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 Schr\"odinger equations and Dirac equations. For \(d\)-dimensional Schr\"odinger-type equations (\(d \geq 2\)), optimal constants of smoothing estimates were obtained by Bez--Saito-...

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Bibliographic Details
Published in:arXiv.org 2024-05
Main Authors: Ikoma, Makoto, Suzuki, Soichiro
Format: Article
Language:English
Subjects:
Constants
Data smoothing
Estimates
Mathematical analysis
Smoothing
Upper bounds
Online Access:Get full text
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Summary:Kato--Yajima smoothing estimates are one of the fundamental results in study of dispersive equations such as Schr\"odinger equations and Dirac equations. For \(d\)-dimensional Schr\"odinger-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 Schr\"odinger-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 Schr\"odinger-type and Dirac equations with radial initial data for any \(d \geq 2\). In addition, we also give optimal constants for the one-dimensional Schr\"odinger-type and Dirac equations.
ISSN:2331-8422