Perturbations of elliptic operators in 1-sided chord-arc domains. Part II: Non-symmetric operators and Carleson measure estimates

Abstract

We generalize to the setting of 1-sided chord-arc domains, that is, to domains satisfying the interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness) and which have an Ahlfors regular boundary, a result of Kenig-Kirchheim-Pipher-Toro, in which Carleson measure estimates for bounded solutions of the equation L u = − div ⁡ ( A ∇ u ) = 0 Lu=-\operatorname {div}(A\nabla u) = 0 with A A being a real (not necessarily symmetric) uniformly elliptic matrix imply that the corresponding elliptic measure belongs to the Muckenhoupt A ∞ A_\infty class with respect to surface measure on the boundary. We present two applications of this result. In the first one we extend a perturbation result recently proved by Cavero-Hofmann-Martell presenting a simpler proof and allowing non-symmetric coefficients. Second, we prove that if an operator L L as above has locally Lipschitz coefficients satisfying certain Carleson measure condition, then ω L ∈ A ∞ \omega _L\in A_\infty if and only if ω L ⊤ ∈ A ∞ \omega _{L^\top }\in A_\infty . As a consequence, we can remove one of the main assumptions in the non-symmetric case of a result of Hofmann-Martell-Toro and show that if the coefficients satisfy a slightly stronger Carleson measure condition the membership of the elliptic measure associated with L L to the class A ∞ A_\infty yields that the domain is indeed a chord-arc domain.