On a Sum of More Complex Product-Type Operators from Bloch-Type Spaces to the Weighted-Type Spaces

Abstract

The aim of the present paper is to completely characterize the boundedness and compactness of a sum operator defined by some more complex products of composition, multiplication, and mth iterated radial derivative operators from Bloch-type spaces to weighted-type spaces on the unit ball. In some applications, the boundedness and compactness of all products of composition, multiplication, and mth iterated radial derivative operators from Bloch-type spaces to weighted-type spaces on the unit ball are also characterized.

1. Introduction

In this section, we provide a detailed introduction to the operators involved and the motivation of the paper.

Let $\mathbb{N}$ be the natural number set, ${\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$, $B(a,r)=\{z\in {\mathbb{C}}^n:|z-a|<r\}$ the open ball in the complex vector space ${\mathbb{C}}^n$ centered at a with radius r and $\mathbb{B}=B(0,1)$. Let $z=(z_1,z_2,\dots ,z_n)$ and $w=(w_1,w_2,\dots ,w_n)$ be two points in ${\mathbb{C}}^n$. Define $\langle z,w\rangle =z_1{\overline{w}}_1+z_2{\overline{w}}_2+\cdots +z_n{\overline{w}}_n$ and ${\left|z\right|}^2=\langle z,z\rangle$.

1.1. Operators Involved in the Paper

Let $\mathrm{\Omega }$ be a domain in ${\mathbb{C}}^n$, $H\left(\mathrm{\Omega }\right)$ the set of all holomorphic functions on $\mathrm{\Omega }$ and $S\left(\mathrm{\Omega }\right)$ the set of all holomorphic self-maps of $\mathrm{\Omega }$. Let $\phi \in S\left(\mathrm{\Omega }\right)$. Associated with $\phi$ is the composition operator $C_{\phi }$, which is defined by $C_{\phi }f=f\circ \phi$ for $f\in H\left(\mathrm{\Omega }\right)$. Let $u\in H\left(\mathrm{\Omega }\right)$. The multiplication operator $M_u$ is defined by $M_{u}f=u·f$ for $f\in H\left(\mathrm{\Omega }\right)$.

If $n=1$, the open unit ball $\mathbb{B}$ becomes the open unit disk $\mathbb{D}$. Let $m\in {\mathbb{N}}_0$. The well-known mth differentiation operator $D^m$ on $H\left(\mathbb{D}\right)$ is defined by

$$\begin{array}{c}{D}^{m}f\left(z\right)=f^{\left(m\right)}\left(z\right),\hfill \end{array}$$

where $f^{\left(0\right)}=f$. If $m=1$, it is reduced to the classical differentiation operator D. As expected, there has been some considerable interest in investigating products of differentiation and other related operators. For example, the most common products

$$\begin{array}{c}{M}_u{C}_{\phi }D,C_{\phi }{M}_{u}D,C_{\phi }D{M}_u,M_{u}D{C}_{\phi },D{M}_u{C}_{\phi },D{C}_{\phi }{M}_u\hfill \end{array}$$

(1)

were extensively studied (see, for example, [1,2,3,4]). One of the reasons why people are interested in the six product-type operators is that people need to obtain further methods and techniques for studying their properties. Some other products containing differentiation operators can also be found in [5,6,7,8] and the related references therein. However, it is easy to see that if one studies the operators in (1) one by one, it will require a commitment of time and energy. In order to surmount this malpractice, the authors in [9] therefore introduced and investigated the following sum operator (for some later and continuous studies, see, for example, [10,11,12])

$$\begin{array}{c}{T}_{u_0,u_1,\phi }=M_{u_0}{C}_{\phi }+M_{u_1}{C}_{\phi }D,\hfill \end{array}$$

(2)

where $u_0,u_1\in H\left(\mathbb{D}\right)$ and $\phi \in S\left(\mathbb{D}\right)$. Sure enough, the operator $T_{u_0,u_1,\phi }$ allows unified research for the operators in (1). More precisely, it follows that

$$\begin{array}{cccccc}\hfill & M_u{C}_{\phi }D=T_{0,u,\phi },\hfill & \hfill & C_{\phi }{M}_{u}D=T_{0,u\circ \phi ,\phi },\hfill & \hfill & M_{u}D{C}_{\phi }=T_{0,u·{\phi }^{\prime },\phi },\hfill \\ \hfill & C_{\phi }D{M}_u=T_{u^{\prime }\circ \phi ,u\circ \phi ,\phi },\hfill & \hfill & D{M}_u{C}_{\phi }=T_{u^{\prime },u·{\phi }^{\prime },\phi },\hfill & \hfill & D{C}_{\phi }{M}_u=T_{(u^{\prime }\circ \phi )·{\phi }^{\prime },(u\circ \phi )·{\phi }^{\prime },\phi }.\hfill \end{array}$$

A very natural way of extending the operators in (1) can be achieved in terms of replacing D by $D^m$. That is,

$$\begin{array}{c}{M}_u{C}_{\phi }{D}^m,C_{\phi }{M}_u{D}^m,C_{\phi }{D}^m{M}_u,M_u{D}^m{C}_{\phi },D^m{M}_u{C}_{\phi },D^m{C}_{\phi }{M}_u.\hfill \end{array}$$

(3)

The significance of this extension is that in overcoming some difficulties such as those caused by ${(f\circ \phi )}^{\left(m\right)}$, some methods and techniques have been excavated. For example, the following famous Faà di Bruno’s formula (see [13]) was used:

$$\begin{array}{c}{(f\circ \phi )}^{\left(m\right)}\left(z\right)=\sum _{k=0}^m{f}^{\left(k\right)}\left(\phi \left(z\right)\right)B_{m,k}({\phi }^{\prime }\left(z\right),\dots ,{\phi }^{(m-k+1)}\left(z\right)),\hfill \end{array}$$

(4)

where

$$\begin{array}{c}{B}_{m,k}:=B_{m,k}(x_1,x_2,\dots ,x_{m-k+1})=\sum \frac{m!}{{\prod }_{i=1}^{m-k-1}{j}_i!}\prod _{i=1}^{m-k-1}{\left(\frac{x_i}{i!}\right)}^{j_i}\hfill \end{array}$$

(5)

is the Bell polynomial, the sum is taken over all non-negative integer sequences $j_1$, $j_2$, …, $j_{m-k+1}$ satisfying ${\sum }_{i=1}^{m-k+1}{j}_i=k$ and ${\sum }_{i=1}^{m-k+1}i{j}_i=m$. In particular, if $k=0$, we have $B_{0,0}=1$ and $B_{m,0}=0$ for $m\in \mathbb{N}$. If $k=1$, then $B_{i,1}=x_i$. If $m=k=i$, then $B_{i,i}=x_1^i$. By using the Faà di Bruno’s formula, the operators in (3) were studied (see, for example, [14,15,16,17]). Motivated by the above-mentioned discussions, one should naturally consider defining an operator such that the operators in (3) can be studied in a unified manner. There may be many people who have the same idea as us. Actually, the authors in [18] introduced the following operator, which achieved the expectations

$$\begin{array}{c}{T}_{u_0,\dots ,u_m,\phi }^m=\sum _{i=0}^m{M}_{u_i}{C}_{\phi }{D}^i,\hfill \end{array}$$

(6)

where $u_0,u_1,\dots ,u_m\in H\left(\mathbb{D}\right)$ and $\phi \in S\left(\mathbb{D}\right)$. It is clear that if $m=1$, the operator in (6) is reduced to the operator in (2). We first see that the operators $M_u{C}_{\phi }{D}^m$ and $C_{\phi }{M}_u{D}^m$ can be easily expressed into forms of the operator $T_{u_0,\dots ,u_m,\phi }^m$, where functions $u_0$, $u_1$, …, and $u_m$ equal what is very simple and clear. Moreover, it seems to be difficult to express other operators in (3). However, we can still do it in terms of replacing $x_j$ with ${\phi }^{\left(j\right)}$ in the Bell polynomial as follows

$$\begin{array}{cc}\hfill & M_u{D}^m{C}_{\phi }=T_{u_0=u·B_{m,0}^{\phi },u_1=u·B_{m,1}^{\phi },\dots ,u_m=u·B_{m,m}^{\phi },\phi }^m,\hfill \\ \hfill & C_{\phi }{D}^m{M}_u=T_{u_0=C_m^0{u}^{\left(m\right)}\circ \phi ,u_1=C_m^1{u}^{(m-1)}\circ \phi ,\dots ,u_m=C_m^{m}u\circ \phi ,\phi }^m,\hfill \\ \hfill & D^m{M}_u{C}_{\phi }=T_{u_0={\sum }_{i=0}^m{C}_m^i{u}^{(m-i)}·B_{i,0}^{\phi },u_1={\sum }_{i=1}^m{C}_m^i{u}^{(m-i)}·B_{i,1}^{\phi },\dots ,u_m=C_m^{m}u·B_{m,m}^{\phi },\phi }^m,\hfill \\ \hfill & D^m{C}_{\phi }{M}_u=T_{u_0={\sum }_{i=0}^m{C}_m^i{(u\circ \phi )}^{(m-i)}·B_{i,0}^{\phi },u_1={\sum }_{i=1}^m{C}_m^i{(u\circ \phi )}^{(m-i)}·B_{i,1}^{\phi },\dots ,u_m=C_m^m(u\circ \phi )·B_{m,m}^{\phi },\phi }^m,\hfill \end{array}$$

where

$$\begin{array}{c}{B}_{i,j}^{\phi }=B_{i,j}({\phi }^{\prime },{\phi }^{″},\dots ,{\phi }^{i-j+1}).\hfill \end{array}$$

One of the natural ways to extend the differentiation operator on domains in ${\mathbb{C}}^n$ is the radial derivative operator defined by

$$\begin{array}{c}\Re f\left(z\right)=\sum _{j=1}^n{z}_j\frac{\partial f}{\partial z_j}\left(z\right).\hfill \end{array}$$

(7)

As expected, the products of the composition, multiplication, and radial derivative operators

$$\begin{array}{c}{M}_u{C}_{\phi }\Re ,C_{\phi }{M}_u\Re ,C_{\phi }\Re M_u,M_u\Re C_{\phi },\Re M_u{C}_{\phi },\Re C_{\phi }{M}_u\hfill \end{array}$$

(8)

were studied (see, for example, [19,20,21]). Correspondingly, the operator in (2) was extended into the following operator in [22], which completed the unified studies of the operators in (8)

$$\begin{array}{c}{T}_{u_0,u_1,u_2,\phi }=M_{u_0}{C}_{\phi }+M_{u_1}{C}_{\phi }\Re +M_{u_2}\Re C_{\phi },\hfill \end{array}$$

(9)

where $u_0,u_1,u_2\in H\left(\mathbb{B}\right)$ and $\phi \in S\left(\mathbb{B}\right)$. Recently, it has been continuously investigated in [23,24,25].

Interestingly, the radial derivative operator can be employed iteratively, that is, if ${\Re }^{m-1}f$ is defined for some $m\in \mathbb{N}\setminus \left\{1\right\}$, then ${\Re }^{m}f$ is naturally defined by ${\Re }^{m}f=\Re \left({\Re }^{m-1}f\right)$. If $m=0$, then we regard that ${\Re }^{0}f=f$. By using the mth iterated radial derivative operator, we obtain the related product-type operators

$$\begin{array}{c}{M}_u{C}_{\phi }{\Re }^m,C_{\phi }{M}_u{\Re }^m,C_{\phi }{\Re }^m{M}_u,M_u{\Re }^m{C}_{\phi },{\Re }^m{M}_u{C}_{\phi },{\Re }^m{C}_{\phi }{M}_u.\hfill \end{array}$$

(10)

The operator $M_u{C}_{\phi }{\Re }^m$ at first written as ${\Re }_{u,\phi }^m$ was introduced and studied in [26]. We still reconsidered the operator in [27,28]. One of the reasons why we reconsider the operator is that we need to obtain more methods and techniques to study its properties. If people consider the fact that $C_{\phi }{M}_u{\Re }^m=M_{u\circ \phi }{C}_{\phi }{\Re }^m$, then the operator $M_u{C}_{\phi }{\Re }^m$ can be regarded as the simplest one in (10). The relatively more simple one in (10) is the operator $C_{\phi }{\Re }^m{M}_u$. From a direct calculation, we obtain that

$$\begin{array}{c}{C}_{\phi }{\Re }^m{M}_u=\sum _{i=0}^m{C}_m^i{M}_{\left({\Re }^{m-i}u\right)\circ \phi }{C}_{\phi }{\Re }^i.\hfill \end{array}$$

(11)

Motivated by (11), we then in [29] directly introduced and characterized the boundedness and compactness of the sum operator

$$\begin{array}{c}{\mathfrak{S}}_{\overrightarrow{u},\phi }^m=\sum _{i=0}^m{M}_{u_i}{C}_{\phi }{\Re }^i.\hfill \end{array}$$

(12)

The boundedness and compactness of the operator were characterized again in [30], and as an application, the same properties of the operator $C_{\phi }{\Re }^m{M}_u$ were also characterized. Here, what we want to emphasize is that the most complicated one in (10) is the operator ${\Re }^m{M}_u{C}_{\phi }$, if you notice that ${\Re }^m{C}_{\phi }{M}_u={\Re }^m{M}_{u\circ \phi }{C}_{\phi }$, which has been investigated very recently in [31].

