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On an integral-type operator from Zygmund-type spaces to mixed-norm spaces on the unit ball. (English) Zbl 1210.32001

Let \(H({\mathbb B})\) be the set of holomorphic functions on the unit ball \({\mathbb B}\) in \({\mathbb C}^n\). Let \(g\in H({\mathbb B}),\, g(0) = 0,\) and \(\varphi\) be a holomorphic self-map of \({\mathbb B}\). Recently, the author introduced [Util. Math. 77, 257–263 (2008; Zbl 1175.47034)] the following operator on \(H({\mathbb B})\): \[ P_{\varphi}^{g} (f)(z) = \int\limits_{0}^{1} f(\varphi(t z)) g(t z) \frac{d t}{t},\; f \in H({\mathbb B}),\; z\in {\mathbb B}. \] Characterization of the boundedness and compactness of this operator as an operator between Zygmund-type space and the mixed-norm space is given.

MSC:

32A10 Holomorphic functions of several complex variables
32A36 Bergman spaces of functions in several complex variables
47G10 Integral operators
47B33 Linear composition operators
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))

Citations:

Zbl 1175.47034
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References:

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