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

Abstract

AbstractLet $$\Omega \subset \mathbb {R}^{n+1}$$ Ω ⊂ R n + 1 , $$n\ge 2$$ n ≥ 2 , be an open set with Ahlfors–David regular boundary satisfying the corkscrew condition. When $$\Omega $$ Ω is quantitatively connected (i.e., it satisfies the Harnack chain condition) it is known that for any real elliptic operator with bounded coefficients, the quantitative absolute continuity of elliptic measures (i.e., its membership to the class $$A_\infty $$ A ∞ ) is equivalent to the fact that all bounded null solutions satisfy Carleson measure estimates. In turn, in the same setting, it is also known that these properties are stable under Fefferman–Kenig–Pipher’s perturbations. Nonetheless, it has been an open problem to show whether, without connectivity, the previous two conditions remain equivalent or whether there is a Fefferman–Kenig–Pipher perturbation theory. In this paper we settle the question of whether, for general real elliptic operators in sets without any connectivity assumption, quantitative absolute continuity of the elliptic measure (expressed via a corona decomposition for the elliptic measure) and Carleson measure type estimates for bounded null solutions are equivalent. We show that any of these conditions is also equivalent to the fact that the Green function is comparable, in the corona sense, to the distance to the boundary. Our characterization has profound consequences. First, we extend Fefferman–Kenig–Pipher’s perturbation result to sets which are not necessarily connected, and this is the first result in this general setting. Second, in the case of the Laplacian, and more generally for $$L^1$$ L 1 -Kenig–Pipher non-symmetric operators with variable coefficients, our conditions characterize the uniform rectifiability of the boundary. Last, our results generalize previous work in settings where quantitative connectivity is assumed.