Global and Local Regularity of Fourier Integral Operators on Weighted and Unweighted Spaces

The authors investigate the global continuity on $L^p$ spaces with $p\in [1,\infty]$ of Fourier integral operators with smooth and rough amplitudes and/or phase functions subject to certain necessary non-degeneracy conditions. In this context they prove the optimal global $L^2$ boundedness result for Fourier integral operators with non-degenerate phase functions and the most general smooth Hormander class amplitudes i.e. those in $S^{m} _{\varrho, \delta}$ with $\varrho , \delta \in [0,1]$. They also prove the very first results concerning the continuity of smooth and rough Fourier integral operators on weighted $L^{p}$ spaces, $L_{w}^p$ with $1< p < \infty$ and $w\in A_{p},$ (i.e. the Muckenhoupt weights) for operators with rough and smooth amplitudes and phase functions satisfying a suitable rank condition.