In this talk, we discuss the “local smoothing” phenomenon for Fourier integral operators with amplitude function belongs to the “symbol class”. We give an overview of the regularity results which have been proven to date. We use harmonic analysis of Hermite functions in the study of Fourier integral operators. Finally, we give an application of the local smoothing estimate to the wave equation and maximal operators. This is a joint work with Prof. P. K. Ratnakumar.