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

Abstract Let Formula Presented, Formula Presented, be a Formula Presented-sided nontangentially accessible domain, that is, a set which is quantitatively open and path-connected. Assume also that Formula Presented satisfies the capacity density condition. Let Formula Presented, Formula Presented be two real (not necessarily symmetric) uniformly elliptic operators in Formula Presented, and write Formula Presented for the respective associated elliptic measures. We establish the equivalence between the following properties: (i) Formula Presented, (ii) L is Formula Presented-solvable for some Formula Presented, (iii) bounded null solutions of L satisfy Carleson measure estimates with respect to Formula Presented, (iv) Formula Presented (i.e., the conical square function is controlled by the nontangential maximal function) in Formula Presented for some (or for all) Formula Presented for any null solution of L, and (v) L is Formula Presented-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, Formula Presented for an arbitrary Borel set Formula Presented). Also, we obtain a qualitative analog of the previous equivalences. Namely, we characterize the absolute continuity of Formula Presented with respect to Formula Presented in terms of some qualitative local Formula Presented estimates for the truncated conical square function for any bounded null solution of L. This is also equivalent to the finiteness Formula Presented-almost everywhere of the truncated conical square function for any bounded null solution of L. As applications, we show that Formula Presented is absolutely continuous with respect to Formula Presented if the disagreement of the coefficients satisfies some qualitative quadratic estimate in truncated cones for Formula Presented-almost everywhere vertex. Finally, when Formula Presented 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 Formula Presented-almost every vertex.