JournalsjemsVol. 17, No. 9pp. 2083–2101

A new function space and applications

  • Jean Bourgain

    Institute for Advanced Study, Princeton, United States
  • Haïm Brezis

    Rutgers University, Piscataway, United States
  • Petru Mironescu

    Université Lyon 1, Villeurbanne, France
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Abstract

We define a new function space BB, which contains in particular BMO, BV, and W1/p,pW^{1/p,p}, 1<p<1 < p < \infty. We investigate its embedding into Lebesgue and Marcinkiewicz spaces. We present several inequalities involving LpL^p norms of integer-valued functions in BB. We introduce a significant closed subspace, B0B_0, of BB, containing in particular VMO and W1/p,pW^{1/p,p}, 1p<1 \le p < \infty. The above mentioned estimates imply in particular that integer-valued functions belonging to B0B_0 are necessarily constant. This framework provides a "common roof" to various, seemingly unrelated, statements asserting that integer-valued functions satisfying some kind of regularity condition must be constant.

Cite this article

Jean Bourgain, Haïm Brezis, Petru Mironescu, A new function space and applications. J. Eur. Math. Soc. 17 (2015), no. 9, pp. 2083–2101

DOI 10.4171/JEMS/551