Weighted Composition Operators and Integral-Type Operators between Weighted Hardy Spaces on the Unit Ball

We study the boundedness and compactness of the weighted composition
operators as well as integral-type operators between weighted Hardy spaces on the unit ball.


Introduction
Let B denote the open unit ball of the n-dimensional complex vector space C n , ∂B its boundary, and let H B denote the space of all holomorphic functions on B. For 0 < p < ∞ and α ≥ 0 we define the weighted Hardy space H p α B as follows: where dσ is the normalized Lebesgue measure on ∂B see, also 1 , as well as 2 , for an equivalent definition of the space .Note that for α 0 the weighted Hardy space becomes the Hardy space H p B .We define the norm • H p α on this space as follows: With this norm H p α B is a Banach space when 1 ≤ p < ∞.For a related space on the unit polydisk; see 3 .In this paper, we investigate two types of operators acting between weighted Hardy spaces.
Let ϕ be a holomorphic self-map of B and u ∈ H B .Then ϕ and u induce a weighted composition operator uC ϕ on H B which is defined by uC ϕ f u f • ϕ .This type of operators has been studied on various spaces of holomorphic functions in C n , by many authors; see, for example, 4 , recent papers 5-17 , and the references therein.
Let g ∈ H D and ϕ be a holomorphic self-map of the open unit disk D in the complex plane.Products of integral and composition operators on H D were introduced by S. Li and S. Stević in a private communication see 18-21 , as well as papers 22 and 23 for closely related operators as follows: 1.3 1.4 In 24 the first author of this paper has extended the operator in 1.4 in the unit ball settings as follows see also 25, 26 .Assume g ∈ H B , g 0 0, and ϕ is a holomorphic self-map of B, then we define an operator on the unit ball as follows: If n 1, then g ∈ H D and g 0 0, so that g z zg 0 z , for some g 0 ∈ H D .By the change of variable ζ tz, it follows that Thus the operator 1.5 is a natural extension of operator J g C ϕ in 1.4 .For related operators see 27-33 as well as the references therein.
In this paper we study the boundedness and compactness of the weighted composition operators as well as the integral-type operator P g ϕ , between different weighted Hardy spaces on the unit ball.
Throughout 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.Moreover, if both a b and b a hold, then one says that a b.

Weighted Composition Operators
This section is devoted to studying weighted composition operators between weighted Hardy spaces.Weighted composition operators between different Hardy spaces on the unit ball were previously studied in 15, 34 , while the composition operators on the unit ball were studied in 35, 36 .For the case of the unit disk see also 37 .Before we formulate the main results in this section we quote several auxiliary results which will be used in the proofs of these ones. where

2.2
Hence we have the desired inequality.
Recall that an f ∈ H B has the homogeneous expansion For the homogeneous expansion of f and any integer j ≥ 1, let and K j I − R j where If f is the identity operator.Note that K j is compact operator on H p α B for each j ∈ N.
Lemma 2.2.If 1 < p < ∞, then R j converges to 0 pointwise in the Hardy space H p B as j → ∞.
Lemma 2.2 and the uniform boundedness principle show that {R j } is an uniformly bounded sequence in H p B .
The following lemma is proved similar to 4, Lemma 3.16 .We omit its proof.

Lemma 2.3. If uC ϕ is bounded from
where denote the essential norm and the operator norm, respectively.Lemma 2.4.Let 0 < p ≤ q < ∞.Suppose that μ is a positive Borel measure on B which satisfies for some positive constant C 1 .Then there exists a positive constant C 2 which depends only on p, q, and the dimension n such that

2.13
Next we assume

2.14
Fix r ∈ 0, 1 and R ∈ 0, 1 , respectively.For ζ ∈ ∂B and t, 0 Since the function f w z , which is defined by 2.11 for this w, satisfies for all z ∈ B ζ, t , we have

