Weighted composition operators between mixed norm spaces and \(H_{\alpha }^{\infty }\) spaces in the unit ball. (English) Zbl 1138.47019

Stević, Stevo

J. Inequal. Appl. 2007, Article ID 28629, 9 p. (2007).

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 \(u\in H(p,q,\phi)\).

Reviewer: Oscar Blasco (Valencia)

MSC:

47B33

Linear composition operators

Keywords:

weighted composition operators; mixed norm spaces in the unit ball

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