Optimal constants of smoothing estimates for Dirac equations with radial data

Abstract

Kato--Yajima smoothing estimates are one of the fundamental results in study of dispersive equations such as Schrödinger equations and Dirac equations. For $d$-dimensional Schrödinger-type equations ($d\ge 2$), 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 Schrödinger-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 Schrödinger-type and Dirac equations with radial initial data for any $d\ge 2$. In addition, we also give optimal constants for the one-dimensional Schrödinger-type and Dirac equations.

Section 4.3 is added