Extension of Functions Preserving Certain Smoothness and Compactness of Embeddings for Spaces of Differentiable Functions

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.