Square function inequality for oscillatory integral operators satisfying homogeneous Carleson-Sjölin type conditions

Abstract

In this paper, we establish an improved variable coefficient version of square function inequality, by which the local smoothing estimate $L_{\hat{I}}^p\pm \to L^p$ for the Fourier integral operators satisfying cinematic curvature condition is further improved. In particular, we establish almost sharp results for $2<p\le 3$ and push forward the estimate for the critical point $p=4$. As a consequence, the local smoothing estimate for the wave equation on the manifold is refined. We generalize the results in \cite{LeVa12, Le18P} to its variable coefficient counterpart. The main ingredients in the argument includes multilinear oscillatory integral estimate \cite{BCT06} and decoupling inequality \cite{BelHicSog18P}.

26 pages. arXiv admin note: text overlap with arXiv:1901.01487