Directional Poincaré inequality on compact Lie groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912632516116480 |
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| author | da Silva, Paulo L. Dattori Kowacs, André Pedroso |
| author_facet | da Silva, Paulo L. Dattori Kowacs, André Pedroso |
| contents | We extend the directional Poincaré inequality on the torus, introduced by Steinerberger in [Ark. Mat. 54 (2016), pp. 555--569], to the setting of compact Lie groups. We provide necessary and sufficient conditions for the existence of such an inequality based on estimates on the eigenvalues of the global symbol of the corresponding vector field. We also prove that such refinement of the Poincaré inequality holds for a left-invariant vector field on a compact Lie group $G$ if and only if the vector field is globally solvable, and extend this equivalence to tube-type vector fields on $\mathbb{T}^1\times G$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_05409 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Directional Poincaré inequality on compact Lie groups da Silva, Paulo L. Dattori Kowacs, André Pedroso Analysis of PDEs Primary: 35A23, 35A01. Secondary: 43A77, 22E30 We extend the directional Poincaré inequality on the torus, introduced by Steinerberger in [Ark. Mat. 54 (2016), pp. 555--569], to the setting of compact Lie groups. We provide necessary and sufficient conditions for the existence of such an inequality based on estimates on the eigenvalues of the global symbol of the corresponding vector field. We also prove that such refinement of the Poincaré inequality holds for a left-invariant vector field on a compact Lie group $G$ if and only if the vector field is globally solvable, and extend this equivalence to tube-type vector fields on $\mathbb{T}^1\times G$. |
| title | Directional Poincaré inequality on compact Lie groups |
| topic | Analysis of PDEs Primary: 35A23, 35A01. Secondary: 43A77, 22E30 |
| url | https://arxiv.org/abs/2510.05409 |