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

TL;DR

This paper improves square function inequalities for Fourier integral operators with Carleson-Sj"olin conditions, leading to sharper local smoothing estimates for wave equations on manifolds, especially for certain p ranges.

Contribution

It extends existing results to variable coefficient settings, providing near-sharp bounds for 2<p≤3 and advancing estimates at p=4 using multilinear oscillatory integrals and decoupling.

Findings

Enhanced local smoothing estimates for Fourier integral operators.

Almost sharp results for 2<p≤3.

Refined estimates for p=4.

Abstract

In this paper, we establish an improved variable coefficient version of square function inequality, by which the local smoothing estimate $L^p_\alpha\rightarrow L^p$ for the Fourier integral operators satisfying cinematic curvature condition is further improved. In particular, we establish almost sharp results for $2<p\leq 3$ 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}.