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Some Banach spaces of Dirichlet series

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Bailleul M., Lefèvre P.

Studia Mathematica

2015

tom 226

nr 1

17-55

The Hardy spaces of Dirichlet series, denoted by $𝓗^{p}$ (p ≥ 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 $𝓐^{p}$ and $ℬ^{p}$. 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 $H^{p}(𝔻)$ embed into Bergman spaces on the unit disk.

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1 Studia Mathematica

1 Bailleul M.

1 Lefèvre P.

1 2015

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Some Banach spaces of Dirichlet series