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

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 $1\le p\le \mathrm{\infty }$. 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 $\mathrm{b}\mathrm{m}\mathrm{o}({\mathbb{R}}^n)$ regularity of wave equations with rough coefficients.