Let , , be a corkscrew domain with Ahlfors–David regular boundary. In this article we prove that is uniformly -rectifiable if every bounded harmonic function on is -approximable or if every bounded harmonic function on satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when and is Ahlfors–David regular. Our results establish a conjecture posed by Hofmann, Martell, and Mayboroda, in which they proved the converse statements. Here we also obtain two additional criteria for uniform rectifiability, one in terms of the so-called estimates and another in terms of a suitable corona decomposition involving harmonic measure.
"Uniform rectifiability from Carleson measure estimates and -approximability of bounded harmonic functions." Duke Math. J. 167 (8) 1473 - 1524, 1 June 2018. https://doi.org/10.1215/00127094-2017-0057