Local and global estimates for hyperbolic equations in Besov-Lipschitz and Triebel-Lizorkin spaces

TL;DR

This paper derives optimal local and global estimates for solutions to linear hyperbolic PDEs within Besov-Lipschitz and Triebel-Lizorkin spaces, utilizing Fourier integral operator bounds across various scales.

Contribution

It provides the first comprehensive framework for hyperbolic equations in these advanced function spaces, covering both Banach and quasi-Banach scales.

Findings

Established optimal estimates for hyperbolic solutions in Besov-Lipschitz spaces.

Extended Fourier integral operator bounds to all scales of these spaces.

Unified local and global analysis for hyperbolic PDEs in advanced function spaces.

Abstract

In this paper we establish optimal local and global Besov-Lipschitz and Triebel-Lizorkin estimates for the solutions to linear hyperbolic partial differential equations. These estimates are based on local and global estimates for Fourier integral operators that span all possible scales (and in particular both Banach and quasi-Banach scales) of Besov-Lipschitz spaces $B^s_{p,q}(\R^n)$, and certain Banach and quasi-Banach scales of Triebel-Lizorkin spaces $F^s_{p,q}(\R^n)$