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

Abstract

Let Ω ⊂ R n+1 , n ≥ 2, be an open set with Ahlfors-David regular boundary satisfying the corkscrew condition. When Ω is connected in some quantitative form (more precisely, it satisfies the Harnack chain condition) one can establish that for any real elliptic operator with bounded coefficients, the quantitative absolute continuity of elliptic measures (i.e., its membership to the class 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 equivalent properties are stable under Fefferman-Kenig-Pipher perturbations. However, without connectivity, not much is known in general and, in particular, 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. First, we extend Fefferman-Kenig-Pipher's perturbation result to non-connected settings. Second, in the case of the Laplacian, these corona decompositions or, equivalently, the partial/weak Carleson measure estimates are meaningful enough to characterize the uniform rectifiability of the boundary of the open set. 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 (partial/weak) Carleson measure estimates. That is, there is a characterization of the uniform rectifiability of the boundary in terms of the properties of the bounded null solutions for operators whose coefficients may not possess any regularity. Third, for Kenig-Pipher operators any of the properties of the characterization is stable under transposition or under symmetrization of the matrices of coefficients. As a result, we obtain that Carleson measure estimates (or its partial/weak form) for bounded null-solutions of nonsymmetric variable operators satisfying an L 1 -Kenig-Pipher condition occur if and only if the boundary of the open set is uniformly rectifiable. Last, our results generalize previous work in settings where quantitative connectivity is assumed since, in that more topologically friendly settings, our conditions are equivalent to the fact that the elliptic measure is a Muckenhoupt weight. ContentsMINGMING CAO, PABLO HIDALGO-PALENCIA, AND JOS É MAR ÍA MARTELL 2. Preliminaries 7 2.1. Notation and definitions 7 2.2. Dyadic grids and sawtooths 11 2.3. PDE estimates 16 2.4. Auxiliary results 17 3. Proof of Theorem 1.1 20 3.1. Proof of (b) =⇒ (c): ω L admits a corona decomposition implies that G L is comparable to the distance to the boundary in the corona sense 20 3.2. Proof of (c) =⇒ (d): G L is comparable to the distance to the boundary in the corona sense implies that L satisfies partial/weak Carleson measu...

Green functions and smooth distances

“…Corona decompositions are a useful and popular tool in the recent literature pertaining to uniformly rectifiable sets, see for instance [3,4,8,9,12,18,33,36,46,47] to cite only a few.…”

“…Proof of Lemma 7. 12 The first step is to replace K u/t by the elliptic measure. Take X 0 ∈ B(x, r ) ∩ and X 1 ∈ \B(x, 4r ) to be two corkscrew points for x at the scale r .…”