On the $A_{\infty }$ condition for elliptic operators in 1-sided nontangentially accessible domains satisfying the capacity density condition

Mingming Cao ; Óscar Domínguez ; José María Martell ; Pedro Tradacete

Forum of Mathematics, Sigma, Tome 10 (2022)

Résumé

Let $\Omega \subset \mathbb {R}^{n+1}$, $n\ge 2$, be a $1$-sided nontangentially accessible domain, that is, a set which is quantitatively open and path-connected. Assume also that $\Omega $ satisfies the capacity density condition. Let $L_0 u=-\mathop {\operatorname {div}}\nolimits (A_0 \nabla u)$, $Lu=-\mathop {\operatorname {div}}\nolimits (A\nabla u)$ be two real (not necessarily symmetric) uniformly elliptic operators in $\Omega $, and write $\omega _{L_0}, \omega _L$ for the respective 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) $\mathcal {S}\mathcal {N}$ (i.e., the conical square function is controlled by the nontangential 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 given by characteristic functions of Borel sets (i.e, $u(X)=\omega _L^X(S)$ for an arbitrary Borel set $S\subset \partial \Omega $).Also, we obtain a qualitative analog of the previous equivalences. Namely, 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 $\omega _{L_0}$-almost everywhere of the truncated conical square function for any bounded null solution of L. As applications, we show that $\omega _{L_0}$ is absolutely continuous with respect to $\omega _L$ if the disagreement of the coefficients satisfies some qualitative quadratic estimate in truncated cones for $\omega _{L_0}$-almost everywhere vertex. Finally, when $L_0$ is either the transpose of L or its symmetric part, we obtain the corresponding absolute continuity upon assuming that the antisymmetric part of the coefficients has some controlled oscillation in truncated cones for $\omega _{L_0}$-almost every vertex.

Détail

DOI : 10.1017/fms.2022.50