Optimal constants of smoothing estimates for Dirac equations with radial data

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.