Optimal forward and reverse estimates of Morawetz and Kato-Yajima type with angular smoothing index

TL;DR

This paper derives optimal constants and characterizes extremisers for smoothing estimates of Schrödinger, Klein-Gordon, and wave equations, incorporating angular regularity and revealing dimension-dependent extremiser behavior.

Contribution

It provides the first sharp forward and reverse smoothing estimates with angular smoothing for these equations, including extremiser characterizations and exact identities in specific cases.

Findings

Optimal constants and extremisers characterized for Schrödinger smoothing estimates.

Extension of sharp smoothing estimates to Klein-Gordon and wave equations.

Exact identities obtained in four dimensions for special smoothing indices.

Abstract

For the solution of the free Schr\"odinger equation, we obtain the optimal constants and characterise extremisers for forward and reverse smoothing estimates which are global in space and time, contain a homogeneous and radial weight in the space variable, and incorporate a certain angular regularity. This will follow from a more general result which permits analogous sharp forward and reverse smoothing estimates and a characterisation of extremisers for the solution of the free Klein-Gordon and wave equations. The nature of extremisers is shown to be sensitive to both the dimension and the size of the smoothing index relative to the dimension. Furthermore, in four spatial dimensions and certain special values of the smoothing index, we obtain an exact identity for each of these evolution equations.