General Toeplitz operators on weighted Bloch-type spaces in the unit ball of ${\mathbb{C}}^n$

Abstract

In this paper, we consider the weighted Bloch-type spaces ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ with $\alpha >0$ and $\beta \ge 0$ in the unit ball of ${\mathbb{C}}^n$. We present some basic properties of the spaces ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$, then we consider the Toeplitz operator $T_{\mu }^{\alpha ,\beta ;\omega }$ acting between ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ spaces, where μ is a positive Borel measure in the unit ball ${\mathbb{B}}_n$. Moreover, we characterize complex measures μ for which the Toeplitz operator $T_{\mu }^{\alpha ,\beta ;\omega }$ is bounded or compact on ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$.

MSC: 47B35, 32A18.

1 Introduction

We start here with some terminology, notations and definitions of various classes of analytic functions defined on the unit ball of ${\mathbb{C}}^n$.

Let ${\mathbb{B}}_n$ be the unit ball of the n-dimensional complex Euclidean space ${\mathbb{C}}^n$. The boundary of ${\mathbb{B}}_n$ is denoted by ${\mathbb{S}}_n$ and is called the unit sphere in ${\mathbb{C}}^n$. Occasionally, we will also need the closed unit ball ${\overline{\mathbb{B}}}_n$. We denote the class of all holomorphic functions on the unit ball ${\mathbb{B}}_n$ by $\mathcal{H}({\mathbb{B}}_n)$. The ball centered at $\mathbf{z}\in {\mathbb{C}}^n$ with radius r is denoted by $B(\mathbf{z},r)$. For $\alpha >-1$, let $d{\nu }_{\alpha }(\mathbf{z})=c_{\alpha }{(1-{|\mathbf{z}|}^2)}^{\alpha }d\nu$, where dν is the normalized Lebesgue volume measure on ${\mathbb{B}}_n$ and $c_{\alpha }=\frac{\mathrm{\Gamma }(n+\alpha +1)}{n!\mathrm{\Gamma }(\alpha +1)}$ (where Γ denotes the gamma function) so that ${\nu }_{\alpha }({\mathbb{B}}_n)\equiv 1$. The surface measure on ${\mathbb{S}}_n$ is denoted by dσ. Once again, we normalize σ so that ${\sigma }_{\alpha }({\mathbb{S}}_n)\equiv 1$. For any $\mathbf{z}=(z_1,z_2,\dots ,z_n),\mathbf{w}=(w_1,w_2,\dots ,w_n)\in {\mathbb{C}}^n$, the inner product is defined by

$$〈\mathbf{z},\mathbf{w}〉=\sum _{k=1}^n{z}_k{\overline{w}}_k.$$

For every point $\mathbf{a}\in {\mathbb{B}}_n$, the Möbius transformation ${\phi }_{\mathbf{a}}:{\mathbb{B}}_n\to {\mathbb{B}}_n$ is defined by

$${\phi }_{\mathbf{a}}(\mathbf{z})=\frac{\mathbf{a}-P_{\mathbf{a}}(\mathbf{z})-S_{\mathbf{a}}{Q}_{\mathbf{a}}(\mathbf{z})}{1-〈\mathbf{z},\mathbf{a}〉},\mathbf{z}\in {\mathbb{B}}_n,$$

where $S_{\mathbf{a}}=\sqrt{1-{|\mathbf{a}|}^2}$, $P_{\mathbf{a}}(\mathbf{z})=\frac{\mathbf{a}〈\mathbf{z},\mathbf{a}〉}{{|\mathbf{a}|}^2}$, $P_0=0$ and $Q_{\mathbf{a}}=I-P_{\mathbf{a}}$. The map ${\phi }_{\mathbf{a}}$ has the following properties that ${\phi }_{\mathbf{a}}(0)=\mathbf{a}$, ${\phi }_{\mathbf{a}}(\mathbf{a})=0$, ${\phi }_{\mathbf{a}}={\phi }_{\mathbf{a}}^{-1}$ and

$$1-〈{\phi }_{\mathbf{a}}(\mathbf{z}),{\phi }_{\mathbf{a}}(\mathbf{w})〉=\frac{(1-{|\mathbf{a}|}^2)(1-〈\mathbf{z},\mathbf{w}〉)}{(1-〈\mathbf{z},\mathbf{a}〉)(1-〈\mathbf{a},\mathbf{w}〉)},$$

where z and w are arbitrary points in ${\mathbb{B}}_n$. In particular,

$$1-{|{\phi }_{\mathbf{a}}(\mathbf{z})|}^2=\frac{(1-{|\mathbf{a}|}^2)(1-{|\mathbf{z}|}^2)}{{|1-〈\mathbf{z},\mathbf{a}〉|}^2}.$$

For $f\in \mathcal{H}({\mathbb{B}}_n)$, the holomorphic gradient of f at z is defined by

$$\mathrm{\nabla }f(\mathbf{z})=(\frac{\partial f}{\partial z_1}(\mathbf{z}),\frac{\partial f}{\partial z_2}(\mathbf{z}),\dots ,\frac{\partial f}{\partial z_n}(\mathbf{z}))$$

and the radial derivative of f at z is defined by

$$\mathfrak{R}f(\mathbf{z})=〈\mathrm{\nabla }f,\overline{\mathbf{z}}〉=\sum _{j=1}^n{z}_j\frac{\partial f(\mathbf{z})}{\partial z_j}.$$

Similarly, the Möbius invariant complex gradient of f at z is defined by

$$\tilde{\mathrm{\nabla }}f(\mathbf{z})=\mathrm{\nabla }(f\circ {\phi }_{\mathbf{z}})(0).$$

For $\alpha >0$, a function $f\in \mathcal{H}({\mathbb{B}}_n)$ is said to belong to the α-Bloch spaces ${\mathcal{B}}^{\alpha }({\mathbb{B}}_n)$ if (see [1])

$$b_{\alpha }(f)({\mathbb{B}}_n)=\underset{\mathbf{z}\in {\mathbb{B}}_n}{sup}|\mathrm{\nabla }f(\mathbf{z})|{(1-{|\mathbf{z}|}^2)}^{\alpha }<\mathrm{\infty }.$$

The little Bloch space ${\mathcal{B}}_0^{\alpha }({\mathbb{B}}_n)$ consists of all $f\in {\mathcal{B}}^{\alpha }({\mathbb{B}}_n)$ such that

$$\underset{|\mathbf{z}|\to 1^{-}}{lim}|\mathrm{\nabla }f(\mathbf{z})|{(1-{|\mathbf{z}|}^2)}^{\alpha }=0.$$

With the norm ${\parallel f\parallel }_{{\mathcal{B}}^{\alpha }}=|f(0)|+b_{\alpha }(f)({\mathbb{B}}_n)$, we know that ${\mathcal{B}}^{\alpha }({\mathbb{B}}_n)$ becomes a Banach space and ${\mathcal{B}}_0^{\alpha }({\mathbb{B}}_n)$ is its closed subspace (see [1]). For $\alpha =1$, the spaces ${\mathcal{B}}^1({\mathbb{B}}_n)$ and ${\mathcal{B}}_0^1({\mathbb{B}}_n)$ become the Bloch and the little Bloch space, respectively (see, for example, [2–5]). Zhu in [5] says that the norm ${\parallel f\parallel }_{\mathcal{B}({\mathbb{B}}_n)}$ is equivalent to

$$|f(0)|+\underset{\mathbf{z}\in {\mathbb{B}}_n}{sup}|\mathfrak{R}f(\mathbf{z})|(1-{|\mathbf{z}|}^2).$$

For $\alpha >-1$ and $0<p<\mathrm{\infty }$, the weighted Bergman space $A_{\alpha }^p({\mathbb{B}}_n)$ consists of holomorphic functions $f\in L^p({\mathbb{B}}_n,d{\nu }_{\alpha })$ such that

$${\parallel f\parallel }_{A_{\alpha }^p({\mathbb{B}}_n)}^p:={\int }_{{\mathbb{B}}_n}{|f(\mathbf{z})|}^{p}d{\nu }_{\alpha }(\mathbf{z})<\mathrm{\infty },$$

that is, $A_{\alpha }^p({\mathbb{B}}_n)=L^p({\mathbb{B}}_n,d{\nu }_{\alpha })\cap \mathcal{H}({\mathbb{B}}_n)$. When the weight $\alpha =0$, we simply write $A^p({\mathbb{B}}_n)$ for $A_0^p({\mathbb{B}}_n)$. In the special case when $p=2$, $A_{\alpha }^2({\mathbb{B}}_n)$ is a Hilbert space. It is well known that for $\alpha >-1$, the Bergman kernel of $A_{\alpha }^2({\mathbb{B}}_n)$ is given by

$$K^{\alpha }(\mathbf{z},\mathbf{w})=\frac{1}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+\alpha +1}},\mathbf{z},\mathbf{w}\in {\mathbb{B}}_n.$$

For $\alpha >-1$, a complex measure μ is such that

$$|{\int }_{{\mathbb{B}}_n}{(1-{|\mathbf{w}|}^2)}^{\alpha }d\mu (\mathbf{w})|=|{\int }_{{\mathbb{B}}_n}d{\mu }_{\alpha }(\mathbf{w})|<\mathrm{\infty }.$$

The general Bergman projection $P_{\alpha }$ is the orthogonal projection of the measure μ from $L^2({\mathbb{B}}_n,d{\nu }_{\alpha })$ into $A_{\alpha }^2({\mathbb{B}}_n)$ defined by

$$P_{\alpha }(\mu )(\mathbf{z})=c_{\alpha }{\int }_{{\mathbb{B}}_n}\frac{{(1-{|\mathbf{w}|}^2)}^{\alpha }}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+\alpha +1}}d\mu (\mathbf{w})=c_{\alpha }{\int }_{{\mathbb{B}}_n}\frac{d{\mu }_{\alpha }(\mathbf{w})}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+\alpha +1}}.$$

