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

Abstract

Let ${\mathrm{HE}}_I$ denote the first Loo-keng Hua domain. In this paper, we obtain many elementary results on ${\mathrm{HE}}_I$ 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 ${\mathrm{HE}}_I$.

1. Introduction

In this paper, the first Loo-Keng Hua domain we need is defined by

$${\mathrm{HE}}_I(N_1,\dots ,N_r;m,n;p_1,\dots ,p_r)=\{{\xi }_j\in {\mathbb{C}}^{N_j},Z\in {\Re }_I(m,n):{\displaystyle \sum _{j=1}^r}{\left|{\xi }_j\right|}^{2p_j}<\det (I-Z{\overline{Z}}^T)\},$$

where ${\xi }_j=({\xi }_{j1},\dots ,{\xi }_{j{N}_j})$, $j=1,\dots ,r$, $\overline{Z}$ denotes the conjugate of the matrix Z, $Z^T$ denotes the transpose of Z, $N_1,\dots ,N_r$, m, n are positive integers, $p_1,\dots ,p_r$ are positive real numbers, and

$${\Re }_I(m,n)=\left\{Z\in {\mathbb{C}}^{m\times n}:I-Z{\overline{Z}}^T>0\right\}$$

denotes the first Cartan domain. Without loss of generality, we assume that $N_j=1$, that is, ${\xi }_j\in \mathbb{C}$, $j=1,2,\dots ,r$, $\xi =({\xi }_1,\dots ,{\xi }_r)$ and ${\parallel \xi \parallel }^2={\displaystyle \sum _{j=1}^r}{\left|{\xi }_j\right|}^{2p_j}$. If no ambiguity can arise, we denote ${\mathrm{HE}}_I(N_1,\dots ,N_r;m,n;p_1,\dots ,p_r)$ by ${\mathrm{HE}}_I$.

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 $\alpha >0$, ${\mathbb{B}}^n=\{z\in {\mathbb{C}}^n:\left|z\right|<1\}$ be the open unit ball of ${\mathbb{C}}^n$ and $H({\mathbb{B}}^n)$ the space of all holomorphic functions on ${\mathbb{B}}^n$. The weighted Zygmund space on ${\mathbb{B}}^n$, usually denoted by ${\mathcal{Z}}^{\alpha }({\mathbb{B}}^n)$, consists of all $f\in H({\mathbb{B}}^n)$ such that

$${\mathcal{Z}}_f:=\underset{z\in {\mathbb{B}}^n}{sup}{(1-|z|}^2{)}^{\alpha }{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k=1}^n}\left|\frac{{\partial }^{2}f}{\partial z_j\partial z_k}(z)\right|<\infty .$$

The quantity ${\mathcal{Z}}_f$ is a seminorm of ${\mathcal{Z}}^{\alpha }({\mathbb{B}}^n)$. Under the norm

$${\parallel f\parallel }_{{\mathcal{Z}}^{\alpha }({\mathbb{B}}^n)}=\left|f(0)\right|+{\displaystyle \sum _{k=1}^n}\left|\frac{\partial f}{\partial z_k}(0)\right|+{\mathcal{Z}}_f,$$

${\mathcal{Z}}^{\alpha }({\mathbb{B}}^n)$ is a Banach space. Actually, there are several equivalent norms on ${\mathcal{Z}}^{\alpha }({\mathbb{B}}^n)$ (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 ${\mathrm{HE}}_I$. We say that $f\in H({\mathrm{HE}}_I)$ is in the weighted Zygmund space on ${\mathrm{HE}}_I$, denoted by ${\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)$, if

$$\begin{array}{cc}\hfill {\mathcal{Z}}_f:& =\underset{(Z,\xi )\in {\mathrm{HE}}_I}{sup}{\left[\det (I-Z{\overline{Z}}^T)-{\parallel \xi \parallel }^2\right]}^{\alpha }{H}_f(Z,\xi )<\infty ,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill H_f(Z,\xi )& ={\displaystyle \sum _{i,k=1}^m}{\displaystyle \sum _{j,l=1}^n}\left|\frac{{\partial }^{2}f}{\partial z_{ij}\partial z_{kl}}(Z,\xi )\right|+{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k=1}^r}\left|\frac{{\partial }^{2}f}{\partial z_{ij}\partial {\xi }_k}(Z,\xi )\right|\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k=1}^r}\left|\frac{{\partial }^{2}f}{\partial {\xi }_k\partial z_{ij}}(Z,\xi )\right|+{\displaystyle \sum _{k,j=1}^r}\left|\frac{{\partial }^{2}f}{\partial {\xi }_j\partial {\xi }_k}(Z,\xi )\right|.\hfill \end{array}$$

${\mathcal{Z}}^{\alpha }({HE}_I)$ is a Banach space under the norm

$${\parallel f\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}=|f(0,0)|+{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}\left|\frac{\partial f}{\partial z_{ij}}(0,0)\right|+{\displaystyle \sum _{k=1}^r}\left|\frac{\partial f}{\partial {\xi }_k}(0,0)\right|+{\mathcal{Z}}_f.$$

Let $\Omega$ be a domain of ${\mathbb{C}}^n$ and $H(\Omega )$ the space of all holomorphic functions on $\Omega$. For $\phi$ a holomorphic self-map of $\Omega$, the composition operator $C_{\phi }$ on the subspaces of $H(\Omega )$ is defined by

$$C_{\phi }f(z)=f(\phi (z)),z\in \Omega .$$

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 $C_{\phi }$ to geometric properties of $\phi$. For example, the first and simplest result in this spirit is an observation: On the Hardy space $H^p(\mathbb{D})$ if $\overline{\phi (\mathbb{D})}\subset \mathbb{D}$ then $C_{\phi }$ 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 $C_{\phi }$ in terms of geometric properties of $\phi$. These works also contribute to the understanding of ${\mathrm{HE}}_I$ 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.

We denote $(Z,\xi )=(z_{11},z_{12},\dots ,z_{mn},{\xi }_1,\dots ,{\xi }_r)$. For $\phi =({\phi }_{11},{\phi }_{12},\dots ,{\phi }_{mn},{\phi }_1,\dots ,{\phi }_r)$ the holomorphic self-map of ${\mathrm{HE}}_I$, we denote

$$\begin{array}{cc}\hfill |{\phi }^{\prime }(Z,\xi ){|}^2& ={\displaystyle \sum _{u,i=1}^m}{\displaystyle \sum _{v,j=1}^n}|\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi ){|}^2+{\displaystyle \sum _{k=1}^r}{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}|\frac{\partial {\phi }_{uv}}{\partial {\xi }_k}(Z,\xi ){|}^2\hfill \\ \hfill & +{\displaystyle \sum _{k=1}^r}{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}|\frac{\partial {\phi }_k}{\partial z_{ij}}(Z,\xi ){|}^2+{\displaystyle \sum _{j,k=1}^r}|\frac{\partial {\phi }_j}{\partial {\xi }_k}(Z,\xi ){|}^2\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill |{\phi }^{″}(Z,\xi ){|}^2& ={\displaystyle \sum _{k,u,i=1}^m}{\displaystyle \sum _{l,v,j=1}^n}|\frac{{\partial }^2{\phi }_{kl}}{\partial z_{uv}\partial z_{ij}}(Z,\xi ){|}^2+{\displaystyle \sum _{k=1}^r}{\displaystyle \sum _{u,i=1}^m}{\displaystyle \sum _{v,j=1}^n}|\frac{{\partial }^2{\phi }_k}{\partial z_{uv}\partial z_{ij}}(Z,\xi ){|}^2\hfill \\ \hfill & +{\displaystyle \sum _{k=1}^r}{\displaystyle \sum _{u,i=1}^m}{\displaystyle \sum _{v,j=1}^n}|\frac{{\partial }^2{\phi }_{uv}}{\partial {\xi }_k\partial z_{ij}}(Z,\xi ){|}^2+{\displaystyle \sum _{l,k=1}^r}{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}|\frac{{\partial }^2{\phi }_l}{\partial {\xi }_k\partial z_{ij}}(Z,\xi ){|}^2\hfill \\ \hfill & +{\displaystyle \sum _{u=1}^r}{\displaystyle \sum _{k,i=1}^m}{\displaystyle \sum _{l,j=1}^n}|\frac{{\partial }^2{\phi }_{kl}}{\partial z_{ij}\partial {\xi }_u}(Z,\xi ){|}^2+{\displaystyle \sum _{k=1}^r}{\displaystyle \sum _{j=1}^r}{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}|\frac{{\partial }^2{\phi }_j}{\partial z_{uv}\partial {\xi }_k}(Z,\xi ){|}^2\hfill \\ \hfill & +{\displaystyle \sum _{i,k=1}^r}{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}|\frac{{\partial }^2{\phi }_{uv}}{\partial {\xi }_i\partial {\xi }_k}(Z,\xi ){|}^2+{\displaystyle \sum _{i,j,k=1}^r}|\frac{{\partial }^2{\phi }_j}{\partial {\xi }_i\partial {\xi }_k}(Z,\xi ){|}^2.\hfill \end{array}$$

In this paper, constants are denoted by C; they are positive and may differ from one occurrence to the next.

2. 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 $f\in H({HE}_I)$ it follows that

