A multi-parameter family of Fourier integral operators

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.