REGULARITY OF THE DIRICHLET PROBLEM FOR ELLIPTIC EQUATIONS WITH SINGULAR DRIFT

is a Carleson measure in a Lipschitz domain Ω ⊂ R, n ≥ 1, (here δ (X) = dist (X, ∂Ω)). If the harmonic measure dωL0 ∈ A∞, then dωL1 ∈ A∞. This is an analog to Theorem 2.17 in [8] for divergence form operators. As an application of this, a new approximation argument and known results we are able to extend the results in [10] for divergence form operators while obtaining totally new results for nondivergence form operators. The theorems are sharp in all cases.


Introduction and Background
Given a bounded Lipschitz domain Ω ⊂ R n+1 , n ≥ 1, and an operator L given by L = div A∇ (divergence fom) or A • ∇ 2 (nondivergence form), the harmonic measure at X ∈ Ω, dω X L , is the unique Borel measure on ∂Ω such that for all continuous functions g ∈ C (∂Ω), is continuous in Ω and it is the unique solution to the Dirichlet problem (1.1) Lu = 0 in Ω u = g on ∂Ω.
Where we assume that the equality Lu = 0 holds in the weak sense for divergence form operators and in the strong a.e.sense for nondivergence form operators.Here A = A (X) is a symmetric (n + 1) × (n + 1) matrix with bounded measurable entries, satisfying a uniform ellipticity condition for some positive constants λ, Λ.In the nondivergence case the entries of the matrix A are assumed to belong to BMO (Ω) with small enough norm.For a given operator L, the harmonic measures dω X L , X ∈ Ω, are regular probability measures which are mutually absolutely continuous with respect to each other.That is, By the Harnack's principle the kernel function k (X, Y, Q) is positive and uniformly bounded in compact subsets of Ω × Ω × ∂Ω.As a consequence, to study differentiability properties of the family dω X L X∈Ω with respect to any other Borel measure dν on ∂Ω, it is enough to fix a point X 0 ∈ Ω and study dω dν , where dω = dω X0 L is referred as the harmonic measure of L on ∂Ω.If there is unique solvability of the continuous Dirichlet problem and a boundary Maximum Principle is available, then the well definition of the harmonic measure follows from Riesz's representation theorem.
Definition 1.1 (Continuous Dirichlet problem -CD).Given an elliptic operator L, we say that the continuous Dirichlet problem is uniquely solvable in Ω, and we say that CD holds for L, if for every continuous function g on ∂Ω, there exists a unique solution u of (1.1), such that u ∈ C 0 Ω W 2,p (Ω) for some 1 ≤ p ≤ ∞.
Remark 1.2.By Theorem 3.2 below [16], a sufficient condition for CD to hold for a nondivergence form operator L = A • ∇ 2 is that there exists ρ > 0 depending on n and the ellipticity constants such that where • BMO(Ω) denotes the BMO norm in Ω (see Definition 3.1 below).It is not known whether or not the continuous Dirichlet problem is uniquely solvable in the case of elliptic nondivergence form operators with just bounded measurable coefficients.On the other hand, even under the restrictions (1.3) for any ρ > 0 it is known [18] that the continuous Dirichlet problem has non-unique "good solutions".That is, for any ρ > 0 there exists A (X) ∈ BMO (Ω) with A BMO(Ω) ≤ ρ and two sequences of C ∞ symmetric matrices A 0,j and A 1,j with the same ellipticity constants as A, such that A ℓ,j (X) → A (X) as j → ∞ for a.e.X, ℓ = 0, 1, and such that for some continuous function g on ∂Ω the solutions u 0,j and u 1,j to the Dirichlet problems L 0,j u 0,j = 0 in Ω u 0,j = g on ∂Ω, and L 1,j u 1,j = 0 in Ω u 1,j = g on ∂Ω, converge uniformly in Ω to different continuous limits u 0 and u 1 .