$$\begin{array}{cc}\hfill & \frac{{\partial }^2(f\circ \phi )}{\partial z_{ij}\partial z_{kl}}(Z,\xi )={\displaystyle \sum _{u,p=1}^m}{\displaystyle \sum _{v,q=1}^n}\frac{{\partial }^{2}f}{\partial V_{uv}\partial V_{pq}}(\phi (Z,\xi ))\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi )\frac{\partial {\phi }_{pq}}{\partial z_{kl}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{\displaystyle \sum _{j=1}^r}\frac{{\partial }^{2}f}{\partial V_{uv}\partial V_j}(\phi (Z,\xi ))\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi )\frac{\partial {\phi }_j}{\partial z_{kl}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}\frac{\partial f}{\partial V_{uv}}(\phi (Z,\xi ))\frac{{\partial }^2{\phi }_{uv}}{\partial z_{ij}\partial z_{kl}}(Z,\xi )+{\displaystyle \sum _{j=1}^r}{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}\frac{{\partial }^{2}f}{\partial V_j\partial V_{uv}}(\phi (Z,\xi ))\frac{\partial {\phi }_j}{\partial z_{ij}}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial z_{kl}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{p=1}^r}{\displaystyle \sum _{k=1}^r}\frac{{\partial }^{2}f}{\partial V_p\partial V_k}(\phi (Z,\xi ))\frac{\partial {\phi }_p}{\partial z_{ij}}(Z,\xi )\frac{\partial {\phi }_k}{\partial z_{kl}}(Z,\xi )+{\displaystyle \sum _{p=1}^r}\frac{\partial f}{\partial V_p}(\phi (Z,\xi ))\frac{{\partial }^2{\phi }_p}{\partial z_{ij}\partial z_{kl}}(Z,\xi ),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \frac{{\partial }^2(f\circ \phi )}{\partial z_{ij}\partial {\xi }_k}(Z,\xi )={\displaystyle \sum _{u,p=1}^m}{\displaystyle \sum _{v,q=1}^n}\frac{{\partial }^{2}f}{\partial V_{uv}\partial V_{pq}}(\phi (Z,\xi ))\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi )\frac{\partial {\phi }_{pq}}{\partial {\xi }_k}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{\displaystyle \sum _{p=1}^r}\frac{{\partial }^{2}f}{\partial V_{uv}\partial V_p}(\phi (Z,\xi ))\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi )\frac{\partial {\phi }_p}{\partial {\xi }_k}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}\frac{\partial f}{\partial V_{uv}}(\phi (Z,\xi ))\frac{{\partial }^2{\phi }_{uv}}{\partial z_{ij}\partial {\xi }_k}(Z,\xi )+{\displaystyle \sum _{p=1}^r}{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}\frac{{\partial }^{2}f}{\partial V_p\partial V_{uv}}(\phi (Z,\xi ))\frac{\partial {\phi }_p}{\partial z_{ij}}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial {\xi }_k}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{p=1}^r}{\displaystyle \sum _{u=1}^r}\frac{{\partial }^{2}f}{\partial V_p\partial V_u}(\phi (Z,\xi ))\frac{\partial {\phi }_p}{\partial z_{ij}}(Z,\xi )\frac{\partial {\phi }_u}{\partial {\xi }_k}(Z,\xi )+{\displaystyle \sum _{p=1}^r}\frac{\partial f}{\partial V_p}(\phi (Z,\xi ))\frac{{\partial }^2{\phi }_p}{\partial z_{ij}\partial {\xi }_k}(Z,\xi ),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \frac{{\partial }^2(f\circ \phi )}{\partial {\xi }_k\partial z_{ij}}(Z,\xi )={\displaystyle \sum _{u,p=1}^m}{\displaystyle \sum _{v,q=1}^n}\frac{{\partial }^{2}f}{\partial V_{uv}\partial V_{pq}}(\phi (Z,\xi ))\frac{\partial {\phi }_{uv}}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_{pq}}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{\displaystyle \sum _{p=1}^r}\frac{{\partial }^{2}f}{\partial V_{uv}\partial V_p}(\phi (Z,\xi ))\frac{\partial {\phi }_{uv}}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_p}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}\frac{\partial f}{\partial V_{uv}}(\phi (Z,\xi ))\frac{{\partial }^2{\phi }_{uv}}{\partial {\xi }_k\partial z_{ij}}(Z,\xi )+{\displaystyle \sum _{p=1}^r}{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}\frac{{\partial }^{2}f}{\partial V_p\partial V_{uv}}(\phi (Z,\xi ))\frac{\partial {\phi }_p}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{p=1}^r}{\displaystyle \sum _{u=1}^r}\frac{{\partial }^{2}f}{\partial V_p\partial V_u}(\phi (Z,\xi ))\frac{\partial {\phi }_p}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_u}{\partial z_{ij}}(Z,\xi )+{\displaystyle \sum _{p=1}^r}\frac{\partial f}{\partial V_p}(\phi (Z,\xi ))\frac{{\partial }^2{\phi }_p}{\partial {\xi }_k\partial z_{ij}}(Z,\xi )\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \frac{{\partial }^2(f\circ \phi )}{\partial {\xi }_j\partial {\xi }_k}(Z,\xi )={\displaystyle \sum _{u,p=1}^m}{\displaystyle \sum _{v,q=1}^n}\frac{{\partial }^{2}f}{\partial V_{uv}\partial V_{pq}}(\phi (Z,\xi ))\frac{\partial {\phi }_{uv}}{\partial {\xi }_j}(Z,\xi )\frac{\partial {\phi }_{pq}}{\partial {\xi }_k}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{\displaystyle \sum _{p=1}^r}\frac{{\partial }^{2}f}{\partial V_{uv}\partial V_p}(\phi (Z,\xi ))\frac{\partial {\phi }_{uv}}{\partial {\xi }_j}(Z,\xi )\frac{\partial {\phi }_p}{\partial {\xi }_k}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}\frac{\partial f}{\partial V_{uv}}(\phi (Z,\xi ))\frac{{\partial }^2{\phi }_{uv}}{\partial {\xi }_j\partial {\xi }_k}(Z,\xi )+{\displaystyle \sum _{p=1}^r}{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}\frac{{\partial }^{2}f}{\partial V_p\partial V_{uv}}(\phi (Z,\xi ))\frac{\partial {\phi }_p}{\partial {\xi }_j}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial {\xi }_k}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{p=1}^r}{\displaystyle \sum _{i=1}^r}\frac{{\partial }^{2}f}{\partial V_p\partial V_i}(\phi (Z,\xi ))\frac{\partial {\phi }_p}{\partial {\xi }_j}(Z,\xi )\frac{\partial {\phi }_i}{\partial {\xi }_k}(Z,\xi )+{\displaystyle \sum _{p=1}^r}\frac{\partial f}{\partial V_p}(\phi (Z,\xi ))\frac{{\partial }^2{\phi }_p}{\partial {\xi }_j\partial {\xi }_k}(Z,\xi ).\hfill \end{array}$$

From Lemma 1, we obtain the evaluation of $H_{f\circ \phi }(Z,\xi )$ as follows.

Lemma 2.

For each $f\in H({HE}_I)$ and $(Z,\xi )\in {HE}_I$, it follows that

$$H_{f\circ \phi }(Z,\xi )\le 8\sqrt{6}C\left(H_f(\phi (Z,\xi ))|{\phi }^{\prime }(Z,\xi ){|}^2+|\nabla f(\phi (Z,\xi ))||{\phi }^{″}(Z,\xi )|\right),$$

where $C=\sqrt{max\{m^2{n}^2,mnr,r^2\}}$ and

$$\nabla f(Z,\xi )=\left(\frac{\partial f(Z,\xi )}{\partial z_{11}},\dots ,\frac{\partial f(Z,\xi )}{\partial z_{mn}},\frac{\partial f(Z,\xi )}{\partial {\xi }_1},\dots ,\frac{\partial f(Z,\xi )}{\partial {\xi }_r}\right).$$

Proof. 

By Lemma 1 and the elementary inequality

$$|{\displaystyle \sum _{j=1}^n}{a}_j{|}^2\le n{\displaystyle \sum _{j=1}^n}{|a_j|}^2,$$

we have

$$\begin{array}{cc}\hfill H_{f\circ \phi }^2(Z,\xi )& =\left({\displaystyle \sum _{i,k=1}^m}{\displaystyle \sum _{j,l=1}^n}\right|\frac{{\partial }^2(f\circ \phi )}{\partial z_{ij}\partial z_{kl}}(Z,\xi )|+{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k=1}^r}|\frac{{\partial }^2(f\circ \phi )}{\partial z_{ij}\partial {\xi }_k}(Z,\xi )|\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k=1}^r}\left|\frac{{\partial }^2(f\circ \phi )}{\partial {\xi }_k\partial z_{ij}}(Z,\xi )\right|+{\displaystyle \sum _{k=1}^r}{\displaystyle \sum _{j=1}^r}\left|\frac{{\partial }^2(f\circ \phi )}{\partial {\xi }_j\partial {\xi }_k}(Z,\xi )\right|{)}^2\hfill \\ \hfill & \le 4[{\left({\displaystyle \sum _{i,k=1}^m}{\displaystyle \sum _{j,l=1}^n}\left|\frac{{\partial }^2(f\circ \phi )}{\partial z_{ij}\partial z_{kl}}(Z,\xi )\right|\right)}^2+{\left({\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k=1}^r}\left|\frac{{\partial }^2(f\circ \phi )}{\partial z_{ij}\partial {\xi }_k}(Z,\xi )\right|\right)}^2\hfill \\ \hfill & +{\left({\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k=1}^r}\left|\frac{{\partial }^2(f\circ \phi )}{\partial {\xi }_k\partial z_{ij}}(Z,\xi )\right|\right)}^2+{\left({\displaystyle \sum _{k=1}^r}{\displaystyle \sum _{j=1}^r}\left|\frac{{\partial }^2(f\circ \phi )}{\partial {\xi }_j\partial {\xi }_k}(Z,\xi )\right|\right)}^2]\hfill \\ \hfill & \le 4m^2{n}^2{\displaystyle \sum _{i,k=1}^m}{\displaystyle \sum _{j,l=1}^n}|\frac{{\partial }^2(f\circ \phi )}{\partial z_{ij}\partial z_{kl}}(Z,\xi ){|}^2+4mnr{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k=1}^r}|\frac{{\partial }^2(f\circ \phi )}{\partial z_{ij}\partial {\xi }_k}(Z,\xi ){|}^2\hfill \\ \hfill & +4mnr{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k=1}^r}|\frac{{\partial }^2(f\circ \phi )}{\partial {\xi }_k\partial z_{ij}}(Z,\xi ){|}^2+4r^2{\displaystyle \sum _{k=1}^r}{\displaystyle \sum _{j=1}^r}|\frac{{\partial }^2(f\circ \phi )}{\partial {\xi }_j\partial {\xi }_k}(Z,\xi ){|}^2\hfill \\ \hfill & \le 24m^2{n}^2{\displaystyle \sum _{i,k,u,p=1}^m}{\displaystyle \sum _{j,l,v,q=1}^n}|\frac{{\partial }^{2}f}{\partial V_{uv}\partial V_{pq}}(\phi (Z,\xi ))\frac{\partial {\phi }_{pq}}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi ){|}^2\hfill \\ \hfill & +24m^2{n}^2{\displaystyle \sum _{i,k,u=1}^m}{\displaystyle \sum _{j,l,v=1}^n}{\displaystyle \sum _{p=1}^r}|\frac{{\partial }^{2}f}{\partial V_{uv}\partial V_p}(\phi (Z,\xi ))\frac{\partial {\phi }_j}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi ){|}^2\hfill \\ \hfill & +24m^2{n}^2{\displaystyle \sum _{i,k,u=1}^m}{\displaystyle \sum _{j,l,v=1}^n}|\frac{\partial f}{\partial V_{uv}}(\phi (Z,\xi ))\frac{{\partial }^2{\phi }_{uv}}{\partial z_{ij}\partial z_{kl}}(Z,\xi ){|}^2\hfill \\ \hfill & +24m^2{n}^2{\displaystyle \sum _{i,k,u=1}^m}{\displaystyle \sum _{j,l,v=1}^n}{\displaystyle \sum _{p=1}^r}|\frac{{\partial }^{2}f}{\partial V_p\partial V_{uv}}(\phi (Z,\xi ))\frac{\partial {\phi }_{uv}}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_j}{\partial z_{ij}}(Z,\xi ){|}^2\hfill \\ \hfill & +24m^2{n}^2{\displaystyle \sum _{i,k=1}^m}{\displaystyle \sum _{j,l=1}^n}{\displaystyle \sum _{u,v=1}^r}|\frac{{\partial }^{2}f}{\partial V_u\partial V_v}(\phi (Z,\xi ))\frac{\partial {\phi }_k}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_j}{\partial z_{ij}}(Z,\xi ){|}^2\hfill \\ \hfill & +24m^2{n}^2{\displaystyle \sum _{i,k=1}^m}{\displaystyle \sum _{j,l=1}^n}{\displaystyle \sum _{p=1}^r}|\frac{\partial f}{\partial V_p}(\phi (Z,\xi ))\frac{{\partial }^2{\phi }_p}{\partial z_{ij}\partial z_{kl}}(Z,\xi ){|}^2\hfill \\ \hfill & +24mnr{\displaystyle \sum _{k=1}^r}{\displaystyle \sum _{i,u,p=1}^m}{\displaystyle \sum _{j,v,q=1}^n}|\frac{{\partial }^{2}f}{\partial V_{uv}\partial V_{pq}}(\phi (Z,\xi ))\frac{\partial {\phi }_{pq}}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi ){|}^2\hfill \\ \hfill & +24mnr{\displaystyle \sum _{i,u=1}^m}{\displaystyle \sum _{j,v=1}^n}{\displaystyle \sum _{k,p=1}^r}|\frac{{\partial }^{2}f}{\partial V_{uv}\partial V_p}(\phi (Z,\xi ))\frac{\partial {\phi }_p}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi ){|}^2\hfill \\ \hfill & +24mnr{\displaystyle \sum _{i,u=1}^m}{\displaystyle \sum _{j,v=1}^n}{\displaystyle \sum _{k=1}^r}|\frac{\partial f}{\partial V_{uv}}(\phi (Z,\xi ))\frac{{\partial }^2{\phi }_{uv}}{\partial z_{ij}\partial {\xi }_k}(Z,\xi ){|}^2\hfill \\ \hfill & +24mnr{\displaystyle \sum _{i,u=1}^m}{\displaystyle \sum _{j,v=1}^n}{\displaystyle \sum _{k,p=1}^r}|\frac{{\partial }^{2}f}{\partial V_p\partial V_{uv}}(\phi (Z,\xi ))\frac{\partial {\phi }_{uv}}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_p}{\partial z_{ij}}(Z,\xi ){|}^2\hfill \\ \hfill & +24mnr{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k,u,v=1}^r}|\frac{{\partial }^{2}f}{\partial V_u\partial V_v}(\phi (Z,\xi ))\frac{\partial {\phi }_v}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_u}{\partial z_{ij}}(Z,\xi ){|}^2\hfill \\ \hfill & +24mnr{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k,p=1}^r}|\frac{\partial f}{\partial V_p}(\phi (Z,\xi ))\frac{{\partial }^2{\phi }_p}{\partial z_{ij}\partial {\xi }_k}(Z,\xi ){|}^2\hfill \\ \hfill & +24mnr{\displaystyle \sum _{i,u,p=1}^m}{\displaystyle \sum _{j,v,q=1}^n}{\displaystyle \sum _{k=1}^r}|\frac{{\partial }^{2}f}{\partial V_{uv}\partial V_{pq}}(\phi (Z,\xi ))\frac{\partial {\phi }_{pq}}{\partial z_{ij}}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial {\xi }_k}(Z,\xi ){|}^2\hfill \\ \hfill & +24mnr{\displaystyle \sum _{i,u=1}^m}{\displaystyle \sum _{j,v=1}^n}{\displaystyle \sum _{k,p=1}^r}|\frac{{\partial }^{2}f}{\partial V_{uv}\partial V_p}(\phi (Z,\xi ))\frac{\partial {\phi }_p}{\partial z_{ij}}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial {\xi }_k}(Z,\xi ){|}^2\hfill \\ \hfill & +24mnr{\displaystyle \sum _{i,u=1}^m}{\displaystyle \sum _{j,v=1}^n}{\displaystyle \sum _{k=1}^r}|\frac{\partial f}{\partial V_{uv}}(\phi (Z,\xi ))\frac{{\partial }^2{\phi }_{uv}}{\partial {\xi }_k\partial z_{ij}}(Z,\xi ){|}^2\hfill \\ \hfill & +24mnr{\displaystyle \sum _{i,u=1}^m}{\displaystyle \sum _{j,v=1}^n}{\displaystyle \sum _{k,p=1}^r}|\frac{{\partial }^{2}f}{\partial V_p\partial V_{uv}}(\phi (Z,\xi ))\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi )\frac{\partial {\phi }_p}{\partial {\xi }_k}(Z,\xi ){|}^2\hfill \\ \hfill & +24mnr{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k,p,u=1}^r}|\frac{{\partial }^{2}f}{\partial V_p\partial V_u}(\phi (Z,\xi ))\frac{\partial {\phi }_u}{\partial z_{ij}}(Z,\xi )\frac{\partial {\phi }_p}{\partial {\xi }_k}(Z,\xi ){|}^2\hfill \\ \hfill & +24mnr{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k,p=1}^r}|\frac{\partial f}{\partial V_p}(\phi (Z,\xi ))\frac{{\partial }^2{\phi }_p}{\partial {\xi }_k\partial z_{ij}}(Z,\xi ){|}^2\hfill \\ \hfill & +24r^2{\displaystyle \sum _{j,k=1}^r}{\displaystyle \sum _{u,p=1}^m}{\displaystyle \sum _{v,q=1}^m}|\frac{{\partial }^{2}f}{\partial V_{uv}\partial V_{pq}}(\phi (Z,\xi ))\frac{\partial {\phi }_{pq}}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial {\xi }_j}(Z,\xi ){|}^2\hfill \\ \hfill & +24r^2{\displaystyle \sum _{j,k,p=1}^r}{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}|\frac{{\partial }^{2}f}{\partial V_{uv}\partial V_p}(\phi (Z,\xi ))\frac{\partial {\phi }_p}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial {\xi }_j}(Z,\xi ){|}^2\hfill \\ \hfill & +24r^2{\displaystyle \sum _{j,k=1}^r}{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}|\frac{\partial f}{\partial V_{uv}}(\phi (Z,\xi ))\frac{{\partial }^2{\phi }_{uv}}{\partial {\xi }_j\partial {\xi }_k}(Z,\xi ){|}^2\hfill \\ & +24r^2{\displaystyle \sum _{j,k,p=1}^r}{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}|\frac{{\partial }^{2}f}{\partial V_p\partial V_{uv}}(\phi (Z,\xi ))\frac{\partial {\phi }_{uv}}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_p}{\partial {\xi }_j}(Z,\xi ){|}^2\hfill \\ & +24r^2{\displaystyle \sum _{i,j,k,p=1}^r}|\frac{{\partial }^{2}f}{\partial V_p\partial V_i}(\phi (Z,\xi ))\frac{\partial {\phi }_i}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_p}{\partial {\xi }_j}(Z,\xi ){|}^2\hfill \end{array}$$

