Compact Sobolev embeddings and torsion functions

For a general open set, we characterize the compactness of the embedding $W^{1,p}_0\hookrightarrow L^q$ in terms of the summability of its torsion function. In particular, for $1\le q<p$ we obtain that the embedding is continuous if and only if it is compact. The proofs crucially exploit a {\it torsional Hardy inequality} that we investigate in details.

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