Non-local Torsion Functions and Embeddings

Given $s\in(0,1)$, we discuss the embedding of $\mathcal{D}_0^{s,p}(\Omega)$ in $L^q(\Omega)$. In particular, for $1\le q<p$ we deduce its compactness on all open sets $\Omega\subset \R^N$ on which it is continuous. We then relate, for all $q$ up the fractional Sobolev conjugate exponent, the continuity of the embedding to the summability of the function solving the fractional torsion problem in $\Omega$ in a suitable weak sense, for every open set $\Omega$. The proofs make use of a non-local Hardy-type inequality in $\mathcal{D}_0^{s,p}(\Omega)$, involving the fractional torsion function as a weight.