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Projections, the weighted Bergman spaces, and the Bloch space. (English) Zbl 0684.47022
The author extends some results of F. Forelli and W. Rudin [Indiana Univ. Math. J. 24, 593-602 (1974; Zbl 0297.47041)] and K. Zhu [J. Funct. Anal. 81, No.2, 260-278 (1988; Zbl 0669.47019)].
Let B be the unit ball in \({\mathbb{C}}^ n\). H(B) denotes the class of holomorphic functions on B. The Bloch space \(\beta\) is the space of all \(g\in H(B)\) for which \[ \| g\|_{\beta}=| g(0)| - \sup_{z\in B}(1-| z|^ 2)| \nabla g(z)| <\infty. \] Let dV denote the normalized volume measure on B, and \(dV_{\alpha}(z)=(1-| z|^ 2)^{\alpha} dV(z)\) \((\alpha >-1)\) is a measure on B. The weighted Bergman space \(A^ p_{\alpha}\) \((1\leq p<\infty)\) is the space \(L^ p(V_{\alpha})\cap H(B).\)
With \(s=\sigma +it\) \((\sigma >-1\), \(-\infty <t<\infty)\) there is associated a kernel \(K_ s=(1-| w|^ 2)^ s(1-z\cdot w)^{-n- 1-s}.\) The kernel \(K_ s\) defines an integral operator \(P_ s.\)
The main results of the paper are the following theorems.
Theorem 1. For \(1\leq p<\infty\), \(P_ s\) is a bounded operator on \(L^ p(V_{\alpha})\) if and only if \(p(1+\sigma)>1+\alpha.\) If \(p(1+\sigma)>1+\alpha\), then \(P_ sf=f\) and \(P_ s\bar f=\overline{f(0)}\) for every \(f\in A^ p_{\alpha}.\)
Theorem 2. For every s, \(P_ s\) is a bounded operator from \(L^{\infty}(B)\) onto \(\beta\). Moreover, \(P_ sC(\bar B)=P_ sC_ 0(B)=\beta_ 0\).
Reviewer: R.Salvi

MSC:
47B38 Linear operators on function spaces (general)
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32A10 Holomorphic functions of several complex variables
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