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.