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

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