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

and Applied Analysis 3 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 φ is a holomorphic self-map of , g ∈ H , and g 0 0. Then, for every f ∈ H it holds [ P g φ ( f )] z f ( φ z ) g z . 2.1 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 p, q > 0, φ is a holomorphic self-map of , g ∈ H , g 0 0, φ is normal, and μ is a weight. Then, the operator P φ : Zμ or Zμ,0 → H p, q, φ is compact if and only if for every bounded sequence fk k∈ ⊂ Zμ or Zμ,0 converging to 0 uniformly on compacts of we have limk→∞‖P φfk‖H p,q,φ 0. The next lemma is folklore and can be found, for example, in 6 one-dimensional case for standard power weights is due to Flett 29, Theorems 6 and 7 . Lemma 2.3. Assume that 0 < p, q < ∞, φ is normal, and m ∈ . Then, the following asymptotic relationship holds for every f ∈ H , ∫1 0 M p q ( f, r )φ r 1 − r dr ∣ ∣f 0 ∣ ∣p ∫1 0 M p q ( f, r ) 1 − r mp φ p r 1 − r dr. 2.2 Lemma 2.4. Assume that μ is normal and f ∈ Zμ. Then, ∣ ∣f z ∣ ∣ ≤ C∥∥f∥∥Zμ ( 1 ∫ |z|


Introduction
A positive, continuous function ν on the interval 0, 1 is called normal 1 if there are δ ∈ 0, 1 and a and b, 0 < a < b such that ν r 1 − r a is decreasing on δ, 1 , lim

2 Abstract and Applied Analysis
If we say that a function ν : → 0, ∞ is normal, we also assume that it is radial, that is, ν z ν |z| , z ∈ .Let μ be a weight.By Z μ Z μ , we denote the class of all f ∈ H such that and call it the Zygmund-type class.The quantity z f is a seminorm.A norm on Z μ can be introduced by f Z |f 0 | z f .Zygmund-type class with this norm will be called the Zygmund-type space.
The little Zygmund-type space on , denoted by Z μ,0 Z μ,0 , is the closed subspace of Z μ consisting of functions f satisfying the following condition For 0 < p, q < ∞, and φ normal, the mixed-norm space H p, q, φ H p, q, φ consists of all functions f ∈ H such that where and dσ is the normalized surface measure on ∂ .For p q, φ r 1 − r 2 α 1 /p , and α > −1, the space is equivalent with the weighted Bergman space A p α .In 2 , the present author has introduced products of integral and composition operators on H as follows see also 3-5 .Assume g ∈ H , g 0 0, and ϕ is a holomorphic self-map of , then we define an operator on H by The operator is an extension of the operator introduced in 6 .Here, we continue to study operator P g ϕ 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 n , see, for example, 3, 6-27 and the references therein.
In this paper, constants are denoted by C; they are positive and may differ from one occurrence to the other.The notation a b means that there is a positive constant C such that a ≤ Cb.If both a b and b a hold, then one says that a b.

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 ϕ is a holomorphic self-map of , g ∈ H , and g 0 0.Then, for every f ∈ H 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 p, q > 0, ϕ is a holomorphic self-map of , g ∈ H , g 0 0, φ is normal, and μ is a weight.Then, the operator P g ϕ : Z μ or Z μ,0 → H p, q, φ is compact if and only if for every bounded sequence f k k∈AE ⊂ Z μ or Z μ,0 converging to 0 uniformly on compacts of we have The next lemma is folklore and can be found, for example, in 6 one-dimensional case for standard power weights is due to Flett 29, Theorems 6 and 7 .
Lemma 2.3.Assume that 0 < p, q < ∞, φ is normal, and m ∈ AE.Then, the following asymptotic relationship holds for every f ∈ H ,
Proof.By Lemma 2.3.1 in 21 applied to Êf we have that Hence, for |z| ≥ 1/2, we have that where then by the mean value property of the function f z − f 0 see 30 , Jensen's inequality, and Parseval's formula, we obtain Êf w 2 dV N w

2.9
From 2.9 and 2.6 , we obtain

2.10
From 2.8 and 2.10 , 2.3 follows, from which by 2.4 the second statement follows.
Lemma 2.5.Assume μ is normal and 2.4 holds.Then, for every bounded sequence f k k∈AE ⊂ Z μ converging to 0 uniformly on compacts of , we have that Proof.From 2.4 , we have that for every ε > 0, there is a δ ∈ 0, min{ε, 1/2} such that Hence, from 2.12 it follows that for each k ∈ AE and |z| ≥ 1 − δ

2.13
From 2.12 and 2.13 , we obtain Letting k → ∞ in this inequality, using the assumption that f k converges to 0 on the compact |w| ≤ 1 − δ, and using the fact that ε is an arbitrary positive number, the lemma follows.

The Boundedness and Compactness of
The boundedness and compactness of the operator P g ϕ : Z μ or Z μ,0 → H p, q, φ are characterized in this section.Theorem 3.1.Assume that p, q > 0, ϕ is a holomorphic self-map of , g ∈ H , g 0 0, φ and μ are normal, and μ satisfies condition 2.4 .Let G z 1 0 g tz dt t .

3.1
Then, the following statements are equivalent: is bounded, then the following asymptotic relations hold:  ≤ C f Z μ G H p,q,φ .

≤
C G H p,q,φ sup z∈ f k z −→ 0, as k −→ ∞, 3.4which along with Lemma 2.2 implies the compactness of P g ϕ : Z μ → H p, q, φ .From 2.4 and by Lemmas 2.3 and 2.4, we have Assume that f k k∈AE ⊂ Z μ is a bounded sequence converging to 0 uniformly on compacts of .Then, by Lemmas 2.1, 2.3, and 2.5, we have