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

TL;DR

This paper extends Carleson measure estimates and epsilon-approximation properties for bounded harmonic functions to domains without Ahlfors regularity, using geometric conditions involving uniformly rectifiable subsets.

Contribution

It generalizes previous results by removing Ahlfors regularity assumptions, establishing these properties under weaker geometric conditions involving uniformly rectifiable boundaries.

Findings

Carleson measure estimates hold without Ahlfors regularity

Epsilon-approximation properties are valid under weaker geometric conditions

Characterizations involve harmonic measure and corona decomposition

Abstract

Let $\Omega$ be a domain in $\mathbb{R}^{d+1}$, $d \geq 1$. In the paper's references [HMM2] and [GMT] it was proved that if $\Omega$ satisfies a corkscrew condition and if $\partial \Omega$ is $d$-Ahlfors regular, i.e. Hausdorff measure $\mathcal{H}^d(B(x,r) \cap \partial \Omega) \sim r^d$ for all $x \in \partial \Omega$ and $0 < r < {\rm diam}(\partial \Omega)$, then $\partial \Omega$ is uniformly rectifiable if and only if (a) a square function Carleson measure estimate holds for every bounded harmonic function on $\Omega$ or (b) an $\varepsilon$-approximation property for all $0 < \varepsilon <1$ for every such function. Here we explore (a) and (b) when $\partial \Omega$ is not required to be Ahlfors regular. We first prove that (a) and (b) hold for any domain $\Omega$ for which there exists a domain $\widetilde \Omega \subset \Omega$ such that $\partial \Omega \subset \partial \widetilde \Omega$ and $\partial \widetilde \Omega$ is uniformly rectifiable. We next assume $\Omega$ satisfies a corkscrew condition and $\partial \Omega$ satisfies a capacity density condition. Under these assumptions we prove conversely that the existence of such $\widetilde \Omega$ implies (a) and (b) hold on $\Omega$ and give further characterizations of domains for which (a) or (b) holds. One is that harmonic measure satisfies a Carleson packing condition for diameters similar to the corona decompositionm proved equivalent to uniform rectifiability in [GMT]. The second characterization is reminiscent of the Carleson measure description of $H^{\infty}$ interpolating sequences in the unit disc.