yields that 


    (∑_τ⊂τ_1∑_T∈_τ, , μ, ν∑_j∈_Tf_T_j_T_τ_1_2^2)^2   =∑_τ⊂τ_1∑_T∈_τ, ,μ,ν∑_j∈_Tf_T_j_T_τ_1_2^4
       ≲ R^2_1∑_τ⊂τ_1∑_T∈_τ, ,μ,ν∑_j∈_Tf_T_j_T_τ_1_4^4.

By summing up all T_τ_1 for τ_1, and observing that μ≲ν, we ultimately obtain

    ∫|(<ref>)|^4≲ν^2∏_k=1^n_r_k^2{f_T_j}_L^4(ℓ^4)^4 .

This leads to (<ref>) and completes the discussion for the case where α_1≥α.




Case II, where _1<. Let us assume _1>R^-1/2. The narrow case _1=R^-1/2 can be proved similarly.  Since _1<, there is only one τ making contribution in (<ref>), so we may discard the information from _. By the definition of _1-broad, we have

    ∫|(<ref>)|^4≲   ∑_τ∑_T∈_τ,, ν∑_τ_1⊂τ∑_τ'_1
    (τ_1,τ_1')≈_1∫|∑_j∈_TS_τ_1,j(f_T_j)|^2|∑_j'∈_TS_τ_1',j'(f_T_j')|^2.

Apply Cauchy-Schwarz's inequality and then Lemma <ref> on the L^4 orthogonality to get 

    ∫|(<ref>)|^4≲   ∑_τ∑_T∈_τ,, ν∑_τ_1⊂τ∑_τ'_1
    (τ_1,τ_1')≈_1∑_j∈_T
    j'∈_T∑_θ⊂τ_1
    θ'⊂τ_1'ν^2∫|S_θ,j(f_T_j)|^2|S_θ',j'(f_T_j')|^2.

For every τ_1 and each tube T with dimensions Rα× R, divide T into Rα_1 × R rectangles {T_τ_1}, oriented in the direction of e(τ_1).
Using the decay property of the kernel S_θ,j, analogous to our approach in the first scenario, we obtain that  

    ∫|(<ref>)|^4≲    ν^2∑_τ∑_T∈_τ, , ν∑_τ_1⊂τ∑_τ'_1
    (τ_1,τ_1')≈_1∑_j∈_T∑_j'∈_T∑_T_τ_1⊂ T
       ∑_θ⊂τ_1∑_θ'⊂τ_1'∫_T_τ_1|S_θ,j(f_T_j_2T_τ_1)|^2|S_θ',j'(f_T_j'_2T_τ_1)|^2.

Since |S_θ, j(f_T_j_2T_τ_1)| is essentially constant on any R^1/2× R-tube with the direction of e(θ) and since (τ_1,τ_1')≳_1,

    ∫_T_τ_1   |S_θ, j(f_T_j_2T_τ_1)|^2|S_θ', j'(f_T_j_2T_τ_1)|^2
       ⪅ R^-2_1^-1S_θ, j(f_T_j_2T_τ_1)_L^2(T_τ_1)^2S_θ', j'(f_T_j_2T_τ_1)_L^2(T_τ_1)^2.

Summing up all θ, j and their minors and noting #{τ_1', (τ_1,τ_1')≈_1}⪅1, we have

    ∫|(<ref>)|^4⪅ν^2/R^2_1∑_τ∑_T∈_τ, , ν∑_τ_1⊂τ
    T_τ_1⊂ T(∑_θ⊂τ_1∩τ∑_j∈_TS_θ, j(f_T_j_2T_τ_1)_L^2(T_τ_1)^2)^2.




Let us fix a T∈_τ, , ν. Partition the T into R^1/2-pseudo balls {Q}. Since T_j is a regular set with factors (_1,…,_n), we see that  #{Q: Q∩ T∩ T_j≠∅}⪅_n R^1/2 for any R^1/2× R tube T by (<ref>) in Lemma <ref>. Let { T} be a collection of R^1/2× R tubes with direction e(θ) that form a finitely overlapping partition of the R× R tube T. Since the kernel of S_θ, j decays rapidly outside an R^1/2× R tube, centered at the origin, with the direction of e(θ), we have

    S_θ, j(f_T_j_2T_τ_1)_L^2(T_τ_1)^2≲∑_ TS_θ, j(f_T_j_2T_τ_1_ T)_L^2(T_τ_1)^2.

From Cauchy-Schwarz's inequality and estimate (<ref>) in Lemma <ref>, it follows that 

    S_θ, j(f_T_j_2T_τ_1_ T)_L^2(T_τ_1)^2⪅_n
         R^1/2∑_Q⊂ TS_θ, j(f_T_j_Q_2T_τ_1)_L^2(T_τ_1)^2.

Combining (<ref>) and (<ref>), we end up with 

    S_θ, j(f_T_j_2T_τ_1)_L^2(T_τ_1)^2⪅_n
         R^1/2∑_Q⊂ TS_θ, j(f_T_j_Q_2T_τ_1)_L^2(T_τ_1)^2.




Let {Y} be a cover of the R× R-tube T using R× R^1/2(log R)^-1-rectangles, whose longer side is parallel to the shorter side of T. Then each Y is quantitatively transverse to T and any T_τ_1. From (<ref>),  we get

    ∑_τ_1⊂τ∑_T_τ_1(∑_θ⊂τ_1∩τ∑_j∈_TS_θ, j(f_T_j_2T_τ_1)^2_L^2(T_τ_1))^2
    ⪅    _n^2^2R∑_τ_1⊂τ∑_T_τ_1(∑_θ⊂τ_1∩τ∑_j∈_T∑_Y⊂ T∑_Q⊂ YS_θ, j(f_T_j_Q_2T_τ_1)_L^2(T_τ_1)^2)^2
    ≲    _n^2^3R^3/2∑_Y⊂ T∑_τ_1⊂τ∑_T_τ_1(∑_θ⊂τ_1∩τ∑_j∈_T∑_Q⊂ YS_θ, j(f_T_j_Q_2T_τ_1)_L^2(T_τ_1)^2)^2

via Cauchy-Schwarz's inequality. Let us fix an R× R^1/2(log R)^-1-rectangle Y now. Partition Y further into R_1× R^1/2(log R)^-1-rectangles Y' (see Figure <ref>) so that

    ∑_Q⊂ YS_θ, j(f_T_j_Q_2T_τ_1)_L^2(T_τ_1)^2=∑_Y'⊂ Y,
    Y'∩ T_τ_1≠∅∑_Q⊂ Y'S_θ, j(f_T_j_Q_2T_τ_1)_L^2(T_τ_1)^2.

Since Y is quantitatively transverse to T_τ_1, there are O(1) Y'⊂ Y such that Y'∩ T_τ_1≠∅. From this observation and (<ref>), we see that 

    ∑_τ_1⊂τ∑_T_τ_1(∑_θ⊂τ_1∩τ∑_j∈_T∑_Q⊂ YS_θ, j(f_T_j_Q_2T_τ_1)_L^2(T_τ_1)^2)^2
    ≲   ∑_τ_1⊂τ∑_T_τ_1∑_Y'⊂ Y,
    Y'∩ T_τ_1≠∅(∑_θ⊂τ_1∩τ∑_j∈_T∑_Q⊂ Y'S_θ, j(f_T_j_Q_2T_τ_1)_L^2(T_τ_1)^2)^2
    ≲   ∑_τ_1⊂τ∑_T_τ_1∑_Y'⊂ Y,
    Y'∩ T_τ_1≠∅(∑_θ⊂τ_1∩τ∑_j∈_T∑_Q⊂ Y'S_θ, j(f_T_j_Q)_2^2)^2.

In the last inequality, we use that the kernel of S_θ, j decays rapidly outside the R_1× R tube, centered at the origin,  with the  direction of e(τ_1).  Again, since there are O(1) many Y'⊂ Y such that Y'∩ T_τ_1≠∅, the double sum of (<ref>) on (T_τ_1, Y') can be bounded by a single sum of Y'. Henceforth, we have  

    (<ref>)≲   ∑_τ_1⊂τ∑_Y'⊂ Y(∑_θ⊂τ_1∩τ∑_j∈_T∑_Q⊂ Y'S_θ, j(f_T_j_Q)_2^2)^2
    ≲   ∑_Y'⊂ Y(∑_τ_1⊂τ∑_θ⊂τ_1∩τ∑_Q⊂ Y'∑_j∈_TS_θ, j(f_T_j_Q)_2^2)^2
    ≲   ∑_Y'⊂ Y(∑_Q⊂ Y'∑_j∈_TS_τ, j(f_T_j_Q)_2^2)^2.






For each S_τ, j(f_T_j_Q)_2^2, (<ref>) gives

    S_τ, j(f_T_j_Q)_2^2⪅^-1 R^-1/2∏_k=1^n-1_r_kf_T_j_Q_2^2.

Plug this back to(<ref>) so that

    (<ref>)   ⪅ ^-2R^-1∏_k=1^n-1_r_k^2∑_Y'⊂ Y(∑_Q⊂ Y'∑_j∈_Tf_T_j_Q_2^2)^2
       ≲^-2R^-1(R^3/2_1^-1)∏_k=1^n-1_r_k^2∑_Y'⊂ Y∑_j∈_Tf_T_j_Y'_4^4
       ≲(^-3R^1/2_1)∏_k=1^n-1_r_k^2∑_j∈_Tf_T_j_Y_4^4

by Cauchy-Schwarz's inequality. 
Putting this back into (<ref>), and subsequently into (<ref>), we arrive at

    ∫|(<ref>)|^4⪅ν^2∏_k=1^n_r_k^2∑_τ∑_T∈_τ, , ν∑_Y⊂ T∑_j∈_Tf_T_j_Y_4^4≲ν^2∏_k=1^n_r_k^2{f_T_j}_L^4(ℓ^4)^4.

Since ≥1, this gives (<ref>) when _1<.  Therefore, we conclude the proposition.







 §.§ Completing the proof of Proposition <ref>



Let's establish Proposition <ref> by utilizing Propositions <ref>, <ref>, and <ref>.
 Define

    I_p(f)^p:=∑_τ∑_T∈_τ, , ν∑_j∈_TS_τ, jf_T_j_p^p.



First, Proposition <ref> yields 

    I_2(f)^2⪅∏_k=1^n_r_k^-1(/R^2)f_2^2.


Subsequently, utilizing (<ref>), Proposition <ref>, and Propositions <ref>, we obtain


    I_2(f)^2
        ⪅(R^2/ν)^-1f_2^2

and (since _τ,⊂_τ, , ν)

    I_4/3(f)^4/3⪅^1/3ν^2/3∏_k=1^n_r_k^2/3f_4/3^4/3

respectively. 


Given that the Fourier multiplier of S_τ,j is confined within a unit ball, we may assume that f is supported in B^2(0,2), a ball centered at the origin with radius 2, implying that |f| is essentially constant within any unit ball. Furthermore, as the kernel of S_τ,j exhibits rapid decay beyond the R-ball centered at the origin, we can infer that f is also supported in an R-ball. Additionally, considering the homogeneity of (<ref>) in Proposition <ref>, we can assume |f|≤ 1 and sup |f|∼ 1. Consequently, f_5/3≳ 1, as |f| essentially remains constant within any unit ball, notably within the unit ball containing x where |f(x)|∼ 1. We partition f as

    f=∑_k≥ 0f_k

where |f_k|∼ 2^-k behaves essentially like a characteristic function. 


Let's tackle each f_k. Multiplying the square of (<ref>) with (<ref>) and the cubic of (<ref>), we obtain


    I_2(f_k)^6I_4/3(f_k)^4⪅(ν/R^2)f_k_2^6f_k_4/3^4≤f_k_5/3^10 ,

since ν≲ R^2 by (<ref>).  Employing Hölder's inequality and (<ref>), we derive 

    I_5/3(f_k)≤ I_2(f_k)^3/5I_4/3(f_k)^2/5⪅f_k_5/3.


If 2^k≥ R^10, given that f is supported in an R-ball, we have f_k_5/3≲ 2^-k/2f_5/3. Hence, by (<ref>),  

    I_5/3(f)≤∑_k≥ 0I_5/3(f_k)   ≤∑_k=0^[10log R]I_5/3(f_k)+∑_k≥ 10log R2^-k/2f_5/3
       ≲(log R)f_5/3⪅f_5/3.


This establishes Proposition <ref> and consequently our main theorem, Theorem <ref>.





§ ENDING REMARK: RULING OUT TAO'S EXAMPLE



Tao's example f(x_1,x_2)=a(R^-1/2x_1)a(x_2)e^ix_1 can be  illustrated by Figure <ref>: On the frequency side, f appears as  a bump function supported on  an R^-1/2×1 vertical rectangle (depicted as the shaded region in  in Figure <ref>) away from the origin. The intersection of this rectangle with any R^-1-annulus A_j of radius t_j∼1  essentially forms  an R^-1/2× R^-1 rectangular cap θ_j. Specifically, if (θ_j,θ_k)≳ R^-1/2, then θ_j,θ_k correspond to different directions. For example, θ_1,θ_2,θ_3 in Figure <ref> give distinct directions. 










To exclude Tao's example, one may pose the following question: If the frequency of the function f is well-localized in a horizontal R^-1/2× 1-rectangle away from the origin, can we verify Conjecture <ref>? We will demonstrate that our approach can address this inquiry.




Let (x_1,x_2)=a(R^-1/2x_1)a(x_2)e^ix_1 be a bump function. Define a family of frequency-localized (with respect to the horizontal rectangle given by ) operators 

    S^_jf:=S_j(∗ f), 1≤ j≤ R.


Let {F_j}_j be a family of disjoint subsets of B_R that each F_j is a union of unit balls. For p=3/2 and any f∈ L^p(^2), we have 

    ∑_jS^_jf_F_j_p⪅f_p.



Notice that the Fourier transform of the kernel of S^_j is supported in an R^-1/2× R^-1-rectangular cap θ_j. Hence the kernel of S^_j is essentially supported on a R^1/2× R-tube with the direction e(θ_j) centered at the origin. For a fixed j, partition B_R into finitely overlapping R^1/2× R-tubes with the direction e(θ_j). Denote this family of R^1/2× R-tubes by _θ_j. For each T∈_θ_j, define

    T_j:=T∩ F_j.

Note that the definition of T_j here agrees with the one in Section <ref>.



By pigeonholing, there exists a subset _θ_j'⊂_θ_j for any j and two numbers ,ν such that

    
  * |T_j|∼ for any T∈⋃_j_θ_j' and any j.
    
  * For a fixed T∈⋃_j_θ_j', #_T∼ν, where _T:={j:|T_j|∼}.
    
  * ∑_jS^_jf_F_j_p⪅∑_j∑_T∈_θ_j'S^_jf_T_j_p.
    
  * ν≲ R^3/2.

