Journal of Functional Analysis 282 (2022) 109376 Contents lists available at ScienceDirect Journal of Functional Analysis www.elsevier.com/locate/jfa Gradient of the single layer potential and quantitative rectifiability for general Radon measures Carmelo Puliatti 1,2 BGSMath and Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Catalonia, Spain a r t i c l e i n f o a b s t r a c t Article history: We identify a set of sufficient local conditions under which Received 9 September 2020 a significant portion of a Radon measure μ on Rn+1 with Accepted 19 December 2021 compact support can be covered by an uniformly n-rectifiable Available online 31 December 2021 set, at the level of a ball B ⊂ Rn+1 such that μ(B) ≈ Communicated by Marianna r(B)n . This result involves a flatness condition, formulated Csornyei in terms of the so-called β1 -number of B, and the L2 (μ|B )- MSC: boundedness, as well as a control on the mean oscillation on 42B37 the ball, of the operator 42B20 35J15 ˆ 28A75 Tμ f (x) = ∇x E(x, y)f (y) dμ(y). Keywords: Singular integrals Here E(·, ·) is the fundamental solution for a uniformly elliptic Rectifiability operator in divergence form associated with an (n +1) ×(n +1) Elliptic measure matrix with Hölder continuous coefficients. This generalizes a Two-phase problems work by Girela-Sarrión and Tolsa for the n-Riesz transform. E-mail address: carmelo.puliatti@ehu.eus. 1 Current address: Departemento de Matemàticas, Universidad del Païs Vasco, Barrio Sarriena S/N 48940 Leioa, Basque Country. 2 The author was supported by 2017-SGR-0395 (Catalonia), MTM-2016-77635-P (MICINN, Spain), MDM- 2014-0445 (MINECO through the María de Maeztu Programme for Units of Excellence in R&D, Spain), and partially supported by IT-1247-19 (Basque Government) and PID2020-114167GB-I00 (MINECO, Spain). https://doi.org/10.1016/j.jfa.2021.109376 0022-1236/© 2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 2 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 The motivation for our result stems from a two-phase problem for the elliptic harmonic measure. © 2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Contents 1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... . . . . 2 2.Preliminaries and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... . . . . 11 3.The existence of principal values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... . . . . 17 3.1. Principal values for rectifiable measures with compact support . . . . . ....... . . . . 17 3.2. Principal values for measures with zero density . . . . . . . . . . . . . . . ....... . . . . 21 4. The Main Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... . . . . 23 5. The modification of the matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... . . . . 27 5.1. The change of variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... . . . . 27 5.2. Reduction of the Main Lemma to the case A(0) = Id . . . . . . . . . . . ....... . . . . 29 6. A first localization lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... . . . . 35 7. The David and Mattila lattice associated with μ and its properties . . . . . . . ....... . . . . 37 8. The Key Lemma, the stopping time condition and a first modification of the measure . . . . . 39 9. Periodization and smoothing of the measure . . . . . . . . . . . . . . . . . . . . . . ....... . . . . 43 10. The localization of T̄ η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... . . . . 47 11. A pointwise inequality and the conclusion of the proof . . . . . . . . . . . . . . . ....... . . . . 52 11.1. A maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... . . . . 54 11.2. The conclusion of the proof of the Key Lemma . . . . . . . . . . . . . . . ....... . . . . 65 12. The two-phase problem for the elliptic measure . . . . . . . . . . . . . . . . . . . . ....... . . . . 67 Declaration of competing interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... . . . . 75 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... . . . . 76 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... . . . . 76 1. Introduction Singular integral operators are deeply related to geometric properties of measures and, in particular, with the concepts of rectifiability and uniform rectifiability. An operator that has been profusely studied in this field is the codimension 1 Riesz transform, which has a wide relevance because it arises as the gradient of the single layer potential associ- ated with the laplacian. In particular, it has also been used to investigate the properties of harmonic measure. For this reason, it is interesting to understand if the results which are known for the Riesz transform generalize to operators defined by gradients of sin- gle layer potentials associated with suitable elliptic PDEs. In this spirit, the aim of the present article is to establish an elliptic equivalent of a quantitative rectifiability theo- rem that Girela-Sarrión and Tolsa proved in [21] for the Riesz transform, which is also a crucial tool in the study of certain non-variational free-boundary problems for harmonic measure. Given a Radon measure μ on Rn+1 , its associated n-dimensional Riesz transform is ˆ x−y Rnμ f (x) = f (y) dμ(y), f ∈ L1loc (μ), |x − y|n+1 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 3 whenever the integral makes sense. Given x ∈ Rn+1 and r > 0, we denote by B(x, r) the open ball of center x and radius r. A Radon measure μ has growth of degree n if there exists a constant C > 0 such that   μ B(x, r) ≤ Crn for all x ∈ Rn+1 , r > 0. (1.1) We call μ n-Ahlfors-David regular (also abbreviated by n-AD-regular or just AD-regular) if there exists C > 0, which is referred to as AD-regularity constant, such that   C −1 rn ≤ μ B(x, r) ≤ Crn for all x ∈ supp μ, 0 < r < diam(supp μ). A set E ⊂ Rn+1 is said n-AD-regular if Hn |E is a n-AD-regular measure, Hn denoting the n-dimensional Hausdorff measure in Rn+1 . Note that the support of an n-AD-regular measure is n-AD-regular. A set E ⊂ Rn+1 is called n-rectifiable if there exists a countable family of Lipschitz functions fj : Rn → Rn+1 such that    Hn E \ fj (Rn ) = 0. j A measure μ is n-rectifiable if it vanishes outside a rectifiable set E and, moreover, it is absolutely continuous with respect to Hn |E . A set E is said uniformly n-rectifiable (or just uniformly rectifiable) if it is n-AD regular and there exist θ, M > 0 such that for all x ∈ E and all r > 0 there is a Lipschitz mapping g from the ball Bn (0, r) ⊂ Rn to Rn+1 with Lip(g) ≤ M such that   Hn E ∩ B(x, r) ∩ g(Bn (0, r)) ≥ θrn . We say that a measure μ is uniformly n-rectifiable if it is n-AD-regular and it vanishes outside of a uniformly n-rectifiable set. Many characterizations of uniformly rectifiable measures are available in the literature (see e.g. the monograph [18]). To the scopes of the present paper, we are particularly interested in those which are formulated in terms of singular integrals. David and Semmes showed in [17] that, under the background n-AD-regularity as- sumption, a measure μ is uniformly n-rectifiable if and only if all the singular integral operators associated to μ with smooth antisymmetric convolution-type kernels are L2 (μ)- bounded. This class includes the n-Riesz transform and it is interesting to understand if the L2 (μ)-boundedness of Rnμ alone suffices to characterize uniform rectifiability. This deep question is usually referred to as David-Semmes problem in codimension 1. It has been solved for n = 1 (or, equivalently, for the Cauchy transform) by Mattila, Melnikov and Verdera in [35] using the so-called Menger curvature of a measure, and by Nazarov, Tolsa and Volverg in [40] for n > 1 via a set of techniques that includes a variational argument and rely on the harmonicity of the codimension 1 Riesz kernel. We 4 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 remark that the David-Semmes problem is formulated, in its more general form, also for higher codimensions, and its full solution is not known yet. The codimension 1 case has various applications, and plays a crucial role in the study of the geometric properties of harmonic measure. In particular, it was used in [6] to prove that mutual absolute continuity of the harmonic measure associated with an open set Ω ⊂ Rn+1 with respect to surface measure Hn in a subset of ∂Ω implies the n-rectifiability of that subset. This answered a problem raised by Bishop (see [13]). The analogous result for elliptic measure relative to an operator in divergence form associated with a uniformly elliptic matrix with Hölder coefficients has been proved in [43], following the ideas of [6], as an application of the characterization of uniform rectifiability via the boundedness of the gradient of single layer potential. The same problem was also tackled in [50] and [8] via alternative techniques, under more restrictive assumptions on the domains and different hypotheses on the coefficients of the uniformly elliptic matrix. Another question proposed by Bishop asks whether, given two disjoint domains Ω1 , Ω2 ⊂ Rn+1 , mutual absolute continuity of their respective harmonic measures implies absolute continuity with respect to surface measure in ∂Ω1 ∩ ∂Ω2 and rectifiability. This is a so-called two-phase problem for harmonic measure and was solved in its full generality in [11]. This work relies on three main tools: a blow-up argument for harmonic measure (see [29] and [9]), a monotonicity formula ([2]) and a quantitative rectifiability criterion (see [21]). In particular, we point out that the theorem by Girela-Sarrión and Tolsa can be interpreted as a higher-dimensional version of previous results by David and Léger, which were formulated in terms of the Menger curvature (see [15] and [31]). Their theorems are of fundamental importance also in other two-phase problems examined in the works [10] and [44]. The goal of the present paper is to identify an analogous criterion in the context of elliptic PDEs in divergence form with Hölder continuous coefficients. Let A = (aij )1≤i,j≤n+1 be an (n + 1) × (n + 1) matrix whose entries aij : Rn+1 → R are measurable functions in L∞ (Rn+1 ). Assume also that there exists Λ > 0 such that Λ−1 |ξ|2 ≤ A(x)ξ, ξ , for all ξ ∈ Rn+1 and a.e. x ∈ Rn+1 , (1.2) A(x)ξ, η ≤ Λ|ξ||η|, for all ξ, η ∈ Rn+1 and a.e. x ∈ Rn+1 . (1.3) We consider the elliptic equation LA u(x) := −div (A(·)∇u(·)) (x) = 0, (1.4) which should be understood in the distributional sense: we say that a function u ∈ 1,2 Wloc (Ω) is a solution of (1.4), or LA -harmonic, in an open set Ω ⊂ Rn+1 if ˆ A∇u · ∇ϕ = 0, for all ϕ ∈ Cc∞ (Ω). C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 5 We denote by EA (x, y), or just by E(x, y) when the matrix A is clear from the context, the fundamental solution for LA in Rn+1 , so that Lx EA (x, y) = δy in the distributional sense, where δy is the Dirac mass at the point y ∈ Rn+1 . For a construction of the fundamental solution under the assumptions (1.2) and (1.3) on the matrix A we refer to ´ [24]. Given a measure μ, the function f (x) = EA (x, y) dμ(y) is usually known as the single layer potential of μ. We define K(x, y) = ∇1 EA (x, y), (1.5) the subscript 1 indicating that we take the gradient with respect to the first variable, and we consider (1.5) as the kernel of the singular integral operator ˆ T μ(x) = K(x, y) dμ(y), for x away from supp(μ). Observe that T μ is the gradient of the single layer potential of μ. Given a function f ∈ L1loc (μ), we set also ˆ Tμ f (x) = T (f μ)(x) = K(x, y)f (y) dμ(y), and, for ε > 0, we consider the ε-truncated version ˆ Tε μ(x) = K(x, y) dμ(y). |x−y|>ε We also write Tμ,ε f (x) = Tε (f μ)(x). We say that the operator Tμ is bounded on L2 (μ) if the operators Tμ,ε are bounded on L2 (μ) uniformly on ε > 0. In the specific case when A is the identity matrix, we have that −LA = Δ and Tμ is the n-dimensional Riesz transform up to a dimensional constant factor. We say that the matrix A is Hölder continuous with exponent α ∈ (0, 1) (or briefly C α continuous), if there exists Ch > 0 such that |aij (x) − aij (y)| ≤ Ch |x − y|α for all x, y ∈ Rn+1 and 1 ≤ i, j ≤ n + 1. (1.6) Under this assumption on the coefficients, the kernel K(·, ·) turns out to be locally of Calderón-Zygmund type (see Lemma 2.1). However we remark that, contrarily to what happens in the case of the kernel of the Riesz transform, in general K(·, ·) is neither homogeneous nor antisymmetric. Under the assumption (1.6) together with uniform ellipticity, it has been shown by Conde-Alonso, Mourgoglou and Tolsa in [14, Theorem 2.5] that Tμ is bounded on L2 (μ) if μ is a uniformly n-rectifiable measure with compact support. Moreover, they also 6 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 proved that, if μ is a non-zero Borel measure whose upper n-density is positive μ-a.e. and the lower n-density vanishes μ-a.e. in Rn+1 , then Tμ is not bounded on L2 (μ). This result was proved for Rnμ by Eiderman, Nazarov and Volverg in [20] and it inspired the variational argument of [40]. We remark that the results of [14] have been recently further extended by Bailey, Morris and Reguera (see [12]) to Schrödinger operators of the form LVA = −divA∇ + V , for A Hölder continuous and the potential V belonging to the reverse Hölder class RHn+1 . Furthermore, Prat, Puliatti, and Tolsa proved in [43] that, under the same assumptions of [14] on LA , an elliptic analogue of the codimension 1 David-Semmes problem holds: if the measure μ is n-AD-regular, has compact support and Tμ is bounded on L2 (μ), then μ is uniformly n-rectifiable. For our applications, it is essential to determine whether Tμ,ε f converges pointwise μ-almost everywhere for ε → 0. In case it does, we denote the limit as pv Tμ f (x) = lim Tμ,ε f (x) ε→0 and we refer to it as the principal value of the integral Tμ f (x). We prove that, anal- ogously to the n-Riesz transform (see [48, Chapter 8] and the references therein), the L2 (μ)-boundedness of Tμ entails the existence of the principal values for general Radon measures with compact support and growth of degree n. In the following statement, M (Rn+1 ) indicates the vector space of Borel real finite measure on Rn+1 , which is a Banach space when endowed with the total variation norm. Theorem 1.1. Let μ be a Radon measure on Rn+1 with compact support and with growth of degree n, i.e. suppose that there is C > 0 such that μ(B(x, r)) ≤ Crn for all x ∈ Rn+1 . Let A be a matrix that satisfies (1.2), (1.3) and (1.6) and assume, moreover, that the gradient of the single layer potential Tμ associated with LA is bounded on L2 (μ). Then: (1) for 1 ≤ p < ∞ and all f ∈ Lp (μ), pv Tμ f (x) exists for μ-a.e. x ∈ Rn+1 ; (2) for all ν ∈ M (Rn+1 ), pv T ν(x) exists for μ-a.e. x ∈ Rn+1 . In light of this result, in the rest of the paper we will often denote the principal value operator simply as T ν, with a slight abuse of notation. We remark that the previous theorem was first proved in the case of the Cauchy transform by Tolsa in [47]. Given a ball B = B(x, r) ⊂ Rn+1 , we denote by r(B) its radius and, for a > 0, by aB its dilation B(x, ar). Multiple notions of density come into play in this paper. For a ball B, we denote μ(B) Θμ (B) = r(B)n C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 7 and, for γ > 0, its smoothened version  Pμ,γ (B) := 2−jγ Θμ (2j B). (1.7) j≥0 We remark that if γ1 ≤ γ2 , then   Pμ,γ2 (B) = 2−jγ2 Θμ (2j B) ≤ 2−jγ1 Θμ (2j B) = Pμ,γ1 (B). j≥0 j≥0 Another notion of density that we need is the pointwise one. In particular, we denote the upper and lower n-densities of μ at x respectively as     μ B(x, r) μ B(x, r) Θ∗μ (x) := lim sup and Θ∗,μ (x) := lim inf . r→0 (2r)n r→0 (2r)n A way to quantify the flatness of a measure at the level of a ball B is in terms of the β1 -coefficients. For an n-plane L we denote ˆ L 1 dist(x, L) L βμ,1 (B) = dμ(x) and βμ,1 (B) = inf βμ,1 (B), r(B)n r(B) L B the infimum being taken over all hyperplanes in Rn+1 . Using a standard notation, given E ⊂ Rn+1 with μ(E) > 0 and f ∈ L1loc (μ) we write ˆ 1 mμ,E (f ) = f dμ μ(E) E for the mean of f with respect to the measure μ on the set E. The main result of the paper is the following. Theorem 1.2. Let n > 1, let μ be a Radon measure on Rn+1 with compact support and consider an open ball B ⊂ Rn+1 . Let C0 , C1 > 0 and let A be a matrix satisfying (1.2), (1.3) and (1.6). Denote by Tμ the gradient of the single layer potential associated with LA and μ. Suppose that μ and B are such that, for some positive λ, δ and ε and some  ∈ (0, 1), the following properties hold α (1) r(B) ≤ λ. (2) C0−1 r(B)n ≤ μ(B) ≤ C0 r(B)n .   (3) Pμ,α (B) ≤ C0 and for all x ∈ B and 0 < r ≤ r(B) we have μ B(x, r) ≤ C0 rn .     (4) Tμ|B is bounded on L2 μ|B with Tμ|B L2 (μ|B )→L2 (μ|B ) ≤ C1 and T χ2B μ ∈   L2 μ|B . (5) βμ,1 (B) ≤ δ. 8 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 (6) We have ˆ 2 T μ(x) − mμ,B (T μ) dμ(x) ≤ εμ(B). B There exists a choice of λ, δ and ε small enough and a proper choice of α  = α (α, n), all possibly depending on C0 and C1 , such that if μ satisfies (1)− · · · −(6), there exists a uniformly n-rectifiable set Γ that covers a big portion of the support of μ inside B. That is to say, there exists τ > 0 such that μ(B ∩ Γ) ≥ τ μ(B). Notice that Theorem 1.2 readily implies that a big piece of μ|B is mutually absolutely continuous with a big piece of Hn |Γ . This is a relevant feature in light of possible appli- cations. At the moment, very few rectifiability criteria for general measures in terms of singular integrals are available in the literature. Our result is a non trivial extension of [21, Theorem 1.1] to more general operators. A prototype for these results can be found in a quantitative version David-Léger theorem [31, Proposition 1.2]. Given x, y, z ∈ R2 , define c(x, y, z) = 0 if the three points are aligned and, otherwise, c(x, y, z) = R(x, y, z)−1 where R(x, y, z) stands for the radius of the circumference containing x, y and z. The theorem asserts that, given a measure μ on R2 with growth of degree 1, and a ball B(x, r) such that μ(B) ≈ r, there exists ε > 0 such that the following holds: if ν := μ|B is such that ˚ c2 (ν) := c(x, y, z)2 dν(x) dν(y) dν(z) ≤ εμ(B), (1.8) then there exists a possibly rotated Lipschitz graph Γ on the plane such that μ(B ∩ Γ) ≥ 99 100 μ(B). The quantity in the left hand side of (1.8) is the so-called Menger curvature of ν and it was introduced in this field by Melnikov in [36], who also showed together with Verdera in [37] that c(ν) is (modulo an error term) comparable to the L2 (ν)-norm of the Cauchy transform of ν. However, Menger curvature cannot be used to link the L2 - boundedness of Rnμ to uniform n-rectifiability, if n > 1. Hence, in a sense, the property (6) in Theorem 1.2 can be interpreted as a suitable substitute of (1.8). The formulation of Theorem 1.2 involves a relatively long list of hypotheses. On the other hand we remark that those assumptions are necessary, natural and, most importantly, optimal for the application to elliptic measure. The assumption (1), namely requiring the ball B to be small enough, represents a relevant conceptual difference with respect to the analogous theorem for the Riesz transform. The locality of our result reflects the non-scale invariant character of the Hölder regularity assumption for the coefficients of the matrix A. This issue is evident also in [43], where the elliptic analogue of the codimension 1 David-Semmes problem requires an additional compactness assumption on the support of the measure. We remark that C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 9 the assumption (1) is particularly relevant for a localization estimate (see Lemma 9.3), and it allows to bound efficiently the error term in the variational argument of Section 11 (see Subsection 11.2). It is also important to mention that the locality of (1) does not affect the applicability of Theorem 1.2 to the two-phase problem for the elliptic measure, which is of qualitative nature. The second part of the requirement (3) excludes a class of measures like, for instance, the area measure. It is verified, for example, if μ has growth of degree n in the sense of (1.1). On the other hand, the assumption Pμ,α (B) ≤ C0 plays a different role and is useful to deal with technical problems. For example, as shown at the end of Section 3, it is important to ensure that T μ(x) − mμ,B (T μ) is defined in a BMO-sense, and it is also essential to obtain proper localization estimates for the gradient of the single layer potential (see Lemma 6.1 and its proof). Furthermore we remark that, in order to show that the integral in the left-hand side of the assumption (6) is well-defined, we also use the existence of principal values. As in the main result of Girela-Sarrión and Tolsa, the hypothesis (5) is important for technical reasons. However, it is not known if is necessary for the result to hold. Indeed, in the case n = 1 covered by the David-Léger theorem, an analogous flatness assumption is not needed: under the assumptions discussed before (1.8), the inequality c2 (μ|B ) ≤ εμ(B) implies the existence of a line L such that βμ,1 L (B) ≤ δ(ε)μ(B), with δ → 0 as ε → 0. However, having to ask βμ,1 (B)  1 does not constitute a problem when applying the theorem to the study of the two-phase problem for elliptic measure, see Section 12. Another difference with respect to [21] is that we could not formulate the theorem in terms of Pμ,1 . Our proof of the theorem shows that a good choice for α is α  = α/2n+1 . It is not clear whether Theorem 1.2 holds with a condition on Pμ,α (B), that seems a more natural homogeneity to assume. The proofs of the rectifiability results for the harmonic measure in [9] and [11] actually rely on the fact that the theorem of Girela-Sarrión and Tolsa holds for α = 1. However, a slight variation on their arguments allows to overcome this technical obstacle. Let us now present an application of Theorem 1.2, which is, in fact, its main moti- vation. Before stating it, recall that if Ω is a Wiener regular set, the elliptic measure p ωL A with pole at p associated with the elliptic operator LA is the probability measure supported on ∂Ω such that, for f ∈ C0 (∂Ω), ˆ p f dωL A = f(p), where f denotes the LA -harmonic extension of f . A large literature is available on the subject. For example, we refer to [22] and [27] (and the references therein) for its definition and basic properties. Theorem 1.3. Let n ≥ 2 and let A be an elliptic matrix satisfying (1.2), (1.3) and (1.6). Let Ω1 , Ω2 ⊂ Rn+1 be two disjoint Wiener-regular domains and, for pi ∈ Ωi , i ∈ {1, 2}, 10 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 pi let ωL A ,i be the respective elliptic measures in Ωi associated with LA and with pole pi . p1 p2 p1 Suppose that E is a Borel set such that ωL |  ωL A ,1 E |  ωL A ,2 E | . Then there exists A ,1 E p1 p1 p2 an n-rectifiable set F ⊂ E with ωL A ,1 (E \ F ) = 0, and such that ω LA ,1 |F and ωLA ,2 |F are mutually absolutely continuous with respect to H |F . n We remark that the generalization of the blow-up methods for the harmonic measure to our elliptic context is contained in the work [8]. Also, the proof of Theorem 1.3 follows closely the path of the work [11]. However, some relevant variations are needed so we decided to sketch the proof at the end of the paper, where we also provide precise references for the reader’s convenience. We finally remark that recently several studies have appeared concerning the con- nection between the geometry of a domain and the properties of its associated elliptic measure, among which we list [1], [5], [23], [25], [26] and [28]. Discussion of the proofs. For the proof of the main theorem we follow the elaborated scheme of [21]. However, there are many delicate obstacles which are not present when dealing with the Riesz transform and that require original approaches. Section 2 is de- voted to settle our notation and to make an overview of the results in PDEs relevant for our work. In particular, we need a rescaled version of some estimate for the gradient of the fundamental solution first proved in the context of homogenization theory, which are indispensable to estimate the behavior of EA (x, y) for big values of |x − y|. In Section 3 we prove Theorem 1.1 separating the case in which μ is rectifiable to the one in which it has zero n-density. This is possible because Prat, Puliatti, and Tolsa proved in [43] that the L2 (Hn |Γ )-boundedness of THn |Γ , Γ ⊂ Rn+1 compact with Hn (Γ) < ∞, implies the rectifiability of Γ, generalizing a result first proved in [41]. Then, in Section 4 we proceed to state the Main Lemma that we use to prove The- orem 1.2. The biggest advantage of this lemma is that the flatness condition on the β1 -number in Theorem 1.2 is replaced by a smallness hypothesis on the α-numbers of Tolsa. The latter are more powerful tools when trying to transfer the flatness estimates to the integrals. Furthermore, we have to show that we can consider the matrix A of the elliptic operator which defines Tμ to be symmetric. We then proceed to discuss, in Section 5, an equivalent formulation of the Main Lemma in terms of an auxiliary elliptic operator which shares more symmetries than LA . In particular, we discuss the construction of a particular auxiliary periodic matrix via a sequence of reflections. In Sections 6, 7, 8 and 9 we recall the definition of the dyadic cells associated with μ as constructed by David and Mattila, and, in an attempt to balance brevity and clarity, we mainly follow the path of the original work for the Riesz transform. However, we remark that these sections are necessary for the sake of the exposition; indeed, they present the core of the contradiction argument for the proof of the Main Lemma and the construction of a periodic auxiliary measure which is needed for the arguments of the remaining sections of the paper. C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 11 The starting point for the crucial variational argument, whose proof occupies Sec- tion 11, is a localization estimate for the potential at the level of a small cube. This is proved in Section 10, along with the existence of the limit of proper smooth truncates of the potential of bounded periodic functions. We emphasize that, again, these proofs rely heavily on the periodicity of the modification of the elliptic matrix. In Section 11 we complete the proof of the Main Lemma via a variational technique. We highlight that one of the most delicate point consists in finding an appropriate variant of a maximum principle in an infinite strip in our elliptic setting. Moreover, the proof in [21] exploits the fact that the map x = (x1 , . . . , xn+1 ) → xn+1 is harmonic. The fact that this function, in general, is not a solution of LA requires a more technical method based on the additional symmetries provided by the modified matrix. In the final Section 12 we sketch the proof of Theorem 1.3, with particular care of highlighting the points which require additional explanations with respect to its harmonic counterpart. 2. Preliminaries and notation We write a  b to denote that there is a constant C > 0 such that a ≤ Cb. To make the dependence of the constant on a parameter t explicit, we write a t b. Also, we say that b  a if a  b and a ≈ b if both a  b and b  a. All the cubes, unless specified, will be considered with their sides parallel to the coordinate axes. Given a cube Q, we denote its side length as (Q) and, for a > 0, we understand aQ as the cube with side length a(Q) and sharing the center with Q. We say that a cube Q has t-thin boundary if μ x ∈ 2Q : dist(x, ∂Q) ≤ λ(Q) ≤ tλμ(2Q) for every λ > 0. Analogously to (1.7), we define   μ(2j Q) Pμ,γ (Q) = 2−jγ Θμ (2j Q) = 2−jγ . (2j Q)n j≥0 j≥0 Given a measure μ and a measurable set E, we denote as μ|E the restriction of μ to E and, for φ : Rn+1 → Rn+1 , we use the notation φμ(E) := μ(φ−1 (E)). An important tool in the study of rectifiability is the so-called α-number introduced by Tolsa in [49]. Let us fix a cube Q ⊂ Rn+1 and consider two Radon measures μ and ν on Rn+1 . A natural way to define a distance between μ and ν is to consider the supremum ˆ dQ (μ, ν) := sup f d(μ − ν), f where f ∈ Lip(Rn+1 ), f Lip ≤ 1 and supp f ⊆ Q. For a n-plane L in Rn+1 , we define 12 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 1 αμL (Q) := inf dQ (μ, cHn |L ). (2.1) (Q)n+1 c≥0 Given a matrix A(·), possibly with variable coefficients, AT (·) indicates its transpose. Also, we write Ln+1 for the Lebesgue measure on Rn+1 . Partial Differential Equations. For any uniformly elliptic matrix A with Hölder continu- ous coefficients, one can show that K(x, y) = ∇1 E(x, y) is locally a Calderón-Zygmund kernel. Lemma 2.1. Let A be an elliptic matrix with Hölder continuous coefficients satisfying (1.2), (1.3) and (1.6). If K(·, ·) is given by (1.5), then it is locally a Calderón-Zygmund kernel. That is, for any given R > 0, (a) |K(x, y)|  |x − y|−n for all x, y ∈ Rn+1 with x = y and |x − y| ≤ R. (b) |K(x, y) −K(x, y  )| +|K(y, x) −K(y  , x)|  |y −y  |α |x −y|−n−α for all y, y  ∈ B(x, R) with 2|y − y  | ≤ |x − y|. (c) |K(x, y)|  |x − y|(1−n)/2 for all x, y ∈ Rn+1 with |x − y| ≥ 1. All the implicit constants in (a), (b) and (c) depend on Λ and A α, while the ones in (a) and (b) depend also on R. The statements above are rather standard. For more details, see [14, Lemma 2.1]. Let ωn denote the surface measure of the unit sphere of Rn+1 . For any elliptic matrix A0 with constant coefficients, we have an explicit expression for the fundamental solution of LA0 , which we denote by Θ(x, y; A0 ). More precisely, Θ(x, y; A0 ) = Θ(x − y; A0 ) with ⎧ ⎪ −1 1 ⎪ ⎪  for n ≥ 2, ⎨ −1 (n − 1)ωn det A0,s (A0,s z · z)(n−1)/2 Θ(z; A0 ) = Θ(z; A0,s ) = (2.2) ⎪ ⎪ 1   ⎪ ⎩  log A−1 0,s z · z for n = 1, 4π det A0,s where A0,s is the symmetric part of A0 , that is, A0,s = 12 (A0 + AT0 ). For more details we refer to [38]. The reason why only the symmetric part of A0 enters (2.2) it that, using Schwarz’s theorem to exchange the order of partial derivatives writing A0 = {aij }i,j , for every appropriate function u we have  1 1 LA0 u = − ∂i (aij ∂j u) = − aij ∂i ∂j u − aij ∂j ∂i u i,j 2 i,j 2 i,j  aij + aji =− ∂i ∂j u = LA0,s u. (2.3) i,j 2 These formal considerations can be made rigorous by standard arguments. C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 13 Differentiating (2.2) we obtain 1 A−1 0,s z ∇Θ(z; A0 ) =  −1 . ωn det A0,s (A0,s z · z)(n+1)/2 The next result is proven in [30, Lemma 2.2]. Lemma 2.2. Let A be an elliptic matrix with Hölder continuous coefficients satisfying (1.2), (1.3) and (1.6). Let also Θ(·, ·; ·) be given by (2.2). Then, for x, y ∈ Rn+1 , 0 < |x − y| ≤ R, (1) |EA (x, y) − Θ(x, y; A(x))|  |x − y|α−n+1 , (2) |∇1 EA (x, y) − ∇1 Θ(x, y; A(x))|  |x − y|α−n , (3) |∇1 EA (x, y) − ∇1 Θ(x, y; A(y))|  |x − y|α−n . Similar inequalities hold if we reverse the roles of x and y and we replace ∇1 by ∇2 . All the implicit constants depend on Λ, A α , and R. The gradient of the fundamental solution in the periodic case. We denote as Λα the set of matrices such that (1.2), (1.3) hold and with α-Hölder coefficients. We say that the matrix A ∈ Λα is -periodic,  > 0, if A(x + z) = A(x) for every z ∈ Zn+1 . For periodic matrices the estimates in Lemma 2.1 turn out to be global. Lemma 2.3 ([30]). Let A ∈ Λα be 1-periodic and let E A be the fundamental solution of LA . Let K(·, ·) is given by (1.5). Then (1) |∇1 E A (x, y)| ≤ c1 |x − y|−n for every x, y ∈ Rn+1 with x = y. (2) | ∇1 E A (x, y) − ∇1 E A (x , y)| + | ∇1 E A (y, x) − ∇1 E A (y, x )| ≤ c2 |x − x |α |x − y|−(n+α) for every x, x , y ∈ Rn+1 such that 2|x − x | ≤ |x − y|. The constants appearing in (1) and (2) are such that c1 ≈n,Λ c2 ≈n,Λ A α. The period of the matrix plays an important role in our construction, so it is useful to rephrase the previous lemma for matrices with a period different from 1. We are interested in studying matrices with small period, so we only consider the case in which it is strictly smaller than 1. Lemma 2.4. Let 0 <  < 1. Let A ∈ Λα be -periodic and let E A be the fundamental solution associated with LA . Then 14 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 (1) |∇1 E A (x, y)| ≤ c1 |x − y|−n for every x, y ∈ Rn+1 with x = y. (2) | ∇1 E A (x, y) − ∇1 E A (x , y)| + | ∇1 E A (y, x) − ∇1 E A (y, x )| ≤ c2 |x − x |α |x − y|−n−α for every x, x , y ∈ Rn+1 such that 2|x − x | ≤ |x − y|. The constants appearing in (1) and (2) are such that c1 ≈n,Λ c2 ≈n,Λ A α. Proof. For  ∈ (0, 1) and all x ∈ Rn+1 we define the rescaled matrix  A(x) := A(x) and we denote by E the fundamental solution of LA . By the definition of fundamental solution, it is not difficult to see that  y) = n ∇1 E A (x, y) ∇1 E(x, for x, y ∈ Rn+1 . (2.4) Moreover,  |A(x)  − A(y)| = |A(x) − A(y)| ≤ α A α |x − y|α ≤ A α |x − y|α ,  so that A α ≤ A α. Applying Lemma 2.3 together with (2.4) we get  −1 x, −1 y)|  −n |−1 x − −1 y|−n = |x − y|−n | ∇1 E A (x, y)| = −n | ∇1 E( for any x, y and  −1 x, −1 y) − ∇1 E( | ∇1 E A (x, y) − ∇1 E A (x , y)| = −n | ∇1 E(  −1 x , −1 y)| |−1 x − −1 x |α |x − x |α  −n = |−1 x − −1 y|n+α |x − y|n+α , for 2|x − x | ≤ |x − y|. The same estimate holds for | ∇1 E A (y, x) − ∇1 E A (y, x )|.  