Weighted Iterated Radial Composition Operators between Some Spaces of Holomorphic Functions on the Unit Ball

and Applied Analysis 3 where dV z is the Lebesgue volume measure on B. Some facts on mixed-norm spaces in various domains in C can be found, for example, in 6–8 see also the references therein . For 0 < p < ∞ the Hardy space H B H consists of all f ∈ H B such that ∥ ∥f ∥ ∥ Hp : sup 0<r<1 (∫ ∂B ∣ ∣f rζ ∣ ∣dσ ζ )1/p < ∞. 1.10 For p 2 the Hardy and the weighted Bergman space are Hilbert. Let φ be a holomorphic self-map of B, u ∈ H B , and m ∈ N0. For f ∈ H B , the weighted iterated radial composition operator is defined by Ru,φ ( f ) z u z Rf ( φ z ) , z ∈ B. 1.11 Note that the operator is the composition of the multiplication, composition and the iterated radial operator, that is Ru,φ Mu ◦ Cφ ◦ R. 1.12 This is one of the product operators suggested by this author to be investigated at numerous talks e.g., in 9 . Note that for m 0 the operator Ru,φ becomes the weighted composition operator see, e.g., 4, 8, 10 . It is of interest to provide function-theoretic characterizations for when φ and u induce bounded or compact weighted iterated radial composition operators on spaces of holomorphic functions. Studying products of some concrete linear operators on spaces of analytic functions attracted recently some attention see, for example, 11–32 as well as the related references therein. Some operators on mixed-norm spaces have been studied, for example, in 8, 10, 11, 16, 25, 26, 29, 33 see also the references therein . Here we study the boundedness and compactness of weighted iterated radial composition operators from mixed-norm spaces to weighted-type spaces on the unit ball for the case m ∈ N. We also calculate the Hilbert-Schmidt norm of the operator on the weighted Bergman-Hilbert space Aα B as well as on the Hardy H 2 B space. 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. 2. Auxiliary Results In this section we quote several lemmas which are used in the proofs of the main results. The next characterization of compactness is proved in a standard way, hence we omit its proof see, e.g., 34 . Lemma 2.1. Assume p, q > 0, φ is a holomorphic self-map of B, u ∈ H B , φ is normal and μ is a weight. Then the operator Ru,φ : H p, q, φ → H∞ μ is compact if and only if for every bounded sequence fk k∈N ⊂ H p, q, φ converging to 0 uniformly on compacts of B as k → ∞, one has lim k→∞ ∥ ∥ ∥Ru,φfk ∥ ∥ ∥ H∞ μ 0. 2.1 4 Abstract and Applied Analysis The following lemma is a slight modification of Lemma 2.5 in 8 and is proved similar to Lemma 1 in 35 . Lemma 2.2. Assume μ is a normal weight. Then a closed set K in H∞ μ,0 is compact if and only if it is bounded and lim |z|→ 1 sup f∈K μ z ∣ ∣f z ∣ ∣ 0. 2.2 The following lemma is folklore and in the next form it can be found in 36 . Lemma 2.3. Assume that 0 < p, q < ∞, φ is normal, and m ∈ N. Then for every f ∈ H B the following asymptotic relationship holds: ∫1 0 M p q ( f, r )φ r 1 − r dr ∣ ∣f 0 ∣ ∣p ∫1 0 M p q ( Rf, r ) 1 − r mp φ p r 1 − r dr. 2.3 Lemma 2.4. Assume that m ∈ N, 0 < p, q < ∞, φ is normal and f ∈ H p, q, φ . Then, there is a positive constant C independent of f such that ∣ ∣Rf z ∣ ∣ ≤ C∥∥f∥∥H p,q,φ |z| φ |z| ( 1 − |z| )n/q m . 2.4 Proof. Let g Rm−1f and z ∈ B. By the definition of the radial derivative, the Cauchy-Schwarz inequality and the Chauchy inequality, we have that ∣ ∣Rg z ∣ ∣ ≤ |z|∣∇g z ∣ ≤ C|z| supB z,1−|z| /4 ∣ ∣g w ∣ ∣ 1 − |z| . 2.5 From 2.3 with m → m − 1 we easily obtain the following inequality see, e.g., 8, Lemma 2.1 : ∣ ∣g z ∣ ∣ ≤ C ∥ ∥f ∥ ∥ H p,q,φ φ |z| ( 1 − |z| )n/q m−1 . 2.6 From 2.5 and 2.6 and the asymptotic relations 1 − |w| 1 − |z|, φ |z| φ |w| , for w ∈ B ( z, 1 − |z| 2 ) , 2.7 inequality 2.4 follows. Abstract and Applied Analysis 5 Lemma 2.5. Letand Applied Analysis 5 Lemma 2.5. Let fa,s z 1 1 − 〈z, a〉 s , z ∈ B. 2.8


