Elliptic measures for Dahlberg-Kenig-Pipher operators: Asymptotically optimal estimates

Abstract

Questions concerning quantitative and asymptotic properties of the elliptic measure corresponding to a uniformly elliptic divergence form operator have been the focus of recent studies. In this setting we show that the elliptic measure of an operator with coefficients satisfying a vanishing Carleson condition in the upper half space is an asymptotically optimal $A_\infty$ weight. In particular, for such operators the logarithm of the elliptic kernel is in the space of (locally) vanishing mean oscillation. To achieve this, we prove local, quantitative estimates on a quantity (introduced by Fefferman, Kenig and Pipher) that controls the $A_\infty$ constant. Our work uses recent results obtained by David, Li and Mayboroda. These quantitative estimates may offer a new framework to approach similar problems.