Square function estimates, BMO Dirichlet problem, and absolute continuity of harmonic measure on lower-dimensional sets
Svitlana Mayboroda Zihui Zhao



S<N, BMO solvability and A_∞ class]Square function estimates, BMO Dirichlet problem, and absolute continuity of harmonic measure on lower-dimensional sets


S. Mayboroda]Svitlana Mayboroda
Svitlana Mayboroda, Department of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
svitlana@math.umn.edu

Z. Zhao]Zihui Zhao
Zihui Zhao, University of Washington, Department of Mathematics, Seattle, WA 98195-4350, USA
zhaozh@uw.edu

The first author was partially supported by the NSF INSPIRE Award DMS-1344235, NSF CAREER Award DMS-1220089, and the Simons Foundation grant 563916, SM. The second author was partially supported by NSF grant numbers DMS-1361823, DMS-1500098 and DMS-1664867.
This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester.


[2010]35J25, 42B37, 31B35.





In the recent work <cit.> G. David, J. Feneuil, and the first author have launched a  program devoted to an analogue of harmonic measure for lower-dimensional sets. A relevant class of partial differential equations, analogous to the class of elliptic PDEs in the classical context, is given by linear degenerate equations with the degeneracy suitably depending on the distance to the boundary. 

The present paper continues this line of research and focuses on the criteria of quantitative absolute continuity of the newly defined harmonic measure with respect to the Hausdorff measure, ω∈ A_∞(σ), in terms of solvability of boundary value problems. The authors establish, in particular, 
square function estimates and solvability of the Dirichlet problem in BMO for domains with lower-dimensional boundaries under the underlying assumption ω∈ A_∞(σ). More generally, it is proved that in all domains with Ahlfors regular boundaries the BMO solvability of the Dirichlet problem is necessary and sufficient for the absolute continuity of the harmonic measure.


[
    [
    ^⋆ E-mail: mailto:Jordan.Collier@csiro.auJordan.Collier@csiro.au

Accepted 2018 February 23. Received 2018 February 23; in original form 2017 June 16
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§ INTRODUCTION


The last decade has seen great advances in understanding of the connections between
analytic, geometric, and PDE properties of sets. One of the central questions in this quest pertains to the necessary and sufficient conditions on the geometry of the domain which guarantee absolute continuity of the harmonic measure ω  with respect to the surface measure σ of the boundary. The interest to this problem begins with the classical 1916  F. and M. Riesz theorem <cit.> which asserts that for a simply connected planar domain with a rectifiable boundary, the harmonic measure is absolutely continuous with respect to the boundary surface measure (see <cit.> for a quantitative version). A local analogue of this result was established in <cit.>, which also showed that absolute continuity may fail in the absence of some topological hypothesis, even for a rectifiable domain. The emerging philosophy is that  the key geometric properties at play are smoothness (or to be precise, rectifiability) and connectedness of the domain.  In higher dimensions, the latter is much trickier, and without any pertinent details we mention that the absolute continuity of the harmonic measure with respect to the boundary surface measure has been proved in Lipschitz graph  domains <cit.>, and later in the so-called chord-arc domains in <cit.>, and more recent achievements in the field have progressively further weakened the underlying geometric hypotheses <cit.>, although the sharp assumptions, particularly in terms of connectedness, are not completely clear yet. 
Meanwhile in the converse direction, the necessary conditions for the absolute continuity of harmonic measure with respect to the Hausdorff measure of the boundary have been obtained in 1-sided chord-arc domains in <cit.> (see also <cit.>), and later in more general domains in <cit.>. As a culmination of this line of work, it was shown without any topological background assumptions that rectifiability is necessary for absolute continuity of the harmonic measure in <cit.>. These results were extended to general elliptic operators and other manifestations of solvability of the Dirichlet boundary value problem in <cit.> to mention only a few: the area is blossoming and we do not aim at a complete listing of the related literature. 



All of these advances heavily rely on the properties of harmonic functions, and as such, do not apply to domains with lower-dimensional boundaries, for instance, a complement of a curve in ^3. In fact, sets of higher co-dimension are not visible by classical Brownian travelers (that is, the probability to hit such a set is zero) and equivalently, by classical harmonic functions. Led by these considerations, 
G. David, J. Feneuil, and the first author have recently launched a  program devoted to a new type of degenerate elliptic PDEs <cit.>, such that the corresponding elliptic measure (still referred to as harmonic measure in the course of this discussion) is not only non-trivial, but absolutely continuous with respect to the Hausdorff  measure in favorable geometric circumstances. The goal of the present paper is to establish equivalence of absolute continuity of harmonic measure to the BMO solvability of the Dirichlet problem on arbitrary Ahlfors-regular domains and in the general class of degenerate elliptic operators, and to prove a technical but very important in many applications roadblock: the square function estimates for solutions. Let us discuss this in more details. 

We shall work in the general context of d-Ahlfors-David regular sets, which are roughly speaking, d-dimensional uniformly at all scales. 
 Let Γ⊂^n be a closed set and d≤ n be an integer. We say Γ is d-Ahlfors  regular if there exists a constant C_0≥ 1 such that for any q∈Γ and r>0,

    C_0^-1 r^d≤ℋ^d(B(q, r)∩Γ) ≤ C_0 r^d,

where ℋ^d is the d-dimensional Hausdorff measure. We shall often denote ℋ^d|_Γ, that is ℋ^d restricted to the set Γ, by σ.


Let Γ be a d-Ahlfors regular set in ^n with d<n-1, and Ω = ^n ∖Γ. Consider the degenerate elliptic operator L=-(A(X)∇) with a real, symmetric n× n matrix A(X) satisfying
	
    A(X) ξ·ζ≤ C_1 |ξ| |ζ| δ(X)^d-n+1 for  X∈Ω and ξ, ζ∈^n,

	
    A(X) ξ·ξ≥ C_1^-1 |ξ|^2 δ(X)^d-n+1 for  X∈Ω and ξ∈^n

	for some C_1 ≥ 1, where δ(X) = (X,Γ). We say a function u in the Sobolev space W_r(Ω) (see the definition in (<ref>)) is a weak solution to Lu=0, if
	
    ∬_Ω A(X) ∇ u ·∇φ  dX = 0,    for any φ∈ C_0^∞(Ω).

The basic elliptic theory of such equations was developed in <cit.>. In particular, it was shown that the Dirichlet problem 

    D{[  Lu=0  in Ω;   u=f on Γ. ].

has a suitably interpreted weak solution for smooth compactly supported (and more general) f on Γ, that such a solution is locally bounded and Hölder continuous in the interior and at the boundary, and finally, that it can be written in terms of the corresponding harmonic measure, and the latter satisfies the usual doubling, non-degeneracy, and change-of-pole conditions. We refer the reader to Section <ref> for details. For now, we only recall that  the harmonic measure is a (family of) positive regular Borel measure(s) ω^X on Γ, X∈Ω, such that, in particular, for any boundary function f∈ C_0^0() the solution to (<ref>) can be written as 

    u(X) = ∫_Γ f dω^X.


	We say the harmonic measure ω is of class A_∞ with respect to the surface measure σ = ℋ^d|_Γ, or simply ω∈ A_∞(σ), if for any ϵ>0, there exists δ=δ(ϵ)>0 such that for any surface ball Δ, any surface ball Δ' ⊂Δ and any Borel set E⊂Δ', we have
	
    σ(E)/σ(Δ') <δω^A(E)ω^A(Δ') <ϵ.

	Here A = A_Δ is a corkscrew point for Δ (see Lemma <ref> for the definition and existence of corkscrew point).

We remark that while the aforementioned basic properties of harmonic measure (existence, doubling, non-degeneracy, change-of-poles etc.) hold in full generality of d-Ahlfors regular sets, d<n-1, the A_∞ property of the harmonic measure is much more delicate and is not expected on very rough domains. 
In particular, already on a planar domain with 1-dimensional boundary rectifiability of the boundary is necessary for ω∈ A_∞(σ). On the other hand, it is not vacuous either, as the authors in <cit.> have proved that for any d<n-1 and Γ a d-dimensional Lipschitz graph with a small Lipschitz constant the harmonic measure is absolutely continuous with respect to the Hausdorff measure for the operator L=-(D(X)^-n+d+1∇) where 

    D(X) = {∫_Γ |X-y|^-d-α d ℋ^d(y) }^-1/α,    X∈Ω,

for some constant α > 0. 
It is easy to see that 
D(X) is equivalent to the Euclidean distance (X,Γ) (and this would even stay true when Γ is an Ahlfors regular set) but not equal.

For any q∈Γ and r>0, we use Δ = Δ(q,r) to denote the surface ball B(q,r)∩Γ, and use T(Δ) := B(q,r)∩Ω to denote the “tent” above Δ.
A function f defined on Γ is a BMO function if
	
    f_BMO := sup_Δ⊂(_Δ |f-f_Δ|^2 dσ)^1/2 <∞.

Here f_Δ denotes the average _Δ f dσ.
 We say that the Dirichlet problem (<ref>) is solvable in BMO if  for any boundary function f∈ C_0^0(), the solution u to (<ref>) given by (<ref>) satisfies a condition that |∇ u|^2 δ(X)^d-n+2 dX is a Carleson measure with norm bounded by a constant multiple of f_BMO^2, that is, 
	
    sup_Δ⊂1/σ(Δ)∬_T(Δ) |∇ u|^2 δ(X)^d-n+2 dX ≤ C f_BMO^2.



One of the main results of the present paper is as follows. 

	Let Γ be a d-Ahlfors regular set in ^n with d<n-1 and Ω = ℝ^n ∖Γ. Consider the operator L=-(A(X)∇) with a real, symmetric n× n matrix A(X) satisfying (<ref>) and (<ref>).
	Then the harmonic measure ω∈ A_∞(σ) if and only if the Dirichlet problem (<ref>) is BMO-solvable.


In co-dimension 1 this has been proved in <cit.> for Lipschitz domains and in <cit.> for uniform domains with Ahlfors regular boundaries. One of the main difficulties 
in our case is to prove an upper bound on the square function by the non-tangential maximal function. The latter, in co-dimension 1, goes back to the work of Dahlberg, Jerison, and Kenig for Lipschitz domains in <cit.>, and their method can be extended to more general sets with the help of preliminary estimates proved in <cit.>. This result, and even more so the method behind it, underpinned many later developments in the subject. To prove it, <cit.> systematically use the harmonic measures of the sawtooth domains to get a good-λ inequality. This technique is not available to us. The sawtooth domain is a domain inside Ω on top of a set E⊂∂Ω = Γ that satisfies some desired properties, and roughly speaking, allows one to exchange local results with global ones. In some sense, it is the use of the sawtooth domains which allows one to exploit the fact that at every scale the A_∞ condition only carries information on a big portion of a boundary ball, rather than the entire boundary ball - a crucial ingredient in this and many other arguments in the theory. In the case of the lower dimensional Γ, however, the boundary of a sawtooth domain may have arbitrarily small/large pieces of dimension d and, simultaneously, pieces of dimension n-1. For that reason, it is not automatically clear if one can make sense of the harmonic measure for the sawtooth domain or to resolve the Dirichlet problem on the sawtooth domain. Instead we are bound to work with the Green function of the entire Ω, and get a good-λ inequality by using various considerations akin to the comparison principle. Needless to say, the geometric arguments for lower-dimensional sets are also very different and the technical side of the present paper ends up surprisingly far from <cit.>, <cit.> and <cit.>. Moreover, since the theory of the lower-dimensional sets is still at its infancy, these technical geometric arguments, e.g. Lemma <ref>, are likely to be useful in many future works.

The formal results in this direction are as follows. For any q∈Γ and α>0, we define the non-tangential cone Γ^α(q) with vertex q and aperture α as 

    Γ^α(q) = {X∈Ω: |X-q| < (1+α) δ(X) },
 
and a truncated cone as

    Γ_r^α(q) = Γ^α(q)∩ B(q,r).

When there is no confusion we drop the super-index α and simply denote them by Γ(q) and Γ_r(q), respectively. We define the non-tangential square function

    Su(q) = ( ∬_Γ(q)  |∇ u|^2 δ(X)^1-d dm(X) )^1/2

and the truncated square function

    S_r u(q) = ( ∬_Γ_r(Q)  |∇ u|^2 δ(X)^1-d dm(X) )^1/2.

We also define the non-tangential maximal function and its truncated analogue

    Nu(q) = sup_X∈Γ(q) |u(X)|,    N_r u(q) = sup_X∈Γ_r(q) |u(X)|.

Given apertures 0< α< α_1 < β, for simplicity we denote Su, S'u as the square function on non-tangential cones of aperture α, α_1, respectively, and denote Nu the non-tangential maximal function of aperture β.

We have:

	Suppose Γ is a d-Ahlfors regular set in ^n with d<n-1, Ω= ^n ∖Γ and  is a collection of dyadic cubes for Γ, see Lemma <ref> for the detail.
	Let u∈ W_r(Ω) be a non-negative solution of Lu=0 such that for some dyadic cube Q∈ and λ>0 there exists q_1 ∈Γ with
	
    S'u(q_1) ≤λ   and    |q_1 - q| ≤ C_2  Q  for all  q∈ Q.

	Then for any X_Q ∉ B(x_Q, 2C_3 ℓ(Q)) and δ sufficiently small, we have
	
    ω^X_Q({ q∈ Q: Su(q) > 2λ, Nu(q) ≤δλ}) ≤ Cδ^2 ω^X_Q( Q )

	Here x_Q, ℓ(Q) are the “center” and “size” of Q, see Lemma <ref>. The constant C>0 depends on the allowable parameters d, n, C_0, C_1, the apertures α, α_1, β, and the given constants C_2, C_3.

If, moreover, ω∈ A_∞(σ), then the good-λ inequality for σ follows and we conclude that 

    Su_L^p(σ)≤ CNu_L^p(σ)

	 for any 1≤ p<∞ and any solution u∈ W_r(Ω) to Lu=0 such that the right hand side is finite.



The paper is structured as follows. In Section <ref> we first state some lemmas proved in <cit.> and prove some preliminary results based off these lemmas. In Section <ref> we prove the above Proposition after a careful analysis of the sawtooth domains, and moreover we prove the upper bound of the square function by the non-tangential maximal function. This is an independdent result and will also be used in Section <ref>, where we prove if the harmonic measure ω is of class A_∞(σ), the Dirichlet problem is BMO-solvable. We prove the converse in Section <ref>, that is, BMO-solvability implies the harmonic measure ω is of class A_∞(σ).



Acknowledgement We would like to thank Guy David for very helpful discussions in connection with this paper.






§ PRELIMINARIES


The ground work for harmonic measures associated to the (degenerate) elliptic operators L on sets of lower dimensions d<n-1 has been laid out in the work of David, Feneuil and Mayboroda, see <cit.>. In this section we state some relevant preliminary results proven in <cit.>; we also prove a few lemmas that follow easily and are needed in later sections. For the convenience of readers familiar with this subject, we point out that the new lemmas we prove here are Lemmas <ref>, <ref> and <ref>. Unless specified otherwise, the constants that appear in the following lemmas would depend only on the allowable constants, namely the dimensions n, d, the Ahlfors regular constant C_0 and the ellipticity constant C_1.

We start with the following notations:

	
  * For any X∈Ω, we denote δ(X) =(X,Γ), the Euclidean distance from X to Γ, and the weight w(X) = δ(X)^d-n+1. 
	
  * We denote
		
    𝒜(X) := 1/w(X) A(X) = δ(X)^n-1-d A(X) .

		By (<ref>) and (<ref>), 𝒜(X) is a uniformly elliptic matrix.
	
  * We define a measure m on Borel sets in ^n by letting m(E) = ∬_E w(X) dm(X). We may write dm(X) = w(X) dX. Since 0<w <∞ a.e. in ^n, m and the Lebesgue measure are mutually absolutely continuous.
	
  * For any q∈Γ and r>0, we use the notation Δ(q,r), or sometimes simply Δ, to denote the surface ball B(q,r) ∩, and T(Δ) to denote the “tent” B(q,r)∩Ω over Δ.
	
  * We denote the surface measure σ = ^d|_.
	
