Estimates for rough Fourier integral and pseudodifferential operators and applications to the boundedness of multilinear operators

TL;DR

This paper establishes sharp boundedness results for rough Fourier integral and pseudodifferential operators on various L^p spaces, and applies these results to improve bounds for multilinear operators in harmonic analysis.

Contribution

It provides new sharp boundedness criteria for rough Fourier integral and pseudodifferential operators, extending to multilinear cases and improving existing results.

Findings

Boundedness of rough Fourier integral operators on L^p spaces.

Enhanced bounds for bilinear pseudodifferential operators.

Results are sharp for certain classes of amplitudes.

Abstract

We study the boundedness of rough Fourier integral and pseudodifferential operators, defined by general rough H\"ormander class amplitudes, on Banach and quasi-Banach $L^p$ spaces. Thereafter we apply the aforementioned boundedness in order to improve on some of the existing boundedness results for H\"ormander class bilinear pseudodifferential operators and certain classes of bilinear (as well as multilinear) Fourier integral operators. For these classes of amplitudes, the boundedness of the aforementioned Fourier integral operators turn out to be sharp. Furthermore we also obtain results for rough multilinear operators.