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Sharp pointwise convergence on the Schrödinger operator along one class of curves

Zhenbin Cao, Changxing Miao

Abstract

Almost everywhere convergence on the solution of Schrödinger equation is an important problem raised by Carleson, which was essentially solved by Du-Guth-Li and Du-Zhang. In this note, we obtain the sharp pointwise convergence on the Schrödinger operator along one class of curves.

Sharp pointwise convergence on the Schrödinger operator along one class of curves

Abstract

Almost everywhere convergence on the solution of Schrödinger equation is an important problem raised by Carleson, which was essentially solved by Du-Guth-Li and Du-Zhang. In this note, we obtain the sharp pointwise convergence on the Schrödinger operator along one class of curves.
Paper Structure (3 sections, 6 theorems, 79 equations)

This paper contains 3 sections, 6 theorems, 79 equations.

Table of Contents

  1. INTRODUCTION
  2. Preparation and basic induction
  3. The proof of Theorem \ref{['th4']}

Key Result

Theorem 1.1

Let $n\geq 2$, $1/2 \leq \alpha <1$ and $\gamma(x,t) \in \mathcal{D}$. For every $f\in H^s(\mathbb{R}^n)$ with $s>\frac{n}{2(n+1)}$, and the range of $s$ is sharp up to the endpoint.

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Proposition 3.1
  • Remark 3.2