Optimal Sobolev inequalities of high order with $L^2$-remainder
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916803035267072 |
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| author | Carletti, Lorenzo Robert, Frédéric |
| author_facet | Carletti, Lorenzo Robert, Frédéric |
| contents | We investigate the validity of the optimal higher-order Sobolev inequality $H_k^2(M^n)\hookrightarrow L^{\frac{2n}{n-2k}}(M^n)$ on a closed Riemannian manifold when the remainder term is the $L^2-$norm. Unlike the case $k=1$, the optimal inequality does not hold in general for $k>1$. We prove conditions for the validity and non-validity that depend on the geometry of the manifold. Our conditions are sharp when $k=2$ and in small dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_17028 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal Sobolev inequalities of high order with $L^2$-remainder Carletti, Lorenzo Robert, Frédéric Analysis of PDEs 35J35, 35G20, 58J05 (Primary) 35J60, 35B44, 35J08 (Secondary) We investigate the validity of the optimal higher-order Sobolev inequality $H_k^2(M^n)\hookrightarrow L^{\frac{2n}{n-2k}}(M^n)$ on a closed Riemannian manifold when the remainder term is the $L^2-$norm. Unlike the case $k=1$, the optimal inequality does not hold in general for $k>1$. We prove conditions for the validity and non-validity that depend on the geometry of the manifold. Our conditions are sharp when $k=2$ and in small dimensions. |
| title | Optimal Sobolev inequalities of high order with $L^2$-remainder |
| topic | Analysis of PDEs 35J35, 35G20, 58J05 (Primary) 35J60, 35B44, 35J08 (Secondary) |
| url | https://arxiv.org/abs/2506.17028 |