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

Abstract

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\uı 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.

45 pages

References in corpus (2)

• A surprising formula for Sobolev norms
• Metric characterization of the sum of fractional Sobolev spaces

Cited by in corpus (1)

• Families of functionals representing Sobolev norms