Weighted Composition–Differentiation Operators Between Bers-Type Spaces on Generalized Hua Domains of the First Kind

Abstract

This paper investigates weighted composition–differentiation operators acting between Bers-type spaces defined on a generalized Hua domain of the first kind. By establishing a key norm inequality for functions in these spaces, we obtain necessary and sufficient conditions for the boundedness and compactness of such operators.

1. Introduction

Let $\gamma =({\gamma }_1,\dots ,{\gamma }_n)$ and $\eta =({\eta }_1,\dots ,{\eta }_n)$ be two points in ${\mathbb{C}}^n$. Their standard inner product is defined by

$$\langle \gamma ,\eta \rangle ={\displaystyle \sum _{j=1}^n}{\gamma }_j{\overline{\eta }}_j.$$

This inner product induces the standard Euclidean norm on ${\mathbb{C}}^n$

$$\parallel \gamma \parallel ={\langle \gamma ,\gamma \rangle }^{{\displaystyle \frac{1}{2}}}={\left({\displaystyle \sum _{j=1}^n}{\left|{\gamma }_j\right|}^2\right)}^{{\displaystyle \frac{1}{2}}}.$$

Now let $r\in \mathbb{N}$, and let $\mathcal{P}=({\mathcal{P}}_1,\dots ,{\mathcal{P}}_r)$ be an r-tuple with ${\mathcal{P}}_i>0$ for every i. For any $\zeta =({\zeta }_1,\dots ,{\zeta }_r)\in {\mathbb{C}}^{n_1}\times \cdots \times {\mathbb{C}}^{n_r}$, we define the $\mathcal{P}$-norm by

$${\parallel \zeta \parallel }_{\mathcal{P}}^2={\displaystyle \sum _{j=1}^r}{\left|{\zeta }_j\right|}^{2{\mathcal{P}}_j}.$$

Correspondingly, for $\zeta =({\zeta }_1,\dots ,{\zeta }_r)$, $\tau =({\tau }_1,\dots ,{\tau }_r)\in {\mathbb{C}}^{n_1}\times \cdots \times {\mathbb{C}}^{n_r}$, we introduce the associated form

$${\langle \zeta ,\tau \rangle }_{\mathcal{P}}={\langle {\zeta }_1,{\tau }_1\rangle }^{{\mathcal{P}}_1}+{\langle {\zeta }_2,{\tau }_2\rangle }^{{\mathcal{P}}_2}+\cdots +{\langle {\zeta }_r,{\tau }_r\rangle }^{{\mathcal{P}}_r}.$$

When $\zeta =\tau$, we have

$${\langle \zeta ,\zeta \rangle }_{\mathcal{P}}={\displaystyle \sum _{j=1}^r}{\langle {\zeta }_j,{\zeta }_j\rangle }^{{\mathcal{P}}_j}={\displaystyle \sum _{j=1}^r}|{\zeta }_j{|}^{2{\mathcal{P}}_j}={\parallel \zeta \parallel }_{\mathcal{P}}^2,$$

which shows that the form is positive definite and compatible with the $\mathcal{P}$-norm.

We proceed to establish the upper bound estimate:

$$\begin{array}{cc}\hfill {|\langle \zeta ,\tau \rangle }_{\mathcal{P}}|& =\left|{\displaystyle \sum _{j=1}^r}{\langle {\zeta }_j,{\tau }_j\rangle }^{{\mathcal{P}}_j}\right|\le {\displaystyle \sum _{j=1}^r}|\langle {\zeta }_j,{\tau }_j\rangle {|}^{{\mathcal{P}}_j}\le {\displaystyle \sum _{j=1}^r}|{\zeta }_j{|}^{{\mathcal{P}}_j}{|{\tau }_j|}^{{\mathcal{P}}_j}\hfill \\ & \le {\left({\displaystyle \sum _{j=1}^r}{\left|{\zeta }_j\right|}^{2{\mathcal{P}}_j}\right)}^{{\displaystyle \frac{1}{2}}}{\left({\displaystyle \sum _{j=1}^r}{\left|{\tau }_j\right|}^{2{\mathcal{P}}_j}\right)}^{{\displaystyle \frac{1}{2}}}={\parallel \zeta \parallel }_{\mathcal{P}}{\parallel \tau \parallel }_{\mathcal{P}}.\hfill \end{array}$$

Let $\mathcal{O}=({\mathcal{O}}_1,\dots ,{\mathcal{O}}_n)\in {\mathbb{C}}^n$ and $\varrho =({\varrho }_1,\dots ,{\varrho }_n)$ with ${\varrho }_j>0$ for $j=1,\dots ,n$. The set

$$\mathbb{P}(\mathcal{O},\varrho )=\left\{(z_1,\dots ,z_n)\in {\mathbb{C}}^n:|z_j-{\mathcal{O}}_j|<{\varrho }_j,j=1,\dots ,n\right\}$$

is called a polydisk (or polydisc) with center $\mathcal{O}$ and radius $\varrho$. In particular, when $\mathcal{O}=0$ and $\varrho =1$ for all j, this is called a unit polydisk, denoted by ${\mathbb{D}}^n$, i.e.,

$${\mathbb{D}}^n=\left\{(z_1,\dots ,z_n)\in {\mathbb{C}}^n:\left|z_j\right|<1,j=1,\dots ,n\right\}.$$

Clearly, when $n=1$, this reduces to an open unit disk, which we denote by $\mathbb{D}$. That is,

$$\mathbb{D}=\left\{z\in \mathbb{C}:\left|z\right|<1\right\}.$$

The ball with center $\mu =({\mu }_1,\dots ,{\mu }_n)\in {\mathbb{C}}^n$ and radius $\rho >0$ is defined by

$$\mathbb{B}(\mu ,\rho )=\left\{(z_1,\dots ,z_n)\in {\mathbb{C}}^n:{\displaystyle \sum _{j=1}^n}{|z_j-{\mu }_j|}^2<{\rho }^2\right\}.$$

In particular, when $\mu =0$ and $\rho =1$, this reduces to a unit ball, which we denote by ${\mathbb{B}}_n$. That is,

$${\mathbb{B}}_n=\left\{(z_1,\dots ,z_n)\in {\mathbb{C}}^n:{\displaystyle \sum _{j=1}^n}{\left|z_j\right|}^2<1\right\}.$$

When $n=1$, this is also a unit disk $\mathbb{D}$.

Regarding the unit disk $\mathbb{D}$, Fatehi and Hammond [1] systematically expounded the definitions of various operators and, building upon this foundation, defined the composition–differentiation operators. In this paper, we extend their framework by generalizing these operators. While earlier work mainly considered individual operators, Zhu and Hu [2] systematically analyzed the properties of finite linear combinations of weighted differential composition operators.

Regarding the unit ball ${\mathbb{B}}_n$, Stević and Ueki [3] mainly discussed two distinct construction forms: those implementing composition followed by differentiation, and those applying differentiation before composition. Meanwhile, they extended the scope of composition–differentiation operators, and characterized the properties of generalized operators. By introducing two new types of operators and leveraging their unique properties, Hagger [4] extended the classical result for Toeplitz operators.

Regarding the unit polydisk ${\mathbb{D}}^n$, Kosiński [5] derived the boundedness of composition operators by introducing an inequality relating the modulus of a function to the modulus of its gradient. Starting from classical compactness theory and a basic research framework of composition operators, Choe [6] further explored the compactness conditions and related criteria for linear combinations of such operators.

For special cases of bounded symmetric domains, relevant studies on the unit disk, unit ball, and polydisk can be found in references [7,8,9,10,11,12,13].

In the 1930s, Cartan [14] completed the foundational classification of irreducible bounded symmetric domains, identifying six distinct families. In this paper, we focus on the Cartan domain of the first kind, which is defined as

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

where $m,n$ are positive integers, $Z^T$ denotes the transpose of Z, and $\overline{Z}$ its entrywise complex conjugate. Regarding ${\Re }_{\mathrm{I}}(m,n)$, Quiroga-Barranco [15] investigated Toeplitz operators with radial-like invariant symbols, characterizing both their boundedness and the commutativity of pairs of such operators associated with distinct invariant symbols. By performing coordinate decomposition on matrices, Jiang [16] derived an inequality between the modulus of a function’s radial derivative and its norm, and thereby characterized the properties of operators on weighted Zygmund spaces. Further studies concerning various Cartan domains can be found in [17,18,19,20].

Since 1998, Yin [21] has systematically constructed five classes of domains, namely Cartan–Hartogs domains, Cartan–Egg domains, Hua domains, generalized Hua domains, and the Hua construction. Among these, the first two are special cases of Hua domains, while the last two are generalizations of Hua domains. Collectively, these families are referred to as Hua domains. We now give the precise definition of the generalized Hua domain of the first kind:

$$\begin{array}{cc} & {\mathrm{GHE}}_{\mathrm{I}}(N_1,N_2,\dots ,N_r;m,n;{\mathcal{P}}_1,{\mathcal{P}}_2,\dots ,{\mathcal{P}}_r;\ell )\hfill \\ & =\left\{{\zeta }_j\in {\mathbb{C}}^{N_j},Z\in {\Re }_{\mathrm{I}}(m,n):{\displaystyle \sum _{j=1}^r}{\left|{\zeta }_j\right|}^{2{\mathcal{P}}_j}<\det {(I-Z{\overline{Z}}^T)}^{\ell },j=1,2,\dots ,r\right\}.\hfill \end{array}$$

Here, $N_1,\dots ,N_r,m,n$ are positive integers, ${\mathcal{P}}_1,\dots ,{\mathcal{P}}_r,\ell >0$, and ${\zeta }_j=({\zeta }_{j1},\dots ,{\zeta }_{j{N}_j}), j=1,\cdots ,r$. When $\ell =m=1$, the generalized Hua domain of the first kind reduces to an ellipsoid. When $\ell =m=1$ and ${\mathcal{P}}_1={\mathcal{P}}_2=\cdots ={\mathcal{P}}_r=1$, the generalized Hua domain of the first kind becomes the unit ball. For simplicity, we set $N_j=1$ for all $j=1,\dots ,r$, so that each corresponding vector ${\zeta }_j$ reduces to a complex scalar in $\mathbb{C}$. For brevity, we write ${\mathrm{GHE}}_{\mathrm{I}}$ to refer to the generalized Hua domain of the first kind throughout the rest of this paper.