The following is the (global) analogue of Lemma 2.2 in the 1-periodic setting. Lemma 2.5. Let A ∈ Λα be 1-periodic. Then for every x, y ∈ Rn+1 , x = y, we have E A (x, y) − Θ(x, y; A(x))  |x − y|α−n+1 ∇1 E A (x, y) − ∇1 Θ(x, y; A(x))  |x − y|α−n ∇1 E A (x, y) − ∇1 Θ(x, y; A(y))  |x − y|α−n , the implicit constants depending on A α and Λ. Similar estimates hold if we replace ∇1 by ∇2 . C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 15 Let us now recall some result from elliptic homogenization. For more details we refer to the work by Avellaneda and Lin [3]. For this purpose, we need to recall the definition of vector of correctors χ and homogenized matrix A0 . Let  > 0 and let A ∈ Λα be a 1-periodic matrix, i.e. A(x + z) = A(x) for every z ∈ Zn+1 . We will denote by χ(x) = (χi (x)), for i ∈ {1, . . . , n + 1} the vector of correctors, which is defined as the solution of the following cell problem ⎧ ⎪ ⎪ ⎨Lχ = div A, χ is 1-periodic, (2.5) ⎪ ⎩´ ⎪ χ(x)dx = 0, [0,1]n+1 where the first condition in (2.5) has to be understood in coordinates as     ∂xi aij ∂xj χh (x) = − ∂xi aih (x), i,j i (aij )i,j being the coefficients of the matrix A. An important fact is that ∇χ ∞ ≤ C, (2.6) the bound C depending only on n, α and A C α . We remark that ∇χ denotes the matrix with variable coefficients whose entries are ∂i χj for i, j = 1, . . . , n +1. Now, if we consider the following family of elliptic operators   Lε := div A(x/ε)∇ · depending on the parameter ε > 0, it can be proved that for any f ∈ L2 (Rn+1 ), the solutions uε ∈ W 1,2 (Rn+1 ) of Lε uε = div f converge weakly in W 1,2 (Rn+1 ) to a function u0 as ε → 0. This function solves the equation L0 u0 := div(A0 ∇u0 ) = div f, where A0 is an elliptic matrix with constant coefficients usually called homogenized ma- trix (see, for example, [45]). Homogenization is a powerful tool to study the fundamental solution of an elliptic equation in divergence form whose associated matrix is periodic and has C α coefficients. The main result that we will use is the following (see [3, Lemma 2] and [30, Lemma 2.5]). 16 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 Lemma 2.6. Let A ∈ Λα . Let us assume that A is 1-periodic. There exists γ ∈ (0, 1) depending on α, A C α and n such that c E A (x, y) − (Id + ∇χ(x))Θ(x, y; A0 )  (2.7) |x − y|n+γ−1 and c ∇1 E A (x, y) − (Id + ∇χ(x))∇1 Θ(x, y; A0 )  , (2.8) |x − y|n+γ where Id denotes the identity matrix and the implicit constants in (2.7) and (2.8) depend just on n, α and A α . The period of the coefficients of A plays a crucial role in these estimates. We will be dealing with matrices with periodicity different from 1, so we need a suitably adapted version of the previous lemma. Let A ∈ Λα be a -periodic matrix. Let us define the 1-periodic matrix  A(x) := A(x) for x ∈ Rn+1 and let χ  defined according  denote the vector of correctors associated with A to (2.5). For  > 0 we define x  χ (x) :=  χ .  Observe that, because of (2.6) there exists C > 0 depending on the n, α and A Cα such that ∇χ ∞ ≤ C. (2.9) Lemma 2.7. Let 0 <  < 1. Let A ∈ Λα be an -periodic matrix. Then there exists γ ∈ (0, 1) and c > 0, both depending just on n, α and A α such that ∇1 E A (x, y) − ∇1 Θ(x, y; A(x)) ≤ cα |x − y|α−n , (2.10) ∇2 E A (x, y) − ∇2 Θ(x, y; A(y)) ≤ cα |x − y|α−n . (2.11) −n−γ ∇1 E A (x, y) − (Id + ∇χ (x))∇1 Θ(x, y; A0 ) ≤ c |x − y| γ , for every x = y. Proof. Let E denote the fundamental solution of the operator LA . As in (2.4), we have  ∇1 E A (x, y) = −n ∇1 E(x/, y/), (2.12) C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 17 so an application of Lemma 2.5 gives ∇1 E A (x, y) − ∇1 Θ(x, y; A(x))  −1 x))| ≤ cα |x − y|α−n . = −n |∇1 E A (−1 x, −1 y) − ∇1 Θ(x, y; A( Using (2.8) and (2.12), we get |∇1 E A (x, y) − (Id + ∇χ (x))∇1 Θ(x, y; A0 )|  = −n |∇1 E(x/, y/) − (Id + ∇ χ(x/))∇1 Θ(x/, y/; A0 )| cn+γ cγ  = , n |x − y|n+γ |x − y|n+γ  where c depends on n, α and A α,  A α ≤ A α. Inequality (2.11) follows as (2.10).  3. The existence of principal values The proof of the existence of principal values can be divided into the study of two different cases: the case in which μ is a rectifiable measure and the one in which μ has zero n-density, i.e. μ(B(x, r)) lim =0 for μ-a.e. x ∈ Rn+1 . r→0 rn Indeed, if μ is a measure on Rn+1 with no point masses and Tμ is bounded on L2 (μ), [43, Theorem 1.2] allows us to write μ = μ0 + μ1 , where μ0 has vanishing upper n-density μ0 -almost everywhere and μ1 is n-rectifiable. See also the argument in [48, Chapter 8] for the case of the Cauchy transform. 3.1. Principal values for rectifiable measures with compact support This subsection follows the scheme of [14, Section 2.2]. The proof of the existence of principal values for Tμ if the measure μ is rectifiable and has compact support relies on the following result. Theorem 3.1. Let μ be a rectifiable measure. Let K ∈ C ∞ (Rn+1 \ {0}) be an odd kernel and homogeneous of degree −n, i.e. K(x) = −K(−x) and K(λx) = λ−n K(x). Assume, for some M = M (n), the further regularity condition |∇j K(x)| n C(j)|x|−n−j for all 0 ≤ j ≤ M and x ∈ Rn+1 \ {0}. Then the operator TK,μ is bounded on L2 (μ) with operator norm TK,μ L2 (μ)→L2 (μ) n K|Sn C M (Rn+1 ) . 18 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 Moreover, the principal value ˆ TK,μ f (x) = lim K(x − y)f (y)dμ(y) ε→0 |x−y|≥ε exists μ-almost everywhere. The proof of the boundedness of TK,μ is due to David and Semmes. The result on principal values was first proved imposing an analogous condition for all j = 0, 1, 2, . . . (for a more detailed exposition we refer, for example, to [33, Chapter 20]). We remark that it has been more recently improved by Mas (see [32, Corollary 1.6]). The previous theorem together with a spherical harmonics expansion of the kernel is the key tool to prove the following result. Lemma 3.1. Let μ be an n-rectifiable measure. There exists M = M (n) such that the following holds. Let b(x, z) be odd in z and homogeneous of degree −n in z, and assume Dzα b(x, z) is continuous and bounded on Rn+1 × Sn , for any multi-index |α| ≤ M . Then for every f ∈ L2 (μ), the limit ˆ Bf (x) = lim b(x, x − y)f (y)dμ(y) ε→0 |x−y|>ε exists for μ-almost every x. Proof. This result is used in [39] (see for example [39, (1.14)]). The proof is a variation of the argument in [39, Proposition 1.2]. For the reader’s convenience we discuss the details below. Let {ϕj,l }j≥1,1≤l≤Nj be an orthonormal basis of L2 (Sn ) consisting of surface spherical harmonics of degree j. Recall that (see [4, (2.12)]) Nj = O(j n−1 ), for j  1. (3.1) Using the homogeneity assumption for b(x, ·) and the orthonormal expansion, we write  z   Nj z  −n b(x, z) = b x, |z| = b(x, ·), ϕj,l L2 (Sn ) ϕj,l |z|−n |z| |z| j≥1 l=1 z  (3.2)  = bj,l (x)ϕj,l |z|−n , |z| j,l where bj,l (x) := b(x, ·), ϕj,l L2 (Sn ) . Since b(x, ·) is an odd function and ϕ2j,l is even for every j, bj,l (x) ≡ 0 for j even. Being b in L∞ (Rn+1 × Sn ) by hypothesis and Hölder’s inequality, we have C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 19 |bj,l (x)| ≤ C(n) b(x, ·) L∞ (Sn ) ϕj,l L2 (Sn ) ≤ C(n) b L∞ (Rn+1 ×Sn ) ≤ C(n). (3.3)   Moreover, recalling that we can suppose j odd, the function K j,l (z) := ϕj,l z/|z| |z|−n satisfies the hypothesis in Theorem 3.1: there exists an harmonic polynomial Pj,l of odd degree j such that ϕj,l (z/|z|) = Pj,l (z)/|z|j , so z  1 ∇ϕj,l  |z| |z| and z  1 z  1 1  j,l (z)  ∇ϕj,l ∇K + ϕj,l  n+1 . |z| |z|n |z| |z|n+1 |z| Analogous estimates hold for higher order derivatives. So, Theorem 3.1 ensures that ˆ TK j,l ,μ f (x) = lim  j,l (x − y)f (y)dμ(y) ≡ lim T  K f (x) (3.4) ε→0 Kj,l ,μ,ε ε→0 |x−y|>ε exists for μ-a.e x. Recall also that by the Theorem 3.1 there exists M = M (n) such that 2 TK j,l ,μ is bounded on L (μ) with operator norm TK j,l ,μ L2 (μ)→L2 (μ)  j,l |Sn  K C M (Sn ) = ϕj,l C M (Sn ) . (3.5) Gathering (3.2), (3.3) and (3.4), to prove the lemma it is enough to show that the dominated convergence theorem applies and, in particular, that  j,l ,μ,ε f (x) ≤ C(x) < ∞, bj (x)TK (3.6) j,l where C(x) does not depend on ε. By Lebesgue differentiation theorem, to prove (3.6) it suffices to show that for every ball B0 ⊂ Rn+1 we have ˆ  j,l ,μ,ε f (x) dμ(x) B0 ,n bj,l (x)TK bj,l ∞ ϕj,l C m (Sn ) f L2 (μ) j,l B j,l,m 0 ≤C f L2 (μ) for some C > 0, where the first inequality above uses the L2 -boundedness (3.5). The smoothness of b implies that (see [46, 3.1.5]) 1 bj,l ∞  3 , j 2 n+2+M 20 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 where the exponent on the right hand side is chosen accordingly to what we need next. Now, recall that the Sobolev space H s (Sn ), s ∈ R can be defined via spherical harmonics expansion. In particular, it is the completion of C ∞ (Sn ) with respect to the norm  n − 1 2s 1/2 v H s (Sn ) := j+ |vj,l |2 , (3.7) 2 j,l where vj,l = v, ϕj,l L2 (Sn ) . For the definition and the properties of this space, we refer for example to [4, Section 3.8] and to [4, Section 6.3] for the relation of (3.7) with that via the restriction of the gradient to the unit sphere. By Sobolev embedding theorem, H s (Sn ) continuously embeds into C(Sn ) for s > n/2. So, choosing s = n2 + 1 and using (3.7) we can estimate  2j + n − 1  n2 +m+1 Dm ϕj,l C(Sn ) n ϕj,l H s+m (Sn ) = . 2 Hence, using (3.1)   M   1 Nj j − 2 n−2−M j 2 +m+1  3 n bj,l ∞ ϕj,l C M (Sn ) n < ∞, m=0 j≥1 j2 j,l j≥1 which concludes the proof.  Theorem 3.2. Let μ be an n-rectifiable measure on Rn+1 with compact support. Let A be a matrix having the properties (1.2), (1.3) and (1.6). Then for every f ∈ L2 (μ) the principal value ˆ Tμ f (x) = lim ∇1 E(x, y)f (y)dμ(y) ε→0 |x−y|>ε exists for μ-almost every x. Proof. Let ε > 0 and denote b(x, z) := ∇1 Θ(z, 0; A(x)). As a consequence of the explicit formula (2.2), it is not difficult to see that each component of b verifies the hypothesis of Lemma 3.1. So, split Tμ,ε as ˆ Tμ,ε f (x) = b(x, x − y)f (y)dμ(y) |x−y|>ε ˆ   + ∇1 E(x, y) − ∇1 Θ(x, y; A(x)) f (y)dμ(y). (3.8) |x−y|>ε The limit for ε → 0 of the first integral in the right hand side of (3.8) exists μ-a.e. because of Lemma 3.1. On the other hand, ∇1 E(x, y) −∇1 Θ(x, y; A(x)) defines an operator which C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 21 is compact on Lp (μ) because of Lemma 2.2, which guarantees that the limit for ε → 0 exists for μ-a.e. x and concludes the proof.  3.2. Principal values for measures with zero density Again, suppose that μ has compact support. A combination of the proof of [34, Theorem 1.4] (see also [48, Theorem 8.10]) and Lemma 2.2 (which has to be used instead of antisymmetry) makes possible to prove that if μ is a Radon measure in Rn+1 with growth of degree n, then for every 1 < p < ∞ and f ∈ Lp (μ), {Tμ,ε f }ε admits a weak limit Tμw f in Lp (μ) as ε → 0. Moreover, the representation formula   Tμw f (x) = lim Tμ f χB(x,r)c (y)dμ(y) (3.9) r→0 B(x,r) holds for μ-almost every x ∈ Rn+1 , giving an explicit way of computing the weak limit. We remark that, in general, we can only infer that formula (3.9) holds if Tμ has an antisymmetric kernel. Let us recall the following theorem by Mattila and Verdera (see [34]), here reported in the formulation of [48, Theorem 8.11]. Theorem 3.3. Let μ be a Radon measure in Rd that has growth of degree n and zero n- dimensional density μ-a.e. Let Tμ be an n-dimensional antisymmetric Calderón-Zygmund operator. Then, for all 1 < p < ∞ and f ∈ Lp (μ), pv Tμ f (x) exists for μ-a.e. x ∈ Rd and coincides with Tμw f (x). Also, for all ν ∈ M (C), pv T ν(x) exists for μ-a.e. x ∈ Rd . This result can be transferred to the gradients of the single layer potential Tμ . Theorem 3.4. Let μ be a Radon measure in Rn+1 that has growth of degree n, zero n- dimensional density and compact support. Suppose that Tμ is a bounded operator from L2 (μ) to L2 (μ). Then, for all 1 < p < ∞ and f ∈ Lp (μ), pv Tμ f (x) exists for μ-a.e. x ∈ Rn+1 and coincides with Tμw f (x). Also, for all ν ∈ M (C), pv T ν(x) exists for μ-a.e. x ∈ Rn+1 . Proof. Let 1 < p < ∞ and f ∈ Lp (μ). We decompose Tμ f into its symmetric and antisymmetric part. That is to say, Tμ f (x) = Tμ(a) f (x) + Tμ(s) f (x), (a) (s) where Tμ is the integral operator with kernel (∇1 E(x, y) −∇1 E(y, x))/2 and Tμ whose kernel is (∇1 E(x, y) + ∇1 E(y, x))/2. We can apply Theorem 3.3 to antisymmetric part (a) (a) Tμ , obtaining that pv Tμ f (x) exists for μ-a.e. x. 22 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 (s) On the other hand, Tμ defines a compact operator on Lp (μ) since ˆ | ∇1 E(x, y) + ∇1 E(y, x)|dμ(y)  diam(supp μ)α , so that the principal values exist. The fact that Tμw f coincides with pv Tμ f a.e. follows from the definition of weak limit together with dominated convergence theorem: ˆ ˆ ˆ  Tμw f g dμ = lim Tμ,ε f g dμ = pv Tμ f g dμ for all g ∈ Lp (μ), ε→0 p being the Hölder conjugate exponent of p.  A remark on the well-posedness of the assumption (6) of Theorem 1.2. Let T, μ and B be as in Theorem 1.2. Let x, y ∈ B and ε > 0 and write   Tε μ(x) − Tε μ(y) = Tμ,ε χ2B (x) − Tμ,ε χ2B (y) + Tμ,ε χRn+1 \2B (x) − Tμ,ε χRn+1 \2B (y) . (3.10) 2 Now observe that, being the operator Tμ|B bounded on L (μ|B ), Theorem 1.1 (2) applies with ν = χ2B μ. So, the first two summands on the right hand side of (3.10) admit a limit as ε → 0 for almost every x, y ∈ B. The limit for ε → 0 of the last summand exists, too. Indeed, since x, y do not belong to Rn+1 \2B, for ε < r(B), ˆ   Tμ,ε χRn+1 \2B (x) − Tμ,ε χRn+1 \2B (y) = ∇1 E(x, z) − ∇1 E(y, z) dμ(y). Rn+1 \2B If we assume α ≤ α in the statement of the main theorem, an application of the Calderón- Zygmund property of the kernel combined with a dyadic decomposition of the domain of integration gives ˆ  ˆ   +∞ 1 ∇1 E(x, z) − ∇1 E(y, z) dμ(z)  |x − y|α dμ(z) j=1 j+1 |x − z|n+α Rn+1 \2B 2 B\2j B ≤ Pμ,α (B) ≤ Pμ,α (B) < +∞. (3.11) In particular, this tells that T μ(x) − T μ(y) exists in the principal value sense for almost every x, y ∈ B. We also want to point out that T μ − mμ,B (T μ) defines an L2 (μ|B )-function. Indeed, for x ∈ B and using (3.11), C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 23 ˆ 1 |T μ(x) − mμ,B (T μ)| ≤ |T μ(x) − T μ(y)|dμ(y) μ(B) B  1/2 ≤ |T (χ2B μ)(x)| + mμ,B |T (χ2B μ)|2 + Pμ,α (B). The right hand side of the previous majorization defines an L2 (μ|B ) function because of the assumptions T (χ2B μ) ∈ L2 (μ|B ) and Pμ,α (B) < +∞ in Theorem 1.2. 4. The Main Lemma As we mentioned in the Introduction, it is convenient to formulate the statement of the main Theorem 1.2 in terms of α-numbers (see (2.1)). This is made possible by a mostly geometric argument: a careful read shows that the same arguments of [21, Section 3] apply to our case. More specifically, in order to prove Theorem 1.2, it suffices to prove the following result, whose proof we shortly outline after the statement for the reader’s convenience. Lemma 4.1 (Main Lemma). Let n > 1 and let C0 , C1 > 0 be some arbitrary con- stants. There exist M = M (C0 , C1 , n) > 0 big enough, λ(C0 , C1 , n) > 0 and ε = ε(C0 , C1 , M, n) > 0 small enough such that if δ = δ(M, C0 , C1 , n) > 0 is sufficiently small, then the following holds. Let μ be a Radon measure in Rn+1 with compact support and Q0 ⊂ Rn+1 a cube centered at the origin satisfying the properties: (1) (M Q0 ) ≤ λ. (2) μ(Q0 ) = (Q0 )n . (3) Pμ,α (M Q0 ) ≤ C0 . (4) For all x ∈ 2Q0 and 0 < r ≤ (Q0 ), Θμ (B(x, r)) ≤ C0 . (5) Q0 has C0 -thin boundary. (6) αμL (3M Q0 ) ≤ δ, for some hyperplane L through the origin. (7) Tμ|2Q0 is bounded on L2 (μ|2Q0 ) with Tμ|2Q0 L2 (μ|2Q0 )→L2 (μ|2Q0 ) ≤ C1 . (8) We have ˆ |T μ(x) − mμ,Q0 (T μ)|2 dμ(x) ≤ εμ(Q0 ). (4.1) Q0 Then there exists some constant τ > 0 and a uniformly n-rectifiable set Γ ⊂ Rn+1 such that μ(Q0 ∩ Γ) ≥ τ μ(Q0 ), (4.2) where the constant τ and the uniform rectifiability constants of Γ depend on all the constants above. 