## Illinois Journal of Mathematics

### 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.

#### Article information

Source
Illinois J. Math., Volume 48, Number 3 (2004), 747-757.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258131050

Digital Object Identifier
doi:10.1215/ijm/1258131050

Mathematical Reviews number (MathSciNet)
MR2114249

Zentralblatt MATH identifier
1113.32002

#### Citation

Cho, Hong Rae; Kwon, Ern Gun. Embedding of Hardy spaces into weighted Bergman spaces in bounded domains with {$C\sp 2$} boundary. Illinois J. Math. 48 (2004), no. 3, 747--757. doi:10.1215/ijm/1258131050. https://projecteuclid.org/euclid.ijm/1258131050