{
  "id": "2502.05882",
  "version": "v1",
  "published": "2025-02-09T12:47:07.000Z",
  "updated": "2025-02-09T12:47:07.000Z",
  "title": "Maximal operators on spaces BMO and BLO",
  "authors": [
    "Grigori A. Karagulyan"
  ],
  "comment": "24 pages. arXiv admin note: text overlap with arXiv:2308.02672",
  "categories": [
    "math.CA"
  ],
  "abstract": "We consider maximal kernel-operators on abstract measure spaces $(X,\\mu)$ equipped with a ball-basis. We prove that under certain asymptotic condition on the kernels those operators maps boundedly BMO(X) into BLO(X), generalizing the well-known results of Bennett-DeVore-Sharpley and Bennett for the Hardy-Littlewood maximal function. As a particular case of such an operator one can consider the maximal function \\begin{equation} M_\\phi f(x)=\\sup_{r>0}\\frac{1}{r^d}\\int_{R^d}|f(t)|\\phi\\left(\\frac{x-t}{r}\\right)dt, \\end{equation} and its non-tangential version. Here $\\phi(x)\\ge 0$ is a bounded spherical function on $R^d$, decreasing with respect to $|x|$ and satisfying the bound \\begin{equation*} \\int_{R^d}\\phi (x)\\log (2+|x|)dx<\\infty. \\end{equation*} We prove that if $f\\in BMO(R^d)$ and $M_\\phi(f)$ is not identically infinite, then $M_\\phi(f)\\in BLO(R^d)$. Our main result is an inequality, providing an estimation of certain local oscillation of the maximal function $M(f)$ by a local sharp function of $f$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-09T12:47:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B25",
      "42B35",
      "43A85"
    ],
    "keywords": [
      "maximal operators",
      "spaces bmo",
      "local sharp function",
      "operators maps boundedly bmo",
      "hardy-littlewood maximal function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}