24 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 Sketch of the proof of Theorem 1.2 via the Main Lemma 4.1. Let σ be a Radon mea- sure on Rn and, given a cube Q ⊂ Rn , we denote n 1   ασR (Q) := inf dQ σ, cHn |Rn . (Q0 )n c≥0 Let D(Rn ) be the family of dyadic cubes in Rn . Let us assume that σ is a finite Radon  > 0 and 0 > 0 we have σ(Q) ≤ C(Q) measure such that, for some C  n for all Q ∈ D(Rn ) and (Q) ≥ 0 . A general geometric argument (see [21, Lemma 3.4]) gives  n  2 (R)n . ασR (3Q)2 (Q)n  C Q∈D(Rn ):Q⊂R,(Q)≥0 This formula and another purely geometric proof allow us, given B and μ as in the statement of Theorem 1.2, for all M  > 10 and δ  , ε > 0 to find a cube Q0 with thin boundary such that 3M  Q0 ⊂ B and dist(3M  Q0 , ∂B) ≥ C0 r(B) for C0 = C0 (C0 , n). The cube Q0 can be constructed in such a way that it satisfies μ(Q0 ) ≥ C0 (Q0 )n and, for δ small enough in the statement of the Main Theorem, αμ (3M  Q0 ) ≤ δ  . Furthermore, ˆ ˆ |T μ(x) − mμ,Q0 (T μ)| dμ(x) ≤ 2 2 |T μ(x) − mμ,B (T μ)|2 dμ(x) Q0 Q0 ≤ 2εμ(B) ≈C0 ,δ εμ(Q0 ). This gives the analogue of [21, Lemma 3.2]. Hence, it is not hard to prove that the n construction of Q0 implies that the measure μ  := (Q 0) μ(Q0 ) μ satisfies the hypotheses (1)- ...-(8) of Lemma 5.5. The details, which we omit for brevity, follow verbatim Section 3 of the paper of Girela-Sarrión and Tolsa, to which we also refer for the whole discussion of the construction sketched here.  The matrix A may have a very general form. In particular, we need some additional argument to overcome the lack of “symmetries” of the matrix with respect to reflections and to periodization (the exact meaning of this sentence will be clear after the reading of Section 5, where we recall how second order PDE’s in divergence form are affected by a change of variable). Indeed, this is a crucial point for our proof to work. A similar problem has been faced in [43]. First, in order to be able to argue via a change of variables, we have to show that we can assume the matrix A to be symmetric. We recall Schur’s test for integral operators with a reproducing kernel. The proof is a standard application of Cauchy-Schwarz’s inequality. Lemma 4.2. Let K : Rn+1 × Rn+1 → Rn+1 be a function such that, for a constant C > 0, we have ˆ |K(x, y)|dμ(x) ≤ C C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 25 and ˆ |K(x, y)|dμ(y) ≤ C. Then the operator T f = K ∗ f is a continuous operator from L2 (μ) to L2 (μ) and T L2 (μ)→L2 (μ) ≤ C. Let A be a matrix as before. We denote by As = (A + AT )/2 its symmetric part and by TμAs its correspondent gradient of the single layer potential. Lemma 4.3. Let Q be a cube in Rn+1 such that, for M > 1, Pμ,α (M Q) ≤ C1 . The (s) operator Tμ|2Q is bounded on L2 (μ|2Q) if and only if Tμ|2Q is bounded on L2 (μ|2Q). In particular    A  Tμ|  2 = Tμ| s  + O((Q)α ). (4.3) 2Q L (μ|2Q)→L2 (μ|2Q) 2Q L2 (μ|2Q)→L2 (μ|2Q) Moreover ˆ 2 T As μ(x)−mμ,Q (T As μ) dμ(x) Q ˆ 2  2 Λ, A α T μ(x) − mμ,Q (T μ) dμ(x) + M α (Q)α + M −α μ(Q). Q (4.4) Proof. Let us first prove (4.3). The identity (2.3) for matrices with constant coefficients leads to ˆ As Tμ| 2Q f (x) = ∇1 E As (x, y)f (y)dμ(y) 2Q ˆ   = ∇1 E As (x, y) − ∇1 Θ(x, y; As (x)) f (y)dμ(y) 2Q (4.5) ˆ ˆ   + ∇1 Θ(x, y; A(x)) − ∇1 E(x, y) f (y)dμ(y) + ∇1 E(x, y)f (y)dμ(y) 2Q 2Q ≡ I + II + Tμ|2Q f (x). To estimate I and II in (4.5) it suffices, then, to invoke Lemma 2.7 and Schur’s test. This finishes the proof of the first part of the lemma. 26 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 Let us now prove (4.4). We split T μ(x) − mμ,Q (T μ)       = T (χM Q μ)(x) − mμ,Q T (χM Q μ) + T (χ(M Q)c μ)(x) − mμ,Q T (χ(M Q)c μ) . (4.6) Let us estimate the two terms in the right hand side separately. Again, as a consequence of (4.5) and Lemma 2.2 we can write    T (χM Q μ) − mμ,Q (T (χM Q μ)) − T As (χM Q μ) + mμ,Q T As (χM Q μ)  M α (Q)α . To bound the second term in the right hand side of (4.6), notice that for x, y ∈ Q standard estimates together with Lemma 2.7 give ˆ |x − y|α Tμ χ(M Q)c (x) − Tμ χ(M Q)c (y)  dμ(z) |x − z|n+α (M Q)c |x − y|α 1 1  α Pμ,α (M Q)  α Pμ,α (M Q)  α , (M Q) M M thus, averaging over y in Q we have   T (χ(M Q)c μ)(x) − mμ,Q T (χ(M Q)c μ)  M −α The same calculations lead to   T As (χ(M Q)c μ)(x) − mμ,Q T As (χ(M Q)c μ)  M −α , so the inequality (4.4) in the statement of the lemma follows by gathering all the previous considerations.  As an immediate application of Lemma 4.3, we can assume in Lemma 4.1 (and hence in Theorem 1.2) that the matrix A is symmetric. In order to see this, let us suppose that we can construct a uniformly n-rectifiable set Γ as in the statement of the Main Lemma 4.1 if the conditions (7) and (8) hold for T As . More specifically, assume that 1 > 0 such that T As is bounded on L2 (μ|2Q ) with there exists C μ|2Q 0 0 As Tμ| L2 (μ|2Q0 )→L2 (μ|2Q0 ) 1 . ≤C 2Q 0 Then, by (4.3) and the condition (1) in the Main Lemma, we have that for λ small enough it holds Tμ|2Q0 L2 (μ|2Q0 )→L2 (μ|2Q0 ) 1 + λα/2 . ≤C C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 27 By (4.4), an analogous consideration holds for the assumption (8) in the Main Lemma. Hence, recalling that λ = λ(C0 , C 1 , n), Γ is the desired uniformly n-rectifiable that satisfies (4.2). Thus, in the rest of the paper we will understand that A is symmetric. Remark 4.1. Arguing as in Lemma 4.3, one could prove that    a  Tμ|  2 2 (μ| = Tμ|  + O((Q)α ), (4.7) 2Q L (μ| 2Q )→L 2Q ) 2Q L2 (μ| 2Q )→L 2 (μ| 2Q ) where T a is the operator corresponding to the antisymmetric part of the kernel K(·, ·), that is to say K a (x, y) = (K(x, y) − K(y, x))/2. Although for the purposes of many a technical parts of the paper it may look natural to work with Tμ| 2Q , we prefer not to make this reduction. Indeed, it would create problems at some crucial stages of the main variational argument. In particular, it would be an obstacle to the application of the maximum principle: it is not clear how to adapt the argument in (11.26) to show that the adjoint operator (T a )∗ (see (11.1) for the definition) solves a proper elliptic equation. We remark that an analogous problem had to be dealt in the papers [14] and [43], where Tμa was not used for the same reasons. Finally, since the inequality (4.7) is never used in the rest of the paper, we prefer to omit its proof. 5. The modification of the matrix 5.1. The change of variable The following lemma deals with how the fundamental solution and its gradient are affected by a change of variable. Lemma 5.1 (see [43], Lemma 5.2). Let φ : Rn+1 → Rn+1 be a locally bilipschitz map and let A ∈ Λα . Let E A be the fundamental solution of LA = − div(A∇·). Set Aφ := | det φ|D(φ−1 )(A ◦ φ)D(φ−1 )T . Then E Aφ (x, y) = E A (φ(x), φ(y)) for x, y ∈ Rn+1 , and ∇1 E Aφ (x, y) = D(φ)T (x) ∇1 E A (φ(x), φ(y)) for x ∈ Rn+1 . Let us state a lemma concerning how the gradient of the fundamental solution trans- forms under a change of variable φ as in Lemma 5.1. We use the notation ˆ Tφ μ(x) := ∇1 E Aφ (x, y)dμ(y). 28 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 Lemma 5.2 (see [43], Lemma 5.3). Let φ : Rn+1 → Rn+1 be a bilipschitz change of variables. For every x ∈ Rn+1 we have Tφ μ(x) = D(φ)T (x)T φμ(φ(x)). A particularly useful change of variable is the one that turns the symmetric part of the matrix at a given point into the identity. For the following statement we refer to [5]. Lemma 5.3. Let Ω ⊂ Rn+1 be an open set, and assume that A is a uniformly elliptic Let As = (A + AT )/2 be the symmetric part of A and for a fixed matrix with real entries.  point y0 ∈ Ω define S = As (y0 ). If  = S −1 (A ◦ S)(·)S −1 , A(·)  is uniformly elliptic, A then A s (z0 ) = Id if z0 = S −1 y0 and u is a weak solution of  = 0 in S −1 (Ω).  = u ◦ S is a weak solution of LA u LA u = 0 in Ω if and only if u We point out that the change of variables ϕ(x) := Sx that we defined in Lemma 5.3 is a linear map and, in particular, a bilipschitz map of Rn+1 to itself, namely there exists C ≥ 1 such that C −1 |x − y| ≤ |ϕ(x) − ϕ(y)| ≤ C|x − y| for x, y ∈ Rn+1 . The bilipschitz constant of ϕ depends on the ellipticity of the matrix A. We need to quantify flatness of images of cubes via maps of the aforementioned type. For a set E ⊂ Rn+1 , we define the α-number in an analogous ways as for cubes. In particular, for any hyperplane L and any measure ν, we denote 1 ανL (E) := inf dE (ν, c Hn |L ). diam(E)n+1 c≥0 This particular notation will be used only in this section. Lemma 5.4. Let ϕ be an affine, bilipschitz change of variables of Rn+1 with bilipschitz constant C ≥ 1. Let L be a hyperplane in Rn+1 . Let Jϕ > 0 be the Jacobian of ϕ. Then, for any Radon measure ν, for any cube Q ⊂ Rn+1 and any constant c ≥ 0 we have that   dQ (ν, c Hn |L ) ≈n,C dϕ(Q) ϕν, c Hn |ϕ(L) . (5.1) Hence, ϕ(L)  ανL (Q) ≈n,C αϕ ν (ϕ(Q) . (5.2) C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 29 Proof. Formula (5.2) is an immediate consequence of (5.1) and the fact that (Q) ≈C diam(ϕ(Q)). Let us prove (5.1). For every c ≥ 0   ϕ c Hn |L = c(ϕ Hn )|ϕ(L) . Indeed for any ϕ Hn |L -measurable set E we have     ϕ(c Hn |L )(E) = c Hn ϕ−1 (E) ∩ L = c Hn ϕ−1 (E ∩ ϕ(L)) = c(ϕ Hn )|ϕ(L) (E). Moreover, as a consequence of the Radon-Nikodym differentiation theorem (see [19, Lemma 1, p. 92]), we have   Hn ϕ−1 (E) = Jϕ Hn (E). So,     dQ (ν, c Hn |L ) ≈C dϕ(Q) ϕν, ϕc Hn |L ≈n,C dϕ(Q) ϕν, c Hn |ϕ(L) , which proves the lemma.  5.2. Reduction of the Main Lemma to the case A(0) = Id Recall that by Lemma 4.3 we can assume A to be a symmetric matrix. Let us begin with a preliminary observation. Let Q0 ⊂ Rn+1 be a cube as in the Main Lemma 4.1 and let us denote S := As (zQ0 )1/2 , where zQ0 is the center of Q0 . We choose the map ϕ so that ϕ(x) = Sx. By Lemma 5.3 we have that Aϕ (ϕ−1 (zQ0 )) = Id. Denoting ν = ϕ−1 μ, the change of variables and Lemma 5.2 give ˆ ˆ 1 1   mν,ϕ−1 (Q0 ) (Tϕ ν) = Tϕ ν d(ϕ−1 μ) = Tϕ ν ϕ−1 (x) dμ(x) ν(ϕ−1 (Q0 )) μ(Q0 ) ϕ−1 (Q0 ) Q0 ˆ 1 = S · T ν(x) dμ(x) = S · mμ,Q0 (T μ). μ(Q0 ) Q0 Thus, ˆ ˆ 2 2 T μ(x) − mμ,Q0 (T μ) dμ(x) = S · (Tϕ ν − mν,ϕ−1 (Q0 ) (Tϕ ν)) dν Q0 ϕ−1 (Q0 ) ˆ 2 ≈ Tϕ ν(x) − mν,ϕ−1 (Q0 ) (Tϕ ν) dν(x). ϕ−1 (Q0 ) 30 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 The implicit constant above depends only on ϕ and, hence, on the ellipticity of the matrix A. Analogously, it is not hard to see that we have Tϕ ν   ≈ Tμ   L2 ν|ϕ−1 (2Q L2 μ|(2Q0 ) . 0) Using these facts and Lemma 5.4, we have that the Main Lemma 4.1 can be restated as follows. Lemma 5.5. Let n > 1 and let C0 , C1 > 0 be some arbitrary constants. There exists M = M (C0 , C1 , n) > 1 big enough, λ(C0 , C1 , n) > 0 small enough and ε = ε(C0 , C1 , M, n) > 0 small enough such that if δ = δ(M, C0 , C1 , n) > 0 is small enough, then the following holds. Let μ be a Radon measure in Rn+1 , Q0 ⊂ Rn+1 a cube centered at the origin and ν := ϕ−1 μ, ϕ being as in the comments before the lemma, satisfying the following properties:   (1) Aϕ ϕ−1 (0) = Id. (2) (M Q0 ) ≤ λ.   (3) ν ϕ−1 (Q0 ) = (Q0 )n .   (4) Pν,α/2 ϕ−1 (M Q0 ) ≤ C0 . (5) For all x ∈ 2Q0 and 0 < r ≤ (Q),  Θμ (B(x, r)) ≤ C0 . (6) Q0 has C0 -thin boundary. ϕ−1 (H)  −1  (7) αν ϕ (3M Q0 ) ≤ δ, where H = {x ∈ Rn+1 : xn+1 = 0}. (8) Tϕ,ν|ϕ−1 (2Q ) is bounded on L2 (ν|ϕ−1 (2Q0 ) ) with 0   Tϕ,ν|  ≤ C1 . ϕ−1 (2Q0 ) L2 (ν|ϕ−1 (2Q )→L2 (ν|ϕ−1 (2Q ) 0) 0) (9) we have ˆ 2   Tϕ ν(x) − mν,ϕ−1 (Q0 ) (Tϕ ν) dν(x) ≤ εν ϕ−1 (Q0 ) . ϕ−1 (Q0 ) Then there exists some constant τ > 0 and a uniformly n-rectifiable set Γ ⊂ Rn+1 such that μ(Q0 ∩ Γ) ≥ τ μ(Q0 ), where the constant τ and the UR constants of Γ depend on all the constants above. The aim of most of the rest of the paper is to provide the proof of this result. In what follows, for the sake of simplicity of the notation, we will assume that A(0) = A(zQ0 ) = Id, which in particular gives that ϕ = Id, μ = ν and Tϕ,μ = Tμ . Indeed, if this is not the case, we should carry the following proofs for the image of cubes via ϕ−1 , C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 31 periodize with respect to the image of a lattice of standard cubes and work with Tϕ instead of T . This would be a merely notational complication that we prefer to avoid to make the arguments more accessible. Reduction to a periodic matrix. The forthcoming lemma shows, roughly speaking, that the local structure of A close to Q0 is what matters to the purposes of Lemma 4.1. An immediate consequence of this fact is that, without loss of generality, we can replace the matrix A with a periodic matrix, provided that the new matrix coincides with A in a suitable neighborhood of the cube Q0 . In what follows, we assume the matrix Ā to have Hölder continuous coefficients of exponent α  < α for technical reasons that will result clearer later on. Lemma 5.6. Let Ā ∈ Λα be such that Ā(x) = A(x) for every x ∈ 2Q0 . Let T̄ denote the gradient of the single layer potential associated with Ā. The operator Tμ|2Q0 is bounded in L2 (μ|2Q0 ) if and only if T̄μ|2Q0 is bounded in L2 (μ|2Q0 ) and   Tμ|2Q0 L2 (μ|2Q0 )→L2 (μ|2Q0 ) = T̄μ|2Q0 L2 (μ|2Q0 )→L2 (μ|2Q0 ) + O (Q0 )α . Moreover we have ˆ |T μ(x) − mμ,Q0 (T μ)|2 dμ(x) Q0 ˆ  2  |T̄ μ(x) − mμ,Q0 (T̄ μ)|2 dμ(x) + (M Q0 )2α + M −α μ(Q0 ), (5.3) Q0 where M is as in the statement of Lemma 4.1 and the implicit constant in (5.3) depends on diam(supp μ). The proof of Lemma 5.6 relies on the fact that Θ(·, ·; A(x)) = Θ(·, ·; Ā(x)) for every x ∈ 2Q0 and it is very similar to the one of Lemma 4.3, so that we omit it. In the rest of the paper, without additional specifications, we will deal with a matrix Ā periodic with period , 2(Q0 ) <   (Q0 ). The definition of the matrix Ā. The construction in the present subsection is dictated by the necessity of having an auxiliary matrix which agrees with A on 2Q0 and has the further properties of being periodic (which is crucial to use the estimates of the theory of homogenization) and of presenting ‘additional symmetries’ with respect to reflections (see the forthcoming Lemma 5.8). For a scheme of this construction we also refer to Fig. 1. Let ej denote the j-th element of the canonical basis of Rn+1 . We denote by ψj : Rn+1 → Rn+1 the map ψj (x) := x + (3(Q0 ) − 2xj )ej , (5.4) 32 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 Fig. 1. A schematization of the construction of Ā at the level of the periodic unit. which corresponds to the reflection across the hyperplane Pj orthogonal to ej and which passes through the point 32 (Q0 )ej . Let 0 < δ < 1/10. Given a matrix B(x) with variable coefficients, we define Bj as Bj = Bψj = D(ψj−1 )(B ◦ ψj )D(ψj−1 )T . (5.5)  as Moreover, we define the matrix B ⎧ ⎪ ⎪ B(x) ⎪ ⎪ ⎨ for dist x, ∂(3Q0 ) ≥ δ(Q0 ), ⎪  B(x) = dist(x,∂(3Q0 ))   (5.6) ⎪ ⎪ B(x) + 1 − dist(x,∂(3Q0 )) Id ⎪ ⎪ δ(Q0 ) δ(Q0 ) ⎩ for dist x, ∂(3Q ) < δ(Q ). ⎪ 0 0 It is also useful to introduce the notation  j (x) = B(x) for xj ≤ 32 (Q0 ), B (5.7) Bj (x) for xj > 32 (Q0 ). Let us apply the previous constructions to the matrix A. First, observe that the matrix  is continuous because Idj = Id and Âj is not necessarily continuous. However, (A) j  ∂(3Q ) ≡ Id. Our aim, now, is to define the final auxiliary matrix Ā by an iteration of A| 0 the construction in (5.7) along every direction and which is followed by a periodization. Before doing so, let us observe that for i, j ∈ {1, . . . , n + 1}, C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 33 i )j (x) = (A (A j )i (x), x ∈ Rn+1 .     This follows directly from (5.5) using the facts that ψi ψj (x) = ψj ψi (x) and that the matrices D(ψi−1 ), D(ψj−1 ) are diagonal. Thus by the linearity of the interpolation in (5.6) we have that         (A)   i j = (A)j i =: (A)i,j , (5.8) so the order of the modifications is not relevant. Let us now construct the matrix Ā in two steps: • For x belonging to the cube of side length 6(Q0 ) centered at the point with coordi- nates 32 (Q0 )(1, . . . , 1) we define  Ā(x) := (A)1,...,n+1 . • By (5.6), the matrix Ā defined in the first step coincide with Id for x belonging to the boundary of the cube with side length 6(Q0 ) and centered at 32 (Q0 )(1, . . . , 1). Hence, Ā admits a continuous and 6(Q0 )-periodic extension to Rn+1 so that   Ā(x) = Ā x + 6k(Q0 ) for every k ∈ Zn+1 . The following holds. Lemma 5.7. The matrix Ā is well-defined and periodic with period 6(Q0 ). Furthermore, for (Q0 ) small enough it is Hölder continuous with exponent α/2n+1 and constant not depending on (Q0 ). Proof. The well-definition of Ā follows from (5.8), and the periodicity holds by construc- tion. We are left with the proof of Hölder continuity. As a first step, we prove that there exists C > 0 such that |Ā(x) − Ā(y)| ≤ C|x − y|α/2 , for x, y ∈ 3Q0 . (5.9) By the construction of Ā, it suffices to verify the condition (5.9) for x, y ∈ {z ∈ 3Q0 : dist(z, ∂(3Q0 )) ≤ δ(Q0 )} with |x − y| ≤ δ(Q0 ). In this case, we denote t := dist(x, ∂(3Q0 )), s := dist(y, ∂(3Q0 )) and we have t  t  s  s  |Ā(x) − Ā(y)| = A(x) + 1− Id − A(y) + 1− Id δ(Q0 ) δ(Q0 ) δ(Q0 ) δ(Q0 ) 34 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 t s ≤ (A(x) − Id) − (A(y) − Id) δ(Q0 ) δ(Q0 ) t−s s ≤ |A(x) − Id| + |A(x) − A(y)| =: I + II. δ(Q0 ) δ(Q0 ) The α-Hölder continuity of A and the choice s ≤ δ(Q0 ) yield II  |x − y|α . On the other hand, the choice |x −y| ≤ δ(Q0 ), the identity A(0) = Id, and the Lipschitz character of the map dist(·, ∂(3Q0 )) give that, for (Q0 )α/2 /δ α ≤ 1 we have |t − s| |t − s|α |x − y|α (Q0 )α/2 I ≤ |x|α  |x|α α  α  |x − y|α/2 ≤ |x − y|α/2 . δ(Q0 ) (δ(Q0 )) δ δα Gathering the above estimates for I and II, we obtain (5.9). An analogous calculation shows that (5.9) holds for x, y ∈ 3Q0 ∪ (3Q0 + 3(Q0 )en+1 ) (see also [43, Lemma 8.1]). By periodicity of the matrix, this is enough to conclude the proof of the lemma. In particular, the exponent α/2n+1 is given by the fact that the proof has to be iterated for all the (n + 1) different directions.  