  We develop subrepresentation inequalities for infinitely degenerate metrics,
and obtain corresponding Poincare and Sobolev inequalities. We then derive
conditions on the degenerate metric under which weak solutions to associated
infinitely degenerate equations with rough coefficients are locally bounded,
satisfy a maximum principle, or are continuous. As an application we obtain
W-hypoellipticity of certain infinitely degenerate quasilinear equations with
smooth coefficients having mild nonlinearities and degeneracies.