$$\begin{array}{cc}& +24r^2{\displaystyle \sum _{j,k,p=1}^r}|\frac{\partial f}{\partial V_p}(\phi (Z,\xi ))\frac{{\partial }^2{\phi }_p}{\partial {\xi }_j\partial {\xi }_k}(Z,\xi ){|}^2\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & \le 16C\left(H_f^2(\phi (Z,\xi ))|{\phi }^{\prime }{(Z,\xi )|}^4+{|\nabla f(\phi (Z,\xi ))|}^2{|{\phi }^{″}(Z,\xi )|}^2\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le 16C{\left(H_f(\phi (Z,\xi ))|{\phi }^{\prime }{(Z,\xi )|}^2+|\nabla f(\phi (Z,\xi ))||{\phi }^{″}(Z,\xi )|\right)}^2,\hfill \end{array}$$

(2)

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 $C{H}_f^2(\phi (Z,\xi )){|{\phi }^{\prime }(Z,\xi )|}^4$, and there are eight terms with the first partial derivatives of f and each term is less than ${C|\nabla f(\phi (Z,\xi ))|}^2{|{\phi }^{″}(Z,\xi )|}^2$. From this, it follows that (2) holds. The proof is completed. □

Next, we obtain the point evaluation estimate for the functions in ${\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)$.

Lemma 3.

There exists a positive constant C independent of $f\in {\mathcal{Z}}^{\alpha }({HE}_I)$ and $(Z,\xi )\in {HE}_I$ such that

$$\begin{array}{c}\hfill |\nabla f(Z,\xi )|\le \frac{{C\parallel f\parallel }_{{\mathcal{Z}}^{\alpha }({HE}_I)}}{{\left(det(I-Z{\overline{Z}}^T)-{\parallel \xi \parallel }^2\right)}^{\alpha }}.\end{array}$$

Proof. 

For each $t\in [0,1]$, we have

$$\begin{array}{c}\hfill {\parallel t\xi \parallel }^2={\displaystyle \sum _{j=1}^r}|t{\xi }_j{|}^{2p_j}\le {\displaystyle \sum _{j=1}^r}|{\xi }_j{|}^{2p_j}={\parallel \xi \parallel }^2<\det (I-Z{\overline{Z}}^T)\le \det (I-t^{2}Z{\overline{Z}}^T).\end{array}$$

(3)

By (3), we obtain

$$\begin{array}{c}\hfill 0<\det (I-Z{\overline{Z}}^T)-{\parallel \xi \parallel }^2\le \det (I-t^{2}Z{\overline{Z}}^T)-{\parallel t\xi \parallel }^2.\end{array}$$

(4)

Since $Z\in {\Re }_I(m,n)$, it follows that $0<\det (I-Z{\overline{Z}}^T)<1$. From this, it is obvious that

$$\begin{array}{c}\hfill 0<\det (I-Z{\overline{Z}}^T)-{\parallel \xi \parallel }^2<1.\end{array}$$

(5)

Then for each $f\in {\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)$ and $(Z,\xi )\in {\mathrm{HE}}_I$, it follows from (4) and (5) that

$$\begin{array}{cc}\hfill \left|\frac{\partial f}{\partial z_{ij}}(Z,\xi )\right|& =|{\int }_0^1\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial f}{\partial z_{ij}}(tZ,t\xi )\right)\mathrm{d}t+\frac{\partial f}{\partial z_{ij}}(0,0)|\hfill \\ \hfill & =|{\int }_0^1\left[{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{z}_{uv}\frac{{\partial }^{2}f}{\partial z_{ij}\partial z_{uv}}(tZ,t\xi )+{\displaystyle \sum _{k=1}^r}{\xi }_k\frac{{\partial }^{2}f}{\partial z_{ij}\partial {\xi }_k}(tZ,t\xi )\right]\mathrm{d}t+\frac{\partial f}{\partial z_{ij}}(0,0)|\hfill \\ \hfill & \le {\int }_0^1\left[{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}\left|\frac{{\partial }^{2}f}{\partial z_{ij}\partial z_{uv}}(tZ,t\xi )\right|+{\displaystyle \sum _{k=1}^r}\left|\frac{{\partial }^{2}f}{\partial z_{ij}\partial {\xi }_k}(tZ,t\xi )\right|\right]\mathrm{d}t+\left|\frac{\partial f}{\partial z_{ij}}(0,0)\right|\hfill \\ \hfill & \le {\parallel f\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{H}\mathrm{E}}_I)}{\int }_0^1\frac{\mathrm{d}t}{{\left[\mathrm{d}et(I-t^{2}Z{\overline{Z}}^T)-{\parallel t\xi \parallel }^2\right]}^{\alpha }}+{\parallel f\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{H}\mathrm{E}}_I)}\hfill \\ \hfill & \le (\frac{1}{{\left[\mathrm{d}\mathrm{e}\mathrm{t}(I-Z{\overline{Z}}^T)-{\parallel \xi \parallel }^2\right]}^{\alpha }}+1){\parallel f\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{H}\mathrm{E}}_I)}\hfill \\ \hfill & \le \frac{{2\parallel f\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{H}\mathrm{E}}_I)}}{{\left[\mathrm{d}\mathrm{e}\mathrm{t}(I-Z{\overline{Z}}^T)-{\parallel \xi \parallel }^2\right]}^{\alpha }},\hfill \end{array}$$

(6)

where we have used the facts of $|z_{uv}|<1$ and $|{\xi }_k|<1$ for $u=1,\dots ,m$, $v=1,\dots ,n$ and $k=1,\dots ,r$, which can be found in later discussions. We also can similarly obtain

$$\begin{array}{c}\hfill \left|\frac{\partial f}{\partial {\xi }_k}(Z,\xi )\right|\le \frac{{2\parallel f\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}}{{\left[\det (I-Z{\overline{Z}}^T)-{\parallel \xi \parallel }^2\right]}^{\alpha }}.\end{array}$$

(7)

By (6) and (7), for $(Z,\xi )\in {\mathrm{HE}}_I$ we have

$$\begin{array}{cc}\hfill |\nabla f(Z,\xi )|& \le {\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}\left|\frac{\partial f}{\partial z_{ij}}(Z,\xi )\right|+\sum _{k=1}\left|\frac{\partial f}{\partial {\xi }_k}(Z,\xi )\right|\hfill \\ \hfill & \le \frac{{C\parallel f\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}}{{\left[\det (I-Z{\overline{Z}}^T)-{\parallel \xi \parallel }^2\right]}^{\alpha }},\hfill \end{array}$$

where $C=2(mn+r)$. The proof is completed. □

By using Lemma 3, we obtain the following result.

