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

TL;DR

This paper establishes well-posedness of second-order hyperbolic equations with rough, time-independent coefficients on Hardy and BMO spaces, extending classical results to less regular coefficients and providing sharp fixed-time regularity results.

Contribution

It proves well-posedness of hyperbolic equations with rough coefficients on Hardy and BMO spaces, extending prior smooth-coefficient results to less regular settings.

Findings

Well-posedness on Hardy spaces for coefficients with $C^{1,1} \, \cap \, C^{r}$ regularity.

Sharp fixed-time $\mathcal{H}^{1}$ and BMO regularity results.

Extension of classical smooth-coefficient results to rough coefficient scenarios.

Abstract

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 $\mathcal{H}^{s,1}_{FIO}(\mathbb{R}^{n})$ and $\mathcal{H}^{s,\infty}_{FIO}(\mathbb{R}^{n})$ 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 $\mathrm{bmo}(\mathbb{R}^{n})$ regularity for such equations, extending work by Seeger, Sogge and Stein in the case of smooth coefficients.