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Strichartz inequalities: some recent developments

Jianhui Li, Zane Kun Li, Po-Lam Yung

Abstract

Strichartz inequalities, originating from Fourier restriction theory, play a central role in the analysis of dispersive partial differential equations. They serve as a cornerstone for many subsequent developments. We survey some of them in memory of Strichartz, highlighting connections to recent developments in Fourier decoupling.

Strichartz inequalities: some recent developments

Abstract

Strichartz inequalities, originating from Fourier restriction theory, play a central role in the analysis of dispersive partial differential equations. They serve as a cornerstone for many subsequent developments. We survey some of them in memory of Strichartz, highlighting connections to recent developments in Fourier decoupling.
Paper Structure (12 sections, 3 theorems, 120 equations, 1 figure)

This paper contains 12 sections, 3 theorems, 120 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Heuristics / White lies
  3. The Fourier transform
  4. The uncertainty principle
  5. Wave packet decomposition
  6. Strichartz inequality for the torus
  7. Initial attempt
  8. Quantifying constructive interference
  9. Constructive interference at a hierarchy of scales
  10. Pruning of wave packets
  11. Remarks
  12. Refined Strichartz inequalities and applications

Key Result

Theorem 4.1

Suppose that $f$ has frequency support in $[0,1]^d$, and $Q_1,Q_2, ...$ are disjoint lattice $R^{1/2}$-cubes in $[0,R]^{d+1}$ so that Suppose that these cubes are arranged in horizontal slabs of the form $\mathbb{R}^{d} \times \{t_0, t_0+R^{1/2}\}$, and that each slab contains $\sim \sigma$ cubes $Q_j$. Let $Y=\cup_{j} Q_j$. Then for any $\varepsilon > 0$,

Figures (1)

  • Figure 1: In this figure we fix $\beta \in P_{1/K}$. On the left we have strips $S$ that are translates of $R^{1/2} K^{-1} \tau_{\beta}^*$. It is made up of $K$ cubes of side length $R^{1/2}$. Each wave packet from any $\theta \in P_{R^{-1/2}}(\beta)$ (pictured in blue) that passes through one of these $K$ cubes passes through all the others. Upon rescaling by a factor of $K$ in the short directions and $K^2$ in the long directions, the rescaled $S$ becomes a cube of side length $R_1^{1/2}$, and the rescaled wave packets becomes one at scale $R_1^{-1/2}$, with $R_1 := R/K^2$. If we consider the convex hull of all wave packets from $\theta \in P_{R^{-1/2}}(\beta)$ that passes through a given strip $S$, we get essentially a box $\square_{\beta}$, which gets rescaled into a cube $B_{R_1}$ of side length $R_1$.

Theorems & Definitions (6)

  • Theorem 4.1: Refined Strichartz inequality, version 1
  • Theorem 4.2: Refined Strichartz inequality, version 2
  • proof : Proof of Theorem \ref{['thm:RSI']}
  • Theorem 4.3: Refined decoupling inequality
  • proof : Proof of Theorem \ref{['thm:RSI2']}
  • proof : Proof of Theorem \ref{['thm:refined_dec']}