UNIFORM RECTIFIABILITY FROM CARLESON MEASURE ESTIMATES AND ε-APPROXIMABILITY OF BOUNDED HARMONIC FUNCTIONS

Let Omega subset of Rn+1, n >= 1, be a corkscrew domain with Ahlfors-David regular boundary. In this article we prove that partial derivative Omega is uniformly n-rectifiable if every bounded harmonic function on Omega is epsilon-approximable or if every bounded harmonic function on Omega satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when Omega = Rn+1 \ E and E 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 S < N estimates and another in terms of a suitable corona decomposition involving harmonic measure.