Carleson perturbations of locally Lipschitz elliptic operators

Abstract: In 1-sided Chord Arc Domains $\Omega$, we show that the $A_\infty$-absolute continuity of the elliptic measure is stable under $L^2$ Carleson perturbations provided that the elliptic operator $L_0=-{\rm div} A_0\nabla$ which is being perturbed satisfies [\sup_{X\in \Omega }{\rm dist}(X,\partial \Omega)|\nabla A_0(X)| <\infty.] $L^2$ Carleson perturbations are are more natural and slightly more general than the Carleson perturbations previously found in the literature (those would be "$L^\infty$ Carleson perturbations"). We decided to limit the article to 1-sided CAD to make it more accessible, but the proof can easily extended to more general setting as long as we have a rich elliptic theory and the domain is 1-sided NTA.