{
  "id": "2109.08014",
  "version": "v1",
  "published": "2021-09-16T14:45:17.000Z",
  "updated": "2021-09-16T14:45:17.000Z",
  "title": "Fractional integration of summable functions: Maz'ya's $Φ$-inequalities",
  "authors": [
    "Dmitriy Stolyarov"
  ],
  "comment": "21 pages, 2 figures",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "We study the inequalities of the type $|\\int_{\\mathbb{R}^d} \\Phi(K*f)| \\lesssim \\|f\\|_{L_1(\\mathbb{R}^d)}^p$, where the kernel $K$ is homogeneous of order $\\alpha - d$ and possibly vector-valued, the function $\\Phi$ is positively $p$-homogeneous, and $p = d/(d-\\alpha)$. Under mild regularity assumptions on $K$ and $\\Phi$, we find necessary and sufficient conditions on these functions under which the inequality holds true with a uniform constant for all sufficiently regular functions $f$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-16T14:45:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional integration",
      "summable functions",
      "inequality holds true",
      "mild regularity assumptions",
      "sufficiently regular functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}