Characterizations of Hardy spaces for Fourier integral operators

Abstract

We prove several characterizations of the Hardy spaces for Fourier integral operators ${\mathcal{H}}_{\mathrm{FIO}}^p({\mathbb{R}}^n)$, for $1<p<\infty$. First we characterize ${\mathcal{H}}_{\mathrm{FIO}}^p({\mathbb{R}}^n)$ 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 ${\mathcal{H}}_{\mathrm{FIO}}^p({\mathbb{R}}^n)$. In particular, we obtain a maximal function characterization and a characterization in terms of vertical square functions.