Let _,ν=⋃_j_θ_j'. Similar to Section <ref>, there exists a factor R^1/2≳≥1, a set of R^1/2-balls _, and a tube set _⊂_,ν such that

    
  * 2B intersects ∼ R^1/2× R-tubes in _,ν for each B∈_.
    
  * |_|⪆|_, ν|
    
  * For each T∈_, we have
    
    ∑_j∈_TS^_j(∑_B∈f_B)_T_j_p^p⪅∑_j∈_TS^_j(∑_B∈_f_B)_T_j_p^p.

    
  * For each T∈_, #{B∈_: B⊂ T} are the same up to a constant multiple.

Therefore, Proposition <ref> implies that

    ∑_T∈_∑_j∈_TS^_j(∑_B∈_f_ B)_T_j_2^2⪅(R^3/2/ν)^-1f_2^2.




On the other hand, we claim that 

    ∑_T∈_, ν∑_j∈_T∫|S^_j(∑_B∈_f_B)_T_j|^4/3⪅^1/3ν^2/3^2/3R^-1f_4/3^4/3.

To prove (<ref>), we follow the proof of Proposition <ref> until (<ref>), with the left-hand side of (<ref>) being replaced by the left-hand side of (<ref>). Since for any function g, S^_jg is essentially constant on any R^1/2× R-tube in the direction of e(θ_j), we obtain the following stronger bound in place of (<ref>):

    S^_j(f_T_j_T_τ_1)_2^2≲ R^-3/2f_T_j_T_τ_1_2^2.

Then, we continue following the proof of Proposition <ref> to conclude (<ref>).



Similar to the argument in Section <ref>, (<ref>) and (<ref>) together yield (<ref>) for p=3/2.



alpha














 Paper Title: On the mean-field limit of the Cucker-Smale model with Random Batch Method.




    
        
        
     
        a]Yuelin Wang sjtu_wyl@sjtu.edu.cn
        a]Yiwen Lin linyiwen@sjtu.edu.cn
[a]School of Mathematical Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, P.R.China 


On the mean-field limit of the Cucker-Smale model with Random Batch Method
    [
    
==========================================================================



§ ABSTRACT 
    In this work, we focus on the mean-field limit of the Random Batch Method (RBM) for the Cucker-Smale model. Different from the classical mean-field limit analysis, the chaos in this model is imposed at discrete time and is propagated to discrete time flux. We approach separately the limits of the number of particles N→∞ and the discrete time interval τ→ 0 with respect to the RBM, by using the flocking property of the Cucker-Smale model and the observation in combinatorics. The Wasserstein distance is used to quantify the difference between the approximation limit and the original mean-field limit. Also, we combine the RBM with generalized Polynomial Chaos (gPC) expansion and proposed the RBM-gPC method to approximate stochastic mean-field equations, which conserves positivity and momentum of the mean-field limit with random inputs.

    
    Keywords: Random Batch Method, mean-field limit, Cucker-Samle model, stochastic Galerkin.






§ INTRODUCTION


    Collective behaviors of many-body systems are ubiquitous in the real world, like  flocking of birds <cit.>, swarming of fishes <cit.>, synchronicity of fireflies <cit.> and pacemaker cells <cit.>. We use the jargon “flocking" to describe the phenomenon in which self-propelled particles organize into an ordered motion using only limited environmental information and simple rules <cit.>. Since there is a great deal of literature on collective behaviors and related models, we refer the readers to <cit.> and the references therein.
    
     
    The Cucker-Smale model, proposed by Cucker and Smale <cit.>, is a famous model of collective behaviors that models phenomenologically  the flocking phenomenon. It is in the form of an N-body second-order system of ordinary differential equations or particle systems  for position and velocity, which resemble Newton's equations.  
    

     In addition, the mean-field limit, as the number of particles N →∞,  is also a prevalent tool for simplifying large particle systems, which makes use of continuum models derived within the framework of mathematical kinetic theory, to approximate collective behaviors. In these years, there has been extensive research on the mean-field limit of particle systems and we refer to <cit.> for survey articles.
     
    As a classical type of N-body systems, in the Cucker-Smale model, each particle interacts with N-1 particles, and thus the computational cost to solve it is 𝒪(N^2) per time step. An efficient algorithm to reduce computational complexity to 𝒪(N) is the Random Batch Method (RBM) proposed by Jin et al. <cit.> in 2020. The RBM indeed constructs a randomly decoupled system with subsystems of interaction between p particles, where the constant p≪ N. In this system, at each time step, the interaction only occurs within small batches with p particles. The choice of batches is random and thus the time-average effect makes it  a good approximation of the original system <cit.>.
    
     
    
    
    An interesting problem is to understand the mean-field limit of the RBM. In 2022, Jin and Li considered the mean-field limit of the RBM on first-order systems with Gaussian noise in <cit.>. 
        
        
        
        
    Inspired by <cit.>, in this paper, we investigate the mean-field limit of the RBM on the Cucker-Smale model as the number of particles tends to infinity. Unlike the first-order system considered in <cit.>, the Cucker-Smale model is a second-order system with interaction coupled with both velocity and position. Technically, moment control needs to be  derived by the asymptotic flock estimate instead of using the contraction property in first-order systems in <cit.>. Specifically, we mainly analyze the path from (b) to (c), and briefly discuss the path from (a) to (d) in the following figure (<ref>). On the upper left corner, Ha et al. give an estimate on (a) in <cit.> and a proof of the existence of mean-field limit on (d) in <cit.>. Their analysis on (d) is uniform in time but implicit in N. In later work <cit.>, an explicit estimate on N is given but only works on finite time. The four models in (<ref>) are represented in Section <ref>.
    
    
    Cucker-Smale system (<ref>) @ > (d)  N→∞ >> mean-field limit (<ref>)
    
             @A (a) τ→ 0 AA  @AA (c) τ→ 0 A
    Random Batch Cucker-Smale model (<ref>) @>> (b)  N→∞ > mean-field limit of RBM (<ref>)


    
   With the justification of (b) and (c),   we employ the RBM, which is an efficient particle-based numerical scheme for the approximation of stochastic mean-field equations of the Cucker-Smale model, inspired by <cit.>. We consider such an application in the stochastic setting, where the interacting force depends on a random variable modeling uncertainties.  This is an important problem in Uncertainty Quantification <cit.>. The method combines the RBM with a generalized Polynomial Chaos (gPC) expansion in the random space and thus we call it the RBM-gPC. It avoids loss of positivity like MCgPC in <cit.>, and moreover preserves the mean velocity during evolution.

    
    The rest of the paper is organized as follows. In Section <ref>, we give a concise introduction to the Cucker-Smale model including notations, assumptions and prerequisites. Section <ref> presents our main results on the analysis of limits (b) and (c) in (<ref>) with helpful discussions on the limits from (a) to (d). Section <ref> and Section <ref> provide the proof details. In Section <ref>, we show some numerical experiments with the RBM-gPC. Finally, Section <ref> is devoted to a summary of our main results and some remaining issues to be explored in the future.
    
    
    
    


  




§ PRELIMINARIES

In this section, we introduce the main models in (<ref>) with assumptions and notations. Also, some basic properties of the models are given.



 §.§ The Cucker-Smale model

Let X_i and V_i be the position and velocity of the i-th particle with unit mass, and ψ(| X_j -X_i|) be the communication weight between the j-th and i-th particles. The Cucker-Smale model reads as the following

    {d/dt X_i(t) =    V_i(t),   i=1,⋯, N,
    d/dt V_i(t) =   κ/N-1∑_j=1^N 
            ψ(|X_j(t)-X_i(t)|)(V_j(t)-V_i(t)),
        .

where κ is the nonnegative coupling strength and ψ satisfies  positivity, boundedness, Lipschitz continuity and mononticity conditions, i.e., there exist positive constants ψ_0, ψ_M  > 0  such that 

    0< ψ_0 ≤ψ(r) ≤ψ_M, ∀ r≥0;   ψ_Lip<∞;   (ψ(r_1)-ψ(r_2))(r_1-r_2)≤ 0,  r_1,r_2 ∈ℝ_+.

Without loss of generality, we set ψ_M = 1 in this paper for convenience. 
Under the condition (<ref>) of ψ, it is observed in <cit.> that the total momentum is conserved as a constant and the total energy is nonincreasing along the Cucker-Smale flow. Actually,

    supposing that {(X_i,V_i)} is the solution of  system (<ref>), then for any t>0, one has
    
    d/dt∑_i=1^N V_i (t) = 0,   d/dt∑_i=1^N |V_i(t)|^2 = -κ/N-1∑_i,jψ(|X_j(t) - X_i(t)|) |V_j (t) - V_i (t) |^2.




A key observation of the Cucker-Smale model, given by Ha et al. in <cit.>, is that the standard deviations of particle phase-space positions are dominated by SDDI (the system of dissipative differential inequalities):

    | d𝒳/dt| ≤𝒱,   d𝒱/dt≤ -ϕ (𝒳) 𝒱,

where (𝒳,𝒱) are nonnegative functions and ϕ is a nonnegtive measurable function. This is a useful tool to analyze the Cucker-Smale model. In this paper, we use a variant  of (<ref>) (Lemma <ref>) in the proof of our main result. One can find its proof in Lemma 3.1 in <cit.>.

    Suppose that two nonnegative Lipschitz functions 𝒳 and 𝒱 satisfy the coupled differential inequalities:
    
    | d𝒳/dt| ≤𝒱,   d𝒱/dt≤ -α𝒱 + γ e^-α t𝒳,    a.e.  t>0,

    where α and γ are positive constants. Then, 𝒳 and 𝒱 satisfy the uniform bound and decay estimates:
    
    
    𝒳(t) ≤2M/α (𝒳(0) + 𝒱(0)),   𝒱(t) ≤ M (𝒳(0) + 𝒱(0)) e^-α t/2,    t≥ 0,

    where M is given by
    
    M:= max{1,2γ/α e} + 8γ/α^3 e^3.





 §.§ The mean-field limit of the Cucker-Smale model


The corresponding evolution for the distribution function f(x, v, t) was first derived by Tadmor, etc. in <cit.> by the BBGKY hierarchy (e.g. <cit.>), which follows

    ∂_t f + v·∇_x f+ ∇_v · [ξ[f]f]=0,

where the operator ξ is defined by

    ξ[f](x,v,t) = κ∫∫ψ(|x-y|)(w-v)f(y,w,t)  dwdy.

It is clear that 

    ∂/∂ t∫∫ vfdxdv =0.

Similar with energy-decreasing in (<ref>) in the microscopic scale, the mean-field limit has the following property: 
 
Let f_0 be compactly supported in ℝ^2d with x = ∫ x f_0 (x,v) dxdv and v = ∫ v f_0 (x,v) dxdv. Define C_x0:=inf{X≥0  |  supp(f_0)⊂ B(x,X) ×ℝ^d } and C_v:=inf{V≥0  |  supp(f_0)⊂ℝ^d × B(v,V)}, where B(x,X) designates the ball of center x and radius X in ℝ^d. Let f_t be the unique solution of  (<ref>) with f(0) = f_0. If the initial data satisfies

    C_v < κ∫_C_x0^∞ψ (2s) ds,

then there exists C_x>0 such that 
    
    supp[f_t]⊂ B(x + tv, C_x) × B(v, C_v e^-κψ(2C_x) t).



    Under  assumption (<ref>) of ψ,  one has ψ(2C_x)≥ψ_0. Actually, the assumption of the lower bound ψ_0 is for the convenience of the proof and can be relaxed if (<ref>) is satisfied by both discrete and continuous systems, guaranteeing the flocking in the Cucker-Smale model.




 §.§ The Wasserstein distance


To introduce the topology of the space of probability measures, here we recall the definitions of the Wasserstein space <cit.>. For a metric space 
 (ℝ^d,ρ) and q≥1, define the Wasserstein space of order q by

    𝒫_q,ρ (ℝ^d):= {μ∈𝒫(ℝ^𝕕); ∫_ℝ^dρ(x,0)^q ν (dx)< ∞},

endowed with the Wasserstein metric 

    W_q,ρ(μ,ν) := ( inf_π∈Π (μ,ν){∫_ℝ^d ×ℝ^dρ(x,y)^q π (dx,dy)})^1/q,

where μ,ν∈𝒫_q (ℝ^d) and Π (μ,ν) is the set of probability measures on ℝ^d ×ℝ^d with marginals μ and ν respectively. The W_q metric, or distance, induces a kind of weak topology and measures the closeness between distributions. We use the notation W_q,p when ρ(x,y):= x-y _p and typically abbreviate W_q = W_q,2. 
For convenience in notation, we simplify · as |·| without ambiguity.


Recall the Jordan decomposition for a signed measure |μ| := μ^+ + μ^-. One can control the W_q distance by the weighted total variation by the following lemma (see e.g. Theorem 6.15 in <cit.>), which helps the proof of our main theorems, i.e., Theorem <ref> and Theorem <ref>, in Section <ref>.

    Let μ and ν be two probability measures on a Polish space (𝔛, d). Let q∈ [1,∞) and x_0 ∈𝔛. Then 
    
    W_q (μ,ν) ≤ 2^1-1/q( ∫ d(x_0,x)^q d |μ-ν|(x) )^1/q.





 §.§ The Random Batch method and its mean-field dynamics


In each sub-time interval [t_k-1,t_k), we set [i]_k as the batch containing the particle i. The RBM-(approximated) system of (<ref>) becomes:

    {∂_t X_i^R(t) =    V_i^R(t),   i=1,⋯, N,
    ∂_t V_i^R(t) =   κ/p-1∑_j∈[i]_kψ(|X_j^R(t)-X_i^R(t)|)(V_j^R(t)-V_i^R(t)),
            
        .

with initial data (X_i^R(0),V_i^R(0)) = (X_i^in,V_i^in), i = 1,⋯,N. See Algorithm <ref>.









In the N→∞ limit, the N-particle system is then reduced to a p-particle
subsystem described by the following system for t∈ [t_k,t_k+1):

    {∂_t X_i(t) =    V_i(t),   i=1,⋯, N,
    ∂_t V_i(t) =   κ/p-1∑_j=1^p 
            ψ(|X_j(t)-X_i(t)|)(V_j(t)-V_i(t)),
        .

with Z_i:=(X_i,V_i) and Z_i (t_k) being i.i.d drawn from f(·; t_k). Here f(·; t_k) is the law of Z_1(t_k^-) for k≥ 1 and f(·;0) equals the initial distribution f_0 . We impose Z_1 (t_k^-) = Z_1 (t_k^+).  Except the individual particle 1, the rest p-1 particles are sampled from an infinite pool of independent particles from particle 1 at each time step t_k. This process becomes the mean-field limit model of the RBM system (<ref>), and one may write the following mean-field limit for the RBM in terms of the probability distribution as shown in Algorithm <ref>, while (<ref>) becomes its microscopic description.










