{
  "id": "2210.08320",
  "version": "v1",
  "published": "2022-10-15T15:38:58.000Z",
  "updated": "2022-10-15T15:38:58.000Z",
  "title": "Nikodym sets and maximal functions associated with spheres",
  "authors": [
    "Alan Chang",
    "Georgios Dosidis",
    "Jongchon Kim"
  ],
  "comment": "37 pages, 2 figures",
  "categories": [
    "math.CA"
  ],
  "abstract": "We study spherical analogues of Nikodym sets and related maximal functions. In particular, we prove sharp $L^p$-estimates for Nikodym maximal functions associated with spheres. As a corollary, any Nikodym set for spheres must have full Hausdorff dimension. In addition, we consider a class of maximal functions which contains the spherical maximal function as a special case. We show that $L^p$-estimates for these maximal functions can be deduced from local smoothing estimates for the wave equation relative to fractal measures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-15T15:38:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nikodym set",
      "full hausdorff dimension",
      "nikodym maximal functions",
      "spherical maximal function",
      "related maximal functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}