The general Bergman projection of the function f is

$$P_{\alpha }f(\mathbf{z})=c_{\alpha }{\int }_{{\mathbb{B}}_n}\frac{f(\mathbf{w}){(1-{|\mathbf{w}|}^2)}^{\alpha }}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+\alpha +1}}d\nu (\mathbf{w})=c_{\alpha }{\int }_{{\mathbb{B}}_n}\frac{f(\mathbf{w})d{\nu }_{\alpha }(\mathbf{w})}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+\alpha +1}}.$$

Let $\omega :(0,1]\to (0,\mathrm{\infty })$ be a right-continuous and nondecreasing function. For a complex measure $\mu ,\alpha >-1$, $\beta \ge 0$, and $f\in L^1({\mathbb{B}}_n,d{\nu }_{\alpha +\beta })$, define weighted general Toeplitz operator as follows:

$$\begin{array}{rcl}{T}_{\mu }^{\alpha ,\beta ;\omega }f(\mathbf{z})& =& \frac{c_{\alpha +\beta }}{{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}{\int }_{{\mathbb{B}}_n}\frac{{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta }f(\mathbf{w})}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+\alpha +\beta +1}}d\mu (\mathbf{w})\\ =& \frac{c_{\alpha +\beta }}{{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}{\int }_{{\mathbb{B}}_n}\frac{f(\mathbf{w})d{\mu }_{\alpha +\beta }(\mathbf{w})}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+\alpha +\beta +1}}.\end{array}$$

Thus $P_{\alpha +\beta ;\omega }(\mu )(\mathbf{z})=T_{\mu }^{\alpha ,\beta ;\omega }(1)(\mathbf{z})$, where 1 stands for a constant function.

Toeplitz operators have been studied extensively on the Bergman spaces by many authors. For references, see [6] and [7]. Boundedness and compactness of the general Toeplitz operators $T_{\mu }^{\alpha }$ on the α-Bloch ${\mathcal{B}}^{\alpha }(\mathbb{D})$ spaces have been investigated in [8] on the unit disk $\mathbb{D}$ for $0<\alpha <\mathrm{\infty }$. Also, in [9], the authors extended the general Toeplitz operator $T_{\mu }^{\alpha }$ to ${\mathcal{B}}^{\alpha }({\mathbb{B}}_n)$ with $1\le \alpha <2$. Recently, in [10], the general Toeplitz operators $T_{\mu }^{\alpha }$ on the analytic Besov $B_p(\mathbb{D})$ spaces with $1\le p<\mathrm{\infty }$ have been investigated. Under a prerequisite condition, the authors characterized a complex measure μ on the unit disk for which $T_{\mu }^{\alpha }$ is bounded or compact on the Besov space $B_p(\mathbb{D})$. For more studies on the Toeplitz operator, we refer to [11–17].

In this paper, we consider the weighted Bloch-type spaces ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ with $\alpha >0$ and $\beta \ge 0$ in the unit ball of ${\mathbb{C}}^n$. We prove a certain integral representation theorem that is used to determine the degree of growth of the functions in the space ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$. It is also proved that the space ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ is a Banach space for each weight $\alpha >0$, $\beta \ge 0$, and the Banach dual of the Bergman space $A^1({\mathbb{B}}_n)$ is ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ for each $\alpha \ge 1$, $\beta \ge 0$. Further, we extend the Toeplitz operator $T_{\mu }^{\alpha ,\beta ;\omega }$ to ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ in the unit ball of ${\mathbb{C}}^n$ and completely characterize the positive Borel measure μ such that $T_{\mu }^{\alpha ,\beta ;\omega }$ is bounded or compact in ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ spaces with $\alpha +\beta \ge 1$.

Throughout the paper, we say that the expressions A and B are equivalent, and write $A\approx B$ whenever there exist positive constants $C_1$ and $C_2$ such that $C_{1}A\le B\le C_{2}A$. As usual, the letter C denotes a positive constant, possibly different on each occurrence. Hereafter, ω stands for a right-continuous and nondecreasing function.

Theorem 1.1 (see [[5], Theorem 1.12])

Suppose b is real and $s>-1$. Then the integrals

$$I_b(\mathbf{z})={\int }_{{\mathbb{S}}_n}\frac{d\sigma (\mathit{\zeta })}{{|1-〈\mathbf{z},\mathit{\zeta }〉|}^{n+b}},\mathbf{z}\in {\mathbb{B}}_n$$

and

$$J_{b,s}(\mathbf{z})={\int }_{{\mathbb{B}}_n}\frac{{(1-{|\mathbf{w}|}^2)}^{s}d\nu (\mathbf{w})}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{n+1+s+b}},\mathbf{z}\in {\mathbb{B}}_n,$$

have the following asymptotic properties.

(1) If $b<0$, then $I_b(\mathbf{z})$ and $J_{b,s}(\mathbf{z})$ are both bounded in ${\mathbb{B}}_n$.

(2) If $b=0$, then

$$I_b(\mathbf{z})\approx I_{b,s}(\mathbf{z})\approx log\frac{1}{1-{|\mathbf{z}|}^2}\mathit{\text{as}}|\mathbf{z}|\to 1^{-1}.$$

(3) If $b>0$, then

$$I_b(\mathbf{z})\approx J_{b,s}(\mathbf{z})\approx {(1-{|\mathbf{z}|}^2)}^{-b}\mathit{\text{as}}|\mathbf{z}|\to 1^{-1}.$$

Lemma 1.1 (see [[5], Lemma 3.3])

Suppose γ is a real constant and $g\in L^1({\mathbb{B}}_n,d\nu )$. If

$$u(\mathbf{z})={(1-{|{\phi }_{\mathbf{a}}(\mathbf{z})|}^2)}^{\beta }{\int }_{{\mathbb{B}}_n}\frac{g(\mathbf{w})d\nu (\mathbf{w})}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{\gamma }},\mathbf{z}\in {\mathbb{B}}_n,$$

then

$$|\tilde{\mathrm{\nabla }}u(\mathbf{z})|\le \sqrt{2}|\gamma |{(1-{|\mathbf{z}|}^2)}^{\frac{1}{2}}{\int }_{{\mathbb{B}}_n}\frac{g(\mathbf{w})d\nu (\mathbf{w})}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{\gamma +\frac{1}{2}}},\mathrm{\forall }\mathbf{z}\in {\mathbb{B}}_n.$$

Let $\beta (\cdot ,\cdot )$ be the Bergman metric on ${\mathbb{B}}_n$. Denote the Bergman metric ball at ${\mathbf{w}}^{(j)}$ by $B({\mathbf{w}}^{(j)},r)=\{\mathbf{z}\in {\mathbb{B}}_n:\beta ({\mathbf{w}}^{(j)},\mathbf{z})<r\}$, where ${\mathbf{w}}^{(j)}\in {\mathbb{B}}_n$ and $r>0$.

Lemma 1.2 (see [[5], Theorem 2.23])

For fixed $r>0$, there is a sequence $\{{\mathbf{w}}^{(j)}\}\in {\mathbb{B}}_n$ such that:

• ${\bigcup }_{j=1}^{\mathrm{\infty }}B({\mathbf{w}}^{(j)},r)={\mathbb{B}}_n$;

• there is a positive integer N such that each $\mathbf{z}\in {\mathbb{B}}_n$ is contained in at most N of the sets $B({\mathbf{w}}^{(j)},2r)$.

The following characterization of Carleson measures can be found in [6], or in [5].

A positive Borel measure μ on the unit ball ${\mathbb{B}}_n$ is said to be a Carleson measure for the Bergman space $A_{\alpha }^p({\mathbb{B}}_n)$ if

$${\int }_{{\mathbb{B}}_n}{|f(\mathbf{z})|}^{p}d{\nu }_{\alpha }(\mathbf{z})\le C{\parallel f\parallel }_{A_{\alpha }^p({\mathbb{B}}_n)}^p,\mathrm{\forall }f\in A_{\alpha }^p({\mathbb{B}}_n).$$

It is well known that a positive Borel measure μ is a Carleson measure if and only if there is a positive constant C such that

$$\underset{{\mathbf{w}}^{(j)}\in {\mathbb{B}}_n}{sup}\frac{\mu (B({\mathbf{w}}^{(j)},r))}{\nu (B({\mathbf{w}}^{(j)},r))}<\mathrm{\infty },$$

where $\{{\mathbf{w}}^{(j)}\}$ is the sequence in Lemma 1.2. If μ satisfies that

$$\underset{j\to \mathrm{\infty }}{lim}\frac{\mu (B({\mathbf{w}}^{(j)},r))}{\nu (B({\mathbf{w}}^{(j)},r))}=0,$$

then μ is called a vanishing Carleson measure.

