Resumen de Carleson measure estimates and ε-approximation for bounded harmonic functions, without Ahlfors regularity assumptions

John Garnett

• Let Ω be a domain in Rd+1, where d≥1. It is known that if Ω satisfies a corkscrew condition and ∂Ω is d-Ahlfors regular, then the following are equivalent:

  (a) a square function Carleson measure estimate holds for bounded harmonic functions on Ω;

  (b) an ε-approximation property holds for all such functions and all 0<ε<1;

  (c) ∂Ω is uniformly rectifiable.

  Here we explore (a) and (b) when ∂Ω is not required to be Ahlfors regular. We first observe that (a) and (b) hold for any domain Ω for which there exists a domain Ω~⊂Ω such that ∂Ω~ is uniformly rectifiable and ∂Ω⊂∂Ω~. We then assume Ω satisfies a corkscrew condition and ∂Ω satisfies a capacity density condition. Under these assumptions, we prove conversely that if (a) or (b) holds for Ω then such a domain Ω~⊃Ω exists. And we give two further characterizations of domains where (a) or (b) holds. The first is that harmonic measure for Ω satisfies a Carleson packing condition with respect to diameters similar to a condition comparing harmonic measures to Hd already known to be equivalent to uniform rectifiability. The second characterization is reminiscent of the Carleson measure description of H∞ interpolating sequences in the unit disc.