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Weighted composition operators from \(\alpha\)-Bloch space to \(H^{\infty}\) on the polydisc. (English) Zbl 1130.47015

Let \(\mathbb D^n\) be the polydisk in \(\mathbb C^n\) and let \(\alpha>0\). A holomorphic function \(f\) in \(\mathbb D^n\) is said to belong to the \(\alpha\)-Bloch class \({\mathcal B}^\alpha(\mathbb D)\) if \[ \| f\|_{\mathcal B^\alpha}:=| f(0)| +\sup_{z\in\mathbb D^n}\sum_{k=1}^n \left| \frac{\partial f}{\partial z_k}(z)\right| (1-| z_k| ^2)^{\alpha}<\infty. \] Let \(\Psi\) be a holomorphic function on \(\mathbb D^n\) and \(\varphi=(\varphi_1,\dots,\varphi_n)\) be a holomorphic selfmap of \(\mathbb D^n\). The authors give necessary and sufficient conditions on \(\Psi\) and \(\varphi\) in order for the weighted composition operators \(\Psi C_\varphi\) to be bounded and compact from \(\mathcal B^\alpha(\mathbb D^n)\) to \(H^\infty(\mathbb D^n)\). The results depend on whether \(0<\alpha<1\), \(\alpha =1\), or \(\alpha>1\). In the case where \(\alpha>1\), the authors \[ \sup_{z\in\mathbb D^n}| \Psi(z)| \sum_{j=1}^n\frac{1}{(1-|\varphi_j(z)|^2)^{\alpha-1}}<\infty. \] The authors also note that one of their results corrects a statement of S.Ohno in [Taiwanese J. Math.5, No.3, 555–563 (2001; Zbl 0997.47025)] on the compactness of weighted composition operators between the Bloch space and \(H^\infty(\mathbb D)\).

MSC:

47B33 Linear composition operators
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
47B38 Linear operators on function spaces (general)

Citations:

Zbl 0997.47025
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References:

[1] Cowen C.C., Composition Operators on Spaces of Analytic Functions (1995) · Zbl 0873.47017
[2] Ohno S., Taiwanese J. Math. 5 pp 555– (2001)
[3] DOI: 10.1007/s101149900028 · Zbl 0967.32007 · doi:10.1007/s101149900028
[4] DOI: 10.1112/blms/12.4.241 · Zbl 0416.32010 · doi:10.1112/blms/12.4.241
[5] DOI: 10.1007/BF02878708 · Zbl 1024.47010 · doi:10.1007/BF02878708
[6] Zhou Z.H., Complex Variables 46 pp 73– (2001)
[7] DOI: 10.1360/03ys9004 · Zbl 1217.32002 · doi:10.1360/03ys9004
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