Introduction
The iterated radial derivative operator R m f is defined inductively by A positive continuous function ν on the interval 0, 1 is called normal 2 if there are δ ∈ 0, 1 and τ and t, 0 < τ < t such that ν r 1 − r τ is decreasing on δ, 1 , lim

1.4
If we say that a function ν : B → 0, ∞ is normal, we also assume that it is radial, that is, Strictly positive continuous functions on B are called weights.
The weighted-type space where μ is a weight see, e.g., 3, 4 as well as 5 for a related class of spaces .
The little weighted-type space For 0 < p, q < ∞, and φ normal, the mixed-norm space H p, q, φ B H p, q, φ consists of all functions f ∈ H B such that where and dσ is the normalized surface measure on ∂B.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 B A p α , which is defined as the class of all f ∈ H B such that where dV z is the Lebesgue volume measure on B. Some facts on mixed-norm spaces in various domains in C n can be found, for example, in 6-8 see also the references therein .For 0 < p < ∞ the Hardy space H p B H p consists of all f ∈ H B such that For p 2 the Hardy and the weighted Bergman space are Hilbert.Let ϕ be a holomorphic self-map of B, u ∈ H B , and m ∈ N 0 .For f ∈ H B , the weighted iterated radial composition operator is defined by Note that the operator is the composition of the multiplication, composition and the iterated radial operator, that is This is one of the product operators suggested by this author to be investigated at numerous talks e.g., in 9 .Note that for m 0 the operator R m u,ϕ becomes the weighted composition operator see, e.g., 4, 8, 10 .It is of interest to provide function-theoretic characterizations for when ϕ and u induce bounded or compact weighted iterated radial composition operators on spaces of holomorphic functions.Studying products of some concrete linear operators on spaces of analytic functions attracted recently some attention see, for example, 11-32 as well as the related references therein.Some operators on mixed-norm spaces have been studied, for example, in 8,10,11,16,25,26,29,33 see also the references therein .
Here we study the boundedness and compactness of weighted iterated radial composition operators from mixed-norm spaces to weighted-type spaces on the unit ball for the case m ∈ N. We also calculate the Hilbert-Schmidt norm of the operator on the weighted Bergman-Hilbert space A 2 α B as well as on the Hardy H 2 B space.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 next characterization of compactness is proved in a standard way, hence we omit its proof see, e.g., 34 .
The following lemma is a slight modification of Lemma 2.5 in 8 and is proved similar to Lemma 1 in 35 .
The following lemma is folklore and in the next form it can be found in 36 .
Lemma 2.3.Assume that 0 < p, q < ∞, φ is normal, and m ∈ N. Then for every f ∈ H B the following asymptotic relationship holds: Proof.Let g R m−1 f and z ∈ B. By the definition of the radial derivative, the Cauchy-Schwarz inequality and the Chauchy inequality, we have that From 2.3 with m → m − 1 we easily obtain the following inequality see, e.g., 8, Lemma 2.1 :

2.6
From 2.5 and 2.6 and the asymptotic relations Then, where and where p m j s , j 2, . . ., m − 1 are nonnegative polynomials for s > 0.
Proof.We prove the lemma by induction.For m 1, which is formula 2.9 with P 1 w w.Assume 2.9 is true for every m ∈ {1, . . ., l}.Taking the radial derivative operator on equality 2.9 with m l, we obtain This section characterizes the boundedness and compactness of R m u,ϕ : Proof.Assume 3.1 holds.Then by Lemma 2.4 for each f ∈ H p, q, φ , we have that Taking the supremum over the unit ball in 3.3 and using 3.1 the boundedness of operator R m u,ϕ : H p, q, φ → H ∞ μ follows and

3.4
Now assume that operator R m u,ϕ : H p, q, φ → H ∞ μ is bounded.By using the test functions that is, for each j 1, . . ., n, holds z j H p,q,φ < ∞.

