Weak product spaces of Dirichlet series
Ole Fredrik Brevig, Karl-Mikael Perfekt

TL;DR
This paper studies weak product spaces of Dirichlet series, revealing that certain subspaces are more manageable and exploring their duals, Hankel forms, and related function space characterizations.
Contribution
It introduces new insights into the structure of weak product spaces of Dirichlet series, especially the subspace $ ext{H}^2_0$, and analyzes their duals and Hankel forms, including a Schur multiplier problem.
Findings
$ ext{H}^2_0 igodot ext{H}^2_0$ does not contain certain infinite-dimensional subspaces.
Existence of bounded Hankel forms with symbols outside $ ext{H}^2$.
Square function characterizations of $ ext{H}^p$ spaces for $0<p< ext{infinity}$.
Abstract
Let denote the space of ordinary Dirichlet series with square summable coefficients, and let denote its subspace consisting of series vanishing at . We investigate the weak product spaces and , finding that several pertinent problems are more tractable for the latter space. This surprising phenomenon is related to the fact that does not contain the infinite-dimensional subspace of of series which lift to linear functions on the infinite polydisc. The problems considered stem from questions about the dual spaces of these weak product spaces, and are therefore naturally phrased in terms of multiplicative Hankel forms. We show that there are bounded, even Schatten class, multiplicative Hankel forms on $\mathscr{H}^2_0…
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