BMO solvability and the $A_{\infty }$ condition for second order parabolic operators

Abstract

We prove that a sharp regularity property ($A_{\infty }$) of parabolic measure for operators in certain time-varying domains is equivalent to a Carleson measure property of bounded solutions. This equivalence was established in the elliptic case by Kenig, Kirchheim, Pipher and Toro, improving an earlier result of Kenig, Dindos and Pipher for solutions with data in BMO. The connection between regularity of the elliptic measure and certain Carleson measure properties of solutions was established in order to study solvability of boundary value problems for non-symmetric divergence form operators (Kenig, Koch, Pipher, and Toro). The extension to the parabolic setting requires an approach to the key estimates of the aforementioned works that primarily exploits the maximum principle. For various classes of parabolic operators ([24]), this criterion also provides an easier route to establish the solvability of the Dirichlet problem with data in $L^p$ for some $p$, and also to quantify these results in several aspects.