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

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 $1<p<\mathrm{\infty }$. 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.