Oscillatory integral decay, sublevel set growth, and the Newton polyhedron

By Michael Greenblatt. 2008

53 citations recorded by the source corpus.

Extending some resolution of singularities methods (Greenblatt in J Funct Anal 255(8):1957–1994, 2008) of the author, a generalization of a well-known theorem of Varchenko (Funct Anal Appl 18(3):175–196, 1976) 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/704fcb192ba17da9c463a61e6249193bdd637b0d

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

A multi-dimensional resolution of singularities with applications to analysis (2010, background): “This gives rise to the following natural question: for n > 1, is there an appropriate generalization C of the class of analytic coordinate systems such that (i) for every Φ ∈ C, the concepts of Newton distance and exponent continue to be meaningful for F ◦ Φ; and (ii) for which (1.5) holds with Cω…”
Estimates for maximal functions associated with hypersurfaces in ℝ3 and related problems of harmonic analysis (2010, background): “Partial results on the value of γ(x, S) can be found in Greenblatt’s articles [14], [12].”