Bounding the decay of oscillatory integrals with a constructible amplitude function and a globally subanalytic phase function

TL;DR

This paper establishes uniform decay bounds for oscillatory integrals with constructible amplitudes and globally subanalytic phases, and demonstrates the integrability of Fourier transforms of such functions.

Contribution

It provides the first uniform decay bounds for a broad class of oscillatory integrals with constructible amplitude and subanalytic phase functions.

Findings

Uniform decay bounds for oscillatory integrals with constructible amplitudes

Fourier transform of continuous, integrable, constructible functions is integrable

Application of bounds to specific classes of functions

Abstract

We call a function "constructible" if it has a globally subanalytic domain and can be expressed as a sum of products of globally subanalytic functions and logarithms of positively-valued globally subanalytic functions. Our main theorem gives uniform bounds on the decay of parameterized families of oscillatory integrals with a constructible amplitude function and a globally subanalytic phase function, assuming that the amplitude function is integrable and that the phase function satisfies a certain natural condition called the hyperplane condition. As a simple application of this theorem, we also show that any continuous, integrable, constructible function of a single variable has an integrable Fourier transform.