{
  "id": "1305.6069",
  "version": "v1",
  "published": "2013-05-26T20:20:46.000Z",
  "updated": "2013-05-26T20:20:46.000Z",
  "title": "Uniform convexity of paranormed generalizations of $L^p$ spaces",
  "authors": [
    "Justyna Jarczyk",
    "Janusz Matkowski"
  ],
  "comment": "30 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "For a measure space $(\\Omega ,\\Sigma ,\\mu)$ and a bijective increasing function $\\varphi :\\left[ 0,\\infty \\right) \\rightarrow \\left[0,\\infty \\right)$ the $L^{p}$-like paranormed ($F$-normed) function space with the paranorm of the form $\\mathbf{p}_{\\varphi}(x)=\\varphi ^{-1}\\left(\\int_{\\Omega}\\varphi \\circ \\left|x\\right|d\\mu \\right)$ is considered. Main results give general conditions under which this space is uniformly convex. The Clarkson theorem on the uniform convexity of $L^{p}$-space is generalized. Under some specific assumptions imposed on $\\varphi$ we give not only a proof of the uniform convexity but also show the formula of a modulus of convexity. We establish the uniform convexity of all finite-dimensional paranormed spaces, generated by a strictly convex bijection $\\varphi$ of $[0, \\infty)$. However, the {\\it a contrario} proof of this fact provides no information on a modulus of convexity of these spaces. In some cases it can be done, even an exact formula of a modulus can be proved. We show how to make it in the case when $S={\\mathbb R}^2$ and $\\varphi$ is given by $\\varphi(t)={\\rm e}^t-1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-26T20:20:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46A16",
      "46E30",
      "47H09",
      "47H10"
    ],
    "keywords": [
      "uniform convexity",
      "paranormed generalizations",
      "strictly convex bijection",
      "function space",
      "exact formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.6069J"
    }
  }
}