Abstract :
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.
