$L^p$ and ${\mathcal{H}}_{FIO}^p$ regularity for wave equations with rough coefficients

Abstract

We consider wave equations with time-independent coefficients that have $C^{1,1}$ regularity in space. We show that, for nontrivial ranges of $p$ and $s$, the standard inhomogeneous initial value problem for the wave equation is well posed in Sobolev spaces ${\mathcal{H}}_{FIO}^{s,p}({\mathbb{R}}^n)$ over the Hardy spaces ${\mathcal{H}}_{FIO}^p({\mathbb{R}}^n)$ for Fourier integral operators introduced recently by the authors and Portal, following work of Smith. In spatial dimensions $n=2$ and $n=3$, this includes the full range $1<p<\infty$. As a corollary, we obtain the optimal fixed-time $L^p$ regularity for such equations, generalizing work of Seeger, Sogge and Stein in the case of smooth coefficients.

arXiv commentTo appear in Pure and Applied Analysis. 58 pages

References in corpus (6)

• Off-singularity bounds and Hardy spaces for Fourier integral operators
• Characterizations of Hardy spaces for Fourier integral operators
• Local smoothing and Hardy spaces for Fourier integral operators
• Rough pseudodifferential operators on Hardy spaces for Fourier integral operators II
• Characterizations of the Hardy space ${\mathcal{H}}_{FIO}^1({\mathbb{R}}^n)$ for Fourier integral operators
• Rough pseudodifferential operators on Hardy spaces for Fourier integral operators

Cited by in corpus (2)

• Local smoothing and Hardy spaces for Fourier integral operators on manifolds
• Function spaces for decoupling