By the definition of ${\mathrm{GHE}}_{\mathrm{I}}$, for any $(Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}$, we have

$${\parallel \zeta \parallel }_{\mathcal{P}}^2={\displaystyle \sum _{j=1}^r}{\left|{\zeta }_j\right|}^{2{\mathcal{P}}_j}<\det {\left(I-Z{\overline{Z}}^T\right)}^{\ell }.$$

Rearranging gives

$$\det {\left(I-Z{\overline{Z}}^T\right)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2>0.$$

(1)

Since $Z\in {\Re }_{\mathrm{I}}(m,n)=\{Z\in {\mathbb{C}}^{mn}:I-Z{\overline{Z}}^T>0\}$, by the singular value decomposition, there exist unitary matrices ${\mathcal{Q}}^{\left(m\right)}$ and ${\mathcal{W}}^{\left(n\right)}$ satisfying $(m\le n)$

$$Z=\mathcal{Q}\left(\begin{array}{ccccccc}{\nu }_1& 0& \cdots & 0& 0& \cdots & 0\\ 0& {\nu }_2& \cdots & 0& 0& \cdots & 0\\ ⋮& ⋮& & ⋮& ⋮& & ⋮\\ 0& 0& \cdots & {\nu }_m& 0& \cdots & 0\end{array}\right)\mathcal{W},$$

with ${\nu }_1\ge {\nu }_2\ge \cdots \ge {\nu }_m\ge 0$. Then

$$Z{\overline{Z}}^T=\mathcal{Q}\left(\begin{array}{cccc}{\nu }_1^2& & & \\ & {\nu }_2^2& & \\ & & \ddots & \\ & & & {\nu }_m^2\end{array}\right){\mathcal{Q}}^{*},$$

where ${\mathcal{Q}}^{*}$ is the conjugate transpose of the unitary matrix $\mathcal{Q}$. So

$$I-Z{\overline{Z}}^T=\mathcal{Q}\left(\begin{array}{cccc}1-{\nu }_1^2& & & \\ & 1-{\nu }_2^2& & \\ & & \ddots & \\ & & & 1-{\nu }_m^2\end{array}\right){\mathcal{Q}}^{*}.$$

The condition $I-Z{\overline{Z}}^T>0$ implies $1-{\nu }_j^2>0$ for all $j=1,\dots ,m$, i.e., $0\le {\nu }_j<1$. Therefore,

$$\det \left(I-Z{\overline{Z}}^T\right)={\displaystyle \prod _{j=1}^m}(1-{\nu }_j^2)\le 1,$$

with equality if and only if all ${\nu }_j=0$, i.e., $Z=0$. Since $\ell >0$, we obtain

$$\det {\left(I-Z{\overline{Z}}^T\right)}^{\ell }\le 1.$$

Also, ${\parallel \zeta \parallel }_{\mathcal{P}}^2\ge 0$, so

$$\det {\left(I-Z{\overline{Z}}^T\right)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\le \det {\left(I-Z{\overline{Z}}^T\right)}^{\ell }\le 1.$$

(2)

In general, except for the unit ball, Hua domains are non-homogeneous domains, and therefore the study of their problems is of great significance. Wang and Su [22] constructed a more general framework for Hua domains, yielding results that hold for all five classes of their classes. Zhang [23] established sufficient conditions ensuring that a Kähler-Ricci soliton must be a Kähler–Einstein metric on the Cartan–Hartogs domain. Wang [24,25] extended the Hua inequality, enabling its application to a broader class of domains. By adopting extended Hua inequalities, Huo [26] explored integral operators defined on Cartan–Hartogs domains and achieved excellent research outcomes. In this paper, we employ this generalized version to characterize the properties of the defined norm. Su and Zhang [27] firstly utilized the properties of matrix determinants on Cartan–Hartogs domains to establish inequalities between two distinct norms, which has been a great inspiration for our research. We further generalize this method and relevant techniques to derive more general forms of inequalities.

Although extensive research has been conducted on Hua domains, the study of differential operators on them remains largely unexplored. To the best of our knowledge, this paper is the first to investigate weighted composition–differentiation operators on ${\mathrm{GHE}}_{\mathrm{I}}$. Building on the work of Zhu [28], we prove Lemma 6, which creatively establishes an inequality between the modulus of the derivative on the unit disk and the norm defined in this paper.

The result of Lemma 6 is then extended from the unit disk to ${\mathrm{GHE}}_{\mathrm{I}}$, yielding Lemma 7. These two lemmas are particularly useful in our study, and we hope they will also be useful to other researchers working on differential operators.

2. Preliminaries

Prior to the below analysis, we introduce the precise definitions of the operators under consideration and the associated Bers-type spaces. We also establish several key lemmas that serve as essential technical foundations for the derivation of the main results of this paper.

Let $\mathfrak{P}$ be a bounded domain in ${\mathbb{C}}^n$, $H\left(\mathfrak{P}\right)$ denote the set of all holomorphic functions on $\mathfrak{P}$, and $H(\mathfrak{P},\mathfrak{P})$ denote the set of all holomorphic self-maps on $\mathfrak{P}$. Then consider $\sigma \in H\left(\mathfrak{P}\right)$ and $\varphi \in H(\mathfrak{P},\mathfrak{P})$. The operator

$$\left(\sigma C_{\varphi }Df\right)\left(z\right):=\sigma \left(z\right)(Df\circ \varphi \left(z\right))$$

is called a weighted composition–differentiation operator, as introduced in [1], for any $f\in H\left(\mathfrak{P}\right),z\in \mathfrak{P}$. Here, for $z=(z_1,z_2,\dots ,z_n)\in \mathfrak{P}$, it follows that

$$Df\left(z\right):={\displaystyle \frac{\partial f\left(z\right)}{\partial z_1}}+{\displaystyle \frac{\partial f\left(z\right)}{\partial z_2}}+\cdots +{\displaystyle \frac{\partial f\left(z\right)}{\partial z_n}}.$$

The Bers-type space on ${\mathrm{GHE}}_{\mathrm{I}}$ is formulated as

$${\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right):=\left\{f\in H\left({\mathrm{GHE}}_{\mathrm{I}}\right):{\parallel f\parallel }_{\alpha ,\infty }<\infty \right\},$$

where

$${\parallel f\parallel }_{\alpha ,\infty }:=\underset{(Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}}{\sup }{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\alpha }\left|f(Z,\zeta )\right|.$$

Our definition of the Bers-type space also applies to the unit ball ${\mathbb{B}}_n$ and the unit disk $\mathbb{D}$, which can be denoted by ${\mathcal{H}}_{\alpha }^{\infty }\left({\mathbb{B}}_n\right)$ and ${\mathcal{H}}_{\alpha }^{\infty }\left(\mathbb{D}\right)$, respectively.

Clearly, ${\parallel ·\parallel }_{\alpha ,\infty }$ is a norm on ${\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$. We next verify the completeness of ${\parallel ·\parallel }_{\alpha ,\infty }$.

Let $\mathcal{A}(Z,\zeta ):=\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2$. Then for any $(Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}$, by inequalities (1) and (2), we have $0<\mathcal{A}(Z,\zeta )\le 1$. Suppose that $\left\{f_j\right\}$ is a Cauchy sequence in ${\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$. Then for any $\epsilon >0$, there exists $K_0>0$ such that for all $p,l>K_0$, we have

$${\parallel f_p-f_l\parallel }_{\alpha ,\infty }=\underset{(Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}}{\sup }{\left[\mathcal{A}(Z,\zeta )\right]}^{\alpha }\left|(f_p-f_l)(Z,\zeta )\right|<\epsilon .$$

(3)

For any compact subset $\mathcal{X}$ of ${\mathrm{GHE}}_{\mathrm{I}}$, there exists ${\delta }_{\alpha }\in (0,1)$, fulfilling

$${\left[\mathcal{A}(Z,\zeta )\right]}^{\alpha }\ge {\delta }_{\alpha },\forall (Z,\zeta )\in \mathcal{X}.$$

From (3), it follows that

$$|f_p(Z,\zeta )-f_l(Z,\zeta )|<{\displaystyle \frac{\epsilon }{{\delta }_{\alpha }}},(Z,\zeta )\in \mathcal{X}.$$

(4)

Then $\left\{f_p\right\}$ converges uniformly to f on every compact subset of ${\mathrm{GHE}}_{\mathrm{I}}$. Consequently, we can find $f\in H\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ with

$$\underset{j\to \infty }{\lim }{f}_j(Z,\zeta )=f(Z,\zeta ),\forall (Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}.$$

Furthermore, letting $l\to \infty$ in (3), we obtain, for all $p>K_0$, that

$${\parallel f_p-f\parallel }_{\alpha ,\infty }=\underset{(Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}}{\sup }{\left[\mathcal{A}(Z,\zeta )\right]}^{\alpha }|f_p(Z,\zeta )-f(Z,\zeta )|\le \epsilon .$$

In particular, set $\mathcal{K}>0$, then for all $p>\mathcal{K}$,

$${\parallel f_p-f\parallel }_{\alpha ,\infty }\le 1.$$

Therefore,

$$\begin{array}{cc}\hfill {\parallel f\parallel }_{\alpha ,\infty }& ={∥f+f_{\mathcal{K}+1}-f_{\mathcal{K}+1}∥}_{\alpha ,\infty }\hfill \\ \hfill & \le {∥f_{\mathcal{K}+1}∥}_{\alpha ,\infty }+{∥f-f_{\mathcal{K}+1}∥}_{\alpha ,\infty }\hfill \\ \hfill & \le {∥f_{\mathcal{K}+1}∥}_{\alpha ,\infty }+1<\infty .\hfill \end{array}$$

Hence, $f\in {\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$, and thus ${\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ is a Banach space.

In the following, $C,C_1,C_2,\dots$ stand for positive constants not depending on the function being discussed, whose values may differ from line to line.

Lemma 1

([29]). Let $\mathcal{S}=\left(s_{ij}\right)$ be an $n\times n$ Hermitian matrix with $\mathcal{S}\ge 0$. Then

$$\det \mathcal{S}\le {\displaystyle \prod _{i=1}^n}{s}_{ii}.$$

(5)

Equality holds if and only if $\mathcal{S}$ is a diagonal matrix.

Lemma 2

([30]). Let ${\alpha }_j\ge -1$ and all ${\alpha }_j$ have the same sign, with $n\ge 2$. Then

$${\displaystyle \prod _{k=1}^n}(1+{\alpha }_j)\ge 1+{\displaystyle \sum _{k=1}^n}{\alpha }_j.$$

(6)

Lemma 3

([26]). Let $0<x\le 1$, $0<y\le 1$, $y\le x$, and let ℑ be a positive integer. Then

$$a-b\le \Im \left(a^{{\displaystyle \frac{1}{\Im }}}-b^{{\displaystyle \frac{1}{\Im }}}\right).$$

(7)

Lemma 4

([28]). If $\alpha >-1$ and $f\in L_{\alpha }^1\left(d{A}_{\alpha }\right)$, then for all $z\in \mathbb{D}$,

$$f\left(z\right)={\int }_{\mathbb{D}}{k}_{\alpha }(z,\tau )f\left(\tau \right)d{A}_{\alpha }\left(\tau \right),$$

where

$$k_{\alpha }(z,\tau )={\displaystyle \frac{1}{{(1-z\overline{\tau })}^{2+\alpha }}},d{A}_{\alpha }\left(z\right)={(\alpha +1)(1-|z|}^2{)}^{\alpha }dA\left(z\right),$$

and $dA\left(z\right)={\displaystyle \frac{1}{\pi }}dxdy={\displaystyle \frac{r}{\pi }}drd\theta$ is the normalized area measure on unit disk $\mathbb{D}$.

