Characterizations of Hardy spaces for Fourier integral operators

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