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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
Full Text:
##### References:
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