Abstract
It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-$A_\infty$ property) of harmonic measure with respect to surface measure, on the boundary of an open set $ \Omega\subset \mathbb{R}^{n+1}$ with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in $\Omega$, with data in $L^p(\partial\Omega)$ for some $p<\infty$. In this paper, we give a geometric characterization of the weak-$A_\infty$ property, of harmonic measure, and hence of solvability of the $L^p$ Dirichlet problem for some finite $p$. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors-David regularity of the boundary) that are natural, and in a certain sense optimal: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors-David bounds); moreover, the examples show that the upper and lower Ahlfors-David bounds are each quantitatively sharp.
Highlights
Azzam’s results are related to those of the present paper, so let us describe them in a bit more detail. He shows that for a domain Ω with n-dimensional Ahlfors-David regular (n-ADR) boundary, harmonic measure is in A∞ with respect to surface measure, if and only if 1) ∂Ω is uniformly rectifiable (n-UR)[1], and 2) Ω is semi-uniform in the sense of Aikawa and Hirata [AH]
We provide two geometric characterizations of such domains, one in terms of uniform rectifiability combined with the weak local John condition, the other in terms of approximation of the boundary in a big pieces sense, by boundaries of Chord-arc subdomains
In the proof of Theorem 1.5, we shall employ a two-parameter induction argument, which is a refinement of the method of “extrapolation” of Carleson measures
Summary
A classical criterion of Wiener characterizes the domains in which one can solve the Dirichlet problem for Laplace’s equation with continuous boundary data, and with continuity of the solution up to the boundary. Azzam’s results are related to those of the present paper, so let us describe them in a bit more detail He shows that for a domain Ω with n-ADR boundary, harmonic measure is in A∞ with respect to surface measure, if and only if 1) ∂Ω is uniformly rectifiable (n-UR)[1], and 2) Ω is semi-uniform in the sense of Aikawa and Hirata [AH]. Given an interior corkscrew condition (which holds automatically in the presence of the doubling property of harmonic measure), and provided that ∂Ω is n-ADR, the A∞ (or even weak-A∞) property of harmonic measure was already known to imply uniform rectifiability of the boundary [HM3] ( the published version appears in [HLMN]; see [MT] for an alternative proof, and a somewhat more general result); as in. We thank the referee for a careful reading of the paper, and for several helpful suggestions that have led us to clarify certain matters, and to make improvements in the presentation
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