Let $\Omega$ be a bounded domain of ${\mathbb{C}}^n$ and $H(\Omega )$ the class of all holomorphic functions on $\Omega$ . Then consider a holomorphic self-map $\varphi$ of $\Omega$ and a function $\psi \in H(\Omega )$. The linear operator is referred to as a weighted composition operator for $f\in H(\Omega )$. If $\psi \left(z\right)\equiv 1$, it reduces to the composition operator, whereas for $\varphi \left(z\right)=z$ it becomes the multiplication operator. For any given holomorphic function f, $\left(\psi C_{\varphi }f\right)\left(z\right)$ represents a generalised composition/multiplication operator. The reader is referred to book[1] for an extensive introduction to the topic.

In this paper, we study the boundedness and the compactness of weighted composition operators from $\alpha$-Bloch spaces ${\mathcal{B}}^{\alpha }$ to Bers-type spaces built on generalised Hua domains of the first kind. On ${\mathrm{GHE}}_{\mathrm{I}}$ the $\alpha$-Bloch space ${\mathcal{B}}^{\alpha }$ consists of all $f\in H\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ such that where

It is clear that ${\mathcal{B}}^{\alpha }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ is a Banach space.

In 1930, Cartan [2] was the the first to characterise the six types of irreducible bounded symmetric domains, which consist of four types of bounded symmetric classical domains, also referred to as Cartan domains, and two exceptional domains whose complex dimension are 16 and 27, respectively. The Cartan domains are defined as follows: where $Z^{\prime }$ denotes the transpose of Z, $\overline{Z}$ denotes the conjugate of Z, and $m,n,p,q$ are positive integers. In 1998, building on the notion of bounded symmetric domains, Yin and Roos constructed a new type of domain called the Cartan-Hartogs domain[3], and Yin introduced the so-called Hua domains[4], which include the Cartan-Hartogs domains, the Cartan-Egg domains, the Hua domains, the generalized Hua domains, and the Hua construction. The generalized Hua domains are defined as follows: where ${\xi }_j=({\xi }_{j1},\cdots ,{\xi }_{j{N}_j}),j=1,\cdots ,r$,${\Re }_{\mathrm{I}}(m,n),{\Re }_{\mathrm{II}}\left(p\right),{\Re }_{\mathrm{III}}\left(q\right),{\Re }_{\mathrm{IV}}\left(n\right)$ denote respectively the Cartan domains of the first type, second type, third type and fourth type, $Z^{\prime }$ denotes the transpose of Z, $\overline{Z}$ denotes the conjugate of Z , $N_1,\cdots ,N_r,m,n,p,q$ are positive integers , and $p_1,\cdots ,p_r$ are positive real numbers. For $k=1,m=1,p_1=\cdots =p_r=1$, the generalized Hua domain of the first kind reduces to the unit ball. Without loss of generality, we may assume that $N_j=1$, then ${\xi }_j\in \mathbb{C},j=1,\cdots ,r,\xi =({\xi }_1,\cdots ,{\xi }_r)$ and ${\parallel \xi \parallel }_p^2={\sum }_{j=1}^r{\left|{\xi }_j\right|}^{2p_j}$. We define

We also write where $|{\xi }_i{|}^{p_i}={\alpha }_i,{\left|t_i\right|}^{p_i}={\beta }_i(i=1,\cdots ,r),\alpha =({\alpha }_1,\cdots ,{\alpha }_r),\beta =({\beta }_1,\cdots ,{\beta }_r).$

For the sake of convenience, the four types of generalized Hua domains will be referred to as ${\mathrm{GHE}}_{\mathrm{I}},{\mathrm{GHE}}_{\mathrm{II}},{\mathrm{GHE}}_{\mathrm{III}},{\mathrm{GHE}}_{\mathrm{IV}}$.

On ${\mathrm{GHE}}_{\mathrm{I}}$, a Bers-type space ${\mathcal{A}}_{\beta }$ consists of all $f\in H\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ such that

It is easy to see that ${\mathcal{A}}_{\beta }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ is a Banach space with norm $\parallel ·\parallel$.

