  We 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 non-degeneracy conditions. We initiate the
investigation of 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 Muckenhoput weights), and establish weighted norm inequalities for
operators with rough and smooth amplitudes and phase functions satisfying a
suitable rank condition. These results are then applied to prove weighted and
unweighted estimates for the commutators of Fourier integral operators with
functions of bounded mean oscillation BMO, then to some estimates on weighted
Triebel-Lizorkin spaces, and finally to global unweighted and local weighted
estimates for the solutions of the Cauchy problem for $m$-th and second order
hyperbolic partial differential equations on $\mathbf{R}^n .$
