Some Banach spaces of Dirichlet series
Volume 226 / 2015
Abstract
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.
