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

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.