Off-singularity bounds and Hardy spaces for Fourier integral operators

Abstract

We define a scale of Hardy spaces H F I O p ( R n ) \mathcal {H}^{p}_{FIO}(\mathbb {R}^n) , p ∈ [ 1 , ∞ ] p\in [1,\infty ] , that are invariant under suitable Fourier integral operators of order zero. This builds on work by Smith for p = 1 p=1 [J. Geom. Anal. 8 (1998), pp. 629–653]. We also introduce a notion of off-singularity decay for kernels on the cosphere bundle of R n \mathbb {R}^n , and we combine this with wave packet transforms and tent spaces over the cosphere bundle to develop a full Hardy space theory for oscillatory integral operators. In the process we extend the known results about L p L^{p} -boundedness of Fourier integral operators, from local boundedness to global boundedness for a larger class of symbols.