Quantities equivalent to the norm of a weighted Bergman space

Let 0 α <∞, 0 < p <∞, and p −α > −2. If f is holomorphic in the unit disc D and if ω is a radial weight function of secure type, then the followings are equivalent: D f (z) p ω(z)dA(z)<∞, D f (z) p−α ∇f (z) α ω(z)dA(z)<∞, 1 0 2π 0 f reiθ p dθ 1−α/p 2π 0 ∇f reiθ p dθ α/p ω(r)r dr <∞. Here ∇f (z) = (1 − |z|2)f (z). Furthermore, if f (0) = 0 and ω is monotone, then three quantities on the left sides are mutually equivalent. This generalizes a classical result of Hardy–Littlewood. © 2007 Elsevier Inc. All rights reserved.