Uniform convergence of the one-dimensional cubic Schrödinger equation

Abstract

In this article, we investigate the uniform convergence problem of the one-dimensional cubic Schrödinger equation. By using the Kato smoothing estimate, the maximal function estimate and the dyadic mixed Lebesgue spaces, we establish the uniform convergence of the one-dimensional cubic Schrödinger equation in $H^s\left(\mathbb{R}\right)(s>\frac{1}{6})$ which is an alternative proof of Theorem 1.1 $(n=1,p=3)$ of Compaan et al. [Pointwise convergence of the Schrödinger flow. Int Math Res Not. 2021;1:596–647].

Keywords:

• Cubic Schrödinger equation
• uniform convergence
• Kato smoothing estimate
• maximal function estimate

AMS Subject Classification:

• 35Q55

Disclosure statement

No potential conflict of interest was reported by the author(s).