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

Abstract

Let $\\Omega$ be a domain in $\\mathbb{R}^{d+1}$, where $d \\geq 1$. It is known that if $\\Omega$ satisfies a corkscrew condition and $\\partial\\Omega$ is $d$-Ahlfors regular, then the following are equivalent: (a) a square function Carleson measure estimate holds for bounded harmonic functions on $\\Omega$; (b) an $\\varepsilon$-approximation property holds for all such functions and all $0 < \\varepsilon < 1$; (c) $\\partial \\Omega$ is uniformly rectifiable. Here we explore (a) and (b) when $\\partial \\Omega$ is not required to be Ahlfors regular. We first observe that (a) and (b) hold for any domain $\\Omega$ for which there exists a domain $\\tilde\\Omega \\subset \\Omega$ such that $\\partial \\tilde \\Omega$ is uniformly rectifiable and $\\partial \\Omega \\subset \\partial \\tilde \\Omega$. We then assume $\\Omega$ satisfies a corkscrew condition and $\\partial \\Omega$ satisfies a capacity density condition. Under these assumptions, we prove conversely that if (a) or (b) holds for $\\Omega$ then such a domain $\\tilde \\Omega \\supset \\Omega$ exists. And we give two further characterizations of domains where (a) or (b) holds. The first is that harmonic measure for $\\Omega$ satisfies a Carleson packing condition with respect to diameters similar to a condition comparing harmonic measures to $\\mathcal{H}^d$ already known to be equivalent to uniform rectifiability. The second characterization is reminiscent of the Carleson measure description of $H^{\\infty}$ interpolating sequences in the unit disc.