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

Abstract

Suppose f f is a holomorphic function on the open unit ball B n {B_n} of C n {{\mathbf {C}}^n} . For 1 ⩽ p > ∞ 1 \leqslant p > \infty and m > 0 m > 0 an integer, we show that f f is in L p ( B n , d V ) {L^p}({B_n},\,dV) (with d V dV the volume measure) iff all the functions ∂ m f / ∂ z α ( | α | = m ) {\partial ^m}f/\partial {z^{\alpha \,}}\;(|\alpha |\, = m) are in L p ( B n , d V ) {L^p}({B_n},\,dV) . We also prove that f f is in the Bloch space of B n {B_n} iff all the functions ∂ m f / ∂ z α ( | α | = m ) {\partial ^m}f/\partial {z^\alpha }\;(|\alpha |\, = m) are bounded on B n {B_n} . The corresponding result for the little Bloch space of B n {B_n} is established as well. We will solve Gleason’s problem for the Bergman spaces and the Bloch space of B n {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 {B_n} . The corresponding results for the polydisc in C n {{\mathbf {C}}^n} are indicated without detailed proof.