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

TL;DR

This paper demonstrates that elliptic measures for certain divergence form operators with vanishing Carleson coefficients are asymptotically optimal $A_ abla$ weights, with the logarithm of the kernel in vanishing mean oscillation, advancing understanding of their asymptotic behavior.

Contribution

It establishes asymptotically optimal $A_ abla$ estimates for elliptic measures under vanishing Carleson conditions, using new quantitative local estimates and recent foundational results.

Findings

Elliptic measure is an asymptotically optimal $A_ abla$ weight.

Logarithm of the elliptic kernel lies in vanishing mean oscillation.

Provides a new quantitative framework for analyzing elliptic measures.

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.