Carleson measure estimates, corona decompositions, and perturbation of elliptic operators without connectivity

TL;DR

This paper develops a corona decomposition approach to characterize Carleson measure estimates and boundary rectifiability for elliptic operators in non-connected domains, extending perturbation results beyond connected settings.

Contribution

It introduces corona decompositions linked to elliptic measure, enabling perturbation and rectifiability results without the need for domain connectivity.

Findings

Corona decomposition characterizes Carleson measure estimates.

Extension of Fefferman-Kenig-Pipher perturbation to non-connected domains.

Boundary is uniformly rectifiable if Carleson measure estimates hold for solutions.

Abstract

Let $\Omega$ be an open set with Ahlfors-David regular boundary satisfying the corkscrew condition. When $\Omega$ is connected in some quantitative form one can establish that for any real elliptic operator with bounded coefficients, the quantitative absolute continuity of elliptic measures is equivalent to the fact that all bounded null solutions satisfy Carleson measure estimates. In turn, in the same setting these equivalent properties are stable under Fefferman-Kenig-Pipher perturbations. However, without connectivity, there is no Fefferman-Kenig-Pipher perturbation result available. In this paper, we work with a corona decomposition associated with the elliptic measure and show that it is equivalent to the fact that bounded null solutions satisfy partial/weak Carleson measure estimates, or to the fact that the Green function is comparable to the distance to the boundary in the corona sense. This characterization has profound consequences. We extend Fefferman-Kenig-Pipher's perturbation to non-connected settings. For the Laplacian, these corona decompositions or, equivalently, the partial/weak Carleson measure estimates are meaningful enough to characterize the uniform rectifiability of the boundary. As a consequence, we obtain that the boundary of the set is uniformly rectifiable if bounded null solutions for any Fefferman-Kenig-Pipher perturbation of the Laplacian satisfy Carleson measure estimates. For Kenig-Pipher operators any of the properties of the characterization is stable under transposition or symmetrization of the matrices of coefficients. As a result, we obtain that Carleson measure estimates for bounded null-solutions of non-symmetric variable operators satisfying an $L^1$-Kenig-Pipher condition occur if and only if the boundary of the open set is uniformly rectifiable. Our results generalize previous work in settings where quantitative connectivity.