The asymptotic behavior of degenerate oscillatory integrals in two dimensions

TL;DR

This paper extends Varchenko's theorem by providing explicit formulas for the decay rate and coefficients of degenerate oscillatory integrals in two dimensions, applicable to smooth phases and involving superadapted coordinates.

Contribution

It introduces explicit formulas for decay rates and coefficients of oscillatory integrals, generalizing previous geometric results to smooth phases with superadapted coordinates.

Findings

Explicit formulas for decay rates and coefficients

Results for sublevel integrals

Extensions to smooth phases

Abstract

A theorem of Varchenko gives the order of decay of the leading term of the asymptotic expansion of a degenerate oscillatory integral with real-analytic phase in two dimensions. His theorem expresses this order of decay in a simple geometric way in terms of its Newton polygon once one is in certain coordinate systems called adapted coordinate systems. In this paper, we give explicit formulas that not only provide the order of decay of the leading term, but also the coefficient of this term. There are three rather different formulas corresponding to three different types of Newton polygon. Analogous results for sublevel integrals are proven, as are analogues for the more general case of smooth phase. The formulas require one to be in certain "superadapted" coordinates. These are a type of adapted coordinate system which we show exists for any smooth phase.