Hardy–Orlicz Spaces of Dirichlet Series: An Interpolation Problem on Abscissae of Convergence. (5th November 2019) - Bailleul, Maxime; Lefèvre, Pascal; Rodríguez-Piazza, Luis
Abstract: The study of Hardy spaces of Dirichlet series denoted by $\mathscr{H}^p$ ($p≥ 1$ ) was initiated in [7 ] when $p=2$ and $p=∞ $, and in [2 ] for the general case. In this paper we introduce the Orlicz version of spaces of Dirichlet series $\mathscr{H}^ψ $ . We focus on the case $ψ =ψ _q(t)=\exp (t^q)-1, $ and we compute the abscissa of convergence for these spaces. It turns out that its value is $\min \{1/q\, , 1/2\}$ filling the gap between the case $\mathscr{H}^∞ $, where the abscissa is equal to $0$, and the case $\mathscr{H}^p$ for $p$ finite, where the abscissa is equal to $1/2$ . The upper-bound estimate relies on an elementary method that applies to many spaces of Dirichlet series. This answers a question raised by Hedenmalm in [6 ].