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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

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)
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