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

[2501.00949] Optimal constants of smoothing estimates for the Dirac equation in arbitrary dimensions Skip to main content Search Submit Donate Log in Mathematics > Analysis of PDEs arXiv:2501.00949 (math) [Submitted on 1 Jan 2025 ( v1 ), last revised 28 Aug 2026 (this version, v3)] Title: Optimal constants of smoothing estimates for the Dirac equation in arbitrary dimensions Authors: Soichiro Suzuki View PDF HTML (experimental) 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 the smoothing estimate associated with a spatial weight $w$ and smoothing function $\psi$ is given by $C = \sup_{k \in \mathbb{N}} \sup_{r > 0} \widetilde{\lambda}_k(r)$, where $\{ \widetilde{\lambda}_k \}$ is a certain sequence of functions defined via integral formulae involving $(w, \psi)$. This is an analogue of a similar result for Schrödinger equations given by Bez, Saito, and Sugimoto (2015), and also extends previous results of Ikoma (2022) and Ikoma and Suzuki (2025) for $d=2, 3$ to arbitrary 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 is well suited to 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 $(w, \psi)$. As it turns out, optimal constants for Dirac equations can be written explicitly in many cases, even when it is impossible for Schrödinger equations. In particular, the classical result of Simon (1992) for Schrödinger equations, which holds when $d \geq 3$ but fails when $d=2$, is true for Dirac equations whenever $d \geq 3$ and remains valid for the massless two-dimensional case. Comments: 34 pages, 3 figures. Revised according to the referee's report Subjects: Analysis of PDEs (math.AP) ; Classical Analysis and ODEs (math.CA) MSC classes: 33C55, 35B65, 35Q41, 42B10 Cite as: arXiv:2501.00949 [math.AP] (or arXiv:2501.00949v3 [math.AP] for this version) https://doi.org/10.48550/arXiv.2501.00949 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Soichiro Suzuki [ view email ] [v1] Wed, 1 Jan 2025 20:24:35 UTC (50 KB) [v2] Tue, 7 Jan 2025 01:44:16 UTC (50 KB) [v3] Fri, 28 Aug 2026 06:34:16 UTC (50 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AP < prev | next > new | recent | 2025-01 Change to browse by: math math.CA References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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