§ THE MAIN RESULT


In this paper, we show the conservation law and the mean-field limit of the Random-Batch Cucker-Smale model. First, we show the conservation of momentum and dissipation of kinetic energy of the  mean-field limit system of RBM in Section <ref>. Then, we analyze the procedure (b) and (c) in Section <ref>. 


Assume that the initial data of Algorithm <ref> and Algorithm <ref> obey the same law f_0(x,v)dxdv. For notational convenience, recall x:= ∫ x f_0(x,v) and v:=∫ vf_0 (x,v); and set z_i:=(x_i,v_i) for each i and 

    f(x,v;t):= ∫_ℝ^2d(p-1) f^(p)(⋯ ; t^-)dx_2 ⋯ dx_p dv_2⋯ dv_p = f_t(x,v).

Note that f is defined on ℝ^2d in Algorithm <ref> and f^(p) is defined on ℝ^2dp. Also, the definition of f is consistent with that in Algorithm <ref> at t=t_k.

In the following, we suppose that f_0 is compactly supported and then there exists a constant R>0 such that supp[f_0]⊂ B(0,R), where B(0,R) is a ball with radius R centered in the origin. Without loss of generality, we set v=0 as a standardization of the initial momentum.




 §.§ Basic characteristics of the mean-field limit system of RBM


Here, we show that the two basic characteristics: conservation of momentum and dissipation of kinetic energy, as observed in the Cucker-Smale model (<ref>) and the RBM system (<ref>), also holds for the RBM mean-field system described in Algorithm <ref>.

 
For t∈ (t_k,t_k+1), it holds
    
    ∂/∂ t∫_ℝ^2d v f(x,v;t) dx dv = 0.

In particular, since v=0, ∫ vf(x,v;t) = 0 for any t∈ [0,T].



By the definition of f, one has

    ∂/∂ t∫_ℝ^2d vf(x,v;t)dxdv =    ∫_ℝ^2d∫_ℝ^2d(p-1) v ∂/∂ t f^(p) (x,·, v, ·;t^-)dxdx_2⋯ dx_p dvdv_2⋯ dv_p
    
            =    ∫_ℝ^2dp v_1 [ -∑_i=1^p ∇_x_i· v_i f^(p)(⋯ ; t) - ∑_i=1^p ∇_v_i·ξ_i f^(p)(⋯ ;t)]dz_1⋯ dz_p
    
            =    κ/p-1∑_j=2^p∫_ℝ^2dpψ (|x_j - x_1|)(v_j-v_1) f^(p)_t(z_1,⋯ ,z_p)dz_1⋯ dz_p
    
            =     0.

The last equality is derived from the fact that ψ (|x_j - x_1|)(v_j-v_1)f^(p)_t is an odd function on (z_j,z_1).





Note that f(x,v;t_k^-) = f(x,v;t_k) = f(x,v;t_k^+). Then by induction one has ∫ vf(x,v;t) = v.


 
For any t∈ (t_k,t_k+1),
    
    ∂/∂ t∫_ℝ^2d |v|^2 f(x,v;t)dxdv ≤ 0.

In particular, M_2^v f_t ≤ M_2^vf_0, where M_2^v f_t:=∫_ℝ^2d |v|^2 f(x,v;t)dxdv. Moreover, since 0<ψ_0≤ψ, it holds
    
    M_2^v f_k+1≤ M_2^v f_0(e^-2κψ_0 τ + 8κ^2 ψ_0/p-1τ^2 )^k .



    
    For any t∈ (t_k,t_k+1),
        
    ∂/∂ t M_2^v f_t =    ∫_ℝ^2dp |v_1|^2 ( -∑_i=1^p ∇_x_i· (v_i f^(p)_t) - ∑_i=1^p ∇_v_i· (ξ_i f^(p)_t) )
    
                    =   ∫_ℝ^2dpκ/p-1∑_j=1^pψ(|x_1-x_j|)(v_j-v_1)· 2v_1 f^(p)_t
    
                    =    -∫_ℝ^2dpκ/p-1∑_j=1^pψ(|x_1-x_j|)|v_j-v_1|^2 f^(p)_t
    ≤    0.

        The last equality is derived by the symmetry of ψ and f_t^(p). By the continuity of f, one has M_2^v f_t ≤ M_2^vf_0. Since ψ has a positive lower bound ψ_0, furthermore, it holds
        
    ∂/∂ t M_2^v f_t ≤    -∫_ℝ^2dpκ/p-1∑ _j=1^p ψ_0 |v_j-v_1|^2 f^(p)_t
    
                    =    -2κψ_0 ∫_ℝ^2d |v|^2 f_t(x,v) dxdv + 2κψ_0 ∫_ℝ^2dp v_1· v_2 f^(p)_t,

        by the symmetry of f^(p)_t. For the second term in (<ref>), one has
        
    ∂/∂ t   ∫_ℝ^2dp v_1 · v_2 f^(p)_t
    
                    =    κ∫_ℝ^2dpv_2/p-1·∑_k 1^p ψ(|x_1-x_k|)(v_k-v_1)f^(p)_t + v_1/p-1·∑_k 2^p ψ(|x_2-x_k|)(v_k-v_2) f^(p)_t
    
                    =    2κ∫_ℝ^2dpv_2/p-1·∑_k 1^p ψ(|x_1-x_k|)(v_k-v_1)f^(p)_t.

        For the last integration in (<ref>), by the indistinguishability of the particles, one has  
        
    ∫_ℝ^2dp v_2·∑_k 1^p ψ(|x_1-x_k|)(v_k-v_1)f^(p)_t
       = ∫_ℝ^2dp v_2·∑_k 1,2^p ψ(|x_1-x_k|)(v_1-v_k)f^(p)_t + ∫_ℝ^2dp v_2·ψ(|x_1-x_2|)(v_2-v_1)f^(p)_t
       = ∫_ℝ^2dp v_2·∑_k 1^p ψ(|x_1-x_k|)(v_1-v_k)f^(p)_t 
                    +∫_ℝ^2dp v_2·ψ(|x_1-x_2|)(v_2-v_1)f^(p)_t
       + ∫_ℝ^2dp v_2·ψ(|x_1-x_2|)(v_2-v_1)f^(p)_t
       = -∫_ℝ^2dp v_2·∑_k 1^p ψ(|x_1-x_k|)(v_k-v_1)f^(p)_t +∫_ℝ^2dp |v_2-v_1|^2 ψ(|x_1-x_2|) f^(p)_t.

        Hence, recalling ψ≤ 1, it holds
        
    ∫_ℝ^2dp v_2·∑_k 1^p ψ(|x_1-x_k|)(v_k-v_1)f^(p)_t
            = 1/2∫_ℝ^2dp |v_2-v_1|^2 ψ(|x_1-x_2|)f^(p)_t≤ 2 M_2^v f_t.

        Then it yields from (<ref>) that 
        
    ∂/∂ t∫_ℝ^2dp v_1 · v_2 f^(p)_t≤4κ/p-1 M_2^vf_t_k-1.

        By combining (<ref>) and (<ref>), it holds
        
    ∂/∂ tM_2^vf_t    ≤ -2κψ_0 M_2^vf_t + 2κψ_04κτ/p-1M_2^vf_t_k-1 +  2κψ_0∫ v_1 · v_2 f_t_k-1^(p)
       =  -2κψ_0 M_2^vf_t + 2κψ_04κτ/p-1M_2^vf_t_k-1
       ≤  -2κψ_0M_2^v f_t + 8κ^2ψ_0/p-1τ M_2^vf_t_k-1.

        By Grönwall's inequality, one has
        
    M_2^v f_k≤ M_2^v f_t_k-1(e^-2κψ_0 τ + 8κ^2ψ_0/p-1τ^2 ) ≤ M_2^v f_0(e^-2κψ_0 τ + 8κ^2 ψ_0/p-1τ^2 )^k .

    
    


    
    
    
    
    
    
    
    
    
    
    
    
    





 §.§ The main theorems



First we discuss the Wasserstein distance between the distribution of the RBM of the N-particle system (<ref>) and the distribution of the N→∞ system (<ref>). 

Define the operator 𝒬_N^(k) on the probability measure space 𝒫(ℝ^2d) as follows. Let Z_i^R(0) be i.i.d drawn from f_0. Corresponding to Algorithm <ref>, we define 

    𝒬_N^(k)(f_0):= Law(Z_1^R(t_k)),

where Law(Z_1^R) means the law of Z_1^R.  Conditioning on a specific sequence of random batches, the particles are not exchangeable. However, when considering the mixture of all possible sequences of random batches, the laws of Z_i^R(t_k) are identical.

In addition, we define 𝒬_∞ as one time step evolution in Algorithm <ref>. Thus one arrives at 

    𝒬_∞^k (f_0) = 𝒬_∞∘⋯∘𝒬_∞(f_0),  (k copies),

which is expected to be the mean field limit of the RBM after k steps. Note that the 𝒬_∞ dynamics can fully determine the probability transition, knowing the marginal distribution of Z_1, while knowing only the marginal distribution is not enough for the dynamics of 𝒬_N^(k). The joint distribution must be known in the latter case. 


Theorem <ref> gives an estimate between 𝒬_∞^k (f_0) and 𝒬_N^(k)(f_0) on the W_q distance. In this theorem, we introduce a quantity ϵ_k∈ℝ, which was first introduced in <cit.>. Plainly speaking, the ϵ_k means the probability that the particle i is not “clean", where a particle i is clean at t_k^- means its batchmates at t≤ t_k were mutually independent and independent to the particle i when they interacted. For better organization, we give the definition of "clean"   and more details in Section <ref>.


    For f_0 with the compact support supp [f_0], it holds that 
    
    W_q(𝒬_∞^k (f_0),𝒬_N^(k) (f_0))≤ C (1+t_k) ϵ_k^1/q,

    with q ∈ [1,∞) and C=C(κ,ψ_0,R). For fixed k, it holds
    
    lim_N→∞ϵ_k =0,

    with ϵ_k≤𝒪(N^-1).







    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    



    Then, we consider the limit dynamics given by the operator 𝒬_∞ and how 𝒬_∞^n (f_0) approximates the dynamics of the Fokker-Planck equation (<ref>), i.e. the procedure (c) in (<ref>). We prove that the W_q distance can be controlled by τ with a smooth initial distribution f_0 with a compact support uniformly in time. 
    
     
        Let f be a solution to the Fokker-Planck  equation (<ref>) with initial condition f(0) = f_0. 
        It holds
        
        
        
        
        the uniform-in-time estimate
        
    sup_n W_q (𝒬_∞^n (f_0), f(nτ))≤ Cτ,

        with q∈[1,∞)and C=C(κ,ψ_0,R).
    

    







 §.§ Some helpful discussions



In addition, we give some  discussions on Lemma <ref> that will be used for the proof of Theorem <ref>, and on the commutation of (<ref>).





  §.§.§ Approximation using PDE analysis


Similar with [<cit.>, Lemma 4.4], one can obtain an 𝒪(τ^2/q) bound on the W_q distance between 𝒬_∞ (f_t_k) and S(f_t_k) in Proposition <ref>. Here S(f_t_k) denotes the law of (<ref>) in Section <ref>.


     Let f be a solution to the Fokker-Planck  equation (<ref>) with initial condition f(0) = f_0 ∈ (C^2∩ W^2,∞)(ℝ^2d). Suppose that the interaction kernel ψ satisfies 
    sup_r≥ 0 (|ψ^'(r)| + |ψ^''(r)|) ≤ϵ_ψ
 with a constant ϵ_ψ. It holds
    
    W_q (𝒬_∞ (f_t_k),S(f_t_k)) ≤ C(κ,ϵ_ψ , R, d) τ^2/q,    q∈ [1,∞).


The proof relies on the PDE analysis on the evolution equations on 𝒬_∞ (f_t_k) and S(f_t_k), using the corresponding operator groups. There are two limitations on this method. One is that more regular assumptions on both the interaction kernel and initial data are needed to guarantee the existence of a unique classical solution f_t ∈ C^2([0,T)×ℝ^2d) for (<ref>). In addition, if one uses the estimate obtained by Lemma <ref> with the total variance bound, the order of τ is 2/q. 
A detailed proof of Proposition <ref> is given in Appendix <ref>. 





  §.§.§ The mean-field limit of the Cucker-Smale system
 

Up to now, we have given a brief analysis of the path ((b),(c)) in the lower right corner of (<ref>). Here, we discuss the upper left corner path in the sense of the Wasserstein distance as well. Denote by 

    μ_t^N:=∑_i=1^N 1/Nδ_z_i(t)
 the empirical measure and μ_t^R the empirical measure derived by the RBM with μ_0^R = μ_0^N.



 
    For the path from (a) to (d) in (<ref>), it holds 
    
    𝔼 W_2 (f_t,μ_t^R)     ≤ C(κ, R,ψ)√(τ/p-1) +
            C(R)·{   N^-1/4     if  d<4,
       N^-1/4√(log(1+N))     if  d=4,
       N^-1/d     if  d>4.
            .



    We first consider the procedure (a). For the first τ→ 0 limit, by the definition of the W_q distance, one has
    
    W_2^2 (μ_t^R, μ_t^N) 
            ≤1/N∑_i=1^N (|X_i^R - X_i|^2 + |V_i^R - V_i|^2).

    Based on [<cit.>, Theorem 3.2], one has
    
    𝔼(1/N∑_i-1^N |V_i^R (t) - V_i (t)|^2)
            ≤ C(1/p-1-1/N-1)τ + Cτ^2 + C(1+τ) exp(-κψ_0 t),

    for some constant C depending on κ, R and ψ.
    Note that
    
    d/dt∑_i=1^N |X_i^R - X_i|^2 ≤
        2√(∑_i=1^N |X_i^R - X_i|^2)√(∑_i=1^N |V_i^R - V_i|^2),

    by the Cauchy-Schwarz inequality and
    
    √(∑_i=1^N |X_i^R - X_i|^2)≤√(∑_i=1^N |V_i^R - V_i|^2) T.

    It thus holds
    
    𝔼 W_2 (μ_t^N,μ_t^R) ≤𝔼1/√(N)√(∑_i=1^N |X_i^R - X_i|^2 + ∑_i=1^N |V_i^R - V_i|^2)
    ≤√(C τ(1/p-1-1/N-1))(1+T).
.

    Next, for the procedure (d), by the estimate of [<cit.>, Corollary 1], one has W_q (μ_t^N,f_t) ≤ C(κ,R)W_q(μ_0^N,f_0). 
    In addition, according to the convergence rate of the empirical measure in [<cit.>, Theorem 1], with f_0 compactly supported, one has 
    
    𝔼 W_q^q (f_0,μ_0^N) ≤ C(R)·{   N^-1/2     if  q>d/2,
       N^-1/2log(1+N)      if  q=d/2,
       N^-q/d     if  q<d/2.
        .