Given two regular Borel measures µ and ω in ∂Ω, dω ∈ A ∞ (dσ) if there exist constants 0 < ε, δ < 1 such that for any boundary ball ∆ = ∆ r (Q) and any Borel set E ⊂ ∆, The relation dω ∈ A ∞ (dµ) is an equivalence relation [15], and any two measures related by the A ∞ property are mutually absolutely continuous with respect to each other.From classic theory of weights, if dω ∈ A ∞ (dµ) then there exists 1 < q < ∞ such that the density h = dω dµ satisfies a reverse Hölder inequality with exponent q: This property is denoted dω ∈ B q (dµ), and dω ∈ B q (dµ) is equivalent to the fact that the Dirichlet problem (1.1) for the operator L is solvable in L p (dµ, ∂Ω), 1 p + 1 q = 1 (see [7] for details).When µ = σ, the Euclidean measure, we write A ∞ for A ∞ (dσ).Definition 1.3 (L p -Dirichlet problem, D p ).Let L be an elliptic operator that satisfies CD and let µ be a doubling measure in ∂Ω.We say that the L p (dµ)-Dirichlet problem is uniquely solvable in Ω, and we write that D p (dµ) holds for L, if for every continuous function g on ∂Ω, the unique solution u of (1.1) satisfies for some constant independent of g.Here N u denotes the nontangential maximal function of u on ∂Ω.When µ = σ is the Lebesgue measure on ∂Ω we simply say that D p holds for L. Definition 1.4 (Carleson measure).Let Ω be an open set in R n+1 and let µ be a nonnegative Borel measure on ∂Ω.For X ∈ ∂Ω and r > 0 denote by △ r (X) = {Z ∈ ∂Ω : |Z − X| < r} and T r (X) = {Z ∈ Ω : |Z − X| < r}.Given a nonnegative Borel measure ν in Ω, we say that ν is a Carleson measure in Ω with respect to µ, if there exist a constant C 0 such that for all X ∈ ∂Ω and r > 0, The infimum of all the constants C 0 such that the above inequality holds for all X ∈ ∂Ω and r > 0 is called the Carleson norm of ν with respect to µ in Ω.For conciseness, we will write ν ∈ C (dµ, Ω) when v is a Carleson measure in Ω, and we denote by ν C(dµ,Ω) its Carleson norm.When µ = σ is the Lebesgue measure on ∂Ω we just say that ν is a Carleson measure in Ω. Definition 1.5.Throughout this work, Q γ (X) denotes a cube centered at X with faces parallel to the coordinate axes and sidelength γ; i.e.
In the remarkable work [7], the authors established a perturbation result relating the harmonic measures of two operators in divergence form.The analogue result was later obtained by the author in [17] for nondivergence form operators.
Theorem 1.6 ([7]- [17]).Let L d,0 = div A 0 ∇ and L d,1 = div A 1 ∇ be two elliptic operators with bounded measurable coefficients in Ω, and let ω d,0 and ω d,1 denote their respective harmonic measures.Let σ be a doubling measure on ∂Ω and suppose that The theorems above were stated in terms of the supremum of the 2 .The above formulation is equivalent.In [8], Theorem 1.6 was extended to elliptic divergence form operators with a singular drift: Theorem 1.7 (Theorem 1.9, Chapter III of [8]).
The results in Theorems 1.6 and 1.7 concern perturbation of elliptic operators.They provide solvability for the L q Dirichlet problem (for some q > 1) for an operator L 1 given that there exists an operator L 0 for which the L p Dirichlet problem is solvable for some p > 1 and the disagreement of their coefficients satisfy the Carleson measure conditions (1.4) and (1.5).In [10], the authors answer a different question: What are sufficient conditions on the coefficients A and b so that a given operator L D = div A∇ + b has unique solutions for the L p -Dirichlet problem for some p > 1? See also [9].

Statement of the results
One of the main results in this work is an analog of Theorem 1.7 for nondivergence form operators.
That is, if the L p -Dirichlet problem is uniquely solvable for L N,0 in Ω, for some 1 < p < ∞, then there exists 1 < q 1 < ∞ such that the L q -Dirichlet problem is uniquely solvable for As an application of this result, Theorem 1.6 (nondivergence case) and a simple averaging of the coefficients argument we obtain an analog to Theorem 1.8.Moreover, this averaging argument and Theorem 1.7 yield an extension of Theorem 1.8 to the case when condition (1.6) is replaced by the weaker assumption (1.4).
