{
  "id": "2201.08222",
  "version": "v1",
  "published": "2022-01-20T15:24:55.000Z",
  "updated": "2022-01-20T15:24:55.000Z",
  "title": "A note on composition operators between weighted spaces of smooth functions",
  "authors": [
    "Andreas Debrouwere",
    "Lenny Neyt"
  ],
  "comment": "9 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "For certain weighted locally convex spaces $X$ and $Y$ of one real variable smooth functions, we characterize the smooth functions $\\varphi: \\mathbb{R} \\to \\mathbb{R}$ for which the composition operator $C_\\varphi: X \\to Y, \\, f \\mapsto f \\circ \\varphi$ is well-defined and continuous. This problem has been recently considered for $X = Y$ being the space $\\mathscr{S}$ of rapidly decreasing smooth functions [1] and the space $\\mathscr{O}_M$ of slowly increasing smooth functions [2]. In particular, we recover both these results as well as obtain a characterization for $X =Y$ being the space $\\mathscr{O}_C$ of very slowly increasing smooth functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-20T15:24:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B33",
      "46E10"
    ],
    "keywords": [
      "composition operator",
      "weighted spaces",
      "slowly increasing smooth functions",
      "real variable smooth functions",
      "weighted locally convex spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}