  Using some resolution of singularities methods of the author, a
generalization of a well-known theorem of Varchenko relating decay of
oscillatory integrals to the Newton polyhedron is proven. They are derived from
analogous results for sublevel integrals, proven here.
  Varchenko's theorem requires a certain nondegeneracy condition on the faces
of the Newton polyhedron on the phase. In this paper, it is shown that the
estimates of Varchenko's theorem also hold for a significant class of phase
functions for which this nondegeneracy condition does not hold. Thus in
problems where one wants to switch coordinates to a coordinate system where
Varchenko's estimates are valid, one has greater flexibility.
  Some additional estimates are also proven for more degenerate situations,
including some too degenerate for the Newton polyhedron to give the optimal
decay in the sense of Varchenko.
