Equivalence between solvability of the Dirichlet and Regularity problem under an L¹ Carleson condition on ₜ A

We study an elliptic operator L: =div (A) on the upper half space. It is known that solvability of the Regularity problem in Ẇ^1, p implies solvability of the adjoint Dirichlet problem in L^p'. Previously, Shen (2007) established a partial reverse result. In our work, we show that if we assume an L¹-Carleson condition on only |ₜ A| the full reverse direction holds. As a result, we obtain equivalence between solvability of the Dirichlet problem (D) ^*₏' and the Regularity problem (R) ₚ under this condition. As a further consequence, we can extend the class of operators for which the Lᵖ Regularity problem is solvable by operators satisfying the mixed L¹-L^ condition. Additionally in the case of the upper half plane, this class includes operators satisfying this L¹-Carleson condition on |ₜ A|.