Absolute continuity of degenerate elliptic measure

Let $\Omega \subset \mathbb{R}^{n+1}$ 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 $\partial \Omega$ with appropriate size conditions. Let $Lu=-\mathrm{div}(A\nabla u)$ 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) $\omega_L \in A_{\infty}(\mu)$, (ii) $L$ is $L^p(\mu)$-solvable for some $p \in (1, \infty)$, (iii) every bounded null solution of $L$ satisfies a Carleson measure estimate with respect to $\mu$, (iv) the conical square function is controlled by the non-tangential maximal function in $L^q(\mu)$ for some (or for all) $q \in (0, \infty)$ for any null solution of $L$, (v) $L$ is $\mathrm{BMO}(\mu)$-solvable, and (vi) every bounded null solution of $L$ is $\varepsilon$-approximable for any $\varepsilon>0$. 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 $L$.