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Lossless Strichartz estimates on the square torus over short time intervals

Connor Quinn

TL;DR

This work proves lossless Strichartz estimates at the critical exponent $q_c = \frac{2(n+1)}{n-1}$ for the Schrödinger equation on the square torus with frequency-localized data on short time windows. The authors develop a multiscale induction on frequency cubes, employing a height-splitting strategy, a broad–narrow analysis, and sharp kernel bounds together with bilinear restriction estimates for the paraboloid to control interactions across scales. They obtain a uniform bound $\|e^{-it\Delta_{\mathbb{T}^{n-1}}}\beta(D_x/\lambda)f\|_{L^{q_c}(\mathbb{T}^{n-1} \times [0, 1/(\lambda \delta)])} \lesssim_{\varepsilon} \|f\|_{2}$ for $\delta \geq \lambda^{-1/(n+1)+\varepsilon}$, extending the regime of lossless control to higher dimensions and improving previous short-time results. The approach clarifies the dispersive behavior on the torus over short intervals and has potential implications for nonlinear Schrödinger analysis on compact manifolds by providing robust, scale-informed estimates.

Abstract

We prove lossless Strichartz estimates at the critical exponent $q_c = \frac{2(n+1)}{n-1}$ on the square torus for the Schrödinger equation with frequency localized initial data on small time windows with length depending on the frequency parameter $λ\gg 1$.

Lossless Strichartz estimates on the square torus over short time intervals

TL;DR

This work proves lossless Strichartz estimates at the critical exponent for the Schrödinger equation on the square torus with frequency-localized data on short time windows. The authors develop a multiscale induction on frequency cubes, employing a height-splitting strategy, a broad–narrow analysis, and sharp kernel bounds together with bilinear restriction estimates for the paraboloid to control interactions across scales. They obtain a uniform bound for , extending the regime of lossless control to higher dimensions and improving previous short-time results. The approach clarifies the dispersive behavior on the torus over short intervals and has potential implications for nonlinear Schrödinger analysis on compact manifolds by providing robust, scale-informed estimates.

Abstract

We prove lossless Strichartz estimates at the critical exponent on the square torus for the Schrödinger equation with frequency localized initial data on small time windows with length depending on the frequency parameter .
Paper Structure (8 sections, 12 theorems, 131 equations)

This paper contains 8 sections, 12 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Notation, Initial Reductions and a Brief Discussion of the Proof
  3. Height Splitting at Scale $\delta_{K}^{\ell - 1}$, for $1 \leq \ell < K$
  4. Large Height Estimates at Scale $\delta_{K}^{\ell-1}$, for $1 \leq \ell < K$
  5. Small Height Estimates at Scale $\delta_{K}^{\ell-1}$, for $1 \leq \ell < K$
  6. Consequences of Bilinear Restriction and Proof of Lemma 5.2
  7. Kernel Estimates
  8. Proof of Theorem 1

Key Result

Theorem 1.1

Let $\varepsilon > 0$. Then if $\delta \geq \lambda^{-\frac{1}{n+1} + \varepsilon}$ we have

Theorems & Definitions (15)

  • Theorem 1.1
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • Theorem 5.3
  • Lemma 5.4
  • Lemma 5.5
  • ...and 5 more