1.2. Motivations of the Paper

When we examine the operator ${\mathfrak{S}}_{\overrightarrow{u},\phi }^m$, we find that it is defined by the operator $M_{u_i}{C}_{\phi }{\Re }^i$, which can be regarded as the simplest operator in (10). Naturally, we can try to extend the definition by using other operators in (10). To this end, in this paper, we introduce the sum operator

$$\begin{array}{c}{S}_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}=M_{u_0}{C}_{\phi }+\sum _{i=1}^k{M}_{u_i}{C}_{\phi }{\Re }^i+\sum _{j=1}^l{M}_{v_j}{\Re }^j{C}_{\phi },\hfill \end{array}$$

(13)

where $u_0,u_1,\dots ,u_k,v_1,\dots ,v_l\in H\left(\mathbb{B}\right)$, $\phi \in S\left(\mathbb{B}\right)$, and $k,l\in \mathbb{N}$. By using the operator, the operators in (10) can be easily expressed into the following forms

$$\begin{array}{cc}\hfill & M_u{C}_{\phi }{\Re }^m=S_{u_0\equiv \cdots \equiv u_{m-1}\equiv 0,u_m=u,v_1\equiv \cdots \equiv v_l\equiv 0,\phi }^{m,l},\hfill \\ \hfill & C_{\phi }{M}_u{\Re }^m=S_{u_0\equiv \cdots \equiv u_{m-1}\equiv 0,u_m=u\circ \phi ,v_1\equiv \cdots \equiv v_l\equiv 0,\phi }^{m,l},\hfill \\ \hfill & C_{\phi }{\Re }^m{M}_u=S_{u_0=C_m^0\left({\Re }^{m}u\right)\circ \phi ,u_1=C_m^1\left({\Re }^{m-1}u\right)\circ \phi ,\dots ,u_m=C_m^{m}u\circ \phi ,v_1\equiv \cdots \equiv v_l\equiv 0,\phi }^{m,l},\hfill \\ \hfill & M_u{\Re }^m{C}_{\phi }=S_{u_0\equiv \cdots \equiv u_k\equiv 0,v_1\equiv \cdots \equiv v_{m-1}\equiv 0,v_m=u,\phi }^{k,m},\hfill \\ \hfill & {\Re }^m{M}_u{C}_{\phi }=S_{u_1\equiv \cdots \equiv u_k\equiv 0,u_0=C_m^0{\Re }^{m}u,v_1=C_m^1{\Re }^{m-1}u,\dots ,v_m=C_m^{m}u,\phi }^{k,m},\hfill \\ \hfill & {\Re }^m{C}_{\phi }{M}_u=S_{u_1\equiv \cdots \equiv u_k\equiv 0,u_0=C_m^0{\Re }^m(u\circ \phi ),v_1=C_m^1{\Re }^{m-1}(u\circ \phi ),\dots ,v_m=C_m^{m}u\circ \phi ,\phi }^{k,m}.\hfill \end{array}$$

(14)

One very obvious major difference between the operators ${\mathfrak{S}}_{\overrightarrow{u},\phi }^m$ and $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}$ is that there are some terms $M_{v_j}{\Re }^j{C}_{\phi }$ in the expression of $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}$. When the jth iterated radial derivative operator ${\Re }^j$ lies between the operators $M_u$ and $C_{\phi }$ in the product $M_u{\Re }^j{C}_{\phi }$, we find that there exist some insurmountable difficulties caused by ${\Re }^j(f\circ \phi )$ (see [31]). We, therefore, guess that there also exist some difficulties in the study of the operator $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}$. Motivated by this, we study this operator from Bloch-type space to weighted-type space in this paper. On the other hand, as far as we know, the operator $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}$ has not been studied so far. This study is considerably interesting to a large number of readers. For example, we will prove that in some sense, the operator $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}$ is bounded or compact from Bloch-type space to weighted-type space if and only if each operator defined in (13) is bounded or compact. This is a very exciting phenomenon, but it may be not right for the general case, that is, from the boundedness of the operator $T=T_1+T_2+\cdots +T_m$, where $T_i$ is a linear operator from Banach spaces X to Y, it cannot deduce the boundedness of the operator $T_i:X\to Y$.

1.3. Bloch-Type and Weighted-Type Spaces

A positive continuous function $\varphi$ on the interval $[0,1)$ is called normal (see [32]), if there are $\lambda \in [0,1)$, a and b ($0<a<b$) such that

$$\begin{array}{c}\frac{\varphi \left(r\right)}{{(1-r)}^a}\mathrm{is}\mathrm{decreasing}\mathrm{on}[\lambda ,1),\underset{r\to 1}{lim}\frac{\varphi \left(r\right)}{{(1-r)}^a}=0;\hfill \\ \frac{\varphi \left(r\right)}{{(1-r)}^b}\mathrm{is}\mathrm{increasing}\mathrm{on}[\lambda ,1),\underset{r\to 1}{lim}\frac{\varphi \left(r\right)}{{(1-r)}^b}=+\infty .\hfill \end{array}$$

The functions $\{\varphi ,\psi \}$ will be called a normal pair, if $\varphi$ is normal and for b in above definition of normal function there exists $\beta >b$ such that

$$\begin{array}{c}\varphi \left(r\right)\psi \left(r\right)={(1-r^2)}^{\beta }.\hfill \end{array}$$

If $\varphi$ is normal, then there exists $\psi$ such that $\{\varphi ,\psi \}$ is a normal pair (see [32]). Note that if $\{\varphi ,\psi \}$ is a normal pair, then $\psi$ is also normal. The purpose of introducing normal pair is to characterize the duality of spaces defined by the normal functions (see, for example, [33,34]). For such a function, the following examples were given in [6]:

$$\mu \left(r\right)={(1-r^2)}^{\alpha },\alpha \in (0,+\infty ),$$

$$\mu \left(r\right)={(1-r^2)}^{\alpha }{\left\{\log 2{(1-r^2)}^{-1}\right\}}^{\beta },\alpha \in (0,1),\beta \in \left. [\frac{\alpha -1}{2}\log 2,0\right],$$

and

$$\mu \left(r\right)={(1-r^2)}^{\alpha }{\left\{\log \log e^2{(1-r^2)}^{-1}\right\}}^{\gamma },\alpha \in (0,1),\gamma \in \left. [\frac{\alpha -1}{2}\log 2,0\right].$$

The following fact can be used to prove that there exist a lot of non-normal functions. It follows from [35] that if $\mu$ is normal, then for each $s\in (0,1)$ there exists a positive constant $C=C\left(s\right)$ such that

$$\begin{array}{c}{C}^{-1}\mu \left(t\right)\le \mu \left(r\right)\le C\mu \left(t\right)\hfill \end{array}$$

(15)

for $0\le r\le t\le r+s(1-r)$. From (15), it is easy to check that the following functions are non-normal

$$\begin{array}{c}\mu \left(r\right)=|sin\left(\log \frac{1}{1-r}\right)|v_{\alpha }\left(r\right)+1\hfill \end{array}$$

and

$$\begin{array}{c}\mu \left(r\right)=|sin\left(\log \frac{1}{1-r}\right)|v_{\alpha }\left(r\right)+\frac{1}{e^{e^{\frac{1}{1-r}}}},\hfill \end{array}$$

where

$$v_{\alpha }\left(r\right)={\left. [(1-r){\left(\log \frac{e}{1-r}\right)}^{\alpha }\right]}^{-1}.$$

From the definition of the normal function, we have that there exists a positive constant $\delta \in (0,1)$ such that for $r\in (\delta ,1)$ it follows that $\varphi \left(r\right)\le {(1-r)}^a$, which shows that ${sup}_{r\in (\delta ,1)}\varphi \left(r\right)\le {(1-\delta )}^a$. Since $\varphi$ is continuous and positive on $[0,\delta ]$, it follows that ${max}_{r\in [0,\delta ]}\varphi \left(r\right)<+\infty$. Therefore, the normal function is bounded on $[0,1)$.

Let $\varphi$ be a normal function. The Bloch-type space ${\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)$ consists of all $f\in H\left(\mathbb{B}\right)$ such that

$$\begin{array}{c}{\parallel f\parallel }_{{\beta }_{\varphi }\left(\mathbb{B}\right)}=\underset{z\in \mathbb{B}}{sup}\varphi \left(\right|z\left|\right)|\Re f\left(z\right)|<+\infty .\hfill \end{array}$$

${\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)$ is a Banach space with the norm

$$\begin{array}{c}{\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}=\left|f\left(0\right)\right|+{\parallel f\parallel }_{{\beta }_{\varphi }\left(\mathbb{B}\right)}.\hfill \end{array}$$

In particular, if $\varphi \left(r\right)=(1-r^2)\log \frac{e}{1-r^2}$, then the space ${\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)$ is the logarithmic Bloch space ${\mathcal{B}}_{\log }\left(\mathbb{B}\right)$. If $\varphi \left(r\right)={(1-r^2)}^{\alpha }$$(\alpha >0)$, then the space ${\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)$ is simplified to the classical weighted Bloch space ${\mathcal{B}}_{Ø}\left(\mathbb{B}\right)$. One can see [36] for some results on the Bloch-type spaces. The operators involved Bloch-type spaces, including Toeplitz operators, composition operators, weighted composition operators, products of composition, multiplication and mth differentiation operators, and so on (see, for example, [37,38,39,40,41]).

A positive and continuous function $\mu$ on $\mathbb{B}$ is said to be weight. Then, the weighted-type space $H_{\mu }^{\infty }\left(\mathbb{B}\right)$ consists all $f\in H\left(\mathbb{B}\right)$ such that

$$\begin{array}{c}{\parallel f\parallel }_{H_{\mu }^{\infty }\left(\mathbb{B}\right)}=\underset{z\in \mathbb{B}}{sup}\mu \left(z\right)\left|f\left(z\right)\right|<+\infty .\hfill \end{array}$$

$H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is a Banach space with the norm ${\parallel ·\parallel }_{H_{\mu }^{\infty }\left(\mathbb{B}\right)}$. In particular, if $\mu \left(z\right)={(1-|z|}^2{)}^{\sigma }$, where $\sigma >0$, then the space $H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is the classical weighted-type space $H_{\sigma }^{\infty }\left(\mathbb{B}\right)$. If $\mu \equiv 1$, then the $H_{\mu }^{\infty }\left(\mathbb{B}\right)$ becomes the well-known bounded holomorphic function space $H^{\infty }\left(\mathbb{B}\right)$. Many operators acting from or to the weighted-type spaces have been investigated (see, for example, [3,7,17,42] and the related references therein). It can be seen that the Bloch-type space and weighted-type space are metric spaces. One can see [43] and the related references therein for getting some profound results of metric spaces.

Let X and Y be two Banach spaces. A linear operator $T:X\to Y$ is bounded if there exists a positive constant K such that ${\parallel Tf\parallel }_Y\le K{\parallel f\parallel }_X$ for all $f\in X$. The operator $T:X\to Y$ is compact if it maps bounded sets into relatively compact sets. The norm ${\parallel T\parallel }_{X\to Y}$ of the operator $T:X\to Y$ is defined by

$${\parallel T\parallel }_{X\to Y}=\underset{{\parallel f\parallel }_X\le 1}{sup}{\parallel Tf\parallel }_Y.$$

As usual, we use the notation $j=\overline{k,l}$ instead of writing $j=k,\dots ,l$, where $k,l\in {\mathbb{N}}_0$ and $k\le l$. Some positive numbers are denoted by C, and they may vary in different situations. The notation $a\lesssim b$ (resp. $a\gtrsim b$) means that there is a normal number C such that $a\le Cb$ (resp. $a\ge Cb$). When $a\lesssim b$ and $b\gtrsim a$, we write $a\asymp b$.

2. Preliminary Results

In this section, we need several elementary results for proving the main results. We first have the following result (see [44]).

Lemma 1.

Let X, Y be Banach spaces of holomorphic functions on $\mathbb{B}$. Suppose that:

(a) 

The point evaluation functionals on X are continuous;

(b) 

The closed unit ball of X is a compact subset of X in the topology of uniform convergence on compact sets;

(c) 

$T:X\to Y$is continuous when X and Y are given the topology of uniform convergence on compact sets.

• Then, the bounded operator $T:X\to Y$ is compact if and only if for every bounded sequence $\left\{f_m\right\}$ in X such that $f_m\to 0$ uniformly on compact sets such as $m\to \infty$, it follows that $\left\{T{f}_m\right\}$ converges to zero in the norm of Y as $m\to \infty$.

We obtain the following characterization of the compactness, which can be proved similar to that in [45], and can also be proved according to Lemma 1. Therefore, we omit the proof.

Lemma 2.

Let ϕ be normal on $[0,1)$, $u_i\in H\left(\mathbb{B}\right)$, $i=\overline{0,k}$, $v_j\in H\left(\mathbb{B}\right)$, $\overline{j=1,l}$, and $\phi \in S\left(\mathbb{B}\right)$, and μ a weight function on $\mathbb{B}$. Then, the bounded operator $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is compact if and only if for any bounded sequence $\left\{f_m\right\}$ in ${\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)$ such that $f_m\to 0$ uniformly on any compact subset of $\mathbb{B}$ as $m\to \infty$, it follows that

$$\begin{array}{c}\underset{m\to \infty }{lim}{\parallel S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}{f}_m\parallel }_{H_{\mu }^{\infty }\left(\mathbb{B}\right)}=0.\hfill \end{array}$$

The next Lemmas 3–5 are needed and obtained from [31].

Lemma 3.

Let $N\in \mathbb{N}$ and $\phi =({\phi }_1,\dots ,{\phi }_n)\in S\left(\mathbb{B}\right)$. Then, for any $z\in \mathbb{B}$ and $f\in H\left(\mathbb{B}\right)$

$$\begin{array}{c}{\Re }^N(f\circ \phi )\left(z\right)=\sum _{j=1}^N\sum _{l_1=1}^n\cdots \sum _{l_j=1}^n\left(\frac{{\partial }^{j}f}{\partial z_{l_1}\partial z_{l_2}\cdots \partial z_{l_j}}\left(\phi \left(z\right)\right)\sum _{k_1,\dots ,k_j}{C}_{k_1,\dots ,k_j}^{\left(N\right)}\prod _{t=1}^j{\Re }^{k_t}{\phi }_{l_t}\left(z\right)\right),\hfill \end{array}$$

(16)

where $k_1+k_2+\cdots +k_j=N$, $j=\overline{1,N}$, and $C_{k_1,k_2,\dots ,k_j}^{\left(N\right)}$ are some positive integers with respect to the positive integers $k_1,k_2,\dots ,k_j$.

