A note on the Schrodinger maximal function

TL;DR

This paper establishes a lower bound on the regularity parameter s needed to control the Schrödinger maximal function for functions in Sobolev spaces, highlighting a fundamental limitation in harmonic analysis.

Contribution

It proves that controlling the Schrödinger maximal function necessitates Sobolev regularity s at least n/(2(n+1)), providing a key insight into the regularity requirements.

Findings

Control of the Schrödinger maximal function requires s ≥ n/(2(n+1))

Lower bound on regularity for maximal function control

Highlights limitations in Schrödinger equation analysis

Abstract

It is shown that control of the Schrodinger maximal functions $\sup_{0<t<1}|e^{it\Delta} f|$ for $f\in H^s(\mathbb R^n)$ requires $s\geq \frac n{2(n+1)}$