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

TL;DR

This paper introduces new Hardy spaces for Fourier integral operators applicable for p<1, extending previous work and establishing their fundamental properties and applications to wave equations with rough coefficients.

Contribution

It defines Hardy spaces $\\mathcal{H}^{p}_{FIO}$ for 0<p<1, extending prior constructions and analyzing their properties, including invariance under Fourier integral operators and duality.

Findings

Established properties of the new Hardy spaces, including interpolation and duality.

Proved invariance of these spaces under Fourier integral operators.

Applied the spaces to analyze regularity of wave equations with rough coefficients.

Abstract

We introduce the Hardy spaces $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ for Fourier integral operators for $0<p<1$, thereby extending earlier constructions for $1\leq p\leq \infty$. 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.