  We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper
half space. It is known that if the matrix $A$ is independent in the
transversal $t$-direction, then we have $\omega\in A_\infty(\sigma)$. In the
present paper we improve on the $t$-independece condition by introducing a
mixed $L^1-L^\infty$ Carleson type condition that only depends on $\partial_t
A$ and show $\omega\in A_\infty(\sigma)$ under this condition. This mixed
$L^1-L^\infty$ Carleson type condition applies to elliptic operators not
covered by the DKP condition, the $t$-independence condition or the Dini
condition.
