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

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