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

TL;DR

This work extends Kato–Yajima smoothing theory to the $d$-dimensional Dirac equation, introducing an abstract criterion for the optimal smoothing constant via $\tilde{\lambda}_k(r)$ and a refined spherical-harmonics decomposition to handle the Dirac operator. It proves that $(2\pi)^{d-1}\bm{C}_{\text{D}}^{(d)}(w,\psi)=\tilde{\lambda}_{*}=\sup_{k\in\mathbb{N}}\esssup_{r>0} \tilde{\lambda}_k(r)$, and provides explicit constants for representative weight–smoothing pairs across all dimensions, including detailed 2D/3D limits and extremiser conditions. The paper also establishes a modified invariant-subspace framework to diagonalise the Dirac symbol, enabling precise computation of optimal constants and highlighting cases where Dirac smoothing behaves differently from Schrödinger smoothing (e.g., lack of nontrivial extremisers for certain masses). The results unify and extend prior 2D/3D Dirac analyses to arbitrary $d\ge2$, with practical implications for dispersive estimates and spectral theory of Dirac operators.

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 $ψ$ is given by $(2π)^{d-1} C = \sup_{k \in \mathbb{N}} \sup_{r > 0} \widetildeλ_k(r)$, where $\{ \widetildeλ_k \}$ is a certain sequence of functions defined via integral formulae involving $(w, ψ)$. This is an analogue of a similar result for Schrödinger 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 $(w, ψ)$. 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 Schrödinger equations.

Figures (13)

• Figure 1: The graph of $(1+r^2)^{1/2} I_{d/2 - 1}(r) K_{d/2 - 1}(r)$.
• Figure 2: The case $d=3$.
• Figure 3: The case $d=4$.
• Figure 4: The case $d=5$.
• Figure 5: The graph of $(1+r^2)^{1/2} ( I_{d/2 - 1}(r) K_{d/2 - 1}(r) + I_{d/2}(r) K_{d/2}(r) )$.

Theorems & Definitions (49)

• Theorem 1.1: BSS2015
• Remark 1.A
• Remark 1.B
• Theorem 1.2: BRS2020
• Corollary 1.3
• Theorem 1.4: Iko2022
• Theorem 1.5
• Theorem 1.6: BS2017, BSS2015
• Theorem 1.7
• Theorem 1.8