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 $|\partial_tA|$ 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.