    Therefore, under the same assumption of Theorem <ref> and Theorem <ref>, for the path from (a) to (d), one has 
    
    𝔼 W_2 (f_t,μ_t^R)    ≤𝔼[W_2 (μ_t^N,μ_t^R) + W_2 (f_t,μ_t^N)]    
             ≤ C(κ,R,ψ)√(τ/p-1) + C(κ,R) 𝔼 W_2 (f_0,μ_0^N)
       ≤ C(κ,R,ψ)√(τ/p-1) + C(κ,R)(𝔼 W_2^2 (f_0,μ_0^N))^1/2.











§ PROOF OF THEOREM <REF>


    In this section, we prove Theorem <ref> by an observation in combinatorics. 
    Recalling the critical quantity ϵ_k we mentioned above Theorem <ref>, we first give the Lemmas <ref>-<ref>, for the proof of Theorem <ref>, and show the mathematical definition of ϵ_k in Section <ref>.
    The proof of Lemmas <ref>-<ref> are shown in Sections <ref>-<ref> respectively.
    
    Finally, Theorem <ref> will be proved in Section <ref> by combining Lemma <ref>, Lemma <ref> and an estimate on the diameters of the compact support in space and velocity, shown in Lemma <ref>.


    
    
        Under the assumption and notation of Theorem <ref>, it holds
            
    |𝒬_∞^k (f_0) - 𝒬_N^(k) (f_0)|_TV≤ 2ϵ_k,

        where ϵ_k is a positive constant  with respect to N.
    
    
    
    
        For any fixed k, with the assumption and notation of Theorem <ref>, it holds that ϵ_k ≤𝒪(N^-1).
    


    




 §.§ Prerequisites

    To give a detailed explanation of ϵ_k in Theorem <ref>, we introduce the concept “clean” in Definition <ref>. First, for each particle i, we define a sequence of lists {L_i^(k)} associated with i, given as follows:
    
        
  * L_i^(0) = {i};
        
  * For k≥ 1, let 𝒞_q^(k-1) be the batch that particle i stays in for t∈[t_k-1,t_k), then
        
    L_i^(k) = ∪_j∈𝒞_q^(k-1) L_j^(k-1).

    
    Here, L_i^(k) can be viewed as the particles that have impacted i for t < t_k. Clearly, a particle i_1∈ L_i^(k) might not have been a batchmate of i. It could have been a batchmate of i_2, and then i_2 was a batchmate of i at some time. The important observation is that if L_i^(k) and L_j^(k) do not intersect for a given sequence of random batches, then particles i and j are independent at t_k^-. Note that we are not claiming all particles in L_j^(k) are independent of those in L_j^(k) at t_k^-. In fact, it is possible that some i_1 ∈ L_i^(k) and j_1 ∈ L_j^(k) are in the same batch on [t_k-1, t_k). However, i_1 and j_1 must be independent at the times when they were added to the batches that eventually impact i,j at t_k^-. Then we are motivated to define the following.
    
    
        We say the particle i is clean on [t_k,t_k+1) if the batch 𝒞_q^(k) that contains i at t_k^+ satisfies the following:
        
            
  * any j∈𝒞_q^(k) is clean at t_k^-;
            
  * any j,ℓ∈𝒞_q^(k) with jℓ, L_j^(k) and L_ℓ^(k) do not intersect.
        
    
    In other words, a particle i is “clean" at t_k^- if its batchmates at t<t_k were mutually independent and independent to i when they interacted.
    
   
    Let A_k denote the set of particles that are clean at t_k^-. Then
    
    A_0 = {1,⋯, N}.

    For k≥ 1, one has 
    
    A_k = { i ∈ A_k-1: i∈𝒞_q^(k-1), ∀ j,ℓ∈𝒞_q^(k-1),  jℓ ,
    
            j∈ A_k-1, ℓ∈ A_k-1,  L_j^(k-1)∩ L_ℓ^(k-1) = ∅}.

    Now we define
    
    ϵ_k := ℙ(1∉ A_k).

    Note that by symmetry, ϵ_k is also the probability that particle i is not clean.









 §.§ Proof of Lemma <ref>


    We first introduce Lemma <ref>.
     
        Consider a fixed sequence of divisions of random batches {𝒞^(ℓ)}_ℓ≤ k-1.
        
            
  * It holds that
                
    |L_i^(k)|≤ p^k,

                and the particle i is clean at t_k^- if and only if the equality holds.
            
  * The distribution of Z_i (Recall Z_i := (X_i,V_i).) for a clean particle i at t_k^- is 𝒬_∞^k (f_0).
        
        Here the symbol |A| means the cardinality of a set A.
    
    
        The proof is a straightforward induction. Here, let us just briefly mention the proof of the second claim. For k=0, the statement is trivial. Now, suppose the statement is true for all k ≤ m-1. We now consider k = m.
        
        For the given sequence of random batches {𝒞^(ℓ)}_ℓ≤ k-1, that a particle i is clean at t_m^- means that on [t_m-1, t_m), the particles in the batch for i are independent of i at t_m-1. By the induction assumption, the distribution of one particle at t_m-1 is given by 𝒬_∞^m-1(f_0). By the independence, the joint distribution of them at t_m-1 is therefore
        
    f^(p) (⋯, t_m-1) = 𝒬_∞^m-1(f_0)^⊗ p.

        From t_m-1 to t_m, the evolution of the joint distribution obeys  equation (<ref>). Hence, at t_m^-, the distribution of particle i is given by 𝒬_∞^m(f_0). 
    



    Now we begin to prove Lemma <ref>.

    
        By Lemma <ref>, one has
        
    𝒬_N^(k) (f_0) = ℙ(1∈ A_k)𝒬_∞^k (f_0)
                + ℙ(1∉ A_k)ν_k,

        for some probability measure ν_k. To see this, consider all possible sequences of random batches. Only the first k divisions of batches (i.e. ones at t_0,⋯, t_k-1) will affect the distribution at t_k. This subsequence (the first k divisions) can take only finitely many values, and let {c^ℓ}_ℓ≤ k-1 be such values. Then, for any E⊂ℝ^d that is Borel measurable,
        
    𝒬_N^(k) (f_0)[E] = ∑_{𝒞^ℓ = c^ℓ , ℓ≤ k-1}ℙ(𝒞^ℓ = c^ℓ ,  ℓ≤ k-1)
                ℙ(Z_1 ∈ E  | 𝒞^ℓ = c^ℓ ,  ℓ≤ k-1 ).

        Lemma <ref> tells us if {c^ℓ}_ℓ≤ k-1 is a value such that particle 1 is clean, then
        
    ℙ(Z_1 ∈ E  | 𝒞^ℓ = c^ℓ ,  ℓ≤ k-1 ) = 𝒬_∞^k (f_0)[E].

        Hence,
        
    𝒬_N^(k) (f_0)[E]    =  ℙ(1∈ A_k)𝒬_∞^k (f_0)[E] 
         + ∑_{𝒞^ℓ = c^ℓ , ℓ≤ k-1},  1∉ A_kℙ(𝒞^ℓ = c^ℓ ,  ℓ≤ k-1)
                ℙ(Z_1 ∈ E  | 𝒞^ℓ = c^ℓ ,  ℓ≤ k-1 )
       = ℙ(1∈ A_k)𝒬_∞^k (f_0)[E] + ℙ(1∉ A_k)ν_k (E),

        
        
        
        
        
        
        with
        
    ν_k (E) = ∑_{𝒞^ℓ = c^ℓ , ℓ≤ k-1},  1∉ A_kℙ(𝒞^ℓ = c^ℓ ,  ℓ≤ k-1)/ℙ(1∉ A_k)ℙ(Z_1 ∈ E  | 𝒞^ℓ = c^ℓ ,  ℓ≤ k-1 ).

        Clearly, ν_k is a convex combination of some conditional marginal distributions of Z_1, each being Law(Z_1) conditioning on a particular sequence of batches for {1∉ A_k}. Hence ν_k is a probability measure. 
        
        By (<ref>), it holds that
        
    |𝒬_∞^k(f_0) - 𝒬_N^(k) (f_0)| ≤ 
                (1-ℙ(1∈ A_k))𝒬_∞^k (f_0) + ℙ(1∉ A_k)ν_k
                = ϵ_k (𝒬_∞^k (f_0) + ν_k).

        Therefore, the total variation distance between the two measures is controlled by
        
    |𝒬_∞^k(f_0) - 𝒬_N^(k) (f_0)|_TV≤ 
                (1-ℙ(1∈ A_k)) + ℙ(1∉ A_k) = 2ϵ_k.

    




    
    
    




 §.§ Proof of Lemma <ref>

    Furthermore, we provide the proof of Lemma <ref> which is a simple variant of [<cit.>, Theorem 3.2].

    
        First of all, clearly, one has
        
    ϵ_0  = 1-1 = 0.

        Now, one can do induction on k. Assume
        
    lim_N→∞ϵ_k = 0,

        and then consider the batches for t_k → t_k+1^-. Assume the batch for particle 1 is 𝒞_q^(k). Denote
        
    B_k = {∀ j, ℓ∈𝒞_q^(k),  jℓ,   L_j^(k)∩ L_ℓ^(k) = ∅}.

        Let ℬ := 𝒞_q^(k)\{1} be the set of other particles that share the same batch with particle 1. Then, by definition of A_k+1, it holds
        
    ℙ (1∈ A_k+1) = ∑_j_1,⋯, j_p-1ℙ(ℬ={j_1,⋯, j_p-1})×
    ℙ(B_k ∩{1∈ A_k}∩_ℓ =1^p-1{j_ℓ∈ A_k} | ℬ= {j_1,⋯, j_p-1}).

         Denote E:= {ℬ={j_1,⋯, j_p-1}}, where we omit the dependence in j_ℓ, 1≤ℓ≤ p-1 for notational convenience. Conditioning on B_k ∩ E (i.e., provided that the event B_k ∩ E happens), whether the particles are clean or not is independent. Hence,
         
    ℙ(B_k ∩{1∈ A_k}∩_ℓ =1^p-1{j_ℓ∈ A_k} |  E) 
    
                 = ℙ(B_k | E))ℙ({1∈ A_k}∩_ℓ = 1^p-1{j_ℓ∈ A_k}|E,B_k)
    
                 = ℙ(B_k|E)Π_ℓ=1^pℙ(j_ℓ∈ A_k|E,B_k),

         where one has set j_p =1. Moreover,
         
    ℙ(j_ℓ∈ A_k|E,B_k) = ℙ({ j_ℓ∈ A_k }∩ E∩ B_k)/ℙ(E∩ B_k) = ℙ({1 ∈ A_k }∩ E∩ B_k)/ℙ(E∩ B_k) = ℙ({1 ∈ A_k }∩ B_k)/ℙ(B_k).

         The second and the last equalities are due to symmetry. For the last equality, ℙ({ℬ = {j_1⋯,j_p-1}}∩ B_k) should be equal for all possible j_1,⋯,j_p-1, and the same  is true for the numerator. This actually is a kind of independence. Hence, eventually due to the fact
         
    ∑_j_1,⋯, j_p-1ℙ (ℬ = {j_1,⋯, j_p-1}) ℙ(B_k | ℬ = {j_1,⋯, j_p-1}) = ℙ (B_k),

        one has
        
    1-ϵ_k+1 = ℙ (1∈ A_k+1) ≥ℙ (B_k) (1-ϵ_k 
                )^p.

        Hence, it suffices to show
        
    lim_N→∞ℙ (B_k) =1.

        To get an estimate for this, we consider the following equivalent way to construct L_i^(k): one starts with L_i ←{i} and repeat the following k times:
        
            
  * Set L_tmp← L_i and A=∅.
            
  * Loop the following until L_tmp is empty.
            
                
  (a) Pick a particle i_1 ∈ L_tmp, then choose p-1 particles from {1,⋯,N}\ A ∪{i_1} denoted by { i_2,⋯ , i_p}.
                
  (b) Set L_i← L_i ∪{ i_2, ⋯, i_p}.
                
  (c) Set A← A ∪{ i_1, i_2, ⋯, i_p}.
                
  (d) Set L_tmp← L_tmp\{ i_1, i_2, ⋯, i_p}.
            
        
        In the above, we are actually looking back from t_k-1. In the j-th iteration, we are constructing batches at t_k-j. Hence, this is an equivalent way to construct L_i^(k).

        Now, we estimate ℙ(B_k) by constructing the lists L_j_ℓ^(k): 1≤ℓ≤ p for j_ℓ∈𝒞_q^(k) using the above procedure. Consider that the lists for j_1,⋯,j_ℓ -1 have been constructed, which have included at most (ℓ -1)p^k particles. Now, for L_j_ℓ^(k) not to intersect with the previous lists, one has to choose particles from {1,⋯,N}\ [∪_z=1^ℓ -1 L_j_z^(k)∪ A ∪{i_1}] in 2.(a) step. Conditioning on the specific choices of L_j_ℓ^(k): 1≤ℓ≤ p and A with 
        
    N_1:= |L_j_ℓ^(k)∪ A∪{i_1}|,   N_2:= |A|,

        this probability is controlled from below by
        
    N- N_1p-1/N-1- N_2p-1≥N- 1-ℓ p^kp-1/N-1p-1.

        Hence, as N→∞,
        
    ℙ(B_k)≥Π_ℓ=1^p [N- 1-ℓ p^kp-1/N-1p-1]^k = 1- 𝒪(N^-1).

        Hence, lim_N→∞ϵ_k+1=0 and the claim follows.
    
    





 §.§ Proof of Theorem <ref>

    


    
    
    Now we can prove Theorem <ref>.
    
    Recall that the TV distance is defined by
    
    |μ - ν|_TV := sup_A∈ℱ |μ(A) - ν(A)|,

    for two probability measures μ, ν defined on certain sample space Ω with events ℱ. 


    We denote the diameters of the compact support in the spatial and velocity variables at time t by 𝒟_X(t) and 𝒟_V(t), respectively, i.e.
    
    𝒟_X(t) := max_i,j| X_i-X_j|,   𝒟_V(t) := max_i,j| V_i-V_j|.

    
    For the RBM system (<ref>), Ha et al. <cit.> gave an estimate on 𝒟_X^R and 𝒟_V^R shown in Lemma <ref>. This estimate works on any fixed sequence of batches and we provide the proof in Appendix <ref> for completeness.
    




        
        Let X^R,V^R be a solution of the RBM model (<ref>) with initial data f_0 satisfying 
        
    𝒟_X(0) + 𝒟_V(0)<∞.

        Then one has
        
    𝒟_V^R (t) ≤𝒟_V(0),   𝒟_X^R (t)≤𝒟_X(0) + 𝒟_V(0)t,   ∀ t ≥ 0.

    



    
    

    
        Denote z:= (x,v), Z^R_i:=(X_i^R,V_i^R), and |z|^q:=|x|^q + |v|^q.
        By Lemma <ref>, taking μ = 𝒬_∞^k (f_0) and ν = 𝒬_N^(k)(f_0), one has
        
    W_q (𝒬_∞^k (f_0),𝒬_N^(k)(f_0)) ≤ 2^1-1/q(∫_ℝ^2d |z|^q d | 𝒬_∞^k (f_0)-𝒬_N^(k)(f_0)|(z))^1/q.


