2008 DG-MP-PDE Seminar - 04

Varchenko described how one can obtain sharp estimates for scalar oscillatory integrals when the Newton polyhedron of the phase satisfies a certain nondegeneracy condition. These estimates can be described in a nice geometric way in terms of this Newton polyhedron. We will describe how these estimates still hold under weaker conditions; the proof uses resolution of singularities methods. We then will describe a class of functions for which the weaker conditions can not be weakened further. Further generalizations will also be discussed.