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
$00$ 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.
Publié le : 2004-07-15
Classification:
32A35,
32A36,
46E15
@article{1258131050,
author = {Cho, Hong Rae and Kwon, Ern Gun},
title = {Embedding of Hardy spaces into weighted Bergman spaces in bounded domains with {$C\sp 2$} boundary},
journal = {Illinois J. Math.},
volume = {48},
number = {3},
year = {2004},
pages = { 747-757},
language = {en},
url = {http://dml.mathdoc.fr/item/1258131050}
}
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., Tome 48 (2004) no. 3, pp. 747-757. http://gdmltest.u-ga.fr/item/1258131050/