The regularity problem with a weaker condition on only the transversal direction

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 the regularity boundary value problem is solvable with data in a Sobolev space. In the present paper we improve on the $t$-independence condition by introducing a mixed $L^1-L^{\mathrm{\infty }}$ condition that only depends on ${\mathrm{\partial }}_{t}A$, the derivative of $A$ in transversal direction. This condition is different from other conditions in the literature and has already been proven to imply solvability of the Dirichlet boundary value problem.