A multi-parameter family of Fourier integral operators

TL;DR

This paper introduces a new class of Fourier integral operators with multi-parameter symbols, proving their boundedness on Hardy spaces and deriving sharp L^p estimates, with applications to wave equations.

Contribution

It defines a novel class of Fourier integral operators with multi-parameter symbols and establishes their boundedness and Sobolev estimates, improving regularity requirements for wave equation solutions.

Findings

Boundedness from H^1 to L^1 for the new operators

Sharp L^p-estimates for the operators

New a priori estimates for wave equations

Abstract

We study a new class of Fourier integral operators defined in R^N. Their symbols are allowed to satisfy a differential inequality with certain multi-parameter characteristic. We prove these operators of order -(N-1)/2 bounded from the classical, atom decomposable H^1-Hardy space to L^1(R^N). As a result, we obtain a sharp L^p-estimate. Simultaneously, a generalized Sobolev Lp-space is introduced. We establish the Sobolev Lp-norm inequality for convolutions with a distribution having singularity on the unit sphere. As an application, we give a new a priori estimate for the solution of wave equations by requiring less regularity on the source term and initial data.