Lemma 4.

Let $w\in \mathbb{B}$, $N\in \mathbb{N}$, $s>0$, $\phi \in S\left(\mathbb{B}\right)$ and

$$\begin{array}{c}{g}_{w,s}\left(z\right)=\frac{1}{{(1-\langle z,w\rangle )}^s},z\in \mathbb{B}.\hfill \end{array}$$

Then

$$\begin{array}{c}{\Re }^N(g_{w,s}\circ \phi )\left(z\right)=\sum _{j=1}^N\left(\prod _{k=0}^{j-1}(s+k)\right)\sum _{k_1,\dots ,k_j}{C}_{k_1,\dots ,k_j}^{\left(N\right)}\frac{{\prod }_{t=1}^j\langle {\Re }^{k_t}\phi \left(z\right),w\rangle }{{(1-\langle \phi \left(z\right),w\rangle )}^{s+j}},\hfill \end{array}$$

(17)

where constants $C_{k_1,k_2,\dots ,k_j}^{\left(N\right)}$ are defined in Lemma 3.

Let

$$\begin{array}{c}{B}_{i,j}\left(\langle \Re \phi \left(z\right),w\rangle \right):=B_{i,j}\left(\langle \Re \phi \left(z\right),w\rangle ,\langle {\Re }^2\phi \left(z\right),w\rangle ,\dots ,\langle {\Re }^{i-j+1}\phi \left(z\right),w\rangle \right).\hfill \end{array}$$

We also have the following version of Lemma 4.

Lemma 5.

Let $N\in \mathbb{N}$ and $\left\{g_{w,s}\right\}$ be the family of functions defined in Lemma 4. Then

$$\begin{array}{c}{\Re }^N(g_{w,s}\circ \phi )\left(z\right)=\sum _{j=1}^N\left(\prod _{k=0}^{j-1}(s+k)\right)\frac{B_{N,j}\left(\langle \Re \phi \left(z\right),w\rangle \right)}{{(1-\langle \phi \left(z\right),w\rangle )}^{s+j}}.\hfill \end{array}$$

(18)

Remark 1.

(i) From Lemmas 4 and 5, we obtain

$$\begin{array}{c}\sum _{k_1,\dots ,k_j}{C}_{k_1,\dots ,k_j}^{\left(N\right)}\prod _{t=1}^j\langle {\Re }^{k_t}\phi \left(z\right),w\rangle =B_{N,j}\left(\langle \Re \phi \left(z\right),w\rangle \right),\hfill \end{array}$$

where $k_1+k_2+\cdots +k_j=N$ and $j=\overline{1,N}$.

(ii) If $\phi =z$, then from [20] we have

$$\begin{array}{c}{\Re }^N{g}_{w,s}\left(z\right)=\sum _{j=1}^N{a}_j^{\left(N\right)}\left(\prod _{k=0}^{j-1}(s+k)\right)\frac{{\langle z,w\rangle }^j}{{(1-\langle z,w\rangle )}^{s+j}},\hfill \end{array}$$

(19)

where the sequences ${\left\{a_j^{\left(N\right)}\right\}}_{j\in \overline{1,N}}$, $N\in \mathbb{N}$, are defined by the relations $a_N^{\left(N\right)}=a_1^{\left(N\right)}=1$ for $N\in \mathbb{N}$ and $a_j^{\left(N\right)}=j{a}_j^{(N-1)}+a_{j-1}^{(N-1)}$ for $2\le j\le N-1,N\ge 3$. Moreover, it is easy to obtain that constants $C_{k_1,\dots ,k_j}^{\left(N\right)}$ satisfy the following conclusion

$$\begin{array}{c}\sum _{k_1,\dots ,k_j}{C}_{k_1,\dots ,k_j}^{\left(N\right)}=a_j^{\left(N\right)}=B_{N,j}(1,1,\dots ,1),\hfill \end{array}$$

(20)

where $k_1+k_2+\cdots +k_j=N$ and $j=\overline{1,N}$.

(iii) Let

$$B_{N,j}\left(\langle z,w\rangle \right):=B_{N,j}(\langle z,w\rangle ,\langle z,w\rangle ,\dots ,\langle z,w\rangle ).$$

From (19) and (20), we obtain the following version of the Formula (19)

$$\begin{array}{c}{\Re }^N{g}_{w,s}\left(z\right)=\sum _{j=1}^N\left(\prod _{k=0}^{j-1}(s+k)\right)\frac{B_{N,j}\left(\langle z,w\rangle \right)}{{(1-\langle z,w\rangle )}^{s+j}}.\hfill \end{array}$$

(21)

The following result is the point-evaluation estimate for the space ${\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)$.

Lemma 6.

Let ϕ be normal on $[0,1)$. Then, there is a positive constant C independent of $f\in {\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)$ and $z\in \mathbb{B}$ such that

$$\begin{array}{c}\left|f\left(z\right)\right|\le C\frac{1-{\left|z\right|}^2}{\varphi \left(\right|z\left|\right)}{\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}.\hfill \end{array}$$

(22)

Proof. 

Theorem 3.1 in [36] shows that $f\in {\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)$ if and only if there is a function $g\in L^{\infty }\left(\mathbb{B}\right)$ such that

$$\begin{array}{c}f\left(z\right)={\int }_{\mathbb{B}}\frac{g\left(w\right)}{\varphi \left(\right|w\left|\right){(1-\langle z,w\rangle )}^{n+t}}d{v}_t\left(w\right),\hfill \end{array}$$

(23)

where $t>max\{b-1,0\}$ and $z\in \mathbb{B}$. Moreover, ${\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}\asymp {\parallel g\parallel }_{\infty }$. From Lemma 2.2 in [46], it follows that

$$\begin{array}{c}\frac{\varphi \left(\right|z\left|\right)}{\varphi \left(\right|w\left|\right)}\le {\left(\frac{1-{\left|z\right|}^2}{1-{\left|w\right|}^2}\right)}^a+{\left(\frac{1-{\left|z\right|}^2}{1-{\left|w\right|}^2}\right)}^b\hfill \end{array}$$

(24)

for $z,w\in \mathbb{B}$, where a and b are the parameters in the definition of the normal function. By (24), we have

$$\begin{array}{cc}\hfill \varphi \left(\right|z\left|\right)\left|f\right(z\left)\right|& \le C\varphi \left(\right|z\left|\right){\int }_{\mathbb{B}}\frac{\left|g\right(w\left)\right|}{{\varphi \left(\right|w\left|\right)|1-\langle z,w\rangle |}^{n+t}}d{v}_t\left(w\right)\hfill \\ \hfill & \le C{\int }_{\mathbb{B}}\frac{\varphi \left(\right|z\left|\right)}{\varphi \left(\right|w\left|\right)}\frac{\left|g\right(w\left)\right|}{{|1-\langle z,w\rangle |}^{n+t}}d{v}_t\left(w\right)\hfill \\ \hfill & \le {C\parallel g\parallel }_{\infty }{\int }_{\mathbb{B}}\frac{{(1-|z|}^2{)}^a{(1-|w|}^2{)}^{t-a}}{{|1-\langle z,w\rangle |}^{n+t}}dv\left(w\right)\hfill \\ \hfill & +{C\parallel g\parallel }_{\infty }{\int }_{\mathbb{B}}\frac{{(1-|z|}^2{)}^b{(1-|w|}^2{)}^{t-b}}{{|1-\langle z,w\rangle |}^{n+t}}dv\left(w\right).\hfill \end{array}$$

(25)

If $a<1$ and $b<1$, from Theorem 1.12 in [47] and (25), then we have

$$\begin{array}{c}\varphi (|z|)\left|f(z)\right|\le C((1-{\left|z\right|}^2{)}^a{+(1-|z|}^2{)}^b{)\parallel g\parallel }_{\infty }\le {C(1-|z|}^2{)\parallel f\parallel }_{{\mathcal{B}}_{\varphi }(\mathbb{B})}.\hfill \end{array}$$

If $a<1$ and $b=1$, since

$$\begin{array}{c}\underset{\left|z\right|\to 1}{lim}{(1-|z|}^2)ln\frac{1}{1-{\left|z\right|}^2}=0,\hfill \end{array}$$

(26)

from Theorem 1.12 in [47] and (25), we have

$$\begin{array}{c}\varphi (|z|)\left|f(z)\right|\le C\left({(1-|z|}^2{)}^a{+(1-|z|}^2)ln\frac{1}{1-{\left|z\right|}^2}\right){\parallel g\parallel }_{\infty }\le {C(1-|z|}^2{)\parallel f\parallel }_{{\mathcal{B}}_{\varphi }(\mathbb{B})}.\hfill \end{array}$$

If $a<1$ and $b>1$, from Theorem 1.12 in [47] and (25), then we have

$$\begin{array}{c}\varphi \left(\right|z\left|\right)\left|f\left(z\right)\right|\le C\left(\right(1-{\left|z\right|}^2{)}^a{+(1-|z|}^2{\left)\right)\parallel g\parallel }_{\infty }\le {C(1-|z|}^2{)\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}.\hfill \end{array}$$

If $a=1$ and $b>1$, from Theorem 1.12 in [47], (25) and (26), then we have

$$\begin{array}{c}\varphi (|z|)\left|f(z)\right|\le C\left({(1-|z|}^2)ln\frac{1}{1-{\left|z\right|}^2}{+(1-|z|}^2)\right){\parallel g\parallel }_{\infty }\le {C(1-|z|}^2{)\parallel f\parallel }_{{\mathcal{B}}_{\varphi }(\mathbb{B})}.\hfill \end{array}$$

If $a>1$ and $b>1$, from Theorem 1.12 in [47] and (25), then we have

$$\begin{array}{c}\varphi \left(\right|z\left|\right)\left|f\left(z\right)\right|\le C\left(\right(1-{\left|z\right|}^2{)+(1-\left|z\right|}^2{\left)\right)\parallel g\parallel }_{\infty }\le {C(1-|z|}^2{)\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}.\hfill \end{array}$$

Combining the above discussions, we obtain

$$\begin{array}{c}\varphi \left(\right|z\left|\right)\left|f\left(z\right)\right|\le {C(1-|z|}^2{)\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}.\hfill \end{array}$$

The proof is finished. □

The following result is an estimate for the higher-order partial derivative of functions in the space ${\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)$.

Lemma 7.

Let $N\in \mathbb{N}$ and ϕ be normal on $[0,1)$. Then, for every multi-index $k=(l_1,\dots ,l_j)$ such that $\left|k\right|=N$, there is a positive constant C independent of $f\in {\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)$ and $z\in \mathbb{B}$ such that

$$\begin{array}{c}\left|\frac{{\partial }^{N}f\left(z\right)}{\partial z_{k_1}^{l_1}\partial z_{k_2}^{l_2}\cdots \partial z_{k_j}^{l_j}}\right|\le \frac{C}{\varphi \left(\right|z\left|\right)(1-{\left|z\right|}^2{)}^{N-1}}{\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}.\hfill \end{array}$$

(27)

Proof. 

From (23), we have

$$\begin{array}{c}\frac{{\partial }^{N}f\left(z\right)}{\partial z_{k_1}^{l_1}\partial z_{k_2}^{l_2}\cdots \partial z_{k_j}^{l_j}}=C{\int }_{\mathbb{B}}\frac{{\overline{w}}_{k_1}^{l_1}{\overline{w}}_{k_2}^{l_2}\cdots {\overline{w}}_{k_j}^{l_j}g\left(w\right)}{\varphi \left(\right|w\left|\right){(1-\langle z,w\rangle )}^{n+N+t}}d{v}_t\left(w\right)\hfill \end{array}$$

(28)

for some $C=C(n,N,t)$ independent of f and z.

Moreover, from Lemma 2.2 in [46], we have that for all $z,w\in \mathbb{B}$

$$\begin{array}{c}\frac{\varphi \left(\right|z\left|\right)}{\varphi \left(\right|w\left|\right)}\le {\left(\frac{1-{\left|z\right|}^2}{1-{\left|w\right|}^2}\right)}^a+{\left(\frac{1-{\left|z\right|}^2}{1-{\left|w\right|}^2}\right)}^b.\hfill \end{array}$$

From this, (28) and Theorem 1.12 in [47], we have

$$\begin{array}{cc}\varphi (|z|)(1-{\left|z\right|}^2{)}^{N-1}\left|\frac{{\partial }^{N}f(z)}{\partial z_{k_1}^{l_1}\partial z_{k_2}^{l_2}\cdots \partial z_{k_j}^{l_j}}\right|\hfill & \le C\varphi (|z|){\int }_{\mathbb{B}}\frac{\left|g(w)\right|(1-{\left|z\right|}^2{)}^{N-1}}{{\varphi (|w|)|1-\langle z,w\rangle |}^{n+N+t}}d{v}_t(w)\hfill \\ \hfill & \le C{\int }_{\mathbb{B}}\frac{\varphi (|z|)}{\varphi (|w|)}\frac{\left|g(w)\right|(1-{\left|z\right|}^2{)}^{N-1}}{{|1-\langle z,w\rangle |}^{n+N+t}}d{v}_t(w)\hfill \\ \hfill & \le {C\parallel g\parallel }_{\infty }{(1-|z|}^2{)}^{N-1}{\int }_{\mathbb{B}}\frac{{(1-|z|}^2{)}^a{(1-|w|}^2{)}^{t-a}}{{|1-\langle z,w\rangle |}^{n+N+t}}dv(w)\hfill \\ \hfill & +{C\parallel g\parallel }_{\infty }{(1-|z|}^2{)}^{N-1}{\int }_{\mathbb{B}}\frac{{(1-|z|}^2{)}^b{(1-|w|}^2{)}^{t-b}}{{|1-\langle z,w\rangle |}^{n+N+t}}dv(w)\hfill \\ \hfill & \le {C\parallel g\parallel }_{\infty }\lesssim {\parallel f\parallel }_{{\mathcal{B}}_{\varphi }(\mathbb{B})}.\hfill \end{array}$$

(29)

The proof is finished. □

Let

$$B_{i,j}\left(\right|\Re \phi \left(z\right)\left|\right):=B_{i,j}\left(|\Re \phi \left(z\right)|,|{\Re }^2\phi \left(z\right)|,\dots ,|{\Re }^{i-j+1}\phi \left(z\right)|\right).$$

Lemma 8.