Remark 2.2.In the divergence form case considered in [8], it was only necessary to assume that condition CD held for just the operator L D,0 , and not both of L D,0 and L D,1 .As mentioned in Remark 1.2, in the nondivergence case there is no well defined notion of solution for operators with measurable coefficients satisfying the hypotheses of Theorem 2.1.Hence, without the CD assumption the harmonic measure for L N,1 would not be defined in general.
To make the statements of the results below more concise, we introduce the following definition.Definition 2.3 (Oscillation).For r > 0, the r-oscillation of a measurable function f (X) (scalar or vector-valued) at a point X, denoted osc r f (X), is given by • ∇ be uniformly elliptic operators in divergence form and nondivergence form, respectively, with bounded measurable coefficient matrix A and drift vector b in a bounded Lipschitz domain Ω.In the nondivergence case we assume that CD holds for L N .Suppose that the coefficients A, b satisfy Then dω LD ∈ A ∞ and dω LN ∈ A ∞ .That is, there exist indexes 1 < p D , p N < ∞ such that the L p -Dirichlet problem is uniquely solvable for L D in Ω, 1 ≤ p ≤ p D , and the L q -Dirichlet problem is uniquely solvable for L N in Ω, 1 ≤ q ≤ p N .
Remark 2.5.The following extensions and generalizations can be obtained (and might be subject for a subsequent work): ( In the case of Theorem 2.4, it can be shown that under vanishing Carleson measure conditions dω LD ∈ A 1 and dω LN ∈ A 1 . 2.1.The theorems are sharp.In [7] it was shown that Theorem 1.6 is sharp for divergence form equations in two fundamental ways (see Theorems 4.11 and 4.2 in [7]).The examples provided in that work were constructed using Beurling-Ahlfors quasiconformal mappings on the half plane R 2 + [2].Quasi-conformal mappings preserve the divergence formstructure of an elliptic operator L D , but when composed with a nondivergence form operator L N the transformed operator has first order drift terms.
The more recent work [10] for divergence form operators (Theorem 1.8) can be applied to obtain regularity of elliptic equation in nondivergence form in the special case that the coefficient matrix satisfies (1.6).
The approximation technique to be introduced in the next section, together with Theorem 2.1, show that Theorem 1.8 can be extended to operators in divergence and nondivergence form with coefficients satisfying the weaker condition (2.2).At the same time, this opens the door to extend the scope of the examples provided in Theorem 1.6 to this wider class of operators.The following theorem is a nondivergence analog to Theorem 4.11 in [7].
2 and such that sup where dσ is the Euclidean measure in ∂ [0, 1] 2 and δ ((x, t)) = t.There exists a coefficients matrix A such that and The above theorem shows that the Carleson measure condition (1.4) in Theorem 1.6 is sharp also in the nondivergence case.In particular, it shows that the main result in [17] is sharp.The proof is a simple application of Theorem 4.11 in [7] and the approximation argument given in the section below (Lemma 3.9).Indeed, by Theorem 4.11 in [7] there exists a coefficients matrix A such that (1) and (2) hold for A and (3) holds for dω LD , with L D = div A∇.For simplicity, let say that two elliptic operators L 0 and L 1 are simultaneously in A ∞ , and write L 0 ≈ L 2 if their respective harmonic measures dω L0 and dω L1 satisfy:

and 2.4 can be applied to show that
See the proof of Theorem 2.1 in Section 4 for a detailed application of this technique.
2.2.Organization of the paper.In the following section we define several function spaces where the coefficients or the solutions to our equations will belong.The basic objects associated to the geometry of Lipschitz domains are also introduced.In this section we also list useful known properties of solutions and the harmonic measure, and we establish some auxiliary results that will allow us to treat the problems locally.In Section 4 we prove Theorem 2.1 by reducing it to a special case (Theorem 2.1).Theorem 2.4 then follows from the result just established and previous theory.Finally, in Section 4.1 we prove Theorem 2.1 by implementing the techniques from [17] (originally adapted from [7]) to this special case.
