JournalsaihpcVol. 36, No. 7pp. 2053–2063

Continuity of composition operators in Sobolev spaces

  • Gérard Bourdaud

    Université de Paris, I.M.J. - P.R.G., Case 7012, 75205 Paris Cedex 13, France
  • Madani Moussai

    Laboratory of Functional Analysis and Geometry of Spaces, M. Boudiaf University of M'Sila, 28000 M'Sila, Algeria
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Abstract

We prove that all the composition operators Tf(g):=fgT_{f}(g): = f \circ g, which take the Adams-Frazier space W_{p}^{m} \cap W\limits^{˙}_{mp}^{1}(\mathbb{R}^{n}) to itself, are continuous mappings from W_{p}^{m} \cap W\limits^{˙}_{mp}^{1}(\mathbb{R}^{n}) to itself, for every integer m2m \geq 2 and every real number 1p<1 \leq p < \infty . The same automatic continuity property holds for Sobolev spaces Wpm(Rn)W_{p}^{m}(\mathbb{R}^{n}) for m2m \geq 2 and 1p<1 \leq p < \infty .

Cite this article

Gérard Bourdaud, Madani Moussai, Continuity of composition operators in Sobolev spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 7, pp. 2053–2063

DOI 10.1016/J.ANIHPC.2019.07.002