© Congli Yang. 2010
Received: 7 May 2010
Accepted: 23 December 2010
Published: 9 January 2011
Let denote the space of all holomorphic functions on the unit ball . This paper investigates the following integral-type operator with symbol , , , , where is the radial derivative of . We characterize the boundedness and compactness of the integral-type operators from general function spaces to Zygmund-type spaces , where is normal function on .
The space is the classical Bergman space (see ), is the classical Besov space , and, in particular, is just the Hardy space . The spaces are spaces, introduced by Aulaskari et al. [4, 5]. Further, , the analytic functions of bounded mean oscillation. Note that is the space of constant functions if . More information on the spaces can be found in [6, 7].
where and . Stević  considered the boundedness of on -Bloch spaces. Lv and Tang got the boundedness and compactness of from to -Bloch spaces for all (see ). Recently, Li and Stević discussed the boundedness of from Bloch-type spaces to Zygmund-type spaces in . For more information about Zygmund spaces, see [12, 13].
2. Some Auxiliary Results
In this section, we quote several auxiliary results which will be used in the proofs of our main results. The following lemma is according to Zhang .
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
The proof of Lemma 2.5 follows by standard arguments (see, e.g., Lemma 3 in ). Hence, we omit the details.
The following lemma is similar to the proof of Lemma 1 in . Hence, we omit it.
3. Main Results and Proofs
Now, we are ready to state and prove the main results in this section.
Then, by  and .
Combing (3.9) and (3.10), we get (3.1).
then by .
By (3.14) and (3.15), in the same way as proving (3.1), we get that (3.3) holds.
From (3.15) and (3.17), we see that (3.4) holds. The proof of this theorem is completed.
which means that (3.18) holds.
which implies that (3.21) holds.
The proof of Theorem 3.3 follows by the proof of Theorem 3.1. So, we omit the details here.
This along with Theorem 3.2 implies that is bounded. Taking the supremum over the unit ball in , letting in (3.46), using the condition ( ), and finally by applying Lemma 2.6, we get the compactness of the operator . This completes the proof of the theorem.
The author wishes to thank Professor Rauno Aulaskari for his helpful suggestions. This research was supported in part by the Academy of Finland 121281.
- Zhao R: On a general family of function spaces. Annales Academiæ Scientiarum Fennicæ Mathematica Dissertationes 1996, (105):56.MathSciNetMATHGoogle Scholar
- Zhao R: On -Bloch functions and VMOA. Acta Mathematica Scientia. Series B 1996, 16(3):349–360.MathSciNetMATHGoogle Scholar
- Zhu KeHe: Operator Theory in Function Spaces, Monographs and Textbooks in Pure and Applied Mathematics. Volume 139. Marcel Dekker, New York, NY, USA; 1990:xii+258.Google Scholar
- Aulaskari R, Lappan P: Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal. In Complex Analysis and Its Applications, Pitman Research Notes in Mathematics Series. Volume 305. Longman Scientific & Technical, Harlow, UK; 1994:136–146.Google Scholar
- Aulaskari R, Xiao J, Zhao RH: On subspaces and subsets of BMOA and UBC. Analysis 1995, 15(2):101–121.MathSciNetView ArticleMATHGoogle Scholar
- Pérez-González F, Rättyä J: Forelli-Rudin estimates, Carleson measures and -functions. Journal of Mathematical Analysis and Applications 2006, 315(2):394–414. 10.1016/j.jmaa.2005.10.034MathSciNetView ArticleMATHGoogle Scholar
- Rättyä J: On some complex spaces and classes. Annales Academiæ Scientiarum Fennicæ Mathematica Dissertationes 2001, (124):73.MathSciNetMATHGoogle Scholar
- Zhu K: Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics. Volume 226. Springer, New York, NY, USA; 2005:x+271.MATHGoogle Scholar
- Stević S: On an integral operator on the unit ball in. Journal of Inequalities and Applications 2005, (1):81–88.MathSciNetMATHGoogle Scholar
- Lv X, Tang X: Extended Cesàro operators from spaces to Bloch-type spaces in the unit ball. Korean Mathematical Society. Communications 2009, 24(1):57–66. 10.4134/CKMS.2009.24.1.057MathSciNetView ArticleMATHGoogle Scholar
- Li S, Stević S: Integral-type operators from Bloch-type spaces to Zygmund-type spaces. Applied Mathematics and Computation 2009, 215(2):464–473. 10.1016/j.amc.2009.05.011MathSciNetView ArticleMATHGoogle Scholar
- Li S, Stević S: Generalized composition operators on Zygmund spaces and Bloch type spaces. Journal of Mathematical Analysis and Applications 2008, 338(2):1282–1295. 10.1016/j.jmaa.2007.06.013MathSciNetView ArticleMATHGoogle Scholar
- Li S, Stević S: Products of Volterra type operator and composition operator from and Bloch spaces to Zygmund spaces. Journal of Mathematical Analysis and Applications 2008, 345(1):40–52. 10.1016/j.jmaa.2008.03.063MathSciNetView ArticleMATHGoogle Scholar
- Zhang XJ: Multipliers on some holomorphic function spaces. Chinese Annals of Mathematics. Series A. Shuxue Niankan. A Ji 2005, 26(4):477–486.MathSciNetMATHGoogle Scholar
- Hu Z: Extended Cesàro operators on mixed norm spaces. Proceedings of the American Mathematical Society 2003, 131(7):2171–2179. 10.1090/S0002-9939-02-06777-1MathSciNetView ArticleMATHGoogle Scholar
- Li SX, Stević S: Riemann-Stieltjes-type integral operators on the unit ball in . Complex Variables and Elliptic Equations 2007, 52(6):495–517. 10.1080/17476930701235225MathSciNetView ArticleMATHGoogle Scholar
- Madigan K, Matheson A: Compact composition operators on the Bloch space. Transactions of the American Mathematical Society 1995, 347(7):2679–2687. 10.2307/2154848MathSciNetView ArticleMATHGoogle Scholar
- Tang X: Extended Cesàro operators between Bloch-type spaces in the unit ball of Cn. Journal of Mathematical Analysis and Applications 2007, 326(2):1199–1211. 10.1016/j.jmaa.2006.03.082MathSciNetView ArticleMATHGoogle Scholar
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