We remark that, being Ā periodic, in the previous lemma there is no need to introduce a radial cut-off for the matrix as in [43]. For the rest of the paper we use the notation  := α/2n+1 . α Properties of EĀ . As a consequence of the definition of Ā and, more specifically, of its periodicity and the fact that by construction Āj (x) = Ā(x) for every x ∈ Rn+1 and j = 1, . . . , n + 1, we have the following. Lemma 5.8. E Ā (x, y) = E Ā (ψj (x), ψj (y)) for j = 1, . . . , n + 1 (5.10) and   E Ā (x, y) = E Ā x + 6k(Q0 ), y + 6k(Q0 ) for k ∈ Zn+1 . By Lemma 2.3, the function K̄ = ∇1 E Ā (·, ·) is (globally) a Calderón-Zygmund kernel. In particular (a) |K̄(x, y)|  |x − y|−n for all x, y ∈ Rn+1 with x = y. C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 35 (b) |K̄(x, y) − K̄(x, y  )| +|K̄(y, x) − K̄(y  , x)|  |y −y  |α |x −y|−n−α for 2|y −y  | ≤ |x −y|. Let T̄μ denote the singular integral operator associated with K̄, ˆ T̄μ f (x) = K̄(x, y)f (y)dμ(y). Lemma 5.6 tells that we can prove the Main Lemma for T̄ instead of T , possibly by slightly worsening the parameters involved. 6. A first localization lemma It is useful to provide a local analogue of the BMO-type estimate (4.1). This is possible because of the smallness of the α-number and the bound for the Pμ,α -density. Also, recall that because of the assumptions in Lemma 4.1, we have μ(M Q0 )  M n μ(Q0 ). In what follows we sketch the proof of the localization of (4.1) for T̄μ , highlighting the differences with respect to the case of the Riesz transform (see [21, Lemma 4.2]). In the rest of the paper we omit to indicate the dependence of the implicit constants in our estimates on C0 and C1 . Lemma 6.1. For δ small enough depending on M , the following inequality holds ˆ   1 |T̄μ χM Q0 |2 dμ  ε + 2α + M 4n+2 δ 1/(4n+4) + (M (Q0 ))2α μ(Q0 ). (6.1) M Q0 Proof. First, observe that ˆ |T̄μ (χM Q0 )|2 dμ Q0 ˆ ≤2 |T̄μ (χM Q0 ) − mμ,Q0 (T̄μ χM Q0 )|2 dμ + 2|mμ,Q0 (T̄μ χM Q0 )|2 μ(Q0 ). (6.2) Q0 Let us estimate the two summands on the right hand side of (6.2) separately. To study the first one, we write ˆ 2 T̄μ χM Q0 − mμ,Q0 (T̄μ χM Q0 ) dμ Q0 ˆ ˆ 2 ≤2 T̄μ χ(M Q0 )c (x) − mμ,Q0 (T̄μ χ(M Q0 )c ) dμ(x) + 2 |T̄ μ − mμ,Q0 (T̄ μ)|2 dμ. Q0 Q0 (6.3) Applying Lemma 2.1, it follows that for x, y ∈ Q0 36 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 ˆ |T̄μ χ(M Q0 )c (x) − T̄μ χ(M Q0 )c (y)| ≤ |K̄(x, z) − K̄(y, z)|dμ(z) (M Q0 )c ˆ 1  |x − y|α dμ(z) |x − z|n+α (M Q0 )c ∞  ˆ 1 |x − y|α 1  |x − y|α  dμ(z)   Pμ,α (M Q0 )  α , j=1 j+1 |x − z|n+α (M Q0 ) α M 2 M Q0 \2j M Q0 being Pμ,α (M Q0 )  1. Then, averaging the previous inequality over the variable y, we get 1 T̄μ χ(M Q0 )c (x) − mμ,Q0 (T̄μ χ(M Q0 )c )  M α and ˆ 1 |T̄μ χ(M Q0 )c (x) − mμ,Q0 (T̄μ χ(M Q0 )c )|2 dμ(x)  μ(Q0 ). M 2α Q0 Recalling that by hypothesis we have ˆ 2 T̄ μ − mμ,Q0 (T̄ μ) dμ ≤ εμ(Q0 ), Q0 we can estimate (6.3) as ˆ  2 1  T̄μ (χ(M Q0 )c ) − mμ,Q0 (T̄μ χM Q0 ) dμ  ε + μ(Q0 ). (6.4) M 2α Q0 An application of Lemma 2.7 together with the antisymmetry of ∇1 Θ(·, ·; Ā(x)) also gives ˆ ˆ 1 mμ,Q0 (T̄μ χQ0 )  |x − y|−n+α dμ(x)dμ(y)  (Q0 )α . (6.5) μ(Q0 ) Q0 Q0 Minor variations of the arguments which prove [21, (4.2)] show that (6.5) |mμ,Q0 (T̄μ χM Q0 )|  |mμ,Q0 (T̄μ χM Q0 \Q0 )| + (Q0 )α  α (6.6)  M 2n+1 δ 1/8(n+1) + M (Q0 ) + (Q0 )α  α  M 2n+1 δ 1/8(n+1) + M (Q0 ) . C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 37 For the sake of brevity we omit the details and we just point out that the presence of the second summand on the right hand side comes from the estimate ˆ    α T̄ ϕHn |H dHn |H  M (Q0 ) (Q0 )n , (6.7) Q0 where ϕ is a proper even C 1 function with 0 ≤ ϕ ≤ 1 and supported on M Q0 \ Q0 . To get the estimate (6.7), we just write ˆ   T̄ ϕHn |H dHn |H Q0 ˆ ˆ 1 1 ≤ K̄(x, y) − ∇1 Θ(x, y; Ā(x)) dHn |H (x)dHn |H (y) 2 2 Q0 M Q 0 ˆ ˆ 1 1 + K̄(x, y) − ∇1 Θ(x, y; Ā(y)) dHn |H (x)dHn |H (y) 2 2 Q0 M Q0 ˆ ˆ 1   + ∇1 Θ(x, y; Ā(x)) + ∇1 Θ(x, y; Ā(y)) dHn |H (x)dHn |H (y) . 2 Q0 M Q0 Then, the third summand is null because of the antisymmetry of its integrand and the first two terms can be estimated via Lemma 2.2. Gathering (6.2), (6.4) and (6.6) we are able to conclude the proof of the lemma.  7. The David and Mattila lattice associated with μ and its properties The dyadic lattice constructed by David and Mattila [16, Theorem 3.2] is a powerful tool in the study of the geometry of Radon measures. Its main properties are listed in the following lemma, that we state for a general Radon measure with compact support. Lemma 7.1 (David and Mattila). Let σ be a compactly supported Radon measure in Rn+1 . Consider two constants K0 > 1 and A0 > 5000K0 and denote W = supp σ. Then there exists a sequence of partitions of W into Borel subsets Q, Q ∈ Dσ,k , which we will refer to as cells, with the following properties: • For each integer k ≥ 0, W is the disjoint union of the cells Q, Q ∈ Dσ,k . If k < l, Q ∈ Dσ,l , and R ∈ Dσ,k , then either Q ∩ R = ∅ or Q ⊂ R.   • For each k ≥ 0 and each cell Q ∈ Dσ,k , there is a ball B(Q) = B zQ , r(Q) such that zQ ∈ W, A−k −k 0 ≤ r(Q) ≤ K0 A0   W ∩ B(Q) ⊂ Q ⊂ W ∩28B(Q) = W ∩ B zQ , 28r(Q) , 38 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 and the balls 5B(Q), Q ∈ Dσ,k are disjoint. • The cells Q ∈ Dσ,k have small boundaries. By this, we mean that for each Q ∈ Dσ,k and each integer l ≥ 0, if we set Nlint := x ∈ Q : dist(x, W \ Q) < A−k−l 0 Nlext (Q) := x ∈ W \ Q : dist(x, Q) < A−k−l 0 and Nl (Q) := Nlint (Q) ∪ Nlext (Q), we get    −3(n+1)−1 −l   σ Nl (Q) ≤ C −1 K0 A0 σ 90B(Q) • Denote by Ddb σ,k the family of cells Q ∈ D σ,k for which     σ 100B(Q) ≤ K0 σ B(Q) . We have that r(Q) = A−k 0 when Q ∈ Dσ,k \ Ddb σ,k and     σ 100B(Q) ≤ K0−1 σ 100l+1 B(Q) (7.1) for all l ≥ 1 with 100l ≤ K0 and Q ∈ Dσ,k \ Ddb σ,k .  Let us denote Dσ := k Dσ,k . Let us choose A0 big enough so that −3(n+1)−1 C −1 K0 1/2 A 0 > A0 > 10. (7.2) Here we list some useful quantities associated with each cell Q ∈ Dσ,k : • J(Q) := k, which may be interpreted as the generation of Q. • (Q) := 56K0 A−k 0 , that we also call side length. Notice that 1 −1 K (Q) ≤ diam(28B(Q)) ≤ (Q) 28 0 and r(Q) ≈ diam(Q) ≈ (Q). • calling zQ the center of Q, we denote BQ := 28B(Q) = B(zQ , 28r(Q)), which in particular gives 1 Q∩ BQ ⊂ Q ⊂ BQ . 28 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 39 We recall, now, some of the properties of the cells in the David and Mattila lattice. The choice in (7.2) implies, for 0 < λ ≤ 1, the estimate     σ {x ∈ Q : dist(x, W \ Q) ≤ λ(Q)} + σ {x ∈ 3.5BQ \ Q : dist(x, Q) ≤ λ(Q)} ≤ cλ1/2 σ(3.5BQ ).  We denote Ddb σ := k≥0 D σ,k and we say that it is the lattice of doubling cells. This db notation is justified by the fact that, for Q ∈ Ddb σ , we have σ(3.5BQ ) ≤ σ(100B(Q)) ≤ K0 σ(B(Q)) ≤ K0 σ(Q). An important feature of the David and Mattila lattice is that every cell Q ∈ Dσ can be covered by doubling cells up to a set of σ-measure zero ([16, Lemma 5.28]). Moreover, if we have two cells R, Q ∈ Dσ with Q ⊂ R and such that every intermediate cell Q  S  R belongs to Dσ \ Ddbσ , we have the control −10n(J(Q)−J(R)−1) σ(100B(Q)) ≤ A0 σ(100B(R)) (7.3) on the decay of the measure. The estimate (7.3) is proved via an iterated application of the inequality σ(100B(Q)) ≤ A−10n 0 σ(100B(Q̂)), (7.4) where Q̂ is the cell from Dσ,J(Q)−1 containing Q (also called parent of Q). We remark that (7.4) follows by (7.1) and a proper choice of A0 and K0 (see [16, Lemma 5.31]). For Q ∈ Dσ , we denote by Dσ (Q) the cells in Dσ which are contained in Q and Dσ (Q) := Dσ (Q) ∩ Ddb db σ . 8. The Key Lemma, the stopping time condition and a first modification of the measure The heart of the proof of Lemma 5.5 consists in providing a control on the abundance of cells with low density (in some sense that we clarify below). The whole construction that we are about to discuss depends on some auxiliary parameter to be chosen properly later in the proof. Definition 8.1 (Low density cells). Let 0 < θ0  1. A cell Q ∈ Dμ is said to be of low density if Θμ (3.5BQ ) ≤ θ0 and it is maximal with respect to the set inclusion. We denote by LD the family of low density cells. 40 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 Most of the rest of the paper deals with the proof of the fact that the low density cells fail to cover a significant portion of Q0 . Lemma 8.1 (Key Lemma). Let ε, δ and M be as in Lemma 4.1. There exists ε0 > 0 such that if M is big enough and θ0 , δ and ε are small enough, then    μ Q0 \ Q ≥ ε0 μ(Q0 ). (8.1) Q∈LD Sketch of the proof of the Main Lemma 5.5 using the Key Lemma (8.1). The proof is analogous to the argument of [21, Section 10]. For the reader’s convenience, we briefly outline the construction of the uniformly n-rectifiable set Γ in the statement of the  Main Lemma. Let us define F := Q0 ∩ supp μ \ Q∈LD Q, the Key Lemma 8.1 gives μ(F ) ≥ ε0 μ(Q0 ). Let σ := μ|Q0 , let Dσ be its associated David-Mattila lattice, and let us define the maximal dyadic operator ˆ 1 MDσ f (x) := sup |f |dσ. Q∈Dσ :x∈Q σ(Q) Q Let F := {x ∈ F : MDσ (χF c )(x) ≤ 1 − (ε0 /2)}. It is not hard to prove that σ(F) ≥ σ(F )/2. We consider a family {Qj }j∈J ⊂ Dσ of maximal cells (with respect to the   inclusion) such that σ(Qj \ F ) > 1 − (ε0 /2) σ(Qj ). For j ∈ J, we denote as Aj the family of maximal doubling cells that cover Qj , and by A0 the family {P ∈ ∪j∈J Aj : σ(P ∩ F ) > 0}. We can now define the main auxiliary measure  ζ = σ|F + Hn |S(Q) , Q∈A0 where S(Q) indicates an n-dimensional sphere with the same center as B(Q) and ra- dius B(Q)/4. Lemma 10.2 in [21] gives that the measure ζ is AD-regular with constant depending only on C0 , θ0 and ε0 . Furthermore, it is routine to check that the proof of [21, Lemma 10.3] yields that T̄ζ is bounded on L2 (ζ) with norm depending on the same parameters and C1 . Hence, we can apply [43, Theorem 1.1], which gives that ζ is uni- formly n-rectifiable. By construction and using the fact that σ(F) ≥ σ(F )/2, Γ := supp ζ satisfies μ(Γ) ≥ μ(F) = σ(F ) ≥ ε0 μ(Q0 )/2, which proves the Main Lemma 5.5.  The rest of the present article (a part from the last section) is devoted to the proof of Lemma 8.1. C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 41 We argue by contradiction: assume that (8.1) does not hold, that is to say    μ Q > (1 − ε0 )μ(Q0 ). (8.2) Q∈LD More specifically, we want to show that a choice of ε0 small enough leads to an absurd. The proof is based on a stopping time argument. Roughly speaking, for Q ∈ LD, we say that a cell R belongs to its associated stopping family if it is a descendant of Q (i.e. R ⊂ Q) and it is sufficiently small. The definition of stopping cells depends on a parameter t, which has to be thought small and that will be appropriately chosen later. Definition 8.2 (Stopping cells). Let Q ∈ LD. Let 0 < t < 1. We say that R ∈ Stop(Q) if the following two conditions are verified and they are maximal with respect to contain- ment: • R ∈ Ddb μ , R ⊂ Q. • (R) ≤ t(Q).  We also denote Stop := Q∈LD Stop(Q) the family of all the stopping cells. Assuming that the stopping cells in Stop(Q) are doubling makes sense in light of the fact that doubling cells cover Q up to a set of μ-measure zero. In particular, this implies that (8.2) is equivalent to    μ Q > (1 − ε0 )μ(Q0 ). Q∈Stop The proof of the Key Lemma 8.1 involves a periodization of the measure μ, which is essentially carried out by replicating μ|Q0 on the horizontal plane according to the periodicity of the matrix Ā. The stopping cells which are close to the boundary of Q0 (in a proper sense) are problematic for our analysis, so we need to rule them out of the construction. In order to do this, we have to show that their contribution to the measure of Q0 is negligible (see (8.4)). We say that P ∈ Bad if P ∈ Stop and 1.1BP ∩ ∂Q0 = ∅. Another technical problem is that Stop may contain infinitely many cells. This second difficulty can be easily overcome considering a finite family of cells, named Stop0 , which contains a big portion of the measure of Stop, e.g.    μ Q > (1 − 2ε0 )μ(Q0 ). (8.3) Q∈Stop0 The rest of the section is devoted to a justification of the last affirmations concerning Bad and the first modification of the measure μ. It is essentially a rewriting of [21, Lemma 42 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 6.2, Lemma 6.3, Lemma 6.4] in our context, in which we highlight the right homogeneity coming from our elliptic setting. The following lemma contains an estimate of the density Pμ,α of the stopping cells in terms of the low density parameter θ0 .  1/(n+α) Lemma 8.2. Let Q ∈ Stop and let t = θ0 . We have  α Θμ (2BQ ) ≤ Pμ,α (2BQ )  θ0n+α . Proof. The first inequality is an immediate consequence of the definition of Pμ,α . To prove the second inequality, we consider the maximal cell R ∈ Dμ such that Q ⊂ R ⊂ R and (R ) ≤ t(R) and write   (Q) α   (Q) α Pμ,α (2BQ )  Θμ (2BP ) + Θμ (2BP )  (P ) (P ) P ∈D σ :Q⊂P ⊂R P ∈D σ :R ⊂P ⊂R   (Q) α  + Θμ (2BP ) + 2−kα Θμ (2k BQ ) (P ) P ∈D σ :R⊂P ⊂Q0 k≥1 =I + II + III + IV. Then, the estimates work as in the case of the work of Girela-Sarrión and Tolsa. In particular, the calculations for [21, Lemma 6.2, estimate of S1 + S2 ] work verbatim and give θ0 I + II  tn Furthermore [21, Lemma 6.2, estimate of S3 + S4 ] gives III + IV  tα , which justifies the choice of t in the statement of the lemma.   