Abstract/Summary

Let H2H2 denote the space of ordinary Dirichlet series with square summable coefficients, and let H20H02 denote its subspace consisting of series vanishing at +∞+∞ . We investigate the weak product spaces H2⊙H2H2⊙H2 and H20⊙H20H02⊙H02 , finding that several pertinent problems are more tractable for the latter space. This surprising phenomenon is related to the fact that H20⊙H20H02⊙H02 does not contain the infinite-dimensional subspace of H2H2 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 H20×H20H02×H02 whose analytic symbols are not in H2H2 . Based on this result we examine Nehari’s theorem for such Hankel forms. We define also the skew product spaces associated with H2⊙H2H2⊙H2 and H20⊙H20H02⊙H02 , 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 HpHp , for 0<p<∞0<p<∞ , 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.