  In a previous paper we considered a class of infinitely degenerate
quasilinear equations and derived a priori bounds for high order derivatives of
solutions in terms of the Lipschitz norm. We now show that it is possible to
obtain bounds just in terms of the supremum norm for a further subclass of such
equations, and we apply the resulting estimates to prove that continuous weak
solutions are necessarily smooth. We also obtain existence, uniqueness and
interior regularity of solutions for the Dirichlet problem with continuous
boundary data.
