Solvability of the Dirichlet problem using a weaker Carleson condition in the upper half plane

We study an elliptic operator $L:=\mathrm{d}\mathrm{i}\mathrm{v}(A\mathrm{\nabla }\cdot )$ on the upper half plane ${\mathbb{R}}_{+}^2$. There are several conditions on the behavior of the matrix $A$ in the transversal $t$-direction that yield $\omega \in A_{\mathrm{\infty }}(\sigma )$. These include the $t$-independence condition, a mixed $L^1-L^{\mathrm{\infty }}$ condition on ${\mathrm{\partial }}_{t}A$, and Dini-type conditions. We introduce an $L^1$ Carleson condition on ${\mathrm{\partial }}_{t}A(x,t)$ that extends the class of elliptic operators for which we have $\omega \in A_{\mathrm{\infty }}(\sigma )$, i.e. solvability of the $L^p$ Dirichlet problem for some $1<p<\mathrm{\infty }$.