Estimates for multiparameter maximal operators of Schr\"odinger type

TL;DR

This paper establishes sharp and near-sharp estimates for multiparameter maximal Schr"odinger operators, extending prior work, analyzing endpoint behavior, and exploring integrability conditions for solutions to the Schr"odinger equation.

Contribution

It introduces new estimates based on kernel integrability, extends existing results, and discusses endpoint behavior and open problems in the context of multiparameter Schr"odinger maximal operators.

Findings

Sharp and almost sharp estimates obtained

Analysis of endpoint behavior conducted

Global integrability conditions for solutions discussed

Abstract

Multiparameter maximal estimates are considered for operators of Schr\"odinger type. Sharp and almost sharp results, that extend work by Rogers and Villarroya, are obtained. We provide new estimates via the integrability of the kernel which naturally appears with a TT* argument and discuss the behavior at the endpoints. We treat in particular the case of global integrability of the maximal operator on finite time for solutions to the linear Schr\"odinger equation and make some comments on an open problem.