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Weighted composition operators between mixed norm spaces and \(H_{\alpha }^{\infty }\) spaces in the unit ball. (English) Zbl 1138.47019
The paper deals with the boundedness and compactness of the weighted composition operator \(uC_\varphi\) between the mixed norm spaces \(H(p,q,\phi)\), \(0<p,q<\infty\), of analytic functions defined on the unit ball of \({\mathbb C}^n\) given by \(\int_0^1 M_p(f,r)^q \frac{\phi^p(r)}{1-r}\,dr\) (where \(M_p(f,r)=(\int_{S}| f(r\xi)|^p\,d\sigma(\xi))^{1/p}\) and \(\phi\) is a normal function) and \(H^\infty_\alpha\), consisting of analytic functions with \(\sup_{| z|<1}(1-| z|^2)^\alpha| f(z)|<\infty\). The main result establishes that the boundedness of \(uC_\varphi\) from \(H(p,q,\phi)\) to \(H^\infty_\alpha\) is equivalent to \[ \sup_{| z|<1}\frac{(1-| z|^2)^\alpha| u(z)|}{\phi(|\varphi(z)| )(1-| \varphi(z)|^2)^{n/q}}<\infty. \] The “little o”-condition is shown to be necessary and sufficient for the compactness. Analogous results hold when replacing \(H^\infty_\alpha\) by \(H^\infty_{\alpha,0}\), adding the condition \(u\in H^\infty_{\alpha,0}\). The author also studies the case \(uC_\varphi\) from \(H^\infty_\alpha\) to \(H(p,q,\phi)\), showing that in the case \(\alpha=1\), the boundedness and compactness are equivalent and hold whenever \(u\in H(p,q,\phi)\).

MSC:
47B33 Linear composition operators
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References:
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