A smoothing property of Schrodinger equations in the critical case

TL;DR

This paper proves a global smoothing property for Schrödinger equations in critical cases, highlighting the importance of operator structure related to geometric properties, with extensions to higher-order and hyperbolic equations.

Contribution

It establishes the critical smoothing estimate for Schrödinger equations in higher dimensions, emphasizing the role of operator structure linked to geometric features.

Findings

Critical smoothing estimate achieved under specific operator structure

Results extended to higher-order operators and hyperbolic equations

Geometric properties influence the smoothing behavior

Abstract

In this paper a global smoothing property of Schrodinger equations is established in the critical case in dimensions two and higher. It is shown that the critical smoothing estimate is attained if the smoothing operator has some structure. This structure is related to the geometric properties of the equations. Results for critical cases for operators of higher orders as well as for hyperbolic equations are also given.