{
  "id": "2212.11463",
  "version": "v1",
  "published": "2022-12-22T03:19:23.000Z",
  "updated": "2022-12-22T03:19:23.000Z",
  "title": "Bilinear maximal functions associated with degenerate surfaces",
  "authors": [
    "Sanghyuk Lee",
    "Kalachand Shuin"
  ],
  "comment": "21 pages, 3 figures",
  "categories": [
    "math.CA"
  ],
  "abstract": "We study $L^{p}\\times L^{q}\\rightarrow L^{r}$-boundedness of (sub)bilinear maximal functions associated with degenerate hypersurfaces. First, we obtain the maximal bound on the sharp range of exponents $p,q,r$ (except some border line cases) for the bilinear maximal functions given by the model surface $\\big\\{(y,z)\\in\\mathbb{R}^{n}\\times \\mathbb{R}^{n}:|y|^{l_{1}}+|z|^{l_{2}}=1\\big\\}$, $(l_{1},l_{2})\\in [1,\\infty)^2$, $n\\ge 2$. Our result manifests that nonvanishing Gaussian curvature is not good enough, in contrast with $L^p$-boundedness of the (sub)linear maximal operator associated to hypersurfaces, to characterize the best possible maximal boundedness. Secondly, we consider the bilinear maximal function associated to the finite type curve in $\\mathbb R^2$ and obtain a complete characterization of the maximal bound. We also prove multilinear generalizations of the aforementioned results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-22T03:19:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B25",
      "42B15",
      "46T30"
    ],
    "keywords": [
      "bilinear maximal function",
      "degenerate surfaces",
      "linear maximal operator",
      "border line cases",
      "finite type curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}