$L^p$ 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.

To appear in Pure and Applied Analysis. 58 pages

References in corpus (3)

• Local smoothing and Hardy spaces for Fourier integral operators
• Rough pseudodifferential operators on Hardy spaces for Fourier integral operators II
• 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