Absolute continuity of degenerate elliptic measure

Let Omega subset of Rn+1 be an open set whose boundary may be composed of pieces of different dimensions. Assume that Omega satisfies the quantitative openness and connectedness, and there exist doubling measures m on Omega and mu on partial derivative Omega with appropriate size conditions. Let Lu=-div(A del u) be a real (not necessarily symmetric) degenerate elliptic operator in Omega. Write omega L for the associated degenerate elliptic measure. We establish the equivalence between the following properties: (i) omega L is an element of A(infinity)(mu), (ii) the Dirichlet problem for L is solvable in L-p(mu) for some p is an element of(1,infinity), (iii) every bounded null solution of L satisfies Carleson measure estimates with respect to mu, (iv) the conical square function is controlled by the non-tangential maximal function in L-q(mu) for all q is an element of(0,infinity) for any null solution of L, and (v) the Dirichlet problem for L is solvable in BMO(mu). On the other hand, we obtain a qualitative analogy of the previous equivalence. Indeed, we characterize the absolute continuity of omega L with respect to mu in terms of local L-2(mu) estimates of the truncated conical square function for any bounded null solution of L. This is also equivalent to the finiteness mu-almost everywhere of the truncated conical square function for any bounded null solution of L. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).