Characterizations of the Hardy space $mathcal{H}_{FIO}^{1}(mathbb{R}^{n})$ for Fourier Integral Operators

The Hardy spaces for Fourier integral operators $mathcal{H}_{FIO}^{p}(mathbb{R}^{n})$, for $1leq pleq infty$, were introduced by Smith in cite{Smith98a} and Hassell et al. in cite{HaPoRo18}. In this article, we give several equivalent characterizations of $mathcal{H}_{FIO}^{1}(mathbb{R}^{n})$, for example in terms of Littlewood--Paley g functions and maximal functions. This answers a question from [Rozendaal,2019]. We also give several applications of the characterizations.