$L^{p}$ and $\mathcal{H}^{p}_{FIO}$ 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 $\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ over the Hardy spaces $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ 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.