Carleson measure estimates and -approximation for bounded harmonic functions, without Ahlfors regularity assumptions
John Garnett
University of California at Los Angeles, USA
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Abstract
Let be a domain in , where . It is known that if satisfies a corkscrew condition and is -Ahlfors regular, then the following are equivalent:
(a) a square function Carleson measure estimate holds for bounded harmonic functions on ;
(b) an -approximation property holds for all such functions and all ;
(c) is uniformly rectifiable.
Here we explore (a) and (b) when is not required to be Ahlfors regular. We first observe that (a) and (b) hold for any domain for which there exists a domain such that is uniformly rectifiable and . We then assume satisfies a corkscrew condition and satisfies a capacity density condition. Under these assumptions, we prove conversely that if (a) or (b) holds for then such a domain exists. And we give two further characterizations of domains where (a) or (b) holds. The first is that harmonic measure for satisfies a Carleson packing condition with respect to diameters similar to a condition comparing harmonic measures to already known to be equivalent to uniform rectifiability. The second characterization is reminiscent of the Carleson measure description of interpolating sequences in the unit disc.
Cite this article
John Garnett, Carleson measure estimates and -approximation for bounded harmonic functions, without Ahlfors regularity assumptions. Rev. Mat. Iberoam. 38 (2022), no. 1, pp. 323–351
DOI 10.4171/RMI/1288