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We establish certain square function estimates for a class of oscillatoryintegral operators with homogeneous phase functions. These results are employedto deduce a refinement of a previous result of Mockenhaupt Seeger and Sogge\cite{MSS-jams} on the local smoothing property for Fourier integral operators,which arise naturally in the study of wave equations on compact Riemannianmanifolds. The proof is an adaptation of the bilinear approach of Tao andVargas \cite{Tao-Vargas-II}, %as well as Garrig\'os and Seeger \cite{GaSe09},and based on bilinear oscillatory integral estimates of Lee \cite{Lee-JFA}.

Improved variable coefficient square functions and local smoothing of Fourier integral operators