A multi-parameter family of Fourier integral operators
Mengmeng Dou, Zipeng Wang, Jiashu Zhang
TL;DR
This work develops a multi-parameter theory for Fourier integral operators with phases homogeneous of degree one and symbols satisfying multi-parameter differential inequalities. By introducing symbol classes ${\bf S}^{-m}_{\rhoup}$ and a generalized Sobolev framework ${\bf L}^p_{\rhoup\,s}$, it extends classical results (Miyachi, Rubin) to a product- or multi-parameter setting and establishes sharp L^p bounds, including a Hardy-space to L^1 bound for order $-{\bf (N-1)\over 2}$. The authors prove Theorem One (boundedness from ${\bf H}^1$ to ${\bf L}^1$ and ${\bf L}^p$-boundedness) and Theorem Two (Sobolev-type estimates for convolutions with $\Omega^{\delta}$ in the multi-parameter Sobolev scale), using a novel two-tier dyadic cone decomposition and a key kernel bound (Lemma One) together with atom decomposition and interpolation. As an application, they obtain an a priori wave equation estimate with reduced regularity on the data, highlighting the practical impact for hyperbolic PDEs with anisotropic or multi-parameter structures. This work broadens the scope of FIO theory to multi-parameter settings and provides sharp tools for analyzing propagation phenomena in complex media.
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.