On an Integral-Type Operator from Zygmund-Type Spaces to Mixed-Norm Spaces on the Unit Ball

The boundedness and compactness of an integral-type operator recently introduced by the author from Zygmund-type spaces to the mixed-norm space on the unit ball are characterized here. 1. Introduction Let be the open unit ball in , its boundary, the normalized volume measure on , and the class of all holomorphic functions on . Strictly positive, bounded, continuous functions on are called weights. For an with the Taylor expansion , let be the radial derivative of , where is a multi-index, and . A positive, continuous function on the interval is called normal [1] if there are and and , such that If we say that a function is normal, we also assume that it is radial, that is, , . Let be a weight. By , we denote the class of all such that and call it the Zygmund-type class. The quantity is a seminorm. A norm on can be introduced by . Zygmund-type class with this norm will be called the Zygmund-type space. The little Zygmund-type space on , denoted by , is the closed subspace of consisting of functions satisfying the following condition For , , and normal, the mixed-norm space consists of all functions such that where and is the normalized surface measure on . For , , and , the space is equivalent with the weighted Bergman space . In [2], the present author has introduced products of integral and composition operators on as follows (see also [3–5]). Assume , , and is a holomorphic self-map of , then we define an operator on by The operator is an extension of the operator introduced in [6]. Here, we continue to study operator by characterizing the boundedness and compactness of the operator between Zygmund-type spaces and the mixed-norm space. For some results on related integral-type operators mostly in , see, for example, [3, 6–27] and the references therein. In this paper, constants are denoted by ; they are positive and may differ from one occurrence to the other. The notation means that there is a positive constant such that . If both and hold, then one says that . 2. Auxiliary Results In this section, we quote several lemmas which are used in the proofs of the main results. The first lemma was proved in [2]. Lemma 2.1. Assume that is a holomorphic self-map of , and . Then, for every it holds The next Schwartz-type characterization of compactness [28] is proved in a standard way (see, e.g., the proof of the corresponding lemma in [11]), hence we omit its proof. Lemma 2.2. Assume , is a holomorphic self-map of , , is normal, and is a weight. Then, the operator is compact if and only if for every bounded sequence converging to 0 uniformly on compacts of we