Optimal Sobolev inequalities in the hyperbolic space
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866911483145748480 |
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| author | Mihula, Zdeněk |
| author_facet | Mihula, Zdeněk |
| contents | We find the optimal function norm on the left-hand side of the $m$th order Sobolev type inequality $\|u\|_{Y(\mathbb{H}^n)} \leq C \|\nabla_g^m u\|_{X(\mathbb{H}^n)}$ in the $n$-dimensional hyperbolic space $\mathbb{H}^n$, $1\leq m < n$. The optimal function norm in the inequality among all rearrangement-invariant function norms is completely characterized. A variety of concrete examples of optimal function norms is provided. The examples include delicate limiting cases, and especially when $m\geq3$, seem to provide new, improved inequalities in these limiting cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_06797 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal Sobolev inequalities in the hyperbolic space Mihula, Zdeněk Functional Analysis 46E30, 46E35 (Primary) 26D10 (Secondary) We find the optimal function norm on the left-hand side of the $m$th order Sobolev type inequality $\|u\|_{Y(\mathbb{H}^n)} \leq C \|\nabla_g^m u\|_{X(\mathbb{H}^n)}$ in the $n$-dimensional hyperbolic space $\mathbb{H}^n$, $1\leq m < n$. The optimal function norm in the inequality among all rearrangement-invariant function norms is completely characterized. A variety of concrete examples of optimal function norms is provided. The examples include delicate limiting cases, and especially when $m\geq3$, seem to provide new, improved inequalities in these limiting cases. |
| title | Optimal Sobolev inequalities in the hyperbolic space |
| topic | Functional Analysis 46E30, 46E35 (Primary) 26D10 (Secondary) |
| url | https://arxiv.org/abs/2305.06797 |