For a given reasonable function $\omega :(0,1]\to (0,\mathrm{\infty })$, the weighted Bloch space ${\mathcal{B}}_{\omega }$ of several complex variables is defined as the set of all analytic functions f on ${\mathbb{B}}_n$ satisfying

$${(1-|\mathbf{z}|)}^{\alpha }|\mathrm{\nabla }f(\mathbf{z})|\le C\omega (1-|\mathbf{z}|),\mathbf{z}\in {\mathbb{B}}_n,\text{ where }\alpha \in (0,\mathrm{\infty }),$$

for some fixed $C=C_f>0$. In the special case where $\omega \equiv 1$, ${\mathcal{B}}_{\omega }$ reduces to the classical Bloch space ℬ in ${\mathbb{C}}^n$. This class of functions extends and generalizes the well known Bloch space. Now, we define the space ${\mathcal{B}}_{\alpha ,\beta ;\omega }({\mathbb{B}}_n)$ in the unit ball ${\mathbb{B}}_n$. For $\alpha >0$ and $\beta \ge 0$, a function $f\in \mathcal{H}({\mathbb{B}}_n)$ is said to belong to the $(\alpha ,\beta ;\omega )$-Bloch space ${\mathcal{B}}_{\alpha ,\beta ;\omega }({\mathbb{B}}_n)$ if

$$b_{\alpha ,\beta ;\omega }(f)({\mathbb{B}}_n)=\underset{\mathbf{a},\mathbf{z}\in {\mathbb{B}}_n}{sup}\frac{{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta }}{{(1-{|{\phi }_{\mathbf{a}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|)}|\mathrm{\nabla }f(\mathbf{z})|<\mathrm{\infty }.$$

The little $(\alpha ,\beta ;\omega )$-Bloch space ${\mathcal{B}}_{\alpha ,\beta ;\omega ,0}({\mathbb{B}}_n)$ is a subspace of ${\mathcal{B}}_{\alpha ,\beta ;\omega }({\mathbb{B}}_n)$ consisting of all $f\in {\mathcal{B}}_{\alpha ,\beta ;\omega }({\mathbb{B}}_n)$ such that

$$\underset{|\mathbf{a}|\to 1^{-}}{lim}\underset{|\mathbf{z}|\to 1^{-}}{lim}\frac{{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta }}{{(1-{|{\phi }_{\mathbf{a}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|)}|\mathrm{\nabla }f(\mathbf{z})|=0.$$

If $\beta =0$, $\omega (1-|\mathbf{z}|)=1$, then we get the α-Bloch space ${\mathcal{B}}^{\alpha }({\mathbb{B}}_n)$ and the little α-Bloch space ${\mathcal{B}}_0^{\alpha }({\mathbb{B}}_n)$. If $\omega (1-|\mathbf{z}|)=1$, $\alpha =1$ and $\beta =0$, then we get the classical Bloch space $\mathcal{B}({\mathbb{B}}_n)$ and ${\mathcal{B}}_0({\mathbb{B}}_n)$. These classes extend the weighted Bloch spaces defined in [18] to the setting of several complex variables.

The logarithmic $(\alpha ,\beta ;\omega )$-Bloch space ${\mathcal{LB}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ is the space of holomorphic functions f such that

$$\underset{\mathbf{a},\mathbf{z}\in {\mathbb{B}}_n}{sup}\frac{{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta }}{{(1-{|{\phi }_{\mathbf{a}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|)}(ln\frac{2}{1-{|\mathbf{z}|}^2})|\mathrm{\nabla }f(\mathbf{z})|<\mathrm{\infty }.$$

Correspondingly, the little logarithmic $(\alpha ,\beta ;\omega )$-Bloch space ${\mathcal{LB}}_{\omega ;0}^{\alpha ,\beta }({\mathbb{B}}_n)$ is a subspace of ${\mathcal{LB}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ consisting of all functions f such that

$$\underset{|\mathbf{a}|\to 1^{-}}{lim}\underset{|\mathbf{z}|\to 1^{-}}{lim}\frac{{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta }}{{(1-{|{\phi }_{\mathbf{a}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|)}(ln\frac{2}{1-{|\mathbf{z}|}^2})|\mathrm{\nabla }f(\mathbf{z})|=0.$$

If $\omega (1-|\mathbf{z}|)=1$ and $\beta =0$, then we get the logarithmic α-Bloch space ${\mathcal{LB}}^{\alpha }({\mathbb{B}}_n)$ and the little logarithmic α-Bloch space ${\mathcal{LB}}_0^{\alpha }({\mathbb{B}}_n)$. If $\omega (1-|\mathbf{z}|)=1$, $\alpha =1$ and $\beta =0$, then we get the logarithmic Bloch space $\mathcal{LB}({\mathbb{B}}_n)$ and ${\mathcal{LB}}_0({\mathbb{B}}_n)$ (see [19]).

2 Holomorphic $(\alpha ,\beta ;\omega )$-Bloch space in the unit ball

In this section, we study the general $(\alpha ,\beta ;\omega )$-Bloch space ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ in the unit ball of ${\mathbb{C}}^n$ by giving some characterizations of $(\alpha ,\beta ;\omega )$-Bloch space, then we present several auxiliary results, which play important roles in the proofs of our main results.

Lemma 2.1 Let $\alpha ,\beta \in (0,\mathrm{\infty })$ and $f\in {\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$. Suppose that

$${\int }_0^1\frac{\omega (1-t|\mathbf{z}|)|\mathbf{z}|dt}{{(1-t^2{|\mathbf{z}|}^2)}^{\alpha +\beta }}<\mathrm{\infty }.$$

(1)

Then

$$|f(z)|\le |f(0)|+{\parallel f\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}.$$

Proof Let $\mathbf{z}\in {\mathbb{B}}_n$, $0\le t<1$ and $f\in {\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$. By the definition of ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ and $|\mathbf{z}|>\frac{1}{2}$, we have that

$$\begin{array}{rcl}|f(\mathbf{z})-f\left(\frac{\mathbf{z}}{2}\right)|& =& \left|{\int }_{\frac{1}{2}}^1〈\mathrm{\nabla }f(t\mathbf{z}),\mathbf{z}〉dt\right|\\ \le & |{\int }_{\frac{1}{2}}^1\mathfrak{R}f(t\mathbf{z})\frac{dt}{t}|\\ \le & b_{\alpha ,\beta ;\omega }(f){\int }_0^1\frac{{(1-{|{\phi }_{\mathbf{a}}(t\mathbf{z})|}^2)}^{\beta }\omega (1-t|\mathbf{z}|)}{{(1-t^2{|\mathbf{z}|}^2)}^{\alpha +\beta }}|\mathbf{z}|dt\\ \le & b_{\alpha ,\beta ;\omega }(f){\int }_0^1\frac{{(1-{|\mathbf{a}|}^2)}^{\beta }\omega (1-t|\mathbf{z}|)}{{|1-〈t\mathbf{z},\mathbf{a}〉|}^{2\beta }{(1-t^2{|\mathbf{z}|}^2)}^{\alpha }}|\mathbf{z}|dt.\end{array}$$

Since $(1-|\mathbf{a}|)\le |1-〈t\mathbf{z},\mathbf{a}〉|$ and $(1-t|\mathbf{z}|)\le |1-〈t\mathbf{z},\mathbf{a}〉|$, $\mathbf{a},\mathbf{z}\in {\mathbb{B}}_n$, we get

$$\begin{array}{rcl}|f(\mathbf{z})-f\left(\frac{\mathbf{z}}{2}\right)|& \le & b_{\alpha ,\beta ;\omega }(f){\int }_0^1\frac{{(1-{|\mathbf{a}|}^2)}^{\beta }\omega (1-|\mathbf{z}|)}{{(1-|\mathbf{a}|)}^{\beta }{(1-t|\mathbf{z}|)}^{\beta }{(1-t^2{|\mathbf{z}|}^2)}^{\alpha }}|\mathbf{z}|dt\\ \le & 4^{\beta }{b}_{\alpha ,\beta }(f){\int }_0^1\frac{\omega (1-t|\mathbf{z}|)|\mathbf{z}|dt}{{(1-t^2{|\mathbf{z}|}^2)}^{\alpha +\beta }},\end{array}$$

from which the result follows. □

Theorem 2.1 For each $0<\alpha ,\beta <\mathrm{\infty }$, $\gamma >-1$ and $f\in \mathcal{H}({\mathbb{B}}_n)$. Then the following conditions are equivalent:

(i) $f\in {\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$;

(ii) The function $\frac{{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta }}{{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}|\mathfrak{R}f(\mathbf{z})|$ is bounded in ${\mathbb{B}}_n$;

(iii) There exists a function $g\in L^{\mathrm{\infty }}({\mathbb{B}}_n)$ such that

$$f(\mathbf{z})={(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|){\int }_{{\mathbb{B}}_n}\frac{g(\mathbf{w})d{\nu }_{\gamma }(\mathbf{w})}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+\alpha +\beta +\gamma }},\mathbf{z}\in {\mathbb{B}}_n.$$

Proof By the Cauchy-Schwarz inequality in ${\mathbb{C}}^n$, we have

$$|\mathfrak{R}f(\mathbf{z})|\le |\mathbf{z}||\mathrm{\nabla }f(\mathbf{z})|\le |\mathrm{\nabla }f(\mathbf{z})|.$$

This proves that (i) ⇒ (ii).

If (ii) holds, then the function

$$g(\mathbf{z})=\frac{c_{\alpha +\beta +\gamma }}{c_{\gamma }}\frac{{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta }}{{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}(f(\mathbf{z})+\frac{\mathfrak{R}f(\mathbf{z})}{n+\alpha +\beta +\gamma })$$

is bounded in ${\mathbb{B}}_n$. For $\mathbf{z}\in {\mathbb{B}}_n$ consider the holomorphic function

$$\begin{array}{rcl}F(\mathbf{z})& =& {\int }_{{\mathbb{B}}_n}\frac{g(\mathbf{w}){(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)d{\nu }_{\gamma }(\mathbf{w})}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+\alpha +\beta +\gamma }}\\ =& {\int }_{{\mathbb{B}}_n}\frac{\omega (1-|\mathbf{w}|)}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+\alpha +\beta +\gamma }}(f(\mathbf{w})+\frac{\mathfrak{R}f(\mathbf{w})}{n+\alpha +\beta +\gamma })d{\nu }_{\alpha +\beta +\gamma }(\mathbf{w}).\end{array}$$

As in the proof of Theorem 7.1 in [5], we have $F=f$.

This shows that (ii) implies (iii). That (iii) implies (i) follows from differentiating under the integral sign and then applying Theorem 1.12 in [5]. □

Theorem 2.2 For each $\alpha >0$, $\beta \ge 0$, $\alpha +\beta >0$ and $s=\alpha +\beta -1$. If $s>-1$, then the Banach dual of $A^1({\mathbb{B}}_n)$ can be identified with ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ (with equivalent norms) under the following integral pairing:

$${〈f,g〉}_s={\int }_{{\mathbb{B}}_n}f(\mathbf{z})\overline{g(\mathbf{z})}d{\nu }_s(\mathbf{z}),f\in A^1({\mathbb{B}}_n),g\in {\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n).$$

(2)

Proof It is easy to see that $1-(\alpha +\beta )+s>-1$. If $g\in {\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$, then by Theorem 2.1, there exists a function $h\in L^{\mathrm{\infty }}({\mathbb{B}}_n)$ such that

$$g(\mathbf{z})=\frac{1}{{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}{\int }_{{\mathbb{B}}_n}\frac{h(\mathbf{w})d{\nu }_{1-(\alpha +\beta )+s}(\mathbf{w})}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+1+s}},\mathbf{z},\mathbf{w}\in {\mathbb{B}}_n,$$

and ${\parallel h\parallel }_{\mathrm{\infty }}\le C{\parallel g\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}$, where C is a positive constant independent of g. By Fubini’s theorem,

$$\begin{array}{rcl}{〈f,g〉}_s& =& {\int }_{{\mathbb{B}}_n}f(\mathbf{z})\overline{h(\mathbf{z})}(1-{|\mathbf{z}|}^2)d{\nu }_{1-(\alpha +\beta )+s}(\mathbf{z})\\ =& c_{1-(\alpha +\beta )+s}{\int }_{{\mathbb{B}}_n}f(\mathbf{z})\overline{h(\mathbf{z})}d\nu (\mathbf{z}).\end{array}$$

Applying Lemma 2.15 in [5] for all $f\in A^1({\mathbb{B}}_n)$, we have

$${\int }_{{\mathbb{B}}_n}|f(\mathbf{z})|d\nu (\mathbf{z})\le {\parallel f\parallel }_{A^1({\mathbb{B}}_n)}.$$

Combining this, we see that

$$\left|{〈f,g〉}_s\right|\le {\parallel h\parallel }_{\mathrm{\infty }}{\parallel f\parallel }_{A^1({\mathbb{B}}_n)}\le C{\parallel g\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}{\parallel f\parallel }_{A^1({\mathbb{B}}_n)}.$$

Conversely, if F is a bounded linear functional on $A^1({\mathbb{B}}_n)$ and $f\in A^1({\mathbb{B}}_n)$, then

$$f_r(\mathbf{z})=\frac{1}{{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}{\int }_{{\mathbb{B}}_n}\frac{f_r(\mathbf{w})d{\nu }_s(\mathbf{w})}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+1+s}}\text{for }0<r<1.$$

It is easy to verify (using the homogeneous expansion of the kernel function) that

$$F(f_r)={\int }_{{\mathbb{B}}_n}{f}_r(\mathbf{w})F_{\mathbf{z}}\left[\frac{1}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+1+s}{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}\right]d{\nu }_s(\mathbf{w}).$$

Define a function g on ${\mathbb{B}}_n$ by

$$\overline{g(\mathbf{w})}=F_{\mathbf{z}}\left[\frac{1}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+1+s}{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}\right].$$

Then

$$F(f_r)={\int }_{{\mathbb{B}}_n}{f}_r(\mathbf{w})\overline{g(\mathbf{w})}d{\nu }_s(\mathbf{w})={〈f,g〉}_s.$$

It remains for us to show that $g\in {\mathcal{B}}_{\omega }^{\alpha ,\beta }$.

We interchange differentiation and the application of F, which can be justified by using the homogeneous expansion of the kernel. The result is

$$\overline{\mathfrak{R}g(\mathbf{w})}=(n+1+s)F_{\mathbf{z}}\left[\frac{1}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+1+s}{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}\right].$$

Since F is bounded on $A^1({\mathbb{B}}_n)$, we have

$$|\mathfrak{R}g(\mathbf{w})|\le \frac{C\parallel F\parallel }{{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}{\int }_{{\mathbb{B}}_n}\frac{d\nu (\mathbf{w})}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{n+2+s}}.$$

An application of Theorem 1.1 for $s+1=\alpha +\beta$ then shows that

$$|\mathfrak{R}g(\mathbf{w})|\le \frac{C\parallel F\parallel }{{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta }{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}.$$

This shows that $g\in {\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ and completes the proof of the theorem. □

Lemma 2.2 If $n>1$, $\alpha +\beta >\frac{1}{2}$, then $f\in {\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ if and only if the function

$$\frac{{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}}{{(1-{|{\phi }_{\mathbf{w}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|)}|\tilde{\mathrm{\nabla }}f(\mathbf{z})|$$

is bounded in ${\mathbb{B}}_n$.

Proof Recall from Lemma 2.14 in [5] that

$$(1-{|\mathbf{z}|}^2)|\mathrm{\nabla }f(\mathbf{z})|\le |\tilde{\mathrm{\nabla }}f(\mathbf{z})|,\mathbf{z}\in {\mathbb{B}}_n.$$

So, the boundedness of

$$\frac{{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}}{{(1-{|{\phi }_{\mathbf{w}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|)}|\tilde{\mathrm{\nabla }}f(\mathbf{z})|$$

implies that of

$$\frac{{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta }}{{(1-{|{\phi }_{\mathbf{w}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|)}|\mathrm{\nabla }f(\mathbf{z})|.$$

On the other hand, if $f\in {\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$, then by Theorem 2.1,

$$f(\mathbf{z})={(1-{|{\phi }_{\mathbf{w}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|){\int }_{{\mathbb{B}}_n}\frac{g(\mathbf{w})d\nu (\mathbf{w})}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+\alpha +\beta }},\mathbf{z}\in {\mathbb{B}}_n,$$

where g is a function in $L^{\mathrm{\infty }}({\mathbb{B}}_n)$. Now we let $f(\mathbf{z})=h(\mathbf{z})u(\mathbf{z})$, where $h(\mathbf{z})={(1-{|{\phi }_{\mathbf{w}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|)$ and

$$u(\mathbf{z})={\int }_{{\mathbb{B}}_n}\frac{g(\mathbf{w})d\nu (\mathbf{w})}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+\alpha +\beta }}.$$

An application of Lemma 1.1 gives

$$|\tilde{\mathrm{\nabla }}u(\mathbf{z})|\le |n+\alpha +\beta |\sqrt{2}{(1-{|\mathbf{z}|}^2)}^{\frac{1}{2}}{\int }_{{\mathbb{B}}_n}\frac{g(\mathbf{w})d\nu (\mathbf{w})}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{n+\alpha +\beta +\frac{1}{2}}},\mathrm{\forall }\mathbf{z}\in {\mathbb{B}}_n.$$

Since $g(\mathbf{z})$ is bounded, by Theorem 1.1 we have

$${\int }_{{\mathbb{B}}_n}\frac{g(\mathbf{w})d\nu (\mathbf{w})}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{n+\alpha +\beta +\frac{1}{2}}}\approx {(1-{|\mathbf{z}|}^2)}^{\frac{1}{2}-(\alpha +\beta )}.$$

So,

$$|\tilde{\mathrm{\nabla }}u(\mathbf{z})|\le C{(1-{|\mathbf{z}|}^2)}^{1-(\alpha +\beta )}.$$

It is easy to check that $\tilde{\mathrm{\nabla }}h(\mathbf{z})=\mathrm{\nabla }(h\circ {\phi }_{\mathbf{z}})(0)=0$.

Using the product rule, we have

$$|\tilde{\mathrm{\nabla }}f(\mathbf{z})|\le |\tilde{\mathrm{\nabla }}h(\mathbf{z})||u(\mathbf{z})|+|h(\mathbf{z})||\tilde{\mathrm{\nabla }}u(\mathbf{z})|\le |\tilde{\mathrm{\nabla }}h(\mathbf{z})||u(\mathbf{z})|$$

and we have

$$\begin{array}{c}|\tilde{\mathrm{\nabla }}f(\mathbf{z})|\le |n+\alpha +\beta |\sqrt{2}{(1-{|\mathbf{z}|}^2)}^{\frac{1}{2}}{(1-{|{\phi }_{\mathbf{a}}(\mathbf{z})|}^2)}^{\beta }{\int }_{{\mathbb{B}}_n}\frac{g(\mathbf{w})d\nu (\mathbf{w})}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{n+\alpha +\beta +\frac{1}{2}}}\hfill \\ \text{for all }\mathbf{z}\in {\mathbb{B}}_n.\hfill \end{array}$$

Hence,

$$\frac{{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}}{{(1-{|{\phi }_{\mathbf{w}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|)}|\tilde{\mathrm{\nabla }}f(\mathbf{z})|$$

is bounded in ${\mathbb{B}}_n$. This completes the proof. □

Lemma 2.3 Let $0<\alpha +\beta \le 2$. Let λ be any real number satisfying the following properties:

• $0\le \lambda \le \alpha +\beta$ if $0<\alpha +\beta <1$;

• $0<\lambda <1$ if $\alpha +\beta =1$;

• $\alpha +\beta -1\le \lambda \le 1$ if $1<\alpha +\beta \le 2$.