Lemma 4.

There exists a positive constant C independent of $f\in {\mathcal{Z}}^{\alpha }({HE}_I)$ and $(Z,\xi )\in {\mathrm{HE}}_I$ such that

$$\begin{array}{c}\hfill |f(Z,\xi )|\le \frac{{C\parallel f\parallel }_{{\mathcal{Z}}^{\alpha }({HE}_I)}}{{\left[det(I-Z{\overline{Z}}^T)-{\parallel \xi \parallel }^2\right]}^{\alpha }}.\end{array}$$

Proof. 

From the proof of Lemma 3, it is easy to see that $|(Z,\xi )|\le \sqrt{mn+r}$. Then, for each $f\in {\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)$ and $(Z,\xi )\in {HE}_I$, we have

$$\begin{array}{cc}\hfill |f(Z,\xi )|& =|f(0,0)+{\int }_0^1\langle \nabla f(tZ,t\xi ),\overline{(Z,\xi )}\rangle \mathrm{d}t|\hfill \\ \hfill & \le |f(0,0)|+{\int }_0^1|\nabla f(tZ,t\xi )||\overline{(Z,\xi )}|\mathrm{d}t\hfill \\ \hfill & \le {\parallel f\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}+C{\int }_0^1\frac{|(Z,\xi )|}{{\left[\mathrm{d}et(I-t^{2}Z{\overline{Z}}^T)-{\parallel t\xi \parallel }^2\right]}^{\alpha }}\mathrm{d}t{\parallel f\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}\hfill \\ \hfill & \le (1+\frac{C\sqrt{mn+r}}{{\left[\det (I-Z{\overline{Z}}^T)-{\parallel \xi \parallel }^2\right]}^{\alpha }})\parallel f{\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}\hfill \\ \hfill & \le \frac{1+C\sqrt{mn+r}}{{\left[\det (I-Z{\overline{Z}}^T)-{\parallel \xi \parallel }^2\right]}^{\alpha }}\parallel f{\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}.\hfill \end{array}$$

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 $C_{\phi }$ on ${\mathcal{Z}}^{\alpha }({HE}_I)$ is compact if and only if for every bounded sequence $\left\{f_k\right\}$ in ${\mathcal{Z}}^{\alpha }({HE}_I)$ such that $f_k\to 0$ uniformly on every compact subset of ${HE}_I$ as $k\to \infty$, it follows that

$$\underset{k\to \infty }{lim}{\parallel C_{\phi }{f}_k\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}=0.$$

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.

Lemma 6.

Let $(V,\zeta )=\phi (Z,\xi )$ for $(Z,\xi )\in {HE}_I$. If

$$\begin{array}{c}\hfill \frac{{\left[det(I-Z{\overline{Z}}^T)-{\parallel \xi \parallel }^2\right]}^{\alpha }}{{\left[det(I-V{\overline{V}}^T)-{\parallel \zeta \parallel }^2\right]}^{\alpha }}|{\phi }^{\prime }(Z,\xi ){|}^2=O(1)\end{array}$$

(8)

as $(V,\zeta )\to \partial {HE}_I$, then

$$\begin{array}{c}\hfill \underset{(Z,\xi )\in {HE}_I}{sup}\frac{{\left[det(I-Z{\overline{Z}}^T)-{\parallel \xi \parallel }^2\right]}^{\alpha }}{{\left[det(I-V{\overline{V}}^T)-{\parallel \zeta \parallel }^2\right]}^{\alpha }}|{\phi }^{\prime }(Z,\xi ){|}^2<\infty .\end{array}$$

Lemma 7.

Let $(V,\zeta )=\phi (Z,\xi )$ for $(Z,\xi )\in {HE}_I$. If

$$\begin{array}{c}\hfill \frac{{\left[det(I-Z{\overline{Z}}^T)-{\parallel \xi \parallel }^2\right]}^{\alpha }}{{\left[det(I-V{\overline{V}}^T)-{\parallel \zeta \parallel }^2\right]}^{\alpha }}|{\phi }^{″}(Z,\xi )|=O(1)\end{array}$$

as $(V,\zeta )\to \partial {HE}_I$, then

$$\begin{array}{c}\hfill \underset{(Z,\xi )\in {HE}_I}{sup}\frac{{\left[det(I-Z{\overline{Z}}^T)-{\parallel \xi \parallel }^2\right]}^{\alpha }}{{\left[det(I-V{\overline{V}}^T)-{\parallel \zeta \parallel }^2\right]}^{\alpha }}|{\phi }^{″}(Z,\xi ){|}^2<\infty .\end{array}$$

Let S be an fixed element in ${\Re }_I(m,n)$. Write

$${\overline{S}}^T=\left(\begin{array}{cccc}{s}_{11}& s_{12}& \cdots & s_{1m}\\ s_{21}& s_{22}& \cdots & s_{2m}\\ \cdots & \cdots & \cdots & \cdots \\ s_{n1}& s_{n2}& \cdots & s_{nm}\end{array}\right).$$

On ${\Re }_I(m,n)$ we define the function

$$A_S(Z)=\det (I-Z{\overline{S}}^T).$$

The following two results were obtained in [21].

Lemma 8.

If

$$Z=\left(\begin{array}{cccc}{z}_{11}& z_{12}& \cdots & z_{1n}\\ z_{21}& z_{22}& \cdots & z_{2n}\\ \cdots & \cdots & \cdots & \cdots \\ z_{m1}& z_{m2}& \cdots & z_{mn}\end{array}\right)\in {\Re }_I(m,n),$$

then

$$A_S(Z)=\left|\begin{array}{cccccc}1-{\displaystyle \sum _{k=1}^n}{s}_{k1}{z}_{1k}& -{\displaystyle \sum _{k=1}^n}{s}_{k2}{z}_{1k}& \cdots & -{\displaystyle \sum _{k=1}^n}{s}_{kj}{z}_{1k}& \cdots & -{\displaystyle \sum _{k=1}^n}{s}_{km}{z}_{1k}\\ ⋮& ⋮& ⋮& ⋮\\ -{\displaystyle \sum _{k=1}^n}{s}_{k1}{z}_{ik}& 1-{\displaystyle \sum _{k=1}^n}{s}_{k2}{z}_{ik}& \cdots & -{\displaystyle \sum _{k=1}^n}{s}_{kj}{z}_{ik}& \cdots & -{\displaystyle \sum _{k=1}^n}{s}_{km}{z}_{ik}\\ ⋮& ⋮& ⋮& ⋮\\ -{\displaystyle \sum _{k=1}^n}{s}_{k1}{z}_{mk}& -{\displaystyle \sum _{k=1}^n}{s}_{k2}{z}_{mk}& \cdots & -{\displaystyle \sum _{k=1}^n}{s}_{kj}{z}_{mk}& \cdots & 1-{\displaystyle \sum _{k=1}^n}{s}_{km}{z}_{mk}\end{array}\right|.$$

For the sake of convenience, we define

$$A_{S,ij}(Z)=\frac{\partial A_S(Z)}{\partial z_{ij}}\mathrm{and}{A}_{S,ij,pq}(Z)=\frac{{\partial }^2{A}_S(Z)}{\partial z_{ij}\partial z_{pq}}.$$

Lemma 9.

For each $Z\in {\Re }_I(m,n)$, it holds

$$A_{S,ij}(Z)=\left|\begin{array}{cccccc}1-{\displaystyle \sum _{k=1}^n}{s}_{k1}{z}_{1k}& -{\displaystyle \sum _{k=1}^n}{s}_{k2}{z}_{1k}& \cdots & -{\displaystyle \sum _{k=1}^n}{s}_{kj}{z}_{1k}& \cdots & -{\displaystyle \sum _{k=1}^n}{s}_{km}{z}_{1k}\\ ⋮& ⋮& ⋮& ⋮\\ -s_{j1}& -s_{j2}& \cdots & -s_{jj}& \cdots & -s_{jm}\\ ⋮& ⋮& ⋮& ⋮\\ -{\displaystyle \sum _{k=1}^n}{s}_{k1}{z}_{mk}& -{\displaystyle \sum _{k=1}^n}{s}_{k2}{z}_{mk}& \cdots & -{\displaystyle \sum _{k=1}^n}{s}_{kj}{z}_{mk}& \cdots & 1-{\displaystyle \sum _{k=1}^n}{s}_{km}{z}_{mk}\end{array}\right|$$

and

$$A_{S,ij,pq}(Z)=\left|\begin{array}{cccccc}1-{\displaystyle \sum _{k=1}^n}{s}_{k1}{z}_{1k}& -{\displaystyle \sum _{k=1}^n}{s}_{k2}{z}_{1k}& \cdots & -{\displaystyle \sum _{k=1}^n}{s}_{kj}{z}_{1k}& \cdots & -{\displaystyle \sum _{k=1}^n}{s}_{km}{z}_{1k}\\ ⋮& ⋮& ⋮& ⋮\\ -s_{q1}& -s_{q2}& \cdots & -s_{qq}& \cdots & -s_{qm}\\ ⋮& ⋮& ⋮& ⋮\\ -s_{j1}& -s_{j2}& \cdots & -s_{jj}& \cdots & -s_{jm}\\ ⋮& ⋮& ⋮& ⋮\\ -{\displaystyle \sum _{k=1}^n}{s}_{k1}{z}_{mk}& -{\displaystyle \sum _{k=1}^n}{s}_{k2}{z}_{mk}& \cdots & -{\displaystyle \sum _{k=1}^n}{s}_{kj}{z}_{mk}& \cdots & 1-{\displaystyle \sum _{k=1}^n}{s}_{km}{z}_{mk}\end{array}\right|.$$

The following result was obtained in [19].

Lemma 10.

There exists a positive constant C independent of Z, $S\in {\Re }_I(m,n)$ such that

$$\begin{array}{c}{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}|A_{S,ij}(Z){|}^2\le C.\end{array}$$

We also need the following two results (see [21]).

Lemma 11.

There exists a positive constant C independent of Z, $S\in {\Re }_I(m,n)$ such that

$$\begin{array}{c}{\displaystyle \sum _{i,p=1}^m}{\displaystyle \sum _{j,q=1}^n}|A_{S,ij,pq}(Z){|}^2\le C.\end{array}$$

Lemma 12.

There exists a positive constant C independent of Z, $S\in {\Re }_I(m,n)$ such that

$$|A_S(Z)|\le C.$$

The following result was obtained in [20,22].

Lemma 13.

If $(Z,\xi )$, $(S,t)\in {HE}_I$, then

$$\begin{array}{c}\hfill \left[A_Z(Z)-{\parallel \xi \parallel }^2\right]\left[A_S(S)-{\parallel t\parallel }^2\right]\le {\left[|A_S(Z)|-\parallel \xi \parallel \parallel t\parallel \right]}^2.\end{array}$$

Let $\alpha \ne 1$ and $(S,t)\in {\mathrm{HE}}_I$. On ${\mathrm{HE}}_I$ define the function

$$f_{(S,t)}(Z,\xi )=\frac{1}{2-2\alpha }\frac{{\left[A_S(S)-{\parallel t\parallel }^2\right]}^{\alpha }}{{[A_S(Z)-{\displaystyle \sum _{j=1}^r}{\xi }_j^{p_j}{\overline{t_j}}^{p_j}]}^{2\alpha -2}}.$$

By using above lemmas, we will see that $f_{(S,t)}\in {\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)$.

