Growth rate of the functions in Bergman type spaces (Q1408795)

Let \(D\) be a bounded domain in \(\mathbb C^n\). For \(0<p, \alpha<\infty\), let \(\|f\|_{p,\alpha}\) be the \(L^p\)-norm with respect to the weighted measure \(dV_\alpha(z)=\delta_D(z)^{\alpha-1}dV(z)\). The authors define the weighted Bergman space \(A^p_\alpha(D)\) consisting of holomorphic functions \(f\) in \(D\) with \(\|f\|_{p,\alpha}<\infty\). For any \(\sigma>0\), let \(A^{-\sigma}(D)\) be the space consisting of holomorphic functions \(f\) in \(D\) with \(\|f\|_{-\sigma}=\sup\{\delta_D(z)^\sigma|f(z)|: z\in D\}<\infty\). If \(D\) has \(C^2\)-boundary, the authors prove the embedding \(A^p_\alpha(D)\subset A^{-(n+\alpha)/p}(D)\). Further they show that the condition of \(C^2\)-smoothness of the boundary of \(D\) is necessary by giving a counter-example of a convex domain with \(C^{1,\lambda}\)-smooth boundary for \(0<\lambda<1\) which does not satisfy the embedding.