  * If B=B(X,r) is a ball and α>0 a constant, we use α B = B(X,α r) to denote the concentric dilation of B. The same notation applies to surface balls αΔ.
  


	Let Γ be a d-Ahlfors regular set in ^n and d<n-1. Then there exists a constant c∈(0,1), that depends only on d, n, C_0, such that for Λ≥ 1 and X_1, X_2 ∈Ω such that δ(X_i) ≥ s and |X_1 -X_2| ≤Λ s, we can find two points Y_i∈ B(X_i, s/2) such that ([Y_1, Y_2],Γ) ≥ cΛ^-d/(n-1-d) s. That is, there is a thick tube in Ω that connects the balls B(X_i, s/2).



	Note that 

    |Y_1 - Y_2| ≤ |Y_1 - X_1| + |X_1 - X_2| + |X_2 + Y_2| < 2Λ s.
 
Let τ = cΛ^-d/(n-1-d) s and Z_1 = Y_1. For 2≤ j ≤ N let Z_j be consecutive points on the line segment [Y_1, Y_2] such that |Z_j - Z_j-1| = τ/3. Then

    (N-1) τ/3≤ |Y_1 - Y_2| < N τ/3.

Combined with (<ref>) we get that the integer

    N ∼|Y_1- Y_2|/τ/3≲Λ^n-1/n-1-d.

Let B_0 = B(X_1, s/2), B_j = B(Z_j, τ/4) for 1≤ j≤ N and B_N+1 = B(X_2, s/2). Clearly B_j ∩ B_j+1≠∅ for all 0≤ j≤ N. Moreover (B_0, Γ), (B_N+1, Γ) ≥ s/2 and for 1≤ j≤ N,

    (B_j,Γ) ≥3/4τ = 3/4 cΛ^-d/n-1-d s,

and

    (B_j,Γ) ≤min{δ(X_1), δ(X_2)} + s/2 + |Y_1 - Y_2| < min{δ(X_1), δ(X_2)} + 3Λ s.




	
		
  * For any θ>0 there exists C_θ>0 such that for any X∈^n and r>0 satisfying δ(X) ≥ (1+θ) r,
			
    C_θ^-1 r^n w(X) ≤ m(B(X,r)) = ∬_B(X,r) w(z) dz ≤ Cr^n w(X).

		
  * There exists C>0 such that for any q∈Γ and r>0,
	
    C^-1 r^d+1≤ m(B(q,r)) = ∬_B(q,r)∩Ω w(z) dz ≤ Cr^d+1.

	


From the above we deduce the following estimate, which will be needed later.
 
Let Γ be d-Ahlfors regular. For any α>-1, we have
	
    ∬_T(2Δ)δ(X)^α dm(X) ≲ r^d+1+α.
 


	The proof is a simple use of Vitali covering.
	For j= 0,1,⋯ let 
	
    T_j = T(2Δ) ∩{x∈Ω: 2^-jr ≤δ(X) < 2^-j+1r},

	
    T_>j = T(2Δ) ∩{x∈Ω: δ(X) < 2^-j+1r}.

	Then
	
    ∬_T(2Δ)δ(X)^α dm(X) = ∑_j=0^∞∬_T_jδ(X)^α dm(X) ≤∑_j=0^∞ (2^-jr)^α m(T_>j).

	
	 
	For every fixed j, we consider a covering of 4Δ by ⋃_q∈ 4Δ B(q, 2^-j+1r/5), from which one can extract a countable Vitali sub-covering 4Δ⊂∪_k B(q_k, 2^-j+1r),	where q_k∈ 4Δ and the balls B_k = B(q_k, 2^-j+1r/5) are pairwise disjoint. The fact that q_k∈ 4Δ = Δ(q_0, 4r) implies 
	
    B_k := B(q_k, 2^-j+1r/5) ⊂ B(q_0,4r + 2^-j+1r/5).

	And the pairwise disjointness of B_k's implies that for every fixed j, there are only finitely many of them. In fact,
	
    ∑_kσ(B_k) = σ( ⋃_k B_k ) ≤σ( Δ(q_0, 4r+ 2^-j+1r/5)) ≲( 4r + 2r/5)^d.
 
	
	Note that σ(B_k) ≈( 2^-j+1r/5 )^d independent of k.
	Let N_j be the number of B_k's, by (<ref>)
	
    N_j·( 2^-j+1r/5)^d≤( 4r + 2r/5)^d ,     thus  N_j≲ 2^jd.

	
	For any X∈ T_>j, let q_X∈ be such that |X-q_X|=δ(X). Then
	
    |q_X-q_0| ≤ |q_X-X| + |X - q_0| < 4r,     i.e.  q_X∈ 4Δ.

	Hence q_X ∈ B(q_k, 2^-j+1r) for some k. Moreover T_>j⊂⋃_k B(q_k, 2· 2^-j+1r).
	Therefore by (<ref>) and (<ref>),
	
    m(T_>j) ≤ N_j ·sup_k m(B(q_k, 2· 2^-j+1r)) ≲ 2^jd(2^-j r)^d+1∼ 2^-jr^d+1.

	Combined with (<ref>) we get
	
    ∬_T(2Δ)δ(X)^α dm(X) ≲∑_j=0^∞ (2^-jr)^α· 2^-jr^d+1 = r^d+1+α∑_j=0^∞ 2^-j(α+1)≲ r^d+1+α.

	The last sum is convergent because α+1 >0.	


Now we define the suitable function spaces.
We denote by C_0^0(Γ) the space of compactly supported continuous functions on Γ, that is, f∈ C_0^0(Γ) if f is defined and continuous on Γ, and there exists a surface ball Δ such that f ⊂Δ. 
We consider the weighted Sobolev space

    W = Ẇ_w^1,2(Ω) = {u∈ L_loc^1(Ω): ∇ u ∈ L^2(Ω, dm) }

and set u_W = ( ∬_Ω |∇ u(X)|^2 dm(X) )^1/2 for u∈ W. 
In fact, it was proved in Lemma 3.3 of <cit.> that since Γ is d-Ahlfors regular with d<n-1,

    W = {u∈ L_loc^1(^n): ∇ u ∈ L^2(^n, dm) }.

We also define a local version of W as follows: Let E⊂^n be an open set, define

    W_r(E) = {u∈ L_loc^1(E): φ u ∈ W  for all φ∈ C_0^∞(E) }.

As observed in <cit.>,

    W_r(E) = {u∈ L^1_loc(E):∇ u ∈ L_loc^2(E,dm) }.

It is easy to see that if E⊂ F are open subsets of ^n, then the function space W_r(F) ⊂ W_r(E).
We set

    H = Ḣ^1/2(Γ) = { g a measurable function on Γ:  ∫_Γ∫_Γ|g(x) - g(y)|^2/|x-y|^d+1 dσ(x) dσ(y) <∞}.

The reader may recognize this is the homogeneous Sobolev space, 

a special case of the Besov spaces.
The authors in <cit.> were able to define a trace operator T: W→ H, see Theorem 3.13 (and Lemma 8.3 for a local version T: W_r(E) → L_loc^1(Γ∩ E)) there.




	Let E⊂Ω be an open set, and let u∈ W_r(E) be a non-negative solution in E. Then for any ϕ∈ C_0^∞(E),
	
    ∬_Ωϕ^2 |∇ u|^2 dm ≤ C ∬_Ω |∇ϕ|^2 u^2 dm,

	where C depends only on n, d and C_1.
	
	In particular, if B is a ball of radius r such that 2B ⊂Ω and u∈ W_r(2B) is a non-negative sub-solution in 2B, then 
	
    ∬_B |∇ u|^2 dm ≤ C r^-2∬_2B u^2 dm.



	(<ref>) holds if we replace 2B by (1+τ)B, τ>0, and in that case the constant C depends on the value of τ.



	
	 	
  * Let B be a ball such that 3B ⊂Ω and let u∈ W_r(3B) be a non-negative solution in 3B. Then 
	 		
    sup_B u ≤ Cinf_B u,

	 		where C depends on n, d and C_1.
	 	
  * Let K be a compact set of Ω and u∈ W_r(Ω) be a non-negative solution in Ω. Then
			
    sup_K u ≤ C_K inf_K u,

			where C_K depends only on n, d, C_0, C_1, (K,Γ) and K.
	 



	Let B⊂^n be a ball centered on Γ of radius r, and let u∈ W_r(2B) be a non-negative subsolution in 2B∖Γ such that Tu=0 a.e. on 2B. Then for any ϕ∈ C_0^∞(2B),
	
    ∬_2Bϕ^2 |∇ u|^2 dm ≤ C ∬_2B |∇ϕ|^2 u^2 dm,

	where C depends on n, d and C_1. In particular (<ref>) implies that
	
    ∬_B |∇ u|^2 dm ≤ C r^-2∬_2B u^2 dm.




	Let p>0. Let B be a ball centered on Γ and u∈ W_r(2B) be a non-negative sub-solution in 2B∖Γ such that Tu = 0 a.e. on 2B. Then
	
    sup_B u ≤ C_p ( 1/m(2B)∬_2B u^p dm )^1/p.




	Let B= B(q,r) be a ball centered on Γ and u∈ W_r(B) be a solution in B such that Tu ≡ 0 on B. There exists β∈ (0,1] such that for any 0<s<r/2,
	
    B(q,s) u ≤ C (s/r)^β( 1/m(B)∬_B |u|^2 dm )^1/2.



We are interested in the solution(s) of the Dirichlet problem (<ref>).

	For any f∈ H, there exists a unique u∈ W such that
	
    {[     Lu = 0       in Ω;     Tu = f a.e. on Γ. ].

	Moreover u_W ≤ C f_H.




	 There exists a bounded linear operator 
	 
    U: C_0^0(Γ) →  C(^n)

	 such that for every f∈ C_0^0(Γ)
	 
	 	
  * the restriction of Uf to Γ is f;
	 	
  * sup_^n Uf = sup_Γ f and ∫_^n Uf = inf_Γ f;
	 	
  * Uf∈ W_r(Ω) and is a solution of L in Ω;
	 	
  * if B is a ball centered on Γ and f≡ 0 on B, then Uf lies in W_r(B);
	 	
  * if f∈ C_0^0(Γ) ∩ H, then Uf ∈ W and is a unique solution of (<ref>).
	 


	Since Uf∈ C(^n), its trace T(Uf) is exactly f. We also remark that C_0^0(Γ) ∩ H is dense in C_0^0(Γ), with the supremum norm.




	For any X∈Ω, there exists a unique positive regular Borel measure ω^X on Γ such that
	
    Uf(X) = ∫_Γ f dω^X,    for any  f∈ C_0^0(Γ).

	Besides, for any Borel set E⊂Γ,
	
    ω^X(E) = sup{ω^X(K): E ⊃ K, K  is compact } = inf{ω^X(V): E⊂ V, V  is open }.

	Moreover, ω^X(Γ) = 1.



	Let E⊂Γ be a Borel set and define the function u_E on Ω by u_E(X) = ω^X(E). Then
	
		
  * if there exists X∈Ω such that u_E(X) = 0, then u_E ≡ 0;
		
  * the function u_E lies in W_r(Ω) and is a solution in Ω;
		
  * if B⊂^n is a ball such that E∩ B = ∅, then u_E ∈ W_r(B) and Tu_E = 0 on B∩Γ.
	


For now we are only able to write down the solution to (<ref>) if the boundary function f∈ C_0^0(Γ), see Lemma <ref>. With the help of the harmonic measure, we prove the following lemma:

	For any function f∈ C_0^0 (Γ) and any Borel set E⊂Γ, the function 
	
    u(X): = ∫_E f dω^X
 
	defined on Ω satisfies the following:
	
		
  * it is continuous in Ω;
		
  * it is a solution of Lu=0 in Ω and lies in W_r(Ω);
		
  * if B⊂^n is an open ball such that E∩ B=∅, then u is continuous in B∩Ω, u can be continuously extended to zero on B∩Γ, and that u∈ W_r(B). 
	



	We note the following:
	
		
  * Compared with Lemma <ref> and Lemma <ref>, this lemma says that fχ_E integrated against the harmonic measure gives rise to a continuous solution, for any Borel set E⊂Γ.
		
  * If the Borel set E is bounded, then the same properties hold for any bounded continuous function f∈ C_b(Γ).
	


	Since the definition (<ref>) is a linear integration, we may assume without loss of generality that f is non-negative. Otherwise we just write f=f_+ - f_-, with f_±∈ C(^n). We first assume that E is an open set, and that ω^X(E)>0 for some X ∈Ω. By Lemma <ref> (i) it follows that ω^X(E)>0 for all X∈Ω. Fix an arbitrary X_0∈Ω. Let K_j be an increasing sequence of compact sets in E, such that ω^X_0(E∖ K_j) < 1/j. By Urysohn's lemma we can construct g_j ∈ C_0^0(Γ) such that χ_K_j≤ g_j ≤χ_E, and without loss of generality we can choose the sequence g_j to be increasing. Note that fg_j ∈ C_0^0(Γ), and hence by Lemma <ref> we may define u_j = U(fg_j) ∈ C^0(Γ). 
	Then
	
    0 ≤ u(X) - u_j(X) = ∫ f( χ_E - g_j ) dω^X ≤ω^X(E∖ K_j) f_L^∞.

	By Lemmas <ref> and <ref>, for any compact subset K in Ω containing X_0, we have 
	
    ω^X(E∖ K_j) ≤ C_K ω^X_0(E∖ K_j)

	holds for every X∈ K. Here the constant C_K only depends on n, d, C_1, (K,Γ) and K, and in particular it is independent of j. Therefore
	
    0 ≤ u(X) - u_j(X) ≤C_K f_L^∞/j,

	namely {u_j} converges uniformly on compact sets of Ω to u, and thus u is continuous on Ω.
	
	Let ϕ∈ C_0^∞(Ω) be arbitrary, we claim that {u_j} has a subsequence, which we relabel, such that
	
    ∇ (ϕ u_j) ⇀∇ (ϕ u)  in  L^2(Ω, w).

	In particular ∇(ϕ u) ∈ L^2(Ω, w) for all ϕ∈ C_0^∞(Ω), and thus u∈ W_r(Ω).
	Indeed, by the interior Caccioppoli inequality (<ref>), we have
	
    ∬_Ω |∇(ϕ u_j)|^2 dm ≤ 2 ∬_Ω( |∇ϕ|^2 u_j^2 + ϕ^2 |∇ u_j|^2 ) dm ≤ C ∬_Ω |∇ϕ|^2 u_j^2 dm.
 
	Recall that u_j → u uniformly on the compact set ϕ, the right hand side of (<ref>) converges to C∬_Ω |∇ϕ|^2 u^2 dm. As a consequence the left hand side of (<ref>) is uniformly bounded in j. Therefore there is a subsequence (which we relabel) such that ∇ (ϕ u_j) converges weakly in L^2(Ω,w) to some function v. By the uniqueness of limit in the distributional sense, we conclude that v=∇(ϕ u), which finishes the proof of the claim (<ref>).
	
	Recall each u_j is a solution of L in Ω. Let φ∈ C_0^∞(Ω) be an arbitrary test function. We choose ϕ∈ C_0^∞(Ω) such that ϕ≡ 1 on φ. In particular ∇(ϕ u) = ∇ u, ∇(ϕ u_j) = ∇ u_j on φ. Thus
	
    ∬_Ω A ∇ u ·∇φ dX = ∬_Ω𝒜∇ u ·∇φ dm     = ∬_Ω𝒜∇ (ϕ u) ·∇φ dm 
        = lim_j→∞∬_Ω𝒜∇ (ϕ u_j ) ·∇φ dm 
        = lim_j→∞∬_Ω𝒜∇ u_j ·∇φ dm  = lim_j→∞∬_Ω A ∇ u_j ·∇φ dX = 0.

	
	If E is not an open set, the proof is similar, and we just need to approximate E from above by open sets. We omit the details here.
	
