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In this paper, we establish an improved variable coefficient version ofsquare function inequality, by which the local smoothing estimate$L^p_\alpha\rightarrow L^p$ for the Fourier integral operators satisfyingcinematic curvature condition is further improved. In particular, we establishalmost sharp results for $2<p\leq 3$ and push forward the estimate for thecritical point $p=4$. As a consequence, the local smoothing estimate for thewave equation on the manifold is refined. We generalize the results in\cite{LeVa12, Le18P} to its variable coefficient counterpart. The mainingredients in the argument includes multilinear oscillatory integral estimate\cite{BCT06} and decoupling inequality \cite{BelHicSog18P}.

Square function inequality for oscillatory integral operators satisfying homogeneous Carleson-Sj\"olin type conditions