{
  "id": "2407.06307",
  "version": "v1",
  "published": "2024-07-08T18:18:47.000Z",
  "updated": "2024-07-08T18:18:47.000Z",
  "title": "Optimal function spaces and Sobolev embeddings",
  "authors": [
    "David Kubíček"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We establish equivalence between the boundedness of specific supremum operators and the optimality of function spaces in Sobolev embeddings acting on domains in ambient Euclidean space with a prescribed isoperimetric behavior. Our approach is based on exploiting known relations between higher-order Sobolev embeddings and isoperimetric inequalities. We provide an explicit way to compute both the optimal domain norm and the optimal target norm in a Sobolev embedding. Finally, we apply our results to higher-order Sobolev embeddings on John domains and on domains from the Maz'ya classes. Furthermore, our results are partially applicable to embeddings involving product probability spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-08T18:18:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E30",
      "46E35",
      "47G10"
    ],
    "keywords": [
      "optimal function spaces",
      "higher-order sobolev embeddings",
      "specific supremum operators",
      "optimal target norm",
      "ambient euclidean space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}