Embedding of Hardy spaces into weighted Bergman spaces in bounded domains with {$C\sp 2$} boundary

Abstract

Let $D$ be a bounded domain in $\mathbb C^n$ with $C^2$ boundary. Let $H^p(D)$ be the Hardy space and $A^{p,\alpha}(D)$ be the space of holomorphic functions which are $L^p$-integrable with respect to the weighted measure $dV_\alpha(z)=\delta_D(z)^{\alpha-1}dV(z)$. We obtain some estimates on the mean growth of $H^p$ functions in $D$. Using these estimates, we can embed the $H^p(D)$ space into $A^{q,\beta}(D)$ for $0<p<q<\infty,\, \beta>0$ satisfying $n/p=(n+\beta)/q$. We also show that the condition of $C^2$-smoothness of the boundary of $D$ is an essential condition 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 result.