Composition Operators on Weighted Zygmund Spaces of the First Loo-keng Hua Domain

: Let HE I denote the first Loo-keng Hua domain. In this paper, we obtain many elementary results on HE I by the continuous and careful discussions. In some applications, we obtain some necessary conditions or sufficient conditions for the boundedness and compactness of the composition operators on weighted Zygmund space defined on HE I .


Introduction
In this paper, the first Loo-Keng Hua domain we need is defined by HE I (N 1 , . . ., N r ; m, n; p 1 , . . ., p r ) = ξ j ∈ C N j , Z ∈ ℜ I (m, n) : where ξ j = (ξ j1 , . . ., ξ jN j ), j = 1, . . ., r, Z denotes the conjugate of the matrix Z, Z T denotes the transpose of Z, N 1 , . . ., N r , m, n are positive integers, p 1 , . . ., p r are positive real numbers, and ℜ I (m, n) = Z ∈ C m×n : I − Z ZT > 0 denotes the first Cartan domain.Without loss of generality, we assume that N j = 1, that is, It is well known that the Bergman kernel function plays an important role in several complex variables.But, for which domains can the Bergman kernel function be computed by explicit formulas?In general, this is a difficult problem.However, Yin and Roos constructed four kinds of domains called the Cartan-Hartogs domains with explicit Bergman kernel functions in 1998.Yin continuously generalized them from that time and constructed four kinds of domains called the Loo-keng Hua domains in 2000.The Loo-keng Hua domains unify the studies of the symmetric classical domains and Egg domains in the theory of several complex variables (see [1]).Besides some special cases (for example, the unit ball), generally speaking, the Loo-keng Hua domains are not transitive (see [2]).
Let α > 0, B n = {z ∈ C n : |z| < 1} be the open unit ball of C n and H(B n ) the space of all holomorphic functions on B n .The weighted Zygmund space on B n , usually denoted by Z α (B n ), consists of all f ∈ H(B n ) such that The quantity Z f is a seminorm of Z α (B n ).Under the norm ) is a Banach space.Actually, there are several equivalent norms on Z α (B n ) (see [3]).
We also usually use this space defined on the unit disk.For such space and some concrete operators, see, for example, [4][5][6][7][8] and the references therein.Now, we try to extend this definition to the domain HE I .We say that f ∈ H(HE I ) is in the weighted Zygmund space on HE I , denoted by Z α (HE I ), if where Z α (HE I ) is a Banach space under the norm Let Ω be a domain of C n and H(Ω) the space of all holomorphic functions on Ω.For φ a holomorphic self-map of Ω, the composition operator C φ on the subspaces of H(Ω) is defined by Here, I will make a brief statement on why people are willing to study composition operators on holomorphic function spaces.The study of composition operators lies at the interface of holomorphic function theory and operator theory.Research on composition operators is part of what can be called "concrete" operator theory.The goal of the study of composition operators will be to answer the questions operator theory demands we ask, questions of boundedness, compactness, spectrum, and so on.The question of the boundedness of composition operators is a basic but subtle one.It has been proved that in the most important classical spaces, such as Hardy spaces, all holomorphic maps of the disk into itself give bounded composition operators, but in function spaces of several variables, the classical spaces have unbounded composition operators, and even the general principles are unclear.Much of the study of the compactness of composition operators has been driven by a desire to relate the compactness of C φ to geometric properties of φ.For example, the first and simplest result in this spirit is an observation: On the Hardy space H p (D) if φ(D) ⊂ D then C φ is compact.All mentioned and unmentioned significance of the study can be found in [9] or in other literature.
Since there is great significance to this research, there have been many studies of composition operators and some extensions of such operators on or between holomorphic function spaces of common domains such as the unit disk, the unit ball, the unit polydisk or the (half) complex plane (see, for example, [9][10][11][12][13][14][15][16] and the references therein).In recent years, there has been a bit of interest in some more complex domains (see, for example, [17,18] for the classical bounded symmetric domain).More recently, Su and his team studied the composition operators from u-Bloch spaces to v-Bloch spaces on the first Cartan-Hartogs domains in [19] and on the first Loo-keng Hua domains in [20], respectively.In these works, they used a lot of methods and techniques in matrix and constructed the complicated functions to characterize the boundedness and compactness of the operator C φ in terms of geometric properties of φ.These works also contribute to the understanding of HE I and its applications in operator theory.
Motivated by the works of [19,20], we consider the composition operators on the weighted Zygmund spaces on the first Loo-keng Hua domain, and some necessary conditions or sufficient conditions for the boundedness and compactness are obtained.One will see that it is more difficult than the cases of the weighted Bloch spaces.We hope that our studies can bring more attention to composition operators on holomorphic function spaces of such domains.

