Uniform Rectifiability and Elliptic Operators Satisfying a Carleson Measure Condition

Hofmann, Steve; Martell, José María; Mayboroda, Svitlana; Toro, Tatiana; Zhao, Zihui

doi:10.1007/s00039-021-00566-4

Cited by 21 publications

(14 citation statements)

References 38 publications

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“…Under such background hypothesis, for the Laplacian, [5,11,19,26] give a characterization of ω ∈ A ∞ (σ) in terms of uniform rectifiablility of the boundary (a quantitative version of rectifiability), which is also equivalent to the fact that Ω satisfies an exterior corkscrew condition and hence Ω is a chord-arc domain. Recently, in [23] (see also [24] and [28]), these characterizations have been extended to the so-called Kenig-Pipher operators, that is, elliptic operators with variable coefficients whose gradient satisfies an L 2 -Carleson condition. On the other hand, in the setting of 1-sided chordarc domains it has been established that for general elliptic operators one can characterize ω L ∈ A ∞ (σ) in terms of the fact that all bounded null solutions satisfy Carleson estimates (cf.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Proof of Theorem 1.6 for the operators in (b). Much as in [23,Corollary 10.3] one can regularize A so that the new matrix A is one of the operators considered in (a) and, moreover, A is a Fefferman-Kenig-Pipher perturbation of A. Thus, Theorem 1.2 and the fact that we have already taken care of the operators in (a) give the desired equivalences.…”

Section: Proof Of Theorem 16mentioning

confidence: 95%

“…This together with Theorem 1.1, implies that harmonic measure, ω −L , admits a corona a decomposition. The same can be done with the operators in (a), because they are a subclass of the Kenig-Pipher operators (see [22,Theorem 7.5]), or similarly to those in (b) much as in [23,Corollary 10.3].…”

Section: Proof Of Theorem 16mentioning

confidence: 99%

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Carleson measure estimates, corona decompositions, and perturbation of elliptic operators without connectivity

Cao¹,

Hidalgo-Palencia²,

Martell³

2022

Preprint

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Let Ω ⊂ R n+1 , n ≥ 2, be an open set with Ahlfors-David regular boundary satisfying the corkscrew condition. When Ω is connected in some quantitative form (more precisely, it satisfies the Harnack chain condition) one can establish that for any real elliptic operator with bounded coefficients, the quantitative absolute continuity of elliptic measures (i.e., its membership to the class A∞) is equivalent to the fact that all bounded null solutions satisfy Carleson measure estimates. In turn, in the same setting it is also known that these equivalent properties are stable under Fefferman-Kenig-Pipher perturbations. However, without connectivity, not much is known in general and, in particular, there is no Fefferman-Kenig-Pipher perturbation result available.In this paper, we work with a corona decomposition associated with the elliptic measure and show that it is equivalent to the fact that bounded null solutions satisfy partial/weak Carleson measure estimates, or to the fact that the Green function is comparable to the distance to the boundary in the corona sense. This characterization has profound consequences. First, we extend Fefferman-Kenig-Pipher's perturbation result to non-connected settings. Second, in the case of the Laplacian, these corona decompositions or, equivalently, the partial/weak Carleson measure estimates are meaningful enough to characterize the uniform rectifiability of the boundary of the open set. As a consequence, we obtain that the boundary of the set is uniformly rectifiable if bounded null solutions for any Fefferman-Kenig-Pipher perturbation of the Laplacian satisfy (partial/weak) Carleson measure estimates. That is, there is a characterization of the uniform rectifiability of the boundary in terms of the properties of the bounded null solutions for operators whose coefficients may not possess any regularity. Third, for Kenig-Pipher operators any of the properties of the characterization is stable under transposition or under symmetrization of the matrices of coefficients. As a result, we obtain that Carleson measure estimates (or its partial/weak form) for bounded null-solutions of nonsymmetric variable operators satisfying an L 1 -Kenig-Pipher condition occur if and only if the boundary of the open set is uniformly rectifiable. Last, our results generalize previous work in settings where quantitative connectivity is assumed since, in that more topologically friendly settings, our conditions are equivalent to the fact that the elliptic measure is a Muckenhoupt weight. ContentsMINGMING CAO, PABLO HIDALGO-PALENCIA, AND JOS É MAR ÍA MARTELL 2. Preliminaries 7 2.1. Notation and definitions 7 2.2. Dyadic grids and sawtooths 11 2.3. PDE estimates 16 2.4. Auxiliary results 17 3. Proof of Theorem 1.1 20 3.1. Proof of (b) =⇒ (c): ω L admits a corona decomposition implies that G L is comparable to the distance to the boundary in the corona sense 20 3.2. Proof of (c) =⇒ (d): G L is comparable to the distance to the boundary in the corona sense implies that L satisfies partial/weak Carleson measu...