Then, for all $\mathbf{z},\mathbf{w}\in {\mathbb{B}}_n$ a holomorphic function $f\in {\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ if and only if

$$\underset{\mathbf{z}\ne \mathbf{w}}{sup}\frac{{(1-{|\mathbf{z}|}^2)}^{\lambda }{(1-|\mathbf{w}|)}^{\alpha +\beta -\lambda }}{{(1-{|{\phi }_{\mathbf{w}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|)}\frac{|f(\mathbf{z})-f(\mathbf{w})|}{|\mathbf{z}-\mathbf{w}|}<\mathrm{\infty }.$$

(3)

Proof Let $f\in {\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$. By a similar proof to the one for Theorem 3.1 in [20], we have

$$|f(\mathbf{z})-f(\mathbf{w})|=\sqrt{n}|\mathbf{z}-\mathbf{w}|{\int }_0^1\left|\mathrm{\nabla }f(t\mathbf{z}-(1-t)\mathbf{w})\right|dt$$

for any $\mathbf{z},\mathbf{w}\in {\mathbb{B}}_n$ with $\mathbf{z}\ne \mathbf{w}$. We know that

$${\parallel f\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}\approx \underset{\mathbf{w},\mathbf{z}\in {\mathbb{B}}_n}{sup}\frac{{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta }}{{(1-{|{\phi }_{\mathbf{w}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|)}|\mathrm{\nabla }f(\mathbf{z})|.$$

Thus, there is a constant $C>0$ such that

$$\frac{|f(\mathbf{z})-f(\mathbf{w})|}{|\mathbf{z}-\mathbf{w}|}\le C{\parallel f\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}{\int }_0^1\frac{{(1-{|{\phi }_{\mathbf{a}}(t\mathbf{z}-(1-t)\mathbf{a})|}^2)}^{\beta }\omega (1-|t\mathbf{z}-(1-t)\mathbf{w}|)}{{(1-{|t\mathbf{z}-(1-t)\mathbf{w}|}^2)}^{\alpha +\beta }}dt.$$

Since $(1-|\mathbf{z}|)\le |1-〈\mathbf{w},\mathbf{z}〉|$ and

$$1-{|t\mathbf{z}+(1-t)\mathbf{w}|}^2\ge 1-|t\mathbf{z}+(1-t)\mathbf{w}|\ge 1-|\mathbf{w}|+(|\mathbf{w}|-|\mathbf{z}|)t,$$

we get

$$\begin{array}{rcl}{(1-{\left|{\phi }_{\mathbf{w}}(t\mathbf{z}-(1-t)\mathbf{w})\right|}^2)}^{\beta }& =& \frac{{(1-{|\mathbf{w}|}^2)}^{\beta }{(1-{|t\mathbf{z}-(1-t)\mathbf{w}|}^2)}^{\beta }}{{|1-〈t\mathbf{z}-(1-t)\mathbf{w},\mathbf{w}〉|}^{2\beta }}\\ \le & \frac{{(1-{|\mathbf{w}|}^2)}^{\beta }{(1-{|t\mathbf{z}-(1-t)\mathbf{w}|}^2)}^{\beta }}{{(1-|\mathbf{w}|)}^{2\beta }}\\ \le & \frac{{(1+|\mathbf{w}|)}^{\beta }{(1-{|t\mathbf{z}-(1-t)\mathbf{w}|}^2)}^{\beta }}{{(1-|\mathbf{w}|)}^{\beta }}\\ \le & \frac{{(1-{|t\mathbf{z}-(1-t)\mathbf{w}|}^2)}^{\beta }}{{(1-|\mathbf{w}|)}^{\beta }}.\end{array}$$

Thus

$$\frac{|f(\mathbf{z})-f(\mathbf{w})|}{|\mathbf{z}-\mathbf{w}|}\le C{\parallel f\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}{\int }_0^1\frac{1}{{(1-|\mathbf{w}|+(|\mathbf{w}|-|\mathbf{z}|)t)}^{\alpha }{(1-|\mathbf{w}|)}^{\beta }}dt.$$

(4)

If $|\mathbf{z}|=|\mathbf{a}|$, then

$$\frac{|f(\mathbf{z})-f(\mathbf{w})|}{|\mathbf{z}-\mathbf{w}|}\le C{\parallel f\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}{\int }_0^1\frac{1}{{(1-|\mathbf{w}|)}^{\alpha +\beta }}dt\le \frac{C{\parallel f\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}}{{(1-{|\mathbf{z}|}^2)}^{\lambda }{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta -\lambda }}.$$

(5)

Now suppose $|\mathbf{z}|\ne |\mathbf{w}|$ as in [21], there is a constant $C>0$ such that this integral in (4) is dominated by

$$\frac{C}{{(1-{|\mathbf{z}|}^2)}^{\lambda }{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta -\lambda }}.$$

Combining with (5), we get that whenever $\mathbf{z}\ne \mathbf{w}$,

$$\frac{|f(\mathbf{z})-f(\mathbf{w})|}{|\mathbf{z}-\mathbf{w}|}\le \frac{C{\parallel f\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}}{{(1-{|\mathbf{z}|}^2)}^{\lambda }{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta -\lambda }}.$$

This proves the necessity. The proof of the sufficiency condition is much akin to the corresponding one in [21], so the proof is omitted. □

Proposition 2.1 Suppose $f\in {\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ and $1\le \alpha +\beta \le 2$. Let λ be any real number satisfying:

• $0<\lambda <1$ if $\alpha +\beta =1$;

• $\alpha +\beta -1\le \lambda \le 1$ if $1<\alpha +\beta \le 2$.

Then

$$\begin{array}{r}\underset{\mathbf{z},\mathbf{w}\in {\mathbb{B}}_n}{sup}\frac{{(1-{|\mathbf{z}|}^2)}^{2\alpha +2\beta -\lambda -1}{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta -\lambda }|f(\mathbf{z})-f(\mathbf{w})|}{{(1-{|{\phi }_{\mathbf{w}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|){|1-〈\mathbf{w},\mathbf{z}〉|}^{2(\alpha +\beta )-(2\lambda +1)}|\mathbf{z}-P_{\mathbf{z}}(\mathbf{w})-S_{\mathbf{z}}{Q}_{\mathbf{z}}(\mathbf{w})|}\\ \le C{\parallel f\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}.\end{array}$$

(6)

Proof Let $\mathbf{z}=0$ in (3), then we have

$${(1-{|\mathbf{w}|}^2)}^{\alpha +\beta -\lambda }\frac{|f(\mathbf{0})-f(\mathbf{w})|}{|\mathbf{w}|}\le C{b}_{\alpha ,\beta ;\omega }(f)({\mathbb{B}}_n),\mathbf{w}\in {\mathbb{B}}_n\mathrm{\setminus }\{\mathbf{0}\}.$$

Now, replacing f by $f\circ {\phi }_{\mathbf{w}}$, we get

$${(1-{|\mathbf{u}|}^2)}^{\alpha +\beta -\lambda }\frac{|f\circ {\phi }_{\mathbf{w}}(\mathbf{0})-f\circ {\phi }_{\mathbf{w}}(\mathbf{u})|}{|\mathbf{u}|}\le C{b}_{\alpha ,\beta ;\omega }(f\circ {\phi }_{\mathbf{w}})({\mathbb{B}}_n),\mathbf{u}\in {\mathbb{B}}_n\mathrm{\setminus }\{\mathbf{0}\}.$$

(7)

Since

$$|\tilde{\mathrm{\nabla }}(f\circ {\phi }_{\mathbf{w}})(\mathbf{z})|=\left|\tilde{\mathrm{\nabla }}f({\phi }_{\mathbf{w}}(\mathbf{z}))\right|,$$

by Lemma 2.2, we obtain that

$$\begin{array}{rcl}{b}_{\alpha ,\beta ;\omega }(f\circ {\phi }_{\mathbf{w}})({\mathbb{B}}_n)& \approx & \underset{\mathbf{z},\mathbf{w}\in {\mathbb{B}}_n}{sup}\frac{{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta -1}}{{(1-{|{\phi }_{\mathbf{w}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|)}|\tilde{\mathrm{\nabla }}(f\circ {\phi }_{\mathbf{w}})(\mathbf{z})|\\ =& \underset{\mathbf{z},\mathbf{w}\in {\mathbb{B}}_n}{sup}\frac{{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta -1}}{{(1-{|{\phi }_{\mathbf{w}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|)}\left|\tilde{\mathrm{\nabla }}f({\phi }_{\mathbf{w}}(\mathbf{z}))\right|\\ =& \underset{\mathbf{z},\mathbf{w}\in {\mathbb{B}}_n}{sup}\frac{{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta -1}{(1-{|{\phi }_{\mathbf{w}}(\mathbf{z})|}^2)}^{\alpha +\beta -1}}{{(1-{|{\phi }_{\mathbf{w}}(\mathbf{z})|}^2)}^{\alpha +\beta -1}{(1-{|{\phi }_{\mathbf{w}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|)}\left|\tilde{\mathrm{\nabla }}f({\phi }_{\mathbf{w}}(\mathbf{z}))\right|.\end{array}$$

Then

$$\begin{array}{rcl}{b}_{\alpha ,\beta ;\omega }(f\circ {\phi }_{\mathbf{z}})({\mathbb{B}}_n)& \le & C{\parallel f\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}\frac{{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta -1}}{{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\alpha +\beta -1}\omega (1-|\mathbf{w}|)}\\ \le & \frac{C{\parallel f\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}}{{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}}.\end{array}$$

