Radial extensions in fractional Sobolev spaces

H. Brezis, P. Mironescu, I. Shafrir

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Given f: ∂(- 1 , 1) n→ R, consider its radial extension Tf(X) : = f(X/ ‖ X‖ ) , ∀X∈[-1,1]n\{0}. Brezis and Mironescu (RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 95:121–143, 2001), stated the following auxiliary result (Lemma D.1). If 0 < s< 1 , 1 < p< ∞ and n≥ 2 are such that 1 < sp< n, then f↦ Tf is a bounded linear operator from Ws,p(∂(- 1 , 1) n) into Ws,p((- 1 , 1) n). The proof of this result contained a flaw detected by Shafrir. We present a correct proof. We also establish a variant of this result involving higher order derivatives and more general radial extension operators. More specifically, let B be the unit ball for the standard Euclidean norm || in Rn, and set Uaf(X):=|X|af(X/|X|), ∀X∈B¯\{0}, ∀f:∂B→R. Let a∈ R, s> 0 , 1 ≤ p< ∞ and n≥ 2 be such that (s- a) p< n. Then f↦ Uaf is a bounded linear operator from Ws,p(∂B) into Ws,p(B).

Original languageEnglish
Pages (from-to)707-714
Number of pages8
JournalRevista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas
Issue number2
StatePublished - 1 Apr 2019


  • Fractional Sobolev spaces
  • Radial extensions
  • Sobolev spaces

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology
  • Computational Mathematics
  • Applied Mathematics


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