A global space-time estimate for dispersive operators through its local estimate

TL;DR

This paper demonstrates that local space-time estimates can imply global estimates for dispersive operators, using Littlewood-Paley theory and a generalized epsilon removal lemma, leading to sharp global estimates for fractional Schrödinger equations.

Contribution

It introduces a method to derive global space-time estimates from local ones for dispersive operators, generalizing Tao's epsilon removal lemma in mixed norms.

Findings

Established a link between local and global space-time estimates for dispersive operators.

Obtained sharp global estimates for fractional Schrödinger equations in R^{2+1}.

Extended epsilon removal techniques to mixed norm settings.

Abstract

We will show that a local space-time estimate implies a global space-time estimate for dispersive operators. In order for this implication we consider a Littlewood-Paley type square function estimate for dispersive operators in a time variable and a generalization of Tao's epsilon removal lemma in mixed norms. By applying this implication to the fractional Schrodinger equation in R^{2+1} we obtain the sharp global space-time estimates with optimal regularity from the previous known local ones.