On the $A_\infty$ condition for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

Abstract

Let $\Omega \subset \mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided non-tangentially accessible domain (i.e., quantitatively open and path-connected) satisfiying the capacity density condition. Let $L_0 u=-\mathrm{div}(A_0 \nabla u)$, $Lu=-\mathrm{div}(A\nabla u)$ be two real uniformly elliptic operators in $\Omega$, with $\omega_{L_0}, \omega_L$ the associated elliptic measures. We establish the equivalence between the following properties: (i) $\omega_L \in A_{\infty}(\omega_{L_0})$, (ii) $L$ is $L^p(\omega_{L_0})$-solvable for some $p\in (1,\infty)$, (iii) bounded null solutions of $L$ satisfy Carleson measure estimates with respect to $\omega_{L_0}$, (iv) the conical square function is controlled by the non-tangential maximal function in $L^q(\omega_{L_0})$ for some (or for all) $q\in (0,\infty)$ for any null solution of $L$, and (v) $L$ is $\mathrm{BMO}(\omega_{L_0})$-solvable. Moreover, in each of the properties (ii)-(v) it is enough to consider the class of solutions $u(X)=\omega_L^X(S)$ with arbitrary Borel sets $S\subset\partial\Omega$. Also, we characterize the absolute continuity of $\omega_{L_0}$ with respect to $\omega_L$ in terms of some qualitative local $L^2(\omega_{L_0})$ estimates for the truncated conical square function for any bounded null solution of $L$. This is also equivalent to the finiteness $\omega_{L_0}$-a.e. of the truncated conical square function for any bounded null solution of $L$. As applications, we show that $\omega_{L_0}\ll\omega_L$ if the disagreement of the coefficients satisfies some qualitative quadratic estimate in truncated cones for $\omega_{L_0}$-a.e. vertex. Finally, when $L_0$ is either the transpose of $L$ or its symmetric part, we obtain the corresponding absolute continuity when the antisymmetric part of the coefficients has some controlled oscillation in truncated cones for $\omega_{L_0}$-a.e. vertex.