Solvability of the Dirichlet problem with data in $L^p$ for some finite $p$ for elliptic operators, such as the Laplacian, amounts to showing that the associated elliptic/harmonic measure satisfies a Reverse Hölder inequality. Under strong connectivity assumptions, it has been proved that such a solvability is equivalent to the fact that that all bounded null-solutions of the operator in question satisfy Carleson measure estimates. In this talk, we will give a historical overview of this theory and present some recent results in collaboration with M. Cao and P. Hidalgo where, without any connectivity, we characterize certain weak Carleson measure estimates for bounded null-solutions in terms of a Corona decomposition for the elliptic measure. This extends the previous theory to non-connected settings where, as a consequence of our method, we establish Fefferman-Kenig-Pipher perturbation results.

The video of this talk is available on the IISc Math Department channel.