Optimal Constants of Smoothing Estimates for the 2D Dirac Equation

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 ≥ 2, the case d ≥ 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.