	Going further, if B⊂^n is an open ball such that E∩ B = ∅, we first prove that u can be continuously extended to zero on Γ∩ B. Take an arbitrary q∈Γ∩ B. Choose r>0 sufficiently small so that B(q,2r) ⊂ B. Consider a function g ∈ C_0^∞(^n) satisfying χ_B(q,r)≤ g ≤χ_B(q,2r). If f∈ C_0^0(Γ), then f(1-g) ∈ C_0^0(Γ). If the Borel set E is bounded and f is only assumed to be bounded continuous, we let φ∈ C_0^∞(^n) be a function such that φ≡ 1 on a compact set containing E and B(q,2r). Then f(1-g)φ∈ C_0^0(Γ).
	 Let 
	
    u(X) := U(f(1-g)φ ) = ∫_ f(1-g)φ dω^X.

	(For simplicity we take φ≡ 1 for case when f∈ C_0^0(Γ).)
	By the positivity of the harmonic measure and the fact that E⊂∖ B(q,2r), we deduce that 0≤ u(X) ≤u(X) for all X∈Ω. Recall by Lemma <ref> that u∈ C(^n), and as X→ q'∈ B(q,r) ∩Γ, the function u(X) → f(1-g)φ(q')=0. By the squeeze theorem u can be continuously extended to zero on B(q,r)∩Γ, and the resulting function, still denoted as u, is continuous in B(q,r).
	
	Now we show that u∈ W_r(B). To this end, let ϕ∈ C_0^∞(B), it suffices to show that ∇( ϕ u ) ∈ L^2(B,w). From Lemma <ref> (iv), Remark <ref> and the boundary Caccioppoli inequality (<ref>), we have
	
    ∬_B |∇ (ϕ u_j)|^2 dM ≤ 2 ∬_B( |∇ϕ|^2 u_j^2 + ϕ^2 | ∇ u_j|^2 ) dm ≤ C ∬_B |∇ϕ|^2 u_j^2 dm.
 
	Recall that u_j → u pointwise on B∖Γ. Since u is continuous on B, u∈ L^2(ϕ, w). Hence by the dominated convergence theorem the right hand side of (<ref>) converges to C∬_B |∇ϕ|^2 u^2 dm. As a consequence the left hand side is uniformly bounded, and thus passing to a subsequence ∇(ϕ u_j) converges weakly in L^2(B,w) to some function v. By the uniqueness of the limit we deduce v= ∇(ϕ u). In particular this implies ∇(ϕ u)∈ L^2(B,w).


As a summary, we can write down the solution of L using the harmonic measure, for the following classes of boundary data: continuous and compactly supported functions f∈ C_0^0(Γ) (see Lemma <ref>), characteristic functions χ_E for Borel sets E⊂Γ (see Lemma <ref>), their products fχ_E (see the above Lemma <ref>), or a linear combination of the above. For the third case, if the Borel set E is bounded, we only need to assume f∈ C_b(Γ).


	There exists M > 1 such that for any q∈Γ and r>0, there exists a point A = A_r(q)∈Ω such that
	
    |A-q| < r,   δ(A) ≥r/M.
 
	This point will be referred to as a corkscrew point hereafter.


	Note that neither Lemma <ref> nor Lemma <ref> is automatically true if d=n-1. In fact in the case of co-dimension 1, people often work with domains that satisfy Harnack chain condition and the existence of corkscrew point at all scales, called uniform domains or 1-sided NTA domains in the literature.




	Let q∈Γ and r>0 be given, and let A= A_r(q) be a corkscrew point as in Lemma <ref>. Let u∈ W_r(B(q,2r)) be a non-negative, non identically zero solution of Lu = 0 in B(q,2r)∩Ω, such that Tu ≡ 0 on Δ(q,2r). Then
	
    u(X) ≤ Cu(A)    for all  X∈ B(q,r).
 


We also recall the following “classical” Poincaré inequality for Sobolev functions.


	Let Γ be a d-Ahlfors regular set in ^n with d<n-1. For any function v∈ W, X∈^n and r>0, let B=B(X,r), then
	
    ( 1/m(B)∬_B |v(Y)-v_B|^2 dm(Y) )^1/2≤ C r ( 1/m(B)∬_B |∇ v(Y)|^2 dm(Y) )^1/2,

	where v_B denotes the average m(B)^-1∫_B v dm.


Suppose Δ= B(q_0,r) ∩Γ is a surface ball.
For any q∈Δ and any j∈ℕ, let 

    Γ_j(q) = Γ(q) ∩(B(q,2^-jr)∖ B(q,2^-j-1r))
  
be a stripe in the cone Γ(q) at height 2^-jr, and 

    Γ_j→ j+m(q) = ⋃_i=j^j+mΓ_i(q) = Γ(q) ∩( B(q, 2^-jr) ∖ B(q, 2^-(j+m)-1r) ) ,
 
be a union of (m+1) stripes. With these notations we can prove a less conventional form of Poincaré inequality, available for solutions with vanishing boundary values.


Suppose that u∈ W_r(Ω) is a non-negative solution of L, Tu=0 on 3Δ and u∈ W_r(B(q_0,3r)).
There exist an aperture α>α and integers m_1, m_2, such that  
for all q ∈Δ, 
	
    ∬_Γ_j^α(q) u^2 dm(X) ≤ C (2^-jr)^2 ∬_Γ_j-m_1 → j+m_2^α(q) |∇ u|^2 dm(X).

The constants m_1, m_2, α and C only depend on n, d, α, C_0, C_1.



	Let B be a ball compactly contained in Ω. Recall that the solution u ∈ W_r(Ω), in particular, φ u ∈ W for φ∈ C_0^∞(Ω) such that φ≡ 1 on B. Apply the above Lemma <ref> to φ u and square both sides, we get

    ∬_B |u(Y)-u_B|^2 dm(Y)  ≤ C r_B^2  ∬_B |∇ u(Y)|^2 dm(Y),


For j∈ℕ, let A_j denote a corkscrew point for B(q,2^-jr), whose existence is guaranteed by Lemma <ref>.
Let m be a large integer whose value is to be determined later. Take X∈Γ_j^α(q), X' = A_j+m, then 

    δ(X) > 1/1+α|X-q| ≥2^-j-1 r/1+α,   δ(X') ≥ 2^-(j+m)r/M,


    |X-X'| ≤ |X-q| + |q-X'| ≤  2^-j r + 2^-(j+m) r ≤ 2^1-j r.
 
Apply Lemma <ref> and Remark <ref> to X, X' with s = 2^-(j+m)r/M and Λ = 2^m+1 M, we can find balls B_0 = B(X,s/2), B_i = B(Z_i, τ/4) with τ = cΛ^-d/(n-1-d)s, B_N+1=B(X',s/2) that form a Harnack chain connecting X to X', and satisfy (<ref>), (<ref>) and (<ref>).
Hence by Lemma 2.3 (i) of <cit.> and (<ref>), (<ref>), we have

    m(B_i) ≥ C^-1( τ/4)^n (B_i,Γ)^d-n+1≳τ^n (Λ s)^d-n+1∼Λ^1-nτ^d+1,

and

    m(B_i) ≤ C(τ/4)^n (B_i,Γ)^d-n+1≲τ^n  τ^d-n+1∼τ^d+1

for all i=0,⋯,N,N+1. 
A simple computation shows B_i+1⊂ 3B_i for all i=1, ⋯ N-1, and B_1 ⊂3/2 B_0, B_N ⊂3/2 B_N+1, if m is sufficiently large.
Therefore for each i=1, ⋯, N-1,

    |u_B_i+1 -u_3 B_i|^2 
    	   ≤( 1/m(B_i+1)∬_B_i+1|u(X)-u_3 B_i| dm(X) )^2 
       ≤1/m(B_i+1)∬_3 B_i |u(X) - u_3 B_i|^2 dm(X) 
       ≲Λ^n-1τ^1-d∬_3B_i |∇ u(Y)|^2 dm(Y)     by (<ref>), (<ref>).

Similarly

    |u_B_i-u_3 B_i|^2 ≲Λ^n-1τ^1-d∬_3 B_i|∇ u(Y)|^2 dm(Y).

Hence

    |u_B_i-u_B_i+1|^2 ≤ CΛ^n-1τ^1-d∬_3 B_i|∇ u(Y)|^2 dm(Y).

A similar argument shows that for the end-point case i=0 or N+1,

    |u_B_i - u_B_i± 1|^2    ≲max{s^1-d, Λ^n-1 s^2 τ^-1-d}∬_3/2 B_i |∇ u(Y)|^2 dm(Y) 
       ∼Λ^n-1 s^2 τ^-1-d∬_3/2 B_i |∇ u(Y)|^2 dm(Y) .

The last line is justified since Λ≫ 1 implies τ≪ s.
Combining this observation, (<ref>), (<ref>) and (<ref>), we get

    ∬_B_0 |u(X) - u_B_N+1|^2 dm(X)    ≲ N ·∬_B_0 |u(X) - u_B_0|^2 dm(X) + N · m(B_0) ∑_i=0^N |u_B_i - u_B_i+1 |^2 
       ≲ NΛ^n-1 s^2 (s/τ)^d+1∬_3/2  B_0 ⋃(⋃_i=1^N 3B_i ) ⋃3/2 B_N+1 |∇ u(Y)|^2 dm(Y) 
       ≤ C' Λ^n-1+d(d+1)/n-1-d + n-1 s^2 ∬_3/2  B_0 ⋃(⋃_i=1^N 3B_i ) ⋃3/2 B_N+1 |∇ u(Y)|^2 dm(Y).

 
On the other hand, by Harnack inequality

    u(X) ≤ C u(X')    for all  X∈ B_N+1 = B(X',s/2).
 
Recall that X' = A_j+m. For any q∈Δ, by the assumption we know that u∈ W_r(B(q,2r)) vanishes on Δ(q,2r). By the boundary Hölder regularity (Lemma <ref>) and boundary Harnack principle (Lemma <ref>) we have

    u(X') ≤ C 2^-mβ u(A_j),

with a constant C independent of j and m. Thus

    u_B_N+1^2 ≲ u^2(X') ≲ 2^-2mβ u^2(A_j) ≲ 2^-2mβ·1/m(B_0)∬_B_0 u^2 dm(X) .
 
The last inequality holds because A_j is a corkscrew point and B_0 = B(X,s/2) for some X∈Γ_j(q).
Combining (<ref>) and (<ref>) we obtain

    ∬_B_0 u^2 dm(X) 
       ≤ 2  m(B_0)(u_B_N+1)^2 + 2∬_B |u(x) - u_B_N+1|^2 dm(X) 
       ≤ A_1 2^-2mβ∬_B_0 u^2 dm(X) + A_2 Λ^n-1+d(d+1)/n-1-d + n-1 s^2 ∬_3/2 B_0 ⋃(⋃_i=1^N 3B_i) ⋃3/2 B_N+1 |∇ u(Y)|^2 dm(Y).

Choose m big enough such that 

    A_1 2^-2m β≤1/2,    as well as  2 ·2^-m/M≤1/2(1+α),

 then we can absorb the first term on the right hand side of (<ref>) to the left. Recall that B_0 = B(X, s/2) for X satisfying (<ref>). The reason for the second assumption in (<ref>) is to guarantee the enlarged ball 3/2 B_0 is compactly contained in Ω. 
 Fix the value of m from now on, thus the value of Λ = 2^m+1/M is also fixed. We get

    ∬_B_0 u^2 dm(X) ≤ C s^2 ∬_3/2 B_0 ⋃(⋃_i=1^N 3 B_i) ⋃3/2 B_N+1 |∇ u(y)|^2 dy,

where s=2^-(j+m)r/M and the constant C depends on d,n,C_0,C_1 (Recall the values of corkscrew constant M and Harnack chain constant c only depend on d,n,C_0,C_1). 
Since B_0 = B(X, s/2) with center X∈Γ_j^α(q), it is a simple exercise to show that given the second assumption of (<ref>), there exists an aperture α_1>α such that

    3/2 B_0 ⊂Γ^α_1_j-1 → j+1(q).

A similar statement holds for 3/2 B_N+1.
Moreover (<ref>) and (<ref>) imply that for i=1, ⋯, N, there exist an aperture α_2>α and an integer m_0 depending on the constants c, M from Lemmas <ref> and <ref>, such that

    3B_i ⊂Γ_j-3 → j+m+m_0 ^α_2(q).

Let α= max{α_1, α_2}.
Combining the above observations with (<ref>) we get

    ∬_B_0 u^2 dm(X) ≤ C s^2 ∬_Γ_j-3 → j+m+m_0 ^α(q)  |∇ u(y)|^2 dy.

	
Consider the covering

    Γ_j^α(q) ⊂⋃_X∈Γ_j^α(q) B( X, s/10).

We can extract a finite Vitali sub-covering {B^k = B(X_k, s/2)}_k such that

    Γ_j^α(q) ⊂⋃_k B^k

and {B^k/5 = B(X_k, s/10)}_k is mutually disjoint. 
Moreover the number of balls B^k's is uniformly bounded by a constant C(n, m, M). Note that (<ref>) holds for all such balls B^k in place of B_0, we deduce

    ∬_Γ_j^α(q) u^2 dm(X) ≤∑_k ∬_B^k u^2 dm(X)  ≤ CC(n,m,M) s^2 ∬_Γ_j-3 → j+m+m_0 ^α(q)  |∇ u(y)|^2 dy.

Since the value of m is fixed, we finish the proof of Lemma <ref>.







	Let λ>1 be given. There exists a constant C_λ >1 such that for any q∈Γ, r>0, and A=A_r(q), a corkscrew point from Lemma <ref>, we have
	
    ω^X(B(q,r)∩Γ) ≥ C_λ^-1  for  X∈ B(q,r/λ),

	
    ω^X(B(q,r)∩Γ) ≥ C_λ^-1  for  X∈ B(A,δ(A)/λ).



In <cit.> the authors also prove the existence, uniqueness and properties of the Green function, that is, formally, a function G defined on Ω×Ω such that for any Y∈Ω,

    {[ LG(·,Y) = δ_Y          in Ω;    G(·,Y) = 0          on Γ ].

where δ_Y is the delta function.


	There exists a constant C≥ 1, such that for any q∈Γ and r>0, Δ = B(q,r)∩Γ and a corkscrew point A = A_r(q), then
	
    C^-1 r^d-1 G(X_0,A) ≤ω^X_0(Δ) ≤ C r^d-1 G(X_0,A)   for  X_0 ∈Ω∖ B(q,2r).




	 For q∈Γ and r>0, we have
	
    ω^X(B(q,2r)∩Γ) ≤ C ω^X(B(q,r)∩Γ)

	for any X∈Ω∖ B(q,4r).




	Let q∈Γ and r>0 be given, and let A = A_r(q) be a corkscrew point as in Lemma <ref>. Let E, F ⊂Δ(q,r) be two Borel subsets of Γ such that ω^A(E) and ω^A(F) are positive. Then 
	
    ω^X(E)/ω^X(F)∼ω^A(E)/ω^A(F),    for any X∈Ω∖ B(q,2r).

	In particular with the choice F=Δ(q,r), 
	
    ω^X(E)/ω^X(Δ(q,r))∼ω^A(E)    for any  X∈Ω∖ B(q,2r).







Let us restate the definition of ω∈ A_∞(σ) and make a few remarks that will become useful later.

	We say the harmonic measure ω is of class A_∞ with respect to the surface measure σ = ℋ^d|_Γ, or simply ω∈ A_∞(σ), if for any ϵ>0, there exists δ=δ(ϵ)>0 such that for any surface ball Δ, any surface ball Δ' ⊂Δ and any Borel set E⊂Δ', we have
	
    σ(E)/σ(Δ') <δω^A(E)ω^A(Δ') <ϵ.

	Here A = A_Δ is a corkscrew point for Δ (see Lemma <ref>).


	
		
  * The reader may recall that the standard definition for A_∞ is that the harmonic measure with a fixed pole, i.e. ω^X_0, satisfies (<ref>). For unbounded boundary Γ though, the standard definition needs to be replaced by its scale-invariant analogue, which is Definition (<ref>). In fact since Γ is unbounded, it is impossible to have ω^X_0∈ A_∞(σ) with a fixed pole X_0, see the comments after Theorem 1.18 of <cit.>.
		
  * The above definition is symmetric: suppose ω∈ A_∞(σ), then we also have σ∈ A_∞(ω) (in a scale-invariant sense), i.e., the smallness of ω^A(E)/ω^A(Δ') implies the smallness of σ(E)/σ(Δ').
		
