{
  "id": "2207.02176",
  "version": "v1",
  "published": "2022-07-05T17:11:38.000Z",
  "updated": "2022-07-05T17:11:38.000Z",
  "title": "Almost everywhere convergence for Lebesgue differentiation processes along rectangles",
  "authors": [
    "Emma D'Aniello",
    "Anthony Gauvan",
    "Laurent Moonens",
    "Joseph M. Rosenblatt"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "In this paper, we study Lebesgue differentiation processes along rectangles $R_k$ shrinking to the origin in the Euclidean plane, and the question of their almost everywhere convergence in $L^p$ spaces. In particular, classes of examples of such processes failing to converge a.e. in $L^\\infty$ are provided, for which $R_k$ is known to be oriented along the slope $k^{-s}$ for $s>0$, yielding an interesting counterpart to the fact that the directional maximal operator associated to the set $\\{k^{-s}:k\\in\\mathbb{N}^*\\}$ fails to be bounded in $L^p$ for any $1\\leq p<\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-05T17:11:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B25",
      "26B05",
      "42B35"
    ],
    "keywords": [
      "rectangles",
      "convergence",
      "study lebesgue differentiation processes",
      "directional maximal operator",
      "euclidean plane"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}