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

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

Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 3, pp. 653-661.

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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

Mots-clés : 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

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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'I.H.P. Analyse non lin\'eaire},
     pages = {653--661},
     publisher = {Elsevier},
     volume = {37},
     number = {3},
     year = {2020},
     doi = {10.1016/j.anihpc.2020.01.004},
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     zbl = {1443.46005},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2020.01.004/}
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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'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 3, pp. 653-661. doi : 10.1016/j.anihpc.2020.01.004. http://www.numdam.org/articles/10.1016/j.anihpc.2020.01.004/

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