Lemma 5

([28]). For any $\alpha >0$, $T>0$, $K>-1$, there exists a constant $C>0$ (depending only on T and K, but independent of t, α, and z) such that for all $z\in \mathbb{D}$, $\alpha \in (-1,K)$, $t\in (0,T)$,

$${\int }_{\mathbb{D}}{\displaystyle \frac{{(1-|\tau |}^2{)}^{\alpha }dA\left(\tau \right)}{|1-z\overline{\tau }{|}^{2+\alpha +t}}}\le {\displaystyle \frac{C\mathrm{\Gamma }(\alpha +1)\mathrm{\Gamma }\left(t\right)}{{(1-|z\left|\right)}^t}}.$$

Lemma 6.

For any $\alpha >0$, $z\in \mathbb{D}$ and $f\in {\mathcal{H}}_{\alpha }^{\infty }\left(\mathbb{D}\right)$, there exists a constant $C>0$ such that

$${(1-|z|}^2{)}^{\alpha +1}|f^{\prime }\left(z\right)|\le C{\parallel f\parallel }_{\alpha ,\infty }.$$

Proof. 

For any $\beta >\alpha >0$, by the fundamental properties of weighted Bergman spaces, we note that ${\mathcal{H}}_{\alpha }^{\infty }\left(\mathbb{D}\right)\subset {\mathcal{H}}_{\beta }^{\infty }\left(\mathbb{D}\right)\subset L^1\left(d{A}_{\beta }\right)$. Since $f\in {\mathcal{H}}_{\alpha }^{\infty }\left(\mathbb{D}\right)$, we have $f\in L^1\left(d{A}_{\beta }\right)$. Then by Lemma 4,

$$f\left(z\right)={\int }_{\mathbb{D}}{\displaystyle \frac{{f\left(\tau \right)(\beta +1)(1-|\tau |}^2{)}^{\beta }}{{(1-z\overline{\tau })}^{2+\beta }}}dA\left(\tau \right).$$

Differentiating with respect to z, we obtain

$$f^{\prime }\left(z\right)={\int }_{\mathbb{D}}{\displaystyle \frac{(2+\beta )(\beta +1)f\left(\tau \right)\overline{\tau }{(1-|\tau |}^2{)}^{\beta }}{{(1-z\overline{\tau })}^{3+\beta }}}dA\left(\tau \right).$$

Therefore,

$$\begin{array}{cc}\hfill {(1-|z|}^2{)}^{\alpha +1}\left|f^{\prime }\left(z\right)\right|& \le {(\beta +2)(\beta +1)(1-|z|}^2{)}^{\alpha +1}{\int }_{\mathbb{D}}\left|{\displaystyle \frac{f\left(\tau \right)\overline{\tau }{(1-|\tau |}^2{)}^{\beta }}{{(1-z\overline{\tau })}^{3+\beta }}}\right|dA\left(\tau \right)\hfill \\ \hfill & \le {(\beta +2)(\beta +1)(1-|z|}^2{)}^{\alpha +1}{\int }_{\mathbb{D}}{\displaystyle \frac{{(1-|\tau |}^2{)}^{\alpha }\left|f\left(\tau \right)\right|(1-{\left|\tau \right|}^2{)}^{\beta }}{{|1-z\overline{\tau }|}^{3+\beta }{(1-|\tau |}^2{)}^{\alpha }}}dA\left(\tau \right)\hfill \\ \hfill & \le {(\beta +2)(\beta +1)\parallel f\parallel }_{\alpha ,\infty }{(1-|z|}^2{)}^{\alpha +1}{\int }_{\mathbb{D}}{\displaystyle \frac{{(1-|\tau |}^2{)}^{\beta -\alpha }}{|1-z\overline{\tau }{|}^{2+(\beta -\alpha )+\alpha +1}}}dA\left(\tau \right).\hfill \end{array}$$

by Lemma 5, there exists $C_1>0$ such that

$${\int }_{\mathbb{D}}{\displaystyle \frac{{(1-|\tau |}^2{)}^{\beta -\alpha }}{|1-z\overline{\tau }{|}^{2+(\beta -\alpha )+\alpha +1}}}dA\left(\tau \right)\le {\displaystyle \frac{C_1\mathrm{\Gamma }(\beta -\alpha +1)\mathrm{\Gamma }(\alpha +1)}{{(1-|z\left|\right)}^{\alpha +1}}}.$$

Therefore,

$${(1-|z|}^2{)}^{\alpha +1}|f^{\prime }{\left(z\right)|\le (\beta +2)(\beta +1)\parallel f\parallel }_{\alpha ,\infty }{(1-|z|}^2{)}^{\alpha +1}·{\displaystyle \frac{C_1\mathrm{\Gamma }(\beta -\alpha +1)\mathrm{\Gamma }(\alpha +1)}{{(1-|z\left|\right)}^{\alpha +1}}}.$$

Using the identity $1-{\left|z\right|}^2=(1-|z\left|\right)(1+|z\left|\right)$, we have

$${(1-|z|}^2{)}^{\alpha +1}={(1-|z\left|\right)}^{\alpha +1}{(1+|z\left|\right)}^{\alpha +1},$$

so that

$${(1-|z|}^2{)}^{\alpha +1}|f^{\prime }{\left(z\right)|\le (\beta +2)(\beta +1)\parallel f\parallel }_{\alpha ,\infty }·{(1+|z\left|\right)}^{\alpha +1}{C}_1\mathrm{\Gamma }(\beta -\alpha +1)\mathrm{\Gamma }(\alpha +1).$$

Since $z\in \mathbb{D}$, we have $\left|z\right|\le 1$, so $1+\left|z\right|\le 2$ and thus ${(1+|z\left|\right)}^{\alpha +1}\le 2^{\alpha +1}$. Therefore,

$$\begin{array}{cc}\hfill {(1-|z|}^2{)}^{\alpha +1}\left|f^{\prime }\left(z\right)\right|& \le (\beta +2)(\beta +1)·2^{\alpha +1}{C}_1\mathrm{\Gamma }(\beta -\alpha +1)\mathrm{\Gamma }(\alpha +1)·{\parallel f\parallel }_{\alpha ,\infty }\hfill \\ & =C·{\parallel f\parallel }_{\alpha ,\infty },\hfill \end{array}$$

where $C=(\beta +2)(\beta +1)C_1·2^{\alpha +1}·\mathrm{\Gamma }(\beta -\alpha +1)·\mathrm{\Gamma }(\alpha +1)$, and we may take $\beta =\alpha +1$. □

Lemma 7.

Let $\alpha >0$, $0<\ell m\le 1$, ${\mathcal{P}}_j$ ($j=1,2,\dots ,r$) be positive integers. Let $\varphi =({\varphi }_{11},{\varphi }_{12},\dots ,{\varphi }_{mn+r})$ be a holomorphic self-map on ${\mathrm{GHE}}_{\mathrm{I}}$. Then for all $f\in {\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$, there exists a constant $C>0$ such that

$${\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{{\displaystyle \frac{\alpha +1}{\ell }}}\left|\sum _{\begin{array}{c}1\le i\le m\\ 1\le j\le n\end{array}}{\displaystyle \frac{\partial f(Z,\zeta )}{\partial z_{ij}}}+{\displaystyle \sum _{s=1}^r}{\displaystyle \frac{\partial f(Z,\zeta )}{\partial {\zeta }_s}}\right|\le C{\parallel f\parallel }_{\alpha ,\infty }.$$

Proof. 

For any $(Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}$, let

$$Z=\left(\begin{array}{cccc}{z}_{11}& z_{12}& \cdots & z_{1n}\\ z_{21}& z_{22}& \cdots & z_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ z_{m1}& z_{m2}& \cdots & z_{mn}\end{array}\right).$$

We first derive the explicit form of $Z{\overline{Z}}^T$ by computing its entries. For any $p,q$ with $1\le p\le m$ and $1\le q\le m$, we have

$${(Z{\overline{Z}}^T)}_{pq}={\displaystyle \sum _{k=1}^n}{z}_{pk}\overline{z_{qk}}.$$

Therefore, $Z{\overline{Z}}^T$ is an $m\times m$ positive semi-definite Hermitian matrix, whose explicit form is

$$Z{\overline{Z}}^T=\left(\begin{array}{cccc}{\displaystyle \sum _{k=1}^n}{\left|z_{1k}\right|}^2& {\displaystyle \sum _{k=1}^n}{z}_{1k}\overline{z_{2k}}& \cdots & {\displaystyle \sum _{k=1}^n}{z}_{1k}\overline{z_{mk}}\\ {\displaystyle \sum _{k=1}^n}{z}_{2k}\overline{z_{1k}}& {\displaystyle \sum _{k=1}^n}{\left|z_{2k}\right|}^2& \cdots & {\displaystyle \sum _{k=1}^n}{z}_{2k}\overline{z_{mk}}\\ ⋮& ⋮& \ddots & ⋮\\ {\displaystyle \sum _{k=1}^n}{z}_{mk}\overline{z_{1k}}& {\displaystyle \sum _{k=1}^n}{z}_{mk}\overline{z_{2k}}& \cdots & {\displaystyle \sum _{k=1}^n}{\left|z_{mk}\right|}^2\end{array}\right).$$

Subtracting this matrix from the $m\times m$ identity matrix I, we obtain the matrix

$$I-Z{\overline{Z}}^T=\left(\begin{array}{cccc}1-{\displaystyle \sum _{k=1}^n}{\left|z_{1k}\right|}^2& -{\displaystyle \sum _{k=1}^n}{z}_{1k}\overline{z_{2k}}& \cdots & -{\displaystyle \sum _{k=1}^n}{z}_{1k}\overline{z_{mk}}\\ -{\displaystyle \sum _{k=1}^n}{z}_{2k}\overline{z_{1k}}& 1-{\displaystyle \sum _{k=1}^n}{\left|z_{2k}\right|}^2& \cdots & -{\displaystyle \sum _{k=1}^n}{z}_{2k}\overline{z_{mk}}\\ ⋮& ⋮& \ddots & ⋮\\ -{\displaystyle \sum _{k=1}^n}{z}_{mk}\overline{z_{1k}}& -{\displaystyle \sum _{k=1}^n}{z}_{mk}\overline{z_{2k}}& \cdots & 1-{\displaystyle \sum _{k=1}^n}{\left|z_{mk}\right|}^2\end{array}\right)$$

Since $Z\in {\Re }_1(m,n)=\{Z\in {\mathbb{C}}^{m\times n}:I-Z{\overline{Z}}^T>0\}$, the matrix $I-Z{\overline{Z}}^T$ is positive definite. In particular, all its diagonal entries are positive, i.e.,

$$1-{\displaystyle \sum _{k=1}^n}{\left|z_{ik}\right|}^2>0,\forall i=1,2,\dots ,m.$$

In particular, for any fixed indices $i,j$, we have

$$|z_{ij}{|}^2\le {\displaystyle \sum _{k=1}^n}{\left|z_{ik}\right|}^2<1.$$

Taking the product over all $i=1,\dots ,m$ and $j=1,\dots ,n$, we obtain

$${\displaystyle \prod _{i=1}^m}{\displaystyle \prod _{j=1}^n}(1-|z_{ij}{{|}^2)}^{\ell }\le 1.$$

(8)

