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

Domínguez, Óscar;Seeger, Andreas;Street, Brian;Van Schaftingen, Jean;Yung, Po-Lam
(2023) Journal of Functional Analysis — Vol. 284, n° 4, p. 109775 (2023)

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Authors
  • Domínguez, Óscar
    Author
  • Seeger, Andreas
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  • Street, Brian
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  • Van Schaftingen, Jeanorcid-logoUCLouvain
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  • Yung, Po-Lam
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Abstract
(en) We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung in the study of the Sobolev space Ẇ1,p. The resulting spaces are identified as a special class of real interpolation spaces of Sobolev-Slobodeckiĭ 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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Domínguez, Ó., Seeger, A., Street, B., Van Schaftingen, J., & Yung, P.-L. (2023). Spaces of Besov-Sobolev type and a problem on nonlinear approximation. Journal of Functional Analysis, 284(4), 109775. https://doi.org/10.1016/j.jfa.2022.109775 (Original work published 2023)