A Characterization of Bergman Spaces on the Unit Ball of ${\mathbb C}^n$. II

http://dx.doi.org/10.4153/CMB-2011-047-6
Canad. Math. Bull. 55(2012), 146-152

  Published:2011-03-23
 Printed: Mar 2012

It has been shown that a holomorphic function $f$ in the unit ball $\mathbb{B}_n$ of ${\mathbb C}_n$ belongs to the weighted Bergman space $A^p_\alpha$, $p>n+1+\alpha$, if and only if the function $|f(z)-f(w)|/|1-\langle z,w\rangle|$ is in $L^p(\mathbb{B}_n\times\mathbb{B}_n,dv_\beta \times dv_\beta)$, where $\beta=(p+\alpha-n-1)/2$ and $dv_\beta(z)= (1-|z|^2)^\beta\,dv(z)$. In this paper we consider the range $0n+1+\alpha$ is particularly interesting.