  In this paper, we consider $L^p$- estimate for a class of oscillatory
integral operators satisfying the Carleson-Sj\"olin conditions with further
convex and straight assumptions. As applications, the multiplier problem
related to a general class of hypersurfaces with nonvanishing Gaussian
curvature, local smoothing estimates for the fractional Schr\"odinger equation
and the sharp resolvent estimates outside of the uniform boundedness range are
discussed.
