Bounds for the optimal constant of the Bakry-Émery $Γ_2$ criterion inequality on $ RP^{d-1}$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913480362164224 |
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| author | Ji, Sehyun |
| author_facet | Ji, Sehyun |
| contents | We prove upper and lower bounds on the optimal constant $Λ_d$ of the Bakry-Émery $Γ_2$ criterion for positive symmetric functions on the unit sphere $S^{d-1}$, which also can be identified as positive functions on the real projective space $RP^{d-1}$. The Bakry-Émery $Γ_2$ criterion inequality was crucially used to prove the monotonicty of the Fisher information for the Landau equation by Guillen and Silvestre recently. Therefore, a better bound on the optimal constant $Λ_d$ expands the range of interaction potentials that exhibits the monotonicity of the Fisher information. In particular, we compute that $Λ_3$ is between $5.5$ and $5.739$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_13954 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds for the optimal constant of the Bakry-Émery $Γ_2$ criterion inequality on $ RP^{d-1}$ Ji, Sehyun Analysis of PDEs Functional Analysis 26D10(Primary) 58J99, 35Q70(Secondary) We prove upper and lower bounds on the optimal constant $Λ_d$ of the Bakry-Émery $Γ_2$ criterion for positive symmetric functions on the unit sphere $S^{d-1}$, which also can be identified as positive functions on the real projective space $RP^{d-1}$. The Bakry-Émery $Γ_2$ criterion inequality was crucially used to prove the monotonicty of the Fisher information for the Landau equation by Guillen and Silvestre recently. Therefore, a better bound on the optimal constant $Λ_d$ expands the range of interaction potentials that exhibits the monotonicity of the Fisher information. In particular, we compute that $Λ_3$ is between $5.5$ and $5.739$. |
| title | Bounds for the optimal constant of the Bakry-Émery $Γ_2$ criterion inequality on $ RP^{d-1}$ |
| topic | Analysis of PDEs Functional Analysis 26D10(Primary) 58J99, 35Q70(Secondary) |
| url | https://arxiv.org/abs/2408.13954 |