2020/06/18 by John L. Garnett, John Garnett, Garnett, John
Mathematics · #Advanced Harmonic Analysis Research #math.CA
paper · pdf · doi:10.48550/arxiv.2006.10682
arxiv created 2020/07/26 · arxiv updated 2020/07/28
Let Ω be a domain in ℝd+1, d ≥ 1. In the paper's references [HMM2] and [GMT] it was proved that if Ω satisfies a corkscrew condition and if ∂ Ω is d-Ahlfors regular, i.e. Hausdorff measure Hd(B(x,r) ∩ ∂ Ω) ∼ rd for all x ∈ ∂ Ω and 0 < r < \rm diam(∂ Ω), then ∂ 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 ε-approximation property for all 0 < ε <1 for every such function. Here we explore (a) and (b) when ∂ 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⊂ Ω such that ∂ Ω⊂ ∂ \widetilde Ω and ∂ \widetilde Ω is uniformly rectifiable. We next assume Ω satisfies a corkscrew condition and ∂ 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∞ interpolating sequences in the unit disc.