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

Chuanwei Gao, Changxing Miao, Jianwei-Urbain Yang

Abstract

In this paper, we establish an improved variable coefficient version of square function inequality, by which the local smoothing estimate $L^p_α\rightarrow L^p$ 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}.

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 for the Fourier integral operators satisfying cinematic curvature condition is further improved. In particular, we establish almost sharp results for and push forward the estimate for the critical point . 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}.

Paper Structure

This paper contains 8 sections, 17 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Reductions
  3. Normalization of the phase function
  4. Parabolic rescaling
  5. Linear v.s. Multilinear square function estimates
  6. $L^3-$variable coefficient square function estimate via multilinear approach
  7. $L^4-$variable coefficient square function estimate via decoupling
  8. Appendix

Key Result

Theorem 1

Suppose $\mathscr F\in I^{\sigma-\frac{1}{4}}(Z,Y;\mathscr C)$ with $Z, Y$ as above, where the canonical relation $\mathscr C$ satisfies the cinematic curvature condition. Then, the following estimate holds where $\sigma(p)$ is defined by

Theorems & Definitions (29)

  • Theorem 1
  • Remark 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 2
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • ...and 19 more