Equivalence between solvability of the Dirichlet and Regularity problem under an $L^1$ Carleson condition on ${\mathrm{\partial }}_{t}A$

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 solvability of the Regularity problem in ${\dot{W}}^{1,p}$ implies solvability of the adjoint Dirichlet problem in $L^{p^{\mathrm{\prime }}}$. Previously, Shen (2007) established a partial reverse result. In our work, we show that if we assume an $L^1$-Carleson condition on only $\mathrm{\mid }{\mathrm{\partial }}_{t}A\mathrm{\mid }$ the full reverse direction holds. As a result, we obtain equivalence between solvability of the Dirichlet problem $(D{)}_{p^{\mathrm{\prime }}}^{\ast }$ and the Regularity problem $(R{)}_p$ under this condition. As a further consequence, we can extend the class of operators for which the $L^p$ Regularity problem is solvable by operators satisfying the mixed $L^1-L^{\mathrm{\infty }}$ condition. Additionally in the case of the upper half plane, this class includes operators satisfying this $L^1$-Carleson condition on $\mathrm{\mid }{\mathrm{\partial }}_{t}A\mathrm{\mid }$.