Acknowledgement.We are grateful to the referee for useful comments and insights that added clarity and elegance to the exposition.In particular we wish to acknowledge the referee's suggestions leading to a simplification of the proof of Theorem 2.4.

Preliminary results
Given a weight w in the Muckenphout class A p (Ω), we denote by L q (Ω, w), 1 ≤ p ≤ q < ∞, the space of measurable functions f such that And for a nonnegative integer k, we define the Sobolev space W k,q (Ω, w) as the space of functions f in L q (Ω, w) such that f has weak derivatives up to order k in L q (Ω, w).Under the assumption w ∈ A p , the space W k,q (Ω, w) is a Banach space and it is also given as the closure of C ∞ 0 (Ω) (smooth functions of compact support in Ω) under the norm see [6], [11].We recall now some definitions.

Definition 3.1 (BMO). Given a locally integrable function
where The space BMO (Ω) of functions of bounded mean oscillation in Ω is given by For ̺ ≥ 0, we let BMO ̺ (Ω) be given by is the space VMO (Ω) of functions of vanishing mean oscillation.We say that a vector or matrix function belongs to a space of scalar functions X if each component belongs to that space X .For example, we sat that the coefficient matrix The following theorem establishes the solvability of the continuous Dirichlet problem for a large class of elliptic operators in nondivergence form.In particular, for such operators the harmonic measure is well defined.

Theorem 3.2 ([16]
).Let L n = A • ∇ 2 be an elliptic operator in nondivergence form in Ω ⊂ R n+1 with ellipticity constants (λ, Λ).There exists a constant ̺ = ̺ (n, λ, Λ), such that if A ∈ BMO ̺ (Ω) then for every g ∈ C (∂Ω) there exists a unique such that L n u = 0 in Ω and u ≡ g on ∂Ω.Moreover, for any subdomain Ω ′ ⋐ Ω and 1 ≤ p < ∞, there exists C = C (n, λ, Λ, p, Ω, dist (Ω ′ , ∂Ω)) such that the above solution satisfies The following comparison principle is the main tool that allows us to treat nondivergence form equations with singular drift.Since we assume that our operator satisfies CD, the result from [1] originally stated in a C 2 domain and for continuous coefficients extends to the more general case stated here.Before stating the theorem, we introduce more notation.
If Ω ⊂ R n+1 is a Lipschitz domain and Q ∈ ∂Ω, r > 0, we define the boundary ball of radius r at Q as The nontangential cone of aperture α and height r at Q, α, r > 0, is defined by Theorem 3.4 (Comparison Theorem for Solutions [1]).Let Ω ⊂ R n+1 be a Lipschitz domain and L = A • ∇ 2 + b • ∇, be a uniformly elliptic operator such that b satisfies and L satisfies CD.There exists a constant C, r 0 > 0 depending only on L and Ω such that if u and v are two nonnegative solutions to Lw = 0 in T 4r (Q), for some Q ∈ ∂Ω, and such that u ≡ v ≡ 0 continuously on Remark 3.5.In [1] the hypothesis on b is δ (X) |b (X)| ≤ η (δ (X)) where η is an non-decreasing function such that η (0) = lim s→0 + η (s) = 0. Nevertheless, the main tools used to obtain Theorem 3.4 were (1) L satisfies property DP (2) a maximum principle (3) Harnack inequality.We assume (1) holds, while (2) and (3) follow as in [1] once we notice that (3.2) implies that b is locally bounded in Ω.Therefore, Theorem 3.4 also holds under our assumptions.
We list now some consequences of the above result that will be useful to us.Lemma 3.6.Let Ω ⊂ R n+1 be a Lipschitz domain and ( ( An important consequence of the properties listed in Lemma 3.6 is the following analog to the "main lemma" in [4].Before stating the result we need to introduce the concept of saw-tooth region.Definition 3.7 (Saw-tooth region).Given a Lipschitz domain Ω and F ⊂ ∂Ω a closed set, a "saw-tooth" region Ω F above F in Ω of height r > 0 is a Lipschitz subdomain of Ω with the following properties: (1) for some 0 < α < β, 0 < c 1 < c 2 and all α < α ′ < α ′′ < β, Γ α ′′ ,c2r (P ) ; (2) ∂Ω ∂Ω F = F ; (3) there exists X 0 ∈ Ω F (the center of Ω F ) such that dist (X 0 , ∂Ω F ) ≈ r; (4) for any P ); (5) Ω F is a Lipschitz domain with Lipschitz constant depending only on that of Ω.