Lemma 14.

Let $\alpha \ne 1$ and $p_j\ge 2$ for $j=1,2,\dots ,r$. Then $f_{(S,t)}\in {\mathcal{Z}}^{\alpha }({HE}_I)$, and there exists a positive constant C such that

$$\underset{(S,t)\in {HE}_I}{sup}{\parallel f_{(S,t)}\parallel }_{{\mathcal{Z}}^{\alpha }({HE}_I)}\le C.$$

Proof. 

We write $c_{\alpha }=1-2\alpha$. By a direct calculation, we have

$$\begin{array}{c}\hfill \frac{\partial f_{(S,t)}}{\partial z_{ij}}(Z,\xi )=\frac{{\left[A_S(S)-{\parallel t\parallel }^2\right]}^{\alpha }}{{[A_S(Z)-{\displaystyle \sum _{j=1}^r}{\xi }_j^{p_j}{\overline{t_j}}^{p_j}]}^{2\alpha -1}}{A}_{S,ij}(Z)\end{array}$$

(9)

and

$$\begin{array}{c}\hfill \frac{\partial f_{(S,t)}}{\partial {\xi }_k}(Z,\xi )={\left[A_S(S)-{\parallel t\parallel }^2\right]}^{\alpha }\frac{p_k{\overline{t_k}}^{p_k}{\xi }_k^{p_k-1}}{{[A_S(Z)-{\displaystyle \sum _{j=1}^r}{\xi }_j^{p_j}{\overline{t_j}}^{p_j}]}^{2\alpha -1}}.\end{array}$$

(10)

By (9) and (10), we see that

$$\begin{array}{cc}\hfill \frac{{\partial }^2{f}_{(S,t)}}{\partial z_{ij}\partial z_{kl}}(Z,\xi )& =c_{\alpha }\frac{{\left[A_S(S)-{\parallel t\parallel }^2\right]}^{\alpha }}{{[A_S(Z)-{\displaystyle \sum _{j=1}^r}{\xi }_j^{p_j}{\overline{t_j}}^{p_j}]}^{2\alpha }}{A}_{S,ij}(Z)A_{S,kl}(Z)\hfill \\ \hfill & +\frac{{\left[A_S(S)-{\parallel t\parallel }^2\right]}^{\alpha }}{{[A_S(Z)-{\displaystyle \sum _{j=1}^r}{\xi }_j^{p_j}{\overline{t_j}}^{p_j}]}^{2\alpha -1}}{A}_{S,ij,kl}(Z),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \frac{{\partial }^2{f}_{(S,t)}}{\partial {\xi }_k\partial z_{ij}}(Z,\xi )& =c_{\alpha }\frac{{\left[A_S(S)-{\parallel t\parallel }^2\right]}^{\alpha }}{{[A_S(Z)-{\displaystyle \sum _{j=1}^r}{\xi }_j^{p_j}{\overline{t_j}}^{p_j}]}^{2\alpha }}{p}_k{\overline{t_k}}^{p_k}{\xi }_k^{p_k-1}{A}_{S,ij}(Z)\hfill \end{array}$$

(12)

and $j\ne k$

$$\begin{array}{cc}\hfill \frac{{\partial }^2{f}_{(S,t)}}{\partial {\xi }_j\partial {\xi }_k}(Z,\xi )& =c_{\alpha }\frac{{\left[A_S(S)-{\parallel t\parallel }^2\right]}^{\alpha }}{{[A_S(Z)-{\displaystyle \sum _{j=1}^r}{\xi }_j^{p_j}{\overline{t_j}}^{p_j}]}^{2\alpha }}{p}_k{p}_j{\overline{t_k}}^{p_k}{\xi }_k^{p_k-1}{\overline{t_j}}^{p_j}{\xi }_j^{p_j-1}.\hfill \end{array}$$

(13)

By (10) and a direct calculation, we also have

$$\begin{array}{cc}\hfill \frac{{\partial }^2{f}_{(S,t)}}{\partial {\xi }_k^2}(Z,\xi )& =c_{\alpha }\frac{{\left[A_S(S)-{\parallel t\parallel }^2\right]}^{\alpha }}{{[A_S(Z)-{\displaystyle \sum _{j=1}^r}{\xi }_j^{p_j}{\overline{t_j}}^{p_j}]}^{2\alpha }}{p}_k^2{\overline{t_k}}^{2p_k}{\xi }_k^{2p_k-2}\hfill \\ \hfill & +\frac{{\left[A_S(S)-{\parallel t\parallel }^2\right]}^{\alpha }}{{[A_S(Z)-{\displaystyle \sum _{j=1}^r}{\xi }_j^{p_j}{\overline{t_j}}^{p_j}]}^{2\alpha -1}}{p}_k(p_k-1){\overline{t_k}}^{p_k}{\xi }_k^{p_k-2}.\hfill \end{array}$$

(14)

Then by (11)–(14) we have

$$\begin{array}{cc}\hfill & H_{f_{(S,t)}}(Z,\xi )={\displaystyle \sum _{i,k=1}^m}{\displaystyle \sum _{j,l=1}^n}|\frac{{\partial }^2{f}_{(S,t)}}{\partial z_{ij}\partial z_{kl}}(Z,\xi )|+{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k=1}^r}|\frac{{\partial }^2{f}_{(S,t)}}{\partial z_{ij}\partial {\xi }_k}(Z,\xi )|\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k=1}^r}|\frac{{\partial }^2{f}_{(S,t)}}{\partial {\xi }_k\partial z_{ij}}(Z,\xi )|+{\displaystyle \sum _{j\ne k,j=1}^r}|\frac{{\partial }^2{f}_{(S,t)}}{\partial {\xi }_j\partial {\xi }_k}(Z,\xi )|+{\displaystyle \sum _{k=1}^r}|\frac{{\partial }^2{f}_{(S,t)}}{\partial {\xi }_k^2}(Z,\xi )|\hfill \\ \hfill & \le \frac{{\left[A_S(S)-{\parallel t\parallel }^2\right]}^{\alpha }}{|A_S(Z)-{\displaystyle \sum _{j=1}^r}{\xi }_j^{p_j}{\overline{t_j}}^{p_j}{|}^{2\alpha }}(c_{\alpha }{\displaystyle \sum _{i,k=1}^m}{\displaystyle \sum _{j,l=1}^n}|A_{S,ij}(Z)||A_{S,kl}(Z)|\hfill \\ \hfill & +|A_S(Z)-{\displaystyle \sum _{j=1}^r}{\xi }_j^{p_j}{\overline{t_j}}^{p_j}|{\displaystyle \sum _{i,k=1}^m}{\displaystyle \sum _{j,l=1}^n}|A_{S,ij,kl}(Z)|\hfill \\ \hfill & +2c_{\alpha }{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k=1}^r}|p_k{\overline{t_k}}^{p_k}{\xi }_k^{p_k-1}||A_{S,ij}(Z)|+c_{\alpha }{\displaystyle \sum _{j\ne k,j=1}^r}|p_k{p}_j{\overline{t_k}}^{p_k}{\xi }_k^{p_k-1}{\overline{t_j}}^{p_j}{\xi }_j^{p_j-1}|\hfill \\ \hfill & +|A_S(Z)-{\displaystyle \sum _{j=1}^r}{\xi }_j^{p_j}{\overline{t_j}}^{p_j}|{\displaystyle \sum _{k=1}^r}|p_k(p_k-1){\overline{t_k}}^{p_k}{\xi }_k^{p_k-2}|)\hfill \\ \hfill & \triangleq \frac{{\left[A_S(S)-{\parallel t\parallel }^2\right]}^{\alpha }}{|A_S(Z)-{\displaystyle \sum _{j=1}^r}{\xi }_j^{p_j}{\overline{t_j}}^{p_j}{|}^{2\alpha }}F(Z,S,\xi ,t).\hfill \end{array}$$

(15)

From Lemmas 10–12, it follows that there exists a positive constant C independent of $(Z,\xi )$, $(S,t)\in {\mathrm{HE}}_I$ such that $F(Z,S,\xi ,t)\le C$. Then from Lemma 13 and (15), we have

$$\begin{array}{cc}\hfill & \underset{(Z,\xi )\in {\mathrm{HE}}_I}{sup}{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }{H}_{f_{(S,t)}}(Z,\xi )\le C.\hfill \end{array}$$

(16)

On the other hand, it is clear that

$$\begin{array}{c}\hfill |f_{(S,t)}(0,0)|\le C\end{array}$$

(17)

and

$$\begin{array}{c}\hfill \frac{\partial f_{(S,t)}}{\partial z_{ij}}(0,0)=\frac{\partial f_{(S,t)}}{\partial {\xi }_k}(0,0)=0.\end{array}$$

(18)

From (16)–(18), the desired result follows. The proof is completed. □

If $\alpha =1$, then on ${\mathrm{HE}}_I$ we consider the function

$$\begin{array}{c}\hfill g_{(S,t)}(Z,\xi )=\left[A_S(S)-{\parallel t\parallel }^2\right]ln\frac{1}{A_S(Z)-{\displaystyle \sum _{j=1}^r}{\xi }_j^{p_j}{\overline{t_j}}^{p_j}}.\end{array}$$

For the function $g_{(S,t)}(Z,\xi )$, we also have the following result, whose proof is similar to that of Lemma 14. So, the proof is omitted.

Lemma 15.

Let $\alpha =1$ and $p_j\ge 2$ for $j=1,2,\dots ,r$. Then $g_{(S,t)}\in {\mathcal{Z}}^{\alpha }({HE}_I)$ and there exists a positive constant C such that

$$\underset{(S,t)\in {HE}_I}{sup}{\parallel g_{(S,t)}\parallel }_{{\mathcal{Z}}^{\alpha }({HE}_I)}\le C.$$

3. Composition Operators on ${\mathcal{Z}}^{\mathit{\alpha }}\mathbf{(}{\mathbf{HE}}_{\mathit{I}}\mathbf{)}$

Now, we begin to study the boundedness and compactness of the composition operators on ${\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)$. We first have the following result for the boundedness.

Theorem 1.

Let φ be a holomorphic self-map of ${HE}_I$, $(W,\zeta )=\phi (Z,\xi )$ and $p_j\ge 2$ for $j=1,2,\dots ,r$. If

$$\begin{array}{c}\hfill \frac{{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }}{{\left[A_W(W)-{\parallel \zeta \parallel }^2\right]}^{\alpha }}|{\phi }^{\prime }(Z,\xi ){|}^2=O(1)\end{array}$$

and

$$\begin{array}{c}\hfill \frac{{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }}{{\left[A_W(W)-{\parallel \zeta \parallel }^2\right]}^{\alpha }}|{\phi }^{″}(Z,\xi )|=O(1)\end{array}$$

as $(W,\zeta )\to \partial {\mathrm{HE}}_I$, then the operator $C_{\phi }$ is bounded on ${\mathcal{Z}}^{\alpha }({HE}_I)$.