Letting $u={\phi }_{\mathbf{w}}(\mathbf{z})$ and $\mathbf{w}\ne \mathbf{z}$ in (7), we obtain

$$\frac{|f(\mathbf{z})-f(\mathbf{w})|}{|{\phi }_{\mathbf{w}}(\mathbf{z})|\omega (1-|\mathbf{z}|){(1-{|{\phi }_{\mathbf{w}}(\mathbf{z})|}^2)}^{\alpha +\beta -\lambda }}\le \frac{C{\parallel f\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}}{{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}}.$$

Since

$$1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2=\frac{(1-{|\mathbf{z}|}^2)(1-{|\mathbf{w}|}^2)}{{|1-〈\mathbf{w},\mathbf{z}〉|}^2}$$

and

$${\phi }_{\mathbf{z}}(\mathbf{w})=\frac{\mathbf{z}-P_{\mathbf{z}}(\mathbf{w})-S_{\mathbf{z}}{Q}_{\mathbf{z}}(\mathbf{w})}{1-〈\mathbf{w},\mathbf{z}〉}.$$

Consequently,

$$\begin{array}{c}\frac{{(1-{|\mathbf{z}|}^2)}^{2\alpha +2\beta -\lambda -1}{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta -\lambda }|f(\mathbf{z})-f(\mathbf{w})|}{\omega (1-|\mathbf{z}|){(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\alpha +\beta -\lambda }{|1-〈\mathbf{w},\mathbf{z}〉|}^{2(\alpha +\beta )-(2\lambda +1)}|\mathbf{z}-P_{\mathbf{z}}(\mathbf{w})-S_{\mathbf{z}}{Q}_{\mathbf{z}}(\mathbf{w})|}\hfill \\ \le C{\parallel f\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}.\hfill \end{array}$$

 □

3 Boundedness of general Toeplitz operators

In this section, we study the boundedness of general Toeplitz operators acting on the weighted Bloch-type spaces ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ in the unit ball of ${\mathbb{C}}^n$.

Theorem 3.1 Let μ be a positive Borel measure on ${\mathbb{B}}_n$. Then we have

(1) if $\alpha +\beta =1$, then $T_{\mu }^{\alpha ,\beta ;\omega }$ is bounded on ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ if and only if $P_{\alpha +\beta -1;\omega }(\mu )\in {\mathcal{LB}}_{\omega }({\mathbb{B}}_n)$ and μ is a Carleson measure;

(2) if $\alpha =\beta =1$, then $T_{\mu }^{\alpha ,\beta ;\omega }$ is bounded on ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ if and only if $P_{\alpha +\beta -1;\omega }(\mu )\in {\mathcal{B}}_{\omega }({\mathbb{B}}_n)\cap {\mathcal{LB}}_{\omega }^2({\mathbb{B}}_n)$ and μ is a Carleson measure;

(3) if $\alpha >1$, $\beta >1$, then $T_{\mu }^{\alpha ,\beta ;\omega }$ is bounded on ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ if and only if $P_{\alpha +\beta -1;\omega }(\mu )\in {\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ and μ is a Carleson measure.

Proof Since the Banach dual of $A^1({\mathbb{B}}_n)$ is ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ under the pairing (2), to prove the boundedness of $T_{\mu }^{\alpha ,\beta ;\omega }$, it suffices to show that

$$\left|{〈f,T_{\mu }^{\alpha ,\beta ;\omega }(g)〉}_{\alpha }\right|\le C{\parallel f\parallel }_{A^1({\mathbb{B}}_n)}{\parallel g\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}$$

for all $f\in A^1({\mathbb{B}}_n)$ and $g\in {\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$, where C is a positive constant that does not depend on f or g.

For $s=\alpha +\beta -1$, by Fubini’s theorem, we get

$$\begin{array}{rcl}{〈f,T_{\mu }^{\alpha ,\beta ;\omega }g〉}_s& =& {\int }_{{\mathbb{B}}_n}f(\mathbf{z})\overline{T_{\mu }^{\alpha ,\beta ;\omega }g(\mathbf{z})}d{\nu }_s(\mathbf{z})\\ =& c_{\alpha +\beta -1}{\int }_{{\mathbb{B}}_n}f(\mathbf{z})\overline{g(\mathbf{z})}{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}d\mu (\mathbf{z}).\end{array}$$

Using the operator $P_{\alpha +\beta ;\omega }$, we have

$$\begin{array}{rcl}{〈f,T_{\mu }^{\alpha ,\beta ;\omega }g〉}_s& =& c_{\alpha +\beta -1}{\int }_{{\mathbb{B}}_n}(I_{\mathbf{z},\mathbf{w};\mathit{\omega }}-P_{\alpha +\beta ;\omega })(f\overline{g})(\mathbf{z}){(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}d\mu (\mathbf{z})\\ +c_{\alpha +\beta -1}{\int }_{{\mathbb{B}}_n}{P}_{\alpha +\beta ;\omega }(f\overline{g})(\mathbf{z}){(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}d\mu (\mathbf{z})\\ =& I_1+I_2,\end{array}$$

where $I_{\mathbf{z},\mathbf{w};\mathit{\omega }}=\frac{I}{{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}$, and I is the identity operator. Also,

$$\begin{array}{c}(I_{\mathbf{z},\mathbf{w};\mathit{\omega }}-P_{\alpha +\beta ;\omega })(f\overline{g})(\mathbf{z})\hfill \\ =\frac{f(\mathbf{z})\overline{g(\mathbf{z})}}{{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}\hfill \\ -\frac{c_{\alpha +\beta }}{{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}{\int }_{{\mathbb{B}}_n}\frac{f(\mathbf{w})\overline{g(\mathbf{w})}{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta }}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+\alpha +\beta +1}}d\nu (\mathbf{w})\hfill \\ =\frac{c_{\alpha +\beta }}{{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}\hfill \\ \times {\int }_{{\mathbb{B}}_n}\frac{(\overline{g(\mathbf{z})}-\overline{g(\mathbf{w})})f(\mathbf{w}){(1-{|\mathbf{w}|}^2)}^{\alpha +\beta }}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+\alpha +\beta +1}{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}d\nu (\mathbf{z}).\hfill \end{array}$$

By Proposition 2.1, we have

$$\begin{array}{rcl}|I_1|& =& c_{\alpha +\beta -1}|{\int }_{{\mathbb{B}}_n}(I_{\mathbf{z},\mathbf{w};\mathit{\omega }}-P_{\alpha +\beta ;\omega })(f\overline{g})(\mathbf{z}){(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}d\mu (\mathbf{z})|\\ =& c_{\alpha +\beta -1}{c}_{\alpha +\beta }|{\int }_{{\mathbb{B}}_n}{\int }_{{\mathbb{B}}_n}\frac{(\overline{g(\mathbf{z})}-\overline{g(\mathbf{w})})f(\mathbf{w}){(1-{|\mathbf{w}|}^2)}^{\alpha +\beta }{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+\alpha +\beta +1}{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}d\nu (\mathbf{w})d\mu (\mathbf{z})|\\ =& c_{\alpha +\beta -1}{c}_{\alpha +\beta }|{\int }_{{\mathbb{B}}_n}f(\mathbf{w}){(1-{|\mathbf{w}|}^2)}^{\alpha +\beta }\\ \times {\int }_{{\mathbb{B}}_n}\frac{(\overline{g(\mathbf{z})}-\overline{g(\mathbf{w})}){(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+\alpha +\beta +1}{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}d\mu (\mathbf{z})d\nu (\mathbf{w})|\\ \le & c_{\alpha +\beta -1}{c}_{\alpha +\beta }{\int }_{{\mathbb{B}}_n}|f(\mathbf{w})|{(1-{|\mathbf{w}|}^2)}^{\lambda }\\ \times {\int }_{{\mathbb{B}}_n}\frac{{(1-{|\mathbf{z}|}^2)}^{2(\alpha +\beta )-\lambda -1}{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta -\lambda }|f(\mathbf{z})-f(\mathbf{w})|}{{|1-〈\mathbf{w},\mathbf{z}〉|}^{2(\alpha +\beta )-(2\lambda +1)}|\mathbf{z}-P_{\mathbf{z}}(\mathbf{w})-S_{\mathbf{z}}{Q}_{\mathbf{z}}(\mathbf{w})|}\\ \times \frac{|\mathbf{z}-P_{\mathbf{z}}(\mathbf{w})-S_{\mathbf{z}}{Q}_{\mathbf{z}}(\mathbf{w})|{(1-{|\mathbf{z}|}^2)}^{\lambda -(\alpha +\beta )}}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{n-(\alpha +\beta )+2\lambda +2}{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}d\mu (\mathbf{z})d\nu (\mathbf{w})\\ \le & C{\int }_{{\mathbb{B}}_n}{\parallel g\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}|f(\mathbf{w})|{(1-{|\mathbf{w}|}^2)}^{\lambda }{\int }_{{\mathbb{B}}_n}\frac{{(1-{|\mathbf{z}|}^2)}^{\lambda -(\alpha +\beta )}}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{n-(\alpha +\beta )+2\lambda +1}}d\mu (\mathbf{z})d\nu (\mathbf{w}).\end{array}$$

Since μ is a Carleson measure, taking $\lambda -(\alpha +\beta )>-1$, then as in [9] or in [[22], Proposition 1.4.10], for fixed $r>0$, we get

$$\begin{array}{c}{\int }_{{\mathbb{B}}_n}\frac{{(1-{|\mathbf{z}|}^2)}^{\lambda -(\alpha +\beta )}}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{n-(\alpha +\beta )+2\lambda +1}}d\mu (\mathbf{z})\hfill \\ \le \sum _{j=1}^{\mathrm{\infty }}\frac{\mu (B({\mathbf{z}}^{(j)},r))}{\nu (B({\mathbf{z}}^{(j)},r))}{\int }_{B({\mathbf{z}}^{(j)},r)}\frac{{(1-{|\mathbf{z}|}^2)}^{\lambda -(\alpha +\beta )}}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{n-(\alpha +\beta )+2\lambda +1}}d\nu (\mathbf{z})\le C.\hfill \end{array}$$

