Comptes Rendus
Mathematical analysis
Superposition with subunitary powers in Sobolev spaces
Comptes Rendus. Mathématique, Volume 353 (2015) no. 6, pp. 483-487.

Let 0<a<1 and set Φ(t)=|t|a, tR. Let 1<p< and n1. We prove that the superposition operator uΦ(u) maps the Sobolev space W1,p(Rn) into the fractional Sobolev space Wa,p/a(Rn). We also investigate the case of more general nonlinearities.

Pour 0<a<1, soit Φ(t)=|t|a, tR. Soient 1<p< et n1. Nous montrons que l'opérateur de superposition uΦ(u) envoie l'espace de Sobolev W1,p(Rn) dans l'espace de Sobolev fractionnaire Wa,p/a(Rn). Nous examinons aussi la superposition avec des non-linéarités plus générales.

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Published online:
DOI: 10.1016/j.crma.2015.03.020
Petru Mironescu 1

1 Université de Lyon, Université Lyon-1, CNRS UMR 5208, Institut Camille-Jordan, 43, bd du 11-Novembre-1918, 69622 Villeurbanne cedex, France
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     title = {Superposition with subunitary powers in {Sobolev} spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {483--487},
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     doi = {10.1016/j.crma.2015.03.020},
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Petru Mironescu. Superposition with subunitary powers in Sobolev spaces. Comptes Rendus. Mathématique, Volume 353 (2015) no. 6, pp. 483-487. doi : 10.1016/j.crma.2015.03.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.03.020/

[1] G. Bourdaud; Y. Meyer Fonctions qui opèrent sur les espaces de Sobolev, J. Funct. Anal., Volume 97 (1991) no. 2, pp. 351-360

[2] H. Triebel Theory of Function Spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, Switzerland, 1983

[3] L. Véron, personal communication.

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