1/(n+α) For the rest of the paper we assume t = θ0 in Definition 8.2. Using the estimates in Lemma 8.2, one can prove that    α μ Q  θ0n+α μ(Q0 ). (8.4) Bad Its proof (see [21, Lemma 6.3] for all the details) relies on a covering argument: let I be an arbitrary finite subset of Bad. An application of Vitali’s covering theorem to the family {1.15BQ }Q∈I gives that there exists a family J ⊂ I such that the cells {1.15BQ }Q∈J are   pairwise disjoint and Q∈J 3(1.15BQ ) covers Q∈I 1.15BQ . Furthermore Q ∈ J ⊂ Bad implies that C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 43 Hn (1.15BQ ∩ ∂Q0 )  r(BQ )n . Lemma 8.2 and the above estimate give       α   α  α μ BP ≤ μ 3.45BP  θ0n+α r(BQ )n  θ0n+α Hn (∂Q0 ) ≈ θ0n+α μ(Q0 ), P ∈I P ∈J Q∈J which readily implies (8.4). First modification of the measure. As already mentioned, for technical purposes it is useful to modify the measure inside Q0 by taking just finitely many stopping cells and getting rid of the cells in Bad. To make the previous statement rigorous, we choose a small parameter 0 < κ0  1 to be fixed later and, after denoting Iκ0 (Q) := {x ∈ Q : dist(x, supp σ \ Q) ≥ κ0 (Q)}, we define the modified measure  μ0 := μ|Qc0 + μ|Iκ0 (Q) . Q∈Stop0 \ Bad Using (8.3) and (8.4), it is not difficult to prove that μ0 differs from μ, in the sense of the total mass, possibly by a very small quantity. Indeed,    α/(n+  α) 1/2 μ − μ0 ≤ 2ε0 + Cθ0 + κ0 μ(Q0 ). (8.5) For this modification to be valid for our purposes, we need the gradient of the single layer potential associated with this measure to satisfy a localization estimate analogous to (6.1). This is easily proved by gathering the L2 (μ|Q0 )-boundedness of T̄μ|Q0 , the estimate (8.5) and the localization estimate (6.1) for μ (see [21, Lemma 6.4]). Lemma 8.3. If δ is chosen small enough (depending on M ), then ˆ |T̄ (χM Q0 μ0 )|2 dμ0 Q0  1    α/(n+  α) 1/2  ε+ + M 4n+2 1/(4n+4) δ + (M (Q0 ))2α + ε 0 + θ + κ μ(Q0 ). M 2α 0 0 9. Periodization and smoothing of the measure The present technical section contains the construction of a suitable auxiliary mea- sure. This is done in two steps: first we need to get rid of the truncation at the level of M (Q0 ) present in the estimate of Lemma 8.3. For this purpose, we replicate the mea- sure periodically by means of horizontal translations. The localization of the gradient 44 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 of the single layer potential associated with this auxiliary measure will make us able to implement a variational argument in Section 11. We remark that a similar localization problem had to be dealt with for the horizontal component of the Riesz transform in [40]. In that case, it was possible to argue introducing an auxiliary measure defined by reflections with respect to the horizontal hyperplane. However, it is not know how to adapt that construction when having to estimate all the components of the transform, so Girela-Sarrión and Tolsa introduced the periodization of the measure. The purpose of the first part of this section is to show how to adapt their construction to our context. Moreover, a priori, the measure μ0 may not be absolutely continuous with respect to the Lebesgue measure on Rn+1 . This would constitute a problem when trying to implement the variational techniques in Lemma 11.1, as the existence of a minimizer of the functional here introduced relies on the boundedness of the density with respect to the measure Ln+1 . This issue can be overcome by introducing a smoothing of the measure. For this reason, we define a second auxiliary measure in (9.6). The periodization. We denote by M := Q0 + zP : zP ∈ 6(Q0 )Zn × {0} the family of disjoint cubes covering H and obtained translating Q0 along the coordinate (horizontal) axes. The factor 6 is chosen in order for this periodization to be coherent with the period of the matrix Ā. Given P ∈ M we denote by zP its center and by TP : Rn+1 → Rn+1 the translation TP (x) := x + zP , so that the periodization of the measure reads   := μ TP μ0 |Q0 . P ∈M  = μ0 . As for the first modification of the Observe that μ0 (∂Q0 ) = 0, which implies χQ0 μ measure, we have to prove the equivalent of the localization Lemma 8.3. Lemma 9.1. Let κ0 , θ0 and ε0 be as in Section 8 and δ as in the Main Lemma. Letting   δ := M n+1 ε0 + θ0  α/(n+  α) 1/2 + κ0 + δ 1/2 , we have   (3M Q0 )  δ. αμH (9.1) Moreover, for C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 45 1  α/(n+  α) ε := ε + + M 4n+2 δ 1/(4n+4) + ε0 + θ0 M 2α + κ + M 2n+2 δ1/(4n+5) + (M (Q0 ))2α 1/2 0 we have ˆ 2 ) d T̄ (χM Q0 μ μ  εμ (Q0 ). (9.2) Q0 Idea of the proof. The geometric inequality (9.1) was proved in [21, Lemma 7.1]. An estimate analogous to (9.2) was proved in [21, Lemma 7.2]. In order to prove it, one  yields μ first observes that the construction of the measure μ |Q0 = μ0 |Q0 , which in turn implies ˆ ˆ  2  2  d T̄ χM Q0 μ μ=  dμ0 T̄ χM Q0 μ Q0 Q0 ˆ ˆ  2  2  T̄ χM Q0 μ0 dμ+ μ − μ0 ) d T̄ χM Q0 ( μ =: I + II Q0 Q0 The first summand can be readily estimated via Lemma 8.3. In order to bound the term 0 |Q0 = μ II let us observe that since, by construction, μ |Q0 and μ0 |(Q0 )c = μ |(Q0 )c , we can write     μ − μ0 ) = T̄ χM Q0 \Q0 ( T̄ χM Q0 ( μ − μ) . Thus, the integral II can be estimated by an approximation of the characteristic function χM Q0 \Q0 via a C 1 cut-off which is compactly supported in M Q0 \ Q0 and using the  at the level of 3M Q0 . We omit further details.  flatness of μ  has polynomial growth: It is not difficult to see that the measure μ (B(x, r))  rn μ for every x ∈ Rn+1 and r > 0. The following lemma contains a technical estimate for a suitably modified version of the density Pμ, α  (2BQ ). Lemma 9.2. For every Q ∈ Stop0 \ Bad the inequality ˆ ˆ  1 α d μ(y)  θ0(n+α)(1+2n) μ(x)d  (Q) μ (9.3) |x − y|n 1.1BQ \Q Q holds. Moreover, the function 46 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376  pμ, α  (x) := χQ Pμ, α  (2BQ ) Q∈Stop0 \ Bad satisfies ˆ  2α p2μ, α μ  θ0(n+α)(1+2  d   α) (Q0 ). μ (9.4) Q0 Remark on the proof. For (9.3) we refer to [21, Lemma 7.3]. In order to prove (9.4) it suffices to follow verbatim the path of (geometric) [21, Lemma 7.4] taking into consid- , which leads to eration the right homogeneity given by α ˆ   2α    θ (n+α) α p2μ, α  d μ  κ̄ + 0 2α + θ0n+α μ (Q0 ), (9.5) κ̄ Q0 where 0 < κ̄ < 1 is a small constant. Inequality (9.5) gives the desired estimate after  2α/[(n+  α)(1+2  α)] making the choice κ̄ = θ0 .  The smoothing. Let us define  μ0 (Q) η0 :=   Hn+1 | 1 B(Q) (9.6) Hn+1 14 B(Q) 4 Q∈Stop0 \ Bad and its periodization  η := TP η0 . P ∈M We remark that, since Stop0 is a finite family, the measures η0 and η both have bounded density with respect to Hn+1 . As specific control on the density is not relevant to the purposes of our proof. The following lemma contains a localization estimate for the potential associated with η. Lemma 9.3. Denoting  2α  2α ε := ε + (Q0 )2α + M n κ−2n−2 0 α (n+α)(1+2 θ0   α)  + θ0(n+α)(1+2n) , we have ˆ |T̄ (χM Q0 η)|2 dη  ε η(Q0 ). Q0 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 47 The presence of the summand (Q0 )2α in ε (already taken into account in ε) to point out that, as in (6.6), the lack of antisymmetry of K̄(·, ·) gives the error term   mμ,Q  T̄μ χQ  (Q)α  (Q0 )α for every Q ∈ Stop0 \ Bad. This contribution is not present in the case of an elliptic matrix with constant coefficients. The rest of the proof is analogous to the one of [21, Lemma 8.1] and all is needed is a careful check that Lemma 9.2 applies and the new homogeneity does not affect the final result. We omit further details. Remark 9.1. Observe that the expressions of δ,  ε and ε all include a summand which depends on ε0 . In particular, the quantities in question are small if ε0 and M (Q0 ) are chosen small enough. Then the choice of ε0  1 (which is possible because we assumed (8.2) to hold) gives the localization for the potentials associated with the auxiliary mea- sures. 10. The localization of T̄ η Let L∞ ∞ M denote the set of functions f ∈ L (η) such that f (x + zP ) = f (x) for every x ∈ Rn+1 and P ∈ M. Let ϕ ∈ C 1 (Rn+1 ) be a non-negative radial function whose support is contained in B(0, 2) and that equals 1 on B(0, 1). For r > 0 and x ∈ Rn+1 let us set ϕr (x) := ϕ(x/r). Observe that ∇ϕ ∞  1. For x, y ∈ Rn+1 we define the regularized kernel  r (x, y) = K̄(x, y)ϕr (x − y) K and its associated operator ˆ Tr (f η)(x) :=  r (x, y)f (y)dη(y) K for f ∈ L∞ M (η), where the integral above is absolutely convergent. We are interested in getting an existence result for the limit pv T̄ (f η)(x) = lim Tr (f η)(x). (10.1) r→∞ For simplicity, we denote the principal value in (10.1) just as T̄ (f η)(x). Lemma 10.1. Let f ∈ L∞ M . The principal value T̄ (f η)(x) exists for every x ∈ R n+1 . Moreover, given any compact set F ⊂ R n+1 , there exist r0 = r0 (F ) > 0 and a constant cF depending on F such that for s > r ≥ r0 48 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376   cF Tr (f η) − Ts (f η)  f ∞, ∞,F rγ where γ ∈ (0, 1) is as in Lemma 2.7. Remark 10.1. Lemma 10.1 implies that the limit in (10.1) converges uniformly on com- pact sets and in supp η. Proof. Recall that we can assume (Q0 ) < 1. Let s > r. Let us denote ν := f η, ϕr,s (x) := ϕr (x) − ϕs (x) for every x ∈ Rn+1 and K  r,s (x, y) := K̄(x, y)ϕr,s (x − y). Because of the periodicity of f and the definition of η, we have  ν= (TP ) (χQ0 ν) P ∈M so that ˆ    Tr (f η)(x) − Ts (f η)(x) =  r,s (x, y)d K (TP ) (χQ0 ν) (y) P ∈M  ˆ (10.2) =  r,s (x, y + zp ) dν(y), K P ∈MQ 0 the last equality being a consequence of K r,s having compact support, which implies that the sum has only finitely many non-zero terms. Let A0 be the homogenized matrix associated with {Lε }ε > 0 and χ be as in Section 2, with  = 6(Q0 ). Recall that ∇χ ∞ 1 (see (2.9)). The matrix A0 is an elliptic matrix whose coefficients are constants and can be expressed in terms of χ and those of A. We denote by Θ(·, ·; A0 ) the fundamental solution of the operator L0 = − div(A0 ∇). We decompose the right hand side of (10.2) as  ˆ K r,s (x, y + zp )dν(y) P ∈MQ 0  ˆ   = K̄(x, y + zP ) − (Id + ∇χ (x))∇1 Θ(x, y + zP ; A0 ) ϕr,s (x − y − zP )dν(y) P ∈MQ 0  ˆ   + Id + ∇χ (x) ∇1 Θ(x, y + zP ; A0 )ϕr,s (x − y − zP )dν(y) P ∈MQ 0 ≡ Ir,s (x) + IIr,s (x). C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 49 Let us observe that since F is compact and y ∈ Q0 , there exists a compact set F such   that ±(x − y) ∈ F, so that if we choose r0 ≥ 2 diam F , both ϕr,s (x − y − zP ) and   ϕr,s (x − y + zP ) vanish for |zP | < r. Moreover, |x − y| ≤ diam F ≤ r/2 ≤ |zP | and |(x − y) − zP | ≈ |(x − y) + zP | ≈ |zP |. Let us now estimate Ir,s (x). As stated in Lemma 2.7, there exist C > 0 and γ ∈ (0, 1) depending only on n and α such that   ∇1 E Ā (x, y + zP ) − Id + ∇χ (x) ∇1 Θ(x, y + zP ; A0 ) ≤ C(Q0 )γ |x − y − zP |−(n+γ) for every x, y ∈ Rn+1 . Then, exploiting the linear growth of η and the considerations on the support of ϕr,s , we get  ˆ  (Q0 )γ d|ν|(y) (P )n+γ |Ir,s (x)|   f ∞ |x − y − zP |n+γ |zP |n+γ P ∈M,|zP |≥rQ P ∈M,|zP |≥r 0 γ f ∞ (Q0 )  γ . (10.3) r  In the last inequality of (10.3) we used the convergence of P ∈M (P )n |zP |−n . We are left with the estimate of IIr,s (x). First, we observe that there exists a compact set F such that both (x − y) and −(x − y) belong to F for all x ∈ F and y ∈ Q0 . Furthermore, P ∈ M if and only if −P ∈ M, by [21, Lemma 8.2, p. 41] we have ∇Θ(x − y − zP ; A0 )ϕr,s (x − y − zP ) − ∇Θ(x − y + zP ; A0 )ϕr,s (x − y + zP ) |x − y| diam (F )  ≤ , (10.4) |zP |n+1 |zP |n+1 for all P ∈ M with |zP | ≥ r. Moreover, the quantity in the left-hand side of the display above vanishes for |zP | < r. Hence, using the antisymmetry of ∇Θ(·; A0 ) and the facts highlighted above, there exists a constant cF > 0 such that  ˆ |IIr,s (x)| ≤ Id + ∇χ ∞ ∇1 Θ(x, y + zP ; A0 )ϕr,s (x − y − zP )dν(y) P ∈MQ 0  ˆ  ∇1 Θ(x, y + zP ; A0 )ϕr,s (x − y − zP )dν(y) (2.9) P ∈M Q 0 ˆ 1  ≤ ∇Θ(x − y − zP ; A0 )ϕr,s (x − y − zP ) 2 P ∈MQ 0 − ∇Θ(x − y + zP ; A0 )ϕr,s (x − y + zP ) d|ν|(y) 50 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376  ˆ |x − y|  d|ν|(y) (10.4) P ∈M:|z |≥r |zP |n+1 P Q 0  |ν|(Q0 ) cF f  diam(F) ∞ ≤ . (10.5) |zP |n+1 r P ∈M:|zP |≥r We conclude the proof of the lemma gathering (10.3), (10.5) and observing that, being γ ∈ (0, 1) and r > 1, r−1 < r−γ .  The measure η is M-periodic and the matrix Ā, by construction, is 6(Q0 )-periodic. This implies that for every f ∈ L∞  M (η) and r > 0, the function Tr (f η) is M-periodic, too. The same holds for pv T (f η). Using Lemma 10.1, the following result is immediate. Corollary 10.1. T̄η is a bounded operator from L∞ ∞ M to LM . For r > 0 big enough and for ∞ every f ∈ LM (η) we have   f ∞ T̄ (f η) − Tr (f η) F . ∞,F rγ Our next intent is to prove the final localization estimate ˆ 2 T̄ η dη  η(Q0 ). (10.6) Q0 We have already proved that for M big enough there exists ε  1 such that ˆ 2 T̄ (χM Q0 η) dη  ε η(Q0 ). (10.7) Q0 Then, in order to prove (10.6), it suffices to use the estimate in the following lemma.  = 6N Lemma 10.2. Let f ∈ L1loc (η) be a M-periodic function and let M  , where N  ≥3 is an odd number. For all x ∈ 2Q0 we have ˆ   1 Q0 )c f η (x)  T̄ χ(M |f |dη. (10.8) γ (Q0 )n M Q0  odd, there exists a subfamily M Proof. Being N  ⊂ M such that  χ( M  Q0 ) c η = TP η  P ∈M  satisfy |zP |  M and whose elements P ∈ M (Q0 ). In particular C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 51 |x − y − zP | ≈ |zp | for x, y ∈ 2Q0 . (10.9) Let r > 0 and x ∈ 2Q0 . Denote ν := f η and observe that there are just finitely many cubes P ∈ M  such that |zP | < r. Arguing as in the proof of Lemma 10.1, this justifies the following writings. ˆ    T r χ( M Q0 )c f η (x) = K̄(x, y)ϕr (x − y)dν(y)  ˆ = K̄(x, y + zP )ϕr (x − y − zP )dν(y) Q P ∈M 0  ˆ   = K̄(x, y + zP ) − (Id + ∇χ (x))∇1 Θ(x, y + zP ; A0 ) ϕr (x − y − zP )dν(y) Q P ∈M 0  ˆ   + Id + ∇χ (x) ∇1 Θ(x, y + zP ; A0 )ϕr (x − y − zP )dν(y) Q P ∈M 0 ≡ Ir (x) + IIr (x) Let us estimate Ir (x). Using (10.9) together with Lemma 2.7 and the estimate |zP |  (Q0 ) for P ∈ M, M  we can write  ˆ (Q0 )γ  ˆ (Q0 )γ |Ir (x)|  dν(y) ≈ dν(y) |x − y − zP |n+γ |zP |n+γ Q P ∈M Q P ∈M 0 0  (Q0 )γ    |ν|(Q0 ) (Q0 )n = |ν|(Q 0 )  (10.10) |zP |n+γ γ (Q0 )n M |zP |n  P ∈M  P ∈M ˆ 1  |f |dη. γ (Q0 )n M Q0 We claim that ˆ 1 |IIr (x)|  |f |dη. (10.11) (Q0 )n M Q0 In order to prove the claim, we observe that Ǩr (·) := ∇Θ(·; A0 )ϕr (·) is an antisymmetric kernel and that P ∈ M  if and only if −P ∈ M.  Thus, we can write ˆ 1    IIr = (Id + ∇χ (x)) Ǩr (zP + (x − y)) − Ǩr (zP − (x − y)) dν(y). 2  P ∈M Q0  we have |zP +(x −y)| ≈ |zP −(x −y)| ≈ |zP |. Let us observe that for x ∈ 2Q0 and P ∈ M Hence, by (2.9) and the Calderón-Zygmund properties of the kernel Ǩr the following inequalities hold: 52 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376  ˆ |x − y|  (Q0 ) |ν|(Q0 ) |IIr |  d|ν|(y)  |ν|(Q0 )  . |zP |n+1 |zP |n  M (Q0 )  P ∈MQ0  P ∈M This proves the claim (10.11). −1 ≤ M The estimates (10.10) and (10.11), together with the observation that M −γ , conclude the proof of the lemma after taking the limit for r → ∞.  Corollary 10.2 (Final localization estimate). We have ˆ  1  2  T̄ η dη  + ε η(Q0 ). M 2γ Q0 Proof. Inequality (10.8) in the case f ≡ 1 reads 1 T̄ (χM Q0 η)  , Mγ so that applying it together with (10.7), we have ˆ ˆ ˆ  1  2 2 2  T̄ η dη  T̄ (χM Q0 η) dη + T̄ (χ(M Q0 )c η) dη  + ε η(Q0 ), M 2γ Q0 Q0 Q0 which finishes the proof.  11. A pointwise inequality and the conclusion of the proof The following lemma implements a variational technique inspired by potential theory that allows to obtain a pointwise inequality for the potential of a proper auxiliary mea- sure. We denote as T̄ ∗ ξ the operator that, given a vector-valued measure ξ,  is defined by ˆ ∗  T̄ ξ(x) := ∇1 Ē(y, x) · dξ(y) (11.1) and which corresponds to the adjoint of T̄ . Lemma 11.1. Suppose that for some 0 < λ ≤ 1 the inequality ˆ |T̄ η|2 dη ≤ λη(Q0 ) (11.2) Q0 holds. Then there is a function b ∈ L∞ (η) such that • 0 ≤ b ≤ 2, C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 53 • b is M-periodic, ´ • Q0 b dη = η(Q0 ), and such that the measure ν = bη satisfies ˆ |T̄ ν|2 dν ≤ λν(Q0 ) (11.3) Q0 and   |T̄ ν|2 (x) + 2T̄ ∗ (T̄ ν)ν (x) ≤ 6λ for ν-a.e. x ∈ Rn+1 . (11.4) Proof. The proof works almost verbatim as that of [21, Lemma 9.1]. We briefly outline it for the reader’s convenience. In particular, we recall that the way to prove (11.4) consists in defining an adapted energy functional ˆ J(a) = λ a L∞ (η) η(Q0 ) + |T̄ (aη)|2 dη, Q0 where a ranges in  ˆ  A = a ∈ L∞ (η) : a ≥ 0, a is M-periodic, and a dη = η(Q0 ) . Q0 Then, observe that 1 ∈ A and (11.2) inf J(a) ≤ J(1) ≤ 2λη(Q0 ). a∈A Thus, since J(a) ≥ λ a L2 ∞(η) η(Q0 ), we have that inf J(a) = inf J(a). a∈A a∈A, a L∞ (η) ≤2 Hence Banach-Alaoglu theorem implies that, possibly passing to a subsequence, aj con- verges weakly * in L∞ (η) to some b ∈ L∞ (η). It is not hard to see that b ∈ A. Then (using the fact that η has bounded density with respect to the Lebesgue measure) one proves that b minimizes J in A and, finally, tests J on the competitors   ν(B) bt := 1 − tχPM (B) b + t b, 0 ≤ t < 1, ν(Q0 ) where B is a ball centered at supp ν ∩ Q0 which is contained in Q0 , and PM (B) :=  R∈M (B + zR ). In this way, we obtain 54 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 ˆ ˆ   |T̄ ν|2 dν + 2 T̄ ∗ (T ν)ν dν ≤ 6λν(B), B B so (11.4) follows by Lebesgue’s differentiation theorem as ν(B) → 0. We further remark that Girela-Sarrión and Tolsa’s calculations do not use the antisymmetry of the kernel of T̄ but just its M-periodicity, which follows by the construction of Ā.  11.1. A maximum principle Let λ, b and ν be as in Lemma 11.1. In order to be able to perform the final argument to get the contradiction, we need to extend the inequality (11.4) out of the support of ν. More precisely, the next step consists in proving that a inequality similar to that provided by Lemma 11.1 holds in a suitable strip. To this purpose, some version of maximum principle is needed. The elliptic setting of the problem makes this procedure slightly more technical then the one adopted by Girela-Sarrión and Tolsa in the case of the Riesz transform. Before presenting the main result of the section, we introduce some notation. We denote by H  the hyperplane  := {x ∈ Rn+1 : xn+1 = 3(Q0 )/2}, H which corresponds to the translate of H that contains the upper face of 3Q0 . Let KS  1 to be chosen later and let S denote the strip  < KS (Q0 )}. S := {x ∈ Rn+1 : dist(x, H) Its boundary ∂S is given by the union of two hyperplanes ∂S+ and ∂S− which lay in the upper and lower half spaces respectively. Let 3   xS± = (Q0 )(1, . . . , 1, 1) ± 0, . . . , 0, KS (Q0 ) . (11.5) 2 For the proof of our next lemma we need to invoke a result on elliptic measure. Suppose that Ω  Rn+1 is an open set with n-AD-regular boundary and consider a point p ∈ Ω. p Let ωΩ denote the elliptic measure on ∂Ω associated with the operator LĀ with pole at p. For the proof of the following standard result we refer to [7, Lemma 2.3]. Lemma 11.2. Let Ω  Rn+1 be open with n-AD-regular boundary with constant CAD . There exists ϑ = ϑ(n, A, CAD ) ∈ (0, 1) such that for every x ∈ ∂Ω and 0 < r < diam Ω, we have  |x − y| ϑ y  ωΩ B(x, r)c ≤ C for y ∈ Ω ∩ B(x, r). (11.6) r C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 55 An application of (11.6) gives a boundary regularity result for LĀ -harmonic functions, see e.g Lemma 2.10 in [5]. Lemma 11.3. Let Ω  Rn+1 be open with n-AD-regular boundary with constant CAD . Let u ≥ 0 be LĀ -harmonic function in B(x, 4r) ∩ Ω and continuous in B(x, 4r) ∩ Ω̄. Suppose, moreover, that u ≡ 0 in ∂Ω ∩ B(x, 4r). Then, extending u by zero in B(x, 4r) \ Ω̄, there exists ϑ = ϑ(n, A, CAD ) ∈ (0, 1) such that u is ϑ-Hölder continuous in B(x, 4r) and, in particular,  dist(y, ∂Ω) ϑ u(y) n,A,CAD sup u for all y ∈ B(x, r). r B(x,2r) Lemma 11.4 (Maximum principle on the strip). Let S be the strip as before and let f be a bounded continuous LĀ -harmonic function on S so that f |∂S ≡ 0. Then f ≡ 0 on S. Proof. Choose R > 100KS and set SR := S ∩ [−R, R]n+1 . For p ∈ S, denote hp := dist(p, ∂S) and let xp be a point that realizes the distance. We choose p far from the “vertical” parts ∂SR \ (∂S+ ∪ ∂S− ) of ∂SR , in particular such that B(xp , R/10) ∩ (∂SR \ p ∂S) = ∅. Let ωR denote the elliptic measure with pole at p associated with LĀ on SR . The family {SR }R is a collection of AD-regular sets whose AD-regularity constants do not depend on R. Then inequality (11.6) implies that there exit two constants C and ϑ, both independent on R, such that  h ϑ p p  p ωR (∂SR \ ∂S) ≤ ωR B(xp , R/10)c ≤ C . R By hypothesis we may assume that f ≤ 1 on ∂SR \ ∂S. Thus, we have ˆ  h ϑ p p P |f (p)| = f dωR ≤ f |∂SR \∂S ∞ ωR (∂SR \ ∂S) ≤ C . (11.7) R The results stated in the lemma follows by passing to the limit in (11.7) for R → ∞.  