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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
##### Keywords:
projections; Bloch space; weighted Bergman space
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