3.9
It is known that L 1 : sup w∈B f w H p,q,φ < ∞ see 8, Theorem 3.3 .From this, using the boundedness of R m u,ϕ : H p, q, φ → H ∞ μ and by Lemma 2.5, we have that for each a ∈ B

3.10
From 3.10 , we have that

3.11
On the other hand, from 3.8 and since φ is normal, we obtain sup

3.13
From 3.4 and 3.13 asymptotic relationship 3.2 follows, finishing the proof of the theorem.

3.16
On the other hand, by 3.10 , we have 3.17 From 3.16 and 3.17 , equality 3.14 easily follows.
Conversely, assume that R m u,ϕ : H p, q, φ → H ∞ μ is bounded and 3.14 holds.From the proof of Theorem 3.1 we know that 3.1 holds.On the other hand, from 3.14 , we have that, for every ε > 0, there is a δ ∈ 0, 1 , such that

Abstract and Applied Analysis 9
Assume f k k∈N is a sequence in H p, q, φ such that sup k∈N f k H p,q,φ ≤ L and f k converges to 0 uniformly on compact subsets of B as k → ∞.Let K {z ∈ B : |ϕ z | ≤ δ}.Then from 3.18 , and by Lemma 2.4, it follows that

3.20
Since f k k∈N converges to zero on compact subsets of B as k → ∞, by Cauchy's estimates it follows that the sequence |∇R m−1 f k | k∈N also converges to zero on compact subsets of B as k → ∞, in particular Using these facts and letting k → ∞ in 3.20 , we obtain that lim sup Since ε is an arbitrary positive number it follows that the last limit is equal to zero.Applying Lemma 2.1, the implication follows.
μ is bounded, and as in the proof of Theorem 3.1, by taking the test functions f j z z j , j 1, . . ., n, we obtain 3.23 .

Abstract and Applied Analysis
Conversely, assume that the operator R m u,ϕ : H p, q, φ → H ∞ μ is bounded and condition 3.23 holds.Then, for each polynomial p, we have from which along with condition 3.23 it follows that R m u,ϕ p ∈ H ∞ μ,0 .Since the set of all polynomials is dense in H p, q, φ , we see that for every f ∈ H p, q, φ there is a sequence of polynomials p k k∈N such that lim k → ∞ f − p k H p,q,φ 0.

3.25
From this and by the boundedness of the operator R m u,ϕ : H p, q, φ → H ∞ μ , we have that 2 n/q m 0.

3.27
Proof.From 3.27 , we see that 3.1 hold.This fact along with 3.3 implies that the set . By taking the supremum in 3.3 over the unit ball in H p, q, φ , using 3.27 and applying Lemma 2.2 we obtain that the operator R m u,ϕ : H p, q, φ → H ∞ μ,0 is compact.If R m u,ϕ : H p, q, φ → H ∞ μ,0 is compact, then by Theorem 3.2, we have that condition 3.14 holds, which implies that for every ε > 0 there is an r ∈ 0, 1 such that As in Theorem 3.3, we have that 3.23 holds.Thus there is a σ ∈ 0, 1 such that If |ϕ z | ≤ r and σ < |z| < 1, then from 3.29 , we obtain Using 3.30 and the fact that from 3.28 , we have  ϕ j z 2β j dv α z .

4.6
We also have that Similar to Theorem 4.1 the following result regarding the case of the Hardy space is proved.We omit the proof.

Let z z 1 z |β|≥0 β a β z β 1. 1 be
, . . ., z n and w w 1 , . . ., w n be points in C n , z, w n k 1 z k w k , and |z| z, z .Let B {z ∈ C n : |z| < 1} be the open unit ball in C n , ∂B its boundary, and H B the class of all holomorphic functions on B. For an f ∈ H B with the Taylor expansion f z |β|≥0 a β z β , let Rf the radial derivative of f, where β β 1 , β 2 , . . ., β n is a multi-index, |β| β 1 • • • β n and z β z 2 Abstract and Applied Analysis

Theorem 4 . 1 .Proof.
Let m ∈ N 0 .Then Hilbert-Schmidt norm of the operator R m u,ϕ on A 2 α , α > −1 is By using the definition of the Hilbert-Schmidt norm and the monotone convergence theorem, we have

Theorem 4 . 2 .
Let m ∈ N 0 .Then, Hilbert-Schmidt norm of the operator R m u,ϕ on H 2 , is