Geodesic
Extension of Functions Preserving Certain Smoothness and Compactness of Embeddings for Spaces of Differentiable Functions
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Studies on function theory and differential equations, Tome 248 (2005), pp. 74-85.

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It is proved that functions from the Sobolev spaces $W_p^l(\Omega )$, where $\Omega \subset \mathbb R^n$ is an arbitrary bounded open set, can be extended from $\Omega $ to $\mathbb R^n$ while preserving certain smoothness in the metric of $L_q$, where $q p$. It is established that an extension that preserves certain smoothness in the metric of $L_p$ is possible if and only if the embedding $W_p^l(\Omega )\subset L_p(\Omega )$ is compact. 
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V. I. Burenkov. Extension of Functions Preserving Certain Smoothness and Compactness of Embeddings for Spaces of Differentiable Functions. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Studies on function theory and differential equations, Tome 248 (2005), pp. 74-85. http://geodesic.mathdoc.fr/item/TM_2005_248_a7/

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