Combining inequalities (5) and (6), we obtain

$$\begin{array}{cc}\hfill \det {(I-Z{\overline{Z}}^T)}^{\ell }& \le {\left(1-{\displaystyle \sum _{k=1}^n}{\left|z_{1k}\right|}^2\right)}^{\ell }\times {\left(1-{\displaystyle \sum _{k=1}^n}{\left|z_{2k}\right|}^2\right)}^{\ell }\times \cdots \times {\left(1-{\displaystyle \sum _{k=1}^n}{\left|z_{mk}\right|}^2\right)}^{\ell }\hfill \\ \hfill & \le {\displaystyle \prod _{k=1}^n}{\left(1-|z_{1k}{|}^2\right)}^{\ell }\times {\displaystyle \prod _{k=1}^n}{\left(1-|z_{2k}{|}^2\right)}^{\ell }\times \cdots \times {\displaystyle \prod _{k=1}^n}{\left(1-|z_{mk}{|}^2\right)}^{\ell }.\hfill \end{array}$$

(9)

By the inequality (9), we have

$$\det {(I-Z{\overline{Z}}^T)}^{\ell }\le {\displaystyle \prod _{i=1}^m}{\displaystyle \prod _{j=1}^n}(1-|z_{ij}{{|}^2)}^{\ell }.$$

Since ${\parallel \zeta \parallel }_{\mathcal{P}}^2\ge 0$, it follows that

$$\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\le {\displaystyle \prod _{i=1}^m}{\displaystyle \prod _{j=1}^n}(1-|z_{ij}{|}^2{)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2.$$

Next, by (8), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \prod _{i=1}^m}{\displaystyle \prod _{j=1}^n}(1-|z_{ij}{|}^2{)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2& \le {\displaystyle \prod _{i=1}^m}{\displaystyle \prod _{j=1}^n}(1-|z_{ij}{|}^2{)}^{\ell }-{\displaystyle \prod _{i=1}^m}{\displaystyle \prod _{j=1}^n}(1-|z_{ij}{|}^2{)}^{\ell }{\parallel \zeta \parallel }_{\mathcal{P}}^2\hfill \\ & ={\displaystyle \prod _{i=1}^m}{\displaystyle \prod _{j=1}^n}(1-|z_{ij}{{|}^2)}^{\ell }\left(1-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right).\hfill \end{array}$$

Since $0<\ell \le 1$ and $0<1-{\parallel \zeta \parallel }_{\mathcal{P}}^2\le 1$, we have

$${\left(1-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right)}^{{\displaystyle \frac{\alpha +1}{\ell }}}\le {\left(1-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right)}^{\alpha +1}.$$

Thus,

$${\displaystyle \prod _{i=1}^m}{\displaystyle \prod _{j=1}^n}(1-|z_{ij}{|}^2{)}^{\alpha +1}·{\left(1-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right)}^{{\displaystyle \frac{\alpha +1}{\ell }}}\le {\displaystyle \prod _{i=1}^m}{\displaystyle \prod _{j=1}^n}(1-|z_{ij}{{|}^2)}^{\alpha +1}·{\left(1-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right)}^{\alpha +1}.$$

Eventually, from inequalities (6) and (7), we can find a constant $C_1$ satisfying

$$\begin{array}{cc}\hfill {(1-\parallel \zeta \parallel }_{\mathcal{P}}^2{)}^{\alpha +1}& ={\left(1-\left(|{\zeta }_1{|}^{2{\mathcal{P}}_1}+|{\zeta }_2{|}^{2{\mathcal{P}}_2}+\cdots +{\left|{\zeta }_r\right|}^{2{\mathcal{P}}_r}\right)\right)}^{\alpha +1}\hfill \\ \hfill & \le {\left(1-|{\zeta }_1{|}^{2{\mathcal{P}}_1}\right)}^{\alpha +1}{\left(1-|{\zeta }_2{|}^{2{\mathcal{P}}_2}\right)}^{\alpha +1}\cdots {\left(1-|{\zeta }_r{|}^{2{\mathcal{P}}_r}\right)}^{\alpha +1}\hfill \\ \hfill & \le {\left[{\mathcal{P}}_1\left(1-|{\zeta }_1{|}^2\right)\right]}^{\alpha +1}{\left[{\mathcal{P}}_2\left(1-|{\zeta }_2{|}^2\right)\right]}^{\alpha +1}\cdots {\left[{\mathcal{P}}_r\left(1-|{\zeta }_r{|}^2\right)\right]}^{\alpha +1}\hfill \\ \hfill & =C_1{\displaystyle \prod _{s=1}^r}{\left(1-|{\zeta }_s{|}^2\right)}^{\alpha +1},\hfill \end{array}$$

(10)

where $C_1={\prod }_{s=1}^r{\mathcal{P}}_s^{\alpha +1}$. Combining all the estimates above, we conclude that

$${\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{{\displaystyle \frac{\alpha +1}{\ell }}}\le C_1\prod _{i,j}(1-|z_{ij}{|}^2{)}^{\alpha +1}{\displaystyle \prod _{s=1}^r}(1-|{\zeta }_s{{|}^2)}^{\alpha +1}.$$

Furthermore, by Lemma 6, we can find constants C and $C_2$ satisfying

$$\begin{array}{cc} & {\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{{\displaystyle \frac{\alpha +1}{\ell }}}\left|\sum _{\begin{array}{c}1\le i\le m\\ 1\le j\le n\end{array}}{\displaystyle \frac{\partial f(Z,\zeta )}{\partial z_{ij}}}+{\displaystyle \sum _{s=1}^r}{\displaystyle \frac{\partial f(Z,\zeta )}{\partial {\zeta }_s}}\right|\hfill \\ & \le C_1\left({\displaystyle \prod _{i=1}^m}{\displaystyle \prod _{j=1}^n}(1-|z_{ij}{{|}^2)}^{\alpha +1}\right)\left({\displaystyle \prod _{s=1}^r}(1-|{\zeta }_s{{|}^2)}^{\alpha +1}\right)\left(\sum _{\begin{array}{c}1\le i\le m\\ 1\le j\le n\end{array}}\left|{\displaystyle \frac{\partial f(Z,\zeta )}{\partial z_{ij}}}\right|+{\displaystyle \sum _{s=1}^r}\left|{\displaystyle \frac{\partial f(Z,\zeta )}{\partial {\zeta }_s}}\right|\right)\hfill \\ & \le C_1\left(\sum _{\begin{array}{c}1\le i\le m\\ 1\le j\le n\end{array}}(1-|z_{ij}{|}^2{)}^{\alpha +1}\left|{\displaystyle \frac{\partial f(Z,\zeta )}{\partial z_{ij}}}\right|+{\displaystyle \sum _{s=1}^r}(1-|{\zeta }_s{{|}^2)}^{\alpha +1}\left|{\displaystyle \frac{\partial f(Z,\zeta )}{\partial {\zeta }_s}}\right|\right)\hfill \\ & \le C_1(mn+r)C_2{\parallel f\parallel }_{\alpha ,\infty }\hfill \\ & ={C\parallel f\parallel }_{\alpha ,\infty },\hfill \end{array}$$

where $C=C_1{C}_2(mn+r)$.

 □

Lemma 8

([26]). Let $(X,\zeta ),(Y,\tau )\in {\mathrm{GHE}}_{\mathrm{I}}$ with $0<\ell m\le 1$. Then

$$\left[\det {(I_m-X{\overline{X}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]+\left[\det {(I_m-Y{\overline{Y}}^T)}^{\ell }-{\parallel \tau \parallel }_{\mathcal{P}}^2\right]\le 2\left|\left|\det {(I_m-X{\overline{Y}}^T)}^{\ell }\right|-{\parallel \zeta \parallel }_{\mathcal{P}}{\parallel \tau \parallel }_{\mathcal{P}}\right|,$$

(11)

and

$$\left[\det {(I_m-X{\overline{X}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]\left[\det {(I_m-Y{\overline{Y}}^T)}^{\ell }-{\parallel \tau \parallel }_{\mathcal{P}}^2\right]\le {\left|\left|\det {(I_m-X{\overline{Y}}^T)}^{\ell }\right|-{\parallel \zeta \parallel }_{\mathcal{P}}{\parallel \tau \parallel }_{\mathcal{P}}\right|}^2.$$

(12)

Equality holds if $(X,\zeta )=(Y,\tau )$.

Lemma 9.

For all $(Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}$ and $f\in {\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$, the following holds:

$$\left|f(Z,\zeta )\right|\le {\displaystyle \frac{{\parallel f\parallel }_{\alpha ,\infty }}{{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\alpha }}}.$$

Proof. 

The definition of ${\parallel ·\parallel }_{\alpha ,\infty }$ directly implies that

$${\parallel f\parallel }_{\alpha ,\infty }=\underset{(Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}}{\sup }{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\alpha }\left|f(Z,\zeta )\right|<\infty .$$

Therefore,

$$\left|f(Z,\zeta )\right|\le {\displaystyle \frac{{\parallel f\parallel }_{\alpha ,\infty }}{{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\alpha }}},$$

which completes the proof. □

Lemma 10.

Let $\varphi =({\varphi }_{11},{\varphi }_{12},\dots ,{\varphi }_{mn+r})$ be a holomorphic self-map on ${\mathrm{GHE}}_{\mathrm{I}}$, and let $\sigma \in H\left({\mathrm{GHE}}_{\mathrm{I}}\right)$. Then the weighted composition–differentiation operator $\sigma C_{\varphi }D:{\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)\to {\mathcal{H}}_{\beta }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ is compact if and only if $\sigma C_{\varphi }D$ is bounded, and for any bounded sequence ${\left\{f_n\right\}}_{n\ge 1}$ in ${\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ that converges uniformly to 0 on every compact subset K of ${\mathrm{GHE}}_{\mathrm{I}}$, we have

$$\underset{n\to \infty }{\lim }{\parallel \sigma C_{\varphi }D{f}_n\parallel }_{\beta ,\infty }=0.$$

Proof. 

Let $\sigma C_{\varphi }D:{\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)\to {\mathcal{H}}_{\beta }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ be a compact operator, and let ${\left\{f_n\right\}}_{n\ge 1}$ be a bounded sequence in ${\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ converging uniformly to 0 on every compact subset of ${\mathrm{GHE}}_{\mathrm{I}}$. Suppose, on the other hand, that $\parallel \sigma C_{\varphi }D{f}_n{\parallel }_{\beta ,\infty }\nrightarrow 0$ as $n\to \infty$. Then we can extract a subsequence ${\{f_{n_j}\}}_{j\ge 1}$ of ${\left\{f_n\right\}}_{n\ge 1}$ satisfying

$$\underset{j\in \mathbb{N}}{\inf }{\parallel \sigma C_{\varphi }D{f}_{n_j}\parallel }_{\beta ,\infty }>0.$$

By the compactness of $\sigma C_{\varphi }D$, we may extract a further subsequence of ${\{f_{n_j}\}}_{j\ge 1}$ (still denoted ${\left\{f_{n_j}\right\}}_{j\ge 1}$ for simplicity) with

$$\underset{j\to \infty }{\lim }{\parallel \sigma C_{\varphi }D{f}_{n_j}-f\parallel }_{\beta ,\infty }=0.$$

From Lemma 9, the following holds:

$$|(\sigma C_{\varphi }D{f}_{n_j}-f)(Z,\zeta )|\le {\displaystyle \frac{\parallel \sigma C_{\varphi }D{f}_{n_j}{-f\parallel }_{\beta ,\infty }}{{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }}}.$$

Now let K be any compact subset of ${\mathrm{GHE}}_{\mathrm{I}}$. Then $\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2$ has a positive lower bound on K. Thus, $\sigma C_{\varphi }D{f}_{n_j}-f\to 0$ uniformly on K. This means that for any $\epsilon >0$, there exists $N_1>0$ with the property that for all $j>N_1$,

$$\left|\sigma (Z,\zeta )D{f}_{n_j}\left(\varphi (Z,\zeta )\right)-f(Z,\zeta )\right|<\epsilon ,$$

for all $(Z,\zeta )\in K$.

