On Fourier integral operators with H\"older-continuous phase

TL;DR

This paper investigates the boundedness of Fourier integral operators with H"older-continuous phases in Lebesgue spaces, revealing decay losses and establishing conditions for continuity, with implications for the Beurling-Helson theorem and time-frequency analysis.

Contribution

It provides the first quantitative analysis of Fourier integral operators with H"older-type singularities, including precise decay estimates and counterexamples.

Findings

Boundedness in L^1 with decay loss depending on H"older exponent

Counterexamples showing loss even for smooth phases

Sufficient conditions for L^2 continuity

Abstract

We study continuity properties in Lebesgue spaces for a class of Fourier integral operators arising in the study of the Boltzmann equation. The phase has a H\"older-type singularity at the origin. We prove boundedness in $L^1$ with a precise loss of decay depending on the H\"older exponent, and we show by counterexamples that a loss occurs even in the case of smooth phases. The results can be seen as a quantitative version of the Beurling-Helson theorem for changes of variables with a H\"older singularity at the origin. The continuity in $L^2$ is studied as well by providing sufficient conditions and relevant counterexamples. The proofs rely on techniques from Time-frequency Analysis.