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Abstract We construct the Neumann function in a 1-sided chord-arc domain (i.e., a uniform domain with an Ahlfors regular boundary), and establish size and Hölder continuity estimates up to the boundary. We then obtain a Kenig-Pipher type theorem, in whichL^psolvability of the Neumann problem is shown to yield solvability inL^qfor 1 <q<p, and in the Hardy spaceH¹, in 2-sided chord-arc domains, under suitable background hypotheses. 

Abstract We prove that if a parabolic Lipschitz (i.e., Lip(1,1/2)) graph domain has the property that its caloric measure is parabolic$$A_{\infty }$$ $A_{\infty }$with respect to surface measure (which property is in turn equivalent to$$\mathrm{L}^{p}$$ $L^p$solvability of the Dirichlet problem for some finite$$p$$ $p$), then the function defining the graph has a half-order time derivative in the space of (parabolic) bounded mean oscillation. Equivalently, we prove that the$$A_{\infty }$$ $A_{\infty }$property of caloric measure implies, in this case, that the boundary is parabolic uniformly rectifiable. Consequently, by combining our result with the work of Lewis and Murray we resolve a long standing open problem in the field by characterizing those parabolic Lipschitz graph domains for which one has$$\mathrm{L}^{p}$$ $L^p$solvability (for some$$p <\infty $$ $p<\infty$) of the Dirichlet problem for the heat equation. The key idea of our proof is to view the level sets of the Green function as extensions of the original boundary graph for which we can prove (local) square function estimates of Littlewood-Paley type. 

Abstract Let Ω ⊂ ℝ n + 1 {\Omega\subset\mathbb{R}^{n+1}} , n ≥ 2 {n\geq 2} , be a 1-sided non-tangentially accessible domain (aka uniform domain), that is, Ω satisfies the interior Corkscrew and Harnack chain conditions, which are respectively scale-invariant/quantitative versions of openness and path-connectedness. Let us assume also that Ω satisfies the so-called capacity density condition, a quantitative version of the fact that all boundary points are Wiener regular. Consider L 0 ⁢ u = - div ⁢ ( A 0 ⁢ ∇ ⁡ u ) {L_{0}u=-\mathrm{div}(A_{0}\nabla u)} , L ⁢ u = - div ⁢ ( A ⁢ ∇ ⁡ u ) {Lu=-\mathrm{div}(A\nabla u)} , two real (non-necessarily symmetric) uniformly elliptic operators in Ω, and write ω L 0 {\omega_{L_{0}}} , ω L {\omega_{L}} for the respective associated elliptic measures. The goal of this program is to find sufficient conditions guaranteeing that ω L {\omega_{L}} satisfies an A ∞ {A_{\infty}} -condition or a RH q {\mathrm{RH}_{q}} -condition with respect to ω L 0 {\omega_{L_{0}}} . In this paper we establish that if the discrepancy of the two matrices satisfies a natural Carleson measure condition with respect to ω L 0 {\omega_{L_{0}}} , then ω L ∈ A ∞ ⁢ ( ω L 0 ) {\omega_{L}\in A_{\infty}(\omega_{L_{0}})} . Additionally, we can prove that ω L ∈ RH q ⁢ ( ω L 0 ) {\omega_{L}\in\mathrm{RH}_{q}(\omega_{L_{0}})} for some specific q ∈ ( 1 , ∞ ) {q\in(1,\infty)} , by assuming that such Carleson condition holds with a sufficiently small constant. This “small constant” case extends previous work of Fefferman–Kenig–Pipher and Milakis–Pipher together with the last author of the present paper who considered symmetric operators in Lipschitz and bounded chord-arc domains, respectively. Here we go beyond those settings, our domains satisfy a capacity density condition which is much weaker than the existence of exterior Corkscrew balls. Moreover, their boundaries need not be Ahlfors regular and the restriction of the n -dimensional Hausdorff measure to the boundary could be even locally infinite. The “large constant” case, that is, the one on which we just assume that the discrepancy of the two matrices satisfies a Carleson measure condition, is new even in the case of nice domains (such as the unit ball, the upper-half space, or non-tangentially accessible domains) and in the case of symmetric operators. We emphasize that our results hold in the absence of a nice surface measure: all the analysis is done with the underlying measure ω L 0 {\omega_{L_{0}}} , which behaves well in the scenarios we are considering. When particularized to the setting of Lipschitz, chord-arc, or 1-sided chord-arc domains, our methods allow us to immediately recover a number of existing perturbation results as well as extend some of them. 

Abstract Let Ω ⊂ ℝ n + 1 {\Omega\subset\mathbb{R}^{n+1}} , n ≥ 2 {n\geq 2} , be a 1-sided non-tangentially accessible domain (also known as uniform domain), that is, Ω satisfies the interior Corkscrew and Harnack chain conditions, which are respectively scale-invariant/quantitative versions of openness and path-connectedness. Let us assume also that Ω satisfies the so-called capacity density condition, a quantitative version of the fact that all boundary points are Wiener regular. Consider two real-valued (non-necessarily symmetric) uniformly elliptic operators L 0 ⁢ u = - div ⁡ ( A 0 ⁢ ∇ ⁡ u )   and   L ⁢ u = - div ⁡ ( A ⁢ ∇ ⁡ u ) L_{0}u=-\operatorname{div}(A_{0}\nabla u)\quad\text{and}\quad Lu=-%\operatorname{div}(A\nabla u) in Ω, and write ω L 0 {\omega_{L_{0}}} and ω L {\omega_{L}} for the respective associated elliptic measures. The goal of this article and its companion[M. Akman, S. Hofmann, J. M. Martell and T. Toro,Perturbation of elliptic operators in 1-sided NTA domains satisfying the capacity density condition,preprint 2021, https://arxiv.org/abs/1901.08261v3 ]is to find sufficient conditions guaranteeing that ω L {\omega_{L}} satisfies an A ∞ {A_{\infty}} -condition or a RH q {\operatorname{RH}_{q}} -condition with respect to ω L 0 {\omega_{L_{0}}} . In this paper, we are interested in obtaininga square function and non-tangential estimates for solutions of operators as before. We establish that bounded weak null-solutions satisfy Carleson measure estimates, with respect to the associated elliptic measure. We also show that for every weak null-solution, the associated square function can be controlled by the non-tangential maximal function in any Lebesgue space with respect to the associated elliptic measure. These results extend previous work ofDahlberg, Jerison and Kenig and are fundamental for the proof of the perturbation results in the paper cited above.