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We study a class of nonlocal functionals in the spirit of the recentcharacterization of the Sobolev spaces $W^{1,p}$ derived by Bourgain, Brezisand 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 characterizationsfor these spaces. We also establish a non-compactness result for sequences andnew (non-)limiting embeddings between Lipschitz and Besov spaces which extendthe previous known results.

A Bourgain-Brezis-Mironescu characterization of higher order Besov-Nikol'skii spaces