       Note that Lemma <ref> degenerates into the original Cucker-Smale model when p=N. Hence for t∈[0,t_1), any {Z_i^(0)(t)}_i=1^p satisfying (<ref>), one obtains
       
    𝒟_V^(0)(t)≤𝒟_V(0),   𝒟_X^(0) (t) ≤𝒟_X (0) + 𝒟_V(0)t.

       Also, by the assumption of f_0, one has v=0, and therefore x=0. Since (<ref>) holds independent of the choice Z_i, defining 
       
    𝖷(t) :=   max{ |x|  |∃ v,  s.t.  f(x,v;t) >0},
    𝖵(t) :=   max{ |v|  |∃ x,  s.t.  f(x,v;t) >0},

       one has for t∈[0,t_1],
       
    𝖵(t)≤ C,   𝖷(t)≤ C(1+t),

       where C is a constant depending on R. 
       Then for t∈[t_1,t_2], for the new subsystem {Z_i^(1)}_i=1^p with initial distribution f_t_1, one can use Lemma <ref> again to get
       
    𝒟_V^(1)(t)   ≤𝒟_V^(1)(0)≤ 2𝖵(t_1) ≤ C, 
    𝒟_X^(1) (t)    ≤𝒟_X^(1) (0) + 𝒟_V^(1)(t-τ)
               ≤ 2𝖷(t_1) + 2𝖵(t_1)(t-τ)≤ C(1+t),

       and (<ref>) holds for t∈[t_1,t_2].
       By induction, for any t, (<ref>) holds. Therefore |𝖷(t_k)|^q + |𝖵(t_k)|^q ≤ C(1+t_k)^q.       
       
       Also, by Lemma <ref>, it holds |Z^R_i|^q≤ C(1+t_k)^q.
       Then for any z∈supp[| 𝒬_∞^k (f_0)-𝒬_N^(k)(f_0)|], |z|^q≤ C(1+t_k)^q. Therefore, 
       
    ∫_ℝ^2d |z|^q d | 𝒬_∞^k (f_0)-𝒬_N^(k)(f_0)|(z) 
               
               
               
               ≤  C(1+t_k)^q | 𝒬_∞^k (f_0)-𝒬_N^(k)(f_0)|_TV.

       
       By Lemma <ref>, one has 
       
    ∫_ℝ^2d |z|^q d | 𝒬_∞^k (f_0)-𝒬_N^(k)(f_0)|(z) ≤ C(1+t_k)^q ϵ_k.

       Hence W_q (𝒬_∞^k (f_0),𝒬_N^(k)(f_0)) ≤ C(1+t_k)ϵ_k^1/q. Then by Lemma <ref>, we finish the proof.
    


    
        The above analysis is not enough to get the mean-field limit independent of τ and for fixed N, ϵ_k→ 1 as k→∞. This is a limitation of the technique in the paper.
    














§ PROOF OF THEOREM <REF>


    Here gives an outline of the proof of Theorem <ref>. First, we show the stability of 𝒬_∞ in Section <ref>. Second, an auxiliary system is introduced in Section <ref> as a bridge to estimate the W_q distance between 𝒬_∞(f_t_k) and f_t_k+1. 
    
    Last,  the proof of Theorem <ref> is given in Section <ref>.




 §.§ Stability of 


In Lemma <ref> and Lemma <ref>, we give the stability analysis on the operator 𝒬_∞.
 
    Suppose two probability measures μ_1,μ_2 satisfying ∫ vdμ_1=∫ vdμ_2, with compact supports supp[μ_1] , supp[μ_2] respectively, where supp[μ_i]⊂ℝ^d× B(0,R_μ), i=1,2, and the constant R_μ >0. Then it holds
    
    W_q (𝒬_∞ (μ_1) ,𝒬_∞ (μ_2)) ≤ CW_q(μ_1,μ_2),    q=[1,∞),

    where C(κ,ψ,R_μ) is a positive constant.



    Set (X_i,V_i)(0)∼μ_1, and (X_i,V_i)(0)∼μ_2, for i=1, ⋯, p. The key to the proof is to make use of Lemma <ref> to get the contraction property of the velocity difference. 
    
    The two steps are as follows.
    

  
Step 1.
    First, we show an auxiliary estimate on 𝒟_V(t). By following the formulation of [<cit.>, Lemma 2.2], denote
    
    Ψ_ij:= ψ(|X_i-X_j|)/p + ( 1- ∑_j=1^p ψ(|X_i-X_j|)/p)δ_ij,

    where δ_ij is the Kronecker delta. Then Ψ_ij is nonnegative and satisfies
    
    Ψ_ij≥ψ(|X_i -X_j |)/p,   ∑_j=1^p Ψ_ij =1,  ∑_j=1^pΨ_ij(V_j-V_i) = ∑_j=1^pψ(|X_i -X_j |)/p(V_j-V_i).

    We denote V_i = (V_i^k) and obtain the following equality
    
    1/2d/dt   |V_i - V_j|^2
            = 1/2d/dt∑_k=1^d|V_i^k - V_j^k|^2
    
            =   ∑_k=1^d(V_i^k - V_j^k)(V_i^k - V_j^k)
    
            =   ∑_k=1^d κ p/p-1 (V_i^k - V_j^k)[
            ∑_m=1^p Ψ_im(V_m^k - V_i^k) - ∑_m=1^p Ψ_jm(V_m^k - V_j^k)]
    
            =    -∑_k=1^d κ p/p-1 (V_i^k - V_j^k)^2
            +  ∑_k=1^d κ (V_i^k - V_j^k)( ∑_m=1^p (Ψ_im-Ψ_jm)V_m^k)
    
            =    -∑_k=1^d κ p/p-1 (V_i^k - V_j^k)^2
       +  ∑_k=1^d κ p/p-1 (V_i^k - V_j^k)[ ∑_m=1^p (Ψ_im-min{Ψ_im,Ψ_jm} + min{Ψ_im,Ψ_jm} - Ψ_jm)V_m^k].

    For a given time t, one can choose indices i_t and j_t satisfying the relation 
    
    𝒟_V(t) = |V_i_t(t) -V_j_t(t)|,

    and thus for each m=1,2,⋯,p,
    
    ∑_k=1^d (V_i_t^k -V_j_t^k)V_j_t^k ≤∑_k=1^d (V_i_t^k -V_j_t^k)V_m^k ≤∑_k=1^d (V_i_t^k -V_j_t^k)V_i_t^k.

    Combining (<ref>) and (<ref>), one obtains
    
    1/2d/dt|V_i_t - V_j_t|^2       
            ≤     -∑_k=1^d κ p/p-1 (V_i_t^k - V_j_t^k)^2
       +  ∑_k=1^d κ p/p-1 (V_i_t^k - V_j_t^k)V_i_t^k ∑_m=1^p (Ψ_im-min{Ψ_im,Ψ_jm}) 
       -  ∑_k=1^d κ p/p-1 (V_i_t^k - V_j_t^k)V_j_t^k ∑_m=1^p (Ψ_jm -min{Ψ_im,Ψ_jm})
    
            =   -∑_m=1^pκ p/p-1min{Ψ_im,Ψ_jm}|V_i_t - V_j_t|^2
    ≤    - κ p/p-1ψ_0 |V_i_t - V_j_t|^2.

    Note that 𝒟_V is Lipschitz continuous with respect to t and thus it is differentiable almost everywhere. Consider the instant t at which 𝒟_V is differentiable. Then,
    
    d/dt𝒟_V^2 ≤d/dt|V_i_t - V_j_t|^2.

    Combining with (<ref>) and (<ref>), one obtains
    
    |d/dt𝒟_X| ≤𝒟_V,   d/dt𝒟_V ≤ -κ p/p-1ψ_0 𝒟_V,

    which is an SDDI. By Lemma <ref>, there exists a positive constant R_μ such that 
    
    𝒟_V(t)≤ 2e^-κψ_0 tR_μ.




    

  
Step 2.
    
    It can be obtained that 
    
    d/dt𝔼 | V_1 - V_1 |^q =  
               1/pd/dt𝔼∑_i=1^p (∑_k=1^d | V_i^k - V_i^k |^2)^q/2,

    with
    
    d/dt𝔼   (∑_k=1^d | V_i^k - V_i^k |^2)^q/2 
            =  𝔼[ q/2(∑_k=1^d | V_i^k - V_i^k |^2)^q/2-1d/dt∑_k=1^d 
    |V_i^k - V_i^k|^2 ]
    
            =    qκ/p-1𝔼[ |V_i - V_i|^q-2∑_j=1^p ψ(|X_j - X_i|)∑_k=1^d(V_j^k - V_i^k - V_j^k + V_i^k)( V_i^k - V_i^k). 
       .  + ∑_j=1^p 
            |V_i - V_i|^q-2(ψ(|X_j - X_i|) - ψ(|X_j - X_i|))
           ∑_k=1^d(V_j^k - V_i^k)( V_i^k - V_i^k) 
           ].

    Rewrite the right-hand side of (<ref>) as
   
    d/dt𝔼 | V_1 - V_1 |^q
           := I_21 + I_22,

   where
   
    I_21:= qκ/p-1𝔼∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) |V_i - V_i|^q-2 (V_j^k - V_i^k - V_j^k + V_i^k)(V_i^k - V_i^k),

   and 
   
    I_22:= qκ/p-1𝔼∑_i,j=1^p ∑_k=1^d
            (ψ(|X_j - X_i|) - ψ(|X_j - X_i|))
    ·|V_i - V_i|^q-2
           ( V_i^k - V_i^k )(V_j^k - V_i^k).

   First estimate I_21 using the index changing trick:
   
    I_21 =    qκ/p-1𝔼∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) |V_i - V_i|^q-2 [(V_j^k -V_j^k)-( V_i^k - V_i^k )](V_i^k - V_i^k)
    
               =    qκ/p-1𝔼[ ∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) |V_i - V_i|^q-2 (V_j^k -V_j^k)(V_i^k - V_i^k).
       .- ∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) |V_i - V_i|^q-2(V_i^k - V_i^k)^2]
    
               =    qκ/p-1𝔼[ ∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) |V_j - V_j|^q-2 (V_j^k -V_j^k)(V_i^k - V_i^k).
       .- ∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) |V_i - V_i|^q-2(V_i^k - V_i^k)^2]
    
               =     -qκ/p-1𝔼[ ∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|). 
       ·[|V_i - V_i|^q-2 (V_i^k -V_i^k) - |V_j - V_j|^q-2 (V_j^k -V_j^k)]
               .(V_i^k - V_i^k)]
    
               =    qκ/2(p-1)𝔼[ ∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) .
       .·[|V_i - V_i|^q-2 (V_i^k -V_i^k) - |V_j - V_j|^q-2 (V_j^k -V_j^k)][(V_j^k - V_j^k)-(V_i^k - V_i^k)]].

    One can use the relation 
    
    ∑_k=1^d[|V_i - V_i|^q-2 (V_i^k -V_i^k) - |V_j - V_j|^q-2 (V_j^k -V_j^k)]
            ·.[(V_j^k - V_j^k)-(V_i^k - V_i^k)]] ≤ 0,

    and the condition (<ref>) of ψ to obtain
    
    I_21≤   qκ/2(p-1)ψ_0 𝔼{∑_i,j=1^p ∑_k=1^d 
            [|V_i - V_i|^q-2 (V_i^k -V_i^k) - |V_j - V_j|^q-2 (V_j^k -V_j^k)]
       · [(V_j^k - V_j^k)-(V_i^k - V_i^k)]}
    
            =    qκ/2(p-1)ψ_0 𝔼[∑_i=1^p ∑_k=1^d 
            |V_i - V_i|^q-2 (V_i^k -V_i^k)∑_j=1^p (V_j^k -V_j^k).
       -  ∑_i=1^p ∑_k=1^d 
             p|V_i - V_i|^q-2 (V_i^k -V_i^k)^2
            -  ∑_j=1^p ∑_k=1^d 
             p|V_j - V_j|^q-2 (V_j^k -V_j^k)^2
       +  .∑_j=1^p ∑_k=1^d 
             |V_j - V_j|^q-2 (V_j^k -V_j^k)∑_i=1^p (V_i^k -V_i^k)].

    By the momentum condition 𝔼∑_i=1^p (V_i^k - V_i^k) = 0, it holds
    
    I_21≤  -p^2qκ/p-1ψ_0 𝔼|V_1 - V_1|^q.


   For I_22, one has
    
    I_22≤   qκ/p-1ψ_Lip𝔼[
                ∑_i,j=1^p ∑_k=1^d | |X_j - X_i| - 
                |X_j - X_i | | V_i - V_i|^q-2|V_i^k - V_i^k| |V_j^k - V_i^k|
                ]
    ≤   qκ/p-1ψ_Lip𝔼[
                ∑_i,j=1^p| |X_j - X_i| - 
                |X_j - X_i | | |V_i - V_i|^q-1 |V_j - V_i|
                ],

    by the Cauchy inequality. Then by the inequality | |a-b| - |c-d| | ≤ |a-c| + |b-d|, one has
   
    I_22≤   qκ/p-1ψ_Lip𝔼[
                ∑_i,j=1^p| |X_i - X_i| + 
                |X_j - X_j | | |V_i - V_i|^q-1 |V_j - V_i|
                ]
    ≤   qκ/p-1ψ_Lip𝒟_V(t) 𝔼[p
                ∑_i=1^p |X_i - X_i| | V_i - V_i |^q-1 + 
                ∑_i,j=1^p |X_j - X_j| | V_i - V_i |^q-1]
    ≤   2q p^2κ/p-1ψ_Lip𝒟_V(t) 𝔼[
                 |X_1 - X_1| | V_1 - V_1 |^q-1],

    by Hölder's inequality. Now by combining (<ref>) and (<ref>) and (<ref>), one has
    
    d/dt𝔼| V_1 - V_1 |^q ≤    -p/p-1qκψ_0 
                𝔼|V_1 - V_1|^q 
       + 2p/p-1qκψ_Lip𝒟_V(t)
                𝔼[|X_1-X_1| |V_1 - V_1|^q-1]
    ≤    -p/p-1qκψ_0 
                𝔼|V_1 - V_1|^q 
       + 2p/p-1qκψ_Lip𝒟_V(t)
                (𝔼[|X_1-X_1|^q])^1/q (𝔼|V_1 - V_1|^q)^1-1/q,

    similarly by Hölder's inequality. Since
    
    d/dt (𝔼   |V_1 - V_1|^q)^1/q = 
                1/q(𝔼|V_1 - V_1|^q)^1/q-1d/dt𝔼|V_1 - V_1|^q
    ≤    -pκ/p-1ψ_0 
                (𝔼[|V_1 - V_1|^q_q])^1/q 
                + 2pκ/p-1ψ_Lip𝒟_V(t)
                (𝔼[|X_1-X_1|^q])^1/q.

    and 
    
    d/dt𝔼|X_1 - X_1|^q =    𝔼q/2(∑_k=1^d |X_1^k - X_1^k|^2)^q/2-1∑_k=1^d(X_1^k - X_1^k)· 2(V_1^k - V_1^k)
    ≤   𝔼 q |X_1 - X_1|^q-1 |V_1 - V_1 |
    ≤    q (𝔼|X_1 - X_1|^q)^1-1/q (𝔼|V_1 - V_1|^q)^1/q,

    thus it holds 
    
    d/dt(𝔼|X_1 - X_1|^q)^1/q = 
            1/q(𝔼|X_1 - X_1|^q)^1/q-1d/dt𝔼|X_1 - X_1|^q
            ≤ (𝔼|V_1 - V_1|^q)^1/q.

