Some Banach spaces of Dirichlet series
Volume 226 / 2015
Abstract
The Hardy spaces of Dirichlet series, denoted by (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.