Conversely, if the operator $C_{\phi }$ is bounded on ${\mathcal{Z}}^{\alpha }({HE}_I)$, then

$$\begin{array}{c}\hfill M_1:=\underset{(Z,\xi )\in {HE}_I}{sup}{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }{H}_{f_{(A,\lambda )}\circ \phi }(Z,\xi )<\infty ,\end{array}$$

$$\begin{array}{c}\hfill M_2:=\underset{(Z,\xi )\in {HE}_I}{sup}\frac{{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }}{{\left[A_W(W)-{\parallel \zeta \parallel }^2\right]}^{\alpha }}H(Z,\xi )<\infty ,\end{array}$$

and

$$\begin{array}{cc}\hfill H(Z,\xi )& ={\displaystyle \sum _{i,k=1}^m}{\displaystyle \sum _{j,l=1}^n}|{\displaystyle \sum _{u,p=1}^m}{\displaystyle \sum _{v,q=1}^n}[A_{W,uv}(W)A_{W,pq}(W)\hfill \\ \hfill & +[A_W(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]A_{W,uv,pq}(W)]\frac{\partial {\phi }_{pq}}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{\displaystyle \sum _{j=1}^r}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}{A}_{W,uv}(W)\frac{\partial {\phi }_j}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +[A_W(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{A}_{W,uv}(W)\frac{{\partial }^2{\phi }_{uv}}{\partial z_{ij}\partial z_{kl}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^r}{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}{A}_{W,uv}(W)\frac{\partial {\phi }_{uv}}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_j}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^r}{\displaystyle \sum _{k=1}^r}{p}_j{p}_k{\zeta }_j^{p_j-1}{\zeta }_k^{p_k-1}{\overline{{\lambda }_j}}^{p_j}{\overline{{\lambda }_k}}^{p_k}\frac{\partial {\phi }_k}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_j}{\partial z_{ij}}(Z,\xi )|\hfill \\ \hfill & +[A_W(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]{\displaystyle \sum _{j=1}^r}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}\frac{{\partial }^2{\phi }_j}{\partial z_{ij}\partial z_{kl}}(Z,\xi )\hfill \\ \hfill & +2{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k=1}^r}|{\displaystyle \sum _{u,p=1}^m}{\displaystyle \sum _{v,q=1}^n}[A_{W,uv}(W)A_{W,pq}(W)\hfill \\ \hfill & +[A_W(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]A_{W,uv,pq}(W)]\frac{\partial {\phi }_{pq}}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{\displaystyle \sum _{j=1}^r}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}{A}_{W,uv}(W)\frac{\partial {\phi }_j}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +[A_W(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{A}_{W,uv}(W)\frac{{\partial }^2{\phi }_{uv}}{\partial z_{ij}\partial z_{kl}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^r}{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}{A}_{W,uv}(W)\frac{\partial {\phi }_{uv}}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_j}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^r}{\displaystyle \sum _{i=1}^r}{p}_j{p}_i{\zeta }_j^{p_j-1}{\zeta }_i^{p_i-1}{\overline{{\lambda }_j}}^{p_j}{\overline{{\lambda }_i}}^{p_i}\frac{\partial {\phi }_i}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_j}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +[A_W(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]{\displaystyle \sum _{j=1}^r}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}\frac{{\partial }^2{\phi }_j}{\partial z_{ij}\partial {\xi }_k}(Z,\xi )|\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k=1}^r}|{\displaystyle \sum _{u,p=1}^m}{\displaystyle \sum _{v,q=1}^n}[A_{W,uv}(W)A_{W,pq}(W)\hfill \\ \hfill & +[A_W(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]A_{W,uv,pq}(W)]\frac{\partial {\phi }_{pq}}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial {\xi }_j}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{\displaystyle \sum _{j=1}^r}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}{A}_{W,uv}(W)\frac{\partial {\phi }_j}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial {\xi }_j}(Z,\xi )\hfill \\ \hfill & +[A_W(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{A}_{W,uv}(W)\frac{{\partial }^2{\phi }_{uv}}{\partial {\xi }_j\partial {\xi }_k}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{p=1}^r}{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}{A}_{W,uv}(W)\frac{\partial {\phi }_{uv}}{\partial {\xi }_j}(Z,\xi )\frac{\partial {\phi }_p}{\partial {\xi }_k}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{p=1}^r}{\displaystyle \sum _{i=1}^r}{p}_j{p}_i{\zeta }_j^{p_j-1}{\zeta }_i^{p_i-1}{\overline{{\lambda }_j}}^{p_j}{\overline{{\lambda }_i}}^{p_i}\frac{\partial {\phi }_i}{\partial {\xi }_j}(Z,\xi )\frac{\partial {\phi }_p}{\partial {\xi }_k}(Z,\xi )\hfill \\ \hfill & +[A_W(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]{\displaystyle \sum _{j=1}^r}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}\frac{{\partial }^2{\phi }_j}{\partial {\xi }_j\partial {\xi }_k}(Z,\xi )|.\hfill \end{array}$$

Proof. 

By Lemmas 2 and 3, for all $f\in {\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)$ and $(Z,\xi )\in {\mathrm{HE}}_I$, we have

$$\begin{array}{cc}\hfill {\mathcal{Z}}_{f\circ \phi }& ={\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }{H}_{f\circ \phi }(Z,\xi )\hfill \\ \hfill & \le C{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }\left(H_f(\phi (Z,\xi ))|{\phi }^{\prime }(Z,\xi ){|}^2+|\nabla f(\phi (Z,\xi ))||{\phi }^{″}(Z,\xi )|\right)\hfill \\ \hfill & =C\frac{{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }}{{\left[A_W(W)-{\parallel \zeta \parallel }^2\right]}^{\alpha }}|{\phi }^{\prime }(Z,\xi ){|}^2{\left[A_W(W)-{\parallel \zeta \parallel }^2\right]}^{\alpha }{H}_f(\phi (Z,\xi ))\hfill \\ \hfill & +C{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }|{\phi }^{″}(Z,\xi )||\nabla f(\phi (Z,\xi ))|\hfill \\ \hfill & \le C\frac{{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }}{{\left[A_W(W)-{\parallel \zeta \parallel }^2\right]}^{\alpha }}|{\phi }^{\prime }(Z,\xi ){|}^2{\parallel f\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}\hfill \\ \hfill & +C\frac{{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }}{{\left[A_W(W)-{\parallel \zeta \parallel }^2\right]}^{\alpha }}|{\phi }^{″}(Z,\xi )|\parallel f{\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}\hfill \\ \hfill & \le {C\parallel f\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}.\hfill \end{array}$$

(19)

By Lemma 4, we have

$$\begin{array}{c}\hfill |f(\phi (0,0))|\le C\parallel f{\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}.\end{array}$$

(20)

From Lemma 3, it follows that

$$\begin{array}{cc}\hfill |\frac{\partial (f\circ \phi )}{\partial z_{ij}}(0,0)|& =|{\displaystyle \sum _{p=1}^m}{\displaystyle \sum _{q=1}^n}\frac{\partial f}{\partial V_{pq}}(\phi (0,0))\frac{{\phi }_{pq}}{\partial z_{ij}}(0,0)+{\displaystyle \sum _{k=1}^r}\frac{\partial f}{\partial V_k}(\phi (0,0))\frac{\partial {\phi }_k}{\partial z_{ij}}(0,0)|\hfill \\ \hfill & \le {\displaystyle \sum _{p=1}^m}{\displaystyle \sum _{q=1}^n}|\frac{\partial f}{\partial V_{pq}}(\phi (0,0))||\frac{{\phi }_{pq}}{\partial z_{ij}}(0,0)|+{\displaystyle \sum _{k=1}^r}|\frac{\partial f}{\partial V_k}(\phi (0,0))||\frac{\partial {\phi }_k}{\partial z_{ij}}(0,0)|\hfill \\ \hfill & \le C\parallel f{\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}.\hfill \end{array}$$

(21)

Similar to (21), we also can obtain

$$\begin{array}{c}\hfill |\frac{\partial (f\circ \phi )}{\partial {\xi }_k}(0,0)|\le C\parallel f{\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}.\end{array}$$

(22)

Then, by (19)–(22) we have

$$\begin{array}{c}\hfill {\parallel C_{\phi }f\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}\le C{\parallel f\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}.\end{array}$$

(23)

(23) shows that $C_{\phi }$ is bounded on ${\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)$.

Conversely, assume that the operator $C_{\phi }$ is bounded on ${\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)$ with

$$\begin{array}{c}\hfill {\parallel C_{\phi }f\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}\le C{\parallel f\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}\end{array}$$

(24)

for all $f\in {\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)$.

First, we consider the case of $\alpha \ne 1$. Let $(P,\lambda )=\phi (S,t)$, where $(S,t)$ is a fixed point in ${\mathrm{HE}}_I$. We will use the function $f_{(P,\lambda )}$ defined by

$$\begin{array}{c}\hfill f_{(P,\lambda )}(Z,\xi )=\frac{1}{2-2\alpha }\frac{{\left[A_P(P)-{\parallel \lambda \parallel }^2\right]}^{\alpha }}{{[A_P(Z)-{\displaystyle \sum _{j=1}^r}{\xi }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]}^{2\alpha -2}}.\end{array}$$

By Lemma 14, we know that $f_{(P,\lambda )}\in {\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)$. By a calculation, we have

$$\begin{array}{cc}\hfill \frac{{\partial }^2{f}_{(P,\lambda )}}{\partial V_{uv}\partial V_{pq}}(\phi (Z,\xi ))& =c_{\alpha }\frac{{\left[A_P(P)-{\parallel \lambda \parallel }^2\right]}^{\alpha }}{{[A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]}^{2\alpha }}{A}_{P,uv}(W)A_{P,pq}(W)\hfill \\ \hfill & +\frac{{\left[A_P(P)-{\parallel \lambda \parallel }^2\right]}^{\alpha }}{{[A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]}^{2\alpha -1}}{A}_{P,uv,pq}(W),\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill \frac{{\partial }^2{f}_{(P,\lambda )}}{\partial V_{uv}\partial V_j}(\phi (Z,\xi ))& =c_{\alpha }\frac{{\left[A_P(P)-{\parallel \lambda \parallel }^2\right]}^{\alpha }{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}}{{[A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]}^{2\alpha }}{A}_{P,uv}(W),\hfill \end{array}$$

(26)

$$\begin{array}{c}\hfill \frac{\partial f_{(P,\lambda )}}{\partial V_{uv}}(\phi (Z,\xi ))=c_{\alpha }\frac{{\left[A_P(P)-{\parallel \lambda \parallel }^2\right]}^{\alpha }}{{[A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]}^{2\alpha -1}}{A}_{P,uv}(W),\end{array}$$

(27)

$$\begin{array}{c}\hfill \frac{{\partial }^2{f}_{(P,\lambda )}}{\partial V_j\partial V_k}(\phi (Z,\xi ))=c_{\alpha }\frac{{\left[A_P(P)-{\parallel \lambda \parallel }^2\right]}^{\alpha }{p}_j{p}_k{\zeta }_j^{p_j-1}{\zeta }_k^{p_k-1}{\overline{{\lambda }_j}}^{p_j}{\overline{{\lambda }_k}}^{p_k}}{{[A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]}^{2\alpha }}(j\ne k),\end{array}$$

(28)

and

$$\begin{array}{c}\hfill \frac{\partial f_{(P,\lambda )}}{\partial V_j}(\phi (Z,\xi ))=\frac{{\left[A_P(P)-{\parallel \lambda \parallel }^2\right]}^{\alpha }{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}}{{[A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]}^{2\alpha -1}}.\end{array}$$

(29)

From (29), it follows that

$$\begin{array}{cc}\hfill \frac{{\partial }^2{f}_{(P,\lambda )}}{\partial V_k^2}(\phi (Z,\xi ))& =c_{\alpha }\frac{{\left[A_P(P)-{\parallel \lambda \parallel }^2\right]}^{\alpha }}{{[A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]}^{2\alpha }}{p}_k^2{\zeta }_k^{2p_k-2}{\overline{\lambda }}_k^{2p_k}\hfill \\ \hfill & +\frac{{\left[A_P(P)-{\parallel \lambda \parallel }^2\right]}^{\alpha }}{{[A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]}^{2\alpha -1}}{p}_k(p_k-1){\zeta }_k^{p_k-2}{\overline{\lambda }}_k^{p_k}.\hfill \end{array}$$

