Weak product spaces of Dirichlet series

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 $\mathscr{H}^2$ denote the space of ordinary Dirichlet series with square summable coefficients, and let $\mathscr{H}^2_0$ denote its subspace consisting of series vanishing at $+\infty$. We investigate the weak product spaces $\mathscr{H}^2\odot\mathscr{H}^2$ and $\mathscr{H}^2_0\odot\mathscr{H}^2_0$, finding that several pertinent problems are more tractable for the latter space. This surprising phenomenon is related to the fact that $\mathscr{H}^2_0\odot\mathscr{H}^2_0$ does not contain the infinite-dimensional subspace of $\mathscr{H}^2$ 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 $\mathscr{H}^2_0 \times \mathscr{H}^2_0$ whose analytic symbols are not in $\mathscr{H}^2$. Based on this result we examine Nehari's theorem for such Hankel forms. We define also the skew product spaces associated with $\mathscr{H}^2\odot\mathscr{H}^2$ and $\mathscr{H}^2_0\odot\mathscr{H}^2_0$, with respect to both half-plane and polydisc differentiation, the latter arising from Bohr's point of view. In the process we supply square function characterizations of the Hardy spaces $\mathscr{H}^p$, for $0 < p < \infty$, from the viewpoints of both types of differentiation. Finally we compare the skew product spaces to the weak product spaces, leading naturally to an interesting Schur multiplier problem.