Transference of local to global $L^2$ maximal estimates for dispersive partial differential equations

TL;DR

This paper presents a simple method to extend local $L^2$ maximal estimates to global ones for dispersive PDEs, avoiding complex wave packet techniques by using scalings and classical oscillatory integral estimates.

Contribution

It introduces an elementary proof technique for transferring local to global estimates in dispersive PDEs, bypassing wave packet methods.

Findings

Successfully transfers local $L^2$ estimates to global estimates

Avoids wave packet techniques in the proof process

Uses scalings and classical oscillatory integral estimates

Abstract

In this paper we give an elementary proof for transference of local to global maximal estimates for dispersive PDEs. This is done by transferring local $L^2$ estimates for certain oscillatory integrals with rough phase functions, to the corresponding global estimates. The elementary feature of our approach is that it entirely avoids the use of the wave packet techniques which are quite common in this context, and instead is based on scalings and classical oscillatory integral estimates.