According to our assumption, ${\left\{f_{n_j}\right\}}_{j\ge 1}$ converges uniformly to 0 on every compact subset of ${\mathrm{GHE}}_{\mathrm{I}}$. According to the Weierstrass theorem, ${\left\{D{f}_{n_j}\right\}}_{j\ge 1}$ also converges uniformly to 0 on every compact subset. Thus, for the above $\epsilon >0$, let $N_2$ be a positive integer with the property that $|D{f}_{n_j}\left(\varphi (Z,\zeta )\right)|<\epsilon$, for all $(Z,\zeta )\in K$ and every $j>N_2$. Let $N=\max \{N_1,N_2\}$ and $\hslash ={\max }_{(Z,\zeta )\in K}\left|\sigma (Z,\zeta )\right|$. Then for all $j>N$, we get

$$\begin{array}{cc}\hfill \left|f\right(Z,\zeta \left)\right|& \le |\sigma (Z,\zeta )D{f}_{n_j}\left(\varphi (Z,\zeta )\right)|+\epsilon \hfill \\ \hfill & \le |D{f}_{n_j}\left(\varphi (Z,\zeta )\right)|·\underset{(Z,\zeta )\in K}{\max }\left|\sigma (Z,\zeta )\right|+\epsilon \hfill \\ \hfill & \le \left(\hslash +1\right)\epsilon ,\forall (Z,\zeta )\in K.\hfill \end{array}$$

Because $\epsilon$ can be taken to be arbitrarily small, we obtain $f(Z,\zeta )\equiv 0$ on K. According to the identity theorem for holomorphic functions, this implies that $f(Z,\zeta )\equiv 0$ throughout ${\mathrm{GHE}}_{\mathrm{I}}$. Therefore,

$$\underset{j\to \infty }{\lim }{\parallel \sigma C_{\varphi }D{f}_{n_j}\parallel }_{\beta ,\infty }=0,$$

contradicting the assumption that ${\inf }_{j\in \mathbb{N}}{\parallel \sigma C_{\varphi }D{f}_{n_j}\parallel }_{\beta ,\infty }>0$.

In the opposite direction, assume ${\left\{f_n\right\}}_{n\ge 1}$ is any bounded sequence in ${\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ so that $\parallel f_n{\parallel }_{\alpha ,\infty }\le D$ for some constant $D>0$ and all n. Then ${\left\{f_n\right\}}_{n\ge 1}$ is uniformly bounded on every compact subset of ${\mathrm{GHE}}_{\mathrm{I}}$. Montel’s theorem guarantees the existence of a subsequence ${\left\{f_{n_j}\right\}}_{j\ge 1}$ of ${\left\{f_n\right\}}_{n\ge 1}$ that converges uniformly to some holomorphic function f on every compact subset of ${\mathrm{GHE}}_{\mathrm{I}}$.

For any fixed $(Z_0,{\zeta }_0)\in {\mathrm{GHE}}_{\mathrm{I}}$, there exists a compact subset $K_{(Z_0,{\zeta }_0)}$ fulfilling $(Z_0,{\zeta }_0)\in K_{(Z_0,{\zeta }_0)}\subset {\mathrm{GHE}}_{\mathrm{I}}$. Since $f_{n_j}\to f$ uniformly on $K_{(Z_0,{\zeta }_0)}$ as $j\to \infty$, there exists $J_0>0$ satisfying $|f_{n_j}(Z,\zeta )-f(Z,\zeta )|<1$, for all $j>J_0$ and $(Z,\zeta )\in K_{(Z_0,{\zeta }_0)}$.

Moreover,

$$\begin{array}{cc}\hfill \left|f\right(Z_0,{\zeta }_0\left)\right|& =|f(Z_0,{\zeta }_0)-f_{n_j}(Z_0,{\zeta }_0)+f_{n_j}(Z_0,{\zeta }_0)|\hfill \\ \hfill & \le |f(Z_0,{\zeta }_0)-f_{n_j}(Z_0,{\zeta }_0)|+|f_{n_j}(Z_0,{\zeta }_0)|.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill & {\left[\det {(I-Z_0{\overline{Z_0}}^T)}^{\ell }-{\parallel {\zeta }_0\parallel }_{\mathcal{P}}^2\right]}^{\alpha }\left|f(Z_0,{\zeta }_0)\right|\hfill \\ \hfill & \le {\left[\det {(I-Z_0{\overline{Z_0}}^T)}^{\ell }-{\parallel {\zeta }_0\parallel }_{\mathcal{P}}^2\right]}^{\alpha }|f(Z_0,{\zeta }_0)-f_{n_j}(Z_0,{\zeta }_0)|\hfill \\ \hfill & +{\left[\det {(I-Z_0{\overline{Z_0}}^T)}^{\ell }-{\parallel {\zeta }_0\parallel }_{\mathcal{P}}^2\right]}^{\alpha }\left|f_{n_j}(Z_0,{\zeta }_0)\right|\hfill \\ \hfill & \le 1+\parallel f_{n_j}{\parallel }_{\alpha ,\infty }\le 1+D.\hfill \end{array}$$

By the arbitrariness of $(Z_0,{\zeta }_0)$, we conclude that for all $(Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}$,

$${\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\alpha }\left|f(Z,\zeta )\right|\le 1+D.$$

This immediately implies that ${\parallel f\parallel }_{\alpha ,\infty }\le 1+D$, and hence

$$\parallel f_{n_j}{-f\parallel }_{\alpha ,\infty }\le \parallel f_{n_j}{\parallel }_{\alpha ,\infty }+{\parallel f\parallel }_{\alpha ,\infty }\le 2D+1.$$

Thus, $f_{n_j}-f\rightrightarrows 0$ on every compact subset of ${\mathrm{GHE}}_{\mathrm{I}}$ as $j\to \infty$. Consequently, we get

$$\underset{j\to \infty }{\lim }{∥\sigma C_{\varphi }D(f_{n_j}-f)∥}_{\beta ,\infty }=\underset{j\to \infty }{\lim }{∥\sigma C_{\varphi }D{f}_{n_j}-\sigma C_{\varphi }Df∥}_{\beta ,\infty }=0.$$

This shows that the operator $\sigma C_{\varphi }D:{\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)\to {\mathcal{H}}_{\beta }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ is compact. □

3. Boundedness of $\mathit{\sigma }{\mathit{C}}_{\mathit{\varphi }}\mathit{D}:{\mathcal{H}}_{\mathit{\alpha }}^{\infty }\left({\mathbf{GHE}}_{\mathbf{I}}\right)\to {\mathcal{H}}_{\mathit{\beta }}^{\infty }\left({\mathbf{GHE}}_{\mathbf{I}}\right)$

Next, we combine Lemma 7 with a suitably constructed test function in order to verify the operator’s bound, thereby establishing the desired estimate.

Theorem 1.

Let $\alpha ,\beta >0$, $0<\ell m\le 1$, and let ${\mathcal{P}}_j$(j = 1,2,…,r)$\in \mathbb{N}$. Suppose $\varphi =({\varphi }_{11},{\varphi }_{12},\dots ,{\varphi }_{mn+r})$ is a holomorphic self-map on ${\mathrm{GHE}}_{\mathrm{I}}$, and $\sigma \in H\left({\mathrm{GHE}}_{\mathrm{I}}\right)$. Let $(Z_{\varphi },{\zeta }_{\varphi })=\varphi (Z,\zeta )$. If

$$M_1=\underset{(Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}}{\sup }{\displaystyle \frac{\left|\sigma (Z,\zeta )\right|{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }}{{\left[\det {(I-Z_{\varphi }{\overline{Z_{\varphi }}}^T)}^{\ell }-{\parallel {\zeta }_{\varphi }\parallel }_{\mathcal{P}}^2\right]}^{\frac{\alpha +1}{\ell }}}}<\infty ,$$

(13)

then $\sigma C_{\varphi }D:{\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)\to {\mathcal{H}}_{\beta }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ is bounded.

For the other direction, if $\sigma C_{\varphi }D:{\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)\to {\mathcal{H}}_{\beta }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ is bounded, then

$$M_2=\underset{(Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}}{\sup }{\displaystyle \frac{\left|\sigma (Z,\zeta )\right|{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }G(Z_{\varphi },{\zeta }_{\varphi })}{{\left[\det {(I-Z_{\varphi }{\overline{Z_{\varphi }}}^T)}^{\ell }-{\parallel {\zeta }_{\varphi }\parallel }_{\mathcal{P}}^2\right]}^{\alpha +1}}}<\infty ,$$

(14)

where

$$G(Z_{\varphi },{\zeta }_{\varphi })=\ell \det {(I-Z_{\varphi }{\overline{Z_{\varphi }}}^T)}^{\ell }\times {\left[\sum _{\begin{array}{c}1\le s\le m\\ 1\le t\le n\end{array}}{\left|tr\left({(I-Z_{\varphi }{\overline{Z_{\varphi }}}^T)}^{-1}{I}_{st}{\overline{Z_{\varphi }}}^T\right)\right|}^2\right]}^{{\displaystyle \frac{1}{2}}}+{\left[{\displaystyle \sum _{j=1}^r}{\mathcal{P}}_j^2{\left|{\zeta }_{{\varphi }_j}\right|}^{4{\mathcal{P}}_j-2}\right]}^{{\displaystyle \frac{1}{2}}}.$$

(15)

Here, $I_{st}$ denotes the $m\times n$ matrix whose $(s,t)$-entry is 1 and all other entries are 0.

Proof. 

