Integration of oscillatory and subanalytic functions

Abstract

We prove the stability under integration and under Fourier transform of a concrete class of functions containing all globally subanalytic functions and their complex exponentials. This paper extends the investigation started in [26] and [8] to an enriched framework including oscillatory functions. It provides a new example of fruitful interaction between analysis and singularity theory.

Uniform analysis on local fields and applications to orbital integrals

“…It seems that this result is new even for fixed F . It compares to a real counterpart about stability under L 2 Fourier transformation of Theorem 8.10 of [5]. .…”

Archimedean Zeta Functions and Oscillatory Integrals

“…For some other works related with complex oscillatory integrals or with the ideas of Malgrange in [64] see for example [89,6,59,71,7,60,3,27,58,47,56,26,8,2,25,88,21,72,73,16,20,80,1], and the references therein.…”

Asymptotics of parameterized exponential integrals given by Brownian motion on globally subanalytic sets

“…In particular, they handle the case x → f (x, y) log(g(x, y)) dy for globally subanalytic f (x, y) and g(x, y). More recently, Cluckers et al [1] have considered oscillatory integrals x → f (x, y)e ig(x,y) for globally subanalytic f (x, y) and g(x, y). These are motivated by Fourier transformation and lead necessarily out of the o-minimal context.…”