Optimal constants of smoothing estimates for the Dirac equation in arbitrary dimensions

Abstract

We give optimal constants of smoothing estimates for the $d$-dimensional free Dirac equation for any $d \geq 2$. Our main abstract theorem shows that the optimal constant $C$ of smoothing estimate associated with a spatial weight $w$ and smoothing function $\psi$ is given by $(2\pi)^{d-1} C = \sup_{k \in \mathbb{N}} \sup_{r > 0} \widetilde{\lambda}_k(r)$, where $\{ \widetilde{\lambda}_k \}$ is a certain sequence of functions defined via integral formulae involving $(w, \psi)$. This is an analogue of a similar result for Schr\"{o}dinger equations given by Bez--Saito--Sugimoto (2015), and also extends previous results of Ikoma (2022) and Ikoma--Suzuki (2024) for $d=2, 3$ to any dimensions $d \geq 2$. In order to prove this, we establish a modified version of the spherical harmonics decomposition of $L^2(\mathbb{S}^{d-1})$, which suits well with the Dirac operator and allows us to find optimal constants. Furthermore, using our abstract theorem, we give explicit values of optimal constants associated with typical examples of $(w, \psi)$. As it turns out, optimal constants for Dirac equations can be written explicitly in many cases, even in the cases that it is impossible for Schr\"{o}dinger equations.