×

Composition operators between generally weighted Bloch spaces of polydisk. (English) Zbl 1138.47018

The paper deals with the boundedness and compactness of the composition operator \(C_\phi\) between the Bloch type spaces \(B^p_{\log}(U^n)\) of analytic functions defined on the polydisc \(U^n\) and given by \[ \sup_{z\in U^n}\sum_{k=1}^n \left| \frac{\partial f}{\partial z_k}(z)\right| (1-| z_k| ^2)^p\log\frac{2}{1-| z_k| ^2}<\infty \] for different values of \(0<p<\infty.\) The main result establishes that the boundedness of \(C_\phi\) from \(B^p_{\log}(U^n)\) to \(B^q_{\log}(U^n)\) is equivalent to \[ \sup_{z\in U^n}\sum_{k,l=1}^n \left| \frac{\partial\phi_l}{\partial z_k}(z)\right|\frac{(1-| z_k|^2)^q}{(1-| \phi_l(z)|^2)^p}\frac{\log\frac{2}{1-| z_k|^2}}{\log\frac{2}{1-|\phi_l(z)|^2}}<\infty. \] The “little o”-condition is shown to be sufficient for the compactness. The same result (without the extra \(\log\)) was proved by D.D. Clahane, S. Stević and Z.-H. Zhou in [“Composition operators between generalized Bloch spaces of the polydisc” (to appear)] and with essentially the same proof.

MSC:

47B33 Linear composition operators
47B38 Linear operators on function spaces (general)
PDFBibTeX XMLCite
Full Text: EuDML EMIS