{
  "id": "2310.00425",
  "version": "v1",
  "published": "2023-09-30T16:10:01.000Z",
  "updated": "2023-09-30T16:10:01.000Z",
  "title": "A note on bilinear spherical maximal functions",
  "authors": [
    "Ankit Bhojak",
    "Surjeet Singh Choudhary",
    "Saurabh Shrivastava",
    "Kalachand Shuin"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "In this article we address endpoint issues for the bilinear spherical maximal functions. We study necessary conditions for the bilinear maximal function, \\[\\mathcal M (f,g)(x)=\\sup_{t>0}\\left|\\int_{\\mathbb S^{1}}f(x-ty)g(x+ty)\\;d\\sigma(y)\\right|\\] to be bounded from $L^{p_1}(\\mathbb R^2)\\times L^{p_2}(\\mathbb R^2)$ to $L^p(\\mathbb R^2)$ and prove sharp results for a linearizarion of $\\mathcal M$. We also obtain borderline restricted weak type estimates for the well studied operator $$\\mathfrak{M}(f,g)(x):=\\sup_{t>0}\\left|\\int_{\\mathbb S^{1}}f(x-ty_1)g(x-ty_2)\\;d\\sigma(y_1,y_2)\\right|,$$ in dimension one and as a consequence, deduce similar estimates for a multilinear analogue of $\\mathfrak{M}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-30T16:10:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B15",
      "42B25"
    ],
    "keywords": [
      "bilinear spherical maximal functions",
      "borderline restricted weak type estimates",
      "bilinear maximal function",
      "study necessary conditions",
      "deduce similar estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}