A stability result for parabolic measures of operators with singular drifts

TL;DR

This paper proves stability of parabolic measures for operators with singular drifts in the upper-half space, establishing quantitative estimates based on oscillation and Carleson measure conditions.

Contribution

It introduces new estimates for parabolic Green functions and a novel Carleson measure criterion for anisotropic A-infinity weights, advancing understanding of parabolic operators with singular drifts.

Findings

Quantitative A-infinity estimates for parabolic measures.

New Green function deviation estimates.

A novel Carleson measure criterion for anisotropic weights.

Abstract

We study the operator \[ \partial_t - \text{div} A \nabla + B \cdot \nabla \] in parabolic upper-half-space, where $A$ is an elliptic matrix satisfying an oscillation condition and $B$ is a singular drift with a Carleson control. Our main result establishes quantitative $A_{\infty}$-estimates for the parabolic measure in terms of oscillation of $A$ and smallness of $B$. The proof relies on new estimates for parabolic Green functions that quantify their deviations from linear functions of the normal variable and on a novel, quantitative Carleson measure criterion for anisotropic $A_{\infty}$-weights.