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

Abstract

We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ 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 $\omega\in A_\infty(\sigma)$. These include the $t$-independence condition, a mixed $L^1-L^\infty$ condition on $\partial_t A$, and Dini-type conditions. We introduce an $L^1$ Carleson condition on $\partial_t A(x,t)$ that extends the class of elliptic operators for which we have $\omega\in A_\infty(\sigma)$, i.e. solvability of the $L^p$ Dirichlet problem for some $1<p<\infty$.