With these provisos, now we state the following: Let Ω be a Lipschitz domain and L = A • ∇ 2 + b • ∇, be a uniformly elliptic operator that satisfies CD.Let F ⊂ ∂Ω be a closed set, and let Ω F be a saw-tooth region above F in Ω.Let ω = ω L,Ω and let ν = ω X0 L,ΩF , where X 0 is the center of Ω F .There exists θ > 0 such that Here θ depends on the Lipschitz character of Ω, but not on E or △.
Given a measurable function f (X) and γ > 0, we recall that the oscillation of f in the cube Q γ (X) (see Definition 1.5) is given by Let Ω be a Lipschitz domain in R n+1 , µ be a doubling measure on ∂Ω, δ (X) = dist (X, ∂Ω), and suppose that g is a measurable function (scalar or vector valued) such that for some constant 0 < α < 1 it satisfies dν = δ (X) osc αδ(X) g (X) 2 dX ∈ C (dµ, Ω) .
The for every Lipschitz subdomain Ω ⊂ Ω, there exists a doubling measure μ on ∂ Ω, with doubling constant depending only on the doubling constant of µ, such that μ = µ in ∂Ω ∂ Ω, and where δ (X) = dist X, ∂ Ω .Moreover, the Carleson norm dν C(dμ, Ω) depends only on the Carleson norm of dν, the doubling constant of µ, and the Lipschitz character of Ω.If µ is the Lebesgue measure on ∂Ω then μ can be taken as the Lebesgue measure on ∂ Ω.
Proof: In the case that dσ is the Euclidean measure dσ on ∂Ω, the result follows as in the proof of Lemma 3.1 in [10].To obtain the Lemma 3.9 for an arbitrary doubling measure σ, given a Lipschitz subdomain Ω ⊂ Ω, we will construct an appropriate extension of σ from ∂Ω ∂ Ω to ∂ Ω.We may assume that Ω = R n+1 + = {(x, t) : x ∈ R n , t > 0}, the general case follows by standard change of variables techniques.For X = (x, t) ∈ R n+1 + , let △ X = {(y, 0) : |x − y| ≤ t} and define M (X) = µ(△X ) σ(△X ) , where σ is the Euclidean measure in R n = ∂Ω.Since µ is doubling, M (X) is continuous in R n+1 + and M → dµ dσ as t → 0 in the weak * topology of measures.Now, for any Borel set E ⊂ ∂ Ω, we define where dσ (X) denotes the Euclidean measure on ∂ Ω.Then obviously μ is an extension of µ from ∂Ω ∂ Ω to ∂ Ω.It remains to check that μ is a doubling measure and that dν ∈ C dμ, Ω .Let △ ⊂ ∂ Ω be a surface ball centered at X 0 = (x 0 , t 0 ) ∈ ∂ Ω and let T (△) ⊂ Ω be the associated Carleson region.Following the ideas in [10], we consider two cases.
+ , for different indexes ι and j, Q i Q j has no interior, and R n+1 be the collection of cubes such that P j △ = ∅, and let X j be an arbitrary point in P j △.Then from the definition of μ and since diam ( the last inequality follows from a simple geometrical argument.From this and (3.3) we have T (△) dν (X) ≤ C μ (△) as wanted.From (3.4) we also have μ (△) ≈ μ (2△) in this case.
The following lemma estates the local character of the regularity of the harmonic measure.

2).
Suppose that b is locally bounded in Ω and it satisfies Then dω L ∈ A ∞ if and only if there exists a finite collection of Lipschitz domains {Ω} N i=1 and compact sets where L i denotes the restriction of L to the subdomain Ω i .
