{
  "id": "2304.03878",
  "version": "v1",
  "published": "2023-04-08T00:29:33.000Z",
  "updated": "2023-04-08T00:29:33.000Z",
  "title": "Discrete logarithmic Sobolev inequalities in Banach spaces",
  "authors": [
    "Dario Cordero-Erausquin",
    "Alexandros Eskenazis"
  ],
  "categories": [
    "math.FA",
    "math.MG"
  ],
  "abstract": "Let $\\mathscr{C}_n=\\{-1,1\\}^n$ be the discrete hypercube equipped with the uniform probability measure $\\sigma_n$. We prove that if $(E,\\|\\cdot\\|_E)$ is a Banach space of finite cotype and $p\\in[1,\\infty)$, then every function $f:\\mathscr{C}_n\\to E$ satisfies the dimension-free vector-valued $L_p$ logarithmic Sobolev inequality $$\\|f-\\mathbb{E} f\\|_{L_p(\\log L)^{p/2}(E)} \\leq \\mathsf{K}_p(E) \\left( \\int_{\\mathscr{C}_n} \\Big\\| \\sum_{i=1}^n \\delta_i \\partial_i f\\Big\\|_{L_p(E)}^p \\, d\\sigma_n(\\delta)\\right)^{1/p}.$$ The finite cotype assumption is necessary for the conclusion to hold. This estimate is the hypercube counterpart of a result of Ledoux (1988) in Gauss space and the optimal vector-valued version of a deep inequality of Talagrand (1994). As an application, we use such vector-valued $L_p$ logarithmic Sobolev inequalities to derive new lower bounds for the bi-Lipschitz distortion of nonlinear quotients of the Hamming cube into Banach spaces with prescribed Rademacher type.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-08T00:29:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "logarithmic sobolev inequality",
      "discrete logarithmic sobolev inequalities",
      "banach space",
      "uniform probability measure",
      "finite cotype assumption"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}