Oscillatory Integral Decay, Sublevel Set Growth, and the Newton   Polyhedron
  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. 
Michael Greenblatt