Characterizations of Bergman spaces and Bloch space in the unit ball of $\textbf {C}^ n$

Abstract:

In this paper we prove that, in the unit ball $B$ of ${{\mathbf {C}}^n}$, a holomorphic function $f$ is in the Bergman space $L_a^p(B),\;0 < p < \infty$, if and only if \[ \int _B {|\tilde \nabla } f(z){|^2}|f(z){|^{p - 2}}{(1 - |z{|^2})^{n + 1}}d\lambda (z) < \infty ,\] where $\tilde \nabla$ and $\lambda$ denote the invariant gradient and invariant measure on $B$, respectively. Further, we give some characterizations of Bloch functions in the unit ball $B$, including an exponential decay characterization of Bloch functions. We also give the analogous results for $\operatorname {BMOA} (\partial B)$ functions in the unit ball.