{
  "id": "2504.19939",
  "version": "v2",
  "published": "2025-04-28T16:09:42.000Z",
  "updated": "2025-05-08T17:24:33.000Z",
  "title": "Stability inequalities with explicit constants for a family of reverse Sobolev inequalities on the sphere",
  "authors": [
    "Tobias König"
  ],
  "comment": "20 pages. Version 2: introduction slightly rewritten, statement of Proposition 3.1 corrected",
  "categories": [
    "math.AP"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-05-08T17:24:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "reverse sobolev inequality",
      "explicit constants",
      "full admissible parameter range",
      "sobolev-type stability inequality",
      "main difficulty"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}