Optimal constants of smoothing estimates for Dirac equations with radial data

TL;DR

This paper determines the exact optimal constants for smoothing estimates of Schrödinger-type and Dirac equations with radial data across all dimensions, extending previous results and addressing open cases.

Contribution

It provides the first complete set of optimal constants for both Schrödinger-type and Dirac equations with radial data in all dimensions, including the challenging higher-dimensional Dirac case.

Findings

Optimal constants for Schrödinger-type equations in all dimensions.

Optimal constants for Dirac equations with radial data in all dimensions.

Extension of previous bounds to exact optimal constants.

Abstract

Kato--Yajima smoothing estimates are one of the fundamental results in study of dispersive equations such as Schr\"odinger equations and Dirac equations. For $d$-dimensional Schr\"odinger-type equations ($d \geq 2$), optimal constants of smoothing estimates were obtained by Bez--Saito--Sugimoto (2017) via the so-called Funk--Hecke theorem. Recently Ikoma (2022) considered optimal constants for $d$-dimensional Dirac equations using a similar method, and it was revealed that determining optimal constants for Dirac equations is much harder than the case of Schr\"odinger-type equations. Indeed, Ikoma obtained the optimal constant in the case $d = 2$, but only upper bounds (which seem not optimal) were given in other dimensions. In this paper, we give optimal constants for $d$-dimensional Schr\"odinger-type and Dirac equations with radial initial data for any $d \geq 2$. In addition, we also give optimal constants for the one-dimensional Schr\"odinger-type and Dirac equations.