Let $N\in \mathbb{N}$, $\phi \in S\left(\mathbb{B}\right)$ and ϕ be normal on $[0,1)$. Then, there exists a positive constant C independent of $f\in {\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)$ and $z\in \mathbb{B}$ such that

$$\begin{array}{c}\left|{\Re }^N(f\circ \phi )\left(z\right)\right|\le C\sum _{j=1}^N\frac{B_{N,j}\left(\right|\Re \phi \left(z\right)\left|\right)}{\varphi \left(\right|z\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}{\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}.\hfill \end{array}$$

Proof. 

From Remark 1 (i), it is obvious that

$$\begin{array}{c}\sum _{k_1,\dots ,k_j}{C}_{k_1,\dots ,k_j}^{\left(N\right)}\prod _{t=1}^j|{\Re }^{k_t}\phi (z)|=B_{N,j}\left(\right|\Re \phi \left(z\right)\left|\right),\hfill \end{array}$$

(30)

where $k_1+k_2+\cdots +k_j=N$ and $j=\overline{1,N}$. Hence, by applying Cauchy–Schwarz inequality, and using Lemmas 3 and 7, we have

$$\begin{array}{cc}\left|{\Re }^N(f\circ \phi )\left(z\right)\right|\hfill & \le \sum _{l_1=1}^n\sum _{l_2=1}^n\cdots \sum _{l_N=1}^n\left|\frac{{\partial }^{N}f}{\partial z_{l_1}\partial z_{l_2}\cdots \partial z_{l_N}}\left(\phi \left(z\right)\right)\right|\sum _{k_1,\dots ,k_N}{C}_{k_1,\dots ,k_N}^{\left(N\right)}\prod _{t=1}^N\left|{\Re }^{k_t}{\phi }_{l_t}\left(z\right)\right|\hfill \\ \hfill & +\sum _{l_1=1}^n\sum _{l_2=1}^n\cdots \sum _{l_{N-1}=1}^n\left|\frac{{\partial }^{N-1}f}{\partial z_{l_1}\partial z_{l_2}\cdots \partial z_{l_{N-1}}}\left(\phi \left(z\right)\right)\right|\sum _{k_1,\dots ,k_{N-1}}{C}_{k_1,\dots ,k_{N-1}}^{\left(N\right)}\prod _{t=1}^{N-1}\left|{\Re }^{k_t}{\phi }_{l_t}\left(z\right)\right|\hfill \\ \hfill & +\cdots +\sum _{l=1}^n\left|\frac{\partial f}{\partial z_l}\left(\phi \left(z\right)\right)\right|\sum _{k_1}{C}_{k_1}^{\left(N\right)}\left|{\Re }^{k_1}{\phi }_l\left(z\right)\right|\hfill \\ \hfill & \le C(\frac{B_{N,N}\left(\right|\Re \phi \left(z\right)\left|\right)}{\varphi \left(\right|z\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{N-1}}{\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}+\frac{B_{N,N-1}\left(\right|\Re \phi \left(z\right)\left|\right)}{\varphi \left(\right|z\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{N-2}}{\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}\hfill \\ \hfill & +\cdots +\frac{B_{N,1}\left(\right|\Re \phi \left(z\right)\left|\right)}{\varphi \left(\right|z\left|\right)}{\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}).\hfill \end{array}$$

(31)

From (31), the desired result follows. □

Remark 2.

If $\phi =z$, then from Lemma 8, we have that there exists a positive constant C independent of $f\in {\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)$ and $z\in \mathbb{B}$ such that

$$\begin{array}{c}\left|{\Re }^{N}f\left(z\right)\right|\le C\sum _{j=1}^N\frac{B_{N,j}\left(\right|z\left|\right)}{\varphi \left(\right|z\left|\right)(1-{\left|z\right|}^2{)}^{j-1}}{\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}\asymp \frac{\left|z\right|}{\varphi \left(\right|z\left|\right)(1-{\left|z\right|}^2{)}^{N-1}}{\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)},\hfill \end{array}$$

where $B_{N,j}\left(\right|z\left|\right)=B_{N,j}\left(\right|z|,|z|,\dots ,|z\left|\right)$.

The next lemma offers an important test function used in the proofs of the main results.

Lemma 9.

Let ϕ be normal on $[0,1)$. Then, for each $t\ge b-1$ and fixed $w\in \mathbb{B}$, the following function is in ${\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)$

$$\begin{array}{c}{f}_{w,t}\left(z\right)=\frac{1-{\left|w\right|}^2}{\varphi \left(\right|w\left|\right)}{\left(\frac{1-{\left|w\right|}^2}{1-\langle z,w\rangle }\right)}^{t+1}.\hfill \end{array}$$

(32)

Moreover,

$$\begin{array}{c}\underset{w\in \mathbb{B}}{sup}{\parallel f_{w,t}\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}\lesssim 1.\hfill \end{array}$$

(33)

Proof. 

Since the definition of function $\varphi$, we have

$$\begin{array}{cc}\varphi \left(\right|z\left|\right)|\Re f_{w,t}\left(z\right)|\hfill & =(t+1)\frac{\varphi \left(\right|z\left|\right)}{\varphi \left(\right|w\left|\right)}\frac{{(1-|w|}^2{)}^{t+1}\left|\langle z,w\rangle \right|}{{|1-\langle z,w\rangle |}^{t+2}}\hfill \\ \hfill & \le (t+1)\frac{\varphi \left(\right|z\left|\right)}{{(1-|z\left|\right)}^{t+1}}\frac{{(1-|w|}^2{)}^{t+1}}{\varphi \left(\right|w\left|\right)}\le C<+\infty .\hfill \end{array}$$

(34)

Therefore, we have that (33) holds. □

Remark 3.

It is obvious that the function defined in (32) satisfies the following estimate

$$\begin{array}{c}|f_{w,t}\left(z\right)|\le \frac{{(1-|w|}^2\left)\right(1+{\left|w\right|)}^t}{\varphi \left(\right|w\left|\right)}\le C\frac{(1-|w{|}^2)}{\varphi \left(\right|w\left|\right)}.\hfill \end{array}$$

This shows that $f_{w,t}$ uniformly converges to zero on any compact subset of $\mathbb{B}$ as $\left|w\right|\to 1$.

Finally, we need the following lemma (see [26]).

Lemma 10.

If $a>0$, then

$$\begin{array}{c}{D}_n\left(a\right)=\left|\begin{array}{cccc}1& 1& \cdots & 1\\ a& a+1& \cdots & a+n-1\\ a(a+1)& (a+1)(a+2)& \cdots & (a+n-1)(a+n)\\ ⋮& ⋮& & ⋮\\ {\displaystyle \prod _{k=0}^{n-2}(a+k)}& {\displaystyle \prod _{k=0}^{n-2}(a+k+1)}& {\displaystyle \cdots }& {\displaystyle \prod _{k=0}^{n-2}(a+k+n-1)}\end{array}\right|=\prod _{k=1}^{n-1}k!.\hfill \end{array}$$

3. The Condition on the Symbols

Let $\phi \left(z\right)=({\phi }_1\left(z\right),\dots ,{\phi }_n\left(z\right))\in S\left(\mathbb{B}\right)$ and ${\Re }^m\phi \left(z\right)=({\Re }^m{\phi }_1\left(z\right),\dots ,{\Re }^m{\phi }_n\left(z\right))$. To characterize the boundedness and compactness of the operator $\Re M_u{C}_{\phi }$, the authors in [19] proposed the condition: there exists a $\lambda \in (0,1)$ such that if $\left|\phi \right(z\left)\right|>\lambda$, then

$$\begin{array}{c}|\Re \phi \left(z\right)|\le \frac{1}{\lambda }\left|\langle \Re \phi \left(z\right),\phi \left(z\right)\rangle \right|.\hfill \end{array}$$

(35)

In the characterization of the boundedness and compactness of the operator $T_{u_0,u_1,u_2,\phi }$, the authors in [25] introduced the condition on symbols $u_1$, $u_2$ and $\phi$: there are $\rho \in (0,1)$ and a positive constant C such that if $\left|\phi \right(z\left)\right|>\rho$, then

$$\begin{array}{c}|u_1\left(z\right)\phi \left(z\right)+u_2\left(z\right)\Re \phi \left(z\right)|\le C\left|\langle u_1\left(z\right)\phi \left(z\right)+u_2\left(z\right)\Re \phi \left(z\right),\phi \left(z\right)\rangle \right|.\hfill \end{array}$$

(36)

The authors in [22] also gave a special relationship such that the symbols $u_1$, $u_2$, and $\phi$ satisfied the condition.

Conditions (35) and (36) hold for all symbols if $n=1$, which shows that it is more complicated for $n>1$. Since $\Re M_u{C}_{\phi }$ can be regarded as the operator $T_{\Re u,0,u,\phi }$, we deduce that condition (36) is reduced to condition (35).

Motivated by previous studies mentioned such as [19,22,25], here we introduce the condition concerning all symbols $\phi$, $u_i$ and $v_j$, $i,j=\overline{1,l}$: there exist $\delta \in (0,1)$ and a positive constant C such that if $z\in K=\{z\in \mathbb{B}:|\phi \left(z\right)|>\delta \}$, then for every $j=\overline{1,l}$

$$\begin{array}{cc}\hfill & \left|\sum _{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right)))\right|\hfill \\ \hfill & \le C\left|\sum _{i=j}^l(u_i(z)B_{i,j}{(|\phi (z)|}^2)+v_i(z)B_{i,j}(\langle \Re \phi (z),\phi (z)\rangle ))\right|,\hfill \end{array}$$

(37)

where

$$\begin{array}{c}{B}_{i,j}\left(\phi \left(z\right)\right):=B_{i,j}(\phi \left(z\right),\phi \left(z\right),\dots ,\phi \left(z\right))\hfill \end{array}$$

and

$$\begin{array}{c}{B}_{i,j}(\Re \phi \left(z\right)):=B_{i,j}(\Re \phi \left(z\right),{\Re }^2\phi \left(z\right),\dots ,{\Re }^{i-j+1}\phi \left(z\right)).\hfill \end{array}$$

Since $B_{1,1}\left(x\right)=x$, the condition (37) is reduced to condition (36) if $l=1$.

Remark 4.

The case of $k=l$ is assumed in the condition (37). If $k\ne l$, for example $k>l$, then by setting $v_{l+1}\equiv v_{l+2}\equiv \cdots \equiv v_k=0$, we see that the condition (37) is equivalent to the following conditions

$$\begin{array}{cc}\hfill & \left|\sum _{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right)))\right|\le C\left|\sum _{i=j}^l(u_i\left(z\right)B_{i,j}{\left(\right|\phi \left(z\right)|}^2)+v_i\left(z\right)B_{i,j}\left(\langle \Re \phi \left(z\right),\phi \left(z\right)\rangle \right))\right|\hfill \end{array}$$

for $j=\overline{1,l}$, and

$$\begin{array}{c}\left|\sum _{i=j}^k{u}_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)\right|\le C|\sum _{i=j}^k{u}_i\left(z\right)B_{i,j}{\left(\right|\phi \left(z\right)|}^2\left)\right|\hfill \end{array}$$

for $j=\overline{l,k}$.

If $l>k$, then by setting $u_{k+1}\equiv u_{k+2}\equiv \cdots \equiv u_l=0$, we also see that the condition (37) is equivalent to the following conditions

$$\begin{array}{cc}\hfill & \left|\sum _{i=j}^k(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right)))\right|\le C\left|\sum _{i=j}^k(u_i\left(z\right)B_{i,j}{\left(\right|\phi \left(z\right)|}^2)+v_i\left(z\right)B_{i,j}\left(\langle \Re \phi \left(z\right),\phi \left(z\right)\rangle \right))\right|\hfill \end{array}$$

for $j=\overline{1,k}$, and

$$\begin{array}{c}\left|\sum _{i=j}^l{v}_i\left(z\right)B_{i,j}(\Re \phi \left(z\right))\right|\le C\left|\sum _{i=j}^l{v}_i\left(z\right)B_{i,j}\left(\langle \Re \phi \left(z\right),\phi \left(z\right)\rangle \right)\right|\hfill \end{array}$$

for $j=\overline{k,l}$.

We need to discuss what kind of symbols can satisfy the condition. Assume $n>1$, then we see that the following example satisfies the condition (37).

Example 1.

Let $\phi \left(z\right)=(z_1,z_2/2,\dots ,z_n/n)$, $u_i\left(z\right)=a_i{z}_1$, and $v_i\left(z\right)=b_i{z}_1$, $i=\overline{1,l}$, where constants $a_i$ and $b_i$ are positive. Then, these symbols satisfy the condition (37).

Proof. 