On the one hand, assume condition (13) holds. For any $f\in {\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$, we proceed as follows:

$${\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }\left|\left(\sigma C_{\varphi }Df\right)(Z,\zeta )\right|={\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }\left|\sigma (Z,\zeta )Df\left(\varphi (Z,\zeta )\right)\right|.$$

Multiplying and dividing by the term ${\left[\det {(I-Z_{\varphi }{\overline{Z_{\varphi }}}^T)}^{\ell }-{\parallel {\zeta }_{\varphi }\parallel }_{\mathcal{P}}^2\right]}^{{\displaystyle \frac{\alpha +1}{\ell }}}$, an application of Lemma 7 gives us

$$\begin{array}{cc}\hfill & {\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }\left|\sigma (Z,\zeta )Df\left(\varphi (Z,\zeta )\right)\right|\hfill \\ \hfill & ={\displaystyle \frac{\left|\sigma (Z,\zeta )\right|{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }}{{\left[\det {(I-Z_{\varphi }{\overline{Z_{\varphi }}}^T)}^{\ell }-{\parallel {\zeta }_{\varphi }\parallel }_{\mathcal{P}}^2\right]}^{\frac{\alpha +1}{\ell }}}}\times {\left[\det {(I-Z_{\varphi }{\overline{Z_{\varphi }}}^T)}^{\ell }-{\parallel {\zeta }_{\varphi }\parallel }_{\mathcal{P}}^2\right]}^{{\displaystyle \frac{\alpha +1}{\ell }}}\left|Df\left(\varphi (Z,\zeta )\right)\right|\hfill \\ \hfill & \le C{\displaystyle \frac{\left|\sigma (Z,\zeta )\right|{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }}{{\left[\det {(I-Z_{\varphi }{\overline{Z_{\varphi }}}^T)}^{\ell }-{\parallel {\zeta }_{\varphi }\parallel }_{\mathcal{P}}^2\right]}^{\frac{\alpha +1}{\ell }}}}{\parallel f\parallel }_{\alpha ,\infty }\hfill \\ \hfill & \le C{M}_1{\parallel f\parallel }_{\alpha ,\infty }.\hfill \end{array}$$

Consequently, for all $(Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}$, we obtain

$$\parallel \sigma C_{\varphi }{Df\parallel }_{\beta ,\infty }=\underset{(Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}}{\sup }{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }\left|\left(\sigma C_{\varphi }Df\right)(Z,\zeta )\right|\le C{M}_1{\parallel f\parallel }_{\alpha ,\infty }.$$

This shows that the operator $\sigma C_{\varphi }D:{\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)\to {\mathcal{H}}_{\beta }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ is bounded.

On the other hand, assume that $\sigma C_{\varphi }D:{\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)\to {\mathcal{H}}_{\beta }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ is a bounded operator. Take any point $(W,\omega )\in {\mathrm{GHE}}_{\mathrm{I}}$ and define a holomorphic test function $f_{(W,\omega )}$ on ${\mathrm{GHE}}_{\mathrm{I}}$ via

$$f_{(W,\omega )}(Z,\zeta ):={\displaystyle \frac{{\left[\det {(I-W{\overline{W}}^T)}^{\ell }-{\parallel \omega \parallel }_{\mathcal{P}}^2\right]}^{\alpha }}{{\left[\det {(I-Z{\overline{W}}^T)}^{\ell }-{\langle \zeta ,\omega \rangle }_p\right]}^{2\alpha }}}.$$

The test function we construct enables us to estimate images under the operator by using the boundedness of the operator. For all $(Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}$, from (12), it follows that

$$\begin{array}{cc} & {\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\alpha }\left|f_{(W,\omega )}(Z,\zeta )\right|\hfill \\ & ={\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\alpha }{\displaystyle \frac{{\left[\det {(I-W{\overline{W}}^T)}^{\ell }-{\parallel \omega \parallel }_{\mathcal{P}}^2\right]}^{\alpha }}{{\left|\det {(I-Z{\overline{W}}^T)}^{\ell }-{\langle \zeta ,\omega \rangle }_p\right|}^{2\alpha }}}\hfill \\ & \le {\displaystyle \frac{{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\alpha }{\left[\det {(I-W{\overline{W}}^T)}^{\ell }-{\parallel \omega \parallel }_{\mathcal{P}}^2\right]}^{\alpha }}{{\left||\det {(I-Z{\overline{W}}^T)}^{\ell }|-|{\langle \zeta ,\omega \rangle }_p|\right|}^{2\alpha }}}\hfill \\ & \le 1.\hfill \end{array}$$

Therefore,

$$\parallel f_{(W,\omega )}{\parallel }_{\alpha ,\infty }=\underset{(Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}}{\sup }{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\alpha }\left|f_{(W,\omega )}(Z,\zeta )\right|\le 1.$$

Taking partial derivatives of the test function, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial f_{(W,\omega )}}{\partial z_{st}}}& ={\displaystyle \frac{{\left[\det {\left(I-W{\overline{W}}^T\right)}^{\ell }-{\parallel \omega \parallel }_{\mathcal{P}}^2\right]}^{\alpha }}{{\left[\det {\left(I-Z{\overline{W}}^T\right)}^{\ell }-{\langle \zeta ,\omega \rangle }_p\right]}^{2\alpha +1}}}·2\alpha \ell \det {(I-Z{\overline{W}}^T)}^{\ell }\mathrm{tr}\left[{(I-Z{\overline{W}}^T)}^{-1}{I}_{st}{\overline{W}}^T\right],\hfill \\ \hfill {\displaystyle \frac{\partial f_{(W,\omega )}}{\partial {\zeta }_j}}& ={\displaystyle \frac{{\left[\det {\left(I-W{\overline{W}}^T\right)}^{\ell }-{\parallel \omega \parallel }_{\mathcal{P}}^2\right]}^{\alpha }}{{\left[\det {\left(I-Z{\overline{W}}^T\right)}^{\ell }-{\langle \zeta ,\omega \rangle }_p\right]}^{2\alpha +1}}}·2\alpha {\mathcal{P}}_j{\zeta }_j^{{\mathcal{P}}_j-1}{\overline{{\omega }_j}}^{{\mathcal{P}}_j},\hfill \end{array}$$

where $1\le s\le m$, $1\le t\le n$, $j=1,\dots ,r$. Therefore,

$$\begin{array}{cc} & |C_{\varphi }D{f}_{(W,\omega )}\left)(Z,\zeta )\right|\hfill \\ & ={\displaystyle \frac{{\left[\det {(I-W{\overline{W}}^T)}^{\ell }-{\parallel \omega \parallel }_{\mathcal{P}}^2\right]}^{\alpha }}{{\left||\det {(I-Z_{\varphi }{\overline{W}}^T)}^{\ell }|-|{\langle {\zeta }_{\varphi },\omega \rangle }_p|\right|}^{2\alpha +1}}}·2\alpha \hfill \\ & \times \left\{\ell \left|\det {(I-Z_{\varphi }{\overline{W}}^T)}^{\ell }\right|\times {\left[\sum _{\begin{array}{c}1\le s\le m\\ 1\le t\le n\end{array}}{\left|tr\left[{(I-Z_{\varphi }{\overline{W}}^T)}^{-1}{I}_{st}{\overline{W}}^T\right]\right|}^2\right]}^{{\displaystyle \frac{1}{2}}}+{\left[{\displaystyle \sum _{j=1}^r}{\mathcal{P}}_j^2{\left|{{\zeta }_{\varphi }}_j^{{\mathcal{P}}_j-1}{\overline{{\omega }_j}}^{{\mathcal{P}}_j}\right|}^2\right]}^{{\displaystyle \frac{1}{2}}}\right\}.\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill \infty & >{∥\sigma C_{\varphi }D∥}_{H_{\alpha }^{\infty }\to H_{\beta }^{\infty }}\ge {∥\sigma C_{\varphi }D{f}_{(W,\omega )}∥}_{\beta ,\infty }\hfill \\ \hfill & =\underset{(Z,\zeta )\in {\mathrm{GHE}}_I}{\sup }{\left[\det {\left(I-Z{\overline{Z}}^T\right)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }\left|\sigma (Z,\zeta )\right|\left|C_{\varphi }D{f}_{(W,\omega )}(Z,\zeta )\right|\hfill \\ \hfill & \ge {\left[\det {\left(I-Z{\overline{Z}}^T\right)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }\left|\sigma (Z,\zeta )\right|·2\alpha ·{\displaystyle \frac{{\left[\det {\left(I-W{\overline{W}}^T\right)}^{\ell }-{\parallel \omega \parallel }_{\mathcal{P}}^2\right]}^{\alpha }}{\left|\det {\left(I-Z_{\varphi }{\overline{W}}^T\right)}^{\ell }\right|-{\left|{〈{\zeta }_{\varphi },\omega 〉}_p\right|}^{2\alpha +1}}}\hfill \\ \hfill & \times \left\{\ell \left|\det {\left(I-Z_{\varphi }{\overline{W}}^T\right)}^{\ell }\right|\times {\left[\sum _{\begin{array}{c}1\le s\le m\\ 1\le t\le m\end{array}}{\left|\mathrm{tr}\left[{(I-Z_{\varphi }{\overline{W}}^T)}^{-1}{I}_{st}{\overline{W}}^T\right]\right|}^2\right]}^{{\displaystyle \frac{1}{2}}}+{\left[{\displaystyle \sum _{j=1}^r}{\mathcal{P}}_j^2{\left|{\zeta }_{\varphi ,j}^{{\mathcal{P}}_j-1}{\overline{\omega }}_j^{{\mathcal{P}}_j}\right|}^2\right]}^{{\displaystyle \frac{1}{2}}}\right\}.\hfill \end{array}$$

Let $(W,\omega )=(Z_{\varphi },{\zeta }_{\varphi })=\varphi (Z,\zeta )$,

$$\begin{array}{cc} & {∥\sigma C_{\varphi }D{f}_{(W,\omega )}∥}_{\beta ,\infty }\hfill \\ & \ge {\left[\det {\left(I-Z{\overline{Z}}^T\right)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }\left|\sigma (Z,\zeta )\right|{\displaystyle \frac{{\left[\det {(I-Z_{\varphi }{\overline{Z_{\varphi }}}^T)}^{\ell }-{\parallel {\zeta }_{\varphi }\parallel }_{\mathcal{P}}^2\right]}^{\alpha }}{{\left||\det {(I-Z_{\varphi }{\overline{Z_{\varphi }}}^T)}^{\ell }|-\parallel {\zeta }_{\varphi }{\parallel }_{\mathcal{P}}^2\right|}^{2\alpha +1}}}·2\alpha \times G(Z_{\varphi },{\zeta }_{\varphi })\hfill \\ & =2\alpha ·{\displaystyle \frac{\left|\sigma (Z,\zeta )\right|{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }}{{\left[\det {(I-Z_{\varphi }{\overline{Z_{\varphi }}}^T)}^{\ell }-{\parallel {\zeta }_{\varphi }\parallel }_{\mathcal{P}}^2\right]}^{\alpha +1}}}G(Z_{\varphi },{\zeta }_{\varphi }).\hfill \end{array}$$

Therefore,

$$\underset{(Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}}{\sup }{\displaystyle \frac{\left|\sigma (Z,\zeta )\right|{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }}{{\left[\det {(I-Z_{\varphi }{\overline{Z_{\varphi }}}^T)}^{\ell }-{\parallel {\zeta }_{\varphi }\parallel }_{\mathcal{P}}^2\right]}^{\alpha +1}}}G(Z_{\varphi },{\zeta }_{\varphi })<\infty .$$

□

For $\alpha >0$, $\ell =m=1$, ${\mathcal{P}}_1=\cdots ={\mathcal{P}}_r=1$, ${\mathrm{GHE}}_{\mathrm{I}}$ reduces to the classical unit ball

$${\mathbb{B}}_{n+r}=\{z\in {\mathbb{C}}^{n+r}{:\left|z\right|}^2<1\}.$$

As an immediate consequence of the main result, we arrive at the following corollary:

Corollary 1.