Proof: If b = 0, the result is an immediate consequence of the "main lemma" in [4] for the divergence case and the analog to the main lemma in the nondivergence case, contained in [5] (see also Lemma 3.8).The case b = 0 then follows from Theorem 1.7 for the divergence case and Theorem 4.1 from next section for the nondivergence case.Indeed, by the mentioned theorems, if L is the operator with drift b and L 0 is the operator with the same second order coefficients but without a drift term, then dω L ∈ A ∞ (dσ) ⇔ dω L0 ∈ A ∞ (dσ).On the other hand, by Lemma 3.9 with g (X) = b (X), the restriction of b to any Lipschitz subdomain Ω ′ ⊂ Ω also satisfies (3.5) in Ω ′ .Hence, by Theorems 1.7 and 4.1, for any doubling measure dσ ′ on ∂Ω ′ , we have ).Now, for Ω i , K i as in the statement of Lemma 3.10, dσ| Ki , the restriction of dσ to the compact set K i , can be extended to a doubling measure dσ i on ∂Ω i with the same doubling constant.Then where dω L|Ωi | Ki denotes the restriction to K i of the harmonic measure of L in Ω i , with a similar definition for dω L0|Ωi | Ki .This shows that Lemma 3.10 in the case b = 0 follows from the case b = 0.

Proofs of the Theorems
We recall that Q r (X) denotes a cube centered at X with sidelength r, and δ (x) denotes the distance of X to the boundary (see Definition 1.5).The proof of Theorem 2.1 relies on the following special case.
i,j=1 and b = (b j ) n+1 j=1 are bounded, measurable coefficients and A satisfy the ellipticity condition (1.2) in a Lipschitz domain Ω.Suppose that CD holds for LN,ℓ , ℓ = 0, 1 in Ω and that We defer the proof of this result (which contains the main substance of Theorem 2.1) to next section.Now we obtain Theorem 2.1 from Theorem 4.1.

Proof of Theorem 2.1:
where A ℓ , b ℓ and L N,ℓ satisfy the hypotheses of Theorem 2.1 for ℓ = 0, 1.Then if dω LN,0 ∈ A ∞ , by Theorem 4.1 it follows that dω L * N,0 ∈ A ∞ where L * N,0 = A 0 • ∇ 2 .By Theorem 1.6 [17] and Theorem 4.1 again, we have that if Theorem 2.4 will follow by reduction to the special case Ω = R n+1 + , the localization given by Lemma 3.10 and an approximation of the coefficients matrix A by appropriate smooth matrices.
We note that because of Theorem 1.7 from [8] in the divergence case and because of Theorem 2.1 in the nondivergence case we might assume b ≡ 0. We will first consider the divergence case, suppose that L D = div A∇ is a uniformly elliptic operator in divergence form with bounded measurable coefficient matrix A satisfying (2.2), i.e.
Since P 0 is an arbitrary point on ∂Ω and ∂Ω is compact, by Lemma 3.10, to prove Theorem 2.4 it is enough to prove that if ω is the harmonic measure for L D in Ω α0 , then where K ⊂ ∂Ω α0 is the compact set given by (4.5) For simplicity, we will write Ω = Ω α0 and Φ = Φ α0 .Since ψ is Lipschitz, it follows that the transformation ρ : Φ → Ω can be extended to a homeomorphism from Φ to Ω and such that the restriction of ρ to ∂Φ is a bi-Lipschitz homeomorphism from ∂Ω to ∂Ω.Indeed, since {η s } s>0 is a smooth approximation of the identity, it easily follows that ρ restricted to ∂Φ is given by (4. 6) , then for some constant C depending only on ∇ y ψ ∞ and n, the following estimate holds for the distance functions δ and δ: To see this, let Let σ be the Lebesgue measure on ∂Φ, then from the definition of μ and the fact that ρ is bi-Lipschitz it easily follows that dμ dσ ≈ 1. Hence we can replace μ by σ in our calculations.Now, if u (x, t) is a solution of We claim that Ã satisfies We will use the following fact: Lemma 4.2.For Ω, Φ, σ, μ, δ, δ and ρ as above, the functions , where σ is the Lebesgue measure on ∂Φ.
