Spaces of Besov-Sobolev type and a problem on nonlinear approximation

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