${\mathcal{H}}^1$ and $\mathrm{b}\mathrm{m}\mathrm{o}$ regularity for wave equations with rough coefficients

We consider second-order hyperbolic equations with rough time-independent coefficients. Our main result is that such equations are well posed on the Hardy spaces ${\mathcal{H}}_{FIO}^{s,1}({\mathbb{R}}^n)$ and ${\mathcal{H}}_{FIO}^{s,\mathrm{\infty }}({\mathbb{R}}^n)$ for Fourier integral operators if the coefficients have $C^{1,1}\cap C^r$ regularity in space, for $r>\frac{n+1}{2}$, where $s$ ranges over an $r$-dependent interval. As a corollary, we obtain the sharp fixed-time ${\mathcal{H}}^1({\mathbb{R}}^n)$ and $\mathrm{b}\mathrm{m}\mathrm{o}({\mathbb{R}}^n)$ regularity for such equations, extending work by Seeger, Sogge and Stein in the case of smooth coefficients.