Therefore,

$$|I_1|\le C{\parallel f\parallel }_{A^1({\mathbb{B}}_n)}{\parallel g\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}.$$

Next considering $I_2$, we have

$$\begin{array}{rcl}|I_2|& =& c_{\alpha +\beta -1}|{\int }_{{\mathbb{B}}_n}{P}_{\alpha +\beta ;\omega }(f\overline{g})(\mathbf{z}){(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}d\mu (\mathbf{z})|\\ =& c_{\alpha +\beta -1}{c}_{\alpha +\beta }|{\int }_{{\mathbb{B}}_n}{\int }_{{\mathbb{B}}_n}\frac{f(\mathbf{w})\overline{g(\mathbf{w})}{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta }{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+\alpha +\beta +1}{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}d\nu (\mathbf{w})d\mu (\mathbf{z})|\\ \le & c_{\alpha +\beta }{\int }_{{\mathbb{B}}_n}|f(\mathbf{w})|{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta }|g(\mathbf{w})|\\ \times \left(\frac{c_{\alpha +\beta -1}}{{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}{\int }_{{\mathbb{B}}_n}\frac{{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}d\mu (\mathbf{z})}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{n+\alpha +\beta +1}}\right)d\nu (\mathbf{w})\\ \le & C{\int }_{{\mathbb{B}}_n}{\parallel f\parallel }_{A^1({\mathbb{B}}_n)}{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta }|g(\mathbf{w})|Q_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})d\nu (\mathbf{w}),\end{array}$$

where

$$Q_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})=\frac{c_{\alpha +\beta -1}}{{(1-{|{\phi }_{\mathbf{z}}(\mathbf{w})|}^2)}^{\beta }\omega (1-|\mathbf{w}|)}{\int }_{{\mathbb{B}}_n}\frac{{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}d\mu (\mathbf{z})}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{n+\alpha +\beta +1}}.$$

As in [9], by simple calculation, we have

$$Q_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})=P_{\alpha +\beta -1;\omega }(\mu )(\mathbf{w})+\frac{1}{n+\alpha +\beta }\mathfrak{R}{P}_{\alpha +\beta -1;\omega }(\mu )(\mathbf{w}).$$

(8)

It is easy to see that

(1) if $\alpha +\beta =1$ and $P_{\alpha +\alpha -1;\omega }(\mu )\in {\mathcal{LB}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$, then

$$(1-{|\mathbf{w}|}^2)Q_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})(ln\frac{2}{1-{|\mathbf{w}|}^2})\in L^{\mathrm{\infty }}({\mathbb{B}}_n);$$

(2) if $\alpha =\beta =1$, $P_{\alpha +\alpha -1;\omega }(\mu )\in {\mathcal{B}}_{\omega }({\mathbb{B}}_n)\cap \mathcal{LB}-{\omega }^2({\mathbb{B}}_n)$, then

$$(1-{|\mathbf{w}|}^2)Q_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})\in L^{\mathrm{\infty }}({\mathbb{B}}_n)$$

and

$${(1-{|\mathbf{w}|}^2)}^2{Q}_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})(ln\frac{2}{1-{|\mathbf{w}|}^2})\in L^{\mathrm{\infty }}({\mathbb{B}}_n);$$

(3) if $\alpha >1$, $\beta >1$, and $P_{\alpha +\alpha -1;\omega }(\mu )\in \mathcal{B}-{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$, then

$${(1-{|\mathbf{w}|}^2)}^{\alpha +\beta }{Q}_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})\in L^{\mathrm{\infty }}({\mathbb{B}}_n).$$

This implies that $|I_2|\le C{\parallel f\parallel }_{A^1({\mathbb{B}}_n)}{\parallel g\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}$. Hence, $T_{\mu }^{\alpha ,\beta ;\omega }$ is a bounded operator on ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ with $\alpha >0$, $\beta \ge 0$.

Conversely, suppose that $T_{\mu }^{\alpha ,\beta ;\omega }$ is a bounded operator on ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$. Take

$$f_{\mathbf{w}}(\mathbf{z})=\frac{{(1-{|\mathbf{w}|}^2)}^t}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+t+1}}\text{for }t>0.$$

It is clear that ${\parallel f_{\mathbf{w}}\parallel }_{A^1({\mathbb{B}}_n)}\le C$. On the other hand, take

$$g_{\mathbf{w}}(\mathbf{z})=\frac{{(1-{|\mathbf{w}|}^2)}^{n+2+t-(\alpha +\beta )}}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+t+1}};{\phi }_{\mathbf{w}}(\mathbf{z})\equiv 1\text{and}\omega (1-|\mathbf{z}|)\equiv 1\text{for }t>0.$$

Then, we have ${\parallel g_{\mathbf{w}}\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}\le C$. Therefore

$$\begin{array}{c}\left|{〈f,T_{\mu }^{\alpha }g〉}_s\right|=c_{\alpha +\beta -1}{(1-{|\mathbf{w}|}^2)}^{n+2+2t-(\alpha +\beta )}{\int }_{{\mathbb{B}}_n}\frac{{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}d\mu (\mathbf{z})}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{2(n+t+1)}}\hfill \\ \le C\parallel T_{\mu }^{\alpha ,\beta ;\omega }\parallel {\parallel f_{\mathbf{w}}\parallel }_{A^1({\mathbb{B}}_n)}{\parallel g_{\mathbf{w}}\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}\le C.\hfill \end{array}$$

Thus,

$${(1-{|\mathbf{w}|}^2)}^{n+2+2t-(\alpha +\beta )}{\int }_{B(\mathbf{w},r)}\frac{{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}d\mu (\mathbf{z})}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{2(n+t+1)}}\le C$$

for every $\mathbf{w}\in {\mathbb{B}}_n$. This implies that

$$\underset{\mathbf{w}\in {\mathbb{B}}_n}{sup}\frac{\mu (B(\mathbf{w},r))}{\nu (B(\mathbf{w},r))}<\mathrm{\infty }.$$

Hence μ is a Carleson measure on ${\mathbb{B}}_n$.

From the proof of the sufficient condition, we find that there exists a constant C such that

$$\begin{array}{rcl}|I_2|& =& c_{\alpha +\beta }|{\int }_{{\mathbb{B}}_n}f(\mathbf{w}){(1-{|\mathbf{w}|}^2)}^{\alpha +\beta }\overline{g(\mathbf{w})Q_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})}d\nu (\mathbf{w})|\\ \le & C{\parallel f\parallel }_{A^1({\mathbb{B}}_n)}{\parallel g\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}.\end{array}$$

This implies that

$$|g(\mathbf{w})Q_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})|{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta }\le C{\parallel g\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}.$$

If $\alpha +\beta =1$, we have

$$|g(\mathbf{w})Q_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})|(1-{|\mathbf{w}|}^2)\le C{\parallel g\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}.$$

Take $g_{\mathbf{w}}(\mathbf{z})=ln\frac{2}{1-〈\mathbf{z},\mathbf{w}〉}$; ${\phi }_{\mathbf{w}}(\mathbf{z})\equiv 1$ and $\omega (1-|\mathbf{z}|)\equiv 1$. It is clear that ${\parallel g_{\mathbf{w}}\parallel }_{{\mathcal{LB}}_{\omega }({\mathbb{B}}_n)}\le C$. Taking $\mathbf{z}=\mathbf{w}$, then

$$|Q_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})|(1-{|\mathbf{w}|}^2)(ln\frac{2}{1-{|\mathbf{w}|}^2})\le C.$$

From (8) we have $P_{\alpha +\beta -1}(\mu )\in {\mathcal{LB}}_{\omega }({\mathbb{B}}_n)$. Let $\alpha =\beta =1$, we have

$$|g(\mathbf{w})Q_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})|{(1-{|\mathbf{w}|}^2)}^2\le C{\parallel g\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}.$$

Take $g_{\mathbf{w}}(\mathbf{z})=\frac{1}{1-〈\mathbf{z},\mathbf{w}〉}+ln\frac{2}{1-〈\mathbf{z},\mathbf{w}〉}$; ${\phi }_{\mathbf{w}}(\mathbf{z})\equiv 1$ and $\omega (1-|\mathbf{z}|)\equiv 1$. It is clear that

$${\parallel g_{\mathbf{w}}\parallel }_{{\mathcal{B}}_{\omega }({\mathbb{B}}_n)\cap {\mathcal{LB}}_{\omega }^2({\mathbb{B}}_n)}\le C.$$

Taking $\mathbf{z}=\mathbf{w}$, then

$$\begin{array}{c}|g(\mathbf{w})Q_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})|{(1-{|\mathbf{w}|}^2)}^2\hfill \\ =(\frac{1}{1-{|\mathbf{w}|}^2}+ln\frac{2}{1-{|\mathbf{w}|}^2})|Q_{\mu }^{\alpha ,\beta }(\mathbf{w})|{(1-{|\mathbf{w}|}^2)}^2\hfill \\ \le |Q_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})|(1-{|\mathbf{w}|}^2)+|Q_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})|{(1-{|\mathbf{w}|}^2)}^2(ln\frac{2}{1-{|\mathbf{w}|}^2})\hfill \\ \le C.\hfill \end{array}$$

By (8), then $P_{\alpha +\beta -1;\omega }(\mu )\in {\mathcal{LB}}_{\omega }^2({\mathbb{B}}_n)$ and $P_{\alpha +\beta -1;\omega }(\mu )\in {\mathcal{B}}_{\omega }({\mathbb{B}}_n)$.