    Note that 𝒟_V≤ 2 e^-κψ_0 t  R_μ by Step 1. 
    Now one knows that both (𝔼|X_1 - X_1|^q)^1/q and (𝔼|V_1 - V_1|^q)^1/q satisfy Lemma <ref> and then there exists a constant C such that
    
    (𝔼|X_1 - X_1|^q)^1/q(t)
            + (𝔼|V_1 - V_1 ^q)^1/q(t)
            ≤ C [(𝔼|X_1 - X_1 |^q)^1/q(0)
            + (𝔼|V_1 - V_1 |^q)^1/q(0)].

    Then, 
    
    W_q (𝒬_∞ (μ_1) ,𝒬_∞ (μ_2)) ≤   (𝔼 |X_1 -X_1|^q (t) + 𝔼 |V_1 -V_1|^q(t))^1/q
    ≤   (𝔼|X_1 - X_1|^q (t))^1/q
             + (𝔼|V_1 - V_1|^q (t))^1/q
    ≤    C [(𝔼|X_1 - X_1|^q)^1/q(0)
            + (𝔼|V_1 - V_1|^q)^1/q(0)]
    ≤   2 C [𝔼 |X_1 -X_1|^q (0) + 𝔼 |V_1 -V_1|^q(0)]^1/q,

    since a^1/q+ b^1/q≤ 2^1-1/q(a+b)^1/q for a≥0, b≥ 0 with q≥ 1. For any ϵ>0, the initial data can be chosen by a coupling such that 
    
    [𝔼 |X_1 -X_1|^q (0) + 𝔼 |V_1 -V_1|^q(0)]^1/q≤ϵ + W_1(μ_1,μ_2).

    Hence by the arbitrariness of ϵ, it holds W_q (𝒬_∞ (μ_1) ,𝒬_∞ (μ_2)) ≤ C W_q(μ_1,μ_2),







    
    For f defined in (<ref>) with R<∞, one has
    
    W_q(𝒬_∞^n-m+1(f_m-1),𝒬_∞^n-m(f_m))
            ≤ C W_q(𝒬_∞(f_m-1), f_m),    q=[1,∞),

    where C is relevant to ψ,κ,R and d.
    
    
        
        By the flocking property shown in Step 1 in Lemma <ref>, one has supp^v[𝒬_∞(f_m-1)]⊂supp^v[f_m-1]. Also, supp^v[f_m-1]⊂supp^v[f_0] by Lemma <ref>. Furthermore, for n-m-1≥ k≥ 0, 𝒬^k+1_∞ (f_m-1) and 𝒬^k_∞ (f_m) have compact supports, where 𝒬^0_∞ (f_m):=f_m.
        
        Similarly with the proof of Proposition <ref>, note that (X_1(kτ),V_1(kτ) and (X_1(kτ), V_1(kτ)) obey 𝒬^k+1_∞ (f_m-1) and 𝒬^k_∞ (f_m) respectively, which satisfy the momentum equality assumption of Lemma <ref>. 
    
        
        Denote 
        
    |Δ X_1|:=(𝔼|X_1 - X_1|^q)^1/q,   |Δ V_1|:=(𝔼|V_1 - V_1|^q)^1/q,   
            |Δ Z_1|:=|Δ X_1|+|Δ V_1|,

        for q≥ 1.
        Hence, by the analysis in Lemma <ref>, (<ref>) and (<ref>) still hold for t∈ [0,(n-m)τ].  
        By Lemma <ref>, one obtains
        
    |Δ Z_1 ((n-m)τ)| ≤ C |Δ Z_1 (0)|.

        By choosing a suitable coupling of the initial data, the proof is finished.
    


    

 §.§ An auxiliary system

    
    We introduce an auxiliary system (<ref>) with law S(f_t_k) to get the estimate on 
    W_q (𝒬_∞ (f_t_k), f_t_k+1) ≤ W_q(S(f_t_k),f_t_k+1) + W_q(S(f_t_k),𝒬_∞ (f_t_k) ).
 

    Rewrite (<ref>) into the following particle system
    
    {d/dtX_i(t) =   V_i(t),  i=1,⋯, N,
    d/dtV_i(t) =   ∫κψ(|x-X_i|)(v-V_i) f_t(x,v)dxdv.
            .


    In order to show the law of (X,V) at time nτ is close to 𝒬_∞^n, we consider the transient system
    
    {d/dtX̂_i(t) =   V̂_i(t),  i=1,⋯, N,
    d/dtV̂_i(t) =   ∫κψ(|x-X̂_i|)(v-V̂_i) f_t_k(x,v)dxdv,
            .

    with (X̂(0),V̂(0))∼f_t_k.
    Denote S(f_t_k):=Law(X̂(τ),V̂(τ)).

    
    Under the same assumption of Lemma <ref>, it holds
        
    supp[S(f_t_k)]⊂
                B(0,C)× B(0,C e^-κψ_0 t_k+1).

    
    
           For S(f_t_k), define
        
    𝖷_k (t) :=   max{ |x(t,x^k,v^k)|  | (x^k,v^k)∈supp[f_t_k]},
    𝖵_k (t) :=   max{ |v(t,x^k,v^k)|  | (x^k,v^k)∈supp[f_t_k]},

        where the pair (x(t,x^k,v^k), v(t,x^k,v^k)) denotes (x(t),v(t)) being the solution of (<ref>) with (x(t_k),v(t_k)) = (x^k,v^k). Note that the maximum is reached because it is assumed that supp[f_t_k] is compact. Denote K_t^x ⊂supp^x [f_t_k] (resp. K_t^v ⊂supp^v [f_t_k]) the set of points (x^k,v^k) such that the maximum is reached in 𝖷_k(t) (resp. in 𝖵_k(t)). By definition, one has |𝖷_k(t)|^2 = |x(t,x^k,v^k)|^2 for every (x^k,v^k)∈ K_t^x. It follows from Danskin's theorem <cit.> and from the fact that ∂_t x(t,x^k,v^k) = v(t,x^k,v^k) that
        
    𝖷_k(t)·𝖷_k(t) = max{ x(t,x^k,v^k) · v(t,x^k,v^k) | (x^k,v^k)∈ K_t^x }.

        Using the Cauchy-Schwarz inequality, one obtains that 
        
    𝖷_k(t) ≤𝖵_k(t).

        Similarly, by  Danskin's theorem again, one has
        
    𝖵_k(t)·𝖵_k(t) = max{ v(t,x^k,v^k) ·∫κψ(|x-x(t,x^k,v^k)|)(v-v(t,x^k,v^k)) f_t_kdxdv
    | (x^k,v^k)∈ K_t^v }.

        Note that by the definition of K_t^v, v(t,x^k,v^k) · (v-v(t,x^k,v^k))≤ 0. It holds that 
        
    𝖵_k(t)·𝖵_k(t)  ≤
                -κψ_0 ∫ v(t,x^k,v^k) · v(t,x^k,v^k) f_t_k dxdv,   (x^k,v^k)∈ K_t^v ,

        since ∫ v f_t_k dxdv = 0 and ∫f_t_kdxdv =1. Then one has 
        
    𝖵_k(t) ≤ -κψ_0 𝖵_k(t),

        and thus (<ref>) holds.
    


    
    Under the same assumption of Lemma <ref>, there exists a positive constant α such that
        
    W_q(S(f_t_k),f_t_k+1)≤ C e^-κψ_0 t_kτ^2,    q∈[1,∞),

        with C=C(κ,ψ_0,ψ_Lip, R).
    
    
    
    
    
    
        Let ℛ(x,v,t) = ∫ψ(|y-x|)(w-v)(f_t(y,w)-f_t_k(y,w))dydw. For t∈ [t_k,t_k+1), by straightforward calculus, it holds
        
        
    | ∂ℛ/∂ t (x,v,t)| =    | ∫ψ(|y-x|)(w-v) ∂f_t/∂ t(y,w)dydw|
    
                    =    | ∫ψ(|y-x|)(w-v) [ - w·∇
                    _y f_t(y,w) ..
       + ..∇_w ·(f_t(y,w)κ∫ψ (|y-y|)(w-w)f_t(y,w)dydw)]dydw |
    
                    =    | ∫[w·∇_y ψ(|y-x|)(w-v) f_t(y,w)..
        - . .dκψ(|y-x|)f_t(y,w)∫ψ (|y-y|)(w-w)f_t(y,w)dydw]dydw |
    ≤   ψ_Lip∫ |w| |v-w| f_t (y,w) dydw 
                    + dκ∫ |w-w| f_t(y,w) f_t (y,w) dydwdydw
    ≤   ψ_Lip∫ (|w| |v| + |w|^2) f_t (y,w) dydw 
                    + dκ∫ 2|w| f_t(y,w) dydw.

        
         By Lemma <ref>, 
         
    |∂ℛ/∂ t(X_1,V_1,t)| ≤ C e^-κψ_0 t.

        
        
        
        
        We first consider q to be an even integer.
        For (X_1,V_1) obeying (<ref>) and (X̂_1,V̂_1) obeying (<ref>), with (X_1,V_1) (t_k) = (X̂_1,V̂_1) (t_k) ∼f_t_k, it holds
        
    d/dt   |V_1 -V̂_1|^2 =  2∑_k=1^d(V^k_1 -V̂^k_1)d/dt(V^k_1 -V̂^k_1)
    
                    =    2κ∑_k=1^d(V^k_1 -V̂^k_1)
                     [∫ψ (|x-X_1|) (v - V_1) f_t_k(x,v) dxdv .
       .- ∫ψ (|x-X̂_1|) (v - V̂_1) f_t_k(x,v) dxdv + ℛ(X_1,V_1)]^(k)
    
                    =     2κ∑_k=1^d(V^k_1 -V̂^k_1)( ∫ψ (|x-X_1|) [(v - V_1)-(v-V̂_1)] f_t_k(x,v) dxdv)^(k)
        + 2κ∑_k=1^d(V^k_1 -V̂^k_1) (∫ [ψ (|x-X̂_1|)- ψ (|x-X_1|)](v - V̂_1) f_t_k(x,v) dxdv ) ^(k)
        + 2κ∑_k=1^d(V^k_1 -V̂^k_1) ℛ(X_1,V_1)^(k)
    ≤     -qψ_0 | V_1 - V̂_1|^2 
                    +2κψ_Lip   |X_1-X̂_1 | ∑_k=1^d
                    (V^k_1 -V̂^k_1)(∫ (v-V̂_1)f_t_k)^(k)
       + 2κ |V_1 - V̂_1 | |ℛ(X_1,V_1)|,
    ≤     -2κψ_0 | V_1 - V̂_1|^2 
                    +2κψ_Lip   |X_1-X̂_1|  |V_1 - V̂_1 |
                    |V̂_1|
       + 2κ |V_1 - V̂_1 | |ℛ(X_1,V_1)|,

        where the superscript (k) means the k-th dimension of vectors. Since ∫ vf_t dxdv = v =0 by (<ref>), and |V̂_1|≤ C e^-κψ_0 t_k by Lemma <ref> and Lemma <ref>, it holds
        
    d/dt |V_1 -V̂_1|^2   
                    ≤   -2κψ_0 | V_1 - V̂_1|^2 
                    +Cκψ_Lip   e^-κψ_0 t_k |X_1-X̂_1|  |V_1 - V̂_1 |
    
                    + 2κ |V_1 - V̂_1 | |ℛ(X_1,V_1)|,

        and
        
    d/dt|V_1 - V̂_1| ≤ -ψ_0 κ |V_1 - V̂_1|
                + Cκψ_Lip   e^-κψ_0 t_k |X_1-X̂_1|
                + κ  |ℛ(X_1,V_1)|.

        Also, with
        
    d/dt | X_1 - X̂_1| ≤ |V_1-V̂_1|,

        one has
        
    d/dt(|X_1-X̂_1| + |V_1-V̂_1| ) ≤ C(|X_1-X̂_1| + |V_1-V̂_1| ) + C|ℛ|.

        By Grönwall's inequality, it holds
        
    |X̂_1-X_1| + |V̂_1-V_1| ≤ C e^-κψ_0 t_kτ^2.

        Then
        
    [𝔼[|X̂_1-X_1|^q + |V̂_1-V_1|^q]]^1/q≤[𝔼[|X̂_1-X_1| + |V̂_1-V_1|]^q]^1/q≤ C e^-κψ_0 t_kτ^2,

        which implies
        
    W_q(S(f_t_k),f_t_k+1)≤ C(κ,ψ_0,ψ_Lip, R ) e^-κψ_0 t_kτ^2.

    



    
    
    
    The estimate on W_q(S(f_t_k),𝒬_∞ (f_t_k) ) is given by Lemma <ref>.
    
    
    
    
        Under the same assumption of Theorem <ref>, it holds
        
    W_q (𝒬_∞ (f_t_k),S(f_t_k)) ≤ C(κ, R) τ^2,    q∈ [1,∞).

    
    
    
    Set (X̂_1,V̂_1) and {(X_i,V_i)} to be the solutions of (<ref>) and (<ref>) respectively. The initial data (X̂_1(0),V̂_1(0)) and {(X_i(0),V_i(0))} follow f_t_k. Take the coupling (X̂_1(0),V̂_1(0)) = (X_1(0),V_1(0)). We introduce {(X_i^E, V_i^E)} as the approximation of {(X_i,V_i)} by the forward Euler method with time step τ. Then, taking the expectation of the selection of {(X_j(0),V_j(0)}_j 1, one has 
    
    d/dt1/2𝔼 |V_1^E - V̂_1|^2 =   ∫κψ (| x- X_1(0)|) (v- V_1(0))·𝔼(V_1^E(t)-V̂_1(t))f_t_k
        - ∫κψ (| x- X̂_1(t)|) (v- V̂_1(t))·𝔼(V_1^E(t)-V̂_1(t))f_t_k
    ≤   κ∫ψ (|x-X_1(0)|) (V̂_1(t)-V_1(0))·𝔼(V_1^E(t)-V̂_1(t))f_t_k
       + κ∫ψ_Lip |X_1(0)-X̂_1(t)| (v-V̂_1(t))·𝔼(V_1^E(t)-V̂_1(t))f_t_k.

