The Bergman spaces, the Bloch space, and Gleason’s problem

Abstract:

Suppose $f$ is a holomorphic function on the open unit ball ${B_n}$ of ${{\mathbf {C}}^n}$. For $1 \leqslant p < \infty$ and $m > 0$ an integer, we show that $f$ is in ${L^p}({B_n}, dV)$ (with $dV$ the volume measure) iff all the functions ${\partial ^m}f/\partial {z^{\alpha }}\;(|\alpha | = m)$ are in ${L^p}({B_n}, dV)$. We also prove that $f$ is in the Bloch space of ${B_n}$ iff all the functions ${\partial ^m}f/\partial {z^\alpha }\;(|\alpha | = m)$ are bounded on ${B_n}$. The corresponding result for the little Bloch space of ${B_n}$ is established as well. We will solve Gleason’s problem for the Bergman spaces and the Bloch space of ${B_n}$ before proving the results stated above. The approach here is functional analytic. We make extensive use of the reproducing kernels of ${B_n}$. The corresponding results for the polydisc in ${{\mathbf {C}}^n}$ are indicated without detailed proof.