Closure of Bergman and Dirichlet spaces in the Bloch norm

Abstract

where dA(z) = dx dy π is the normalized Lebesgue area measure on D. In this definition we understand that the sum does not exist if n = 0. Throughout this paper ω satisfies ω̂(z) = ́ 1 |z| ω(s) ds > 0 for all z ∈ D, for otherwise Aω,n = H(D). We write A p ω = A p ω,0 and D ω = A p ω,1 for the weighted Bergman and Dirichlet spaces, respectively. As usual, Aα and D p α denote the classical weighted Bergman and Dirichlet spaces induced by the standard radial weight ω(z) = (1−|z|), where −1 < α < ∞. For f ∈ H(D) and 0 < r < 1, set