Semi-Uniform Domains and the A∞ Property for Harmonic Measure

Abstract

We study the properties of harmonic measure in semi-uniform domains. Aikawa and Hirata showed in [AH08] that, for John domains satisfying the capacity density condition (CDC), the doubling property for harmonic measure is equivalent to the domain being semi-uniform. Our first result removes the John condition by showing that any domain satisfying the CDC whose harmonic measure is doubling is semiuniform. Next, we develop a substitute for some classical estimates on harmonic measure in nontangentially accessible domains that works in semi-uniform domains.We also show that semi-uniform domains with uniformly rectifiable boundary have big pieces of chord-arc subdomains. We cannot hope for big pieces of Lipschitz subdomains (as was shown for chord-arc domains by David and Jerison [DJ90]) due to an example of Hrycak, which we review in the appendix.Finally, we combine these tools to study the A ∞ -property of harmonic measure. For a domain with Ahlfors-David regular boundary, it was shown by Hofmann and Martell that the A ∞ property of harmonic measure implies uniform rectifiability of the boundary [HM15, HLMN17] . Since A ∞ -weights are doubling, this also implies the domain is semi-uniform. Our final result shows that these two properties, semi-uniformity and uniformly rectifiable boundary, also imply the A ∞ property for harmonic measure, thus classifying geometrically all domains for which this holds.

Plenty of big projections imply big pieces of Lipschitz graphs

“…For (T1) and (T4), some further pruning will be needed. First, from (4.7), (4.5), and the assumption μ(E Q 0 ) 1 2 μ(Q 0 ), we infer that…”

“…This means that UR sets do not necessarily have PBP, or at least bounds for n-UR constants do not imply bounds for PBP constants. The details of Hrycak's construction are contained in the appendix of Azzam's paper [1], but they can also be outlined in a few words: pick n := −1 . Sub-divide I 0 := [0, 1] × {0} ⊂ R 2 into n segments I 1 , .…”

“…. ⊂ D with the following properties: 1 4 μ(Q(T j )) for all j 0. (T2) E Q 0 ∩ Q(T j ) ⊂ ∪Leaves(T j ) for all j 0.…”

Geometric Measure Theory–Recent Applications

“…Similar results hold on chord arc domains (these are NTA domains for which the surface measure to the boundary is Ahlfors regular; that is, the surface measure of a ball centered on the boundary and of radius grows like ) (see [24], [59]). The relationship between quantitative absolute continuity properties of harmonic measure with respect to surface measure and the regularity of the boundary (also expressed in quantitative terms) is now very well understood, see for example [3,6,7,15,[34][35][36].…”

Square function estimates, the BMO Dirichlet problem, and absolute continuity of harmonic measure on lower-dimensional sets

“…The emerging philosophy is that the key geometric properties at play are smoothness (or to be precise, rectifiability) and connectedness of the domain. In higher dimensions, the latter is much trickier, and without any pertinent details we mention that the absolute continuity of the harmonic measure with respect to the boundary surface measure has been proved in Lipschitz graph domains [Da], and later in the so-called chord-arc domains in [DJ, Se], and more recent achievements in the field have progressively further weakened the underlying geometric hypotheses [BL,Ba,HM1,AHMNT,Mo,ABaHM,ABoHM,Az,HM2], although the sharp assumptions, particularly in terms of connectedness, are not completely clear yet. Meanwhile in the converse direction, the necessary conditions for the absolute continuity of harmonic measure with respect to the Hausdorff measure of the boundary have been obtained in 1-sided chord-arc domains in [HMU] (see also [AHMNT]), and later in more general domains in [MT, HLMN].…”