On the condition for elliptic operators in 1-sided nontangentially accessible domains satisfying the capacity density condition

Let Omega subset of Rn+1, n >= 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 = - div(A(0)del u), Lu = - div(A del u) be two real (not necessarily symmetric) uniformly elliptic operators in Omega, and write at omega(L0),omega(L) for the respective associated elliptic measures. We establish the equivalence between the following properties: (i) omega(L) is an element of A(infinity)(omega(L0)), (ii) L is L-P (omega(L0))-solvable for some p is an element of (1, infinity), (iii) bounded null solutions of L satisfy Carleson measure estimates with respect to omega(L0), (iv) S < Ar (i.e., the conical square function is controlled by the nontangential maximal function) in L-q (omega(L0)) for some (or for all) q is an element of (0, infinity) for any null solution of L, and (v) L is BMO(omega(L0))-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(X)(L)(S) (S) for an arbitrary Borel set S subset of partial derivative Omega).
The authors would like to thank the anonymous referees for their comments to improve the presentation of the paper. All the authors acknowledge financial support from MCIN/AEI/ 10.13039/501100011033 grants FJC2018-038526-I (first authors), CEX2019-000904-S (first, third and last authors), PID2019-107914GB-I00 (first and third authors), MTM2017-84058- P (second author) and PID2020-116398GB-I00 (last author). The third author also acknowledges that the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT.