Research Article On a New Integral-Type Operator from the Weighted Bergman Space to the Bloch-Type Space on the Unit Ball

We introduce an integral-type operator, denoted by 𝑃𝑔𝜑, 
on the space of holomorphic functions on the unit ball 𝔹⊂ℂ𝑛, which is an extension of the product of composition and 
integral operators on the unit disk. The operator norm of 
𝑃𝑔𝜑 from the weighted Bergman space 𝐴𝑝𝛼(𝔹) to the 
Bloch-type space ℬ𝜇(𝔹) or the little Bloch-type 
space ℬ𝜇,0(𝔹) is calculated. The compactness 
of the operator is characterized in terms of inducing functions 
𝑔 and 𝜑. Upper and lower bounds for the essential norm 
of the operator 𝑃𝑔𝜑∶𝐴𝑝𝛼(𝔹)→ℬ𝜇(𝔹), 
when 𝑝>1, are also given.


Introduction
Let B be the open unit ball in the complex vector space C n , S ∂B its boundary, D the open unit disk in the complex plane C, dV z the Lebesgue measure on B, dV α z c α 1 − |z| 2 α dV z , where α > −1 and where the constant c α is chosen such that V α B 1, dσ the normalized rotation invariant measure on S that is, σ S 1 , H B the class of all holomorphic functions on the unit ball and H ∞ H ∞ B the space of all bounded holomorphic functions on B with the norm 1  we will also use the notation A p −1 for the Hardy space H p .A positive continuous function φ on 0, 1 is called normal see 2 if there is δ ∈ 0, 1 and a and b, 0 < a < b such that

1.10
From now on if we say that a function μ : B → 0, ∞ is normal, we will also assume that it is radial, that is, where μ is normal.For μ z 1 − |z| 2 β , β > 0, we obtain the weighted space H ∞ β H ∞ β B see, e.g., 3-5 .
The little weighted space The class of all f ∈ H B such that where μ is normal, is called the Bloch-type space, and is denoted by the Bloch-type space becomes a Banach space.The little Bloch-type space B μ,0 is a subspace of The α-Bloch space B α is obtained for μ z 1 − |z| 2 α , α ∈ 0, ∞ see, e.g., 6-11 .For α 1 the space B 1 B becomes the classical Bloch space.
Let ϕ be a holomorphic self-map of B. For any f ∈ H B , the composition operator is defined by It is of interest to provide function theoretic characterizations when ϕ induces bounded or compact composition operators on spaces of holomorphic functions.For some classical results in the topic see, e.g., 12 .For some recent results see, for example, 3-5, 7, 13-23 and the references therein.
Let g ∈ H D and ϕ be a holomorphic self-map of D. For f ∈ H D , products of integral-type and composition operator are defined as follows: 1.17 When ϕ z z, operators in 1.17 are reduced to the integral operator introduced in 24 .For some other results on the operator; see, for example, 25, 26 , and related references therein.Some results on related integral-type operators on spaces of holomorphic functions in C n can be found, for example, in 27-41 see also the references therein .
In 42 , among other results, we proved the following theorem regarding the boundedness of the operator J g C ϕ : One of the interesting questions is to extend operators in 1.17 in the unit ball settings and to study their function theoretic properties on spaces of holomorphic functions on the unit ball in terms of inducing functions.
Assume that g ∈ H B , g 0 0, and ϕ is a holomorphic self-map of B. We introduce the following important integral-type operator on the space of holomorphic functions on B: First note that when n 1, the operator is reduced to an operator of the form as the second operator in 1.17 .Indeed, since g ∈ H D and g 0 0, it follows that g z zg 0 z , z ∈ D for some g 0 ∈ H D .By using this fact and the change of variables ζ tz, we obtain Hence operator 1.19 is a natural extension of the second operator in 1.17 .Now we formulate the following big research project related to the operator P g ϕ .
Research project 1.Let X and Y be two Banach spaces of holomorphic functions on the unit ball in C n e.g., the weighted Bergman space A p α , the Bloch-type space B μ , the Hardy space H p space, the weighted space H ∞ μ , the Besov space B p , BMOA etc. Characterize the boundedness, compactness, essential norms, and other operator theoretic properties of the operator P g ϕ : X → Y in terms of function theoretic properties of inducing functions ϕ and g.
Another interesting question is to find the exact value of the norm of operators on spaces of holomorphic functions.Majority of papers in the area only find asymptotics of the operator norm of certain linear operators on some spaces of holomorphic functions.There are a few papers which calculate the operator norm of these operators.Recently in 4 we calculated operator norm of the weighted composition operator uC ϕ mapping the Bloch space B to the weighted space H ∞ μ , which motivates us to find the norms of weighted composition and other closely related operators between various spaces of holomorphic functions.

