Solvability of the Dirichlet problem for a new class of elliptic operators

Abstract

We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot )$ on the upper half space. It is known that if the matrix $A$ is independent in the transversal $t$-direction, then we have $\omega \in A_{\infty }(\sigma )$. In the present paper we improve on the $t$-independence condition by introducing a mixed $L^1-L^{\infty }$ Carleson type condition that only depends on ${\partial }_{t}A$ and show $\omega \in A_{\infty }(\sigma )$ under this condition. This condition is different from other conditions in the literature. In the case of the upper half plane, we obtain the improvement that an $L^1$-Carleson condition on $|{\partial }_{t}A|$ implies $\omega \in A_{\infty }(\sigma )$. In particular, this condition is similar to an $L^1$-version of the DKP condition with derivative in only the transversal direction.