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 the smoothing estimate associated with a spatial weight $w$ and smoothing function $\psi$ is given by $C = \sup_{k \in \mathbb{N}} \sup_{r > 0} \widetilde{\lambda}_k(r)$, 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, and Sugimoto (2015), and also extends previous results of Ikoma (2022) and Ikoma and Suzuki (2025) for $d=2, 3$ to arbitrary 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 is well suited to 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 when it is impossible for Schr\"{o}dinger equations. In particular, the classical result of Simon (1992) for Schr\"{o}dinger equations, which holds when $d \geq 3$ but fails when $d=2$, is true for Dirac equations whenever $d \geq 3$ and remains valid for the massless two-dimensional case.