  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
$L$, and (v) $L$ is $\mathrm{BMO}(\mu)$-solvable. 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 $L$.
