BMO solvability with singular drifts on ample sawtooth domains implies $L^p$ solvability

TL;DR

This paper demonstrates that BMO solvability in certain sawtooth domains for elliptic operators with singular drifts implies the weak A_infty condition for elliptic measure, leading to L^p solvability of the Dirichlet problem.

Contribution

It establishes a link between BMO solvability and L^p solvability for elliptic operators with singular drifts on ample sawtooth domains.

Findings

BMO solvability implies weak A_infty condition for elliptic measure.

Weak A_infty condition leads to L^p solvability of the Dirichlet problem.

Results extend to more general Lipschitz domains.

Abstract

For a linear elliptic operator with a singular drift that satisfies a finite Carleson measure condition, we prove that there exist `ample' sawtooth domains of the unit ball $B(0,1)\subset \R^{n+1}$ so that a BMO solvability assumption in these sawtooth subdomains implies that the elliptic measure satisfies the weak $A_\infty$ condition with respect to the surface measure on this `ample' sawtooth domain. This is a quantifiable absolute continuity condition, which is equivalent to saying the $L^p$ Dirichlet problem is solvable for some $1<p<\infty$. Such singular drifts have been considered in the literature in the context of perturbative $L^p$ Dirichlet solvability problems, by Hofmann-Lewis and Kenig-Pipher. By an ample sawtooth domain, we mean a sawtooth domain whose boundary coincides with the boundary of the unit ball, except for an arbitrarily small fraction. The methods can be naturally extended to show the result for more general bounded Lipschitz domains.