  * In particular, the assumption (<ref>) implies that ω^A ≪σ when restricted to Δ. We denote the Radon-Nikodym derivative by k^A = dω^A/d σ. Since both ω^A and σ are Radon measures, we have

    k^A(q) = lim_Δ'=Δ(q,r) r→ 0ω^A(Δ')/σ(Δ'),    for σ-a.e.  q∈Δ.

 Moreover since σ is doubling, by standard harmonic analysis techniques (see <cit.> for example for the proof) (<ref>) implies that k^A satisfies a reverse Hölder inequality: there are constants r_0>1, C>0 such that for all r∈ (1,r_0),
			
    ( _Δ|k^A|^r dσ)^1/r≤ C _Δ k^A dσ.
 
			The constants r_0 and C only depend on the constants characterizing the A_∞ property (<ref>); in particular, they are independent of Δ and A.
	

			
Recall that one of our main goals is to prove Theorem <ref>, which states the equivalence between ω∈ A_∞(σ) and the BMO solvability of the Dirichlet problem. We make a few preliminary remarks.

Note that (, σ) is a space of homogeneous type. By John-Nirenberg inequality for space of homogeneous type, we may also use any L^p norm (1≤ p<∞) in the definition (<ref>), and the resulting BMO norms are all equivalent. See <cit.> and <cit.>. Also it is easy to see that if f∈ L^∞(), then f is a BMO function with f_BMO≤√(2)f_L^∞.



We observe that the Carleson measure norm of |∇ u|^2δ(X) dm(X) is in some sense equivalent to the integral of the truncated square function.
Suppose Δ=Δ(q_0,r) is an arbitrary surface ball. For any X∈ T(Δ), we define Δ^X={q∈: X∈Γ(q)}. Let q_X∈ be a point such that |X-q_X|=δ(X). Then 

    Δ(q_X, αδ(X))⊂Δ^X⊂Δ(q_X, (α+2)δ(X)).
 
Since Γ is d-Ahlfors regular, (<ref>) implies σ(Δ^X) ≈δ(X)^d. Thus

    ∬_T(Δ)u dm(X)    ≈∬_T(Δ) |∇ u|^2 δ(X)^1-dσ(Δ^X ) dm(X) 
        =∬_T(Δ) |∇ u|^2 δ(X)^1-d∫_Δ^X dσ(q) dm(X).
 
Changing the order of integration, on one hand we get an upper bound

    ∬_T(Δ) |∇ u|^2 δ(X)^1-d∫_Δ^X dσ(q) dm(X)    ≤∫_|q-q_0|<(α+2)r∬_Γ_(α+1) r(q) |∇ u|^2 δ(X)^1-d dm(X)dσ
       ≤∫_(α+2)Δ|S_(α+1) ru|^2dσ.

On the other hand, we get a lower bound

    ∬_T(Δ) |∇ u|^2 δ(X)^1-d∫_Δ^X dσ(q) dm(X)    ≥∫_|q-q_0|<r/2∬_Γ_r/2(q) |∇ u|^2 δ(X)^1-d dm(X)dσ
       ≥∫_1/2Δ |S_r/2 u|^2 dσ.

Therefore for any q_0∈,

    sup_Δ=Δ(q_0,s) 
     s>01/σ(Δ)∬_T(Δ)u dm(X) ≈sup_Δ=Δ(q_0,r) 
     r>01/σ(Δ)∫_Δ |S_r u|^2 dσ.
 







§ BOUND OF THE SQUARE FUNCTION BY THE NON-TANGENTIAL MAXIMAL FUNCTION


The goal of this section is to prove:


	Let Γ be a d-Ahlfors regular set in ^n with an integer d≤ n-1, and let ω be the harmonic measure of the domain Ω=	^n ∖Γ.
	 Suppose ω∈ A_∞(σ), then
	
    Su_L^p(σ)≤ CNu_L^p(σ)

	 for any 1≤ p<∞ and any solution u∈ W_r(Ω) to Lu=0 such that the right hand side is finite. Here the constant C>0 depends on the allowable parameters d, n, C_0, C_1, the aperture α and the A_∞ constant(s).

It suffices to prove (<ref>) for non-negative harmonic functions u, because otherwise, we just split u= u_+ - u_- and use the linearity of L and the triangle inequality.
Before starting to prove the theorem we need to recall some notation and preliminary results.


	Let Γ⊂^n be a d-Ahlfors regular set. Then there exist constants a_0, A_1, γ>0, depending only on d, n and C_0, such that for each k∈ℤ, there is a collection of Borel sets (“dyadic cubes”)
	
    𝔻_k : = {Q_j^k ⊂Γ: j∈𝒥_k },

	where 𝒥_k denotes some index set depending on k, satisfying the following properties.
	
		
  * Γ = ⋃_j∈𝒥_k Q_j^k for each k∈ℤ.
		
  * If m≥ k then either Q_i^m ⊂ Q_j^k or Q_i^m ∩ Q_j^k = ∅.
		
  * For each pair (j,k) and each m<k, there is a unique i∈𝒥_m such that Q_j^k ⊂ Q_i^m.
		
  * Q_j^k ≤ A_1 2^-k.
		
  * Each Q_j^k contains some surface ball Δ(x_j^k, a_0 2^-k) := B(x_j^k, a_0 2^-k) ∩Γ.
		
  * ℋ^d ( { q∈ Q_j^k: (q, Γ∖ Q_j^k) ≤ρ 2^-k}) ≤ A_1 ρ^γℋ^d (Q_j^k), for all (j, k) and all ρ∈ (0,a_0).
	
	We shall denote by 𝔻 = 𝔻(Γ) the collection of all relevant Q_j^k, i.e.
	
    = ⋃_k _k.



	
		
  * For a dyadic cube Q∈, we let k(Q) denote the “dyadic generation” to which Q belongs, i.e. we set k(Q)=k if Q∈_k. We also set its “length” ℓ(Q) = 2^-k(Q). Thus ℓ(Q) = 2^-k(Q)∼ Q.
		
  * Properties (iv) and (v) imply that for each cube Q∈, there is a point x_Q ∈Γ such that 
			
    Δ(x_Q, r_Q) ⊂ Q ⊂Δ(x_Q, C_2 r_Q),

			where r_Q = a_0 2^-k(Q)∼ Q and C_2 = A_1/a_0.
		


Now we define sawtooth domains following the definitions of Hofmann and Martell, see for example <cit.>, <cit.> and <cit.>.
Since Ω is an open set, it has a Whitney decomposition, that is, a collection of closed “Whitney” boxes in Ω, denoted by = (Ω), which form a covering of Ω with pairwise non-overlapping interiors and satisfy

    4 I ≤(4I, ) ≤(I,) ≤ 40  I,    for any  I∈,

and also 

    1/4 I_1 ≤ I_2 ≤ 4 I_1

whenever I_1 and I_2 in  touch. (See <cit.> for reference.) Let X_I denote the center of I and ℓ(I) the side length of I, then I ∼ℓ(I). We also write k(I) = k if ℓ(I) = 2^-k. 

Let 𝔻 be a collection of dyadic cubes for the Ahlfors regular set Γ, as in Lemma <ref>.
For any dyadic cube Q∈, pick two parameters η≪ 1 and K≫ 1, and define

    _Q^0 : = {I∈: η^1/4ℓ(Q) ≤ℓ(I) ≤ K^1/2ℓ(Q), (I,Q) ≤ K^1/2ℓ(Q)}.
 
Let X_Q denote a corkscrew point for the surface ball Δ(x_Q, r_Q/2). We can guarantee that X_Q is in some I∈_Q^0 provided we choose η small enough and K large enough. For each I∈_Q^0, by Lemma <ref> and the discussions after that, there is a Harnack chain connecting X_I to X_Q, we call it ℋ_I. By the definition of _Q^0 we may construct this Harnack chain so that it consists of a bounded number of balls (depending on the values of η, K), and stays a distance at least cη^n-1/4(n-1-d)ℓ(Q) away from  (see (<ref>)). We let _Q denote the set of all J∈ which meet at least one of the Harnack chains ℋ_I, with I∈_Q^0, i.e.

    _Q := {J∈: there exists  I∈_Q^0  for which ℋ_I ∩ J ≠∅}.

Clearly _Q^0 ⊂_Q. Besides, it follows from the construction of the augmented collections _Q and the properties of the Harnack chains (in particular (<ref>) and (<ref>)) that there are uniform constants c and C such that

    cη^n-1/4(n-1-d)ℓ(Q) ≤ℓ(I) ≤ CK^1/2ℓ(Q),   (I,Q) ≤ CK^1/2ℓ(Q)

for any I∈_Q. In particular once η, K is fixed, for any Q∈ the cardinality of _Q is uniformly bounded, which we denote by N_0.

Next we choose a small parameter θ∈ (0,1) so that for any I∈, the concentric dilation I^* = (1+θ)I still satisfies the Whitney property

    I ∼ I^* ∼(I^*,) ∼(I,).

Moreover by taking θ small enough we can guarantee that (I^*, J^*) ∼(I,J) for every I, J∈, I^* meets J^* if and only if ∂ I meets ∂ J and that 1/2 J∩ I^*=∅ for any distinct I, J∈. In what follows we will need to work with further dilations I^** = (1+2θ) I or I^*** = (1+4θ)I etc.. (We may need to take θ even smaller to make sure the above properties also hold for I^**, I^*** etc..)
Given an arbitrary Q∈, we may define an associated Whitney region U_Q, U_Q^* as follows

    U_Q: = ⋃_I∈_Q I^*,    U_Q^*: = ⋃_I∈_Q I^**.


Let _Q = {Q'∈: Q' ⊂ Q}. For any Q∈ and any family ℱ= {Q_j} of disjoint cubes in _Q∖{Q}, we define the local discretized sawtooth relative to  by

    _,Q : = _Q ∖⋃_Q_j∈_Q_j.
 
We also define the local sawtooth domain relative to  by

    Ω_,Q:= ( ⋃_Q'∈_,Q U_Q'),   Ω_,Q^*:= ( ⋃_Q'∈_,Q U_Q'^*)

For convenience we set

    _,Q := ⋃_Q'∈_,Q_Q',

so that in particular, we may write

    Ω_,Q = (⋃_I∈_,Q I^* ),   Ω_,Q^* = (⋃_I∈_,Q I^**).

We will need further fattened sawtooth domain Ω_,Q^** etc. whose definitions follow the same lines as above. We remark that by (<ref>), there is a constant C_3 depending on K, θ such that

    Ω_ℱ,Q⊂ B(x_Q, C_3 ℓ(Q)) ∩Ω

for any Q∈ and collection of maximal cubes ℱ, where x_Q is the “center” of Q as in (<ref>).

Finally, to work with sawtooth domains, it is more natural to use a discrete dyadic version of the approach region rather than the standard non-tangential cone defined in (<ref>): for every q∈, we define the dyadic non-tangential cones as

    Γ_d(q) = ⋃_Q∈: Q∋ q U_Q,   Γ_d(q) = ⋃_Q∈: Q∋ q U_Q^***

where we use Γ_d to denote a cone with bigger “aperture” or fattened region;
we also define the local dyadic non-tangential cones as

    Γ_d^Q(q) = ⋃_Q'∈_Q : Q'∋ q U_Q',   Γ_d^Q (q) = ⋃_Q'∈_Q : Q'∋ q U_Q'^***.


We claim that given an aperture α>0, there exists K (in the definition (<ref>)) sufficiently large such that the standard non-tangential cone Γ^α(q) ⊂Γ_d(q) for all q∈; and vice versa, for fixed values of η, K and the dilation constant θ, there exists α_1>0 such that the dyadic cone Γ_d(q) ⊂Γ^α_1(q) for all q∈.
For any X∈Γ^α(q), let I be a Whitney box such that X∈ I^*. By (<ref>) we know ℓ(I) ∼δ(X).
Let Q be a cube containing q with length ℓ(Q) = ℓ(I). Then

    (I,Q) ≤ |X-q| < (1+α) δ(X) ≤ C (1+α) ℓ(I) = C(1+α)ℓ(Q).
 
If K is sufficiently large so that K^1/2≥ C(1+α), then (<ref>) and ℓ(I) = ℓ(Q) implies that I∈_Q^0. By the definition (<ref>) it follows that X∈Γ_d(q).
In particular, since Γ^α(q) is open, we also have Γ^α(q) ⊂Γ_d(q).
On the other hand, suppose X∈Γ_d(q), by definition (<ref>) X is contained in some I^*= (1+θ)I for a Whitney box I∈_Q and dyadic cube Q containing q. Then by (<ref>),

    |X-q| ≤ I^* + (I,Q) +  Q ≤ C(K,θ) ℓ(Q),


    δ(X) ∼ℓ(I) ≥ C(η) ℓ(Q).

Therefore there exists α_1 sufficiently large, depending on the values of η, K, θ, such that

    |X-q| < (1+α_1)δ(X),

i.e. X∈Γ^α_1(q).
 We summarize that now we have

    Γ^α(q) ⊂Γ_d(q) ⊂Γ_d(q) ⊂Γ^α_1(q),    for all q∈.

Clearly α_1>α. Moreover, there exists β>α_1 depending on η, K, θ such that the fattened dyadic non-tangential cone

    Γ_d(q) ⊂Γ^β(q)    for all q∈.

From now on we fix the values of η, K, θ and β> α_1>α>0.

Let F= Q∖⋃_Q_j∈ Q_j and suppose it is not empty. We claim that

    (⋃_q∈ FΓ^Q_d(q) ) ⊂Ω_, Q⊂Ω_, Q⊂Ω_,Q^***⊂⋃_q∈ FΓ^Q_d(q).

In fact, for any q∈ F, it is clear that q is in some Q'∈_,Q; and by (<ref>), the definition of Ω_,Q, we have the first incluement. On the other hand any X∈Ω_, Q^*** belongs to some U_Q'^*** with Q'∈_,Q, and thus X∈Γ_d^Q(q) for arbitrary q∈ Q'. By the definition of _,Q, we know Q'∩ F≠∅, so by taking q∈ Q'∩ F we get X∈⋃_q∈ FΓ^Q_d(q).

For N sufficiently large, we augment the collection of maximal cubes  by adding all dyadic cubes in 𝔻 of size smaller than or equal to 2^-Nℓ(Q), and we denote by ^N a collection consisting of all maximal cubes of the above augmented collection. In particular Q'∈_^N,Q if and only if Q'∈_,Q and ℓ(Q') > 2^-Nℓ(Q). By doing this we guarantee that the sawtooth domain Ω_^N,Q is compactly contained in Ω (roughly speaking (Ω_^N,Q, Ω^c) ∼ 2^-Nℓ(Q)).
Similar to Lemma 4.44 of <cit.> we can construct a smooth cutoff function of Ω_^N,Q:


	There exists ψ_N∈ C_0^∞(^n) such that
	
		
  * χ_Ω_^N,Q^*≲ψ_N ≤χ_Ω_^N,Q^**;
		
  * sup_X∈Ω |∇ψ_N (X)|δ(X) ≲ 1;
		
  * We abbreviate 𝒲_ℱ^N,Q as 𝒲_N and set Σ = ∂Ω_^N,Q^*,
			 
    𝒲_N^Σ = {I∈𝒲_N: there exists  J∈𝒲∖𝒲_N  with ∂ I ∩∂ J ≠∅}.
 Then
			
    ∇ψ_N ≡ 0  in ⋃_I∈∖ I^***.

		
  * For each I∈_N, let Q_I denote a cube in _^N,Q such that I∈_Q_I. Suppose ω is the harmonic measure with pole X_0 and X_0 satisfies (X_0, Ω_^N,Q^***) ≳ℓ(Q). Then
			
    ∑_I∈𝒲_N^Σω(Q_I) ≲ω(Q)

			with a constant depending on η, K, a_0, C_1, d and the Ahlfors regular constant of Γ.
	


	
		
  * We remark that the construction of ψ_N and the proof of its properties (i), (ii), (iii) are higher codimensional analogues of Lemma 4.44 of <cit.>. However we prove (iv) instead of the second estimate in their (4.46), because we will need to prove a good-λ inequality for the harmonic measure, instead of the surface measure. Since harmonic measure could have much worse decay properties than the surface measure, not to mention that  and ∂Ω_^N,Q are objects of different dimensions, proving (iv) requires a different argument.
		
