  We study a class of nonlocal functionals in the spirit of the recent
characterization of the Sobolev spaces $W^{1,p}$ derived by Bourgain, Brezis
and Mironescu. We show that it provides a common roof to the description of the
$BV(\mathbb{R}^N)$, $W^{1,p}(\mathbb{R}^N)$, $B_{p,\infty}^s(\mathbb{R}^N)$ and
$C^{0,1}(\mathbb{R}^N)$ scales and we obtain new equivalent characterizations
for these spaces. We also establish a non-compactness result for sequences and
new (non-)limiting embeddings between Lipschitz and Besov spaces which extend
the previous known results.
