# zbMATH — the first resource for mathematics

The Bergman spaces, the Bloch space, and Gleason’s problem. (English) Zbl 0657.32002
Let $$B_ n$$ be the open unit ball in $${\mathbb{C}}^ n$$ with dV the normalized volume measure on $$B_ n$$. The Bergman space $$L^ p_ a(B_ n)$$ is the closed subspaces of $$L^ p(B_ n,dV)$$ consisting of holomorphic functions. The Bloch space $${\mathcal B}(B_ n)$$ is the space of holomorphic functions f on $$B_ n$$ with the property that $$(1-| z|^ 2)| \nabla f(z)|$$ is bounded on $$B_ n$$, where $$\nabla f$$ is the analytic gradient of f. For any integer $$m\geq 1$$ and $$| \alpha | =m$$, we define $$T_{\alpha}f(z)=(1-| z|^ 2)^ m\partial^ mf(z)/\partial z^{\alpha}$$ for holomorphic functions f on $$B_ n$$. In this paper, a generalized version of Gleason’s problem for the Bergman space and the Bloch space is studied for the first-order and higher order derivatives. For the main results of this paper, the author shows that $$f\in L^ p_ a(B_ n)$$, $$1\leq p<\infty$$, iff $$T_{\alpha}f\in L^ p(B_ n,dV)$$ and also proves that $$f\in {\mathcal B}(B_ n)$$ iff $$T_{\alpha}f\in L^{\infty}(B_ n,dV)$$. Functional analytic approach is introduced in the study with extensive use of reproducing kernel of $$B_ n$$. These results generalize some known results on the disc in $${\mathbb{C}}$$. The corresponding characterizations for the Bergman spaces $$L^ p_ a(D^ n)$$, $$1\leq p<\infty$$, of the polydisc $$D^ n$$ are also indicated with a result given for the bidisc.
Reviewer: S.H.Tung

##### MSC:
 32A10 Holomorphic functions of several complex variables 46E15 Banach spaces of continuous, differentiable or analytic functions 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.) 47B38 Linear operators on function spaces (general)
##### Keywords:
kernel function; Bergman space; Bloch space; Gleason’s problem
Full Text: