Structure of the Lipschitz free p-spaces Fp(Zd) and Fp(Rd) for 0<p≤1

  1. Albiac, Fernando 1
  2. Ansorena, José L. 2
  3. Cúth, Marek 3
  4. Doucha, Michal 4
  1. 1 Department of Mathematics, Statistics and Computer Sciences–InaMat2, Universidad Pública de Navarra, Campus de Arrosadía, 31006, Pamplona, Spain
  2. 2 Department of Mathematics and Computer Sciences, Universidad de La Rioja, 26004, Logroño, Spain
  3. 3 Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, 186 75, Praha 8, Czech Republic
  4. 4 Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67, Praha 1, Czech Republic
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Revue:
Collectanea mathematica

ISSN: 0010-0757

Année de publication: 2022

Volumen: 73

Fascículo: 3

Pages: 337-357

Type: Article

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DOI: 10.1007/S13348-021-00322-9 DIALNET GOOGLE SCHOLAR

D'autres publications dans: Collectanea mathematica

Dépôt institutionnel: lockAccès ouvert Editor

Résumé

Our aim in this article is to contribute to the theory of Lipschitz free p-spaces for 0<p≤1 over the Euclidean spaces Rd and Zd. To that end, we show that Fp(Rd) admits a Schauder basis for every p∈(0,1], thus generalizing the corresponding result for the case p=1 by Hájek and Pernecká (J Math Anal Appl 416(2):629–646, 2014, Theorem 3.1) and answering in the positive a question that was raised by Albiac et al. in (J Funct Anal 278(4):108354, 2020). Explicit formulas for the bases of Fp(Rd) and its isomorphic space Fp([0,1]d) are given. We also show that the well-known fact that F(Z) is isomorphic to ℓ1 does not extend to the case when p<1, that is, Fp(Z) is not isomorphic to ℓp when 0<p<1.

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