Weak products of Dirichlet functions

Abstract

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 H1-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 [3] 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 Maz'ya and Verbitsky [17]. Thus, we will establish a precise connection between the results of [3] and [17].