On an integral-type operator between bloch-type spaces
Introduction
Let be the unit ball of , the open unit disk in , the class of all holomorphic functions on B. For , letrepresent the radial derivative of f.
For , recall that the -Bloch space , is the space consisting of all functions such thatUnder the above norm, is a Banach space. When , we get the classical Bloch space . For more information of the Bloch space and the -Bloch space (see, e.g., [3], [43] and the references therein).
A positive continuous function on is called normal, if there exist positive numbers s and and such that [26]
For a normal function , an is said to belong to the Bloch-type space , if(see, e.g., [39]). The Bloch-type space is a Banach space with the normLet denote the subspace of consisting of those for whichThis space is called the little Bloch-type space. When , the induced spaces and become the -Bloch space and the little -Bloch space .
Let be a holomorphic self-map of B. The composition operator is defined byIt is interesting to provide a function theoretic characterization of when induces a bounded or compact composition operator on various spaces. Recall that a linear operator is said to be bounded if the image of a bounded set is a bounded set, while a linear operator is compact if it takes bounded sets to sets with compact closure. The book [4] contains plenty of information on this topic. For some recent results in this research area, see, e.g., [3], [5], [6], [20], [30], [32], [40], [41], [42], [46] and the references therein.
Let and be a holomorphic self-map of . Products of integral and composition operators on were introduced by Li and Stević (see, [10], [16], [17], [23], as well as [18], [31] for a related operator) as follows:andOperators in (2), (3) are extensions of the following classical integral operator:which was introduced in [25].
One of the interesting questions has been to extend operators in (2), (3) in the unit ball settings and to study their function theoretic properties between spaces of holomorphic functions on the unit ball in terms of inducing functions.
If is such that and is a holomorphic self-map of B, then in [35] Stević introduced the following operator on the unit ball:and has shown that it is a natural extension of the operator in (3). In [34] Stević proposed a big research project regarding the operator. Some further results in this direction can be found in [36], [37]. A particular case of the operator in (4) was independently introduced in [45] and studied in [44].
For and the operator is reduced to operator , so called, extended Cesàro operator or the Riemann–Stieltjes operator, which was studied, e.g., in [1], [2], [7], [8], [9], [11], [12], [13], [14], [15], [19], [21], [22], [27], [39] (see also the references therein). Some related integral-type operators can be found, e.g., in [2], [28], [29], [33], [38].
In this paper, we study the boundedness and compactness of the operator between Bloch-type spaces. Our main results are natural extensions of some results in [37]. As a corollary, we obtain some characterizations for the boundedness and compactness of the extended Cesàro operator between Bloch-type spaces.
Throughout this paper C will denote constants in this paper, they are positive and may differ from one occurrence to the other. means that there is a positive constant C such that . Moreover, if both and hold, then one says that .
Section snippets
Auxiliary results
In order to prove our main results, we need some auxiliary results which are incorporated in the following lemmas. The following lemma can be found, for example, in [39], [42]. Lemma 1 Assume that is a normal function on . If , thenfor some C independent of f.
The following lemma can be proved similar to the proof of Lemma 1 in [24]. We omit the details. Lemma 2 Assume that is a normal function on . A closed set K in is compact if and only if it is bounded and
Main results and their proofs
Now we are in a position to state and prove our main results in this paper. Theorem 1 Assume that is a holomorphic self-map of B, and are normal functions on . Then the following statements are equivalent. is bounded; is bounded;
Moreover, when is bounded, the following relationship:holds.
Proof
. Assume that (7) holds. Note that for every . For any
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