Growth rate of the functions in Bergman type spaces

For 0 < p,α < ∞, let f p,α be the Lp-norm with respect the weighted measure dVα(z) = δD(z)α−1 dV (z). We define the weighted Bergman space A pα (D) consisting of holomorphic functions f with f p,α < ∞. For any σ > 0, let A−σ (D) be the space consisting of holomorphic functions f in D with f −σ = sup{δD(z)σ |f (z)|: z ∈ D} <∞. If D has C2 boundary, then we have the embedding A pα (D) ⊂ A−(n+α)/p(D). We show that the condition of C2-smoothness of the boundary of D is necessary by giving a counter-example of a convex domain with C1,λ-smooth boundary for 0<λ<1 which does not satisfy the embedding.  2003 Elsevier Inc. All rights reserved.