2.16
By the same argument, the function f w R z gives the following estimate:

2.17
Now we need to prove that there exists a positive constant C such that

2.19
Hence for C max{4 q α n /p , C n 9 q α n /p }, we have the inequality in 2.18 .
For f ∈ H p α B the dilate function f R belongs to the ball algebra, and so f R is in the Hardy space H p B .Hence Lemma 2.4 gives for some positive constant C and all R ∈ 0, 1 .This implies that and so we have for all R ∈ 0, 1 .By Lemma 2.1 we have

2.23
This completes the proof.
The following proposition is proved in a standard way; see, for example, the proofs of the corresponding results in 4, 32, 33, 39 .Hence we omit its proof.Proposition 2.6.Let 0 < p, q < ∞ and α, β ≥ 0. Suppose that u ∈ H B and ϕ is a holomorphic selfmap of B which induce the bounded operator uC ϕ : In the proof of Theorem 2.8, we need the following lemma.
for some positive constant C α, p, n which depends on α, p, and n.This inequality implies that the family {f w } w∈B is also bounded in A

2.26
Proof.To prove a lower estimate

2.28
This inequality and 2.12 give the lower estimate for

2.29
Then we can choose R 0 ∈ 0, 1 such that By the same argument as in the proof of inequality 2.20 in Theorem 2.5, we obtain that where the positive constant C is independent of r, R and a positive integer j.Since f R is in the ball algebra, Lemma 2.2 gives

2.32
Combining this with 2.31 , we have and so we have

2.35
Since ε > 0 is arbitrary, this estimate and Lemma 2.3 imply

2.36
which completes the proof.
Remark 2.9.In the above proof, we used Lemma 2.2.This lemma required the assumption 1 < p < ∞.Hence we cannot have an upper estimate for uC ϕ e,H p α B → H q β B in the case 0 < p ≤ 1.However, Proposition 2.6 shows that the compactness of uC ϕ : 2.37

Integral-Type Operators
Here we study the boundedness and compactness of the integral-type operators P g ϕ between weighted Hardy spaces on the unit ball.
For f ∈ H B with the Taylor expansion f z |γ|≥0 a γ z γ , let Rf z |γ|≥0 |γ|a γ z γ be the radial derivative of f.
The following lemma was proved in 24 see also 25 .
Lemma 3.1.Assume that ϕ is a holomorphic self-map of B, g ∈ H B and g 0 0. Then for every f ∈ H B one holds A positive continuous function ω on the interval 0, 1 is called normal 40 if there is a δ ∈ 0, 1 and a and b, 0 < a < b such that ω r 1 − r a is decreasing on δ, 1 and lim where Proof.The proof of the lemma in the case 1 ≤ q ≤ ∞ can be found in 27, Theorem 2 .However, due to an overlook, the proof for the case q ∈ 0, 1 has a gap.Hence we will give a correct proof here in the case.We may assume that f 0 0, otherwise we can consider the functions h z f z − f 0 .Also we may assume that δ 0, to avoid some minor technical difficulties.
By 27, Lemma 1 , for each fixed q ∈ 0, 1 , there is a positive constant C depending only on q and the dimension n such that for every r ∈ 0, 1 and f ∈ H B such that f 0 0.
From 3.5 and the fact that ω is normal, we have

3.6
By 40, page 291, Lemma 6 there exists a positive constant C such that for every r ∈ 0, 1 .Combining this with 3.6 , we have

3.8
The reverse inequality is proved by the following inequality: and the fact that ω r ω 1 r/2 for ω normal see 27 .Hence, we obtain the result for the case m 1.
For m ≥ 2 it should be only noticed that 1 − r m ω r is still normal, that R m f 0 0 for every integer m ≥ 1, and use the method of induction.Theorem 3.3.Let 0 < p ≤ q < ∞ and α, β > 0. Suppose that g ∈ H B with g 0 0 and ϕ is a holomorphic self-map of B. Then P g ϕ :