The lemma above follows from the fact that This property is discussed in [10], and it can be obtained as an application of the characterization of A ∞ in terms of Carleson measures given in [7].Then (4.9) follows by applying (4.3), (4.7) and Lemma 4.2 to (4.10).We recall that Φ = Φ α0 where Φ α is given by Φ α = {y : where I is the (n + 1)× (n + 1) identity matrix.It follows that Ã * (Y ) is an elliptic matrix function, with the same ellipticity constants as Ã.
The measure σ extends trivially from ∂Φ ∂R n+1 + to ∂R n+1 + , we dub this extension (which is just the Euclidean measure) dσ * .With this definitions, because of (4.9), for Y = (y, t) ∈ R n+1 + , Ã * satisfies (4.12) : is a Carleson measure in R n+1 + with respect to σ * ).Where Q γ (Y ) is the cube centered at Y with faces parallel to the coordinate axes and sidelength γt.Now we will construct a smooth approximation of Ã * via an n+1 dimensional approximate identity.Let where ϕ ∈ C ∞ 0 R n+1 is supported in the ball or radius α < 1 at the origin (α to be chosen later) and ϕ = 1.Since P t 1 ≡ 1 we have ∇ x,t P t 1 ≡ 0 and therefore Ã * * = P t Ã * satisfies (Y ) and C is a universal constant (this constant depends on the Lipschitz norm of ϕ, which we can assume only depends on the dimension n).From (4.12) it follows that , hence an application of Theorem 1.7 and the"main lemma" in [4] (see Lemma 3.10), implies that ω| K ∈ A ∞ (dω * | K ), where ω is the harmonic measure of L D restricted to the compact set K given by (4.5).This, in turn, implies that ω| K ∈ A ∞ (dσ| K ) and proves (4.4), hence Theorem 2.4, for the divergence case.
(X) and let W = ρ −1 (Z), then from the definitions of Ã and A * * we have From (4.7) and the fact that ρ is bi-Lipschitz, and since ρ Applying the proof of Lemma 3.9 to δ (W ) from the second property in (4.13) it follows that for some 0 < c < 1 where µ is the Lebesgue measure on ∂Φ α 0 3 ; (4.14) then follows from the fact that ρ is bi-Lipschitz.Now, by the product rule of differentiation Applying the chain rule in each term, we see that ∇A * * satisfies (4.15) because of the first property in (4.13), (4.11), and the boundedness of |∇ Y ρ|.

Proof of Theorem 4.1
In the spirit of [7] (see also [17]) we will obtain Theorem 4.1 as a consequence of the following perturbation result.G 0 (X) sup is a Carleson measure in Ω with respect to dω LN,0 on ∂Ω with Carleson norm bounded by ε 0 , i.e., G 0 (X) sup Where B 2 dω LN,0 denotes the reverse Hölder class of dω LN,0 with exponent 2.
We defer the proof of Theorem 5.1 to the next subsection, and prove now Theorem 4.1, we follow the argument in [7].Let △ r (Q) be the boundary ball △ r (Q) = {P ∈ ∂Ω : |Q − P | < r}, and denote by T r (Q) the Carleson region in Ω associated to △ r (Q), T r (Q) = {X ∈ Ω : |X −Q|< r}.By Lemma 3.10 we may assume that b (X) ≡ 0 if δ (X) > r 0 for some fixed (small) r 0 > 0. To prove Theorem 4.1 it is enough to show that if ω1 = ω LN,1 with LN,1 as in the statement of the theorem, then for all Q ∈ ∂Ω, For Q ∈ ∂Ω, r > 0, α > 0, let Γ α,r (Q) be a nontangential cone of fixed aperture α and height r, i.e.
