THE CLOSURE OF DIRICHLET SPACES IN THE BLOCH SPACE

If 0 < p < infinity and alpha > -1, the space of Dirichlet type D-alpha(p) 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(alpha)(p). Of special interest are the spaces D-p-1(p) (0 < p < infinity) and the analytic Besov spaces B-p = D-p-2(p) (1 < p < infinity). Let B denote the Bloch space. It is known that the closure of B-p ( 1 < p < infinity) in the Bloch norm is the little Bloch space B-0. A description of the closure in the Bloch norm of the spaces H-p boolean AND 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-alpha(p) boolean AND B (1 <= p < infinity, alpha > -1). In particular, we prove that for all p = 1 the closure of the space D-p-1(p) boolean AND B coincides with that of H-2 boolean AND 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-alpha(p) boolean AND B.