{
  "id": "1702.00602",
  "version": "v1",
  "published": "2017-02-02T10:06:35.000Z",
  "updated": "2017-02-02T10:06:35.000Z",
  "title": "Unions of cubes in $\\mathbb{R}^{n}$, combinatorics in $\\mathbb{Z}^{n}$ and the John-Nirenberg and John-Strömberg inequalities",
  "authors": [
    "Michael Cwikel"
  ],
  "comment": "23 pages",
  "categories": [
    "math.FA",
    "math.MG"
  ],
  "abstract": "Suppose that the $d$-dimensional unit cube $Q$ is the union of three disjoint \"simple\" sets $E$, $F$ and $G$ and that the volumes of $E$ and $F$ are both greater than half the volume of $G$. Does this imply that, for some cube $W$ contained in $Q$. the volumes of $E\\cap W$ and $F\\cap W$ both exceed $s$ times the volume of $W$ for some absolute positive constant $s$? Here, by \"simple\" we mean a set which is a union of finitely many dyadic cubes. We prove that an affirmative answer to this question would have deep consequences for the important space $BMO$ of functions of bounded mean oscillation introduced by John and Nirenberg. The notion of a John-Str\\\"omberg pair is closely related to the above question, and the above mentioned result is obtained as a consequence of a general result about these pairs. We also present a number of additional results about these pairs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-02T10:06:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30H35",
      "42B35",
      "46E35"
    ],
    "keywords": [
      "john-strömberg inequalities",
      "combinatorics",
      "john-nirenberg",
      "dimensional unit cube",
      "additional results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}