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The dichotomy of Nikodym sets and local smoothing estimates for wave equations

Mingfeng Chen, Shaoming Guo

Abstract

We show that Nikodym sets and local smoothing estimates for linear wave equations form a dichotomy: If Nikodym sets for a family of curves exist, then the related maximal operator is not bounded on $L^p(\mathbb{R}^2)$ for any $p<\infty$; if Nikodym sets do not exist, then local smoothing estimates hold, and the related maximal operator is bounded on $L^p(\mathbb{R}^2)$ for some $p<\infty$. Whenever the maximal operator is bounded on $L^p(\mathbb{R}^2)$ for some $p<\infty$, we also determine the sharp exponent for $L^p(\mathbb{R}^2)$ bounds.

The dichotomy of Nikodym sets and local smoothing estimates for wave equations

Abstract

We show that Nikodym sets and local smoothing estimates for linear wave equations form a dichotomy: If Nikodym sets for a family of curves exist, then the related maximal operator is not bounded on for any ; if Nikodym sets do not exist, then local smoothing estimates hold, and the related maximal operator is bounded on for some . Whenever the maximal operator is bounded on for some , we also determine the sharp exponent for bounds.
Paper Structure (34 sections, 14 theorems, 631 equations)

This paper contains 34 sections, 14 theorems, 631 equations.

Table of Contents

  1. Introduction
  2. Strongly degenerate condition
  3. Vertical Newton distance
  4. Strongly degenerate conditions and Nikodym sets
  5. Strongly degenerate functions
  6. Nikodym sets
  7. Examples
  8. Examples for the algorithm of determining the critical exponent
  9. Examples for the sharpness of
  10. Resolution of singularities
  11. A standard reduction
  12. Structures of bad regions
  13. An iteration algorithm
  14. Sharp bounds: Vertices-dominating
  15. The dominating monomial satisfies .
  16. ...and 19 more sections

Key Result

Theorem 1.2

Let $\gamma(v, \theta)$ be an analytic function near the origin. Then $\gamma$ is not strongly degenerate at the origin if and only if there exists $\epsilon>0$ such that for some $p<\infty$, and for all smooth functions $a$ supported in $\mathbb{B}_{\epsilon}$.

Theorems & Definitions (36)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3: Vertical Newton distance
  • Remark
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['231122lemma2_1']}
  • Lemma 2.2
  • Remark
  • ...and 26 more