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 $Lu=-mathrm{div}(A abla u)$ 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 $L$.
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