(30)

By using (25)–(30) and Lemma 1, we have

$$\begin{array}{cc}\hfill & \frac{{\partial }^2(f_{(P,\lambda )}\circ \phi )}{\partial z_{ij}\partial z_{kl}}(Z,\xi )={\displaystyle \sum _{u,p=1}^m}{\displaystyle \sum _{v,q=1}^n}\frac{{\partial }^2{f}_{(P,\lambda )}}{\partial V_{uv}\partial V_{pq}}(\phi (Z,\xi ))\frac{\partial {\phi }_{pq}}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{\displaystyle \sum _{j=1}^r}\frac{{\partial }^2{f}_{(P,\lambda )}}{\partial V_{uv}\partial V_j}(\phi (Z,\xi ))\frac{\partial {\phi }_j}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}\frac{\partial f_{(P,\lambda )}}{\partial V_{uv}}(\phi (Z,\xi ))\frac{{\partial }^2{\phi }_{uv}}{\partial z_{ij}\partial z_{kl}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^r}{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}\frac{{\partial }^2{f}_{(P,\lambda )}}{\partial V_j\partial V_{uv}}(\phi (Z,\xi ))\frac{\partial {\phi }_{uv}}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_j}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^r}{\displaystyle \sum _{k=1}^r}\frac{{\partial }^2{f}_{(P,\lambda )}}{\partial V_j\partial V_k}(\phi (Z,\xi ))\frac{\partial {\phi }_k}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_j}{\partial z_{ij}}(Z,\xi )+{\displaystyle \sum _{j=1}^r}\frac{\partial f_{(P,\lambda )}}{\partial V_j}(\phi (Z,\xi ))\frac{{\partial }^2{\phi }_j}{\partial z_{ij}\partial z_{kl}}(Z,\xi )\hfill \\ \hfill & =\frac{{\left[A_P(P)-{\parallel \lambda \parallel }^2\right]}^{\alpha }}{{[A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]}^{2\alpha }}\{c_{\alpha }{\displaystyle \sum _{u,p=1}^m}{\displaystyle \sum _{v,q=1}^n}[A_{P,uv}(W)A_{P,pq}(W)\hfill \\ \hfill & +(A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j})A_{P,uv,pq}(W)]\frac{\partial {\phi }_{pq}}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +c_{\alpha }{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{\displaystyle \sum _{j=1}^r}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}{A}_{P,uv}(W)\frac{\partial {\phi }_j}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +(A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}){\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{A}_{uv}(W,A)\frac{{\partial }^2{\phi }_{uv}}{\partial z_{ij}\partial z_{kl}}(Z,\xi )\hfill \\ \hfill & +c_{\alpha }{\displaystyle \sum _{j=1}^r}{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}{A}_{P,uv}(W)\frac{\partial {\phi }_{uv}}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_j}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +c_{\alpha }{\displaystyle \sum _{j\ne k,j,k=1}^r}{p}_j{p}_k{\zeta }_j^{p_j-1}{\zeta }_k^{p_k-1}{\overline{{\lambda }_j}}^{p_j}{\overline{{\lambda }_k}}^{p_k}\frac{\partial {\phi }_k}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_j}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +c_{\alpha }{\displaystyle \sum _{k=1}^r}{p}_k^2{\zeta }_k^{2p_k-2}{\overline{\lambda }}_k^{2p_k}\frac{\partial {\phi }_k}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_k}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +(A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}){\displaystyle \sum _{k=1}^r}{p}_k(p_k-1){\zeta }_k^{p_k-2}{\overline{\lambda }}_k^{p_k}\frac{\partial {\phi }_k}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_k}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +(A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}){\displaystyle \sum _{j=1}^r}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}\frac{{\partial }^2{\phi }_j}{\partial z_{ij}\partial z_{kl}}(Z,\xi )\}.\hfill \end{array}$$

(31)

Similarly, we obtain

$$\begin{array}{cc}\hfill & \frac{{\partial }^2(f_{(P,\lambda )}\circ \phi )}{\partial z_{ij}\partial {\xi }_k}(Z,\xi )=\frac{{\left[A_P(P)-{\parallel \lambda \parallel }^2\right]}^{\alpha }}{{[A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]}^{2\alpha }}\{c_{\alpha }{\displaystyle \sum _{u,p=1}^m}{\displaystyle \sum _{v,q=1}^n}[A_{P,uv}(W)A_{P,pq}(W)\hfill \\ \hfill & +(A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j})A_{P,uv,pq}(W)]\frac{\partial {\phi }_{pq}}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +c_{\alpha }{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{\displaystyle \sum _{j=1}^r}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}{A}_{P,uv}(W)\frac{\partial {\phi }_j}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +(A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}){\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{A}_{P,uv}(W)\frac{{\partial }^2{\phi }_{uv}}{\partial z_{ij}\partial z_{kl}}(Z,\xi )\hfill \\ \hfill & +c_{\alpha }{\displaystyle \sum _{j=1}^r}{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}{A}_{P,uv}(W)\frac{\partial {\phi }_{uv}}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_j}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +c_{\alpha }{\displaystyle \sum _{u\ne v,u,v=1}^r}{p}_u{p}_v{\zeta }_v^{p_u-1}{\zeta }_u^{p_v-1}{\overline{{\lambda }_u}}^{p_u}{\overline{{\lambda }_v}}^{p_v}\frac{\partial {\phi }_u}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_v}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +c_{\alpha }{\displaystyle \sum _{u=1}^r}{p}_u^2{\zeta }_u^{2p_u-2}{\overline{\lambda }}_u^{2p_u}\frac{\partial {\phi }_u}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_u}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +(A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}){\displaystyle \sum _{u=1}^r}{p}_u(p_u-1){\zeta }_u^{p_u-2}{\overline{\lambda }}_u^{p_u}\frac{\partial {\phi }_u}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_u}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +(A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}){\displaystyle \sum _{j=1}^r}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}\frac{{\partial }^2{\phi }_j}{\partial z_{ij}\partial {\xi }_k}(Z,\xi )\}\hfill \end{array}$$

(32)

and

$$\begin{array}{cc}\hfill & \frac{{\partial }^2(f_{(P,\lambda )}\circ \phi )}{\partial {\xi }_j\partial {\xi }_k}(Z,\xi )=\frac{{\left[A_P(P)-{\parallel \lambda \parallel }^2\right]}^{\alpha }}{{[A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]}^{2\alpha }}\{c_{\alpha }{\displaystyle \sum _{u,p=1}^m}{\displaystyle \sum _{v,q=1}^n}[A_{P,uv}(W)A_{P,pq}(W)\hfill \\ \hfill & +(A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j})A_{P,uv,pq}(W)]\frac{\partial {\phi }_{pq}}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial {\xi }_j}(Z,\xi )\hfill \\ \hfill & +c_{\alpha }{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{\displaystyle \sum _{j=1}^r}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}{A}_{P,uv}(W)\frac{\partial {\phi }_j}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial {\xi }_j}(Z,\xi )\hfill \\ \hfill & +(A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}){\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{A}_{P,uv}(W)\frac{{\partial }^2{\phi }_{uv}}{\partial {\xi }_j\partial {\xi }_k}(Z,\xi )\hfill \\ \hfill & +c_{\alpha }{\displaystyle \sum _{p=1}^r}{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}{A}_{P,uv}(W)\frac{\partial {\phi }_{uv}}{\partial {\xi }_j}(Z,\xi )\frac{\partial {\phi }_p}{\partial {\xi }_k}(Z,\xi )\hfill \\ \hfill & +c_{\alpha }{\displaystyle \sum _{u\ne v,u,v=1}^r}{p}_u{p}_v{\zeta }_v^{p_u-1}{\zeta }_u^{p_v-1}{\overline{{\lambda }_u}}^{p_u}{\overline{{\lambda }_v}}^{p_v}\frac{\partial {\phi }_u}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_v}{\partial {\xi }_j}(Z,\xi )\hfill \\ \hfill & +c_{\alpha }{\displaystyle \sum _{u=1}^r}{p}_u^2{\zeta }_u^{2p_u-2}{\overline{\lambda }}_u^{2p_u}\frac{\partial {\phi }_u}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_u}{\partial {\xi }_j}(Z,\xi )\hfill \\ \hfill & +(A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}){\displaystyle \sum _{u=1}^r}{p}_u(p_u-1){\zeta }_u^{p_u-2}{\overline{\lambda }}_u^{p_u}\frac{\partial {\phi }_u}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_u}{\partial {\xi }_j}(Z,\xi )\hfill \\ \hfill & +(A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}){\displaystyle \sum _{j=1}^r}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}\frac{{\partial }^2{\phi }_j}{\partial {\xi }_j\partial {\xi }_k}(Z,\xi )\}.\hfill \end{array}$$

(33)

Then by (31)–(33), we have

$$\begin{array}{cc}\hfill & {\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }{H}_{f_{(P,\lambda )}\circ \phi }(Z,\xi )=\frac{{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }{\left[A_P(P)-{\parallel \lambda \parallel }^2\right]}^{\alpha }}{{[A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]}^{2\alpha }}\hfill \\ \hfill & \times \{{\displaystyle \sum _{i,k=1}^m}{\displaystyle \sum _{j,l=1}^n}|c_{\alpha }{\displaystyle \sum _{u,p=1}^m}{\displaystyle \sum _{v,q=1}^n}[A_{P,uv}(W)A_{P,pq}(W)\hfill \\ \hfill & +(A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j})A_{P,uv,pq}(W)]\frac{\partial {\phi }_{pq}}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{\displaystyle \sum _{j=1}^r}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}{A}_{P,uv}(W)\frac{\partial {\phi }_j}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +[A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{A}_{P,uv}(W)\frac{{\partial }^2{\phi }_{uv}}{\partial z_{ij}\partial z_{kl}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^r}\sum _{\begin{array}{c}1\le u\le m\\ 1\le v\le n\end{array}}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}{A}_{uv}(W,A)\frac{\partial {\phi }_{uv}}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_j}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{j\ne k,j,k=1}^r}{p}_j{p}_k{\zeta }_j^{p_j-1}{\zeta }_k^{p_k-1}{\overline{{\lambda }_j}}^{p_j}{\overline{{\lambda }_k}}^{p_k}\frac{\partial {\phi }_k}{\partial z_{kl}}(Z,\xi )\frac{\partial {\phi }_j}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +[A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]{\displaystyle \sum _{j=1}^r}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}\frac{{\partial }^2{\phi }_j}{\partial z_{ij}\partial z_{kl}}(Z,\xi )|\hfill \\ \hfill & +2{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k=1}^r}|{\displaystyle \sum _{u,p=1}^m}{\displaystyle \sum _{v,q=1}^n}[A_{P,uv}(W)A_{P,pq}(W)\hfill \\ \hfill & +[A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]A_{P,uv,pq}(W)]\frac{\partial {\phi }_{pq}}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{\displaystyle \sum _{j=1}^r}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}{A}_{P,uv}(W)\frac{\partial {\phi }_j}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +[A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{A}_{P,uv}(W)\frac{{\partial }^2{\phi }_{uv}}{\partial z_{ij}\partial z_{kl}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^r}{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}{A}_{P,uv}(W)\frac{\partial {\phi }_{uv}}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_j}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^r}{\displaystyle \sum _{i=1}^r}{p}_j{p}_i{\zeta }_j^{p_j-1}{\zeta }_i^{p_i-1}{\overline{{\lambda }_j}}^{p_j}{\overline{{\lambda }_i}}^{p_i}\frac{\partial {\phi }_i}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_j}{\partial z_{ij}}(Z,\xi )\hfill \\ \hfill & +[A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]{\displaystyle \sum _{j=1}^r}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}\frac{{\partial }^2{\phi }_j}{\partial z_{ij}\partial {\xi }_k}(Z,\xi )|\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k=1}^r}|{\displaystyle \sum _{u,p=1}^m}{\displaystyle \sum _{v,q=1}^n}[A_{P,uv}(W)A_{P,pq}(W)\hfill \\ \hfill & +[A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]A_{P,uv,pq}(W)]\frac{\partial {\phi }_{pq}}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial {\xi }_j}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{\displaystyle \sum _{j=1}^r}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}{A}_{P,uv}(W)\frac{\partial {\phi }_j}{\partial {\xi }_k}(Z,\xi )\frac{\partial {\phi }_{uv}}{\partial {\xi }_j}(Z,\xi )\hfill \\ \hfill & +[A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{A}_{P,uv}(W)\frac{{\partial }^2{\phi }_{uv}}{\partial {\xi }_j\partial {\xi }_k}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{p=1}^r}{\displaystyle \sum _{u=1}^m}{\displaystyle \sum _{v=1}^n}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}{A}_{P,uv}(W)\frac{\partial {\phi }_{uv}}{\partial {\xi }_j}(Z,\xi )\frac{\partial {\phi }_p}{\partial {\xi }_k}(Z,\xi )\hfill \\ \hfill & +{\displaystyle \sum _{p=1}^r}{\displaystyle \sum _{i=1}^r}{p}_j{p}_i{\zeta }_j^{p_j-1}{\zeta }_i^{p_i-1}{\overline{{\lambda }_j}}^{p_j}{\overline{{\lambda }_i}}^{p_i}\frac{\partial {\phi }_i}{\partial {\xi }_j}(Z,\xi )\frac{\partial {\phi }_p}{\partial {\xi }_k}(Z,\xi )\hfill \\ \hfill & +[A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}]{\displaystyle \sum _{j=1}^r}{p}_j{\zeta }_j^{p_j-1}{\overline{{\lambda }_j}}^{p_j}\frac{{\partial }^2{\phi }_j}{\partial {\xi }_j\partial {\xi }_k}(Z,\xi )|\}\hfill \\ \hfill & =\frac{{\left(A_Z(Z)-{\parallel \xi \parallel }^2\right)}^{\alpha }{\left(A_P(P)-{\parallel \lambda \parallel }^2\right)}^{\alpha }}{{(A_P(W)-{\displaystyle \sum _{j=1}^r}{\zeta }_j^{p_j}{\overline{{\lambda }_j}}^{p_j})}^{2\alpha }}H(Z,\xi )\hfill \\ \hfill & \le \parallel C_{\phi }{f}_{(P,\lambda )}{\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}.\hfill \end{array}$$

