Characterizations of Bergman spaces and Bloch space in the unit ball of 𝐶ⁿ

Abstract

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