It is easy to see that ${\Re }^i\phi \left(z\right)=\phi \left(z\right)$ for each $i=\overline{1,l}$. Hence, we obtain that

$$\begin{array}{cc}\hfill & |\sum _{i=j}^l(u_i(z)B_{i,j}{(|\phi (z)|}^2)+v_i(z)B_{i,j}(\langle \Re \phi (z),\phi (z)\rangle ))|=\sum _{i=j}^l|z_1||\langle (a_i+b_i)B_{i,j}(\phi (z)),\phi (z)\rangle |\hfill \\ \hfill & =\sum _{i=j}^l|z_1||(a_i+b_i)B_{i,j}(\phi (z))||\phi (z){|}^j\ge {\delta }^j\sum _{i=j}^l|z_1||(a_i+b_i)B_{i,j}(\phi (z))|\hfill \\ \hfill & ={\delta }^j|\sum _{i=j}^l(u_i(z)B_{i,j}(\phi (z))+v_i(z)B_{i,j}(\Re \phi (z)))|,\hfill \end{array}$$

which implies that (37) holds. □

Except for the above example, we also see that if $n=1$, all symbols satisfy the condition.

Proposition 1.

If $n=1$, all symbols φ, $u_i$ and $v_i$, $i=\overline{1,l}$ satisfy the condition (37).

Proof. 

Since

$$\begin{array}{cc}\hfill & \left|\sum _{i=j}^l(u_i\left(z\right)B_{i,j}{\left(\right|\phi \left(z\right)|}^2)+v_i\left(z\right)B_{i,j}\left(\langle \Re \phi \left(z\right),\phi \left(z\right)\rangle \right))\right|\hfill \\ \hfill & =\left|〈\sum _{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right))),\phi {\left(z\right)}^j〉\right|,\hfill \end{array}$$

we have

$$\begin{array}{cc}\hfill & \left|\sum _{i=j}^l(u_i\left(z\right)B_{i,j}{\left(\right|\phi \left(z\right)|}^2)+v_i\left(z\right)B_{i,j}\left(\langle \Re \phi \left(z\right),\phi \left(z\right)\rangle \right))\right|\hfill \\ \hfill & =\left|\sum _{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right)))\right|{\left|\phi \left(z\right)\right|}^j\hfill \\ \hfill & \ge {\delta }^j\left|\sum _{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right)))\right|,\hfill \end{array}$$

which implies that (37) holds for all symbols $\phi$, $u_i$ and $v_i$, $i=\overline{1,l}$. □

For $n>1$, it is difficult, but we still give the following result.

Proposition 2.

Let $\phi \in S\left(\mathbb{B}\right)$ and $u_i,v_i\in H\left(\mathbb{B}\right)$ for each $i=\overline{1,l}$. Then, the following statements hold.

(i) If ${\sum }_{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right))$ and $\phi {\left(z\right)}^j$ are linearly dependent for each $z\in K$, $j=\overline{1,l}$, then the condition (37) holds;

(ii) If $v_i\equiv 0$ for $i=\overline{1,l}$, then the condition (37) holds.

Proof. 

(i) If ${\sum }_{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right))$ and $\phi {\left(z\right)}^j$ are linearly dependent for each $z\in K$, $j=\overline{1,l}$, we have

$$\begin{array}{cc}\hfill & \left|\sum _{i=j}^l(u_i\left(z\right)B_{i,j}{\left(\right|\phi \left(z\right)|}^2)+v_i\left(z\right)B_{i,j}\left(\langle \Re \phi \left(z\right),\phi \left(z\right)\rangle \right))\right|\hfill \\ \hfill & =\left|〈\sum _{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right))),\phi {\left(z\right)}^j〉\right|\hfill \\ \hfill & =\left|\sum _{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right)))\right|{\left|\phi \left(z\right)\right|}^j\hfill \\ \hfill & \ge {\delta }^j\left|\sum _{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right)))\right|,\hfill \end{array}$$

which implies that (37) holds.

(ii) If $v_i\equiv 0$ for $i=\overline{1,l}$, we have

$$\begin{array}{c}\left|\sum _{i=j}^l{u}_i\left(z\right)B_{i,j}{\left(\right|\phi \left(z\right)|}^2\left)\right|=|\sum _{i=j}^l{u}_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)\right|{\left|\phi \left(z\right)\right|}^j\ge {\delta }^j\left|\sum _{i=j}^l{u}_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)\right|,\hfill \end{array}$$

which implies that (37) holds. □

4. Boundedness and Compactness of the Operator $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$

We now begin to characterize the boundedness of the operator $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$. We first consider the case of $k=l$.

Theorem 1.

Assume that (37) is satisfied, $k,l\in \mathbb{N}$, $k=l$, $u_0\in H\left(\mathbb{B}\right)$, $u_i,v_i\in H\left(\mathbb{B}\right)$, $i=\overline{1,l}$, ϕ normal on $[0,1)$, $\phi \in S\left(\mathbb{B}\right)$ and μ a weight function on $\mathbb{B}$. Then, the operator $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded if and only if

$$\begin{array}{c}{I}_0:=\underset{z\in \mathbb{B}}{sup}\frac{\mu \left(z\right)|u_0\left(z\right)\left|\right(1-{\left|\phi \left(z\right)\right|}^2)}{\varphi \left(\right|\phi \left(z\right)\left|\right)}<+\infty \hfill \end{array}$$

(38)

and

$$\begin{array}{c}{I}_j:=\underset{z\in \mathbb{B}}{sup}\frac{\mu \left(z\right)|{\sum }_{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right)))|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}<+\infty \hfill \end{array}$$

(39)

for $j=\overline{1,l}$.

Moreover, if the operator $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded, then the following asymptotic relationship holds

$$\begin{array}{c}\parallel S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}{\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)}\asymp \sum _{j=0}^l{I}_j.\hfill \end{array}$$

(40)

Proof. 

Suppose that (38) and (39) hold. From Lemma 6, Lemma 7 and Remark 1 (i), we have

$$\begin{array}{cc}\hfill & \mu \left(z\right)|u_0\left(z\right)f\left(\phi \left(z\right)\right)+\sum _{i=1}^l{u}_i\left(z\right){\Re }^{i}f\left(\phi \left(z\right)\right)+\sum _{j=1}^l{u}_j\left(z\right){\Re }^j(f\circ \phi )\left(z\right)|\hfill \\ \hfill & \le \mu \left(z\right)|u_0\left(z\right)f\left(\phi \left(z\right)\right)|+\mu \left(z\right)|\sum _{i=1}^l\left(u_i\left(z\right){\Re }^{i}f\left(\phi \left(z\right)\right)+v_i\left(z\right){\Re }^i(f\circ \phi )\left(z\right)\right)|\hfill \\ \hfill & =\mu \left(z\right)|u_0\left(z\right)f\left(\phi \left(z\right)\right)|+\mu \left(z\right)|\sum _{i=1}^l\sum _{j=1}^i(u_i\left(z\right)\sum _{l_1=1}^n\cdots \sum _{l_j=1}^n\left(\frac{{\partial }^{j}f}{\partial z_{l_1}\partial z_{l_2}\cdots \partial z_{l_j}}\left(\phi \left(z\right)\right)\sum _{k_1,\dots ,k_j}{C}_{k_1,\dots ,k_j}^{\left(i\right)}\prod _{t=1}^j{\phi }_{l_t}\left(z\right)\right)\hfill \\ \hfill & +v_i\left(z\right)\sum _{l_1=1}^n\cdots \sum _{l_j=1}^n\left(\frac{{\partial }^{j}f}{\partial z_{l_1}\partial z_{l_2}\cdots \partial z_{l_j}}\left(\phi \left(z\right)\right)\sum _{k_1,\dots ,k_j}{C}_{k_1,\dots ,k_j}^{\left(i\right)}\prod _{t=1}^j{\Re }^{k_t}{\phi }_{l_t}\left(z\right)\right)\left)\right|\hfill \\ \hfill & =\mu \left(z\right)|u_0\left(z\right)f\left(\phi \left(z\right)\right)|+\mu \left(z\right)|\sum _{j=1}^l\sum _{i=j}^l(u_i\left(z\right)\sum _{l_1=1}^n\cdots \sum _{l_j=1}^n\left(\frac{{\partial }^{j}f}{\partial z_{l_1}\partial z_{l_2}\cdots \partial z_{l_j}}\left(\phi \left(z\right)\right)\sum _{k_1,\dots ,k_j}{C}_{k_1,\dots ,k_j}^{\left(i\right)}\prod _{t=1}^j{\phi }_{l_t}\left(z\right)\right)\hfill \\ \hfill & +v_i\left(z\right)\sum _{l_1=1}^n\cdots \sum _{l_j=1}^n\left(\frac{{\partial }^{j}f}{\partial z_{l_1}\partial z_{l_2}\cdots \partial z_{l_j}}\left(\phi \left(z\right)\right)\sum _{k_1,\dots ,k_j}{C}_{k_1,\dots ,k_j}^{\left(i\right)}\prod _{t=1}^j{\Re }^{k_t}{\phi }_{l_t}\left(z\right)\right)\left)\right|\hfill \\ \hfill & \lesssim \mu \left(z\right)|u_0\left(z\right)f\left(\phi \left(z\right)\right)|+\mu \left(z\right)\sum _{j=1}^l\sum _{l_1=1}^n\cdots \sum _{l_j=1}^n\left|\frac{{\partial }^{j}f}{\partial z_{l_1}\partial z_{l_2}\cdots \partial z_{l_j}}\left(\phi \left(z\right)\right)\right|\left|\sum _{i=j}^l\sum _{k_1,\dots ,k_j}{C}_{k_1,\dots ,k_j}^{\left(i\right)}\right(u_i\left(z\right)\prod _{t=1}^j{\phi }_{l_t}\left(z\right)\hfill \\ \hfill & +v_i\left(z\right)\prod _{t=1}^j{\Re }^{k_t}{\phi }_{l_t}\left(z\right)\left)\right|\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \lesssim \frac{\mu \left(z\right)|u_0\left(z\right)\left|\right(1-{\left|\phi \left(z\right)\right|}^2)}{\varphi \left(\right|\phi \left(z\right)\left|\right)}{\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}+\sum _{j=1}^l\frac{\mu \left(z\right)\left|{\displaystyle \sum _{i=j}^l}{\displaystyle \sum _{k_1,\dots ,k_j}}{C}_{k_1,\dots ,k_j}^{\left(i\right)}\left(u_i\left(z\right)\phi {\left(z\right)}^j+v_i\left(z\right){\displaystyle \prod _{t=1}^j}{\Re }^{k_t}\phi \left(z\right)\right)\right|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}{\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}\hfill \\ \hfill & =\frac{\mu \left(z\right)|u_0\left(z\right)\left|\right(1-{\left|\phi \left(z\right)\right|}^2)}{\varphi \left(\right|\phi \left(z\right)\left|\right)}{\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}+\sum _{j=1}^l\frac{\mu \left(z\right)\left|{\displaystyle \sum _{i=j}^l}\left(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right))\right)\right|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}{\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}\hfill \\ \hfill & =\left(I_0+\sum _{j=1}^l{I}_j\right){\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}=\sum _{j=0}^l{I}_j{\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}.\hfill \end{array}$$

From this, it follows that

$$\begin{array}{c}\parallel S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}{f\parallel }_{H_{\mu }^{\infty }\left(\mathbb{B}\right)}\le \left(C\sum _{j=0}^l{I}_j\right){\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}.\hfill \end{array}$$

(41)

By taking the supremum in inequality (41) over the unit ball in the space ${\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)$, using conditions (38) and (39), we have that the operator $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded. Moreover, from (41) and the definition of operator norm, we have

$$\begin{array}{c}\parallel S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}{\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)}\le C\sum _{j=0}^l{I}_j.\hfill \end{array}$$

(42)

Now, suppose that $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded. Then, there exists a positive constant C independent of $f\in {\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)$ such that

$$\begin{array}{c}\parallel S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}{f\parallel }_{H_{\mu }^{\infty }\left(\mathbb{B}\right)}\le C{\parallel f\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}.\hfill \end{array}$$

(43)

By using test function $f\left(z\right)=1\in {\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)$, we have

$$\begin{array}{c}K:=\underset{z\in \mathbb{B}}{sup}\mu \left(z\right)\left|u_0\left(z\right)\right|<+\infty .\hfill \end{array}$$

(44)

By using test function $f_k\left(z\right)=z_k^j\in {\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)$, $k=\overline{1,n}$ and $j=\overline{1,l}$, from (44) and the boundedness of $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$, we have

$$\begin{array}{c}\mu \left(z\right)|u_0\left(z\right){\phi }_k{\left(z\right)}^j+\sum _{i=j}^l\left(u_i\left(z\right)B_{i,j}\left({\phi }_k\left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re {\phi }_k\left(z\right))\right)|<+\infty \hfill \end{array}$$

(45)

for each $j\in \{1,2,\dots ,l\}$. Using (44), (45) and the triangle inequality and the fact $\left|\phi \right(z\left)\right|\le 1$, we have

$$\begin{array}{cc}\hfill & \underset{z\in \mathbb{B}}{sup}\mu \left(z\right)\left|\sum _{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right)))\right|\hfill \\ \hfill & =\underset{z\in \mathbb{B}}{sup}\mu \left(z\right)\sqrt{{\displaystyle \sum _{k=1}^n}|{\displaystyle \sum _{i=j}^l}(u_i\left(z\right)B_{i,j}\left({\phi }_k\left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re {\phi }_k\left(z\right))){|}^2}\hfill \\ \hfill & \le C+\underset{z\in \mathbb{B}}{sup}\mu \left(z\right)\sqrt{{\displaystyle \sum _{k=1}^n}{\left|u_0\left(z\right){\phi }_k{\left(z\right)}^j\right|}^2}\hfill \\ \hfill & \le C+\underset{z\in \mathbb{B}}{sup}\mu \left(z\right)|u_0{\left(z\right)\left|\right|\phi \left(z\right)|}^j\hfill \\ \hfill & \le C+K<+\infty .\hfill \end{array}$$

