## 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:

 [1] Cowen CC, MacCluer BD: Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, Fla, USA; 1995:xii+388. [2] Shields, AL; Williams, DL, Bonded projections, duality, and multipliers in spaces of analytic functions, Transactions of the American Mathematical Society, 162, 287-302, (1971) [3] Ohno, S, Weighted composition operators between[inlineequation not available: see fulltext.] and the Bloch space, Taiwanese Journal of Mathematics, 5, 555-563, (2001) · Zbl 0997.47025 [4] Li S, Stević S: Weighted composition operators between and -Bloch spaces in the unit ball. to appear in Taiwan Journal of Mathematics to appear in Taiwan Journal of Mathematics · Zbl 1160.47308 [5] Sharma, AK; Sharma, SD, Weighted composition operators between Bergman-type spaces, Communications of the Korean Mathematical Society, 21, 465-474, (2006) · Zbl 1160.47308 [6] Clahane, DD; Stević, S, Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball, Journal of Inequalities and Applications, 2006, 11 pages, (2006) · Zbl 1131.47018 [7] Zhu, X, Weighted composition operators between[inlineequation not available: see fulltext.] and Bergman type spaces, Communications of the Korean Mathematical Society, 21, 719-727, (2006) · Zbl 1160.47028 [8] Rudin, W, Function theory in the unit ball of ℂ\^{}{n}, No. 241, xiii+436, (1980), Berlin, Germany [9] Stević, S, On generalized weighted Bergman spaces, Complex Variables, 49, 109-124, (2004) · Zbl 1053.47020 [10] Madigan, K; Matheson, A, Compact composition operators on the Bloch space, Transactions of the American Mathematical Society, 347, 2679-2687, (1995) · Zbl 0826.47023
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.