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A multi-parameter cinematic curvature

Mingfeng Chen, Shaoming Guo, Tongou Yang

TL;DR

This work develops a multi-parameter cinematic curvature framework for maximal operators and proves $L^p$ bounds under a robust cinematic curvature condition, with small-amplitude localization. By integrating Zahl’s local smoothing results, the authors obtain quantitative $L^p$ decay across frequency pieces and show the bound holds for $p>p_d$ with $p_d$ effectively sharpened to $d+1$ in conjunction with Zahl’s work. The ellipse-maximal operator serves as a central example, for which an $L^3$ bound and a frequency-localized local smoothing estimate are established (the latter yielding $L^p$ bounds for $p>3$). A key technical tool is a reduction algorithm that decomposes the phase into normal forms and applies decoupling, both in translation-invariant and variable-coefficient contexts, ultimately achieving the claimed $L^p$-boundedness results and contributing toward a multi-parameter local smoothing theory aligned with Zahl's conjecture.

Abstract

We state a multi-parameter cinematic curvature condition, and prove $L^p$ bounds for related maximal operators. In particular, we verify a local smoothing conjecture of Zahl.

A multi-parameter cinematic curvature

TL;DR

This work develops a multi-parameter cinematic curvature framework for maximal operators and proves bounds under a robust cinematic curvature condition, with small-amplitude localization. By integrating Zahl’s local smoothing results, the authors obtain quantitative decay across frequency pieces and show the bound holds for with effectively sharpened to in conjunction with Zahl’s work. The ellipse-maximal operator serves as a central example, for which an bound and a frequency-localized local smoothing estimate are established (the latter yielding bounds for ). A key technical tool is a reduction algorithm that decomposes the phase into normal forms and applies decoupling, both in translation-invariant and variable-coefficient contexts, ultimately achieving the claimed -boundedness results and contributing toward a multi-parameter local smoothing theory aligned with Zahl's conjecture.

Abstract

We state a multi-parameter cinematic curvature condition, and prove bounds for related maximal operators. In particular, we verify a local smoothing conjecture of Zahl.
Paper Structure (16 sections, 7 theorems, 253 equations)

This paper contains 16 sections, 7 theorems, 253 equations.

Table of Contents

  1. Introduction
  2. More connections to Zahl's work Zah23
  3. Maximal operators along ellipses
  4. An $L^3$ bound
  5. A local smoothing estimate
  6. Proof of Corollary \ref{['231127corollary1_4']}
  7. Normal forms
  8. Reduction algorithm
  9. The first step of the algorithm
  10. The second step of the algorithm
  11. Outputs of the whole algorithm
  12. Final reduction
  13. Proof of Proposition \ref{['230329prop2_2']}
  14. Variable coefficient maximal operators
  15. Reduction algorithm
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $d\ge 3$. Let $\gamma(\theta; {\bf v}): \mathbb{R}\times \mathbb{R}^{d-1}\to \mathbb{R}$ be a smooth function that satisfies the $(d-1)$-parameter cinematic curvature condition as in Y_230330nondegenerate. Then there exists $p_d>0$ depending only on $d$ such that for every $p>p_d$ and every smooth bump function $\chi(\theta; {\bf v})$ that is supported in a sufficiently small neighborhood of

Theorems & Definitions (23)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Definition 5.1: Normal forms
  • Proposition 5.2
  • proof : Proof of Theorem \ref{['230329theorem1_1']} by assuming Proposition \ref{['230329prop2_2']}.
  • Claim 6.1
  • proof : Proof of Claim \ref{['221031claim3_1']}
  • ...and 13 more