  * Note that in (iv), the choice of Q_I may not be unique. Suppose both Q_I, Q_I are cubes in _^N,Q such that I∈_Q_I and I∈_Q_I. By the construction of _Q's and in particular (<ref>), we know
	
    ℓ(Q_I) ∼ℓ(I) ∼ℓ(Q_I),   (Q_I, Q_I) ≲ℓ(Q_I)

	with constants depending on η, K. Since harmonic measure is doubling, we have
	
    C_1 ω(Q_I) ≤ω(Q_I) ≤ C_2 ω(Q_I)

	with constants only depending on the doubling constant and η, K. That is to say, for different choices of Q_I the left hand side of (<ref>) differs at most by a constant multiple. But once we associate a cube Q_I to I, the choice will be fixed.
	





	The proof of (i) is a modification of the proof from <cit.> in higher codimensions. We recall that given I, any closed dyadic cube in ^n, we set I^** = (1+2θ)I and I^*** = (1+4θ)I. Let us introduce I^**= (1+3θ)I so that
	
    I^**⊊⊊⊂ I^***.

	Given I_0 = [-1/2,1/2]^n⊂^n, we fix ϕ_0∈ C_0^∞(^n) such that χ_I_0^**≤ϕ_0≤χ_I_0^** and |∇ϕ_0|≲ 1, with the implicit constant depending on θ. For every I∈ we set ϕ_I= ϕ_0((·-X_I)/ℓ(I)) where X_I is the center of I, so that ϕ_I∈ C_0^∞(^n), χ_I^**≤ϕ_I ≤χ_ and |∇ϕ_I| ≲ 1/ℓ(I). Let Φ(X):= ∑_I∈ϕ_I(X) for every X∈Ω. Since for each compact subset of Ω, the previous sum has finitely many non-vanishing terms, we have Φ∈ C^∞_loc(Ω). Also 0 ≤Φ(X) ≲ C_θ since the family {}_I∈ has bounded overlap. Hence we can set Φ_I = ϕ_I/Φ and one can easily see that Φ_I∈ C_0^∞(^n), C_θ^-1χ_I^**≤Φ_I ≤χ_ and |∇Φ_I|≲ 1/ℓ(I). Recall the definition of _N = _^N,Q in (<ref>), we set
	
    ψ_N(X) = ∑_I∈_NΦ_I(X) = ∑_I∈_Nϕ_I(X)∑_I∈ϕ_I(X),    X∈Ω.

	We first note that the number of terms in the sum defining ψ_N is bounded depending on N. Indeed if Q'∈_^N,Q then Q' ∈_Q and 2^-Nℓ(Q) < ℓ(Q')≤ℓ(Q), which implies _^N,Q has finite cardinality with bounded depending only the Alhfors regular constant and N. Also by construction _Q has cardinality depending only in the allowable parameters η, K. Hence #_N ≤ C_N<∞. This and the fact that each Φ_I∈ C_0^∞(^n) yield that ψ_N∈ C_0^∞(^n). Moreover
	
    ψ_N ⊂⋃_I∈_N = ⋃_Q'∈_^N,Q⋃_I∈_Q⊂( ⋃_Q'∈_^N,Q U_Q'^**) = Ω_^N,Q^**.

	This and the definition of ψ_N immediately gives ψ_N ≤χ_Ω_^N,Q^**. On the other hand if X∈Ω_^N,Q^* then there exists I∈_N such that X∈ I^**, in which case ψ_N(X) ≥Φ_I(X) ≥ C_θ^-1. This completes the proof of (i).
	
	To obtain (ii) we note that for every X∈Ω
	
    |∇ψ_N(X)| ≤∑_I∈_N |∇Φ_I(X)| ≲∑_I∈1/ℓ(I)χ_(X) ≲1/δ(X),

	where we have used that if X∈ then ℓ(I) ∼δ(I) and also that the family {}_I∈ has bounded overlap.
	
	Now we turn to (iii). Fix I∈_N∖_N^Σ and X∈ I^***, and set _X = {J∈: ϕ_J(X) ≠ 0}. We first note that _X ⊂_N. Indeed if ϕ_J(X) ≠ 0 then X∈J^**. Hence X∈ I^***∩ J^*** and our choice of θ gives that ∂ I meets ∂ J, this in turn implies that J∈_N since I∉_N^Σ. All these imply
	
    ψ_N(X) = ∑_J∈_Nϕ_J(X)∑_J∈ϕ_J(X) = ∑_J∈_N∩_Xϕ_J(X)∑_J∈∩_Xϕ_J(X) = ∑_J∈_N∩_Xϕ_J(X)∑_J∈_N∩_Xϕ_J(X) = 1.

	Hence ψ_N|_I^***≡ 1 for every I∈_N∖_N^Σ. This and the bounded overlap of the family {I^***}_I∈_N immediately give that ∇ψ_N ≡ 0 in ⋃_I∈_N∖_N^Σ I^***.
	
	Finally, it remains to prove the most difficult property, (iv).
	For any I∈_N^Σ, by definition there exists some J_I ∈∖_N such that ∂ I ∩∂ J_I ≠∅. Roughly speaking, this is to say that I is a Whitney box living in the “boundary” of Ω_^N,Q^*.
	 Thus pick any Q'_I ∈ such that _Q'_I contains J_I, we know Q'_I ∉_^N,Q, that is, either Q'_I ∈_Q_j for some Q_j∈^N, or Q'_I ∉_Q. We classify I∈_N^Σ based on which category its associated cube Q'_I lives in: We denote 
	 
    Σ_j = {I∈_N^Σ: Q'_I ∈_Q_j} for any  Q_j∈^N,

	 and 
	 
    Σ_0 = {I∈_N^Σ:  Q'_I ∉_Q}.
 
	 (Note that for each I∈_N^Σ, we associate it to a unique Q'_I, even though the choice itself is not unique.)
	Recall (<ref>) we have ℓ(I)∼ℓ(J_I). Moreover by the definition of _Q and (<ref>),
	
    ℓ(Q'_I) ∼ℓ(J_I) ∼ℓ(I) ∼ℓ(Q_I)

	and
	
    (Q_I,Q'_I) ≤(Q_I, I) + (I,J_I) + (J_I,Q'_I) ≲ℓ(Q_I) + ℓ(Q'_I) ≲ℓ(Q'_I).

	By similar argument as in remark <ref> (2) and the doubling property of harmonic measure, we have ω(Q_I) ∼ω(Q'_I) for any I∈_N^Σ, with a uniform constant depending on η, K. Therefore to prove (<ref>) it suffices to show
	
    ∑_I∈𝒲_N^Σω(Q'_I) ≲ω(Q).

	
	 We claim that for any Q_j∈^N,
	
    ∑_I∈Σ_j ω(Q'_I) ≲ω(Q_j).

	Recall that all such Q'_I's live in _Q_j.
	For each k∈ℕ we denote Σ_j^k = {I∈Σ_j: ℓ(Q'_I) = 2^-kℓ(Q_j) }.
	Since Q_I ∈_^N,Q, Q_j ∈^N, we always have Q_j ∩ Q_I = ∅, so by (<ref>)
	
    (Q'_I, (Q_j)^c ) ≤(Q'_I, Q_I) ≲ℓ(Q'_I) = 2^-kℓ(Q_j).

	That is, the smaller Q'_I is, the closer it is to the “boundary” of Q_j. The Q'_I's of different generations are very far from being disjoint, however we will sum up the ω(Q'_I)'s by swapping them for the harmonic measure of mutually disjoint cubes.
	By (<ref>), for ρ sufficiently small there is an integer k_1= k_1(ρ) such that for any integer k≥ k_1,
	
    ⋃_k'≥ k⋃_I∈Σ_j^k' Q'_I  ⊂{q∈ Q_j: (q, (Q_j)^c ) ≤ρ/2ℓ(Q_j) }.
 
	In fact by choosing k_1 slightly bigger, we can even guarantee that for any integer k≥ k_1,
	
    ⋃_k'≥ k⋃_I∈Σ_j^k' Q'_I ⊂⋃_i∈ℐ_k Q_j^i  ⊂{q∈ Q_j: (q, (Q_j)^c ) ≤ρ/2ℓ(Q_j) },

	where {Q_j^i}_i∈ℐ_k is the collection of all dyadic cubes in _Q_j of length 2^-kℓ(Q_j) such that Q_j^i ⊂{q∈ Q_j: (q,(Q_j)^c) ≤ρℓ(Q_j)/2 }. By Lemma <ref> (v) (vi) the index set ℐ_k has finite cardinality and #ℐ_k ≤ C2^kd. (A priori the set ℐ_k could be empty, in which case (<ref>) just means there is no Q'_I corresponding to any I∈⋃_k'≥ kΣ_j^k'. This case is easy to deal with.)
	
	On the other hand by Lemma <ref>, as long as we fix ρ∈ (0,a_0) satisfying A_1 ρ^γ<1, the set {q∈ Q_j: (q,(Q_j)^c) > ρ/2ℓ(Q_j) } is not empty; moreover, there is an integer k_2 sufficiently large such that for each k≥ k_2 we can find a cube Q_j such that ℓ(Q_j ) = 2^-kℓ(Q_j) and
	
    Q_j ⊂{q∈ Q_j: (q,(Q_j)^c) > ρ/2ℓ(Q_j) }.

	We may think of Q_j as sitting in the “center” of Q_j, and all Q'_I's in a ρ/2-boundary layer of Q_j. Let k_0 = max{k_1, k_2}, and let N_1 denote the (maximal) number of Q'_I's with ℓ(Q'_I) = 2^-k_0ℓ(Q_j). By (<ref>) and Lemma <ref> (vi), N_1 is uniformly bounded by a constant depending on a_0, A_1, ρ, k_0 and d.  Moreover by the doubling property of ω, each such Q'_I satisfies 
	
    ω(Q'_I) ≤ω(Q_j) ≤ C(k_0)ω(Q_j),

	with the constant C(k_0) depending on k_0 as well as the doubling constant of ω.
	Recall that for each Q'_I, the number of all possible I's corresponding to it is uniformly bounded by C(N_0). 
	Therefore
	
    ∑_I∈Σ_j^k_0ω(Q'_I) ≤ C(N_0) ∑_Q'_I: ℓ(Q'_I) = 2^-k_0ℓ(Q_j)ω(Q'_I) ≤  C(N_0) N_1 C(k_0) ω(Q_j).

	Now for any I∈Σ_j^k with k=1,⋯, k_0-1, again by the doubling property of harmonic measure we have ω(Q'_I) ≤ C(k_0) ω(Q_j). By Lemma <ref> (iv) (v), the total number of Q'_I's in _Q_j such that ℓ(Q'_I)=2^-kℓ(Q_j) with k=1,⋯, k_0-1 is uniformly bounded by a constant depending only on k_0, a_0, C_1, d and the Ahlfors regular constant of Γ. Thus the total number of I's in Σ_j^k with k=1, ⋯, k_0-1 is also uniformly bounded. Therefore combining with  (<ref>), we get
	
    ∑_k=1^k_0∑_I∈Σ_j^kω(Q'_I) ≲ω(Q_j).estimate-k_0

	
	For future generations, we recall (<ref>), which says all the Q'_I's corresponding to some I∈Σ_j^k with k≥ k_0 are contained in ⋃_i∈ℐ_k_0 Q_j^i. 
	The following proof is illustrated in the (idealized) Figure <ref>, where each label denotes the cube near it enclosed or shaded by the same color. 
	
	Consider any cube Q'= Q_j^i for an arbitrary i∈ℐ_k_0.
	Apply the above argument to Q' in place of Q_j, we can find a cube Q' = Q^i_j ∈_Q' with length ℓ(Q') = 2^-k_0ℓ(Q')= 2^-2k_0ℓ(Q_j) sitting in the “center” of Q', in the sense that
	
    Q^i_j ⊂{q∈ Q': (q,(Q')^c) > ρ/2ℓ(Q') };

	and all future generations satisfy
	
    ⋃_k≥ 2 k_0⋃_I∈Σ_j^k Q'_I ∈_Q' Q'_I ⊂⋃_i_2∈ℐ_k_0Q_j^i i_2 ⊂{q∈ Q': (q, (Q')^c) ≤ρ/2ℓ(Q') },

	where {Q_j^i i_2}_i_2∈ℐ_k_0 is the collection of all dyadic cubes of length 2^-k_0ℓ(Q') = 2^-2k_0ℓ(Q_j) that is completely contained in {q∈ Q'=Q_j^i: (q, (Q')^c) ≤ρℓ(Q')/2}. (The index set for i_2 may not be the same as the index set for i, but their cardinalities are uniformly bounded by C2^k_0 d, so we abuse the notation here and simply assume they are the same.)
	Moreover we can get an analogous estimate of (<ref>):
	
    ∑_k=k_0+1 ^2k_0∑_I∈Σ_j^k  Q'_I ∈_Q'ω(Q'_I) ≲ω(Q^i_j).

	Summing up (<ref>) over all cubes Q' ∈{Q_j^i}_i∈ℐ_k_0, recall (<ref>) we get
	
    ∑_k=k_0+1 ^2k_0∑_I∈Σ_j^k ω(Q'_I) ≲∑_i∈ℐ_k_0ω(Q_j^i).

	
	Since {Q_j^i}_i∈ℐ_k_0 is a collection of cubes in the same generation, they are mutually disjoint, and their sub-cubes {Q_j^i}_i∈ℐ_k_0 are also mutually disjoint.
	Hence
	
    ∑_k=k_0+1 ^2k_0∑_I∈Σ_j^k ω(Q'_I) ≲∑_i∈ℐ_k_0ω(Q_j^i) = ω(  _i∈ℐ_k_0Q_j^i ) .estimate-2k_0

	Moreover, recall the second inclument of (<ref>) and (<ref>), each Q_j^i is disjoint from Q_j, so we can add up (<ref>) and (<ref>) with ease. 
	We can repeat this argument iteratively: for any l∈ℕ we apply the argument to cube Q'=Q_j^i_1 i_2 ⋯ i_l with i_1,⋯,i_l∈ℐ_k_0 to get an analogous estimate of (<ref>), then we sum up over the index sets and get
	
    ∑_k=lk_0+1^(l+1)k_0∑_I∈Σ_j^kω(Q'_I) ≲∑_i_1, ⋯,i_l ∈ℐ_k_0ω(Q_j^i_1⋯ i_l) = ω( _i_1,⋯,i_l∈ℐ_k_0Q_j^i_1⋯ i_l).estimate-(l+1)k_0

	Most significantly for us, for each l∈ℕ the union of cubes on the right hand side of (<ref>) is disjoint from all the cubes from all previous summations. Therefore we conclude that
	
    ∑_k=1^∞∑_I∈Σ_j^kω(Q'_I) ≲ω(_l∈ℕ( _i_1,⋯,i_l∈ℐ_k_0Q_j^i_1⋯ i_l) ) ≤ω(Q_j).

	It is trivial to see ∑_I∈Σ_j^0ω(Q'_I) ≲ω(Q_j), so
	
    ∑_I∈Σ_jω(Q'_I) =  ∑_k∈ℕ∑_I∈Σ_j^kω(Q'_I) ≲ω(Q_j).

	Since the maximal cubes Q_j in ℱ^N are mutually disjoint and contained in Q, we have
	
    ∑_Q_j∈^N∑_I∈Σ_jω(Q'_I) ≲∑_Q_j∈^Nω(Q_j) ≤ω(Q).

	
	Now we consider I∈Σ_0, which by definition means Q'_I ∉_Q. Recall (<ref>) and (<ref>), and that ℓ(I) ≤ C ℓ(Q) for all I∈_N = _^N,Q, we have
	
    ℓ(Q'_I) ∼ℓ(I) ≤ C ℓ(Q),   (Q_I, Q'_I )≲ℓ(Q'_I) ≤ C ℓ(Q).

	In particular since Q_I∈_Q, we have
	
    (Q'_I, Q) ≤(Q'_I, Q_I) ≲ℓ(Q'_I) ≤ C ℓ(Q).