3.11
The assumption g 0 0 implies P g ϕ f 0 0, and Lemma 3.1 shows R P g ϕ f gC ϕ f.Hence we obtain sup 0<r<1 1 − r β M q q P g ϕ f, r sup 0<r<1 1 − r β q M q q gC ϕ f, r , 3.12 and so we obtain . This implies that the boundedness of P g ϕ : is a necessary and sufficient condition for the boundedness of P Theorem 3.5.Let 0 < p ≤ q < ∞ and α, β > 0. Suppose that g ∈ H B with g 0 0 and ϕ is a holomorphic self-map of B which induce the bounded operator P g ϕ : Proof.First we assume that condition 3.14 holds.Take a bounded sequence {f j } j∈N ⊂ H p α B which converges to 0 uniformly on compact subsets of B. Theorem 2.8 and the remark in Section 2 show that gC ϕ : 3.15 From 3.15 and since P g ϕ f j q H q β gC ϕ f j q H q β q , we have that P g ϕ f j q H q β → 0 as j → ∞.By Proposition 3.4, we see that is compact.To prove the necessity of the condition in 3.14 , we consider the family of test functions f w which is defined by 2.11 .Hence we have → 0 as |w| → 1 − .This fact along with 3.16 implies the condition in 3.14 , finishing the proof of the theorem.Theorem 3.6.Let 1 < p ≤ q < ∞ and α, β > 0. Suppose that g ∈ H B with g 0 0 and ϕ is a holomorphic self-map of B which induce the bounded operator 3.17 Proof.To prove a lower estimate, we take an arbitrary compact operator K : H p α B → H q β B .Since Lemma 2.7 implies that the family of functions f w defined by 2.11 converges to 0 weakly in H p α B as |w| → 1 − , we obtain

3.18
Combining this with 3.16 , we have which is a lower estimate.By some modification of Lemma 2.3 and the application of Lemmas 3.1 and 3.2, we get

3.20
As in the proof of Theorem 2.8, we obtain lim inf

3.21
and so we have an upper estimate for

The Case
When p ∞ and α > 0, we define the weighted-type space H ∞ α B as follows: In this case, the operator norm Proof.By the definition of the space so it follows from Lemma 3.1 and Lemma 3.2 that Hence we obtain Now we prove the reverse inequality.For w ∈ B, we put Note that f w ∈ H ∞ α,0 B for each w ∈ B and moreover sup w∈B f w H ∞ α ≤ 1.When ϕ z / 0, we have for all t ∈ 0, 1 .Letting t → 1 − in 4.8 , we have

4.9
For the constant function 1 ∈ H ∞ α,0 B we obtain Inequality 4.10 shows that the estimate in 4.9 also holds when ϕ z 0. Hence, from 4.9 we obtain which along with the obvious inequality 4.12 completes the proof of the theorem.
For the compactness of , we can also prove the following proposition which is similar to Proposition 2.6.Proposition 4.2.Let α, β > 0. Suppose that g ∈ H B with g 0 0 and ϕ is a holomorphic self-map of B which induce the bounded operator

4.13
In particular, 4.14 Proof.First we consider the family {f w } w∈B where

4.15
We can easily check that f w ∈ H ∞ α,0 B , f w H ∞ α ≤ 1 for all w ∈ B and f w → 0 uniformly on compact subsets of B as |w| → 1 − .Hence 40, page 296, Theorem 2 implies that f w → 0 weakly in H ∞ α,0 B as |w| → 1 − .If ϕ ∞ < 1, then as in the proof of 26, Theorem 3 it can be seen that the operator On the other hand, the limit in 4.13 is vacuously equal to zero, from which the result follows in this case.If ϕ ∞ 1, then take a sequence {ϕ z j } j∈N in B with |ϕ z j | → 1 as j → ∞ and put F j z f ϕ z j z for each j ∈ N. Then {F j } j∈N is a bounded sequence in H ∞ α,0 B and {F j } j∈N converges to 0 weakly in H ∞ α,0 B , as j → ∞.Hence for every compact operator

