arXiv subjects

Makoto Ikoma

Publications and source records attributed to Makoto Ikoma.

2 recordsLinked to original sources

arXiv2025

Optimal constants of smoothing estimates for the 3D Dirac equation

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 Schrödinger equation and the Dirac equation, and improve a result by Ben-Artzi and Umeda (2021).

math.AP↗

arXiv2024

Optimal constants of smoothing estimates for Dirac equations with radial data

Kato--Yajima smoothing estimates are one of the fundamental results in study of dispersive equations such as Schrödinger equations and Dirac equations. For $d$-dimensional Schrödinger-type equations ($d \geq 2$), optimal constants of smoothing estimates were obtained by Bez--Saito--Sugimoto (2017) via the so-called Funk--Hecke theorem. Recently Ikoma (2022) considered optimal constants for $d$-dimensional Dirac equations using a similar method, and it was revealed that determining optimal constants for Dirac equations is much harder than the case of Schrödinger-type equations. Indeed, Ikoma obtained the optimal constant in the case $d = 2$, but only upper bounds (which seem not optimal) were given in other dimensions. In this paper, we give optimal constants for $d$-dimensional Schrödinger-type and Dirac equations with radial initial data for any $d \geq 2$. In addition, we also give optimal constants for the one-dimensional Schrödinger-type and Dirac equations.

math.AP↗