Uniform rectifiability from Carleson measure estimates and ε-approximability of bounded harmonic functions

Abstract

Let Ω ⊂ R n+1 , n ≥ 1, be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that ∂Ω is uniformly n-rectifiable if every bounded harmonic function on Ω is ε-approximable or if every bounded harmonic function on Ω satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when Ω = R n+1 \ E and E is Ahlfors-David regular. Our results solve a conjecture posed by Hofmann, Martell, and Mayboroda in a recent work where they proved the converse statements. Here we also obtain two additional criteria for uniform rectifiability. One is given in terms of the so-called "S < N " estimates, and another in terms of a suitable corona decomposition involving harmonic measure.

Uniform rectifiability and ε-approximability of harmonic functions in L p

“…This follows from the recent results of S. Bortz and the second author [4]. Hence, combining our results with the results in [4,13,18] gives us the following characterization theorem: Theorem 1.6. -Suppose that E ⊂ R n+1 is an n-dimensional ADR set and let Ω := R n+1 \ E. The following conditions are equivalent:…”

“…It has been used to e.g. explore the absolute continuity properties of elliptic measures [15,22] and, very recently, give a new characterization of uniform rectifiability [13,18].…”

“…Recently, the first author, Martell and Mayboroda showed that if E is a UR set of codimension 1, then every bounded harmonic function in R n+1 \ E is ε-approximable for every ε ∈ (0, 1) [18]. After this, it was shown by Garnett, Mourgoglou and Tolsa that ε-approximability of bounded harmonic functions implies uniform rectifiability for n-ADR sets [13]. This characterization result was then generalized for a class of elliptic operators by Azzam, Garnett, Mourgoglou and Tolsa [1].…”

$\varepsilon $-approximability of harmonic functions in $L^p$ implies uniform rectifiability

“…In the proof of the corona decomposition for harmonic measure in [GMT], εapproximability is used only to prove a packing condition for the low density cubes [GMT,Lemma 3.7]. Thus, we actually have:…”

“…By [GMT,Proposition 5.1], Theorem 1.2 is enough to imply Theorem 1.1. We provide some context to the results herein.…”