Weak products of Dirichlet functions

For a Hilbert space HH of functions let H⊙HH⊙H be the space of weak products of functions in HH, i.e. all functions h   that can be written as h=∑i=1∞figi for some fi,gi∈Hfi,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)f∈Hol(D) with ∫D|f′|2dA<∞ and let DhDh be the harmonic Dirichlet space, which consists of functions of the form f+g¯ for f,g∈Df,g∈D. In this paper we show that every real-valued function in Dh⊙DhDh⊙Dh is a single product of two functions in DhDh and that the Cauchy projection is a bounded operator from Dh⊙DhDh⊙Dh onto D⊙DD⊙D. It follows that D⊙DD⊙D consists exactly of the H1H1-functions whose real and imaginary parts are single products of DhDh-functions. The dual space of D⊙DD⊙D was characterized by Arcozzi, Rochberg, Sawyer and Wick in [3] and the result implies the characterization of the dual of Dh⊙DhDh⊙Dh. We will show that the characterization of the dual of Dh⊙DhDh⊙Dh also follows from results of Maz'ya and Verbitsky [17]. Thus, we will establish a precise connection between the results of [3] and [17].