  The local $L^2$-mapping property of Fourier integral operators has been
established in H\"ormander \cite{H} and in Eskin \cite{E}. In this paper, we
treat the global $L^2$-boundedness for a class of operators that appears
naturally in many problems. As a consequence, we will improve known global
results for several classes of pseudo-differential and Fourier integral
operators, as well as extend previous results of Asada and Fujiwara \cite{AF}
or Kumano-go \cite{Ku}. As an application, we show a global smoothing estimate
to generalized Schr\"odinger equations which extends the results of Ben-Artzi
and Devinatz \cite{BD}, Walther \cite{Wa}, and \cite{Wa2}.
