  It was recently shown that the harmonic measure is absolutely continuous with
respect to the Hausdorff measure on a domain with an $n-1$ dimensional
uniformly rectifiable boundary, in the presence of now well understood
additional topological constraints. The topological restrictions, while mild,
are necessary, as the counterexamples of C. Bishop and P. Jones show, and no
analogues of these results have been available for higher co-dimensional sets.
  In the present paper we show that for any $d<n-1$ and for any domain with a
$d$-dimensional uniformly rectifiable boundary the elliptic measure of an
appropriate degenerate elliptic operator is absolutely continuous with respect
to the Hausdorff measure of the boundary. There are no topological or
dimensional restrictions contrary to the aforementioned results.
