We will explain the proof of a Carleson measure estimate on solutions of parabolic equations with real measurable time-dependent coefficients that implies that the parabolic measure is an $A_\infty$ weight.

This corresponds to the parabolic analog of a recent result by Hofmann, Kenig, Mayboroda and Pipher for elliptic equations. Our proof even simplifies theirs. As is well known, the $A_\infty$ property implies that   $L^p$ Dirichlet problem is well-posed. An important ingredient of the proof is a Kato square root property for parabolic operators on the boundary, which can be seen as a consequence of certain square function estimates applicable to Neumann and regularity problems.  All this is  joint work with Moritz Egert and Kaj Nyström.