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In this paper, we establish an improved variable coefficient version of the square function inequality, by which the local smoothing estimateL  α  p   →   L  p{L^{p}_{\alpha}\to L^{p}}    for the Fourier integral operators satisfying cinematic curvature condition is further improved. In particular, we establish almost sharp results for2  &lt;  p  ⩽  3{2&lt;p\leqslant 3}    and push forward the estimate for the critical pointp  =  4{p=4}   . As a consequence, the local smoothing estimate for the wave equation on the manifold is refined. We generalize the results in [S. Lee and A. Vargas, On the cone multiplier inℝ  3\mathbb{R}^{3}   , J. Funct. Anal. 263 2012, 4, 925–940; J. Lee, A trilinear approach to square function and local smoothing estimates for the wave operator, preprint 2018,  https://arxiv.org/abs/1607.08426v5 ] to its variable coefficient counterpart. The main ingredients in the argument includes multilinear oscillatory integral estimate [J. Bennett, A. Carbery and T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math. 196 2006, 2, 261–302] and decoupling inequality [D. Beltran, J. Hickman and C. D. Sogge, Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds, Anal. PDE 13 2020, 2, 403–433].

Square function inequality for a class of Fourier integral operators satisfying cinematic curvature conditions