The doubling property of the elliptic measure, for elliptic operators with drifts satisfying an average diverging condition

Abstract: We show doubling of the elliptic measure corresponding to the operator with an elliptic principal term and a drift that diverges, on average on Whitney cubes, like the inverse distance to the boundary, with a small constant. Essentially a small Carleson constant assumption on the drift, this generalizes earlier results with the hypothesis of pointwise smallness of such a drift. This relates to recent perturbative results of rough Dirichlet solvability in domains with drifts or potentials that satisfy a Carleson measure condition, which have also been considered earlier by Hofmann-Lewis and Kenig-Pipher. While we work in 1-sided chord arc domains, these results are new even for the half-space. In the process, we also prove Hardy inequalities in such domains with Alhfors-David regular boundary, using a stopping time argument.