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arXiv · 2112.05539

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

Óscar Domínguez·Andreas Seeger·Brian Street·Jean Van Schaftingen·Po-Lam Yung
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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\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.

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BibTeXRIS

Óscar Domínguez, Andreas Seeger, Brian Street, Jean Van Schaftingen, Po-Lam Yung. 2021-12-10. Spaces of Besov-Sobolev type and a problem on nonlinear approximation. https://doi.org/10.1016/j.jfa.2022.109775

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