  We study fractional variants of the quasi-norms introduced by Brezis, Van
Schaftingen, and Yung in the study of the Sobolev space $\dot W^{1,p}$. The
resulting spaces are identified as a special class of real interpolation spaces
of Sobolev-Slobodecki\u{\i} spaces. We establish the equivalence between
Fourier analytic definitions and definitions via difference operators acting on
measurable functions. We prove various new results on embeddings and
non-embeddings, and give applications to harmonic and caloric extensions. For
suitable wavelet bases we obtain a characterization of the approximation spaces
for best $n$-term approximation from a wavelet basis via smoothness conditions
on the function; this extends a classical result by DeVore, Jawerth and Popov.
