Optimal constants of smoothing estimates for the 3D Dirac equation

TL;DR

This paper determines the optimal smoothing constants and extremisers for the 3D Dirac equation by simplifying the eigenvalue problem using a specialized spherical harmonics basis, extending previous 2D results.

Contribution

It provides the first explicit calculation of optimal constants and extremisers for the 3D Dirac equation's smoothing estimates, overcoming previous computational challenges.

Findings

Explicit optimal constants for 3D Dirac smoothing estimates

Construction of a special spherical harmonics basis simplifying eigenvalue calculations

Establishment of equivalence between Schrödinger and Dirac smoothing estimates

Abstract

Recently, Ikoma (2022) considered optimal constants and extremisers for the $2$-dimensional Dirac equation using the spherical harmonics decomposition. Though its argument is valid in any dimensions $d \geq 2$, the case $d \geq 3$ remains open since it leads us to too complicated calculation: determining all eigenvalues and eigenvectors of infinite dimensional matrices. In this paper, we give optimal constants and extremisers of smoothing estimates for the $3$-dimensional Dirac equation. In order to prove this, we construct a certain orthonormal basis of spherical harmonics. With respect to this basis, infinite dimensional matrices actually become block diagonal and so that eigenvalues and eigenvectors can be easily found. As applications, we obtain the equivalence of the smoothing estimate for the Schr\"{o}dinger equation and the Dirac equation, and improve a result by Ben-Artzi and Umeda (2021).