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 is a Muckenhoupt weight. In turn, any of these

conditions occurs if and only if the gradient of every bounded null solution

satisfies a Carleson measure estimate. In this talk, we will consider a

qualitative analog of the latter equivalence showing that one can characterize

the absolute continuity of surface measure with respect to elliptic measure in

terms of the finiteness almost every where of the conical square function for

any bounded null solution. This result is obtained in the context of 1-sided

chord-arc domains, that is, sets which are quantitatively open and connected

with a boundary which is Ahlfors regular.

Joint work with Mingming Cao and Andrea Olivo.