JournalsaihpcVol. 37, No. 3pp. 653–661

Atomic decompositions, two stars theorems, and distances for the Bourgain–Brezis–Mironescu space and other big spaces

  • Luigi D'Onofrio

    Dipartimento di Scienze e Tecnologie, Università degli Studi di Napoli “Parthenope”, Centro Direzionale Isola C4, 80100 Napoli, Italy
  • Luigi Greco

    Dipartimento di Ingegneria Elettrica e delle Tecnologie dell'Informazione, Università degli Studi di Napoli “Federico II”, Via Claudio 21, 80125 Napoli, Italy
  • Carlo Sbordone

    Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Via Cintia, 80126 Napoli, Italy
  • Roberta Schiattarella

    Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Via Cintia, 80126 Napoli, Italy
  • Karl-Mikael Perfekt

    Department of Mathematics and Statistics, University of Reading, Reading RG6 6AX, United Kingdom
Atomic decompositions, two stars theorems, and distances for the Bourgain–Brezis–Mironescu space and other big spaces cover
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Abstract

Given a Banach space E with a supremum-type norm induced by a collection of operators, we prove that E is a dual space and provide an atomic decomposition of its predual. We apply this result, and some results obtained previously by one of the authors, to the function space B\mathscr{B} introduced recently by Bourgain, Brezis, and Mironescu. This yields an atomic decomposition of the predual B\mathscr{B}_{⁎}, the biduality result that B0=B\mathscr{B}_{0}^{⁎} = \mathscr{B}_{⁎} and B=B\mathscr{B}_{⁎}^{⁎} = \mathscr{B}, and a formula for the distance from an element fBf \in \mathscr{B} to B0\mathscr{B}_{0}.

Cite this article

Luigi D'Onofrio, Luigi Greco, Carlo Sbordone, Roberta Schiattarella, Karl-Mikael Perfekt, Atomic decompositions, two stars theorems, and distances for the Bourgain–Brezis–Mironescu space and other big spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 3, pp. 653–661

DOI 10.1016/J.ANIHPC.2020.01.004