  Motivated by applications to stochastic differential equations, an extension
of H\"{o}rmander's hypoellipticity theorem is proved for second-order
degenerate elliptic operators with non-smooth coefficients. The main results
are established using point-wise Bessel kernel estimates and a weighted Sobolev
inequality of Stein and Weiss. Of particular interest is that our results apply
to operators with quite general first-order terms.