(46)

Let $w\in \mathbb{B}$ and $d_k=k+1$. For each $j\in \{1,2,\dots ,l\}$ and constants $c_k=c_k^{\left(j\right)}$, $k=\overline{0,l}$, let

$$\begin{array}{c}{h}_w^{\left(j\right)}\left(z\right)=\sum _{k=0}^l{c}_k^{\left(j\right)}{f}_{w,k}\left(z\right),\hfill \end{array}$$

(47)

where $f_{w,k}$ is defined in Lemma 9. By Lemma 9, we have

$$\begin{array}{c}{L}_j=\underset{w\in \mathbb{B}}{sup}{\parallel h_w^{\left(j\right)}\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}<+\infty .\hfill \end{array}$$

(48)

From (43), (48), Lemma 5 and Remark 1 (iii), we have

$$\begin{array}{cc}\hfill & L_j\parallel S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}{\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)}\ge {\parallel S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}{h}_{\phi \left(w\right)}^{\left(j\right)}\parallel }_{H_{\mu }^{\infty }\left(\mathbb{B}\right)}\hfill \\ \hfill & =\underset{z\in \mathbb{B}}{sup}\mu \left(z\right)|u_0\left(z\right)h_{\phi \left(w\right)}^{\left(j\right)}\left(\phi \left(z\right)\right)+\sum _{i=1}^l\left(u_i\left(z\right){\Re }^i{h}_{\phi \left(w\right)}^{\left(j\right)}\left(\phi \left(z\right)\right)+v_i\left(z\right){\Re }^i(h_{\phi \left(w\right)}^{\left(j\right)}\circ \phi )\left(z\right)\right)|\hfill \\ \hfill & \ge \mu \left(w\right)|u_0\left(w\right)h_{\phi \left(w\right)}^{\left(j\right)}\left(\phi \left(w\right)\right)+\sum _{i=1}^l\left(u_i\left(w\right){\Re }^i{h}_{\phi \left(w\right)}^{\left(j\right)}\left(\phi \left(w\right)\right)+v_i\left(w\right){\Re }^i(h_{\phi \left(w\right)}^{\left(j\right)}\circ \phi )\left(w\right)\right)|\hfill \\ \hfill & =\mu \left(w\right)|u_0\left(w\right)h_{\phi \left(w\right)}^{\left(j\right)}\left(\phi \left(w\right)\right)+\sum _{i=1}^l\left(u_i\left(w\right)\sum _{k=0}^l{c}_k{\Re }^i{f}_{\phi \left(w\right),k}\left(\phi \left(w\right)\right)+v_i\left(w\right)\sum _{k=0}^l{c}_k{\Re }^i(f_{\phi \left(w\right),k}\circ \phi )\left(w\right)\right)|\hfill \\ \hfill & =\mu \left(w\right)|u_0{\left(w\right)(1-|\phi \left(w\right)|}^2)\frac{c_0+c_1+\cdots +c_l}{\varphi \left(\right|\phi \left(w\right)\left|\right)}\hfill \\ \hfill & +\sum _{i=1}^l\left(u_i\left(w\right)B_{i,1}{\left(\right|\phi \left(w\right)|}^2)+v_i\left(w\right)B_{i,1}\left(\langle \Re \phi \left(w\right),\phi \left(w\right)\rangle \right)\right)\frac{(d_0{c}_0+\cdots +d_l{c}_l)}{\varphi \left(\right|\phi \left(w\right)\left|\right)}\hfill \\ \hfill & +\cdots \hfill \\ \hfill & +\sum _{i=j}^l\left(u_i\left(w\right)B_{i,j}{\left(\right|\phi \left(w\right)|}^2)+v_i\left(w\right)B_{i,j}\left(\langle \Re \phi \left(w\right),\phi \left(w\right)\rangle \right)\right)\frac{(d_0\cdots d_{j-1}{c}_0+\cdots +d_l\cdots d_{l+j-1}{c}_l)}{\varphi \left(\right|\phi \left(w\right)\left|\right)(1-{\left|\phi \left(w\right)\right|}^2{)}^{j-1}}\hfill \\ \hfill & +\cdots \hfill \\ \hfill & +\left(u_l\left(w\right)B_{l,l}\left(\phi \left(w\right)\right)+v_l\left(w\right)B_{l,l}\left(\langle \Re \phi \left(w\right),\phi \left(w\right)\rangle \right)\right)\frac{(d_0\cdots d_{l-1}{c}_0+\cdots +d_l\cdots d_{2l-1}{c}_l)}{\varphi \left(\right|\phi \left(w\right)\left|\right)(1-{\left|\phi \left(w\right)\right|}^2{)}^{l-1}}|.\hfill \end{array}$$

(49)

Since $d_k>0$, $k=\overline{0,l}$, by Lemma 10, we have the following linear equations

$$\left(\begin{array}{cccc}{\displaystyle 1}& {\displaystyle 1}& \cdots & {\displaystyle 1}\\ {\displaystyle d_0}& {\displaystyle d_1}& \cdots & {\displaystyle d_l}\\ {\displaystyle ⋮}& ⋮& \ddots & ⋮\\ {\displaystyle \prod _{k=0}^{j-1}{d}_k}& {\displaystyle \prod _{k=0}^{j-1}{d}_{k+1}}& \cdots & {\displaystyle \prod _{k=0}^{j-1}{d}_{k+l}}\\ {\displaystyle ⋮}& ⋮& \ddots & ⋮\\ {\displaystyle \prod _{k=0}^{l-1}{d}_k}& {\displaystyle \prod _{k=0}^{l-1}{d}_{k+1}}& \cdots & {\displaystyle \prod _{k=0}^{l-1}{d}_{k+l}}\end{array}\right)\left(\begin{array}{c}{c}_0\\ c_1\\ ⋮\\ \\ c_j\\ \\ ⋮\\ \\ c_l\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ ⋮\\ \\ 1\\ \\ ⋮\\ \\ 0\end{array}\right).$$

(50)

From (49), (50) and (37), we have

$$\begin{array}{cc}{L}_j{\parallel S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)}\hfill & \ge \underset{z\in K}{sup}\frac{\mu \left(z\right)|{\sum }_{i=j}^l(u_i\left(z\right)B_{i,j}{\left(\right|\phi \left(z\right)|}^2)+v_i\left(z\right)B_{i,j}\left(\langle \Re \phi \left(z\right),\phi \left(z\right)\rangle \right)\left)\right|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}\hfill \\ \hfill & =\underset{z\in K}{sup}\frac{\mu \left(z\right)|\langle {\sum }_{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right))),\phi {\left(z\right)}^j\rangle |}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}\hfill \\ \hfill & \gtrsim \underset{z\in K}{sup}\frac{\mu \left(z\right)|{\sum }_{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right)))|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}.\hfill \end{array}$$

(51)

On the other hand, from (46), we have

$$\begin{array}{cc}\hfill & \underset{z\in \mathbb{B}\setminus K}{sup}\frac{\mu \left(z\right)|{\sum }_{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right)))|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}\hfill \\ \hfill & \le \underset{z\in \mathbb{B}}{sup}\frac{\mu \left(z\right)|{\sum }_{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right)))|}{{max}_{\left|z\right|\le \delta }\varphi \left(z\right){(1-{\delta }^2)}^{j-1}}<+\infty .\hfill \end{array}$$

(52)

From (51) and (52), we find that (39) holds for $j=\overline{1,l}$.

For constants $c_k=c_k^{\left(0\right)}$, $k=\overline{0,l}$, let

$$\begin{array}{c}{h}_w^{\left(0\right)}\left(z\right)=\sum _{k=0}^l{c}_k^{\left(0\right)}{f}_{w,k}\left(z\right).\hfill \end{array}$$

(53)

By Lemma 9, we know that $L_0={sup}_{w\in \mathbb{B}}{\parallel h_w^{\left(0\right)}\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}<+\infty$. From this, (49), (50) and Lemma 10, we obtain

$$\begin{array}{c}{L}_0{\parallel S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)}\ge \frac{\mu \left(z\right)|u_0\left(z\right)\left|\right(1-{\left|\phi \left(z\right)\right|}^2)}{\varphi \left(\right|\phi \left(z\right)\left|\right)}.\hfill \end{array}$$

(54)

Hence, we have that $I_0<+\infty$. Moreover, we have

$$\begin{array}{c}C\parallel S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}{\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)}\ge \sum _{j=0}^l{I}_j.\hfill \end{array}$$

(55)

From (42) and (55), we obtain (40). The proof is completed. □

The following result gives a sufficient condition for the boundedness of the operator $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ for $k=l$. It does not need to satisfy the condition (37).

Corollary 1.

Let $k,l\in \mathbb{N}$, $k=l$, $u_0\in H\left(\mathbb{B}\right)$, $u_i,v_i\in H\left(\mathbb{B}\right)$, $i=\overline{1,l}$, ϕ normal on $[0,1)$, $\phi \in S\left(\mathbb{B}\right)$ and μ a weight function on $\mathbb{B}$. If

$$\begin{array}{c}\underset{z\in \mathbb{B}}{sup}\frac{\mu \left(z\right)|u_0\left(z\right)\left|\right(1-{\left|\phi \left(z\right)\right|}^2)}{\varphi \left(\right|\phi \left(z\right)\left|\right)}<+\infty \hfill \end{array}$$

and

$$\begin{array}{c}\underset{z\in \mathbb{B}}{sup}\frac{\mu \left(z\right)|{\sum }_{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right)))|}{{(1-|\phi \left(z\right)|}^2{)}^{j-1}}<+\infty \hfill \end{array}$$

for $j=\overline{1,l}$, then the operator $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded.

If we consider some special symbols, we can obtain the following interesting results. For example, if we let $v_j\equiv 0$, $j=\overline{1,l}$, then the operator $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}$ is reduced to the operator ${\mathfrak{S}}_{\overrightarrow{u},\phi }^k$, that is,

$${\mathfrak{S}}_{\overrightarrow{u},\phi }^k=\sum _{i=0}^k{M}_{u_i}{C}_{\phi }{\Re }^i.$$

Then, from Theorem 3.2 in [30], we can obtain similarly the following result, which is right without any additional conditions on the symbols.

Theorem 2.

The operators $M_{u_i}{C}_{\phi }{\Re }^i:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$, $i=\overline{0,k}$, are bounded operator if and only if ${\mathfrak{S}}_{\overrightarrow{u},\phi }^k:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded and

$$\begin{array}{c}\mu \left(z\right)|u_i\left(z\right)\left|\right|\phi \left(z\right)|<+\infty \hfill \end{array}$$

(56)

for each $i=\overline{1,k}$.

Moreover, if we consider $u_i\equiv 0$, $i=\overline{1,k}$, then the operator $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}$ becomes the following operator, denoted by $S_{\overrightarrow{v},\phi }^l$. Namely,

$$S_{\overrightarrow{v},\phi }^l=\sum _{j=0}^l{M}_{v_j}{\Re }^j{C}_{\phi }.$$

For this special case, the condition (37) becomes: there exist $\delta \in (0,1)$ and two positive constants $C_1$ and $C_2$ such that if $z\in K=\{z\in \mathbb{B}:|\phi \left(z\right)|>\delta \}$, then

$$\begin{array}{c}\left|{\Re }^j\phi \left(z\right)\right|\le C_1\left|\langle {\Re }^j\phi \left(z\right),\phi \left(z\right)\rangle \right|\le C_2|\langle \Re \phi \left(z\right),\phi \left(z\right)\rangle {|}^j\hfill \end{array}$$

(57)

for every $j=\overline{1,l}$.

Then, from Remark 4.1 in [31], we have the following interesting result.

Theorem 3.

Assume that (57) is satisfied. Then, the operator $S_{\overrightarrow{v},\phi }^l:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded if and only if the operators $M_{v_j}{\Re }^j{C}_{\phi }:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$, $j=\overline{0,l}$, are bounded.

Remark 5.

The boundedness can be discussed similarly for two cases of $k>l$ and $k<l$. Here, we omit.

We next begin to consider the compactness of the operator $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ only for $k=l$.

Theorem 4.

Assume that (37) is satisfied, $k,l\in \mathbb{N}$, $k=l$, $u_0\in H\left(\mathbb{B}\right)$, $u_i,v_i\in H\left(\mathbb{B}\right)$, $i=\overline{1,l}$, ϕ normal on $[0,1)$, $\phi \in S\left(\mathbb{B}\right)$ and μ a weight function on $\mathbb{B}$. Then, the operator $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is compact if and only if the operator $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded,

$$\begin{array}{c}\underset{\left|\phi \right(z\left)\right|\to 1}{lim}\frac{\mu \left(z\right)|u_0\left(z\right)\left|\right(1-{\left|\phi \left(z\right)\right|}^2)}{\varphi \left(\right|\phi \left(z\right)\left|\right)}=0\hfill \end{array}$$

(58)

and

$$\begin{array}{c}\underset{\left|\phi \right(z\left)\right|\to 1}{lim}\frac{\mu \left(z\right)|{\sum }_{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right)))|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}=0\hfill \end{array}$$

(59)

for $j=\overline{1,l}$.

Proof. 

