Some Banach spaces of Dirichlet series

Abstract

The Hardy spaces of Dirichlet series denoted by H p (p ≥ 1) have been studied in [12] when p = 2 and in [3] for the general case. In this paper we study some L p -generalizations of spaces of Dirichlet series, particularly two families of Bergman spaces denoted A p and 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 B p spaces have properties similar to the classical Bergman spaces of the unit disk while the A p spaces have a different behavior.

Composition operators on Bohr-Bergman spaces of Dirichlet series

“…We shall use the the Rademacher-Menchov Lemma on the space L 2 (D ∞ , ν α ). Hence we let e n (χ) = [d(n)] α/2 χ(n) and c n = a n [d(n)] −α/2 n −s , for f of the form (3). When σ > 0, the Rademacher-Menchov Lemma implies that f χ (s) converges for almost every χ ∈ D ∞ .…”

“…These inequalities are obtained in [3] by an iterative process similar to the one used in [12] from the following result: For 1 ≤ p < ∞ and F ∈ H(D),…”

Contractive inequalities for Bergman spaces and multiplicative Hankel forms

“…Very recently, a study of Bergman spaces of Dirichlet series A p begun in [3]. In analogy with the Hardy spaces of Dirichlet series, A p is constructed from the corresponding Bergman space, A p (D ∞ ).…”

Composition operators on weighted Bergman spaces of Dirichlet series

“…When dµ(σ) = 2e −2σ dσ, these spaces are simply denoted by A p . It is shown in [1] that they are spaces of convergent Dirichet series on C 1/2 .…”