Non-local Torsion functions and Embeddings

Given $s\in (0,1)$, we discuss the embedding of ${\mathcal{D}}_0^{s,p}(\mathrm{\Omega })$ in $L^q(\mathrm{\Omega })$. In particular, for $1\le q<p$ we deduce its compactness on all open sets $\mathrm{\Omega }\subset {\mathbb{R}}^N$ 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 $\mathrm{\Omega }$ in a suitable weak sense, for every open set $\mathrm{\Omega }$. The proofs make use of a non-local Hardy-type inequality in $\mathcal D^{s,p}_0(\Omega)$, involving the fractional torsion function as a weight.