Some Banach spaces of Dirichlet series

Maxime Bailleul ¹   ; Pascal Lefèvre ¹

¹ Univ Lille-Nord-de-France UArtois Laboratoire de Mathématiques de Lens EA 2462 Fédération CNRS Nord-Pas-de-Calais FR 2956 F-62 300 Lens, France

Studia Mathematica, Tome 226 (2015) no. 1, pp. 17-55

Résumé

The Hardy spaces of Dirichlet series, denoted by $ \mathcal {H}^p$ ($p\geq 1$), have been studied by Hedenmalm et al. (1997) when $p=2$ and by Bayart (2002) in the general case. In this paper we study some $L^p$-generalizations of spaces of Dirichlet series, particularly two families of Bergman spaces, denoted $ \mathcal {A}^p$ and $ \mathcal {B}^p$. Each could appear as a “natural” way to generalize the classical case of the unit disk. We recover classical properties of spaces of analytic functions: boundedness of point evaluation, embeddings between these spaces and “Littlewood–Paley” formulas when $p=2$. Surprisingly, it appears that the two spaces have a different behavior relative to the Hardy spaces and that these behaviors are different from the usual way the Hardy spaces $H^p({\mathbb D })$ embed into Bergman spaces on the unit disk.

Détail

DOI : 10.4064/sm226-1-2

Keywords: hardy spaces dirichlet series denoted mathcal geq have studied hedenmalm bayart general paper study p generalizations spaces dirichlet series particularly families bergman spaces denoted mathcal nbsp mathcal each could appear natural generalize classical unit disk recover classical properties spaces analytic functions boundedness point evaluation embeddings between these spaces littlewood paley formulas surprisingly appears spaces have different behavior relative hardy spaces these behaviors different usual hardy spaces mathbb embed bergman spaces unit disk