Suppose ϕ is a holomorphic self-map on ${\mathbb{B}}_{n+r}$, and $\sigma \in H\left({\mathbb{B}}_{n+r}\right)$. Then $\sigma C_{\varphi }D:{\mathcal{H}}_{\alpha }^{\infty }\left({\mathbb{B}}_{n+r}\right)\to {\mathcal{H}}_{\beta }^{\infty }\left({\mathbb{B}}_{n+r}\right)$ is bounded if and only if

$$\underset{z\in {\mathbb{B}}_{n+r}}{\sup }{\displaystyle \frac{\left|\sigma \left(z\right)\right|(1-{\left|z\right|}^2{)}^{\beta }}{{(1-|\varphi \left(z\right)|}^2{)}^{\alpha +1}}}<\infty .$$

4. Compactness of $\mathit{\sigma }{\mathit{C}}_{\mathit{\varphi }}\mathit{D}:{\mathcal{H}}_{\mathit{\alpha }}^{\infty }\left({\mathbf{GHE}}_{\mathbf{I}}\right)\to {\mathcal{H}}_{\mathit{\beta }}^{\infty }\left({\mathbf{GHE}}_{\mathbf{I}}\right)$

In this section, Lemma 7 is also applied to prove compactness of the operator, which we verify by discussing the properties of the constructed test functions.

Theorem 2.

Suppose $\alpha ,\beta >0$, $0<\ell m\le 1$, and let $\varphi =({\varphi }_{11},{\varphi }_{12},\dots ,{\varphi }_{mn+r})$ be a holomorphic self-map of ${\mathrm{GHE}}_{\mathrm{I}}$, $\sigma \in H\left({\mathrm{GHE}}_{\mathrm{I}}\right)$. Denote $(Z_{\varphi },{\zeta }_{\varphi })=\varphi (Z,\zeta )$, and let ${\mathcal{P}}_j$ ($j=1,2,\dots ,r$) be positive integers. If $\sigma \in {\mathcal{H}}_{\beta }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ and

$$\underset{\varphi (Z,\zeta )\to \partial {\mathrm{GHE}}_{\mathrm{I}}}{\lim }{\displaystyle \frac{\left|\sigma (Z,\zeta )\right|{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }}{{\left[\det {(I-Z_{\varphi }{\overline{Z_{\varphi }}}^T)}^{\ell }-{\parallel {\zeta }_{\varphi }\parallel }_{\mathcal{P}}^2\right]}^{\frac{\alpha +1}{\ell }}}}=0,$$

(16)

then $\sigma C_{\varphi }D:{\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)\to {\mathcal{H}}_{\beta }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ is compact.

For the other direction, if $\sigma C_{\varphi }D:{\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)\to {\mathcal{H}}_{\beta }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ is compact, then $\sigma \in {\mathcal{H}}_{\beta }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ and

$$\underset{\varphi (Z,\zeta )\to \partial {\mathrm{GHE}}_{\mathrm{I}}}{\lim }{\displaystyle \frac{\left|\sigma (Z,\zeta )\right|{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }}{{\left[\det {(I-Z_{\varphi }{\overline{Z_{\varphi }}}^T)}^{\ell }-{\parallel {\zeta }_{\varphi }\parallel }_{\mathcal{P}}^2\right]}^{\alpha +1}}}G(Z_{\varphi },{\zeta }_{\varphi })=0,$$

(17)

where $G(Z_{\varphi },{\zeta }_{\varphi })$ is the same as (15).

Proof. 

Assuming that (16) holds, then the supremum condition

$$\underset{(Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}}{\sup }{\displaystyle \frac{\left|\sigma (Z,\zeta )\right|{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }}{{\left[\det {(I-Z_{\varphi }{\overline{Z_{\varphi }}}^T)}^{\ell }-{\parallel {\zeta }_{\varphi }\parallel }_{\mathcal{P}}^2\right]}^{\frac{\alpha +1}{\ell }}}}<\infty$$

(18)

holds. By virtue of Theorem 1, the operator $\sigma C_{\varphi }D:{\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)\to {\mathcal{H}}_{\beta }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ is bounded. Choose a bounded sequence ${\left\{f_j\right\}}_{j\ge 1}$ converging uniformly to zero on every compact subset of ${\mathrm{GHE}}_{\mathrm{I}}$. Thus, one can find $J_1>0$, for which $\parallel f_j{\parallel }_{\alpha ,\infty }\le J_1$ holds for all $j=1,2,\cdots$. In view of (16), for any $\epsilon >0$, there exists $\delta \in (0,1)$, whenever $\mathrm{dist}(\varphi (Z,\zeta ),\partial {\mathrm{GHE}}_{\mathrm{I}})<\delta$, then

$${\displaystyle \frac{\left|\sigma (Z,\zeta )\right|{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }}{{\left[\det {(I-Z_{\varphi }{\overline{Z_{\varphi }}}^T)}^{\ell }-{\parallel {\zeta }_{\varphi }\parallel }_{\mathcal{P}}^2\right]}^{\frac{\alpha +1}{\ell }}}}<\epsilon .$$

We now apply Lemma 7 to estimate the main term:

$$\begin{array}{cc}\hfill & {\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }\left|\left(\sigma C_{\varphi }D{f}_j\right)(Z,\zeta )\right|\hfill \\ \hfill & ={\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }|\sigma (Z,\zeta )·\left(C_{\varphi }D{f}_j\right)(Z,\zeta )|\hfill \\ \hfill & ={\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }\left|\sigma (Z,\zeta )\right||D{f}_j\left(\varphi (Z,\zeta )\right)|\hfill \\ \hfill & ={\displaystyle \frac{\left|\sigma (Z,\zeta )\right|{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }}{{\left[\det {(I-Z_{\varphi }{\overline{Z_{\varphi }}}^T)}^{\ell }-{\parallel {\zeta }_{\varphi }\parallel }_{\mathcal{P}}^2\right]}^{\frac{\alpha +1}{\ell }}}}{\left[\det {(I-Z_{\varphi }{\overline{Z_{\varphi }}}^T)}^{\ell }-{\parallel {\zeta }_{\varphi }\parallel }_{\mathcal{P}}^2\right]}^{{\displaystyle \frac{\alpha +1}{\ell }}}\left|D{f}_j\left(\varphi (Z,\zeta )\right)\right|\hfill \\ \hfill & \le C{\displaystyle \frac{\left|\sigma (Z,\zeta )\right|{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }}{{\left[\det {(I-Z_{\varphi }{\overline{Z_{\varphi }}}^T)}^{\ell }-{\parallel {\zeta }_{\varphi }\parallel }_{\mathcal{P}}^2\right]}^{\frac{\alpha +1}{\ell }}}}{\parallel f_j\parallel }_{\alpha ,\infty }\hfill \\ \hfill & \le C{J}_1\epsilon .\hfill \end{array}$$

Additionally, let

$$E_{\delta }:=\{\varphi (Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}:\mathrm{dist}(\varphi (Z,\zeta ),\partial {\mathrm{GHE}}_{\mathrm{I}})\ge \delta \}.$$

Then $E_{\delta }$ is a compact subset of ${\mathrm{GHE}}_{\mathrm{I}}$. For the above $\epsilon$, we can find $N_{\epsilon }>0$ so that $|f_j\left(\varphi (Z,\zeta )\right)|<\epsilon$, for all $j>N_{\epsilon }$ whenever $(Z,\zeta )\in E_{\delta }$. Since $\sigma \in {\mathcal{H}}_{\beta }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$, it follows from the Weierstrass theorem that $|D{f}_j\left(\varphi (Z,\zeta )\right)|<\epsilon$, for sufficiently large j and all $(Z,\zeta )\in E_{\delta }$. Therefore, for the above $\epsilon$, then the following holds:

$$\begin{array}{cc}\hfill & {\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }\left|\left(\sigma C_{\varphi }D{f}_j\right)(Z,\zeta )\right|\hfill \\ \hfill & ={\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }\left|\sigma (Z,\zeta )·\left(C_{\varphi }D{f}_j\right)(Z,\zeta )\right|\hfill \\ \hfill & ={\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }\left|\sigma (Z,\zeta )\right||D{f}_j\left(\varphi (Z,\zeta )\right)|\hfill \\ \hfill & \le {\parallel \sigma \parallel }_{\beta ,\infty }\epsilon .\hfill \end{array}$$

Combining the inequalities above, we obtain

$$\parallel \sigma C_{\varphi }D{f}_j{\parallel }_{\beta ,\infty }=\underset{(Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}}{\sup }{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }\left|\left(\sigma C_{\varphi }D{f}_j\right)(Z,\zeta )\right|\to 0\mathrm{as}j\to \infty .$$

Finally, using Lemma 10, we conclude that $\sigma C_{\varphi }D:{\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)\to {\mathcal{H}}_{\beta }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ is compact.

Meanwhile, since every compact operator between Banach spaces is automatically bounded, it follows that if $\sigma C_{\varphi }D:{\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)\to {\mathcal{H}}_{\beta }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ is compact, then it is also bounded. Let $f=z_{11}$, and clearly $f\in {\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$. Then there exists a constant C satisfying that

$${\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }\left|\sigma (Z,\zeta )\right|={\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }\left|\left(\sigma C_{\varphi }Df\right)(Z,\zeta )\right|\le C{\parallel f\parallel }_{\alpha ,\infty }<\infty .$$

This indicates $\sigma \in {\mathcal{H}}_{\beta }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$. Take into account the sequence $(W^i,{\omega }^i)=\varphi (Z^i,{\zeta }^i)$ subject to $\varphi (Z^i,{\zeta }^i)\to \partial {\mathrm{GHE}}_{\mathrm{I}}$ as $i\to \infty$. In the absence of such a sequence, $\varphi (Z^i,{\zeta }^i)$ cannot approach the boundary of ${\mathrm{GHE}}_{\mathrm{I}}$ at all. In this case, the limit in condition (17) is vacuously satisfied, since there are no points along which the expression could fail to tend to zero. Furthermore, we construct the following sequence of test functions:

$$f_i(Z,\zeta ):={\displaystyle \frac{{\left[\det {(I-W^i{\overline{W^i}}^T)}^{\ell }-{\parallel {\omega }^i\parallel }_{\mathcal{P}}^2\right]}^{\alpha }}{{\left[\det {(I-Z{\overline{W^i}}^T)}^{\ell }-{\langle \zeta ,{\omega }^i\rangle }_p\right]}^{2\alpha }}},i=1,2,\cdots .$$

For one thing, the test function is bounded in the norm of ${\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$; for another, it tends to zero near the boundary of the domain. For all $(Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}$, by (12), we have

$$\begin{array}{cc}\hfill {\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\alpha }\left|f_i(Z,\zeta )\right|& ={\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\alpha }{\displaystyle \frac{{\left[\det {(I-W^i{\overline{W^i}}^T)}^{\ell }-{\parallel {\omega }^i\parallel }_{\mathcal{P}}^2\right]}^{\alpha }}{{\left|\det {(I-Z{\overline{W^i}}^T)}^{\ell }-{\langle \zeta ,{\omega }^i\rangle }_p\right|}^{2\alpha }}}\hfill \\ & \le {\displaystyle \frac{{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\alpha }{\left[\det {(I-W^i{\overline{W^i}}^T)}^{\ell }-{\parallel {\omega }^i\parallel }_{\mathcal{P}}^2\right]}^{\alpha }}{{\left|\det {(I-Z{\overline{W^i}}^T)}^{\ell }-\left|{\langle \zeta ,{\omega }^i\rangle }_p\right|\right|}^{2\alpha }}}\hfill \\ & \le 1.\hfill \end{array}$$

