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

D'Onofrio, Luigi; Greco, Luigi; Perfekt, Karl-Mikael; Sbordone, Carlo; Schiattarella, Roberta

doi:10.1016/j.anihpc.2020.01.004

D'Onofrio, Luigi ¹ ; Greco, Luigi ² ; Perfekt, Karl-Mikael ³ ; Sbordone, Carlo ⁴ ; Schiattarella, Roberta ⁴

¹Dipartimento di Scienze e Tecnologie, Università degli Studi di Napoli “Parthenope”, Centro Direzionale Isola C4, 80100 Napoli, Italy
²Dipartimento di Ingegneria Elettrica e delle Tecnologie dell'Informazione, Università degli Studi di Napoli “Federico II”, Via Claudio 21, 80125 Napoli, Italy
³Department of Mathematics and Statistics, University of Reading, Reading RG6 6AX, United Kingdom
⁴Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Via Cintia, 80126 Napoli, Italy

Annales de l'Institut Henri Poincaré. C, Analyse non linéaire, Tome 37 (2020) no. 3, pp. 653-661

Résumé

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 $B$ introduced recently by Bourgain, Brezis, and Mironescu. This yields an atomic decomposition of the predual $B_{⁎}$ , the biduality result that $B_{0}^{⁎} = B_{⁎}$ and $B_{⁎}^{⁎} = B$ , and a formula for the distance from an element $f \in B$ to $B_{0}$ .

MR Zbl

DOI : 10.1016/j.anihpc.2020.01.004

Keywords: Dual and predual, Bourgain-Brezis-Mironescu space, Atomic decomposition

Affiliations des auteurs :

D'Onofrio, Luigi ¹ ; Greco, Luigi ² ; Perfekt, Karl-Mikael ³ ; Sbordone, Carlo ⁴ ; Schiattarella, Roberta ⁴

¹ Dipartimento di Scienze e Tecnologie, Università degli Studi di Napoli “Parthenope”, Centro Direzionale Isola C4, 80100 Napoli, Italy
² Dipartimento di Ingegneria Elettrica e delle Tecnologie dell'Informazione, Università degli Studi di Napoli “Federico II”, Via Claudio 21, 80125 Napoli, Italy
³ Department of Mathematics and Statistics, University of Reading, Reading RG6 6AX, United Kingdom
⁴ Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Via Cintia, 80126 Napoli, Italy

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     author = {D'Onofrio, Luigi and Greco, Luigi and Perfekt, Karl-Mikael and Sbordone, Carlo and Schiattarella, Roberta},
     title = {Atomic decompositions, two stars theorems, and distances for the {Bourgain{\textendash}Brezis{\textendash}Mironescu} space and other big spaces},
     journal = {Annales de l'Institut Henri Poincar\'e. C, Analyse non lin\'eaire},
     pages = {653--661},
     year = {2020},
     publisher = {Elsevier},
     volume = {37},
     number = {3},
     doi = {10.1016/j.anihpc.2020.01.004},
     mrnumber = {4093624},
     zbl = {1443.46005},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2020.01.004/}
}

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AU  - Greco, Luigi
AU  - Perfekt, Karl-Mikael
AU  - Sbordone, Carlo
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%A Greco, Luigi
%A Perfekt, Karl-Mikael
%A Sbordone, Carlo
%A Schiattarella, Roberta
%T Atomic decompositions, two stars theorems, and distances for the Bourgain–Brezis–Mironescu space and other big spaces
%J Annales de l'Institut Henri Poincaré. C, Analyse non linéaire
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D'Onofrio, Luigi; Greco, Luigi; Perfekt, Karl-Mikael; Sbordone, Carlo; Schiattarella, Roberta. Atomic decompositions, two stars theorems, and distances for the Bourgain–Brezis–Mironescu space and other big spaces. Annales de l'Institut Henri Poincaré. C, Analyse non linéaire, Tome 37 (2020) no. 3, pp. 653-661. doi: 10.1016/j.anihpc.2020.01.004

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