Characterizations of Hardy spaces for Fourier integral operators

Abstract

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