Now, we prove an existence result on the infinite strip S. Lemma 11.5. There exists a function fS : S̄ → R such that: (1) fS is LĀ -harmonic in the strip S and continuous in S̄. (2) fS is M-periodic. (3) fS (x) = ±1 on ∂S± and fS (x) = 0 for x ∈ H. Proof. Let k ∈ N, k ≥ 100KS and denote Sk = S ∩ [−k, k]n+1 . We define the continuous functions fk on ∂Sk as 56 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 xn+1 − 32 (Q0 ) fk (x) = . KS (Q0 ) In particular, observe that fk (x) = ±1 for x ∈ ∂S± and   fk (x) = −fk x1 , . . . , xn , −xn+1 + 3(Q0 ) ,  i.e. it is antisymmetric with respect to H. Define uk be the LĀ -harmonic function such that uk |∂Sk = fk , whose existence is guaranteed by the continuity of fk and the AD-regularity of Sk . Our aim is to prove that, a part from possibly considering a proper subsequence, uj converges uniformly in the compact subsets of Sk , for every k to an LĀ -harmonic function in S. We claim that there exist γ ∈ (0, 1) and Ck > 0 such that |uj (x) − uj (y)| ≤ Ck |x − y|γ for x, y ∈ S̄k , j ≥ k + 2. (11.8) Assume that (11.8) holds. As a consequence of Ascoli-Arzelà’s theorem together with a standard diagonalization argument, there is a function fS so that uk converges to fS uniformly on the compact subsets of S. The LĀ -harmonicity of fS is a consequence of Caccioppoli’s estimate (cfr. [22, Theorem 3.77]). To prove (2), define v = (6(Q0 ), 0, . . . , 0) and observe that, being the matrix Ā M- periodic and since fS is constant on ∂S± , the function f (x) = fS (x) − fS (x + v ) satisfies the hypothesis of Lemma 11.4. So, f ≡ 0 and  fS is M-periodic.   To prove (3), first observe that A(x) = Aφ x1 , . . . , xn , −xn+1 + 3(Q0 ) , where φ is the function that maps a point to its reflected with respect to H  and Aφ is defined as in (5.5). Then we can apply again Lemma 11.4 to   f(x) = fS (x) + fS x1 , . . . , xn , −xn+1 + 3(Q0 ) , which is LĀ -harmonic and vanishes on ∂S. We are left with the proof of the claim (11.8). By Lemma 11.3, there exists ϑ ∈ (0, 1) depending only on n, Ā and the AD-regularity of ∂Ω (hence independent both on j and k) such that uj is ϑ-Hölder continuous in the set {x ∈ Ω̄ : dist(x, ∂Ω) ≤ 2(Q0 )}. Being uj ∞ ≤ 2 for every j, by De Giorgi-Nash interior estimates we can infer that there exists γk independent on j such that, for every j ≥ k + 2, uj is γk -Hölder continuous in {x ∈ Ωk+1 : dist(x, ∂Ω) > (Q0 )}. Gathering the interior and the boundary regularity of uj proves (11.8).   we have the By the previous lemma, Lemma 11.3 and the fact that fS ≡ 0 on H, estimate  dist(y, H)  ϑ |fS (y)|  , for  ≤ 10(Q0 ). y ∈ S with dist(y, H) KS (Q0 ) C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 57 Let us define the auxiliary function FS (x) := fS (x)T̄ ν(xS+ ). Observe that FS |∂S± ≡ ±T̄ ν(xS+ ). The rest of the present section is devoted to the proof of the following, which may be regarded as an approximated maximum principle on S. Lemma 11.6 (Pointwise bound for the potential on the strip). For x ∈ S we have   1 1  K α |T̄ ν(x) − FS (x)|2 + 4T̄ ∗ (T̄ ν)ν (x)  λ1/2 +  α S + + (CS (Q0 )) + , KS2α KSϑ CS where CS is a constant chosen so that CS  KS . Before proving this lemma, we need some auxiliary result. Lemma 11.7. Let xS+ and xS− as in (11.5). Then: (1) For x ∈ ∂S+ , dist(x, xS+ )  (Q0 ) we have the estimate 1 |T̄ ν(x) − T̄ ν(xS+ )|  . (11.9) KSα The analogous estimate holds in x ∈ ∂S− replacing xS+ with xS− . (2) The difference of −T̄ ν(xS+ ) and T̄ ν(xS− ) can be estimated as 1 |T̄ ν(x+ ) + T̄ ν(xS− )|  . KSα  ≥ 2(Q0 ) we have (3) For x with dist(x, H)   T̄ ∗ (T̄ ν)ν (x)  λ1/2 . (11.10) Proof. Let us begin with the proof of (1). Because of the M-periodicity of T̄ ν, we can assume without loss of generality that xH ∈ [−3(Q0 ), 3(Q0 )]n × {0}, xH denoting the projection of x on H. We claim that for P ∈ M and y ∈ Q0 we have (Q0 )α |K̄(x, y + zP ) − K̄(xS+ , y + zP )|  . (KS (Q0 ))n+α + |zP |n+α This follows from the (global) Calderón-Zygmund estimates for K̄(·, ·) once we observe that |x − xS+ |  |x − y − zP | ≈ KS (Q0 ) + |zP |. So, for r > 0, standard calculations give 58 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376  ˆ (Q0 )α Tr ν(x) − Tr ν(xS+ )  dν(y) (KS (Q0 ))n+α + |zP |n+α P ∈MQ 0  (Q0 )n+α (Q0 )n+α 1 =    = α . (KS (Q0 ))n+α + |zP |n+α (KS (Q0 ))n+α K P ∈M S Being this estimate independent on the choice of r, in the limit for r → ∞ we have (11.9). The proof of the analogous estimate for xS− is identical, so we omit it and we go to the proof of (2). Denote by x∗ the reflection of the point x across x0 = 32 (Q0 )(1, . . . , 1), i.e. x∗ = 2x0 − x. By the specific choice of x0 , this transformation can be obtained via a composition of the reflections ψj ’s with respect to the hyperplanes passing through x0 which we defined in (5.4): x∗ = ψ1 ◦ · · · ◦ ψn+1 (x). Moreover, 3   (xS+ )∗ = 3(Q0 )(1, . . . , 1) − (Q0 )(1, . . . , 1) − 0, . . . , 0, KS (Q0 ) = xS− . (11.11) 2 Thus, an immediate application of Lemma 5.1 and (5.10) gives that, for y ∈ Q0 , K̄(xS+ , y + zP ) = −K̄(xS− , y ∗ + zP∗ ), P ∈ M. (11.12) Observe that |y + zP − (y ∗ − zP∗ )| ≤ |y − y ∗ | + |zP − (−zP∗ )|  (Q0 ) which, combined with Lemma 2.4, gives K̄(xS+ , y + zP ) + K̄(xS− , y − zP ) (11.12) = |K̄(xS+ , y + zP ) − K̄(x∗S− , y ∗ − zP∗ )| (11.13)  α (Q0 ) =|K̄(xS+ , y + zP ) − K̄(xS+ , y ∗ − zP∗ )|  , (KS (Q0 ))n+α + |zP |n+α where the second equality uses (11.11) (with −zP = z−P replacing zP ). Taking r > 0 and applying (11.13), we have  (Q0 )n+α 1 Tr ν(xS + ) + Tr ν(xS − )   α (KS (Q0 ))n+α + |zP |n+α K P ∈M S C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 59 which, taking the limit for r → ∞, proves (2). We are left with the proof of (3). Set σ = (T̄ ν)ν and observe that this measure is M- periodic. So, without loss of generality, we can assume that xH ∈ [−3(Q0 ), 3(Q0 )]n × {0}. Let r > 0 and, denoting by A0 the homogenized matrix associated with Ā, by χ the vector of correctors and  = 6(Q0 ), write  ˆ Tr σ(x) =  r (y + zP , x)dσ(y) K P ∈MQ 0  ˆ   = K̄(y + zP , x) − (Id + ∇χ (y + zP ))∇1 Θ(y + zP , x; A0 ) ϕr (x − y − zP )dσ(y) P ∈MQ 0  ˆ + (Id + ∇χ (y + zP ))∇1 Θ(y + zP , x; A0 )ϕr (x − y − zP )dσ(y) P ∈MQ 0 ≡ Ir + IIr . Recalling that ∇χ ∞  1 and using Lemma 2.7, we can proceed with the following estimates  ˆ (Q0 )γ  ˆ (Q0 )γ |Ir |  d|σ|(y)  d|σ|(y) |x − y − zP |n+γ  + |zP |)n+γ (dist(x, H) P ∈MQ P ∈M Q0 0 (11.14) (Q0 ) |σ|(Q0 ) n+γ (Q0 )n+γ |σ|(Q0 ) |σ|(Q0 )    ,  (dist(x, H) + |zP |)n+γ (Q0 )n  dist(x, H) (Q0 ) γ n (Q0 )n  ≥ 2(Q0 ). We claim that where the last inequality holds because we assumed dist(x, H) |σ|(Q0 ) |IIr |  . (11.15) (Q0 ) In order to prove this, let us first observe that the antisymmetry of ∇Θ(·; A0 ) and the periodicity of χ yield ˆ 1   IIr = ∇χ (zP + y)∇Θ(zP + (y − x); A0 )ϕr (x − y + zP ) 2 P ∈MQ 0  − ∇χ (−zP + y)∇Θ(zP − (y − x); A0 )ϕr (x − y − zP ) dν(y) ˆ (11.16) 1   = ∇χ (zP + y) ∇Θ(zP + (y − x); A0 )ϕr (x − y + zP ) 2 P ∈MQ 0  − ∇Θ(zP − (y − x); A0 )ϕr (x − y − zP ) dν(y). Let us define Ǩr (·) := ∇Θ(·; A0 )ϕr (·). We claim that, for x ∈ Q0 ∩ H such that dist(x, H) ≥ (Q0 ) 60 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 dist(x, H) |Ǩr (zP + (x − y)) − Ǩr (zP − (x − y))|  for y ∈ Q0 . (11.17) (dist(x, H) + |zP |)n+1 The proof of the formula above can be conducted splitting the cases |zP | ≤ 2|x − y| and |zP | > 2|x − y|, and coupling the Calderón-Zygmund properties of Ǩr with simple geometric considerations. The details can be found in [21, (8.25)]. Gathering (11.16), (11.17) and (2.9) we obtain  ˆ |IIr |  ∇χ ∞ |Ǩr (zP + (x − y)) − Ǩr (zP − (x − y))|d|ν|(y) P ∈MQ 0  ˆ dist(x, H)  d|ν|(y) (dist(x, H) + |zP |)n+1 P ∈MQ 0 dist(x, H)  (P )n |ν|(Q0 ) = |ν|(Q0 ) n  n+1  , (P ) (P )n P ∈M dist(x, H) + |zP | which concludes the proof of (11.15) and, thus, the lemma. It is possible to prove this estimate analogously to the case of the Riesz transform. We omit its proof in order not to make the presentation too lengthy. We remark that the calculations that lead to (11.15) solely relies on the Calderón-Zygmung property of the kernel and some geometric considerations that are independent on its specific expression. We refer to [21, (8.20)] for more details. Hence, gathering (11.14), (11.15) and passing to the limit on r, we get ˆ ∗   1 T̄ (T̄ ν)ν (x)  |T̄ ν|dν. (Q0 )n Q0 Then, recalling (11.3), the growth of ν and using Cauchy-Schwarz’s inequality, ˆ ∗   1  1/2 T̄ (T̄ ν)ν (x)  |T̄ ν|2 dν ν(Q0 )  λ1/2 , (Q0 )n Q0 which finishes the proof of (3).  The following result is a direct consequence of Lemma 11.7. Corollary 11.1. For x ∈ ∂S 1 |T̄ ν(x) − FS (x)|2  , (11.18) KS2α where the implicit constant does not depend on S. C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 61 Another result which is needed for the application of the maximum principle is the estimate of |FS (x)| for x close to the support of the measure ν.  ≤ Lemma 11.8. Let KS be an odd natural number, KS ≥ 3. For x ∈ Rn+1 with dist(x, H) 10(Q0 ) we have 1 |FS (x)|  . KSϑ Proof. Because of the Hölder continuity of fS , we can write  dist(x, H)  ϑ |T̄ ν(xS+ )| |FS (x)|  |T̄ ν(xS+ )|  . KS (Q0 ) KSϑ So, to prove the lemma, it suffices to show that |T̄ ν(xS+ )| ≤ C for some constant C > 0 not depending on KS . Recall now that ν = b η. Applying Lemma 10.2 with M = 6KS and f = b to the point 0 ∈ 2Q0 , we have the estimate ˆ   1 T̄ χ(6KS Q0 )c ν (0)  |b| dη  1, (11.19) (6KS )γ (Q0 )n Q0 where the implicit constant in the last inequality does not depend on KS . Now, we observe that the (global) Calderón-Zygmund properties of K̄ and the fact that |xS+ |  KS (Q0 ) imply ˆ     T̄ χ(6KS Q0 )c ν (0) − T̄ χ(6KS Q0 )c ν (xS+ )  K̄(0, y) − K̄(xS+ , y) dν(y) (6KS Q0 )c ˆ |xS+ |α  dν(y)  1. (|y| + |xS+ |)n+α (6KS Q0 )c (11.20) Then, by (11.19), (11.20) and the triangle inequality, we have   T̄ χ(6KS Q0 )c ν(xS+ )       (11.21) ≤ T̄ χ(6KS Q0 )c ν (0) + T̄ χ(6KS Q0 )c ν (0) − T̄ χ(6KS Q0 )c ν (xS+ )  1. Moreover, since dist(xS+ , supp ν)n  KS (Q0 ) and estimating the kernel via Lemma 2.4, 62 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 ˆ ˆ   1 T̄ χ6KS Q0 ν (xS+ ) ≤ |K̄(xS+ , y)|dν(y)  dν(y) |xS+ − y|n 6KS Q0 6KS Q0 ν(6KS Q0 ) KSn (Q0 )n    1. (11.22) dist(xS+ , supp ν)n dist(xS+ , supp ν)n Thus, gathering (11.21) and (11.22) we obtain     |T̄ ν(xS+ )| ≤ T̄ χ6KS Q0 ν (xS+ ) + T̄ χ(6KS Q0 )c ν(xS+ )  1, which proves the lemma.  In order to be able to use the previous lemma, from now on we assume without loss of generality KS ≥ 3 and we suppose it to be an odd number. Observe that for x ∈ supp ν, Lemma 11.8 and (11.4) give 2 sup T̄ ν(x) − FS (x) + 4T̄ ∗ ((T̄ ν)ν)(x) x∈supp ν ≤ sup 2 T̄ ν(x)|2 + 4T̄ ∗ ((T̄ ν)ν)(x) + 2|FS (x)|2 (11.23) x∈supp ν 1 ≤ 12λ + 2|FS (x)|2  λ + . KSϑ Moreover, by (11.10) and (11.18), 2 1 sup T̄ ν(x) − FS (x) + 4T̄ ∗ ((T̄ ν)ν)(x)  + λ1/2 x∈∂S KS2α which, together with (11.23) brings us to 2 1 1 sup T̄ ν(x) − FS (x) + 4T̄ ∗ ((T̄ ν)ν)(x)  λ1/2 + + ϑ. (11.24) x∈∂S∪supp ν KS2α KS Finally, we provide the proof of Lemma 11.6.   Proof. We recall that Ā = ĀT . Let g ∈ L∞ (S; Rn+1 ). We claim that T̄ ∗ g Ln+1 is a LĀT -harmonic (vector valued) function. This would imply the maximum principle     sup T̄ ∗ g Ln+1 (x) = sup T̄ ∗ g Ln+1 (x). (11.25) x∈S x∈∂S∩supp ν Observe that, because of Lemma 11.5, the same equality holds with FS (x) in place of T̄ ∗ (g Ln+1 )(x). Let ϕ ∈ Cc∞ (S \ supp g ) be a test function. To prove the claim, apply the definition of T̄ ∗ together with Fubini’s theorem together with the fact that Ē(x, y) = E AT (y, x): C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 63 ˆ ˆ ˆ  ĀT ∇T̄ ∗ (g Ln+1 ) · ∇ϕ = ĀT ∇x ∇y Ē(y, x) · g (y)dy · ∇ϕ(x)dx ˆ ˆ  = ∇y ĀT ∇x Ē(y, x) · ∇ϕ(x)dx · g (y)dy ˆ ˆ  (11.26) = ∇y ĀT ∇x E ĀT (x, y) · ∇ϕ(x)dx · g (y)dy ˆ = ∇ϕ · g = 0. Notice that for every z ∈ Rn+1 we have the elementary relation |z|2 = sup 2e, z − β 2 , β≥0,e∈Sn so that, choosing z = T̄ ν(x) − FS (x), it reads 2 T̄ ν(x) − FS (x) = sup 2e, T̄ ν(x) − 2e, FS (x) − β 2 . (11.27) β≥0,e∈Sn We want to show that the argument of the supremum in the right hand side of (11.27) differs from a LĀ -harmonic function possibly by a small term. This will allow to apply the maximum principle on the strip and to finish the proof. For a fixed e ∈ Sn and x ∈ supp ν, we split   e, T̄ ν(x) = −T̄ ∗ (νe)(x) + T̄ ∗ (νe)(x) + e, T̄ ν(x) and consider that, claiming that the dominated convergence theorem applies, ˆ   T̄ ∗ (νe)(x) + e, T̄ ν(x) = lim K r (x, y) + K  r (y, x) · e dν(y). (11.28) r→∞ To prove that the previous identity holds, set CS  KS to be chosen later. By the triangle inequality, the antisymmetry of ∇1 Θ(x, y; Ā(x)) and the linear growth of ν, we have ˆ  r (x, y) + K K  r (y, x) dν(y) |x−y| CS (Q0 ). As before, by the periodicity of MS we can assume that xH ∈ [−3(Q0 ), 3(Q0 )]n × {0}. Hence, using arguments analogous to the ones in Lemma 11.7, it is possible to prove that for y ∈ Q0 and zP such that |x − y − zP | > CS (Q0 ), we have (KS (Q0 ))α K̄(x, y + zP ) + K̄(x, y − zP )  , |zP |n+α + |x|n+α hence, calling MS the subset of P ∈ M such that |x − y − zP | > CS (Q0 ) for every y ∈ Q0 , we have  ˆ  α |K  r (x, y − zP )|dν(y)  KS .  r (x, y + zP ) + K (11.30) CS P ∈MSQ 0 Analogously, one can prove  ˆ  α |K  r (y − zP , x)|dν(y)  KS ,  r (y + zP , x) + K (11.31) CS P ∈MSQ 0 so, gathering (11.29), (11.30) and (11.31) and letting r → ∞, we can use the dominated convergence theorem and we estimate (11.28) as  K α T̄ ∗ (νe)(x) + e, T̄ ν(x) |  (CS (Q0 ))α + S . (11.32) CS We are now ready to proceed with the calculations for the maximum principle. Indeed, taking x ∈ S, an application of (11.27) and (11.32) gives 2 T̄ ν(x) − FS (x) + 4T̄ ∗ ((T̄ ν)ν)(x) = sup 2e, T̄ ν(x) − 2e, FS (x) − β 2 + T̄ ∗ ((T̄ ν)ν)(x) β≥0,e∈Sn  sup −2T̄ ∗ (νe)(x) − 2e, FS (x) − β 2 + T̄ ∗ ((T̄ ν)ν)(x) β≥0,e∈Sn  K α + (CS (Q0 ))α + S . CS Then, using the maximum principle (11.25) we have |T̄ ν(x) − FS (x)|2 + 4T̄ ∗ ((T ν)ν)(x)    K α  sup 2 − T̄ ∗ νe + (T̄ ν)ν (x) − 2e, FS (x) − β 2 + (CS (Q0 ))α + S β≥0,e∈S n CS   ≤ sup sup −2T̄ ∗ νe + (T̄ ν)ν (z) − 2e, FS (z) − β 2 z∈∂S∪supp ν β≥0,e∈Sn C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 65  K α + (CS (Q0 ))α + S . CS So, another application of (11.27) and (11.32) concludes the proof of the lemma. Indeed, recalling the estimate (11.24) on ∂S ∪ supp ν, |T̄ ν(x) − FS (x)|2 + 4T̄ ∗ ((T ν)ν)(x)  sup sup 2e, T̄ ν(x) − 2e, FS (x) − β 2 z∈∂S∪supp ν β≥0,e∈Sn  K α + T̄ ∗ ((T̄ ν)ν)(x) + (CS (Q0 ))α + S CS  K α |T̄ ν(z) − FS (z)|2 + 4T̄ ∗ ((T̄ ν)ν)(z) + (CS (Q0 ))α + S  sup z∈∂S∪supp ν CS 1 1  K α + ϑ + (CS (Q0 ))α + S  λ1/2 + 2  α .  