- Research Article
- Open Access
© 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.
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