	If ℓ(Q'_I) ≥ℓ(Q), then 
	
    ℓ(Q'_I) ∼ℓ(Q),   (Q'_I,Q) ≤ Cℓ(Q).

	There are finitely many such Q'_I's and by the doubling property of harmonic measure ω(Q'_I) ∼ω(Q).
	If ℓ(Q'_I) < ℓ(Q), let Q_0∈ be the cube containing Q'_I with length ℓ(Q_0) = ℓ(Q). By the assumption Q'_I ∉_Q, we know Q_0 is disjoint from Q. On the other hand (<ref>) implies
	
    (Q_0,Q) ≤(Q'_I, Q) ≤ Cℓ(Q),

	that is, Q_0 is a sibling (i.e. of the same generation) of Q in a Cℓ(Q)-neighborhood of Q. There are finitely many such Q_0's. Moreover
	
    (Q'_I, (Q_0)^c) ≤(Q'_I, Q) ≲ℓ(Q'_I).

	So if ℓ(Q'_I) ≪ℓ(Q), we can guarantee that Q'_I lies in the ρ/2-boundary layer of Q_0: Q'_I ⊂{q∈ Q_0: (q,(Q_0)^c) ≤ρℓ(Q_0)/2}. Apply the same argument to Q_0 in place of Q_j, we get 
	
    ∑_I∈Σ_0 Q'_I ∈_Q_0ω(Q'_I) ≲ω(Q_0) ∼ω(Q).

	Summing up (<ref>) over all (finitely many) Q_0's satisfying (<ref>), we get
	
    ∑_I∈Σ_0ω(Q'_I) ≲ω(Q).

	Finally we combine (<ref>) and (<ref>) and conclude that
	
    ∑_I∈_N^Σω(Q'_I) = ∑_Q_j∈^N∑_I∈Σ_jω(Q'_I) + ∑_I∈Σ_0ω(Q'_I) ≲ω(Q).

	Therefore
	
    ∑_I∈_N^Σω(Q'_I) ≲ω(Q).

	



Now that all the preparatory work has been done, we proceed to sketch the basic idea for the proof of Theorem <ref>. It is well known in harmonic analysis that the proof of Su_L^p(σ)≤ C Nu_L^p(σ) can be reduced to the proof of a certain good-λ inequality measured by σ. We first prove Proposition <ref>, which is a good-λ inequality measured by ω; then we use the assumption ω∈ A_∞(σ) to obtain the desired good-λ inequality for σ.

Recall that we use Su, S'u, S”u to denote the square functions on standard non-tangential cones of aperture α,α_1, β, respectively, and Nu non-tangential maximal function on cones of aperture β, where β>α_1>α are fixed apertures (see the discussion before Lemma <ref>).
Also recall from (<ref>) that for any collection ℱ of dyadic cubes, the sawtooth domain Ω_ℱ,Q⊂ B(x_Q, C_3 ℓ(Q)) ∩Ω. In fact, by choosing a slightly bigger constant C_3 we can also guarantee Ω_, Q^***⊂ B(x_Q, C_3 ℓ(Q)) ∩Ω. 




	For simplicity we denote ω = ω^X_Q.  
	Let E = {q∈ Q:Su(q)> 2λ, Nu(q)≤δλ} and F={q∈ Q: Nu(q) ≤δλ}. If F is empty, then the left hand side of (<ref>) is zero, and there is nothing to prove. So we assume F≠∅.
	Note that Nu(q) is a continuous function, so Q∖ F={q∈ Q: Nu(q) >δλ} is relatively open in Q. We run a stopping time procedure for the descendants of Q, and stop at Q' ∈_Q whenever Nu(q) > δλ for all q∈ Q'. We denote the collection of all maximal cubes by _2 = {Q_j}⊂_Q ∖{Q}. 
	We claim that they form a partition:

    Q∖ F = {q∈ Q: Nu(q) > δλ} = ⋃_Q_j∈_2 Q_j.
 
Clearly by construction ∪_Q_j∈_2 Q_j is contained in the set on the left. For any q_0 ∈ Q such that Nu(q_0)>δλ, since the set {q∈: Nu(q)> δλ} is open, Q∖ F ≠ Q and the cubes in  are nested, there exists a small cube Q'∈_Q ∖{Q} containing q_0, such that Nu(q)>δλ for all q∈ Q'. By the stopping time procedure, either Q'∈_2, or Q' is contained in some cube Q_j∈_2. Hence q_0 ∈ Q' ⊂⋃_Q_j∈_2 Q_j, and we prove the claim (<ref>).
	Recall (<ref>), which we rewrite here:
	
    ( ⋃_q∈ FΓ^Q_d(q) ) ⊂Ω__2,Q⊂Ω__2,Q⊂Ω__2,Q^***⊂⋃_q∈ FΓ^Q_d(q).

	We claim that |u(X)|≤δλ for all X∈Ω__2,Q^***. In fact, by (<ref>) and (<ref>) we know that every X∈Ω__2,Q^*** is contained in some Γ^Q_d(q) ⊂Γ^β(q) for some q∈ F. 
	Since Nu(q) = sup_X∈Γ^β(q) |u(X)| ≤δλ for q∈ F, we get |u(X)| ≤δλ.
	
	
	Step 1.
	Recall the assumption that S^'u(q_1) ≤λ for some q_1 satisfying |q_1 - q| ≤ C_2 Q for all q∈ Q. Denote r= Q. We claim that for any τ>0 there exists δ>0 sufficiently small such that the truncated square function S^_τ ru(q)>λ for any q∈ E. 
	
	Fix q∈ E.
	Recall that Su(q) > 2λ for q∈ E. We denote U = Γ^α(q) ∖ B(q,τ r), then we aim to show
	
    ∬_U |∇ u(X)|^2 δ(X)^1-d dm(X) ≤ 3λ^2.

	 Let U_1 = Γ^α(q) ∖ B(q,t r) for a constant t>τ to be chosen later, and U_2 = Γ^α(q) ∩( B(q,t r) ∖ B(q,τ r) ). Then U= U_1 ∪ U_2.
	A simple computation shows that
	
    U_1 = Γ^α(q) ∖ B(q,t r) ⊂Γ^α_1(q_1)

	if the apertures satisfy
	
    (1+α) ( 1+ C_2/t) ≤ 1+α_1,

	that is, if t is sufficiently large such that
	
    α + C_2 (1+α)/t≤α_1.

	Therefore
	
    ∬_U_1 |∇ u(X)|^2 δ(X)^1-d dm(X) ≤∬_Γ^α_1(q_1) |∇ u(X)|^2 δ(X)^1-d dm(X) = S'u(q_1)^2 ≤λ^2.

	Let Γ_j(q) = Γ^α(q) ∩( B(q, 2^j τ r) ∖ B(q,2^j-1τ r) ) for j=1,2,⋯, then 
	
    U_2 ⊂⋃_j:2^j-1τ r < trΓ_j(q).

	Each Γ_j(q) can be covered by a finite union (depending on n) of balls B_j,k with radius r_j,k∼_α 2^j τ r. Let B^*_j,k denote a slight fattening of B_j,k such that we still have B^*_j,k⊂Γ^β(q), then by Lemma <ref> (i) m(B^*_j,k) ∼ r_j,k^d+1∼ (2^j τ r)^d+1. Then
	
    ∬_U_2 |∇ u(X)|^2 δ(X)^1-d dm(X)     = ∑_2^j-1τ r< tr∬_Γ_j(q) |∇ u(X)|^2 δ(X)^1-d dm(X) 
       ∼_α, β∑_2^j-1τ r< tr  (2^j τ r)^1-d∑_1≤ k≤ C(n)∬_B_j,k |∇ u(X)|^2  dm(X) 
       ≲∑_2^j-1τ r< tr  1≤ k≤ C(n) (2^j τ r)^-1-d∬_B_j,k^* |u(X)|^2  dm(X) 
       ≲(δλ)^2 ∑_2^j-1τ r< tr (2^j τ r)^-1-d m(B^*_j,k) 
       ≲(δλ)^2 log_2( t/τ) 
        < 2λ^2,

	if δ is sufficiently small depending on the values of t, τ and α, β. Therefore (<ref>) holds, and thus for any q∈ E,
	
    |S_τ r u(q)|^2     =∬_Γ^α(q) ∩ B(q,τ r)  |∇ u(X)|^2 δ(X)^1-d dm(X) 
        = ∬_Γ^α(q) ∖ U |∇ u(X)|^2 δ(X)^1-d dm(X) 
        > λ^2.

	
	Step 2.
	Combining (<ref>) with E⊂ F we get
	
    λ^2 ω(E) ≤∫_E |S_τ ru(q)|^2 dω(q)    ≤∫_F∬_Γ^α_τ r(q) |∇ u(X)|^2 δ(X)^1-d dm(X) dω(q).

	By (<ref>) we have
	
    Γ_τ r^α(q) ⊂Γ_d(q) ⊂Γ_d(q)
 
	for any q∈ Q.
	In particular if X belongs to the left hand side of (<ref>), then X∈ U_Q' for some dyadic cube Q' containing q. Moreover 
	
    δ(X) ≤ |X-q| < τ r = τ Q ∼τℓ(Q).

	 By the definition of U_Q' and (<ref>), we have 
	 
    δ(X) ≳ cη^n-1/4(n-1-d)ℓ(Q') .
 	
	 By combining (<ref>), (<ref>) and choosing τ small enough depending on η, we can guarantee that ℓ(Q') < 2ℓ(Q). Since Q'∩ Q∋ q, by property (ii) of Lemma <ref> we know Q'∈_Q. Hence Γ_τ r^α(q) ⊂Γ_d^Q(q). Again since Γ_τ r^α(q) is an open set, we also have Γ_τ r^α(q) ⊂Γ_d^Q(q).
	  Therefore
	
    ⋃_q∈ FΓ^α_τ r(q) ⊂⋃_q∈ F( Γ^Q_d(q) ) ⊂( ⋃_q∈ FΓ^Q_d(q) )

	Applying Fubini's theorem to the right hand side of (<ref>), we conclude that it is bounded by
	
    ∬_(⋃_p∈ FΓ^Q_d(p) ) |∇ u(X)|^2 δ(X)^1-dω({q∈ F: X∈Γ^Q_d(q)}) dm(X).

	For any p∈ F and any X∈Γ_d^Q(p), we have X∈ I ∈𝒲_Q' for a cube Q' containing p and in __1,Q. Thus |X-q| ∼ℓ(Q')∼ℓ(I) ∼δ(X). Since the family {I^*}_I∈ has bounded overlap and harmonic measure ω has pole at X_Q, we conclude by Lemma <ref> that
	
    ω({q∈ F: X∈Γ^Q_d(q)}) ∼ω(⋃_Q'∈_Q ℓ(Q')∼δ(X)∼(X,Q') Q' ) ∼ G(X_Q,X) δ(X)^d-1.

	Combining (<ref>), (<ref>), (<ref>) and (<ref>) and using (<ref>), we get
	
    λ^2 ω(E) ≲∬_Ω__2, Q  |∇ u(X)|^2 G(X_Q,X) dm(X)     = ∬_Ω__2, Q  |∇ u(X)|^2 G(X_Q,X) w(X) dX 
       ≲∬_Ω__2, Q A∇ u ·∇ u G dX.

	Here we abbreviate G(X) = G(X_Q,X) when there is no ambiguity as to what the pole is. Recall that the pole X_Q ∉ B(x_Q, 2C_3 ℓ(Q)), and similar to (<ref>) we may choose the dilation constant θ small enough so that Ω__2, Q^***⊂ B(x_Q, 3/2 C_3ℓ(Q)). They guarantee that X_Q ∉Ω__2, Q^***, and moreover (X_Q, Ω__2, Q^***) ≳ℓ(Q). Hence G(X) is harmonic in the fat sawtooth domain Ω__2, Q^***.
	
	Step 3.
	Next we are going to prove
	
    ∬_Ω__2, Q A∇ u ·∇ u G dX ≲ (δλ)^2 ω(Q).

	Recall the discussion before Lemma <ref>, we can augment _2 by adding all dyadic cubes of lengths smaller or equal to 2^-Nℓ(Q), and denote by _2^N the collection of maximal cubes giving rise to the aforementioned augmented collection. We claim that
	
    ∬_Ω__2^N, Q A∇ u ·∇ u G dX ≲ (δλ)^2 ω(Q)

	with a constant independent of N. Thus by passing N→∞ we obtain (<ref>).

	Recall that in Lemma <ref>, we construct a smooth cut-off function ψ_N such that χ_^*≲ψ_N ≤χ_^**. Hence
	
    ∬_ A∇ u ·∇ u G dX ≤∬_^n A∇ u ·∇ u G ψ_N dX
 
	Since u, G ∈ W_r(^**) ∩ L^∞(^**), we have uGψ_N, u^2 ψ_N ∈ W_0^1,2(^**). In particular they can be approximated by smooth functions in C_0^∞(^**) ⊂ C_0^∞(Ω). In the sawtooth region ^** we have -(A∇ u) = -(A∇ G) = 0, thus
	
	

    ∬_^n A∇ u ·∇ u G ψ_N dX 
    
    	 =    ∬_^n A∇ u ·∇(u G ψ_N ) - 1/2 A∇ (u^2) ·∇ (Gψ_N) dX  
    
    	 =     0 - 1/2∬_^n A ∇ (Gψ_N) ·∇ (u^2) dX 
    
    	 =     - 1/2( ∬_^nψ_N A∇ G ·∇ (u^2) +  G A∇ψ_N ·∇(u^2) dX ) 
    
    	 =     -1/2( ∬_^n A∇ G ·∇ (u^2 ψ_N) - u^2 A∇ G ·∇ψ_N + 2 uG A ∇ u ·∇ψ_N dX ) 
    
    	 =    1/2∬_^n u^2 A∇ G ·∇ψ_N dX - ∬_^n uG A∇ u ·∇ψ_N dX 
    
    	 =:    1/2 I - II,

where we use the symmetry of A and the equation -(A∇ u) = 0 in the second equality, and -(A∇ G) = 0 in the second to last equality.
We first estimate the second term.
By (<ref>), the contribution to the integral II only comes from Whitney boxes I∈_N^Σ. Recall the harmonic function u is non-negative and we use X_I to denote the center of Whitney box I. By Lemma <ref> (ii), Hölder inequality, estimate of the weight (<ref>), interior Cacciopoli inequality (<ref>), Harnack inequality (<ref>) and (<ref>), we have

    |II|    ≤∑_I∈u(X_I)G(X_I)/ℓ(I)∬_I^*** |∇ u| dm 
       ≤∑_I∈u(X_I)G(X_I)/ℓ(I)· m(I^***) ( _I^*** |∇ u|^2 dm )^1/2
       ≲∑_I∈ u(X_I)G(X_I) ℓ(I)^d-1( _I^**** |u|^2 dm )^1/2
       ≲∑_I∈ u(X_I)^2 G(X_I) ℓ(I)^d-1
       ∼∑_I∈ u(X_I)^2 ω(Q_I),

where Q_I is defined as in Lemma <ref> (iv). Using the estimate |u(X)| ≤δλ for all X∈^*** and (<ref>), we have

    |II| ≲∑_I∈ u(X_I)^2 ω(Q_I) ≲(δλ)^2 ω(Q).

Similarly,

    |I| ≤∑_I∈𝒲_N^Σu(X_I)^2/ℓ(I)∬_I^*** |∇ G| dm ≲∑_I∈ u(X_I)^2 ω(Q_I) ≲(δλ)^2 ω(Q).

We finish the proof of (<ref>) by combining (<ref>), (<ref>) and (<ref>).

Finally we combine (<ref>) and (<ref>), and get

    λ^2 ω(E) ≲(δλ)^2 ω(Q).

And thus 

    ω(E) ≤ Cδ^2 ω(Q).

This finishes the proof of the good-λ inequality for ω.


We will also need the following auxiliary fact:  

	For any apertures 0<α<α' and any function u∈ W_r(Ω), let Su and S u denote the square function with aperture α and α' respectively.
	Suppose S u<∞ for σ-almost every q∈, then the set {q∈: Su(q) >λ} is open for every λ>0.
 
The proof is  similar in spirit to that of Lemma 4.6 in <cit.>.


