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A Note on Almost Everywhere Convergence Along Tangential Curves to the Schrödinger Equation Initial Datum

Javier Minguillón

Abstract

In this short note, we give an easy proof of the following result: for $ n\geq 2, $ $\underset{t\to0}{\lim} \,e^{itΔ}f\left(x+γ(t)\right) = f(x) $ almost everywhere whenever $ γ$ is an $ α- $Hölder curve with $ \frac12\leq α\leq 1 $ and $ f\in H^s(\mathbb{R}^n) $, with $ s > \frac{n}{2(n+1)} $. This is the optimal range of regularity up to the endpoint.

A Note on Almost Everywhere Convergence Along Tangential Curves to the Schrödinger Equation Initial Datum

Abstract

In this short note, we give an easy proof of the following result: for almost everywhere whenever is an Hölder curve with and , with . This is the optimal range of regularity up to the endpoint.
Paper Structure (3 sections, 4 theorems, 23 equations)

This paper contains 3 sections, 4 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Intermediate results
  3. Proof of Theorem \ref{['theorem_Theorem 1.3. from Du Zhang modified']}

Key Result

Theorem 1.1

Let $n \geq 1$. Fix $\frac{1}{2}\leq \alpha\leq 1$ and $\tau\geq1$. For any $\varepsilon>0$, there exists a positive constant $C_{\varepsilon,\tau}$ such that, for every $\gamma\in\Gamma_{\tau}^\alpha$, holds for all $f$$\in H^{\frac{1}{2(n+1)}+\epsilon }(\mathbb R^n).$

Theorems & Definitions (8)

  • Theorem 1.1
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 2.1: Corollary 1.7 in DuZhang2019SharpEstimatesSchrodinger
  • Theorem 2.2
  • proof : Proof of Theorem \ref{['theorem_Cor1.7 of Du Zhang modified']}
  • proof : Proof of Theorem \ref{['theorem_Theorem 1.3. from Du Zhang modified']}.