    By Lemma <ref> and Lemma <ref>, one has 
    
    | v-V̂_1(t)|≤ Ce^-κψ_0 t_k.

    Note that 
    
    |X_1(0)-X̂_1(t)|≤ Ce^-κψ_0 t_kτ

    and 
    
    |V̂_1(t)-V_1(0)|≤ Ce^-κψ_0 t_kτ.

    Take the expectation of (X_1(0),V_1(0)), one has
    
    d/dt𝔼 |V_1^E - V̂_1|^2
             ≤ Ce^-κψ_0 t_kτ√(𝔼|V_1^E - V̂_1|^2).

    Hence
    
    𝔼 |V_1^E - V̂_1|^2 ≤ Ce^-2κψ_0 t_kτ^4.

    Similar with (<ref>), one has
    
    𝔼 |X_1^E - X̂_1|^2 ≤ Ce^-2κψ_0 t_kτ^6.

    Now consider
    
    d/dt1/2𝔼|V^E_1(t) -V_1(t)|^2 =   𝔼(V^E_1(t) -V_1(t))κ/p-1∑_j=1^p[ψ(X_j(0)-X_1(0))(V_j(0)-V_1(0)) 
       - ψ(X_j(t) - X_1(t))(V_j(t) - V_1(t))]
    ≤   𝔼κ(V^E_1 -V_1)[ψ_Lip|X_2(0)-X_1(0)- X_2(t) + X_1(t))(V_2(0)-V_1(0)) 
       + ψ(X_2(t) - X_1(t))(V_2(0) - V_1(0) - V_2(t) + V_1(t))].

    By Lemma <ref> and Lemma <ref>, one knows that |V_j(t)|≤ e^-κψ_0 t_k, for any j and t∈ [0,τ]. Hence it's easy to obtain  
    
    d/dt𝔼 |V_1^E - V_1|^2
             ≤ Ce^-κψ_0 t_kτ√(𝔼|V_1^E - V_1|^2),

    and 
    
    𝔼 |V_1^E - V_1|^2 ≤ Ce^-2κψ_0 t_kτ^4,   𝔼 |X_1^E - X̂_1|^2 ≤ Ce^-2κψ_0 t_kτ^6.

    Then, by the triangular inequality of Wasserstein distance, one has
    
    W_2 (𝒬_∞ (f_t_k),S(f_t_k)) ≤ C(κ, R)e^-κψ_0 t_kτ^2.

    The estimate of W_q is similar by calculating 
    
    d/dt(𝔼|V_1^E - V̂_1|^q)^1/q=1/q(𝔼|V_1^E - V̂_1|^q)^1/q-1d/dt𝔼|V_1^E - V̂_1|^q ≤  C(κ, R)e^-κψ_0 t_kτ.

        
    
    


    

 §.§ Proof of Theorem <ref>

    Now we prove Theorem <ref> by combining the above lemmas.
    
    
    By the triangular inequality, it holds
    
    W_q(𝒬_∞^n (f_0), f(nτ))  ≤∑_m=1^n W_q (𝒬_∞^n-m+1 (f_m-1),𝒬_∞^n-m(f_m)).

    And then, by the stability result in Lemma <ref>, it holds
    
    ∑_m=1^n W_q (𝒬_∞^n-m+1 (f_m-1),𝒬_∞^n-m(f_m))
            ≤∑_m=1^n C W_q(𝒬_∞ (f_m-1), f_m).

    Again by the triangular inequality, one has
    
    W_q(𝒬_∞ (f_m-1), f_m) ≤ W_q(𝒬_∞ (f_m-1), S(f_m-1))
            + W_q(S(f_m-1), f_m).

    Then, by Lemma <ref> and Lemma <ref>, we have
    
    ∑_m=1^n W_q (𝒬_∞^n-m+1 (f_m-1),𝒬_∞^n-m(f_m))
            ≤∑_m=1^n  C e^-κψ_0 (m-1)ττ^2 ≤  C τ,

    where the constant C is independent of n.
        One completes the proof.
    
    
    
    
        Although our analysis is based on W_q,2 distance, the main results in Section <ref> also work on W_q,p, p∈ [1,∞), by noting that
    
    x _p≤ d^1-1/q+1/px _q,    x∈ℝ^d,    1<p,q<∞.

    







§ APPLICATION: THE RBM-GPC


In this section, we show a simple application for the mean-field limit analysis for the Cucker-Smale model combined with the RBM. 

Based on the above analysis, and inspired by Carrillo's work <cit.>, we can construct the numerical scheme for the approximation of stochastic mean-field models of swarming which preserve the nonnegativity of the solution. 

Let us consider the stochastic mean-field equation for the distribution function f(θ, x, v; t):

    ∂_t f + v·∇_x f = ∇_v · [ℋ[f]f],

where

    ℋ[f](θ,x,v;t) = κ∫∫ψ(|x-y|,θ)(v-w)f(θ,y,w;t)dwdy,

with ψ satisfying  positivity, boundedness, Lipschitz continuity and mononticity conditions: there exists a positive constant ψ_M  > 0  such that 

    0≤ψ(r,θ) ≤ψ_M, ∀ r≥0,   ψ_Lip<∞,


    (ψ(r_1,θ)-ψ(r_2,θ))(r_1-r_2)≤ 0,   r_1,r_2 ∈ℝ_+.

Here, θ is a random variable from a measurable space (Ω,ℱ) to (I_Θ,ℬ_ℝ), with a probability distribution π (θ), with I_Θ⊂ℝ and ℬ_ℝ the Borel set.

It is easy to see that the total mass and momentum of (<ref>) is conserved in time.
Note that if we directly apply the classical stochastic Galerkin method to (<ref>), the solution might lose its positivity. However, we can make use of the microscopic dynamics to approximate f. Our method is to consider the corresponding Cucker-Smale model with random input θ on the interaction kernel ψ:

    {∂_t x_i(t,θ) =    v_i(t,θ),  i=1,⋯, N,
    ∂_t v_i(t,θ) =   κ/N-1∑_j=1^N 
            ψ(|x_j(t,θ)-x_i(t,θ)|,θ)(v_j(t,θ)-v_i(t,θ)).
        .


Denote {Φ_k(θ)}_j=0^∞ to be the gPC basis with respect to π, which means that Φ_k is a polynomial of degree k, and
these polynomials form an orthonormal basis of L^2(Ω,ℱ,π). 
We approximate the position and the velocity of i-th agent as follows:

    x_i(θ,t)≈ x^K_i = ∑_k=0^K^kx̂_i(t)Φ_k(θ),
           v_i(θ,t)≈ v^K_i = ∑_k=0^K^kv̂_i(t)Φ_k(θ),

where

    ^kx̂_i=∫ x_i(θ,t)Φ_k(θ)dπ,   ^kv̂_i=∫ v_i(θ,t)Φ_k(θ)dπ.

We obtain the following system

    {d/dt^ℓx̂_i (t) =   ^ℓv̂_i (t),
    d/dt^ℓv̂_i (t) =   κ/N-1∑_j=1^N∑_k=0^K e_ℓ k^ij(^kv̂_j(t)-^kv̂_i(t)),
           l = 0,1,…,K..

where

    e_ℓ k^ij=1/|Φ_ℓ|^2∫ψ(|x_j-x_i|,θ) Φ_k(θ)Φ_ℓ(θ)dπ.

Then, we design Algorithm <ref>, on the generalized Polynomial Chaos expansion in the random space based on the RBM:




    Although being derived from different numerical methods (the RBM and the Monte Carlo method), Algorithm <ref> is similar with the Monte Carlo gPC (MCgPC) proposed in <cit.> at first glance. Both can reduce the total cost of (<ref>) to 𝒪(KN). The main difference between them is that the RBM-gPC preserves the momentum during evolution. Actually, since the RBM updates information in the unit of batches, (instead, the MCgPC updates information of each particle independently at per time step.) for any t∈[t_m,t_m+1], the total sum of velocities is conserved in each batch by calculus:
    
    d/dt∑_i∈𝒞_q^ℓv̂_i (t) = κ/p-1∑_i,j∈𝒞_q∑_k=0^K e_ℓ k^ij(^kv̂_j(t)-^kv̂_i(t))=0,

    for l=0,⋯,K. Furthermore, it holds 
    
    ∑_i=1^N ^ℓv̂_i (t) = ∑_q=1^N/p∑_i∈𝒞_q^ℓv̂_i (t) = ∑_q=1^N/p∑_i∈𝒞_q^ℓv̂_i (t_m) = ∑_q=1^N/p∑_i∈𝒞_q^ℓv̂_i (t_0) = ∑_i=1^N ^ℓv̂_i (0).

    










    We use 
    x_i^K,R(t,θ):= ∑_k=0^K^kx̂_i(t)Φ_k(θ),   v_i^K,R(t,θ):=∑_k=0^K^kv̂_i(t)Φ_k(θ)
 to denote the approximation of position and velocity obtained by Algorithm <ref>. The kinetic distribution f^R_N,K(θ,x,v,t) is recovered from the empirical density function 
    f^R_N,K(θ,x,v,t) = 1/N∑_i=1^N δ(x-x_i^K,R(t,θ))⊗δ(v-v_i^K,R(t,θ)).
 Common approximations methods for Dirac delta distributions can be used when needed. For example, we fix an upper and lower bound for the (x,v)-space, to restrict our computational domain in a cube space and discretize it in uniform cells {C_l}_l=1^N_l of size |C_l|=h^2d, where h is the edge length of each cell. One can reconstruct the distribution f^R_N,K by histograms counting the number of particles belonging to each cell. Then the discrete distribution function f^R_N,K reads
    
    f^R_N,K(θ,x,v,t) = 1/|C_l|(1/N∑_i=1^N 𝕀(x_i^K,R(t,θ),v_i^K,R(t,θ)∈ C_l))·(∑_l=1^N_l𝕀((x,v)∈ C_l)),

    where 𝕀(·) is the indicator function.
    Clearly, the positivity of the distribution function is preserved. One can also refer to a brief review, by Lee et al. <cit.>, on the numerical methods for regularized Dirac delta functions.








    With respect to the stochastic variable θ, the dynamics of the N-particle system achieves spectral accuracy as we adopt the generalized polynomial chaos (gPC) basis <cit.>, provided that we have a smooth dependence of the particle solution from the random field.

    For any given θ and K, denote the distribution function of the N-particle system (<ref>) by f_N,K. 
    
    If one uses the reconstruction (<ref>), it holds that
        
    𝔼 W_2 (f^R_N,K,f^R_N,K) ≤(𝔼1/N∑_i=1^N |(x_i^R,v_i^R) - c_l(i)|^2 )^1/2,

        where c_l(i) is the center of the cell C_l containing (x_i^R,v_i^R). Since h is the uniform edge length, |(x_i^R,v_i^R) - c_l(i)|^2 ≤∑^d (h/2)^2 for any i. Combining with the estimate of W_2 (f_N,K,f^R_N,K) in Section <ref>, it holds that
    
    
    
    
        
    𝔼 W_2 (f_N,K,f^R_N,K)     ≤𝒪(√(τ/p-1)) +
           {   𝒪(N^-1/4)     if  d<2,
       𝒪(N^-1/4√(log(1+N)))     if  d=2,
       𝒪(N^-1/d)     if  d>2,
            .
            + 𝒪(√(d)h).







 §.§ Numerical experiments

Here we present some numerical tests for the RBM-gPC. In the following, we take the Legendre gPC base.



  
The space homogeneous case
We consider the space-independent case in the one-dimensional setting:

    ∂_t f(θ,v;t) = ∂_v [ K(θ)(v-u)f(θ,v;t)],

with u = ∫ vf(v)dv and K(θ)=0.5 + 0.01θ, θ∼𝒰(0,1). The long time solution is a Dirac delta δ(v-u) provided K(θ)>0, see <cit.>. We take the Legendre polynomial basis. The gPC approximation is given for h=0,⋯,M by 

    ∂_t f̂_h (v;t) = 1/|Φ_h|^2∂_v [ ∑_k=0^M (v-u) ℋ_hkf̂_t_k(v;t)],

where

    ℋ_hk = ∫ K(θ)Φ_h (θ) Φ_k (θ) d g (θ),   f̂_h (v;t) = ∫ f(θ,v;t) Φ_h (θ) d g (θ).

We consider an initial density function f_0(v) by

    f_0(v) = β[exp(-(v-μ)^2/2σ^2) + exp(-(v+μ)^2/2σ^2)],   σ^2=0.1,  μ = 0.5,

with β>0 a normalization constant. Discrete samples of the initial data of the ODE system are obtained from f_0(v). Denote the expected temperature of the system 𝒯 by 

    𝒯(f) = ∫ (v-u)^2 f(θ,v;t)dvdg(θ).

In Figure <ref>, we show the evolution of the expected density toward the Dirac delta function δ(v-u), u=0 for the RBM-gPC scheme at different times with p=2 and N=10^4.

In Figure <ref>, we take T=0.5, K=3, dt=10^-2 and dv=10^-2. Time integration is performed through a 4th order Runge-Kutta method. In Figure <ref>, we evaluate the mean square error (MSE) for the expected temperature 

    MSE_T := 1/n_m∑_i=1^n_m|𝒯(f_ref) - 𝒯(f_rbm^(i)) |^2,

as N grows with p=2, where 𝒯(f_ref) denotes the reference expected temperature obtained by (<ref>) through a central difference scheme, and 𝒯(f_rbm^(i)) denotes the expected temperature obtained by the RBM-gPC in the i-th turn. We have applied n_m=100 for random initialization and batch splits. 
Figure <ref> shows the difference of TV distance between the expected reference distribution and the expected distribution by the RBM-gPC:

    Err_TV := | 𝔼_θ f_ref - 𝔼_θ1/n_m∑_i=1^n_m f_rbm^(i)|_L^1,

with n_m =100 as well.












Also, Figure <ref> and Figure <ref> show the decay of MSE_T as the batch size p and the time step dt, respectively. The reference distribution of Figure <ref> is obtained by the RBM-gPC with N=2^8, p = 2^8, and dt = 10^-2, and of Figure <ref> is obtained with N=2^8, p = 2, and dt = 10^-2. We still take n_m=100.






  
Stochastic 1D Cucker-Smale dynamics
In this test, we consider the 1D Cucker-Smale dynamics. Let us consider the initial distribution as the following bivariate and bimodal distribution

    f_0(x,v) = 1/2πσ_xσ_vexp(-x^2/2σ_v ^2)[exp(-(v+1)^2/2σ_x ^2)+ exp(-(v-1)^2/2σ_x^2)],

with σ_x^2 =0.5, σ_v^2=0.2. Our initial data for particle positions and velocities are
sampled from f_0. The stochastic interactions are given by

    H(θ; |x_i - x_j|) = 1/(1+|x_i - x_j|^2)^γ(θ),

with γ (θ) = 0.1+ 0.05 θ, θ∼𝒰[-1,1]. The results are obtained through the RBM-gPC scheme with dt = 10^-2, N=10^4, batch size p =2, and K=3. The expected distribution is reconstructed by  in Matlab in the domain [-3,3]×[-3,3] with 100 gridpoints in both space and velocity. Figure <ref> presents the evolution over the time interval t∈[0,4] of the expected distribution. 






