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Superposition operator in Sobolev spaces on domains

Author: Denis A. Labutin
Journal: Proc. Amer. Math. Soc. 128 (2000), 3399-3403
MSC (1991): Primary 46E35; Secondary 47H30
Published electronically: May 11, 2000
MathSciNet review: 1676320
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Abstract: For an arbitrary open set $\Omega\subset \mathbb{R}^n$ we characterize all functions $G$ on the real line such that $G\circ u\in W^{1,p}(\Omega)$ for all $u\in W^{1,p}(\Omega)$. New element in the proof is based on Maz'ya's capacitary criterion for the imbedding $ {W^{1,p}(\Omega)\hookrightarrow L^\infty(\Omega)}$.

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Additional Information

Denis A. Labutin
Affiliation: Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra 0200, ACT, Australia

Keywords: Sobolev spaces, superposition operator
Received by editor(s): August 1, 1998
Received by editor(s) in revised form: January 22, 1999
Published electronically: May 11, 2000
Additional Notes: This work was supported by the Russian Foundation for Basic Research grant 96-01-00243.
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society

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