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

TL;DR

This paper determines the best possible constants in smoothing estimates for the free Dirac equation across all dimensions, extending previous results and providing explicit formulas for many cases.

Contribution

It introduces a general theorem for optimal smoothing constants for the Dirac equation in arbitrary dimensions, extending prior work and enabling explicit calculations.

Findings

Derived explicit formulas for optimal constants in smoothing estimates.

Extended known results from 2D and 3D to all dimensions ≥ 2.

Provided a modified spherical harmonics decomposition suited for the Dirac operator.

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.