Li, Haiying; Liu, Peide Weighted composition operators between \(H^{\infty }\) and generally weighted Bloch spaces on polydisks. (English) Zbl 1236.47019 Int. J. Math. 21, No. 5, 687-699 (2010). Summary: Let \(U^{n}\) be the unit polydisk of \(\mathbb{C}^{n}\), \(\varphi (z) = (\varphi _{1}(z),\varphi _{2}(z),\dots ,\varphi _{n}(z))\) be a holomorphic self-map of \(U^{n}\) and \(\psi \) be a holomorphic function on \(U^{n}\). \(H^{\infty }(U^{n})\) is the space of all bounded holomorphic functions on \(U^{n}\) and by a generally weighted Bloch space we mean \(B^{\alpha }_{\log }(U^n) = \{f \in H(U^n) : \text{sup}_{z \in U^n} \sum ^n_{k=1} |\frac{\partial f}{\partial z_k}(z)|(1-|z_k|^2)^{\alpha } \log \frac{2}{1-|z_k|^2} < +\infty \}\). We give necessary and sufficient conditions for the boundedness and compactness of the weighted composition operator \(\psi C_{\varphi }\) between \(H^{\infty}(U^{n})\) and \(B^{\alpha }_{\log }(U^n)\). Cited in 1 Document MSC: 47B33 Linear composition operators 46E15 Banach spaces of continuous, differentiable or analytic functions 32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) Keywords:holomorphic self-map; weighted composition operator; \(H^{\infty }\); generally weighted Bloch space PDFBibTeX XMLCite \textit{H. Li} and \textit{P. Liu}, Int. J. Math. 21, No. 5, 687--699 (2010; Zbl 1236.47019) Full Text: DOI References: [1] Cima J. A., Proc. Amer. Math. Soc. 91 pp 217– [2] Cowen C. C., Composition operators on spaces of analytic functions (1995) · Zbl 0873.47017 [3] Li H., J. Inequal. Pure Appl. Math. 8 pp 1– [4] Li H., Acta. Math. Sci. Ser. A Chin. Ed. 29 pp 1634– [5] DOI: 10.15352/bjma/1240336427 · Zbl 1163.47019 · doi:10.15352/bjma/1240336427 [6] DOI: 10.1090/S0002-9947-1995-1273508-X · doi:10.1090/S0002-9947-1995-1273508-X [7] DOI: 10.1090/S0002-9939-04-07617-8 · Zbl 1056.32005 · doi:10.1090/S0002-9939-04-07617-8 [8] Rikio Y., Arch. Math. 78 pp 310– [9] DOI: 10.1007/978-1-4612-0887-7 · doi:10.1007/978-1-4612-0887-7 [10] DOI: 10.1007/s101149900028 · Zbl 0967.32007 · doi:10.1007/s101149900028 [11] Timoney R., Bull. London. Math. Soc. 37 pp 241– [12] Timoney R., J. R. Angew. Math. 319 pp 1– [13] DOI: 10.1007/BF02878708 · Zbl 1024.47010 · doi:10.1007/BF02878708 [14] Zhou Z. H., Sci. China Ser. A 46 pp 73– [15] Zhu K., Spaces of Holomorphic functions in the unit ball (2005) · Zbl 1067.32005 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.