  In this article, we give an alternative and simpler proof of the solvability
of the regularity problem - i.e. the Dirichlet problem with boundary data in
$W^{1,p}$ - for uniformly elliptic operators on $\mathbb R^n_+$ satisfying a
(possibly large) Carleson condition. We also slightly enlarge the class of
operators for which the regularity problem is solvable and gives the analogue
result for weighted uniformly elliptic operators on $\mathbb R^n \setminus
\mathbb R^d$, $d<n-1$.
