The Hardy spaces ${\mathcal{H}}_{FIO}^p({\mathbb{R}}^n)$ for Fourier integral operators for $p<1$

Abstract

We introduce the Hardy spaces ${\mathcal{H}}_{FIO}^p({\mathbb{R}}^n)$ for Fourier integral operators for $0<p<1$, thereby extending earlier constructions for $1\le p\le \infty$. We then establish various properties of these spaces, including their behavior under complex interpolation and duality, and their invariance under Fourier integral operators. We also obtain Sobolev embeddings, equivalent characterizations, and a molecular decomposition. These spaces are used in the companion article arXiv:2502.02511 to determine the sharp ${\mathcal{H}}^1({\mathbb{R}}^n)$ and $\mathrm{bmo}({\mathbb{R}}^n)$ regularity of wave equations with rough coefficients.

31 pages. Originally part of arXiv:2502.02511. This part of the manuscript was split off and turned into a separate article