The $A_\infty$ condition, $\varepsilon$-approximators, and Varopoulos extensions in uniform domains

Abstract: Suppose that $\Omega \subset\mathbb R^{n+1}$, $n\geq1$, 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 $\partial\Omega$ in the sense that $\omega_L \in A_\infty(\sigma)$ if and only if any bounded solution $u$ to $Lu = 0$ in $\Omega$ is $\varepsilon$-approximable for any $\varepsilon \in (0,1)$. By $\varepsilon$-approximability of $u$ we mean that there exists a function $\Phi = \Phi^\varepsilon$ such that $|u-\Phi|{L^\infty(\Omega)} \le \varepsilon|u|{L^\infty(\Omega)}$ and the measure $\widetilde{\mu}_\Phi$ with $d\widetilde{\mu} = |\nabla \Phi(Y)| \, dY$ is a Carleson measure with $L^\infty$ control over the Carleson norm. As a consequence of this approximability result, we show that boundary $\operatorname{BMO}$ 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 $\operatorname{BMO}$ 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.