Function spaces related to the Dirichlet space

Abstract

We introduce two spaces of holomorphic functions on the disk that play a role in the Dirichlet space theory similar to the roles of H 1 and BMO in the classical Hardy space theory. We develop some basic function-and operator-theoretic results for those spaces.The Hardy space H 2 is the Hilbert space of holomorphic functions on the unit disk for which

Hankel operators and invariant subspaces of the Dirichlet space

“…The paper [3, pp. 22-24] contains an excellent motivation for the study of weak products and a summary of results about such spaces, also see [4,19]. Here we just mention that if H is the Hardy space H 2 of the unit disc D, then it follows from the Riesz factorization that H 2 H 2 = H 1 with equality of norms, and in fact every h ∈ H 2 H 2 can be written as a single product of functions in H 2 .…”

“…Similarly, it turns out that L 2 a L 2 a = L 1 a (see [7]). Furthermore, the papers [4,14,19] contain a number of results about D D.…”

Bilinear forms on weighted Besov spaces

“…As it happens in the unweighted case (see for instance [13], [9], [3]), from the equivalences between (i) and (ii) in Theorem 1.2, we obtain the following duality result for weak products. Theorem 1.5.…”

“…is a Carleson measure for B p s (dµ t ), that is, if and only if the embedding [14], [1], [13], [3], [7], [8] and the references therein).…”

Some Hilbert spaces related with the Dirichlet space

“…D linking spaces with the same index. It might be interesting to compare this sequence with the sequence of Banach spaces related to the Dirichlet spaces studied in [3]. Note that for d 3 the reproducing kernel of HS d is continuous up to the boundary.…”