Sobolev spaces revisited

  • Haïm Brezis

    Rutgers University, Piscataway, USA; Technion, Haifa, Israel; Sorbonne Universités, Paris, France
  • Andreas Seeger

    University of Wisconsin, Madison, USA
  • Jean Van Schaftingen

    Université Catholique de Louvain, Louvain-la-Neuve, Belgium
  • Po-Lam Yung

    The Australian National University, Canberra, Australia
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We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on n, using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis, and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies, and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo–Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the norm of functions in .

Cite this article

Haïm Brezis, Andreas Seeger, Jean Van Schaftingen, Po-Lam Yung, Sobolev spaces revisited. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 33 (2022), no. 2, pp. 413–437

DOI 10.4171/RLM/976