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 $Lu=-{\rm div}(A\nabla u) = 0$ with $A$ being a real (not necessarily symmetric) uniformly elliptic matrix, imply that the corresponding elliptic measure belongs to the Muckenhoupt $A_\infty$ 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 $L$ as above has locally Lipschitz coefficients satisfying certain Carleson measure condition then $\omega_L\in A_\infty$ if and only if $\omega_{L^\top}\in A_\infty$. 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 $L$ to the class $A_\infty$ yields that the domain is indeed a chord-arc domain.

中文翻译：

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单边弦弧域中椭圆算子的扰动。第二部分：非对称算子和 Carleson 测度估计

我们推广到单边弦弧域的设置，即满足内部 Corkscrew 和 Harnack Chain 条件的域（这些分别是开放性和路径连通性的尺度不变/定量版本）并且具有 Ahlfors规则边界，Kenig-Kirchheim-Pipher-Toro 的结果，其中 Carleson 测量方程 $Lu=-{\rm div}(A\nabla u) = 0$ 的有界解的估计，其中 $A$ 是实数（不一定是对称的）一致椭圆矩阵，意味着对应的椭圆测度属于 Muckenhoupt $A_\infty$ 类关于边界上的表面测度。我们展示了这个结果的两个应用。在第一个中，我们扩展了最近由 Cavero-Hofmann-Martell 证明的扰动结果，提出了一个更简单的证明并允许非对称系数。第二，我们证明，如果上述算子$L$具有满足某些Carleson测度条件的局部Lipschitz系数，则$\omega_L\in A_\infty$当且仅当$\omega_{L^\top}\in A_\infty$。因此，我们可以去除 Hofmann-Martell-Toro 结果的非对称情况下的主要假设之一，并表明如果系数满足稍强的 Carleson 测度条件，则与 $L 相关的椭圆测度的成员资格$ 到类 $A_\infty$ 产生域确实是一个和弦弧域。