Superposition operator in Sobolev spaces on domains

Labutin, Denis

doi:10.1090/S0002-9939-00-05421-6

Superposition operator in Sobolev spaces on domains
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by Denis A. Labutin

Proc. Amer. Math. Soc. 128 (2000), 3399-3403

DOI: https://doi.org/10.1090/S0002-9939-00-05421-6

Published electronically: May 11, 2000

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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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