$(p, q)$-Sobolev inequality and Nash inequality on forward complete Finsler metric measure manifolds
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866912210290212864 |
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| author | Cheng, Xinyue Ni, Qihui |
| author_facet | Cheng, Xinyue Ni, Qihui |
| contents | In this paper, we carry out in-depth research centering around the $(p, q)$-Sobolev inequality and Nash inequality on forward complete Finsler metric measure manifolds under the condition that ${\rm Ric}_{\infty} \geq -K$ for some $K \geq 0$. We first obtain a global $p$-Poincaré inequality on such Finsler manifolds. Based on this, we can derive a $(p, q)$-Sobolev inequality. Furthermore, we establish a global optimal $(p, q)$-Sobolev inequality with a sharp Sobolev constant. Finally, as an application of the $p$-Poincaré inequality, we prove a Nash inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17877 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $(p, q)$-Sobolev inequality and Nash inequality on forward complete Finsler metric measure manifolds Cheng, Xinyue Ni, Qihui Differential Geometry 53C60, 53B40, 53C21 In this paper, we carry out in-depth research centering around the $(p, q)$-Sobolev inequality and Nash inequality on forward complete Finsler metric measure manifolds under the condition that ${\rm Ric}_{\infty} \geq -K$ for some $K \geq 0$. We first obtain a global $p$-Poincaré inequality on such Finsler manifolds. Based on this, we can derive a $(p, q)$-Sobolev inequality. Furthermore, we establish a global optimal $(p, q)$-Sobolev inequality with a sharp Sobolev constant. Finally, as an application of the $p$-Poincaré inequality, we prove a Nash inequality. |
| title | $(p, q)$-Sobolev inequality and Nash inequality on forward complete Finsler metric measure manifolds |
| topic | Differential Geometry 53C60, 53B40, 53C21 |
| url | https://arxiv.org/abs/2501.17877 |