{
  "id": "2407.14052",
  "version": "v1",
  "published": "2024-07-19T06:22:32.000Z",
  "updated": "2024-07-19T06:22:32.000Z",
  "title": "Maz'ya's $Φ$-inequalities on domains",
  "authors": [
    "Dmitriy Stolyarov"
  ],
  "comment": "15 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "We find necessary and sufficient conditions on the function $\\Phi$ for the inequality $$\\Big|\\int_\\Omega \\Phi(K*f)\\Big|\\lesssim \\|f\\|_{L_1(\\mathbb{R}^d)}^p$$ to be true. Here $K$ is a positively homogeneous of order $\\alpha - d$, possibly vector valued, kernel, $\\Phi$ is a $p$-homogeneous function, and $p=d/(d-\\alpha)$. The domain $\\Omega\\subset \\mathbb{R}^d$ is either bounded with $C^{1,\\beta}$ smooth boundary for some $\\beta > 0$ or a halfspace in $\\mathbb{R}^d$. As a corollary, we describe the positively homogeneous of order $d/(d-1)$ functions $\\Phi\\colon \\mathbb{R}^d \\to \\mathbb{R}$ that are suitable for the bound $$\\Big|\\int_\\Omega \\Phi(\\nabla u)\\Big|\\lesssim \\int_\\Omega |\\Delta u|.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-19T06:22:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "inequality",
      "sufficient conditions",
      "smooth boundary",
      "positively homogeneous",
      "homogeneous function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}