Some Elementary Lemmas
From some direct calculations, we obtain Lemma 1, which gives the expressions of partial derivatives of composition functions.Lemma 1.Let φ be a holomorphic self-map of HE I .Then for f ∈ H(HE I ) it follows that and From Lemma 1, we obtain the evaluation of H f •φ (Z, ξ) as follows.
Lemma 2. For each f ∈ H(HE I ) and (Z, ξ) ∈ HE I , it follows that where C = max{m 2 n 2 , mnr, r 2 } and Proof.By Lemma 1 and the elementary inequality we have where C = max{24m 2 n 2 , 24mnr, 24r 2 }.Now, we explain how we obtain the inequality (2).We see that in (1), there are sixteen terms with the second partial derivatives of f and each term is less than 4 , and there are eight terms with the first partial derivatives of f and each term is less than From this, it follows that (2) holds.The proof is completed.
Next, we obtain the point evaluation estimate for the functions in Z α (HE I ).
By using Lemma 3, we obtain the following result.Proof.From the proof of Lemma 3, it is easy to see that |(Z, ξ)| ≤ √ mn + r.Then, for each f ∈ Z α (HE I ) and (Z, ξ) ∈ HE I , we have The proof is completed.
In order to characterize the compactness, we need the following result, which is similar to Proposition 3.11 in [9].Therefore, the proof is omitted.Lemma 5. Let φ be the holomorphic self-map of HE I .Then the bounded operator C φ on Z α (HE I ) is compact if and only if for every bounded sequence { f k } in Z α (HE I ) such that f k → 0 uniformly on every compact subset of HE I as k → ∞, it follows that Since the proofs are essential, the same as that of Lemmas 2.4 and 2.5 in [19], we omit the proofs of the next two results.
Let S be an fixed element in ℜ I (m, n).
On ℜ I (m, n) we define the function The following two results were obtained in [21].
For the sake of convenience, we define The following result was obtained in [19].
Lemma 10.There exists a positive constant C independent of Z, S ∈ ℜ We also need the following two results (see [21]).The following result was obtained in [20,22].
Let α ̸ = 1 and (S, t) ∈ HE I .On HE I define the function By using above lemmas, we will see that f (S,t) ∈ Z α (HE I ).Lemma 14.Let α ̸ = 1 and p j ≥ 2 for j = 1, 2, . . ., r.Then f (S,t) ∈ Z α (HE I ), and there exists a positive constant C such that Proof.We write c α = 1 − 2α.By a direct calculation, we have and By ( 9) and ( 10), we see that A S,ij (Z)A S,kl (Z) and j ̸ = k By (10) and a direct calculation, we also have Then by ( 11)-( 14) we have A S,ij (Z) A S,kl (Z) A S,ij,kl (Z) From Lemmas 10-12, it follows that there exists a positive constant C independent of (Z, ξ), (S, t) ∈ HE I such that F(Z, S, ξ, t) ≤ C. Then from Lemma 13 and (15), we have sup On the other hand, it is clear that and From ( 16)-( 18), the desired result follows.The proof is completed.
If α = 1, then on HE I we consider the function For the function g (S,t) (Z, ξ), we also have the following result, whose proof is similar to that of Lemma 14.So, the proof is omitted.

Composition
Operators on Z α (HE I ) Z α (HE I ) Z α (HE I ) Now, we begin to study the boundedness and compactness of the composition operators on Z α (HE I ).We first have the following result for the boundedness.
Conversely, if the operator C φ is bounded on Z α (HE I ), then A W,uv (W)A W,pq (W) A W,uv (W)A W,pq (W) A W,uv (W)A W,pq (W) Proof.By Lemmas 2 and 3, for all f ∈ Z α (HE I ) and (Z, ξ) ∈ HE I , we have By Lemma 4, we have From Lemma 3, it follows that Similar to (21), we also can obtain Then, by ( 19)-( 22) we have (23) shows that C φ is bounded on Z α (HE I ).
Conversely, assume that the operator C φ is bounded on Z α (HE I ) with for all f ∈ Z α (HE I ).
First, we consider the case of α ̸ = 1.Let (P, λ) = φ(S, t), where (S, t) is a fixed point in HE I .We will use the function f (P,λ) defined by By Lemma 14, we know that f (P,λ) ∈ Z α (HE I ).By a calculation, we have A P,uv (W)A P,pq (W)  A P i (P i ) − ∥λ i ∥ 2 α H(S i , t i ) = 0.
The proof is completed.
Funding: This work was supported by the Sichuan Science and Technology Program (2022ZYD0010).

Lemma 3 .
There exists a positive constant C independent of f ∈ Z α (HE I ) and (Z, ξ) ∈ HE I such that

Lemma 11 .Lemma 12 .
There exists a positive constant C independent of Z, S ∈ ℜ I (m, n) such that m ∑ i,p=1 n ∑ j,q=1 A S,ij,pq (Z) 2 ≤ C.There exists a positive constant C independent of Z, S ∈ ℜ I (m, n) such that |A S (Z)| ≤ C.