Assume that $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is compact. It is obvious that $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded. If ${\parallel \phi \parallel }_{\infty }<1$, then it is clear that (58) and (59) are true. Therefore, we suppose that ${\parallel \phi \parallel }_{\infty }=1$. Let $\left\{z_m\right\}$ be a sequence in $\mathbb{B}$ such that $\left|\phi \right(z_m\left)\right|\to 1$ as $m\to \infty$ and $h_m^{\left(j\right)}=h_{\phi \left(z_m\right)}^{\left(j\right)}$, where $h_w^{\left(j\right)}$ are defined in (47) for a fixed $j\in \{1,2,\dots ,l\}$. Then, we have that ${sup}_{m\in \mathbb{N}}{\parallel h_m^{\left(j\right)}\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}<+\infty$. By Remark 3, we have that $h_m^{\left(j\right)}\to 0$ uniformly on any compact subset of $\mathbb{B}$ as $m\to \infty$. Hence, by Lemma 2 we obtain

$$\begin{array}{c}\underset{m\to \infty }{lim}{\parallel S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}{h}_m^{\left(j\right)}\parallel }_{H_{\mu }^{\infty }\left(\mathbb{B}\right)}=0.\hfill \end{array}$$

(60)

From (51), for sufficiently large m, we have that

$$\begin{array}{cc}\hfill & \frac{\mu \left(z_m\right)\left|{\sum }_{i=j}^l(u_i\left(z_m\right)B_{i,j}\left(\phi \left(z_m\right)\right)+v_i\left(z_m\right)B_{i,j}(\Re \phi \left(z_m\right)))\right|}{\varphi \left(\right|\phi \left(z_m\right)\left|\right)(1-|\phi \left(z_m\right){{|}^2)}^{j-1}}\le {\parallel S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}{h}_m^{\left(j\right)}\parallel }_{H_{\mu }^{\infty }\left(\mathbb{B}\right)}.\hfill \end{array}$$

(61)

Taking $m\to \infty$ in (61), by using (60), we have that (59) holds for $j=\overline{1,l}$.

Furthermore, let $h_m^{\left(0\right)}=h_{\phi \left(z_m\right)}^{\left(0\right)}$, where $h_w^{\left(0\right)}$ is defined in (53). Then, we also have that ${sup}_{m\in \mathbb{N}}{\parallel h_m^{\left(0\right)}\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}<+\infty$ and $h_m^{\left(0\right)}\to 0$ uniformly on any compact subset of $\mathbb{B}$ as $m\to \infty$. Hence, by Lemma 2 we have

$$\begin{array}{c}\underset{m\to \infty }{lim}{\parallel S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}{h}_m^{\left(0\right)}\parallel }_{H_{\mu }^{\infty }\left(\mathbb{B}\right)}=0.\hfill \end{array}$$

(62)

From (54), we have

$$\begin{array}{c}\frac{\mu \left(z_m\right)|u_0\left(z_m\right)\left|\right(1-|\phi \left(z_m\right){|}^2)}{\varphi \left(\right|\phi \left(z_m\right)\left|\right)}\le {\parallel S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}{h}_m^{\left(0\right)}\parallel }_{H_{\mu }^{\infty }\left(\mathbb{B}\right)}.\hfill \end{array}$$

(63)

Letting $m\to \infty$ in (63) and using (62), we have that (58) holds.

Now, assume that $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded. From (44) and (46), we have

$$\begin{array}{c}\mu \left(z\right)|u_0\left(z\right)|\le C<+\infty \hfill \end{array}$$

(64)

and

$$\begin{array}{c}\mu \left(z\right)\left|\sum _{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right)))\right|\le C<+\infty \hfill \end{array}$$

(65)

for all $z\in \mathbb{B}$. On the other hand, from (58) and (59), we have that for arbitrary $\epsilon >0$, there is a $\delta \in (0,1)$ such that on K

$$\begin{array}{c}\frac{\mu \left(z\right)|u_0\left(z\right)\left|\right(1-{\left|\phi \left(z\right)\right|}^2)}{\varphi \left(\right|\phi \left(z\right)\left|\right)}<\epsilon .\hfill \end{array}$$

(66)

and

$$\begin{array}{c}\frac{\mu \left(z\right)|{\sum }_{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right)))|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}<\epsilon .\hfill \end{array}$$

(67)

Assume that $\left\{f_s\right\}$ is a sequence such that ${sup}_{s\in \mathbb{N}}{\parallel f_s\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}\le M$ and $f_s\to 0$ uniformly on any compact subset of $\mathbb{B}$ as $s\to \infty$. Then, by Lemmas 3, 6, and 7 and (64)–(67), we have

$$\begin{array}{cc}\hfill & \parallel S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}{f}_s{\parallel }_{H_{\mu }^{\infty }\left(\mathbb{B}\right)}\hfill \\ \hfill & =\underset{z\in \mathbb{B}}{sup}\mu \left(z\right)|u_0\left(z\right)f\left(\phi \left(z\right)\right)+\sum _{i=1}^l\left(u_i\left(z\right){\Re }^{i}f\left(\phi \left(z\right)\right)+v_i\left(z\right){\Re }^i(f\circ \phi )\left(z\right)\right)|\hfill \\ \hfill & =\underset{z\in K}{sup}\mu \left(z\right)|u_0\left(z\right)f\left(\phi \left(z\right)\right)+\sum _{i=1}^l\left(u_i\left(z\right){\Re }^{i}f\left(\phi \left(z\right)\right)+v_i\left(z\right){\Re }^i(f\circ \phi )\left(z\right)\right)|\hfill \\ \hfill & +\underset{z\in \mathbb{B}\setminus K}{sup}\mu \left(z\right)|u_0\left(z\right)f\left(\phi \left(z\right)\right)+\sum _{i=1}^l\left(u_i\left(z\right){\Re }^{i}f\left(\phi \left(z\right)\right)+v_i\left(z\right){\Re }^i(f\circ \phi )\left(z\right)\right)|\hfill \\ \hfill & \le C\underset{z\in K}{sup}\frac{\mu \left(z\right)|u_0\left(z\right)\left|\right(1-{\left|\phi \left(z\right)\right|}^2)}{\varphi \left(\right|\phi \left(z\right)\left|\right)}{\parallel f_s\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}\hfill \\ \hfill & +C\underset{z\in K}{sup}\frac{\mu \left(z\right)|{\sum }_{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right)))|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}{\parallel f_s\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)}\hfill \\ \hfill & +\underset{z\in \mathbb{B}\setminus K}{sup}\mu \left(z\right)|u_0\left(z\right)\left|\right|f_s\left(\phi \left(z\right)\right)|\hfill \\ \hfill & +\underset{z\in \mathbb{B}\setminus K}{sup}\sum _{j=1}^l\mu \left(z\right)\left|\sum _{i=j}^l(u_i\left(z\right)B_{i,j}\left(\phi \left(z\right)\right)+v_i\left(z\right)B_{i,j}(\Re \phi \left(z\right)))\right|\underset{\{l_1,l_2,\dots ,l_j\}}{max}\left|\frac{{\partial }^j{f}_s}{\partial z_{l_1}\partial z_{l_2}\cdots \partial z_{l_j}}\left(\phi \left(z\right)\right)\right|\hfill \\ \hfill & \le CM\epsilon +C\underset{\left|w\right|\le \delta }{sup}\sum _{j=0}^l\underset{\{l_1,l_2,\dots ,l_j\}}{max}\left|\frac{{\partial }^j{f}_s}{\partial z_{l_1}\partial z_{l_2}\cdots \partial z_{l_j}}\left(w\right)\right|.\hfill \end{array}$$

(68)

Since $f_s\to 0$ uniformly on any compact subset of $\mathbb{B}$ as $s\to \infty$, by Cauchy’s estimates, we also have that $\frac{{\partial }^j{f}_s}{\partial z_{l_1}\partial z_{l_2}\cdots \partial z_{l_j}}\to 0$ uniformly on any compact subset of $\mathbb{B}$ as $s\to \infty$. From this and using the fact that $\{w\in \mathbb{B}:|w|\le \delta \}$ is a compact subset of $\mathbb{B}$, by letting $s\to \infty$ in inequality (68), we obtain

$$\begin{array}{c}\underset{s\to \infty }{lim\; sup}{\parallel S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}{f}_s\parallel }_{H_{\mu }^{\infty }\left(\mathbb{B}\right)}\le CM\epsilon .\hfill \end{array}$$

Since $\epsilon$ is an arbitrary positive number, it follows that

$$\begin{array}{c}\underset{s\to \infty }{lim}{\parallel S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}{f}_s\parallel }_{H_{\mu }^{\infty }\left(\mathbb{B}\right)}=0.\hfill \end{array}$$

(69)

From (69) and Lemma 2, the operator $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is compact. □

From Theorem 3.4 in [30] and Remark 4.2 in [31], we have the following interesting results.

Theorem 5.

The operator ${\mathfrak{S}}_{\overrightarrow{u},\phi }^k:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is compact and (56) holds if and only if the operators $M_{u_i}{C}_{\phi }{\Re }^i:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$, $i=\overline{0,k}$ are compact.

Theorem 6.

Assume that (57) is satisfied. Then, the operator $S_{\overrightarrow{v},\phi }^l:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is compact if and only if the operators $M_{v_j}{\Re }^j{C}_{\phi }:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$, $j=\overline{0,l}$ are compact.

5. Some Applications

As some applications of the results in Part 4, we can characterize the boundedness and compactness of the operators $M_u{C}_{\phi }{\Re }^m$, $C_{\phi }{M}_u{\Re }^m$, $C_{\phi }{\Re }^m{M}_u$, $M_u{\Re }^m{C}_{\phi }$, ${\Re }^m{M}_u{C}_{\phi }$, and ${\Re }^m{C}_{\phi }{M}_u:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$. More specifically, all results of this section are obtained from the relationships in (14). Since

$$M_u{C}_{\phi }{\Re }^m=S_{u_0\equiv \cdots \equiv u_{m-1}\equiv 0,u_m=u,v_1\equiv \cdots \equiv v_l\equiv 0,\phi }^{m,l},$$

the following corollaries come from Proposition 2 (ii), Theorems 1 and 4.

Corollary 2.

Let $m\in \mathbb{N}$, $u\in H\left(\mathbb{B}\right)$, ϕ normal on $[0,1)$, $\phi \in S\left(\mathbb{B}\right)$ and μ a weight on $\mathbb{B}$. Then, the operator $M_u{C}_{\phi }{\Re }^m:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded if and only if

$$\begin{array}{c}{L}_j:=\underset{z\in \mathbb{B}}{sup}\frac{\mu \left(z\right)\left|u\left(z\right)\right||B_{m,j}\left(\phi \left(z\right)\right)|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}<+\infty .\hfill \end{array}$$

for $j=\overline{1,m}$.

Moreover, if the operator $M_u{C}_{\phi }{\Re }^m:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded, then the following asymptotic relationship holds

$$\begin{array}{c}\parallel M_u{C}_{\phi }{\Re }^m{\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)}\asymp \sum _{j=1}^m{L}_j.\hfill \end{array}$$

Corollary 3.

Let $m\in \mathbb{N}$, $u\in H\left(\mathbb{B}\right)$, ϕ normal on $[0,1)$, $\phi \in S\left(\mathbb{B}\right)$ and μ a weight on $\mathbb{B}$. Then, the operator $M_u{C}_{\phi }{\Re }^m:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is compact if and only if the operator $M_u{C}_{\phi }{\Re }^m:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded and

$$\begin{array}{c}\underset{\left|\phi \right(z\left)\right|\to 1}{lim}\frac{\mu \left(z\right)\left|u\left(z\right)\right||B_{m,j}\left(\phi \left(z\right)\right)|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}=0\hfill \end{array}$$

for $j=\overline{1,m}$.

Since

$$C_{\phi }{M}_u{\Re }^m=S_{u_0\equiv \cdots \equiv u_{m-1}\equiv 0,u_m=u\circ \phi ,v_1\equiv \cdots \equiv v_l\equiv 0,\phi }^{m,l},$$

the following corollaries come from Proposition 2 (ii), Theorems 1 and 4.

Corollary 4.

Let $m\in \mathbb{N}$, $u\in H\left(\mathbb{B}\right)$, ϕ normal on $[0,1)$, $\phi \in S\left(\mathbb{B}\right)$ and μ a weight on $\mathbb{B}$. Then, the operator $C_{\phi }{M}_u{\Re }^m:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded if and only if

$$\begin{array}{c}{M}_j:=\underset{z\in \mathbb{B}}{sup}\frac{\mu \left(z\right)\left|u\left(\phi \left(z\right)\right)\right||B_{m,j}\left(\phi \left(z\right)\right)|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}<+\infty \hfill \end{array}$$

for $j=\overline{1,m}$.

Moreover, if the operator $C_{\phi }{M}_u{\Re }^m:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded, then the following asymptotic relationship holds

$$\begin{array}{c}\parallel C_{\phi }{M}_u{\Re }^m{\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)}\asymp \sum _{j=1}^m{M}_j.\hfill \end{array}$$

Corollary 5.

Let $m\in \mathbb{N}$, $u\in H\left(\mathbb{B}\right)$, ϕ normal on $[0,1)$, $\phi \in S\left(\mathbb{B}\right)$ and μ a weight on $\mathbb{B}$. Then, the operator $C_{\phi }{M}_u{\Re }^m:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is compact if and only if the operator $C_{\phi }{M}_u{\Re }^m:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded and

$$\begin{array}{c}\underset{\left|\phi \right(z\left)\right|\to 1}{lim}\frac{\mu \left(z\right)\left|u\left(\phi \left(z\right)\right)\right||B_{m,j}\left(\phi \left(z\right)\right)|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}=0\hfill \end{array}$$

for $j=\overline{1,m}$.

Since

$$C_{\phi }{\Re }^m{M}_u=S_{u_0=C_m^0\left({\Re }^{m}u\right)\circ \phi ,u_1=C_m^1\left({\Re }^{m-1}u\right)\circ \phi ,\dots ,u_m=C_m^{m}u\circ \phi ,v_1\equiv \cdots \equiv v_l\equiv 0,\phi }^{m,l},$$

the following results hold from Proposition 2 (ii), Theorems 1 and 4.

Corollary 6.