Thus,

$$\parallel f_i{\parallel }_{\alpha ,\infty }=\underset{(Z,\zeta )\in {\mathrm{GHE}}_{\mathrm{I}}}{\sup }{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\alpha }\left|f_i(Z,\zeta )\right|\le 1.$$

This implies that $\left\{f_i\right\}$ possesses boundedness relative to ${\parallel ·\parallel }_{\alpha ,\infty }$, so $f_i\in {\mathcal{H}}_{\alpha }^{\infty }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ for $i=1,2,\dots$. By (11), we also have

$$\begin{array}{cc}\hfill |f_i(Z,\zeta )|& ={\displaystyle \frac{{\left[\det {(I-W^i{\overline{W^i}}^T)}^{\ell }-{\parallel {\omega }^i\parallel }_{\mathcal{P}}^2\right]}^{\alpha }}{{\left|\det {(I-Z{\overline{W^i}}^T)}^{\ell }-{\langle \zeta ,{\omega }^i\rangle }_p\right|}^{2\alpha }}}\hfill \\ & \le {\displaystyle \frac{{\left[\det {(I-W^i{\overline{W^i}}^T)}^{\ell }-{\parallel {\omega }^i\parallel }_{\mathcal{P}}^2\right]}^{\alpha }}{{\left|\det {(I-Z{\overline{W^i}}^T)}^{\ell }-\left|{\langle \zeta ,{\omega }^i\rangle }_p\right|\right|}^{2\alpha }}}\hfill \\ & \le {\displaystyle \frac{{\left[\det {(I-W^i{\overline{W^i}}^T)}^{\ell }-{\parallel {\omega }^i\parallel }_{\mathcal{P}}^2\right]}^{\alpha }}{{\left\{\frac{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]+\left[\det {(I-W^i{\overline{W^i}}^T)}^{\ell }-{\parallel {\omega }^i\parallel }_{\mathcal{P}}^2\right]}{2}\right\}}^{2\alpha }}}\hfill \end{array}$$

$$\begin{array}{cc} & \le {\displaystyle \frac{2^{2\alpha }{\left[\det {(I-W^i{\overline{W^i}}^T)}^{\ell }-{\parallel {\omega }^i\parallel }_{\mathcal{P}}^2\right]}^{\alpha }}{{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{2\alpha }}}.\hfill \end{array}$$

Since $(W^i,{\omega }^i)\to \partial {\mathrm{GHE}}_{\mathrm{I}}$, as $i\to \infty$, we have $\det {(I-W^i{\overline{W^i}}^T)}^{\ell }-{\parallel {\omega }^i\parallel }_{\mathcal{P}}^2\to 0$. Consider a compact subset $E\subset {\mathrm{GHE}}_{\mathrm{I}}$. For $(Z,\zeta )\in E$, the quantity $\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2$ is bounded below by a positive constant, so $\left\{f_i(Z,\zeta )\right\}$ converges uniformly to 0 on every compact subset of ${\mathrm{GHE}}_{\mathrm{I}}$ as $i\to \infty$. By Lemma 10, it follows that $\parallel \sigma C_{\varphi }D{f}_i{\parallel }_{\beta ,\infty }\to 0$.

Taking partial derivatives of the test function, for each i we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial f_i}{\partial z_{st}}}& ={\displaystyle \frac{{\left[\det {\left(I-W^i{\overline{W^i}}^T\right)}^{\ell }-{\parallel {\omega }^i\parallel }_{\mathcal{P}}^2\right]}^{\alpha }}{{\left[\det {\left(I-Z{\overline{W^i}}^T\right)}^{\ell }-{\langle \zeta ,{\omega }^i\rangle }_p\right]}^{2\alpha +1}}}·2\alpha \ell \det {(I-Z{\overline{W^i}}^T)}^{\ell }\mathrm{tr}\left[{(I-Z{\overline{W^i}}^T)}^{-1}{I}_{st}{\overline{W^i}}^T\right],\hfill \\ \hfill {\displaystyle \frac{\partial f_i}{\partial {\zeta }_j}}& ={\displaystyle \frac{{\left[\det {\left(I-W^i{\overline{W^i}}^T\right)}^{\ell }-{\parallel {\omega }^i\parallel }_{\mathcal{P}}^2\right]}^{\alpha }}{{\left[\det {\left(I-Z{\overline{W^i}}^T\right)}^{\ell }-{\langle \zeta ,{\omega }^i\rangle }_p\right]}^{2\alpha +1}}}·2\alpha {\mathcal{P}}_j{\zeta }_j^{{\mathcal{P}}_j-1}{\overline{{\omega }_j^i}}^{{\mathcal{P}}_j},\hfill \end{array}$$

where $1\le s\le m$, $1\le t\le n$, $j=1,\dots ,r$, and $i=1,2,\cdots$.

By substituting the partial derivatives of $f_i$ into the expression for the operator $C_{\varphi }D$, we derive the following estimate for $\parallel \sigma C_{\varphi }D{f}_i{\parallel }_{\beta ,\infty }$:

$$\begin{array}{cc}\hfill & \parallel \sigma C_{\varphi }D{f}_i{\parallel }_{\beta ,\infty }\hfill \\ \hfill & =\underset{(Z,\zeta )\in {\mathrm{GHE}}_I}{\sup }{\left[\det {\left(I-Z{\overline{Z}}^T\right)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }\left|\sigma (Z,\zeta )\right|\left|C_{\varphi }D{f}_i(Z,\zeta )\right|\hfill \\ \hfill & =\underset{\varphi (Z,\zeta )\in {\mathrm{GHE}}_1}{\sup }{\left[\det {\left(I-Z{\overline{Z}}^T\right)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }\left|\sigma (Z,\zeta )\right|\hfill \\ \hfill & \times {\displaystyle \frac{{\left[\det {(I-W^i{\overline{W^i}}^T)}^{\ell }-{\parallel {\omega }^i\parallel }_{\mathcal{P}}^2\right]}^{\alpha }}{{\left|\det {(I-Z_{\varphi }{\overline{W^i}}^T)}^{\ell }-{\langle {\zeta }_{\varphi },{\omega }^i\rangle }_p\right|}^{2\alpha +1}}}·2\alpha \hfill \\ \hfill & \times \left\{\ell \left|\det {(I-Z_{\varphi }{\overline{W^i}}^T)}^{\ell }\right|\times {\left[\sum _{\begin{array}{c}1\le s\le m\\ 1\le t\le n\end{array}}{\left|tr\left[{(I-Z_{\varphi }{\overline{W^i}}^T)}^{-1}{I}_{st}{\overline{W^i}}^T\right]\right|}^2\right]}^{{\displaystyle \frac{1}{2}}}+{\left[{\displaystyle \sum _{j=1}^r}{\mathcal{P}}_j^2{\left|{{\zeta }_{\varphi }}_j^{{\mathcal{P}}_j-1}{\overline{{\omega }_j^i}}^{{\mathcal{P}}_j}\right|}^2\right]}^{{\displaystyle \frac{1}{2}}}\right\}.\hfill \end{array}$$

Taking $(W^i,{\omega }^i)=\varphi (Z^i,{\zeta }^i)$,

$$\begin{array}{c}\hfill \parallel \sigma C_{\varphi }D{f}_i{\parallel }_{\beta ,\infty }\ge {\displaystyle \frac{|\sigma (Z^i,{\zeta }^i)|{\left[\det {\left(I-Z^i{\overline{Z^i}}^T\right)}^{\ell }-{\parallel {\zeta }^i\parallel }_{\mathcal{P}}^2\right]}^{\beta }}{{\left[\det {\left(I-Z_{\varphi }^i{\overline{Z_{\varphi }^i}}^T\right)}^{\ell }-{\parallel {\zeta }_{\varphi }^i\parallel }_{\mathcal{P}}^2\right]}^{\alpha +1}}}\times G(Z_{\varphi }^i,{\zeta }_{\varphi }^i),\end{array}$$

where

$$G(Z_{\varphi }^i,{\zeta }_{\varphi }^i)=\ell \left|\det {(I-Z_{\varphi }^i{\overline{Z_{\varphi }^i}}^T)}^{\ell }\right|\times {\left[\sum _{\begin{array}{c}1\le s\le m\\ 1\le t\le n\end{array}}{\left|tr\left[{(I-Z_{\varphi }^i{\overline{Z_{\varphi }^i}}^T)}^{-1}{I}_{st}{\overline{Z_{\varphi }^i}}^T\right]\right|}^2\right]}^{{\displaystyle \frac{1}{2}}}+{\left[{\displaystyle \sum _{j=1}^r}{\mathcal{P}}_j^2{\left|{{\zeta }_{\varphi }^i}_j\right|}^{4{\mathcal{P}}_j-2}\right]}^{{\displaystyle \frac{1}{2}}}.$$

Since $\parallel \sigma C_{\varphi }D{f}_i{\parallel }_{\beta ,\infty }\to 0$, as $i\to \infty$, it follows that

$$\underset{\varphi (Z,\zeta )\to \partial {\mathrm{GHE}}_{\mathrm{I}}}{\lim }{\displaystyle \frac{\left|\sigma (Z,\zeta )\right|{\left[\det {(I-Z{\overline{Z}}^T)}^{\ell }-{\parallel \zeta \parallel }_{\mathcal{P}}^2\right]}^{\beta }}{{\left[\det {(I-Z_{\varphi }{\overline{Z_{\varphi }}}^T)}^{\ell }-{\parallel {\zeta }_{\varphi }\parallel }_{\mathcal{P}}^2\right]}^{\alpha +1}}}G(Z_{\varphi },{\zeta }_{\varphi })=0.$$

□

By considering the same case as in Corollary 1, we obtain the following corollary on the unit ball, which gives the necessary and sufficient conditions for compactness.

Corollary 2.

Suppose ϕ is a holomorphic self-map on ${\mathbb{B}}_{n+r}$, and $\sigma \in H\left({\mathbb{B}}_{n+r}\right)$. Then $\sigma C_{\varphi }D:{\mathcal{H}}_{\alpha }^{\infty }\left({\mathbb{B}}_{n+r}\right)\to {\mathcal{H}}_{\beta }^{\infty }\left({\mathbb{B}}_{n+r}\right)$ is a compact operator if and only if

$$\underset{\varphi \left(z\right)\to \partial {\mathbb{B}}_{n+r}}{\lim }{\displaystyle \frac{\left|\sigma \left(z\right)\right|(1-{\left|z\right|}^2{)}^{\beta }}{{(1-|\varphi \left(z\right)|}^2{)}^{\alpha +1}}}=0.$$