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

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

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