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Proof Of Theorem 16mentioning

confidence: 95%

See 1 more Smart Citation

Carleson measure estimates, corona decompositions, and perturbation of elliptic operators without connectivity

Cao¹,

Hidalgo-Palencia²,

Martell³

2022

Preprint

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“…Indeed, in a first attempt to generalize this to the class of Kenig-Pipher operators, Hofmann, the third author of the present paper, and Toro [31] were able to consider variable coefficients whose gradient satisfies some L 1 -Carleson condition (in turn, stronger than the one in [41]). The general case, on which the operators are in the optimal Kenig-Pipher-class (that is, the gradient of the coefficients satisfies an L 2 -Carleson condition) has been recently solved by Hofmann et al [30].…”

Section: Introductionmentioning

confidence: 99%

On the $A_\infty$ condition for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

Cao¹,

Domínguez²,

Martell³

et al. 2021

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Let Ω ⊂ R n+1 , n ≥ 2, be a 1-sided non-tangentially accessible domain, that is, a set which is quantitatively open and path-connected. Assume also that Ω satisfies the capacity density condition. Let L 0 u = − div(A 0 ∇u), Lu = − div(A∇u) be two real (not necessarily symmetric) uniformly elliptic operators in Ω, and write ω L 0 , ω L for the respective associated elliptic measures. We establish the equivalence between the following properties:bounded null solutions of L satisfy Carleson measure estimates with respect to ω L 0 , (iv) S < N (i.e., the conical square function is controlled by the non-tangential maximal function) in L q (ω L 0 ) for some (or for all) q ∈ (0, ∞) for any null solution of L, and (v) L is BMO(ω L 0 )-solvable. Moreover, in each of the properties (ii)-(v) it is enough to consider the class of solutions given by characteristic functions of Borel sets (i.e, u(X) = ω X L (S ) for an arbitrary Borel set S ⊂ ∂Ω). Also, we obtain a qualitative analog of the previous equivalences. Namely, we characterize the absolute continuity of ω L 0 with respect to ω L in terms of some qualitative local L 2 (ω L 0 ) estimates for the truncated conical square function for any bounded null solution of L. This is also equivalent to the finiteness ω L 0 -almost everywhere of the truncated conical square function for any bounded null solution of L. As applications, we show that ω L 0 is absolutely continuous with respect to ω L if the disagreement of the coefficients satisfies some qualitative quadratic estimate in truncated cones for ω L 0 -almost everywhere vertex. Finally, when L 0 is either the transpose of L or its symmetric part, we obtain the corresponding absolute continuity upon assuming that the antisymmetric part of the coefficients has some controlled oscillation in truncated cones for ω L 0 -almost every vertex.

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“…Later, Kenig, Koch, Pipher and Toro [30] applied "approximation to more general elliptic boundary value problems and proved that on any Lipschitz domain elliptic harmonic measure is A 1 equivalent to boundary surface measure. Further connections between "-approximation, Carleson measure estimates, square functions, maximal functions, and A 1 conditions for elliptic measures have been obtained on Lipschitz domains by several authors, including [13,19,28,29,33], and then on domains with Ahlfors regular boundaries by [3,[20][21][22][23], and most recently by [2,4,5,18,24,25].…”

Section: Introductionmentioning

confidence: 99%

Carleson measure estimates and $\varepsilon$-approximation for bounded harmonic functions, without Ahlfors regularity assumptions

Garnett

2021

Rev. Mat. Iberoam.

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Let be a domain in R d C1 , where d 1. It is known that if satisfies a corkscrew condition and @ is d -Ahlfors regular, then the following are equivalent:(a) a square function Carleson measure estimate holds for bounded harmonic functions on I (b) an "-approximation property holds for all such functions and all 0 < " < 1I (c) @ is uniformly rectifiable.Here we explore (a) and (b) when @ is not required to be Ahlfors regular. We first observe that (a) and (b) hold for any domain for which there exists a domain z such that @ z is uniformly rectifiable and @ @ z . We then assume satisfies a corkscrew condition and @ satisfies a capacity density condition. Under these assumptions, we prove conversely that if (a) or (b) holds for then such a domain z exists. And we give two further characterizations of domains where (a) or (b) holds. The first is that harmonic measure for satisfies a Carleson packing condition with respect to diameters similar to a condition comparing harmonic measures to H d already known to be equivalent to uniform rectifiability. The second characterization is reminiscent of the Carleson measure description of H 1 interpolating sequences in the unit disc.

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Uniform Rectifiability and Elliptic Operators Satisfying a Carleson Measure Condition

Cited by 21 publications

References 38 publications

Carleson measure estimates, corona decompositions, and perturbation of elliptic operators without connectivity

Carleson measure estimates, corona decompositions, and perturbation of elliptic operators without connectivity

On the $A_\infty$ condition for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

Carleson measure estimates and $\varepsilon$-approximation for bounded harmonic functions, without Ahlfors regularity assumptions

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