(34)

By replacing $(Z,\xi )$ by $(S,t)$ in (34), we obtain

$$\begin{array}{c}\hfill \underset{(S,t)\in {\mathrm{HE}}_I}{sup}\frac{{\left[A_S(S)-{\parallel t\parallel }^2\right]}^{\alpha }}{{\left[A_P(P)-{\parallel \lambda \parallel }^2\right]}^{\alpha }}H(S,t)<\infty .\end{array}$$

If $\alpha =1$, let $(P,\lambda )=\phi (S,t)$, where $(S,t)$ is a fixed point in ${\mathrm{HE}}_I$. We will use the function $g_{(P,\lambda )}$ defined by

$$\begin{array}{c}\hfill g_{(P,\lambda )}(Z,\xi )=\left[A_P(P)-{\parallel \lambda \parallel }^2\right]ln\frac{1}{A_P(Z)-{\displaystyle \sum _{j=1}^r}{\xi }_j^{p_j}{\overline{{\lambda }_j}}^{p_j}}.\end{array}$$

For the same reason, it can be proved for this case, and the details are omitted. The proof is completed. □

Next, we study the compactness of the operator $C_{\phi }$ on ${\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)$.

Theorem 2.

Let φ be a holomorphic self-map of ${HE}_I$, $(W,\zeta )=\phi (Z,\xi )$ and $p_j\ge 2$ for $j=1,2,\dots ,r$. If

$$\begin{array}{c}\hfill \underset{(W,\zeta )\to \partial {HE}_I}{lim}\frac{{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }}{{\left[A_W(W)-{\parallel \zeta \parallel }^2\right]}^{\alpha }}|{\phi }^{\prime }(Z,\xi ){|}^2=0\end{array}$$

(35)

and

$$\begin{array}{c}\hfill \underset{(W,\zeta )\to \partial {HE}_I}{lim}\frac{{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }}{{\left[A_W(W)-{\parallel \zeta \parallel }^2\right]}^{\alpha }}|{\phi }^{″}(Z,\xi )|=0,\end{array}$$

(36)

then the operator $C_{\phi }$ is compact on ${\mathcal{Z}}^{\alpha }({HE}_I)$.

Conversely, if the operator $C_{\phi }$ is compact on ${\mathcal{Z}}^{\alpha }({HE}_I)$, then

$$\begin{array}{c}\hfill \underset{(W,\zeta )\to \partial {HE}_I}{lim}\frac{{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }}{{\left[A_W(W)-{\parallel \zeta \parallel }^2\right]}^{\alpha }}H(Z,\xi )=0.\end{array}$$

(37)

Proof. 

In order to prove that the operator $C_{\phi }$ is compact on ${\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)$, by Lemma 5 we only need to prove that, if $\left\{f_i\right\}$ is a sequence in ${\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)$ such that ${sup}_{i\in \mathbb{N}}{\parallel f_i\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}\le M$ and $f_i\to 0$ uniformly on any compact subset of ${\mathrm{HE}}_I$ as $i\to \infty$, then

$$\underset{i\to \infty }{lim}{\parallel C_{\phi }{f}_i\parallel }_{{\mathcal{Z}}^{\alpha }({HE}_I)}=0.$$

We first observe that the conditions (35) and (36) imply that for every $\epsilon >0$, there exists an $\sigma >0$, such that for any $(Z,\xi )\in K=\{(Z,\xi )\in {\mathrm{HE}}_I:\mathrm{dist}(\phi (Z,\xi ),\partial {\mathrm{HE}}_I)<\sigma \}$

$$\begin{array}{c}\hfill \frac{{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }}{{\left[A_W(W)-{\parallel \zeta \parallel }^2\right]}^{\alpha }}|{\phi }^{\prime }(Z,\xi ){|}^2<\epsilon \end{array}$$

(38)

and

$$\begin{array}{c}\hfill \frac{{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }}{\left[A_W(W)-{\parallel \zeta \parallel }^2\right]{)}^{\alpha }}|{\phi }^{″}(Z,\xi )|<\epsilon .\end{array}$$

(39)

For such $\epsilon$ and $\sigma$, by using (38) and (39), Lemmas 2 and 3, we have

$$\begin{array}{cc}\hfill {\mathcal{Z}}_{f_i\circ \phi }& =\underset{(Z,\xi )\in {HE}_I}{sup}{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }{H}_{f_i\circ \phi }(Z,\xi )\hfill \\ \hfill & \le (\underset{(Z,\xi )\in K}{sup}+\underset{(Z,\xi )\in {\mathrm{HE}}_I\setminus K}{sup}){\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }{H}_{f_i\circ \phi }(Z,\xi )\hfill \\ \hfill & \le CM\underset{(Z,\xi )\in K}{sup}\frac{{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }}{{\left[A_W(W)-{\parallel \zeta \parallel }^2\right]}^{\alpha }}|{\phi }^{″}(Z,\xi )|\hfill \\ \hfill & +CM\underset{(Z,\xi )\in K}{sup}\frac{{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }}{{\left[A_W(W)-{\parallel \zeta \parallel }^2\right]}^{\alpha }}|{\phi }^{\prime }(Z,\xi ){|}^2\hfill \\ \hfill & +C{R}_1\underset{(Z,\xi )\in {\mathrm{HE}}_I\setminus K}{sup}|\nabla f_i(\phi (Z,\xi ))|+C{R}_2\underset{(Z,\xi )\in {\mathrm{HE}}_I\setminus K}{sup}{H}_{f_i}(\phi (Z,\xi )),\hfill \end{array}$$

(40)

where

$$R_1=\underset{(Z,\xi )\in {\mathrm{HE}}_I\setminus K}{sup}{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }|{\phi }^{″}(Z,\xi )|$$

and

$$R_2=\underset{(Z,\xi )\in {\mathrm{HE}}_I\setminus K}{sup}{\left[A_Z(Z)-{\parallel \xi \parallel }^2\right]}^{\alpha }|{\phi }^{\prime }(Z,\xi ){|}^2.$$

It is clear that $R_1$, $R_2<\infty$. Since $\left\{f_i\right\}$ converges to zero uniformly on any compact subset of ${\mathrm{HE}}_I$ as $i\to \infty$ implies that $\left\{H_{f_i}\right\}$ and $\{\nabla f_i\}$ also do as $i\to \infty$. From this, (40) and (21), we obtain

$$\begin{array}{c}\hfill \underset{i\to \infty }{lim}{\mathcal{Z}}_{f_i\circ \phi }=\underset{i\to \infty }{lim}{f}_i(\phi (0,0))=\underset{j\to \infty }{lim}\frac{\partial (f_i\circ \phi )}{\partial z_{ij}}(0,0)=\underset{j\to \infty }{lim}\frac{\partial (f_i\circ \phi )}{\partial {\xi }_k}(0,0)=0.\end{array}$$

(41)

From (41), it follows that

$$\begin{array}{c}\hfill \underset{i\to \infty }{lim}{\parallel C_{\phi }{f}_i\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}=0,\end{array}$$

(42)

which shows that the operator $C_{\phi }$ is compact on ${\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)$.

Conversely, assume that the operator $C_{\phi }$ is compact on ${\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)$. Then the operator $C_{\phi }$ is bounded on ${\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)$. Consider a sequence $\left\{(P_i,{\lambda }_i)\right\}=\left\{\phi (S_i,t_i)\right\}$ in ${\mathrm{HE}}_I$ such that $\phi (S_i,t_i)\to \partial {\mathrm{HE}}_I$ as $i\to \infty$. If such a sequence does not exist, then condition (37) obviously holds. Using this sequence, we define the function sequence $f_i(Z,\xi )=f_{(P_i,{\lambda }_i)}(Z,\xi )$ for $\alpha \ne 1$ (if $\alpha =1$, consider $g_i(Z,\xi )=g_{(P_i,{\lambda }_i)}(Z,\xi )$), where $f_{(P_i,{\lambda }_i)}$ is the function $f_{(P,\lambda )}$ replaced $(P,\lambda )$ by $(P_i,{\lambda }_i)$ in the proof of Theorem 1. By Lemma 13, we see that the sequence $\left\{f_i\right\}$ is uniformly bounded in ${\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)$. It is easy to see that $\left\{f_i\right\}$ converges to zero uniformly on any compact subset of ${\mathrm{HE}}_I$ as $i\to \infty$. So, by Lemma 5, ${lim}_{i\to \infty }{\parallel C_{\phi }{f}_i\parallel }_{{\mathcal{Z}}^{\alpha }({\mathrm{HE}}_I)}=0$. From this and similar to the proof of Theorem 1 (here the details are omitted), we have

$$\begin{array}{c}\hfill \underset{i\to \infty }{lim}\frac{{\left[A_{S_i}(S_i)-{\parallel t_i\parallel }^2\right]}^{\alpha }}{{\left[A_{P_i}(P_i)-{\parallel {\lambda }_i\parallel }^2\right]}^{\alpha }}H(S_i,t_i)=0.\end{array}$$

The proof is completed. □