{
  "id": "1402.0528",
  "version": "v5",
  "published": "2014-02-03T21:32:49.000Z",
  "updated": "2016-08-29T15:52:20.000Z",
  "title": "ODE to $L^p$ norms",
  "authors": [
    "Jarno Talponen"
  ],
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "abstract": "In this paper we relate the geometry of Banach spaces to the theory of differential equations, apparently in a new way. We will construct Banach function space norms arising as weak solutions to ordinary differential equations of first order. This provides as a special case a new way of defining varying exponent $L^p$ spaces, different from the Orlicz type approach. We explain heuristically how the definition of the norm by means of the particular ODE is justified. The resulting class of spaces includes the classical $L^p$ spaces as a special case. We present an ODE-free means of defining the norms investigated.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-07-30T10:46:56.000Z",
      "title": "ODE representation for varying exponent $L^p$ norm",
      "abstract": "In this paper we relate the geometry of Banach spaces to the theory of differential equations, apparently in a new way. We will construct Banach function space norms arising as weak solutions to ordinary differential equations of first order. This provides as a special case a new way of defining varying exponent $L^p$ spaces, different from the Orlicz type approach. We explain heuristically how the definition of the norm by means of the particular ODE is justified. The resulting class of spaces includes the classical $L^p$ spaces as a special case. It turns out that the duality of these spaces behaves in an anticipated way, same as the uniform convexity and uniform smoothness. We study the arising duality of the ODEs and also the duality of their solutions. We present an ODE-free means of defining the norms investigated. Extensions of the definitions to several directions are discussed at the end.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2016-08-29T15:52:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E30",
      "46B10",
      "34A12",
      "31B10"
    ],
    "keywords": [
      "varying exponent",
      "ode representation",
      "function space norms arising",
      "construct banach function space norms",
      "special case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.0528T"
    }
  }
}