Research project 2.
Let X and Y be two Banach spaces of holomorphic functions as in Research project 1. Calculate the operator norm of P g ϕ : X → Y in terms of inducing functions ϕ and g.
In this paper, among other results, we will calculate the operator norm of P g ϕ : A p α B → B μ B .We will also characterize the boundedness, compactness, and the essential norm of the operator.These results partially solve problems posed in the above research projects.
Throughout the paper, C denotes a positive constant not necessarily the same at each occurrence.The notation A B means that there is a positive constant C such that A/C ≤ B ≤ CA.

Auxiliary results
In this section, we give several auxiliary results, which are used in the proofs of the main results.
The following result can be found in 44 .For closely related results, see also 11, 45-52 and the references therein.
The following lemma can be proved similar to 53, Lemma 1 .
Lemma 2.4.Suppose that μ is normal.A closed set K in B μ,0 is compact if and only if it is bounded and The following lemma is related to 32, Lemma 1 and 34, Lemma 2 .
Lemma 2.5.Assume that f, g ∈ H B and g 0 0. Then Proof.Since the function f ϕ z g z is holomorphic and g 0 0, it has the Taylor expansion in the following form α / 0 a α z α .Then as claimed.
from which it follows that Now we prove the reverse inequality.For w ∈ B fixed, set We have that f w A p α 1, for each w ∈ B. For α > −1 this fact is well known.The proof for the case α −1 could be less known, and we give a proof of it for the lack of a specific reference and for the benefit of the reader.Let z rζ, ζ ∈ S, then we have where we have used the following formula see, e.g., 1 5.1 Proof.First assume that the operator 1 is vacuously satisfied.Hence, assume that ϕ ∞ 1 and assume to the contrary that 5.1 does not hold.Then there is a sequence z k k∈N satisfying the condition |ϕ

5.2
For w ∈ B fixed, set On the other hand, by Lemma 2.5 and 5.2 , we obtain for every k ∈ N, which contradicts with 5.4 .Now assume that 5.1 holds.Then for every ε > 0 there is an r ∈ 0, 1 such that when On the other hand, since the operator Then by Lemma 2.1 and 5.6 , for r < |ϕ z | < 1, we obtain From 5.7 and 5.8 , it follows that P g ϕ h k B μ → 0 as k → ∞, from which the compactness of the operator P g ϕ : A p α → B μ follows.

Compactness of the operator
This section characterizes the compactness of the operator from which the result follows in this case.Now assume ϕ ∞ 1.By using the test functions F k z f ϕ z k z , k ∈ N, defined in 5.3 we obtain that condition 5.1 holds, which implies that for every ε > 0, there is an r ∈ 0, 1 such that for r < |ϕ z | < 1, condition 5.6 holds.

6.4
From 5.6 and 6.4 , condition 6.1 follows.Now assume that condition 6.1 holds.Then the quantity M in Theorem 3.1 is finite.Using this fact and the following inequality Let X and Y be Banach spaces, and let L : X → Y be a bounded linear operator.The essential norm of the operator L : X → Y , denoted by L e,X→Y , is defined as follows: where • X→Y denote the operator norm.
From this definition and since the set of all compact operators is a closed subset of the set of bounded operators, it follows that operator L is compact if and only if L e,X→Y 0.
In this section, we study the essential norm of the operator P Proof.Assume that ϕ z k k∈N is a sequence in B such that |ϕ z k | → 1 as k → ∞.Note that the sequence f ϕ z k k∈N where f w is defined in 3.4 is such that f ϕ z k A p α 1 for each k ∈ N and it converges to zero uniformly on compacts of B. From this and by 11, Theorems 2.12 and 4.50 , it follows that f ϕ z k → 0 weakly in A p α , as k → ∞ here we use the condition p > 1 .Hence, for every compact operator K : Thus, for every such sequence and for every compact operator K : A p α → B μ , we have that

