The A∞ Condition, ε-Approximators, and Varopoulos Extensions in Uniform Domains

Abstract

Suppose that Omega subset of Rn+1, n >= 1, is a uniform domain with n-Ahlfors regular boundary and L is a (not necessarily symmetric) divergence form elliptic, real, bounded operator in Omega. We show that the corresponding elliptic measure omega(L) is quantitatively absolutely continuous with respect to surface measure of partial derivative Omega in the sense that omega(L) is an element of A(infinity)(sigma) if and only if any bounded solution u to Lu = 0 in Omega is epsilon-approximable for any epsilon is an element of (0, 1). By epsilon-approximability of u we mean that there exists a function Phi = Phi(epsilon) such that parallel to u - Phi parallel to(L infinity(Omega)) <= epsilon parallel to u parallel to(L infinity(Omega)) and themeasure (mu) over tilde (Phi) with d (mu) over tilde = vertical bar del Phi(Y)vertical bar dY is a Carleson measure with L-infinity control over the Carleson norm. As a consequence of this approximability result, we show that boundary BMO functions with compact support can have Varopoulos-type extensions even in some sets with unrectifiable boundaries, that is, smooth extensions that converge non-tangentially back to the original data and that satisfy L-1-type Carleson measure estimates with BMO control over the Carleson norm. Our result complements the recent work of Hofmann and the third named author who showed the existence of these types of extensions in the presence of a quantitative rectifiability hypothesis.