The closure of Dirichlet spaces in the Bloch space

Abstract

If 0 < p < ∞ and α > −1, the space of Dirichlet type D α consists of those functions f which are analytic in the unit disc D and have the property that f ′ belongs to the weighted Bergman space A α . Of special interest are the spaces Dp p−1 (0 < p < ∞) and the analytic Besov spaces B = Dp p−2 (1 < p < ∞). Let B denote the Bloch space. It is known that the closure of B (1 < p < ∞) in the Bloch norm is the little Bloch space B0. A description of the closure in the Bloch norm of the spaces H ∩B has been given recently. Such closures depend on p. In this paper we obtain a characterization of the closure in the Bloch norm of the spaces D α ∩ B (1 ≤ p < ∞, α > −1). In particular, we prove that for all p ≥ 1 the closure of the space Dp p−1 ∩ B coincides with that of H ∩ B. Hence, contrary with what happens with Hardy spaces, these closures are independent of p. We apply these results to study the membership of Blaschke products in the closure in the Bloch norm of the spaces D α ∩ B.