	If q∈ is such that S'u(q) >λ, then there exists η>0 so that

    ∬_Γ^α(q) ∖ B(q,η) |∇ u|^2 δ(X)^1-d dm(X) > ( Su(q) + λ/2)^2.

We claim that there exists ϵ>0 such that for any p∈Δ(q,ϵη), we have

    ∬_Γ^α(p) ∖ B(p,η) |∇ u|^2 δ(X)^1-d dm(X) > λ^2,
 
and therefore Su(p)> λ.

We observe that

    |∬_Γ^α(q) ∖ B(q,η) |∇ u|^2 δ(X)^1-d dm(X)- ∬_Γ^α(p) ∖ B(p,η) |∇ u|^2 δ(X)^1-d dm(X)| 
               ≤∬_D |∇ u|^2 δ(X)^1-d dm(X),

where D= (Γ^α(q) ∖ B(q,η) ) ( Γ^α(p) ∖ B(p,η) ) is the set difference. It suffices to show that the integral ∬_D |∇ u|^2 δ(X)^1-d dm(X) is sufficiently small, if we choose ϵ sufficiently small.

Suppose that X∈Γ^α(q) ∖ B(q,η), then |X-q| < (1+α)δ(X) and |X-q|≥η. Thus δ(X) > η/1+α. If moreover X∉Γ^α(p) ∖ B(p,η) and p∈ B(q,ϵη), then |X-q| > (1+α) (1-ϵ) δ(X). By symmetry, we need to study sets of the form

    V_q = {X∈Ω: |X-q| ≥η,  (1+α)(1-ϵ) δ(X) < |X-q| < (1+α) δ(X) },


    V_p = {X∈Ω: |X-p| ≥η,  (1+α)(1-ϵ) δ(X) < |X-p| < (1+α) δ(X) }.

Without loss of generality we may assume S'u(q)<∞. If not, by the assumption that S'u<∞ almost everywhere, we can always find q' ∈Δ(q, ϵη/2) such that S'u(q')<∞, and in particular p∈Δ(q,ϵη) ⊂Δ(q',2ϵη). In this case we just replace q by q', and ϵ by 2ϵ. 
Moreover, if ϵ < 1/4, we have that

    V_q ∪ V_p ⊂ V_ϵ := {X∈Ω: |X-q| ≥η/2,  (1+α)(1-ϵ)^2δ(X) < |X-q| < (1+α)1-ϵ/1-2ϵδ(X) }.

Note that for given α'>α, by choosing ϵ sufficiently small we can guarantee (1+α) 1-ϵ/1-2ϵ≤ 1+α'. Thus V_ϵ⊂Γ^α'(q) ∖ B (q, η/2)=:V_0, and as ϵ tends to zero, the set V_ϵ decreases to an empty set. Moreover, 

    ∬_V_0 |∇ u|^2 δ(X)^1-d dm(X) ≤∬_Γ^α'(q) |∇ u|^2 δ(X)^1-d dm(X) =|S'u(q)|^2 <∞,

hence by the continuity of measure from above, we deduce that

    ∬_V_ϵ |∇ u|^2 δ(X)^1-d dm(X) ↘ 0.

In particular, by choosing ϵ sufficiently small, we can guarantee

    ∬_D |∇ u|^2 δ(X)^1-d dm(X) ≤∬_V_ϵ |∇ u|^2 δ(X)^1-d dm(X) < ( Su(q)+λ/2)^2 - λ^2

Combining (<ref>) with (<ref>), we conclude the proof of the claim (<ref>).



Now we set out to complete the
0.08in


We first prove the theorem assuming that S'u_L^p(σ) is finite. 

Under this assumption, we have that S”u_L^p(σ)∼S'u_L^p(σ)∼Su_L^p(σ). For reference, see Proposition 4 of <cit.>. (The stated proof in <cit.> is for the upper half plane, but the argument goes through for Ahlfors regular sets of higher codimension.)
Therefore by a standard argument, the proof of (<ref>) can be reduced to the following good-λ inequality: For any ϵ>0 sufficiently small, we can find δ= δ(ϵ)>0 such that for all λ>0,
	
    σ({ q∈: Su(q) > 2λ, Nu(q) ≤δλ}) ≤ϵσ( {q∈: S'u(q) > λ}),

	and δ→ 0 as ϵ→ 0.
If {q∈:S'u(q)>λ} is empty, (<ref>) is trivial, so we assume the set is not empty. 
We apply Lemma <ref> with apertures 0<α_1<β. Since S”u_L^p(σ)∼S'u_L^p(σ) <∞, in particular S”u(q)<∞ almost everywhere. Therefore {q∈: S'u(q)>λ} is open. 
We also remark that the set {q∈: S'u(q) >λ} has finite σ-measure, and moreover

    σ({q∈: S'u(q) >λ}) ≤1/λ^p∫_S'u(q) >λ |S'u|^p dσ≤S'u_L^p(σ)^p/λ^p <∞.

In particular, for any dyadic cube Q∈ completely contained in {q∈: S'u(q) >λ} 

    ℓ(Q)^d∼σ(Q) ≤σ({q∈: S'u(q) >λ}) ≤S'u_L^p(σ)/λ^p,

so its length has a uniform upper bound (albeit depending on the value of λ). Recall that ℓ(Q) ∼ 2^-k(Q), and suppose k_0 ∈ℤ is such that

    2^-k_0 d≳S'u_L^p(σ)/λ^p,
 
with a sufficiently large implicit constant.
Then by (<ref>), any cube Q_0 in _k_0 can not be completely contained in {q∈: S'u(q) >λ}. 

 
We run a stopping time procedure as follows: For each Q_0 ∈_k_0, we traverse all its descendants, and stop whenever we find a cube Q∈_Q_0 such that S'u(q) > λ for all q∈ Q. Let _1={Q_l} be the collection of all stopping cubes in ⋃_Q_0 ∈_k_0_Q_0. Similar to the proof of (<ref>), we can show that they form a partition:

    {q∈: S'u(q) >λ} = ⋃_Q_l ∈_1 Q_l.
 
Note that the assumption Su(q)>2λ clearly implies S'u(q)>λ, namely 

    {q∈: Su(q)>2λ}⊂{q∈: S'u(q)>λ} = ⋃_Q_l∈_1 Q_l.
 
Therefore to prove (<ref>), it suffices to localize and show that

    σ({ q∈ Q : Su(q) > 2λ, Nu(q) ≤δλ}) ≤ϵσ( Q )    for any  Q=Q_l∈_1.

 
 


Recall that by (<ref>), every Q∈ is contained in a surface ball Δ(x_Q,C_2 r_Q).
Let X'_Q denote a corkscrew point for B(x_Q, C_2 r_Q). Recall Definition <ref> of ω∈ A_∞(σ) and Remark <ref> (ii) right afterwards. Assuming ω∈ A_∞(σ), then to prove (<ref>) it suffices to show

    ω^X'_Q({q∈ Q: Su(q)>2λ, Nu(q) ≤δλ}) ≤ C(δ) ω^X'_Q(Q),

with a constant C(δ) independent of Q and λ, and that C(δ)→ 0 as δ→ 0.
Recall that for any collection ℱ of dyadic cubes, there is a constant C_3 such that Ω_, Q^***⊂ B(x_Q, C_3 ℓ(Q)) ∩Ω. Let X_Q be a corkscrew point for B(x_Q, 2C_3 Mℓ(Q)), then

    |X_Q - x_Q| ≥δ(X_Q) ≥ 2C_3 ℓ(Q).

Thus X_Q ∉ B(x_Q, 2C_3 ℓ(Q)), and in particular X_Q ∉Ω_,Q^***.
  Moreover, there is a Harnack chain of finite length (depending only on M, C_2 and C_3) connecting X_Q to X'_Q; in particular the harmonic measures ω^X_Q(E) ∼ω^X'_Q(E) for any Borel set E⊂ Q. Therefore the proof of (<ref>) is equivalent to the proof of

    ω^X_Q({ q∈ Q: Su(q) > 2λ, Nu(q) ≤δλ}) ≤ C(δ) ω^X_Q( Q ).


Recall that Q=Q_l∈_1 is a maximal cube with respect to the stopping criterion {S'u(q)>λ}. By maximality the parent of Q, denoted by Q, contains at least one point q_1 ∉{q ∈: S'u(q)>λ}, that is, S'u(q_1)≤λ. For any q∈ Q we have

    |q_1-q| ≤Q≤ A_1 2^-k(Q) = A_1 2^-(k(Q)-1)≤A_1/a_0 Q.
 
Therefore for any maximal cube, we may use Lemma <ref>, with constant C_2 = A_1/a_0, to conclude the desired estimate (<ref>).



All the above arguments show that if we know a priori S'u_L^p(σ) is finite, we can prove Su_L^p(σ)≲Nu_L^p(σ). If we do not have this a priori information, then for κ sufficiently small we let

    _κ = {Q∈: κ≤ℓ(Q) ≤ 1/κ},


    Ω_κ = ⋃_Q∈_κ U_Q,   Ω^*_κ = ⋃_Q∈_κ U_Q^*,   Ω^**_κ = ⋃_Q∈_κ U_Q^**  etc.
  
and define the κ-approximate non-tangential cones as 

    Γ^α_κ(q) = Γ^α(q) ∩Ω_κ,   Γ_κ^α_1 = Γ^α_1(q) ∩Ω_κ,   Γ^β_κ(q) = Γ^β(q) ∩Ω_κ^***,
 
define the κ-approximate dyadic non-tangential cones as

    Γ_d,κ(q) = Γ_d(q) ∩Ω_κ = ⋃_Q∈^κ: Q∋ q U_Q,   Γ_d,κ(q) = Γ_d(q) ∩Ω^***_κ.

In this regime we have the following inclusions analogous to (<ref>) and (<ref>):

    Γ^α_κ(q) ⊂Γ_d,κ(q) ⊂Γ^α_1_κ(q),   Γ_d,κ(q) ⊂Γ^β_κ(q).

Moreover, the κ-approximate local non-tangential cones 

    Γ^Q_d,κ(q) = Γ_d^Q(q) ∩Ω_κ =  ⋃_Q' ∈_Q ∩^κ: Q'∋ q U_Q',   Γ^Q_d,κ(q) = Γ_d^Q ∩Ω_κ^***

satisfy the following inclusions analogous to (<ref>):

    ⋃_q∈ FΓ^Q_d,κ(q) ⊂Ω_,Q∩Ω_κ⊂Ω_,Q∩Ω_κ⊂Ω_,Q^***∩Ω_κ^***⊂⋃_q∈ FΓ_d,κ^Q(q),

for any dyadic cube Q and collection of maximal cubes Γ⊂_Q ∖{Q}, under the assumption that F= Q ∖⋃_Q_j∈ Q_j is not empty.
We then define the κ-approximate square functions S_κ u, S'_κ u and non-tangential maximal function N_κ u accordingly, as integrals defined on the κ-approximate non-tangential cones instead of standard non-tangential cones.
Since N_κ u(q) ≤ Nu(q) for all q∈, we have N_κ u_L^p(σ)≤Nu_L^p(σ) <∞. By the interior Cacciopoli inequality (<ref>) and that β>α_1>α, we have

    S_κ u(q) ≤ S'_κ u(q) ≲ C(κ) N_κ u(q),

and thus

    S'_κ u_L^p(σ)≲ C(κ) N_κ u_L^p(σ)≤  C(κ) N u_L^p(σ) <∞.
 
We can not let κ go to zero in (<ref>) since the upper bound in the right hand side depends on κ (in fact C(κ) →∞ as κ→ 0). However, since S'_κ u_L^p(σ) is finite, we can apply the previous arguments and prove that S_κ u_L^p(σ)≲N_κ u_L^p(σ), with a constant independent of κ. Hence

    S_κ u_L^p(σ)≲N_κ u_L^p(σ)≤ CN u_L^p(σ)

with a constant C independent of κ. Therefore we can safely let κ go to zero and conclude that

    Su_L^p(σ) = lim sup_κ→ 0S_κ u_L^p(σ)≤ C Nu_L^p(σ).

This finishes the proof of Theorem <ref>.







§ Ω∈ A_∞(Σ) IS EQUIVALENT TO BMO-SOLVABILITY




 §.§ From ω∈ A_∞(σ) to L^p-solvability

 

Assume ω∈ A_∞(σ), then there exist some p_0 ∈ (1, ∞) such that the elliptic problem (<ref>) is L^p-solvable for all p ∈ (p_0, ∞), in the sense that there exists a universal constant C>0 such that for any f∈ C_0^0(Γ) and any Borel set E⊂Γ,
	the solution u(X) = ∫_E f dω^X satisfies the estimate Nu_L^p(σ)≤ Cfχ_E _L^p(σ).	


	For a bounded set E, it suffices to assume that f∈ C_b(Γ).



	We first treat the case when E=Γ.
	Let q∈ and denote for any p>1
	
    ℳ_p f(q) = sup_Δ∋ q( _Δ |f|^p dσ)^1/p<∞.
 
	We claim
	
    |u(X)| ≤ C ℳ_p f(q)    for any  X∈Γ(q).

	Hence Nu(q) ≤ C ℳ_p f(q), and thus by the L^p-boundedness (p>1) of Hardy-Littlewood maximal function (see <cit.> for spaces of homogeneous type and <cit.>)
	
    Nu_L^p(σ)≤ C ℳf _L^p(σ)≲f_L^p(σ).

	
	In fact, let X∈Γ(q) be fixed and Δ = Δ(q, (1+α)δ(X) ). For j∈ℕ let Δ_j = 2^j Δ, and set Δ_-1 = ∅. We have
	
    u(X) = ∫ f dω^X = ∑_j=0^∞∫_Δ_j∖Δ_j-1 f d ω^X.

	For each j ∈ℕ let A_j denote a corkscrew point for Δ_j. Recall Definition <ref> of ω∈ A_∞(σ) and the discussion after that, in particular (<ref>) and (<ref>). We have that for each j, the Radon-Nikodym derivative
	
    k^A_j (q') = dω^A_j/dσ (q') = lim_Δ' → q'ω^A_j(Δ')/σ(Δ')
 
	satisfies a reverse Hölder inequality 
	
    ( _Δ_j|k^A_j|^r dσ)^1/r≤ C _Δ_j k^A_j dσ

	for all r∈ (1,r_0),	with uniform constants r_0>1 and C>0. 
	For any j≥ 2 and any surface ball Δ' ⊂Δ_j ∖Δ_j-1, by the Hölder regularity of solutions near the boundary (see Lemma <ref>), we have
	
    ω^X(Δ') ≲ 2^-jβω^A_j-2(Δ') ∼ 2^-jβω^A_j(Δ').

	Hence for any q'∈Δ_j ∖Δ_j-1,
	
    k^X(q') =  lim_Δ' → q'ω^X(Δ')/σ(Δ') =  lim_Δ' → q' Δ' ⊂Δ_j∖Δ_j-1ω^X(Δ')/σ(Δ')≲ 2^-jβlim_Δ' ∋ q' Δ' ⊂Δ_j∖Δ_j-1ω^A_j(Δ')/σ(Δ') = 2^-jβ k^A_j(q').

	Therefore by (<ref>), (<ref>), and Hölder inequality for conjugates 1/p+1/r=1 with r∈ (1,r_0), we obtain
	
    |u(X)| ≤∑_j=0^∞∫_Δ_j∖Δ_j-1 |f k^X| d σ   ≲∑_j=0^∞ 2^-jβ∫_Δ_j |f| k^A_j d σ
       ≤∑_j=0^∞ 2^-jβσ(Δ_j) ( _Δ_j |f|^p dσ)^1/p( _Δ_j | k^A_j|^r d σ)^1/r
       ≲∑_j=0^∞ 2^-jβσ(Δ_j) ( _Δ_j |f|^p dσ)^1/p( _Δ_j k^A_j d σ) 
       ≤∑_j=0^∞ 2^-jβℳ_p f(q) ω^A_j(Δ_j) 
       ≲ℳ_p f(q),

	thus we finish proving the claim (<ref>) for any p∈ (p_0, ∞), where p_0 is the conjugate of r_0. Note that we never use the continuity or compact support of f, and replacing f by fχ_E we can repeat the same argument with no change. The assumption that E is bounded or f has compact support guarantees we still have a priori finite integrability in (<ref>).