The boundedness and the compactness of weighted composition operators on (or between) spaces of holomorphic functions on various domains received a large attention. Wang and Liu [5] studied the boundedness and the compactness of the weighted composition operators on the Bers-type space on the open unit disc, whereas Zhou and Xu [6] characterised the boundedness and the compactness of the weighted composition operators between $\alpha$-Bloch space and $\beta$-Bloch space, and Li [11] investigated the boundedness and the compactness of the weighted composition operators from Hardy space to Bers-type space, Zhu [19] characterised the boundedness and compactness of $D_{\varphi ,u}^n:\mathcal{B}\to H_{\alpha }^{\infty }$. For the unit poly-disk, Li and Stević [7][8] presented some necessary and sufficient conditions for the boundedness and the compactness of the weighted composition operators between $H^{\infty }$ and $\alpha$-Bloch space, whereas for the open unit ball, Li and Stević [9] studied the boundedness and the compactness of the weighted composition operators between $H^{\infty }$ and Bloch space [see also [14]-[17]].

Jiang[10] has charaterised the boundedness and the compactness of the weighted composition operators on the Bers-type space on the Hua domains. On the other hand, the boundedness and the compactness of the weighted composition operators from $\alpha$-Bloch to ${\mathcal{A}}_{\beta }$ have not been studied in details. In this paper, we obtain some necessary conditions and sufficient conditions for the boundedness and the compactness of the weighted composition operators from $\alpha$-Bloch to ${\mathcal{A}}_{\beta }$ on generalised Hua domain of the first kind by using a generalisation of Hua’s inequalities.

Lemma 2.1. Let $\beta >0$, then

Lemma 2.2.  Let $0<a\le 1,0<b\le 1$ and $b\le a$, q is a positive integer, then

Lemma 2.3. (see[12]) Let $x\ge -1$, if $0<\alpha \le 1$, then

if $\alpha <0$ or $\alpha >1$, then

Lemma 2.4. (see[12]) Let $a_k\ge 0,k=1,2,\cdots ,m$, then

Lemma 2.5. (see[12]) Let $a_k\in \mathbb{C}$, if $p\ge 1$, then

If $0<p<1$, then

Lemma 2.6. (see[13]) Let

be an $m\times n$ matrix $(m\le n)$. Then, there exist an $m\times m$ unitary matrix U and an $n\times n$ unitary matrix V such that

and

Lemma 2.7. (see[13]) Let

satisfying

Then, there exists a square matrix P such that

and the minimum value is obtained for $U=\Theta P$ and $V=I$, where

.

Lemma 2.8. (see[12] Minkowski inequality of integration formula ) Let $a_k,b_k\ge 0,k=1,2\cdots ,n$, then

Lemma 2.9. Let $p_i$$(i=1,2,\cdots ,r)$ be positive integers, $0<km\le 1$, and $t\in [0,1]$, then

Lemma 2.10. Let us consider $0<mk\le 1$, some positive intergers $p_j$$(j=1,2,\cdots ,r)$, $t\in [0,1],(Z,\xi )\in {\mathrm{GHE}}_{\mathrm{I}}$, $q=\max \{p_1,p_2,\cdots ,p_r\}$. Then, the following inequality holds

Lemma 2.11. Given $0<km\le 1$, $p_j$ some positive integers $(j=1,2,\cdots ,r)$, $\forall (Z,\xi )\in {\mathrm{GHE}}_{\mathrm{I}}$,$q=\max \{p_1,p_2,\cdots ,p_r\}$ and f a holomorphic function on ${\mathcal{B}}^{\alpha }$$\left({\mathrm{GHE}}_{\mathrm{I}}\right)$, then there exists a constant C such that

Lemma 2.12. Let $\varphi =({\varphi }_{11},{\varphi }_{12}\dots {\varphi }_{mn+r})$ be a holomorphic self-map of ${\mathrm{GHE}}_{\mathrm{I}}$ and $\psi \in H\left({\mathrm{GHE}}_{\mathrm{I}}\right)$. The weighted composition operator $\psi C_{\varphi }:{\mathcal{B}}^{\alpha }\left({\mathrm{GHE}}_{\mathrm{I}}\right)\to {\mathcal{A}}_{\beta }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ is compact if and only if $\psi C_{\varphi }$ is bounded and for any bounded sequence ${\left\{f_n\right\}}_{n\ge 1}$ in ${\mathcal{B}}^{\alpha }\left({\mathrm{GHE}}_{\mathrm{I}}\right)$ converging to 0 uniformly on compact subsets of ${\mathrm{GHE}}_{\mathrm{I}}$, $\parallel \psi C_{\varphi }{f}_n{\parallel }_{{\mathcal{A}}_{\beta }}\to 0$ as $n\to \infty$.