For a fixed α 0 > 0 to be determined later, let E r (Q) be given by Fix α 0 , r 0 , such that Γ α0,r0 (Q) ⊂ T 2r0 (Q) for all Q ∈ ∂Ω.Then, letting σ be the Lebesgue measure on ∂Ω, by Fubini's theorem, the hypothesis ) and the doubling property of dσ, we have That is, the average in △ r0 (Q) of E 2 b,r0 is bounded.Hence, there exists a closed set ) and E b,r0 (P ) ≤ C for all P ∈ F .Let Ω F be a "saw-tooth" region above F in Ω as given in Definition 3.7, and let b * (X) be given by b The drift b * so defined satisfies E b * ,r0 (P ) ≤ C 0 for all P ∈ △ r0 (Q).We claim that if C 0 is small enough then the operators we only need to check the Carleson measure condition (5.1) near △ 2r0 (Q).More precisely, we will show that for all s < r 0 /2 and P ∈ △ 2r0 (Q), where G 0 (X, Y ) is the Green's function for L 0 in Φ, G 0 (Y ) = G 0 (X 0 , Y ) (where X 0 is the center of Φ) and ω 0 = ω X0 L0,Φ in ∂Φ.As in [17] (see Lemmas 2.8 and 2.14 there), we have where and G (X) = G X, X , with G (Z, X) the Green's function for L 0 in a fixed domain T ⋑ Ω and X ∈ T \Ω is a fixed point away from Ω.
5.1.Proof of Theorem 5.1.The proof of this results closely follows the steps in [17].We will sketch the main steps and refer the reader to [17] for the technical details omitted here.First, by standard arguments the problem is reduced to treating the case in which Ω is the unit ball B = B 1 (0) (this is justified as far as the methods are preserved under bi-Lipschitz transformation).For simplicity, we will write ω 0 = ω LN,0 and ω 1 = ω LN,1 .To see that dω 1 ∈ B 2 (dω 0 ) it is equivalent to prove that if u 1 is a solution of the Dirichlet problem LN,1 where g is continuous in ∂B, then (5.7) where N u is the nontangential maximal function (with some fixed aperture α > 0) of u.We let u 0 be the solution to LN,0 u 0 = 0 in B u 0 = g on ∂B.
We will only prove this in some detail for i = 1.Even though i = 1 is allegedly the simplest case of the three, its proof captures the significant differences with the proof of Lemma 3.2 in [17] (the analog to Lemma 5.2 here), so the other two cases follow in a similar manner as in [17].

Lemma 3 . 10 .
Let L be an elliptic operator in divergence form or nondivergence form with drift b in a Lipschitz domain Ω; i.e.L = A•∇+ b•∇ or L =A • ∇ 2 + b • ∇ where A satisfies the ellipticity condition (1. The other inequality in (4.7) follows in a similar manner.The Lebesgue measure σ on ∂Ω induces a doubling measure μ on ∂Φ by the relation μ (E) = σ (ρ (E)) , for any Borel set E ⊂ ∂Φ.

( 4 . 13 ) 2 t
tE * (Y ) 2 ∈ C dσ * , R n+1 dX ∈ C dσ * , R n+1 + .Moreover, from the definitions it is easy to check that Ã * * is elliptic with the same ellipticity constants as Ã.Let now L * D = div Y Ã * ∇ Y and L * * D = div Y Ã * * ∇ Y .By the Carleson measure property of Ẽ * * , and Lemma 3.9, L * D and L * * D satisfy the hypotheses of Theorem 1.7 in Φ (with respect to the measure σ), therefore, if ω * and ω * * denote the harmonic measures of L * D and L * * D in Φ, respectively, we have that ω * ∈ A ∞ ⇔ ω * * ∈ A ∞ .On the other hand, because of the Carleson measure property of Ẽ * , and Lemma 3.9, L * * D satisfies the hypotheses of Theorem 1.8 in Φ; and therefore ω * * ∈ A ∞ .From what we just proved it follows that ω * ∈ A ∞ .Let L * D denote the pull-back of L * D from Φ to Ω through the mapping ρ.Since the mapping ρ : ∂Φ → ∂Ω is bi-Lipschitz, and if ω * denotes the harmonic measure of L * D in Ω, then ω * ∈ A ∞ .This follows directly from the definitions of the A ∞ class, the harmonic measure ω * and L * D .On the other hand, the operator L * D coincides with L D = div X A∇ X in ρ Φ α 0 3