 §.§ Proof of the BMO-solvability



	Assume that ω∈ A_∞(σ). For any f∈ C_0^0(Γ), let u = Uf∈ W_r(Ω) be a solution to Lu=0 given by Lemmas <ref> and <ref>. Then |∇ u|^2 δ(X)dm(X) is a Carleson measure, and moreover 

    sup_Δ⊂1/σ(Δ )∬_T(Δ )|∇ u|^2 δ(X) dm(X) ≤ Cf_BMO(σ) ^2.




	Fix an arbitrary surface ball Δ=Δ(q_0,r). Let α>0. Denote the constant c=max{α+2, 12} and let =cΔ = Δ(q_0, cr) be a concentric dilation. 
We denote the average f_ = _ f dσ. 
Let 

    f_1= (f-f_)χ_,    f_2 = (f-f_)χ_∖,    f_3 = f_,

and for any X∈Ω let

    u_1(X) = ∫_ f_1 dω^X = ∫_(f-f_) dω^X,


    u_2(X) = ∫_ f_2 dω^X =  ∫_∖( f-f_) dω^X = ∫_∖ f dω^X  - f_ω^X(∖),


    u_3 ≡ f_.

By Lemmas <ref>, <ref>, <ref> and <ref>, they are solutions to L, and u_1, u_2 can be continuously extended to ∖ and , respectively. Moreover

    (u_1 + u_2 + u_3 ) (X) = ∫_ f dω^X = Uf(X) = u(X).

Clearly the Carleson measure of the constant function u_3 is trivial. 

Apply Theorem <ref> to f_1 and u_1 we get Nu_1_L^p(σ)≤ Cf_1_L^p(σ)<∞. Combined with Theorem <ref>, we get

    S u_1_L^p(σ)≲Nu_1_L^p(σ)≲f_1_L^p(σ) = (∫_ |f-f_|^p dσ)^1/p
 
for any p∈ (p_0,∞).
By (<ref>) and (<ref>)

    ∬_T(Δ)u_1 dm(X) ≤ C∫_(α+2)Δ |S_(α+1) r u_1|^2 dσ

Recall that = cΔ⊃ (α+2)Δ, thus

    ∬_T(Δ)u_1 dm(X)    ≤ C ∫_ |S_(α+1) r u_1|^2 dσ
       ≤ C σ()^1-2/p(∫_ |S u_1|^p dσ)^2/p
       ≤ C σ()^1-2/pS u_1_L^p(σ)^2,
 
for any p> max{2,p_0}.
Combining (<ref>) and (<ref>) we get

    ∬_T(Δ)u_1 dm(X) ≤ Cσ(Δ) f_BMO(σ)^2<∞.


Turning to the estimate for u_2, let {I_k}⊂ be a collection of dyadic Whitney boxes that intersect of T(Δ) (recall the properties of Whitney decomposition  in (<ref>)). On each Whitney box I_k, we have by the interior Cacciopoli inequality (<ref>)

    ∬_I_ku_2 dm(X)    ≲ℓ(I_k)∬_I_k |∇ u_2|^2 dm(X) 
       ≲ℓ(I_k)·1/ℓ(I_k)^2∬_I_k^* |u_2(X)|^2 dm(X) 
       ≲∬_I_k^*|u_2(X)|^2/δ(X) dm(X),

Recall I_k^* = (1+θ) I_k is the dilation of I_k satisfying (<ref>).
Then summing up we get

    ∬_T(Δ)u_2 dm(X)    ≲∑_k ∬_ I_k^*|u_2(X)|^2/δ(X) dm(X) 
       ≲∬_T(3/2Δ)|u_2(X)|^2/δ(X) dm(X).

In the last line we use the finite overlap of {I_k^*}, and the fact that by taking θ sufficiently small, we can ensure that I^*_k ⊂ T(3/2Δ) for all I_k intersects T(Δ).
Recall that 3/2Δ= Δ(q_0, 3/2r) and T(3/2Δ) denotes B(q_0, 3/2 r)∩Ω.

Let f_2^± denote the positive and negative part of f_2, and let u_2^± = ∫_∖ f_2^± dω^X ≥ 0. There is a technical issue that f_2^±∉ C_0^0(Γ), however by splitting u_2^± as follows,

    u_2^+(X) = ∫_{f≥ f_}∖ f dω^X - f_ω^X ( {f≥ f_}∖),


    u_2^-(X) = -∫_{f< f_}∖ f dω^X + f_ω^X ( {f < f_}∖),

we can confirm by combining Lemmas <ref> and <ref> that u_2^±∈ W_r(Ω) are indeed legitimate solutions of L, and they can be continuously extended to  by zero.
By the linearity of integration, we have u_2 = ∫_ f_2 dω^X =  u_2^+ - u_2^-.  
Let v(X) : = u_2^+(X) + u_2^-(X), again by linearity we have

    v(X) = ∫_ |f_2| dω^X = ∫_∖ |f-f_| dω^X.
 
Thus |u_2(X)| ≤ v(X) for all X∈Ω. Moreover by the properties of u_2^±, we know that v∈ W_r(Ω) is a solution of L, Tv = 0 on  and that v∈ W_r(B(q_0,cr)). (Recall that =cΔ =  B(q_0,cr) ∩.)
We claim that

    v(X) ≤ C f_BMO(σ)   for all  X∈ T(6Δ).
	
	By the definition (<ref>), the function v vanishes on . Note that ⊃ 12Δ by the choice of , v∈ W_r(B(q_0,12r)) is a non-negative solution in T(12Δ) and Tv ≡ 0 on 12Δ. Let A be a corkscrew point for T(12Δ), by the boundary Harnack inequality (<ref>)
	
    v(X) ≤ C v(A),  for all  X∈ T( 6Δ).

	For any j∈ℕ, let A_j be a corkscrew point for the surface ball 2^j.
	Similar to (<ref>), we get
	
    v(A)    ≲∑_j=1^∞ 2^-jβ∫_2^j∖ 2^j-1 |f-f_| k^A_j dσ
       ≤∑_j=1^∞ 2^-jβ(_2^j |f-f_|^p dσ)^1/p( _2^j| k^A_j|^r dσ)^1/rσ(2^j) 
       ≲∑_j=1^∞ 2^-jβf_BMO(σ)ω^A_j(2^j) 
       ≲f_BMO(σ).

	Here p is a conjugate to r. We conclude the proof of (<ref>). 
	
	Next, we show a finer estimate based off (<ref>), which is
	
    v(X) ≤ C(δ(X)r)^βf_BMO(σ)   for all  X∈ T(3/2Δ),
  
	where β∈ (0,1] is the exponent from Lemma <ref>.
	To this end, for any X∈ T(3/2Δ), let q_X be a boundary point such that |X-q_X|=δ(X). Note that
	
    |X-q_X|=δ(X) ≤ |X-q_0| < 3/2r ,

	i.e. X∈ B(q_X,3r/2)∩Ω. Note also
	
    |q_X-q_0| ≤ |q_X-X|+|X-q_0| <3r/2 + 3r/2 =3r,

	so B(q_X,3r)⊂ B(q_0, 6r).
	Since ⊃ 6Δ⊃Δ(q_X, 3r), v∈ W_r(B(q_X,3r)) is a non-negative solution in B(q_X, 3r)∩Ω and Tv ≡ 0 on Δ(q_X, 3r). By the boundary Hölder regularity (<ref>) and the first part of this lemma (<ref>), we conclude
	
    v(X)    ≲( |X-q_X|/3r)^β( 1/m(B(q_X,3r))∬_B(q_X,3r)∩Ω |v|^2 dm )^1/2
       ≲( δ(X)/r)^βsup_T(6Δ) v   ≲( δ(X)/r)^βf_BMO(σ).


Combining (<ref>) and (<ref>), we get

    ∬_T(Δ)u_2 dm(X) ≲f_BMO(σ)^2r^2β( ∬_T (3/2Δ)δ(X)^2β - 1 dm(X) ) .
 
Since 2β-1>-1, we can use Lemma <ref> with exponent α=2β-1 to get

    ∬_T(Δ)u_2 dm(X) ≲ r^df_BMO(σ)^2 ≲σ(Δ)f_BMO(σ)^2.
 
Combining (<ref>) and (<ref>) finishes the proof.






 §.§ From BMO-solvability to ω∈ A_∞(σ)


In this subsection, we prove the other half of Theorem <ref>:

	Assume that for any f∈ C_0^0(Γ), the solution u = Uf∈ W_r(Ω) given by Lemmas <ref> and <ref> satisfies the property that |∇ u|^2 δ(X)dm(X) is a Carleson measure with 

    sup_Δ⊂1/σ(Δ )∬_T(Δ )|∇ u|^2 δ(X) dm(X) ≤ Cf_BMO(σ) ^2.

Then ω∈ A_∞(σ), with the implicit constant depending on d, n, C_0, C_1 and the above constant C.


Let us start with proving the following Lemma.

	Suppose the Dirichlet problem (<ref>) is BMO-solvable. Then any non-negative function f∈ C_0^0(Γ) whose supporte is contained in a surface ball Δ satisfies
	
    ∫_Δf dω^A ≤ C f_BMO(σ).
 
	Here A is a corkscrew point for Δ.





Since f∈ C_0^0(Γ) is a non-negative function, by Lemma <ref> u= Uf∈ W_r(Ω) is a non-negative solution of L. Suppose Δ has radius r. Consider another surface ball Δ'=B(q',r) ∩Γ of the same radius r and which is 2r-distance away from Δ. Thus in particular, Tu = 0 on 3Δ' and that u∈ W_r(B(q',3r)), by Lemma <ref> (i) and (iv).
Applying the BMO-solvability assumption to u=Uf and the surface ball Δ', we have

    ∬_T(Δ')u dm(X)≤ Cσ(Δ')f_BMO(σ)^2


We have shown in (<ref>) that

    ∬_T(Δ')u dm(X) ≳∫_Δ'/2 |S_r/2 u|^2 dσ,

where S_r/2u is the truncated square function of aperture α>α, whose value is determined in Lemma <ref> and only depends on n, d, C_0, C_1 and α.
In order to get a lower bound of the square function S_r/2u, we decompose the non-tangential cone Γ_r/2(q) into stripes as in (<ref>) and use the Poincaré-type inequality proved in Lemma <ref> for surface ball Δ'.
Let m_1, m_2 be integers determined in Lemma <ref>. We obtain

    |S_r/2 u|^2 (q)     = ∬_Γ^α_r/2 (q) |∇ u|^2 δ(X)^1-d dm(X) 
       ≥1/m_1+m_2∑_j=m_1+1^∞∬_Γ^α_j-m_1 → j + m_2(q) |∇ u|^2 δ(X)^1-d dm(X) 
       ≳∑_j=m_1+1^∞(2^-jr)^1-d∬_Γ^α_j-m_1→ j+m_2  (q) |∇ u|^2 dm(X) 
       ≳∑_j=m_1+1^∞ (2^-jr)^1-d· (2^-jr)^-2∬_Γ_j^α(q) u^2 dm(X) 
       ≳∑_j=m_1+1 ^∞ u^2(A_j),

where A_j∈Γ_j(q) is a corkscrew point at the scale 2^-jr.
In the last inequality, we use the interior corkscrew condition, as each stripe of cone Γ_j(q) contains a ball of radius comparable to 2^-j-1r (as long as α is chosen to be big, say α>2M, where M is the corkscrew constant). Moreover,

    ∑_j=m_1+1 ^∞ u^2(A_j) ≥ u^2(A_m_1) ≳ u^2(A_1).

Recall for any q∈Δ', the point A_1= A_1(q) is a corkscrew point of B(q,2^-1r). 
Let A' be the corkscrew point for T(Δ'/2), by Lemma <ref> and Harnack inequality, u(A') ≈ u(A_1). 
Therefore

    |S_r/2u|^2 (q) ≳ u^2(A_1) ≳ u^2(A'),   for any  q∈Δ' .

Combining this with (<ref>) and (<ref>), we get

    σ(Δ')f_BMO(σ)^2 ≳∫_Δ'/2 |S_r/2u|^2 dσ≳σ(Δ'/2)u^2(A') ≳σ(Δ')u^2(A') ,

and thus

    u(A') ≲f_BMO(σ).
 
Let A be a corkscrew point for Δ. Since Δ and Δ' have the same radius r and they are 2r-distance apart, we have u(A) ∼ u(A'). By assumption f is supported on Δ, hence

    u(A) = ∫_Δ f dω^A.

The lemma follows by combining (<ref>) and (<ref>).




With that at hand, we pass to the
0.08in


By the change of pole formula in Lemma <ref> and Harnack inequality, to prove ω∈ A_∞(σ) and in particular (<ref>), it suffices to show:
For any ϵ>0 fixed, we can find η= η(ϵ), such that for any Borel set E ⊂Δ,

    σ(E)/σ(Δ) <η  implies  ω^A (E)/ω^A (Δ) < ϵ.
 
Here Δ is a surface ball and A is a corkscrew point for Δ.
In fact, since σ and ω are regular Borel measures, we may assume E is an open subset of Δ.

Recall from Lemma <ref> that 

    ω^A(Δ) ≥ C^-1
 for some C>1. Thus to show ω^A(E)/ω^A(Δ)<ϵ it suffices to show ω^A(E) < C^-1ϵ.
Let δ>0 be a small constant to be determined later, we define a function

    f(x) = max{ 0, 1+δlog M_σχ_E(x)}
 
where M_σ is the Hardy-Littlewood maximal function with respect to σ.
Similar to Section 5.3 of <cit.>, f satisfies

	
  * 0≤ f≤ 1, and f≡ 1 on the open set E;
	
  * f_BMO(σ)≤ Aδ, where A is a constant independent of E;
	
  * If 
		
    σ(E)/σ(Δ) < η(δ) ∼ e^-1/δ,
  
		then f is supported in 2Δ.



Next we use a mollification argument to approximate f by continuous functions. Let φ be a radially symmetric smooth function on ℝ^n such that φ = 1 on B_1/2, φ⊂ B_1 and 0≤φ≤ 1. Let 

    φ_ϵ(z) = 1/ϵ^dφ(z/ϵ),    f_ϵ(x) = ∫_y∈ f(y) φ_ϵ (x-y) dσ(y)/∫_y∈(x-y)dσ(y) for  x∈ .
 
Then these f_ϵ's satisfy the following properties:

	
  * each f_ϵ is continuous, and is supported in 3Δ;
	
  * there is a constant C (independent of ϵ) such that f_ϵ_BMO(σ)≤ C f_BMO(σ);
	
  * f(x) ≤lim inf_ϵ→ 0 f_ϵ(x) for all x in their support 3Δ. 
 
The proof of the above properties is a slight modification of Appendix A of <cit.>: here the mollifier {φ_ϵ} is an approximation of identity of dimension d, instead of dimension n-1. The proof uses standard mollification arguments and the Ahlfors regularity of . Moreover, the proof of the last property also uses  the precise definition of f in (<ref>).

Let A' be a corkscrew point with respect to 3Δ. The last property and Fatou's lemma imply

    ∫_3Δ f(x) dω^A'(x) ≤∫_3Δlim inf_ϵ→ 0 f_ϵ(x) dω^A'(x) ≤lim inf_ϵ→ 0∫_3Δ f_ϵ(x) dω^A'(x) .
 
Since each f_ϵ is non-negative, continuous and supported on 3Δ, we apply Lemma <ref> and get

    ∫_3Δ f_ϵ(x) dω^A'(x) ≤ C f_ϵ_BMO(σ)≤ C' f_BMO(σ).
 
Combining (<ref>) and (<ref>), we get

    ∫_3Δ f(x) dω^A'(x)≤ C' f_BMO(σ)≤ C”δ.

On the other hand, since f≥χ_E

    ∫_3Δ f(x) dω^A'(x) ≥ω^A'(E) ≳ω^A(E).

The last inequality follows from the Harnack inequality and the fact that A, A' are corkscrew points to surface balls Δ, 3Δ respectively.
 Therefore ω^A(E) ≤ Cδ as long as the condition (<ref>), i.e. σ(E)/σ(Δ)<η is satisfied. In other words, ω∈ A_∞(σ).
















9

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