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

Abstract

Let $\Omega$ be a domain in ${\mathbb{R}}^{d+1}$, where $d\ge 1$. 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 $\epsilon$-approximation property holds for all such functions and all $0<\epsilon <1$;
(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.