Let $m\in \mathbb{N}$, $u\in H\left(\mathbb{B}\right)$, ϕ normal on $[0,1)$, $\phi \in S\left(\mathbb{B}\right)$ and μ a weight on $\mathbb{B}$. Then, the operator $C_{\phi }{\Re }^m{M}_u:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded if and only if

$$\begin{array}{c}{N}_0:=\underset{z\in \mathbb{B}}{sup}\frac{\mu \left(z\right)|\left({\Re }^{m}u\right)\left(\phi \left(z\right)\right)\left|\right(1-|\phi \left(z\right){|}^2)}{\varphi \left(\right|\phi \left(z\right)\left|\right)}<+\infty \hfill \end{array}$$

and

$$\begin{array}{c}{N}_j:=\underset{z\in \mathbb{B}}{sup}\frac{\mu \left(z\right)|{\sum }_{i=j}^m\left({\Re }^{m-i}u\right)\left(\phi \left(z\right)\right)B_{i,j}\left(\phi \left(z\right)\right)|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}<+\infty \hfill \end{array}$$

for $j=\overline{1,m}$.

Moreover, if the operator $C_{\phi }{\Re }^m{M}_u:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded, then the following asymptotic relationship holds

$$\begin{array}{c}\parallel C_{\phi }{\Re }^m{M}_u{\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)}\asymp \sum _{j=0}^m{N}_j.\hfill \end{array}$$

Corollary 7.

Let $m\in \mathbb{N}$, $u\in H\left(\mathbb{B}\right)$, ϕ normal on $[0,1)$, $\phi \in S\left(\mathbb{B}\right)$ and μ a weight on $\mathbb{B}$. Then, the operator $C_{\phi }{\Re }^m{M}_u:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is compact if and only if the operator $C_{\phi }{\Re }^m{M}_u:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded,

$$\begin{array}{c}\underset{\left|\phi \right(z\left)\right|\to 1}{lim}\frac{\mu \left(z\right)|\left({\Re }^{m}u\right)\left(\phi \left(z\right)\right)\left|\right(1-|\phi \left(z\right){|}^2)}{\varphi \left(\right|\phi \left(z\right)\left|\right)}=0\hfill \end{array}$$

and

$$\begin{array}{c}\underset{\left|\phi \right(z\left)\right|\to 1}{lim}\frac{\mu \left(z\right)|{\sum }_{i=j}^m\left({\Re }^{m-i}u\right)\left(\phi \left(z\right)\right)B_{i,j}\left(\phi \left(z\right)\right)|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}=0\hfill \end{array}$$

for $j=\overline{1,m}$.

Since

$$M_u{\Re }^m{C}_{\phi }=S_{u_0\equiv \cdots \equiv u_k\equiv 0,v_1\equiv \cdots \equiv v_{m-1}\equiv 0,v_m=u,\phi }^{k,m}$$

and the condition (37) is reduced to the following condition

$$\begin{array}{c}\left|\sum _{j=1}^m{B}_{m,j}(\Re \phi \left(z\right))\right|\le C|\sum _{j=1}^m{B}_{m,j}\left(\langle \Re \phi \left(z\right),\phi \left(z\right)\rangle \right)\left)\right|,\hfill \end{array}$$

(70)

we obtain the next results from Theorems 1 and 4.

Corollary 8.

Assume that (70) is satisfied, $m\in \mathbb{N}$, $u\in H\left(\mathbb{B}\right)$, ϕ normal on $[0,1)$, $\phi \in S\left(\mathbb{B}\right)$ and μ a weight on $\mathbb{B}$. Then, the operator $M_u{\Re }^m{C}_{\phi }:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded if and only if

$$\begin{array}{c}{\tilde{L}}_j:=\underset{z\in \mathbb{B}}{sup}\frac{\mu \left(z\right)\left|u\left(z\right)\right||B_{m,j}(\Re \phi \left(z\right))|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}<+\infty \hfill \end{array}$$

for $j=\overline{1,m}$.

Moreover, if the operator $M_u{\Re }^m{C}_{\phi }:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded, then the following asymptotic relationship holds

$$\begin{array}{c}\parallel M_u{\Re }^m{C}_{\phi }{\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)}\asymp \sum _{j=1}^m{\tilde{L}}_j.\hfill \end{array}$$

Corollary 9.

Assume that (70) is satisfied, $m\in \mathbb{N}$, $u\in H\left(\mathbb{B}\right)$, ϕ normal on $[0,1)$, $\phi \in S\left(\mathbb{B}\right)$ and μ a weight on $\mathbb{B}$. Then, the operator $M_u{\Re }^m{C}_{\phi }:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is compact if and only if the operator $M_u{\Re }^m{C}_{\phi }:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded and

$$\begin{array}{c}\underset{\left|\phi \right(z\left)\right|\to 1}{lim}\frac{\mu \left(z\right)\left|u\left(z\right)\right||B_{m,j}(\Re \phi \left(z\right))|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}=0\hfill \end{array}$$

for $j=\overline{1,m}$.

Since

$${\Re }^m{M}_u{C}_{\phi }=S_{u_1\equiv \cdots \equiv u_k\equiv 0,u_0=C_m^0{\Re }^{m}u,v_1=C_m^1{\Re }^{m-1}u,\dots ,v_m=C_m^{m}u,\phi }^{k,m}$$

and the condition (37) is reduced to the following condition

$$\begin{array}{c}\left|\sum _{i=j}^m{C}_m^i\left({\Re }^{m-i}u\right)\left(z\right)B_{i,j}(\Re \phi \left(z\right))\right|\le C\left|\sum _{i=j}^m{C}_m^i\left({\Re }^{m-i}u\right)\left(z\right)B_{i,j}\left(\langle \Re \phi \left(z\right),\phi \left(z\right)\rangle \right)\right|\hfill \end{array}$$

(71)

for $j=\overline{1,m}$. We have the following corollaries from Theorems 1 and 4.

Corollary 10.

Assume that (71) is satisfied, $m\in \mathbb{N}$, $u\in H\left(\mathbb{B}\right)$, ϕ normal on $[0,1)$, $\phi \in S\left(\mathbb{B}\right)$ and μ a weight on $\mathbb{B}$. Then, the operator ${\Re }^m{M}_u{C}_{\phi }:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded if and only if

$$\begin{array}{c}{\tilde{M}}_0:=\underset{z\in \mathbb{B}}{sup}\frac{\mu \left(z\right)|\left({\Re }^{m}u\right)\left(z\right)\left|\right(1-|\phi \left(z\right){|}^2)}{\varphi \left(\right|\phi \left(z\right)\left|\right)}<+\infty \hfill \end{array}$$

and

$$\begin{array}{c}{\tilde{M}}_j:=\underset{z\in \mathbb{B}}{sup}\frac{\mu \left(z\right)|{\sum }_{i=j}^m{C}_m^i\left({\Re }^{m-i}u\right)\left(z\right)B_{i,j}(\Re \phi \left(z\right))|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}<+\infty \hfill \end{array}$$

for $j=\overline{1,m}$.

Moreover, if the operator ${\Re }^m{M}_u{C}_{\phi }:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded, then the following asymptotic relationship holds

$$\begin{array}{c}\parallel {\Re }^m{M}_u{C}_{\phi }{\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)}\asymp \sum _{j=0}^m{\tilde{M}}_j.\hfill \end{array}$$

Corollary 11.

Assume that (71) is satisfied, $m\in \mathbb{N}$, $u\in H\left(\mathbb{B}\right)$, ϕ normal on $[0,1)$, $\phi \in S\left(\mathbb{B}\right)$ and μ a weight on $\mathbb{B}$. Then, the operator ${\Re }^m{M}_u{C}_{\phi }:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is compact if and only if the operator ${\Re }^m{M}_u{C}_{\phi }:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded,

$$\begin{array}{c}\underset{\left|\phi \right(z\left)\right|\to 1}{lim}\frac{\mu \left(z\right)|\left({\Re }^{m}u\right)\left(z\right)\left|\right(1-|\phi \left(z\right){|}^2)}{\varphi \left(\right|\phi \left(z\right)\left|\right)}=0\hfill \end{array}$$

and

$$\begin{array}{c}\underset{\left|\phi \right(z\left)\right|\to 1}{lim}\frac{\mu \left(z\right)|{\sum }_{i=j}^m{C}_m^i\left({\Re }^{m-i}u\right)\left(z\right)B_{i,j}(\Re \phi \left(z\right))|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}=0\hfill \end{array}$$

for $j=\overline{1,m}$.

Since

$${\Re }^m{C}_{\phi }{M}_u=S_{u_1\equiv \cdots \equiv u_k\equiv 0,u_0=C_m^0{\Re }^m(u\circ \phi ),v_1=C_m^1{\Re }^{m-1}(u\circ \phi ),\dots ,v_m=C_m^{m}u\circ \phi ,\phi }^{k,m}$$

and the condition (37) is reduced to the following condition

$$\begin{array}{c}\left|\sum _{i=j}^m{C}_m^i{\Re }^{m-i}(u\circ \phi )\left(z\right)B_{i,j}(\Re \phi \left(z\right))\right|\le C\left|\sum _{i=j}^m{C}_m^i{\Re }^{m-i}(u\circ \phi )\left(z\right)B_{i,j}\left(\langle \Re \phi \left(z\right),\phi \left(z\right)\rangle \right)\right|\hfill \end{array}$$

(72)

for $j=\overline{1,m}$. we obtain the following corollaries from Theorems 1 and 4.

Corollary 12.

Assume that (72) is satisfied, $m\in \mathbb{N}$, $u\in H\left(\mathbb{B}\right)$, ϕ normal on $[0,1)$, $\phi \in S\left(\mathbb{B}\right)$, and μ a weight on $\mathbb{B}$. Then, the operator ${\Re }^m{C}_{\phi }{M}_u:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded if and only if

$$\begin{array}{c}{\tilde{N}}_0:=\underset{z\in \mathbb{B}}{sup}\frac{\mu \left(z\right)|{\Re }^m(u\circ \phi )\left(z\right)\left|\right(1-{\left|\phi \left(z\right)\right|}^2)}{\varphi \left(\right|\phi \left(z\right)\left|\right)}<+\infty \hfill \end{array}$$

and

$$\begin{array}{c}{\tilde{N}}_j:=\underset{z\in \mathbb{B}}{sup}\frac{\mu \left(z\right)|{\sum }_{i=j}^m{C}_m^i{\Re }^{m-i}(u\circ \phi )\left(z\right)B_{i,j}(\Re \phi \left(z\right))|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}<+\infty \hfill \end{array}$$

for $j=\overline{1,m}$.

Moreover, if the operator ${\Re }^m{C}_{\phi }{M}_u:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded, then the following asymptotic relationship holds

$$\begin{array}{c}\parallel {\Re }^m{C}_{\phi }{M}_u{\parallel }_{{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)}\asymp \sum _{j=0}^m{\tilde{N}}_j.\hfill \end{array}$$

Corollary 13.

Assume that (72) is satisfied, $m\in \mathbb{N}$, $u\in H\left(\mathbb{B}\right)$, ϕ normal on $[0,1)$, $\phi \in S\left(\mathbb{B}\right)$ and μ a weight on $\mathbb{B}$. Then, the operator ${\Re }^m{C}_{\phi }{M}_u:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is compact if and only if the operator ${\Re }^m{C}_{\phi }{M}_u:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ is bounded,

$$\begin{array}{c}\underset{\left|\phi \right(z\left)\right|\to 1}{lim}\frac{\mu \left(z\right)|{\Re }^m(u\circ \phi )\left(z\right)\left|\right(1-{\left|\phi \left(z\right)\right|}^2)}{\varphi \left(\right|\phi \left(z\right)\left|\right)}=0\hfill \end{array}$$

and

$$\begin{array}{c}\underset{\left|\phi \right(z\left)\right|\to 1}{lim}\frac{\mu \left(z\right)|{\sum }_{i=j}^m{C}_m^i{\Re }^{m-i}(u\circ \phi )\left(z\right)B_{i,j}(\Re \phi \left(z\right))|}{\varphi \left(\right|\phi \left(z\right)\left|\right)(1-{\left|\phi \left(z\right)\right|}^2{)}^{j-1}}=0\hfill \end{array}$$

for $j=\overline{1,m}$.

6. Conclusions

In this paper, we define the sum operator

$$\begin{array}{c}{S}_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}=M_{u_0}{C}_{\phi }+\sum _{i=1}^k{M}_{u_i}{C}_{\phi }{\Re }^i+\sum _{j=1}^l{M}_{v_j}{\Re }^j{C}_{\phi }\hfill \end{array}$$

on some subspaces of $H\left(\mathbb{B}\right)$, where $u_0,u_1,\dots ,u_k,v_1,\dots ,v_l\in H\left(\mathbb{B}\right)$, $\phi \in S\left(\mathbb{B}\right)$, and $k,l\in \mathbb{N}$. We completely characterized the boundedness and compactness of the operator $S_{\overrightarrow{u},\overrightarrow{v},\phi }^{k,l}:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ in terms of the behaviors of the symbols $u_j$, $v_j$, and $\phi$. As an application, the corresponding results of the operators $M_u{C}_{\phi }{\Re }^m$, $C_{\phi }{M}_u{\Re }^m$, $C_{\phi }{\Re }^m{M}_u$, $M_u{\Re }^m{C}_{\phi }$, ${\Re }^m{M}_u{C}_{\phi }$, ${\Re }^m{C}_{\phi }{M}_u:{\mathcal{B}}_{\varphi }\left(\mathbb{B}\right)\to H_{\mu }^{\infty }\left(\mathbb{B}\right)$ are obtained. This paper can be viewed as a continuation and extension of the work of [30,31]. We hope that the study can attract more people’s attention to such operators.