# Characterizations of Hardy spaces for Fourier integral operators

## TL;DR

This paper develops new characterizations of Hardy spaces associated with Fourier integral operators, linking them to classical $L^p$ norms and providing tools like maximal functions and square functions for analysis.

## Contribution

It introduces novel characterizations of $ ext{H}^{p}_{FIO}$ spaces in terms of $L^p$ norms of frequency localizations, extending classical $L^p$ characterizations to these Hardy spaces.

## Key findings

- Characterization of $ ext{H}^{p}_{FIO}$ via $L^p$ norms of frequency localizations
- Derivation of maximal function characterization for $ ext{H}^{p}_{FIO}$
- Development of vertical square function characterization for $ ext{H}^{p}_{FIO}$

## Abstract

We prove several characterizations of the Hardy spaces for Fourier integral operators $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$, for $1<p<\infty$. First we characterize $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ in terms of $L^{p}(\mathbb{R}^{n})$-norms of parabolic frequency localizations. As a corollary, any characterization of $L^{p}(\mathbb{R}^{n})$ yields a corresponding version for $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$. In particular, we obtain a maximal function characterization and a characterization in terms of vertical square functions.