{
  "id": "2411.11470",
  "version": "v1",
  "published": "2024-11-18T11:13:01.000Z",
  "updated": "2024-11-18T11:13:01.000Z",
  "title": "Sobolev Versus Homogeneous Sobolev Extension",
  "authors": [
    "Pekka Koskela",
    "Riddhi Mishra",
    "Zheng Zhu"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper, we study the relationship between Sobolev extension domains and homogeneous Sobolev extension domains. Precisely, we obtain the following results. 1- Let $1\\leq q\\leq p\\leq \\infty$. Then a bounded $(L^{1, p}, L^{1, q})$-extension domain is also a $(W^{1, p}, W^{1, q})$-extension domain. 2- Let $1\\leq q\\leq p<q^\\star\\leq \\infty$ or $n< q \\leq p\\leq \\infty$. Then a bounded domain is a $(W^{1, p}, W^{1, q})$-extension domain if and only if it is an $(L^{1, p}, L^{1, q})$-extension domain. 3- For $1\\leq q<n$ and $q^\\star<p\\leq \\infty$, there exists a bounded domain $\\Omega\\subset\\mathbb{R}^n$ which is a $(W^{1, p}, W^{1, q})$-extension domain but not an $(L^{1, p}, L^{1, q})$-extension domain for $1 \\leq q <p\\leq n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-18T11:13:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "30L99"
    ],
    "keywords": [
      "bounded domain",
      "homogeneous sobolev extension domains",
      "relationship"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}