{
  "id": "2403.14317",
  "version": "v1",
  "published": "2024-03-21T11:39:51.000Z",
  "updated": "2024-03-21T11:39:51.000Z",
  "title": "On $C^1$ Whitney extension theorem in Banach spaces",
  "authors": [
    "Michal Johanis",
    "Luděk Zajíček"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Our note is a complement to recent articles \\cite{JS1} (2011) and \\cite{JS2} (2013) by M. Jim\\'enez-Sevilla and L. S\\'anchez-Gonz\\'alez which generalise (the basic statement of) the classical Whitney extension theorem for $C^1$-smooth real functions on $\\mathbb R^n$ to the case of real functions on $X$ (\\cite{JS1}) and to the case of mappings from $X$ to $Y$ (\\cite{JS2}) for some Banach spaces $X$ and $Y$. Since the proof from \\cite{JS2} contains a serious flaw, we supply a different more transparent detailed proof under (probably) slightly stronger assumptions on $X$ and $Y$. Our proof gives also extensions results from special sets (e.g. Lipschitz submanifolds or closed convex bodies) under substantially weaker assumptions on $X$ and $Y$. Further, we observe that the mapping $F\\in C^1(X;Y)$ which extends $f$ given on a closed set $A\\subset X$ can be, in some cases, $C^\\infty$-smooth (or $C^k$-smooth with $k>1$) on $X\\setminus A$. Of course, also this improved result is weaker than Whitney's result (for $X=\\mathbb R^n$, $Y=\\mathbb R$) which asserts that $F$ is even analytic on $X\\setminus A$. Further, following another Whitney's article and using the above results, we prove results on extensions of $C^1$-smooth mappings from open (``weakly'') quasiconvex subsets of $X$. Following the above mentioned articles we also consider the question concerning the Lipschitz constant of $F$ is $f$ is a Lipschitz mapping. We also present a new observation on the limitation of the possible validity of Whitney's extension theorem for $C^2$-smooth functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-21T11:39:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46G05",
      "46T20"
    ],
    "keywords": [
      "banach spaces",
      "smooth real functions",
      "whitneys extension theorem",
      "classical whitney extension theorem",
      "lipschitz constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}