4.18
Combining this with the estimate

4.19
Next we prove an upper estimate.Assume that {r l } l∈N ⊂ 0, 1 is a sequence which increasingly converges to 1.For this {r l } l∈N , we define the operators defined by As in the proof of 26, Theorem 3 , Proposition 4.2 shows that P Discrete Dynamics in Nature and Society and fix ε > 0. Then we can choose R ∈ 0, 1 such that

4.25
Since the boundedness of P

4.27
On the other hand, the monotonicity of M ∞ f, r shows

4.28
Thus we have Letting l → ∞ and ε → 0, we have

Lemma 2 . 7 .
Let 1 < p < ∞, α ≥ 0, and f w be the family of test functions defined in 2.11 .Then f w → 0 weakly in H p α B as |w| → 1 − .Proof.The family {f w } w∈B is bounded in H p α B and f w → 0 uniformly on compact subsets of B as |w| → 1 − .By the definitions of the space H p α B and the norm • H p α , we see that H p α B is a subspace of the weighted Bergman space A p α B and

pα
B .Note also that the family converges to 0 uniformly on compact subsets of B as |w| → 1 − .Hence f w → 0 weakly in A p α B as |w| → 1 − .In order to prove that f w → 0 weakly in H p α B as |w| → 1 − , we take an arbitrary bounded linear functional Λ on H p α B .By the Hahn-Banach theorem, Λ can be extended to a bounded linear functional Λ on A p α B so that Λ f w Λ f w for all w ∈ B. Since f w → 0 weakly in A p α B as |w| → 1 − , we have Λ f w Λ f w → 0 as |w| → 1 − , and so

6 . 3 . 4 .
completes the proof.The next proposition is proved similar to Proposition 2.Proposition Let 0 < p, q < ∞, and α, β > 0. Suppose that g ∈ H B with g 0 0 and ϕ is a holomorphic self-map of B which induce the bounded operator P is compact if and only if for every bounded sequence {f j } j∈N in H p α B which converges to 0 uniformly on compact subsets of B, {P g ϕ f j } j∈N converges to 0 in H q β B .

3 . 16 for
all w ∈ B. Since {f w } w∈B is a bounded sequence in H p α B and f w → 0 uniformly on compact subsets of B as |w| → 1 − , the compactness of P

1 −|<1 1 − |z| β 1 g z 1 − ϕ z α ≤ 2 M 2 ε . 4 . 29 From 4 .
|z| β 1 g z f ϕ z − f r l ϕ z ≤ 2 supR<|ϕ z 23 , 4.27 , 4.29 , and the compactness of P g r l ϕ , we obtain ∂B and t > 0. By the estimate 2.16 , we see that the inequality 2.18 is true for all t ∈ 0, t R .Thus we assume t > t R .By the same argument as in 36, pages 241-242, proof of Theorem 1.1 , we see that the inequality 2.17 shows that there exists a positive constant C n which depends only on the dimension n such that test functions f w defined in 2.11 .The family {f w } w∈B is bounded in H p α B , say by L, and f w → 0 uniformly on compact subsets of B as |w| → 1 − .Thus by Lemma 2.7 we have that f w → 0 weakly in is compact if and only if for every bounded sequence {f j } j∈N B is bounded, we see that g ∈ H ∞ β 1,0 B .By a standard argument as in the proof of 26, Corollary 3 , we have Let α, β > 0. Suppose that g ∈ H B with g 0 0 and ϕ is a holomorphic self-map of B such thatP g ϕ : H ∞ α B or H ∞ α,0 B → H ∞ β,0 B is bounded.Then P Hence we obtain the following characterization for the compactness of the operator P g ϕ :H ∞ α B or H ∞ α,0 B → H ∞ β,0 B .Corollary 4.4.