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We define a scale of Hardy spaces $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$,$p\in[1,\infty]$, that are invariant under suitable Fourier integral operatorsof order zero. This builds on work by Smith for $p=1$. We also introduce anotion of off-singularity decay for kernels on the cosphere bundle of$\mathbb{R}^{n}$, and we combine this with wave packet transforms and tentspaces over the cosphere bundle to develop a full Hardy space theory foroscillatory integral operators. In the process we extend the known resultsabout $L^{p}$-boundedness of Fourier integral operators, from local boundednessto global boundedness for a larger class of symbols.

Off-singularity bounds and Hardy spaces for Fourier integral operators