  In nice environments, such as Lipschitz or chord-arc domains, it is
well-known that the solvability of the Dirichlet problem for an elliptic
operator in $L^p$, for some finite $p$, is equivalent to the fact that the
associated elliptic measure belongs to the Muckenhoupt class $A_\infty$. In
turn, any of these conditions occurs if and only if the gradient of every
bounded null solution satisfies a Carleson measure estimate. This has been
recently extended to much rougher settings such as those of 1-sided chord-arc
domains, that is, sets which are quantitatively open and connected with a
boundary which is Ahlfors-David regular.
  In this paper, we work in the same environment and consider a qualitative
analog of the latter equivalence showing that one can characterize the absolute
continuity of the surface measure with respect to the elliptic measure in terms
of the finiteness almost everywhere of the truncated conical square function
for any bounded null solution. As a consequence of our main result
particularized to the Laplace operator and some previous results, we show that
the boundary of the domain is rectifiable if and only if the truncated conical
square function is finite almost everywhere for any bounded harmonic function.
Also, we obtain that for two given elliptic operators $L_1$ and $L_2$, the
absolute continuity of the surface measure with respect to the elliptic measure
of $L_1$ is equivalent to the same property for $L_2$ provided the disagreement
of the coefficients satisfy some quadratic estimate in truncated cones for
almost everywhere vertex. Finally for the case on which $L_2$ is either the
transpose of $L_1$ or its symmetric part we show the equivalence of the
corresponding absolute continuity upon assuming that the antisymmetric part of
the coefficients has some controlled oscillation in truncated cones for almost
every vertex.
