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Extended Cesàro operators on mixed norm spaces. (English) Zbl 1054.47023
Given a function \(g\) holomorphic on the unit ball \(B\) of \(\mathbb C^n\), let \({\mathcal R} g=\sum_{j=1}^n z_j{{\partial g}\over {\partial z_j}}\) be the radial derivative of \(g\). The extended Cesàro operator \(T_g\) with symbol \(g\) is then defined by \[ T_gf(z)=\int_0^1 f(tz) {\mathcal R}g(tz)\, dt,\qquad z\in B, \] for functions \(f\) holomorphic on \(B\). In this paper, the author considers these operators acting on certain weighted mixed norm spaces \(H_{p,q}(\varphi)\). Roughly speaking, the space \(H_{p,q}(\varphi)\) is the space of all holomorphic functions on \(B\) whose \(q\)-integral means belong to \(L^p\) with respect to certain weights involving \(\varphi\). These mixed norm spaces contain the well-known weighted Bergman spaces as special cases.
The author shows that \(T_g\) is bounded (resp. compact) on \(H_{p,q}(\varphi)\) for some (resp. all) \(p,q\in(0,\infty]\) if and only if \(g\) is a (resp. little) Bloch function. This result extends earlier works byJ. H. Shi and G. P. Ren [Proc. Am. Math. Soc. 126, 3553–3560 (1998; Zbl 0905.47019)] and Z. Xiao [Acta Math. Sin., New Ser. 14, 647–654 (1998; Zbl 0927.30030)].

MSC:
47B38 Linear operators on function spaces (general)
32A36 Bergman spaces of functions in several complex variables
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