  
Stochastic 2D Cucker-Smale dynamics
Figure <ref> shows the evolution over the time interval t∈ [0,4] of the 2D Cucker-Smale nonhomogeneous mean-field model. For the initial distribution, we consider uniformly distributed N=10^4 particles on a 2D annulus with a circular counterclockwise motion

    f_0 (x,v) = 1/|C|χ (x∈ C)δ( v - k∧ x/|x|),

being C: = {x∈ℝ^2: 0.5≤ |x_1-x_2|≤ 1 } and k the fundamental unit vector of the z-axis, where |C| means the cardinality of set C.
The interaction kernel shares the same form with (<ref>).
The evolution shows the flocking phenomenon in the top row and that the mean velocity is conserved in the bottom row of Figure <ref>. The results are obtained through the RBM-gPC scheme with dt = 10^-2, N=10^4, batch size p =2, and K=3. The reconstruction follows the same method as in the 1D case.








§ CONCLUSION

In conclusion, we give an analysis on the mean-field limit of the RBM for the Cucker-Smale model and obtain a long-time approximation. In addition, as a simple application, we propose a generalized Polynomial Chaos based Random Batch Method (RBM-gPC) as a numerical method to simulate the mean-field limit dynamics of the Cucker-Smale model with random inputs.

This paper focuses on the separate limits of the number of particles N and the time step τ but not a uniform limit of (N,τ). For future work, it will be interesting to investigate the mixture of particles as time evolves, and to study on the uniform limit. Also, numerically, an error estimate on this model applied with the particle method (corresponding to a new model) can be an interesting topic to study.







§ ACKNOWLEDGEMENT

We thank Prof. Shi Jin and Prof. Lei Li for detailed and valuable comments and advice on the manuscript. This work is supported by the NSFC grant (No. 12031013, 12201404), the Shanghai Municipal Science and Technology Key Project (No. 22JC1402300), and Project supported by the National Science Foundation for International Senior Scientists (No. 12350710181).



    
    

§ PROOF OF LEMMA <REF>

    The proof is given by Lemma 2.4 in <cit.>.
    
        For the first assertion, we claim that the relative velocities are non-increasing in time. Let t∈ [t_m-1, t_m) be given. Then one can choose time-dependent indices k and ℓ such that 
        
    |V_k^R (t) - V_ℓ^R (t)| = 𝒟_V^R(t).

        Then for such k and ℓ, one has 
        
    d/dt | V_k^R(t) -V_ℓ^R(t)|^2 =    
                2(V_k^R(t) - V_ℓ^R (t)) ·d/dt(V_k^R(t) - V_ℓ^R (t))
    
                =    2κ/p-1∑_j∈ [k]_mψ (|X_j^R - X_k^R|)(V_j^R - V_k^R)·(V_k^R - V_ℓ^R)
       - 2κ/p-1∑_j∈[i]_mψ (|X_j^R - X_ℓ^R|)(V_j^R - V_ℓ^R)·(V_k^R - V_ℓ^R).

        In order to show that the right-hand side of (<ref>) is not positive, we use the maximality of | V_k^R(t) -V_ℓ^R(t)| at time t. Since
        
    | V_k^R(t) -V_ℓ^R(t)| ≥ | V_j^R(t) -V_ℓ^R(t)|,    j = 1,⋯,N,

        one has
        
    (V_j^R(t)-V_k^R (t))·(V_k^R(t)-V_ℓ^R (t)) =
                - ((V_k^R-V_ℓ^R) - (V_j^R-V_ℓ^R))· (V_k^R - V_ℓ^R)
    ≤ - | V_k^R(t) -V_ℓ^R(t)|^2 + | V_j^R(t) -V_ℓ^R(t)|| V_k^R(t) -V_ℓ^R(t)|≤ 0,    j= 1,⋯,N.

        Similarly, one has 
        
    (V_j^R(t)-V_ℓ^R (t))· (V_k^R(t)-V_ℓ^R (t)) ≥ 0,    j=1,⋯,N.

        Hence, | V_k^R(t) -V_ℓ^R(t)|^2 is non-increasing in time.

        Then by the definition od 𝒟_X^R and 𝒟_V^R, one has 
        
    d/dt𝒟_X^R(t)≤𝒟_V^R(t) ≤𝒟_V^R(0),    a.e.  t>0.

        This yields the second assertion on 𝒟_X^R.
    
    
    
    

    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
        
    
    
    
    
    
    

    
    
    
    
    
    
    
    
    
    
    
    
    
    

    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
        
    
    
    
    
    
    
    
    
    
    
    
    


    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    

    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    


    




    
    
    
    
    

§ DETAILS ON PROPOSITION <REF>


    

  
Fixed-time regularity of f. First we check the regularity of f by Lemma <ref>, which is needed in the proof of Proposition <ref>.
    
    
    For the flocking kinetic model (<ref>), suppose that the initial probability measure f_0 ∈ (C^2∩ W^2,∞)(ℝ^2d) is compactly supported, and the interaction kernel ψ satisfies 
    sup_r≥ 0 (|ψ^'(r)| + |ψ^''(r)|) ≤ϵ_ψ
 with a constant ϵ_ψ. Then, for any T∈ (0,∞), there exists a unique classical solution f_t ∈ C^2([0,T)×ℝ^2d) for (<ref>). 

    The proof stem of Lemma <ref> is generally a smoothness check for f inspired by in [<cit.>, Theorem 3.1]. The main difference with <cit.> is that here we need better regularity for the operator (ℒ^*)^2 and (ℒ_k^*)^2 in Lemma <ref>.
    
    Local existence follows from [<cit.> Theorem 3.1] by the standard fixed point argument. One only needs to obtain a C^2 bound of f to conclude the global existence of classical solutions. One can express the non-conservative kinetic model (<ref>) in terms of the non-linear transport operator 𝒯:=∂_t + v·∇_x + ξ [f] ·∇_v,
    
    𝒯f = -f∇_v ·ξ[f].

    We claim that there exists a positive constant C(d,κ,ϵ_ψ, R) such that
    
    | 𝒯f| ≤ C |f|,

    
    | 𝒯(∂_x_if) | ≤ C (|f| + |∇_v f| + |∂_x_if)|,

    
    | 𝒯(∂_v_i f) | ≤ C (|∇_v f| + |∂_x_if|),

    
    | 𝒯(∂_x_i x_jf) | ≤ C (  |∂_x_if| + |∂_x_jf| + |∂_x_i x_jf| +|∇_v f| + |∇_v∂_x_if| + |∇_v∂_x_jf|),

    
    | 𝒯(∂_x_i v_jf) | ≤ C ( |∂_x_i x_jf| +|∇_v f| + |∂_v_j∂_x_if| + |∇_v∂_v_jf| + |∇_v∂_x_if|),


    
    | 𝒯(∂_v_i v_jf) | ≤ C ( |∂_x_i v_jf| + |∂_x_j∂_v_if| +  |∂_v_j∂_v_if|).


    
    To verify these inequalities, note that -∇_v·ξ [f] = κ d ∫ψ (|y-x|)f(y,w,t)dydw and thus |∇_v·ξ [f]| ≤κ d. By direct calculus, one obtains the following estimates:
 

    For (<ref>),
    
    | 𝒯f| ≤| -κf∇_v ·∫ψ(|y-x|) (w-v) f(y,w,t) dydw| ≤κ d |f|;

    for (<ref>),
    
    | 𝒯(∂_x_if) |    ≤| - (∂ _x_iξ [f]) ·∇_v f -  (∂_x_i∇_v ·ξ [f])f  -  (∇_v ·ξ [f])∂ _x_if|
       ≤κϵ_ψ (√(M_2^v f_0) + 2R) |∇_v f| + κ d ϵ_ψ |f| + κ d |∂_x_if|)
       ≤ C (|f| + |∇_v f| + |∂_x_if)|;

    For (<ref>),
    
    |𝒯∂_v_if | ≤ |-∂_x_if| + κ |∇_v f| + κ d |∂_v_if|
            ≤ C (|∇_v f| + |∂_x_if|).

    The estimate on (<ref>)-(<ref>) is similar and we omit it for brevity.
    
    Now, let F(t) measure the W^2,∞-norm of f,
    
    F := f(·,t) _W^2,∞ .

    The above inequalities imply
    
    d/dtF(t) ≤ CF(t).

    Then we end up with the energy bound by Grönwall's inequality
    F(t)≤ F(0) e^Ct.
    Equipped with this a priori W^2,∞ estimate, standard continuation principle yields a global extension of local classical solutions. 




The following is a simple prerequisite for Proposition <ref>.

    
For any g_1∈ L^1(ℝ^2d),g_2∈ L^1(ℝ^2dp), the operators ℒ_k^* and ℒ^* defined by (<ref>) and (<ref>) respectively, it holds
    
    ∫ | e^τℒ_k^*g_1| dz ≤∫ | g_1| dz,

    
    ∫|  e^τℒ^*  g_2  | dz_1 ⋯ dz_p ≤∫|  g_2  | dz_1 ⋯ dz_p.


     
    
    One only needs to consider g_1≥ 0 since there is composition g_1 = g_1^+ -g_1^-. Also, since e^τℒ_k^*cg_1 = ce^τℒ_k^*g_1 and e^τℒ^*cg_2 = ce^τℒ^*g_2 for any constant c, one can set g_1 be a probability density function. Then,
    
    ∫ |e^τℒ_k^*g_1| dz = ∫ e^τℒ_k^*g_1 dz,

    since e^τℒ_k^*g_1 is another probability density function derived from g_1 by the dynamics (<ref>). It's clear that
    
    ∫ e^τℒ_k^*g_1 dz = 
        ∫∫_0^τ∂/∂ tg_1(t)dtdz + ∫ g_1 dz = ∫ g_1dz,

    where g_1(t) denotes the solution of (<ref>) with initial datum g_1. For the general g_1, 
    
    ∫ | e^τℒ_k^*g_1| dz ≤∫ | e^τℒ_k^*g_1^+| dz
             + ∫ | e^τℒ_k^*g_1^-| dz
             =  ∫ | g_1^+| dz + ∫ | g_1^-| dz =  ∫ | g_1| dz.

    The proof of (<ref>) is similar.





    
    

  
Proof of Proposition <ref>.
    It's clear that S = e^τℒ_k^*f_t_k, where 
    
    ℒ_k^* := -  (v ·∇_x) · -
        ∇_v ·( ∫κψ(|y-x|)(w-v)f_t_k(y,w)dydw ·).

    Define the operator of (<ref>) as
    
    ℒ^* := -∑_i=1^p (v_i∇_x_i)·  - ∑_i=1^p∇_v_i· (ξ_i ·).

    Then one has
    
    𝒬_∞ (f_t_k) =   ∫ e^τℒ^*Π_i=1^p f_t_k(x_i,v_i)dx_2⋯ dx_pdv_2⋯ dv_p 
    
            =   f_t_k (x_1,v_1) + τ∫ℒ^* Π_i=1^p f_t_k (x_i,v_i)dx_2⋯ dx_pdv_2⋯ dv_p 
       + ∫_0^τ (τ -s) ∫ (ℒ^*)^2 e^sℒ^*Π_i=1^p f_t_k (x_i,v_i)dx_2⋯ dx_pdv_2⋯ dv_p ds,

    where
    
    
    ∫ℒ^* Π_i=1^p f_t_k (z_i) dz_2⋯ dz_p 
                =     -∑_i=1^p ∫ (v_i ∇_x_i)·Π_j=1^p f_t_k (z_j) - ∇ _v_i· (ξ_i Π_j=1^p)f_t_k (z_j)dz_2⋯ dz_p)
    
                =     - (v_1 ∇_x_1) ·f_t_k (z_1) - ∇_v_1·∫κψ (|y-x_1|)(w-v_1) f_t_k (y,w) dydw.

    
    Note that S(f_t_k) = f_t_k + τℒ_k^*f_t_k + ∫_0^τ (τ -s)(ℒ_k^*)^2 e^sℒ_k^*f_t_k ds, then
    
        
    | S(f_t_k) - 𝒬_∞ (f_t_k) |(z_1)≤∫ _0^τ (τ-s) [ |(ℒ_k^*)^2 e^sℒ_k^*f_t_k(z_1)|  +  | ∫ (ℒ^*)^2 e^sℒ^*Π_i=1^p f_t_k (z_i) dz_2⋯ dz_p | ]ds.

    
    
    As shown in Lemma <ref>, f_t_k∈ C^2(z), hence it holds 
    
    ∫ |(ℒ_k^*)^2 f_t_k| dz≤ C ,  ∫| (ℒ^*)^2 Π _i=1^p f_t_k (z_i)| dz_2 ⋯ dz_p≤ C,
 
    where C=C(κ,ϵ_ψ ,R,d). By Lemma <ref>, taking g_1 = (ℒ_k^*)^2  f_t_k and g_2 = (ℒ^*)^2 Π _i=1^p f_t_k (z_i), one has
    
    ∫ |(ℒ_k^*)^2 e^τℒ_k^*f_t_k| dz ≤∫ |(ℒ_k^*)^2 f_t_k| dz,

    and
    
    ∫| (ℒ^*)^2 e^τℒ^*Π _i=1^p f_t_k (z_i)| dz_1 ⋯ dz_p ≤∫| (ℒ^*)^2 Π _i=1^p f_t_k (z_i)| dz_1 ⋯ dz_p,

    Using the exchangability of ℒ^* and e^τℒ^*, with ℒ^*=ℒ_k^* or ℒ^*. 
    
    Then one has |S(f_t_k) - 𝒬_∞ (f_t_k)|_TV≤ C τ^2. 
    
    For S(f_t_k), by Lemma <ref>, it holds
    
    supp[S_∞(f_t_k)]⊂
            B(0,C)× B(0,C e^-κψ_0 t).


    For the support of 𝒬_∞(f_t_k), by Lemma <ref>, one knows that 
    
    supp[𝒬_∞(f_t_k)]⊂
            B(0,C)× B(0,C e^-κψ_0 t_k).

    
    Hence for any z ∈supp|S(f_t_k) - f_t_k|, it holds |z|≤ C.
    Then, by Lemma <ref> and applying similar analysis on Lemma <ref> Step 1, one has 
    
    W_q (𝒬_∞ (f_t_k),S(f_t_k)) ≤ C(κ,ϵ_ψ , f_0,d) τ^2/q.















plain





 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.
	
	
	[
    [
    August 5, 2024
==================


		
		
	
	
	
	
	

§ 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.
	
	
	
	
	
	
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