Weak Products of Dirichlet Functions

For a Hilbert space H of functions let H H be the space of weak products of functions in H, i.e. all functions h that can be written as h = ∑∞ i=1 figi for some fi, gi ∈ H with ∑∞ i=1 ‖fi‖‖gi‖ <∞. Let D denote the Dirichlet space of the unit circle ∂D, i.e. the nontangential limits of functions f ∈ Hol(D) with ∫ D |f ′|2dA < ∞ and let Dh be the harmonic Dirichlet space, which consists of functions of the form f + g for f, g ∈ D. In this paper we show that every real-valued function in Dh Dh is a single product of two functions in Dh and that the Cauchy projection is a bounded operator from Dh Dh onto D D. It follows that D D consists exactly of the H-functions whose real and imaginary parts are single products of Dh-functions. The dual space of D D was characterized by Arcozzi, Rochberg, Sawyer and Wick in [ARSW10] and the result implies the characterization of the dual of Dh Dh. We will show that the characterization of the dual of Dh Dh also follows from results of Mazya and Verbitsky, [MV02]. Thus, we will establish a precise connection between the results of [ARSW10] and [MV02]. We also prove some general results about weak product spaces of analytic functions. For example, we show that H H ⊆ H(k), where H(k) is the space of analytic functions whose reproducing kernel is the square of the reproducing kernel of H. Furthermore, we will show that some of the above mentioned results for the Dirichlet space D hold for the superharmonically weighted Dirichlet spaces D(μ).