Some Banach spaces of Dirichlet series

The Hardy spaces of Dirichlet series denoted by ${\mathcal{H}}^p$ ($p\ge 1$) have been studied in [12] when p = 2 and in [3] for the general case. In this paper we study some Lp-generalizations of spaces of Dirichlet series, particularly two families of Bergman spaces denoted ${\mathcal{A}}^p$ and ${\mathcal{B}}^p$. We recover classical properties of spaces of analytic functions: boundedness of point evaluation, embeddings between these spaces and “Littlewood-Paley” formulas when p = 2. We also show that the ${\mathcal{B}}^p$ spaces have properties similar to the classical Bergman spaces of the unit disk while the ${\mathcal{A}}^p$ spaces have a different behavior.