KS KS CS 11.2. The conclusion of the proof of the Key Lemma To simplify the notation, set 1 1  K α α S Err(KS , CS , (Q0 )) := 2  α + ϑ + (CS (Q0 )) + . KS K S CS Notice that if x ∈ 2Q0 , Lemma 11.6 together with Lemma 11.8 allows to majorize |T̄ ν(x)|2 as 2 T̄ ν(x)  |T̄ ν(x) − FS (x)|2 + |FS (x)|2 + 4T̄ ∗ ((T̄ ν)ν)(x) − 4T̄ ∗ ((T̄ ν)ν)(x)  λ1/2 + Err(KS , CS , (Q0 )) + |FS (x)|2 − T̄ ∗ ((T̄ ν)ν)(x) (11.33) ∗ λ 1/2 + Err(KS , CS , (Q0 )) − T̄ ((T̄ ν)ν)(x). Let ϕ be a smooth function such that χQ0 ≤ ϕ ≤ χ2Q0 and ∇ϕ ∞  (Q0 )−1 . Set ψ := ĀT ∇ϕ and observe that it verifies ˆ T̄ ∗ [ψ Ln+1 ](x) = T̄ ∗ [ĀT ∇ϕ Ln+1 ](x) = ∇1 E Ā (y, x) · ĀT (y)∇ϕ(y)dy ˆ = Ā(y) ∇1 E Ā (y, x) · ∇ϕ(y)dy = ϕ(x), the last equality being a consequence of the definition of fundamental solution. The choice of ϕ ≥ χQ0 , together with Cauchy-Schwarz’s inequality, gives ˆ ˆ ˆ ν(Q0 ) ≤ ϕdν = T̄ ∗ (ψLn+1 )dν = T̄ ν · ψdLn+1 ˆ 1/2  ˆ 1/2 (11.34) ≤ |T̄ ν| |ψ|dL 2 n+1 |ψ|dL n+1 . 66 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 Now, observe that ψ ∞ ≤ ĀT ∞ ∇ϕ ∞  (Q0 )−1 (11.35) and ˆ 1 |ψ|dLn+1  Ln+1 (2Q0 )  (Q0 )n . (11.36) (Q0 ) We claim that ˆ |T̄ ν|2 |ψ|dLn+1  (Q0 )n . Applying (11.33) and (11.36), we can write ˆ |T̄ ν|2 |ψ|dLn+1 ˆ ˆ      λ1/2 + Err(KS , CS , (Q0 )) |ψ|dLn+1 + T̄ ∗ (T̄ ν)ν |ψ|dLn+1 ˆ    λ1/2 + Err(KS , CS , (Q0 )) |ψ|dLn+1 ˆ ˆ     (11.37) + T̄ ∗ χ(30Q0 )c (T̄ ν)ν |ψ|dLn+1 + T̄ ∗ χ30Q0 (T̄ ν)ν |ψ|dLn+1 ˆ      λ1/2 + Err(KS , CS , (Q0 )) (Q0 )n + T̄ ∗ χ(30Q0 )c (T̄ ν)ν |ψ|dLn+1 ˆ   + T̄ ∗ χ30Q0 (T̄ ν)ν |ψ|dLn+1   = λ1/2 + Err(KS , CS , (Q0 )) (Q0 )n + I + II, where I and II are defined by the last equality.  = 30. In particular, recalling (11.3), The estimate for I is an application of (10.8) with M ˆ   1 T̄ ∗ χ(30Q0 )c (T̄ ν)bη) (x)  |(T̄ ν)b|dη (Q0 )n Q0 1/2  ˆ 1/2 ν(Q0 ) ν(Q0 ) ≤ |T̄ ν|2 dν ≤λ1/2 , (Q0 )n (Q0 )n Q0 which, together with (11.36), implies I  λ1/2 ν(Q0 ). (11.38) For the estimate of II, recall that |K̄(x, y)|  |x − y|−n . This and (11.35) imply C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 67 ˆ ˆ   1 1 (Q0 ) T̄ |ψ|Ln+1 (x) = K̄(x, y)|ψ|(y)dy  dy  = 1. (Q0 ) |x − y|n (Q0 ) 2Q0 Then, by Cauchy-Schwarz’s inequality, the periodicity of T̄ ν and the localization (11.3), ˆ  ˆ 1/2 II ≤ n+1 T̄ (|ψ|L ) · T̄ νdν  ν(Q0 )1/2 |T̄ ν|2 dν  λ1/2 ν(Q0 ). (11.39) 30Q0 30Q0 So, gathering (11.34), (11.37), (11.38) and (11.39), we have  1/2 ν(Q0 )  Err(KS , CS , (Q0 )) + λ1/2 ν(Q0 ). (11.40) Choosing KS big enough, KS /CS small enough, CS (Q0 ) and λ small enough, we have Err(KS , CS , (Q0 )) + λ1/2  1, so (11.40) brings us to the contradiction ν(Q0 )  ν(Q0 ). This proves the Key Lemma and, hence, completes the proof of Theorem 1.2. 12. The two-phase problem for the elliptic measure To the purpose of the application to the study of the elliptic measure, it is useful to reformulate Theorem 1.2 under slightly different hypothesis. The proof of the following closely resembles that of [9, Theorem 3.3]. Theorem 12.1. Let μ be a Radon measure in Rn+1 and let B ⊂ Rn+1 be a ball with μ(B) > 0. Assume that, for some constants C0 , C1 > 0 and 0 < λ, δ, τ  1 the following conditions hold: (1) r(B) ≤ λ. (2) Pμ,α (B) ≤ C0 Θμ (B). (3) L There is some n-plane L through the center of B such that βμ,1 (B) ≤ δΘμ (B). (4) There is GB ⊂ B such that for all x ∈ GB   μ B(x, r) sup + T∗ (χ2B μ)(x) ≤ C1 Θμ (B). 0 0 such that, if δ, τ and λ are small enough (depending on C0 and C1 ), there is a uniformly n-rectifiable set Γ such that μ(B ∩ Γ) ≥ ϑμ(B). The proof in the case A ≡ Id is based on a T b theorem for suppressed kernels by Nazarov, Treil and Volberg. To replicate the proof of Azzam, Mourgoglou and Tolsa in the elliptic context, we define the suppressed kernel associated with K(·, ·) as  |x − y|2   Φ (x, y) = χ K  K(x, y), Φ(x)Φ(y) where χ : [0, +∞) → [0, 1] is a smooth, vanishes identically in [0, 1/2] and equals 1 in [1, +∞) and Φ is a 1-Lipschitz function to be chosen as in the proof of [9]. Then, one can split 1  1  K(x, y) = K(x, y) + K(y, x) + K(x, y) − K(y, x) = K (s) (x, y) + K (a) (x, y), 2 2 apply the T b theorem for suppressed kernels (see also [48, Section 5.12] and the references therein) to the antisymmetric part of K and exploit the L2 -boundedness of the symmetric part guaranteed by the freezing technique of Lemma 2.2. We leave to the interested reader to check that there is no further difficulty in the proof Theorem 12.1. The rest of the present section is devoted to show how to apply Theorem 12.1 to prove the two-phase problem for the elliptic measure.  After possibly splitting the set E, we can assume diam E ≤ 10 1 min diam Ω1 ,  diam Ω2 . We choose the poles pi , i = 1, 2 such that pi ∈ Ωi ∩ 2B  \ B,  where B  is a ball centered at E with radius r(B)  = 2 diam E. We are going to apply Theorem 12.1 to the measure ω1 : we are going to prove that we can find an n-rectifiable set F ⊂ E such that ω1 |F  Hn |F  ω1 |F . In particular, we can suppose that Ω1 is such that   Hn+1 B ∩ Ω1 ≈ r(B).  (12.1) By the so-called Bourgain’s estimates (see [43, Section 12] for the statement in the elliptic case and [6] for a proof in the case A ≡ Id) together with (12.1), we can infer that there exists δ0 such that    ≈ 1, ω1 2δ −1 B for 0 < δ < δ0 .  > 0 and i = 1, 2. We say that a ball B is a-Pωi ,γ -doubling if Let a, γ Pωi ,γ (B) ≤ aΘωi (B). The following lemma is important for the applicability of the doubling condition. C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 69 Lemma 12.1. Let γ  ∈ (0, 1). Let Ω1 , Ω2 be Wiener regular domains in Rn+1 and let E ⊂ ∂Ω1 ∩ ∂Ω2 be a set on which ω1 |E  ω2 |E  ω1 |E . Then there exists a constant γ , n) big enough such that for ω1 |E -almost every x ∈ Rn+1 we can find a sequence a = a( of a-Pωi ,γ -doubling balls B(x, ri ) with ri → 0 as i → ∞. Proof. Let i = 1, 2. Let m ∈ Z, m ≥ 1 and denoting Zm := x ∈ ∂Ωi : for all j ≥ m, B(x, 2−j ) is not a-Pωi ,γ -doubling it suffices to prove that ωi |E (Zm ) = 0 for every m. Arguing as in [11, Lemma 6.1] we have that, for x ∈ Zm , we can estimate the elliptic measure of B(x, r) as ωi (B(x, r)) ≤ C(m)rn+γ for r ≤ 2−m . Then ω|E (A) ≤ ω(A) ≤ C(m)Hn+γ (A) for any A ⊂ Zm . We recall that the dimension of ω|E can be defined as dim ω|E := inf s : ∃ F ⊂ ∂Ω s.t. Hs (F ) = 0 and ω|E (F ∩ K) = ω|E (∂Ω ∩ K) ∀K ⊂ Rn+1 compact First let us bound dim ω|E from below. Let F ⊂ ∂Ω be such that Hn+γ (F ) = 0. For K ⊂ Zm compact and such that ω|E (K) > 0, we have ω|E (F ∩ K) ≤ C(m)Hn+γ (F ∩ K) = 0. This in turn implies dim ω|E ≥ n + γ . (12.2) Conversely, [8] gives that dim ω|E = n, which gathered with (12.2) tells that n≥n+γ . Being γ > 0, this brings to a contradiction and, in particular, this proves that ω(Zm) = 0 for every m.  Let i = 1, 2. Denote by ui (·) = Gi (pi , ·) the Green function associated with Ωi with pole at pi . We understand that ui is extended by zero to Ωci . As a corollary of [5, Theorem 1.5], which was formulated under weaker assumptions on the regularity of the matrix A, we can state the following monotonicity formula. Lemma 12.2 (Monotonicity formula). Let Ωi and ui be as above and let R > 0. Suppose that As (ξ) = Id for ξ ∈ ∂Ω1 ∩ ∂Ω2 . Then, setting 70 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376  ˆ   ˆ  1 |∇u1 (y)|2 1 |∇u2 (y)|2 γ(ξ, r) = dy · dy , r2 |y − ξ|n−1 r2 |y − ξ|n−1 B(ξ,2r) B(ξ,2r) we have that, for some c > 0, α −r α ) γ(ξ, r) ≤ γ(ξ, s)ec(s <∞ for 0 < r ≤ s < R. We remark that Azzam, Garnett, Mourgoglou and Tolsa proved their result under the hypothesis A(ξ) = Id. However, the same proof works under our assumption.3 The following lemma is crucial to prove the elliptic variant version of the blowups. Lemma 12.3. Let Ω1 be a Wiener regular domain and denote by ω1 = ω1p1 its associated elliptic measure with pole at p1 ∈ Ω1 . Let B be a ball centered at ∂Ω1 and such that p1 ∈/ 10B. Assuming that ω1 (8B) ≤ Cω1 (δ0 B) and Hn+1 (B \ Ω1 ) ≥ C −1 r(B)−1 , we have Hn+1 (Ω1 ∩ 2δ0 B)  r(B)n+1 . (12.3) Moreover Hn+1 (2δ0 B \ Ω1 ) ≈ Hn+1 (2δ0 B \ Ω2 ) ≈ r(B)n+1 . (12.4) Proof. Denote r = r(B). Let us first prove (12.3). Consider a smooth function ϕ ≥ 0 such that ϕ ≡ 1 on δ0 B and supp ϕ ⊂ 2δ0 B. In particular, suppose that ϕ ∞  (δ0 r)−1 . Then, recalling that, by the properties of Green’s function and being x1 outside of the support of ϕ, ˆ ˆ ϕdω1 = − AT ∇u1 · ∇ϕ, we use the ellipticity of the matrix A and write ˆ ˆ ω1 (2δ0 B) ≤ ϕdω1 ≤ |∇u1 · A∇ϕ| ˆ ˆ ˆ 1  |∇u1 ||∇ϕ| = |∇u1 ||∇ϕ|  |∇u1 |. δ0 r Ω1 ∩2δ0 B Ω1 ∩2δ0 B Then applying, in order, Hölder’s and Caccioppoli’s inequalities, 3 It suffices to define the matrix D in [5, Appendix A.1] as D = A(ξ) − A and observe that LA(ξ) = LAs (ξ) = Id. C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 71 ˆ ˆ 1 Hn+1 (Ω1 ∩ 2δ0 B)1/2  1/2 |∇u1 | ≤ |∇u1 |2 δ0 r δ0 r Ω1 ∩2δ0 B 2δ0 B ˆ H n+1 (Ω1 ∩ 2δ0 B)1/2 1  1/2  |u1 |2 , δ0 r δ0 r 4δ0 B so (δ0 r)(n+1)/2 ω1 (2δ0 B)  Hn+1 (Ω1 ∩ 2δ0 B)1/2 sup |u1 |. (δ0 r)2 4δ0 B At this point, recalling that (see [43, Section 12]) ω1 (8B) sup u1 (y)  , y∈4δ0 B rn−1 we have (δ0 r)(n−3)/2 ω1 (δ0 B)  Hn+1 (Ω1 ∩ 2δ0 B)1/2 ω1 (8B) (δ0 r)n−1 which, since we suppose ω1 (8B) ≤ Cω1 (δ0 B), concludes the proof of (12.3). The second estimate in the statement of the lemma is a direct application of the first one (see also [11, Lemma 3.4]).  The following lemma provides the connection between the function γ in Lemma 12.2 and elliptic measure. Lemma 12.4. Let i = 1, 2 and Ωi , pi be as above. Let 0 < R < mini dist(pi , ∂Ωi ). Then, for 0 < r < R/4 and ξ ∈ ∂Ω1 ∩ ∂Ω2 we have ω1 (B(ξ, r)) ω2 (B(ξ, r)  γ(ξ, 2r)1/2 . (12.5) rn rn Moreover, if r < δ0 R/8 and ωi (B(ξ, 8r))  ωi (B(ξ, δo r)), ω1 (B(ξ, 16δ0−1 )) ω2 (B(ξ, 16δ0−1 )) γ(ξ, r)1/2  · . (12.6) rn rn The proof of (12.5) is analogous to that for the harmonic measure in [29]. The proof of (12.6) is an application of Caccioppoli’s inequality together with Lemma 12.3 (see also [11, Lemma 3.5]). The blowup technique for the elliptic measure developed in [8] is crucial to prove the next lemma. We remark that the authors formulated this result under more general assumptions on the matrix A then the ones of the present work. 72 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 Lemma 12.5. Let Ω1 , Ω2 and E be as above. Let ε < 1/100 and, for m ≥ 1, define Em as the set of ξ ∈ E such that for all ξ ∈ E, 0 < r < 1/m and i = 1, 2 the following properties hold: (E1) ωi (B(ξ, 2r)) ≤ m ωi (B(ξ, r)). (E2) Hn+1 (B(ξ, r) ∩ Ωi ) ≥ m1 n+1 r . −n (E3) βω1 ,1 (B(ξ, r)) < εr ω1 (B(ξ, r)). The sets Em cover E up to a set of ω1 -measure 0, i.e.    ω1 E \ Em = 0. m≥1 The proof follows by known results in the literature. However, we think that it may be useful to the reader to dispose of precise references. Sketch of the proof. Set  ω1 (E ∩ B(ξ, r)) ω2 (E ∩ B(ξ, r))  E ∗ = ξ ∈ E : lim = lim =1 . r→0 ω1 (B(ξ, r)) r→0 ω2 (B(ξ, r)) One can see that ωi (E \ E ∗ ) = 0, i = 1, 2. Now, for ξ ∈ E ∗ , set h(ξ) = dω1 dω2 (ξ), Λ = ξ ∈ E ∗ : 0 < h(ξ) < ∞ and Γ = {ξ ∈ Λ : ξ is a Lebesgue point for h with respect to ω1 }. By Lebesgue differentiation theorem, ωi (E \ Γ) = ωi (E ∗ \ Γ) for i = 1, 2. Then, in order to prove the lemma it suffices to show that for ω1 -almost every ξ ∈ Γ: (P1) ω1 is locally doubling, i.e. ω1 (B(ξ, 2r)) lim sup < ∞. r→0 ω1 (B(ξ, r)) (P2) For i = 1, 2 Hn+1 (B(ξ, r) ∩ Ωi ) lim inf >0 r→0 rn+1 (P3) We have the flatness estimate rn lim βω1 ,1 (B(ξ, r)) = 0. r→0 ω1 (B(ξ, r)) C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 73 The condition (P1) holds because of the flatness of the tangents Tan(ωi , ξ), see [8, The- orem 1.3], which is known to imply the locally doubling condition ([42, Corollary 2.7]). The property (P2) follows by the arguments in [11] together with (12.4). To prove (P3), it suffices to argue as in the end of [11, Section 5].  Now consider m ≥ 1 such that ωi (Em ). Lemma 12.6. Let δ > 0. For ω1 -almost every x ∈ Em there is rx > 0 such that, given an a-Pγ ,ω1 -doubling ball B(x, r) with r ≤ rx , there exits a set Gm (x, r) ⊂ Em ∩ B(x, r) such that ω1 (B(z, t)) ω1 (B(x, r))  for every z ∈ Gm (x, r), 0 < t ≤ 2r. tn rn In particular, ω1 (B(x, r) \ Gm (x, r)) ≤ δω1 (B(x, r)), (12.7) mδ the set of points where (12.7) is verified, we have and, if we denote by E   m,δ = 0. ω1 Em \ E This lemma can be proved arguing as in [11, Lemma 6.2] and more precisely combining the locally doubling property of the elliptic measure ensured by the blowup argument together with Lemma 12.4. We also point out that their argument relies on the monotonicity formula of Alt, Caffarelli and Friedman. So, to prove it in the elliptic case we have to invoke Lemma 12.2, whose hypothesis include the assumption As (x) = Id. This, of course, is not true in general. However, one can argue via the change of variable in Lemma 5.3 to achieve this property. For a more detailed treatment of how the elliptic measure varies under that transformation we refer to [5, Corollary 2.5]. We omit further details. From now on fix γ =α . The following lemma contains an estimate of the potential of ω1 which is needed to recollect the property (4) in Theorem 12.1. Lemma 12.7 (cfr. [11, Lemma 6.3]). Let 0 < c  1 to be chosen small enough. For m ≥ 1 and δ > 0, let Em,δ and rx be as in the previous lemma. Consider x0 ∈ E m,δ 0 and take   0 < r0 < min rx0 , 1/m, dist(p1 , ∂Ω1 ) . Assume, moreover, that B0 = B(x0 , r) is an a-Pω1 ,α -doubling ball. Then, for all x ∈ Gm (x0 , r0 ) we have T∗ (χ2B0 ω1 )(x)  Θω1 (B0 ). 74 C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 Proof. Suppose As (x0 ) = Id. Indeed, if this is not the case, one can argue via a change of variable as mentioned before. Also, without loss of generality, we can consider only the case r ≤ r0 /4. Let ε > 0. The proof relies on the estimates for the smoothened potential ˆ Tε ω1 (z) := K(z, y)ϕε (z − y)dω1 (y), z ∈ Rn+1 , where ϕ : Rn+1 → [0, 1] is a smooth radial function whose support is contained in Rn+1 \B(0, 1), equals 1 on Rn+1 \B(0, 2) and ϕε denotes the dilate ϕε (z) = ϕ(ε−1 z). Now take x ∈ Gm (x0 , r0 ) and considering r ≤ r0 /4 and define ˆ vr (z) = E(p1 , z) − E(z, y)ϕr (x − y)dω1 (y), z ∈ Rn+1 \[supp(ϕr (x − ·)ω1 ) ∪ {p1 }]. (12.8) Recall that As (x0 ) = Id and that Θ(·; A(x0 )) = Θ(·; As (x0 )). On the same range of z of (12.8) we consider ˆ v̄r (z) = Θ(p1 − z; Id) − Θ(z − y; Id)ϕr (x − y)dω1 (y). As in [11, Lemma 6.3], to prove the lemma it suffices to show the validity of the estimate |Tr ω1 (x) − Tr0 /4 ω1 (x)|  Θω1 (B0 ). To this purpose, observe that |Tr ω1 (x) − Tr0 /4 ω1 (x)| = |∇vr (x) − ∇vr0 /4 (x)| ˆ   = ∇1 E(x, y) ϕr (x − y) − ϕr0 /4 (x − y) dω1 (y) . Now, using Lemma 2.2 and the Hölder continuity of A, it is not difficult (recall that r0 ≤ 1) to prove that r0α 1 | ∇1 E(x, y) − ∇1 Θ(x − y; Id)|  ≤ , |x − y|n |x − y|n which in turn implies |Tr ω1 (x) − Tr0 /4 ω1 (x)| ˆ    Θω1 (B0 ) + ∇1 Θ(x − y; Id) ϕr (x − y) − ϕr0 /4 (x − y) dω1 (y) = |∇v̄r (x) − ∇v̄r0 /4 (x)| + Θω1 (B0 ). (12.9) C. Puliatti / Journal of Functional Analysis 282 (2022) 109376 75 We claim that |v̄r (x) − v̄r0 /4 (x)|  Θω1 (B0 ), which would conclude the proof. To show this, notice that functions v̄r and v̄r0 /4 are harmonic outside supp(ϕr (x − ·)ω1 ) ∪ {p1 }, hence in particular in B(x, r). Then, an application of the mean value property gives 1 |∇v̄r (x) − ∇v̄r0 /4 (x)|  |v̄r (z) − v̄r0 /4 (z)|dz. (12.10) r B(x,r) Another application of the freezing argument together with the C α -continuity of A proves |v̄r (z) − v̄r0 /4 (z) − vr (z) − vr0 /4 (z)|  r0α r Θω1 (B0 ), z ∈ B(x, r) that, gathered with (12.9) and (12.10) gives 1 |Tr ω1 (x) − Tr0 /4 ω1 (x)|  Θω1 (B0 ) + |vr (z) − vr0 /4 (z)|dz r B(x,r) 1 1 ≤ Θω1 (B0 ) + |vr (z)|dz + |vr0 /4 (z)|dz. r r B(x,r) B(x,r) From this point on, the proof is analogous to that in [11].  The proof of Theorem 1.3 follows verbatim the footprints of that of [9] and [11]. More m,δ and r0 as in Lemma 12.7, we split the set Gm (x0 , r0 ) as a precisely, taking x0 ∈ E union of   m (x0 , r0 ) = x ∈ Gm (x0 , r0 ) : lim Θω1 B(x, r) = 0 Gzd r→0 and m (x0 , r0 ) = Gm (x0 , r0 ) \ Gm (x0 , r0 ). Gpd zd Then, using Lemma 12.7, the elliptic analogue of [11, Lemma 6.5] and Theorem 12.1, it is possible to infer that ω1 (Gzd m (x0 , r0 )) = 0. 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