7.3
Taking the infimum in 7.3 over the set of all compact operators K : A p α → B μ , we obtain from which the first inequality follows.
In the sequel we prove the second inequality.Assume that r l l∈N is a sequence which increasingly converges to 1. Consider the operators defined by We prove that these operators are compact.Indeed, since |r l ϕ z | ≤ r l < 1, it follows that condition 5.1 in Theorem 5.1 is vacuously satisfied, from which the claim follows.
Recall that g ∈ H ∞ μ .Let ρ ∈ 0, 1 be fixed for a moment.Employing Lemma 2.1, and using the fact

When p ≥ 1 ,
p > 0, α > −1, consists of all f ∈ H B such that the weighted Bergman space with the norm • A p α becomes a Banach space.If p ∈ 0, 1 , it is a Frechet space with the translation invariant metric d

Corollary 3 . 2 . 4 .Theorem 4 . 1 .Theorem 5 . 1 .
of 3.8 does not depend on the space B μ we may replace it by B μ,0 the second equality in 3.1 also holds.Assume that p > 0, α ≥ −1, g ∈ H B , μ is normal, and ϕ is a holomorphic self-map of B. Then P g ϕ : A p α → B μ is bounded if and only if μ is bounded, then 3.10 follows from Theorem 3.1.If 3.10 holds, then the boundedness of P g ϕ : A p α → B μ follows from 3.3 .The boundedness of the operator P g ϕ : A p α → B μ,0Here we characterize the boundedness of the operatorP g ϕ : A p α → B μ,0 .Assume that p > 0, α ≥ −1, g ∈ H B , μ is normal, and ϕ is a holomorphic self-map of B. Then P g ϕ : A p α → B μ,0 is bounded if and only if P g ϕ : A p α → B μ is bounded and g ∈ H ∞ μμ is bounded and g ∈ H ∞ μ,0 .Then, for each polynomial p, we have μ z RP g ϕ p z μ z g z p ϕ z ≤ μ z g z p ∞ −→ 0, as |z| −→ 1 4.1 from which it follows that P g ϕ p ∈ B μ,0 .Since the set of all polynomials is dense in A p α , we have that for every f ∈ A p α there is a sequence of polynomials p k k∈N such that lim k→∞ f − p k A p α μ,0 .Since B μ,0 is a closed subset of B μ , the boundedness of P g ϕ : A p α → B μ,0 follows.Now assume that P g ϕ : A p α → B μ,0 is bounded.Then clearly P g ϕ : A p α → B μ is bounded.Taking the test function f z 1 ∈ A p α , we obtain g ∈ H ∞ μThis section is devoted to studying of the compactness of the operator P g ϕ : A p α → B μ .We prove the following result.Assume that p > 0, α ≥ −1, g ∈ H B , μ is normal, ϕ is a holomorphic self-map of B, and the operator P g ϕ : A p α → B μ is bounded.Then the operator P g ϕ : A p α → B μ is compact if and only if

. 3 where
f w is defined in 3.4 .Recall that f w A p α 1, for each w ∈ B. Then F k A p α 1, k ∈ Nand it is easy to see that F k → 0 uniformly on compacts of B as k → ∞.Hence, by Lemma 2.2 μ,0 is bounded.Then the operator P g ϕ : A p α → B μ,0 is compact if and only if μ,0 is bounded and as in Theorem 4.1 we have

. 6 which 9 If α > − 1 ,≤ C ρ 1 − 11 If α − 1 ,
follows by using the triangle inequality for the norm, the monotonicity of the integral meansf ϕ z − f r l ϕ z .7.then by using the mean value theorem, the subharmonicity of the partial derivatives of f and Lemma 2.3, we haveI l ≤ sup r l −→ 0 as l −→ ∞.7.then applying in 7.10 the known fact that for each compact K ⊂ B,sup w∈K ∇f w ≤ C f p , 7.12for some C independent of f see 11 , we obtain that 7.11 also holds in this case.Letting l → ∞ in 7.8 , using 7.11 , and then letting ρ → 1, the second inequality in 7.2 follows, finishing the proof of the theorem.Motivated by Theorem 7.1, we leave the following open problem.Open problem 1. Find the exact value of the essential norm of the operator P g ϕ : A p α → B μ .
For p > 0 the Hardy space H p H p B consists of all f ∈ H B such that p dσ ζ < ∞. 1.5It is well known that for every f ∈ H p the radial limitf * ζ : lim and ϕ is a holomorphic self-map of B. Then the operator P p α converging to zero uniformly on compacts of B, one has