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Weighted composition operators between \(H^{\infty }\) and generally weighted Bloch spaces on polydisks. (English) Zbl 1236.47019

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 \(B^{\alpha }_{\log }(U^n)\).

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))
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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
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