# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
  Cowen C.C., Composition Operators on Spaces of Analytic Functions (1995) · Zbl 0873.47017  Ohno S., Taiwanese J. Math. 5 pp 555– (2001)  DOI: 10.1007/s101149900028 · Zbl 0967.32007 · doi:10.1007/s101149900028  DOI: 10.1112/blms/12.4.241 · Zbl 0416.32010 · doi:10.1112/blms/12.4.241  DOI: 10.1007/BF02878708 · Zbl 1024.47010 · doi:10.1007/BF02878708  Zhou Z.H., Complex Variables 46 pp 73– (2001)  DOI: 10.1360/03ys9004 · Zbl 1217.32002 · doi:10.1360/03ys9004
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.