The bergman spaces, the bloch space, and gleason’s problem

Suppose is a holomorphic function on the open unit ball [Fourmula Present] an integer, we show that/is in Lp(Bn, dV) (with dV the volume measure) iff all the functions [Fourmula Present]. We also prove that/is in the Bloch space of Bn iff all the functions [Fourmula Present] are bounded on Bn. The corresponding result for the little Bloch space of Bn is established as well. We will solve Gleason’s problem for the Bergman spaces and the Bloch space of Bn before proving the results stated above. The approach here is functional analytic. We make extensive use of the reproducing kernels of Bn. The corresponding results for the polydisc in Cn are indicated without detailed proof.