  The present paper pioneers the study of the Dirichlet problem with $L^q$
boundary data for second order operators with complex coefficients in domains
with lower dimensional boundaries, e.g., in $\Omega := \mathbb R^n \setminus
\mathbb R^d$ with $d<n-1$. Following the first results of Guy David and the two
first authors, the article introduces an appropriate degenerate elliptic
operator and show that the Dirichlet problem is solvable for all $q>1$ provided
that the coefficients satisfy the small Carleson norm condition.
  Even in the context of the classical case $d=n-1$, (the analogues of) our
results are new. The conditions on the coefficients are more relaxed than the
previously known ones (most notably, we do not impose any restrictions
whatsoever on the first $n-1$ rows of the matrix of coefficients) and the
results are more general. We establish local rather than global estimates
between the square function and the non-tangential maximal function and,
perhaps even more importantly, we establish new Moser-type estimates at the
boundary and improve the interior ones.
