Stability inequalities with explicit constants for a family of reverse Sobolev inequalities on the sphere
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915278229602304 |
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| author | König, Tobias |
| author_facet | König, Tobias |
| contents | We prove a stability inequality associated to the reverse Sobolev inequality on the sphere $\mathbb S^n$, for the full admissible parameter range $s - \frac{n}{2} \in (0,1) \cup (1,2)$. To implement the classical proof of Bianchi and Egnell, we overcome the main difficulty that the underlying operator $A_{2s}$ is not positive definite.
As a consequence of our analysis and recent results from Gong et al. (arXiv:2503.20350 [math.AP]), the case $s - \frac{n}{2} \in (1,2)$ remarkably constitutes the first example of a Sobolev-type stability inequality (i) whose best constant is explicit and (ii) which does not admit an optimizer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19939 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability inequalities with explicit constants for a family of reverse Sobolev inequalities on the sphere König, Tobias Analysis of PDEs We prove a stability inequality associated to the reverse Sobolev inequality on the sphere $\mathbb S^n$, for the full admissible parameter range $s - \frac{n}{2} \in (0,1) \cup (1,2)$. To implement the classical proof of Bianchi and Egnell, we overcome the main difficulty that the underlying operator $A_{2s}$ is not positive definite. As a consequence of our analysis and recent results from Gong et al. (arXiv:2503.20350 [math.AP]), the case $s - \frac{n}{2} \in (1,2)$ remarkably constitutes the first example of a Sobolev-type stability inequality (i) whose best constant is explicit and (ii) which does not admit an optimizer. |
| title | Stability inequalities with explicit constants for a family of reverse Sobolev inequalities on the sphere |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.19939 |