Weak Product Spaces of Dirichlet Series

Let (Formula presented.) denote the space of ordinary Dirichlet series with square summable coefficients, and let (Formula presented.) denote its subspace consisting of series vanishing at (Formula presented.). We investigate the weak product spaces (Formula presented.) and (Formula presented.), finding that several pertinent problems are more tractable for the latter space. This surprising phenomenon is related to the fact that (Formula presented.) does not contain the infinite-dimensional subspace of (Formula presented.) 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 (Formula presented.) whose analytic symbols are not in (Formula presented.). Based on this result we examine Nehari’s theorem for such Hankel forms. We define also the skew product spaces associated with (Formula presented.) and (Formula presented.), 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 (Formula presented.), for (Formula presented.), 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.