The global wave front set of tempered oscillatory integrals with inhomogeneous phase functions

TL;DR

This paper investigates the singularities of tempered oscillatory integrals with inhomogeneous phase functions, providing a global wave front set analysis that extends traditional homogeneity constraints, with applications to quantum field theory and Fourier integral operators.

Contribution

It introduces a framework for analyzing the wave front set of oscillatory integrals without requiring phase function homogeneity, broadening the understanding of their singularities and decay properties.

Findings

Describes singularities via lack of smoothness and decay at infinity.

Characterizes stationary points including boundary elements in compactification.

Applies results to quantum field theory and Fourier integral operators.

Abstract

We study certain families of oscillatory integrals $I_\varphi(a)$, parametrised by phase functions $\varphi$ and amplitude functions $a$ globally defined on $\mathbb{R}^d$, which give rise to tempered distributions, avoiding the standard homogeneity requirement on the phase function. The singularities of $I_\varphi(a)$ are described both from the point of view of the lack of smoothness as well as with respect to the decay at infinity. In particular, the latter will depend on a version of the set of stationary points of $\varphi$, including elements lying at the boundary of the radial compactification of $\mathbb{R}^d$. As applications, we consider some properties of the two-point function of a free, massive, scalar relativistic field and of classes of global Fourier integral operators on $\mathbb{R}^d$, with the latter defined in terms of kernels of the form $I_\varphi(a)$.