When $\alpha ,\beta >1$, taking $g_{\mathbf{w}}(\mathbf{z})={(1-〈\mathbf{z},\mathbf{w}〉)}^{1-(\alpha +\beta )}$, we have ${\parallel g_{\mathbf{w}}\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}\le C$.

From Lemma 2.1, we get

$$|Q_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})|{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta }\le C\text{for }\mathbf{w}\in {\mathbb{B}}_n.$$

By (8) it is obvious that $P_{\alpha +\beta -1;\omega }(\mu )\in {\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$.

This completes the proof of Theorem 3.1. □

4 Compactness of general Toeplitz operators

In this section, we study the compactness of Toeplitz operators on the weighted Bloch-type spaces ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ in the unit ball of ${\mathbb{C}}^n$. We need the following lemma.

Lemma 4.1 Let $0<\alpha <\mathrm{\infty }$, $0\le \beta <\mathrm{\infty }$ and $T_{\mu }^{\alpha ,\beta ;\omega }$ be a bounded linear operator from ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ into ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$. When $0<\alpha <1$, $0\le \beta <1$ and $\alpha +\beta <1$, then $T_{\mu }^{\alpha ,\beta ;\omega }$ is compact if and only if

$$\underset{j\to \mathrm{\infty }}{lim}{\parallel T_{\mu }^{\alpha ,\beta ;\omega }{f}_j\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}=0,$$

whenever $(f_j)$ is a bounded sequence in ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ that converges to 0 uniformly on ${\overline{\mathbb{B}}}_n$.

Proof This lemma can be proved by Montel’s theorem and Lemma 2.1. □

Theorem 4.1 Let μ be a positive Borel measure on ${\mathbb{B}}_n$. We have the following:

(1) if $\alpha +\beta =1$, then $T_{\mu }^{\alpha ,\beta ;\omega }$ is compact on ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ if and only if $P_{\alpha +\beta -1;\omega }(\mu )\in {\mathcal{LB}}_{\omega ;0}({\mathbb{B}}_n)$ and μ is a vanishing Carleson measure;

(2) if $\alpha =\beta =1$, then $T_{\mu }^{\alpha ,\beta ;\omega }$ is compact on ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ if and only if $P_{\alpha +\beta -1;\omega }(\mu )\in {\mathcal{B}}_{\omega ;0}({\mathbb{B}}_n)\cap {\mathcal{LB}}_{\omega ;0}^2({\mathbb{B}}_n)$ and μ is a vanishing Carleson measure;

(3) if $\alpha >1$, $\beta >1$, then $T_{\mu }^{\alpha ,\beta ;\omega }$ is compact on ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ if and only if $P_{\alpha +\beta -1}(\mu )\in {\mathcal{B}}_{\omega ;0}^{\alpha ,\beta }({\mathbb{B}}_n)$ and μ is a vanishing Carleson measure.

Proof For $\alpha +\beta \ge 1$, let $(g_j)$ be a sequence in ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ satisfying ${\parallel g_j\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}\le 1$ and $g_j$ converges to 0 uniformly as $j\to \mathrm{\infty }$ on ${\overline{\mathbb{B}}}_n$. Suppose $f\in A^1({\mathbb{B}}_n)$. By duality, we have that $T_{\mu }^{\alpha ,\beta ;\omega }$ is compact on ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ if and only if

$$\underset{j\to \mathrm{\infty }}{lim}\underset{{\parallel f\parallel }_{A^1({\mathbb{B}}_n)}\le 1}{sup}\left|〈f,T_{\mu }^{\alpha ,\beta ;\omega }(g_j)〉\right|=0.$$

Similarly, as in the proof of Theorem 3.1 for $s=\alpha +\beta -1$, we have

$$\begin{array}{rcl}{〈f,T_{\mu }^{\alpha ,\beta ;\omega }{g}_j〉}_s& =& c_{\alpha +\beta -1}{\int }_{{\mathbb{B}}_n}f(\mathbf{z})\overline{g_j(\mathbf{z})}{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}d\mu (\mathbf{z})\\ =& c_{\alpha +\beta -1}{\int }_{{\mathbb{B}}_n}[(I_{\mathbf{z},\mathbf{w};\omega }-P_{\alpha +\beta ;\omega })(f{\overline{g}}_j)](\mathbf{z}){(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}d\mu (\mathbf{z})\\ +c_{\alpha +\beta -1}{\int }_{{\mathbb{B}}_n}{P}_{\alpha +\beta ;\omega }(f{\overline{g}}_j)(\mathbf{z}){(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}d\mu (\mathbf{z})\\ =& J_1+J_2.\end{array}$$

For fixed $0<\epsilon <1$, since μ is a vanishing Carleson measure, there exists $0<\eta <1$ such that

$${(1-{|\mathbf{z}|}^2)}^{\lambda }{\int }_{{\mathbb{B}}_n\mathrm{\setminus }\eta {\mathbb{B}}_n}\frac{{(1-{|\mathbf{w}|}^2)}^{\lambda -(\alpha +\beta )}}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{n+2\lambda +1-(\alpha +\beta )}}d\mu (\mathbf{w})<\epsilon ,$$

where $\eta {\mathbb{B}}_n=\{\mathbf{z}\in {\mathbb{C}}^n,|\mathbf{z}|<\eta \}$ and $\lambda -(\alpha +\beta )>-1$. For a positive constant $0<\delta <1$, as in the proof of Theorem 3.1, by Proposition 2.1, we obtain

$$\begin{array}{rcl}|J_1|& =& c_{\alpha +\beta -1}{c}_{\alpha +\beta }|{\int }_{{\mathbb{B}}_n}{\int }_{{\mathbb{B}}_n}\frac{(\overline{g_j(\mathbf{z})}-\overline{g_j(\mathbf{w})})f(\mathbf{w}){(1-{|\mathbf{w}|}^2)}^{\alpha +\beta }{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+\alpha +1}{(1-{|{\phi }_{\mathbf{w}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|)}d\nu (\mathbf{w})d\mu (\mathbf{z})|\\ =& c_{\alpha +\beta -1}{c}_{\alpha +\beta }|{\int }_{{\mathbb{B}}_n}f(\mathbf{w}){(1-{|\mathbf{w}|}^2)}^{\alpha +\beta }\\ \times {\int }_{{\mathbb{B}}_n}\frac{(\overline{g_j(\mathbf{z})}-\overline{g_j(\mathbf{w})}){(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+\alpha +1}{(1-{|{\phi }_{\mathbf{w}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|)}d\mu (\mathbf{z})d\nu (\mathbf{w})|\\ \le & c_{\alpha +\beta -1}{c}_{\alpha +\beta }{\int }_{{\mathbb{B}}_n\mathrm{\setminus }\delta {\mathbb{B}}_n}|f(\mathbf{w})|{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta }\\ \times {\int }_{{\mathbb{B}}_n}\frac{|g_j(\mathbf{z})-g_j(\mathbf{w})|{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{n+\alpha +1}{(1-{|{\phi }_{\mathbf{w}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|)}d\mu (\mathbf{z})d\nu (\mathbf{w})\\ +c_{\alpha +\beta -1}{c}_{\alpha +\beta }{\int }_{\delta {\mathbb{B}}_n}|f(\mathbf{w})|{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta }\\ \times {\int }_{{\mathbb{B}}_n}\frac{|g_j(\mathbf{z})-g_j(\mathbf{w})|{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{n+\alpha +1}{(1-{|{\phi }_{\mathbf{w}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|)}d\mu (\mathbf{z})d\nu (\mathbf{w})\\ =& L_1+L_2.\end{array}$$

Since $g_j\to 0$ as $j\to \mathrm{\infty }$ on compact subsets of ${\mathbb{B}}_n$, we can choose j big enough so that

$$|f(\mathbf{w})|{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta }<\epsilon .$$

Therefore,

$$\begin{array}{rcl}{L}_2& \le & \epsilon C{\int }_{\delta {\mathbb{B}}_n}{\int }_{{\mathbb{B}}_n}\frac{|g_j(\mathbf{z})-g_j(\mathbf{w})|{(1-{|\mathbf{z}|}^2)}^{\alpha +\beta -1}}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{n+\alpha +1}{(1-{|{\phi }_{\mathbf{w}}(\mathbf{z})|}^2)}^{\beta }\omega (1-|\mathbf{z}|)}d\mu (\mathbf{z})d\nu (\mathbf{w})\\ \le & \epsilon C{\parallel g_j\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}.\end{array}$$

Now, taking δ such that $1-{[\epsilon {(1-\eta )}^{n+1+\lambda }]}^{\frac{1}{\lambda }}\le \delta <1$, then

$$\begin{array}{rcl}{L}_1& \le & C{\parallel g_j\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}{\int }_{{\mathbb{B}}_n\mathrm{\setminus }\delta {\mathbb{B}}_n}|f(\mathbf{w})|{(1-{|\mathbf{w}|}^2)}^{\lambda }{\int }_{{\mathbb{B}}_n}\frac{{(1-{|\mathbf{z}|}^2)}^{\lambda -(\alpha +\beta )}}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{n+2\lambda +1-(\alpha +\beta )}}d\mu (\mathbf{z})d\nu (\mathbf{w})\\ \le & C{\int }_{{\mathbb{B}}_n\mathrm{\setminus }\delta {\mathbb{B}}_n}|f(\mathbf{w})|{(1-{|\mathbf{w}|}^2)}^{\lambda }{\int }_{{\mathbb{B}}_n\mathrm{\setminus }\eta {\mathbb{B}}_n}\frac{{(1-{|\mathbf{z}|}^2)}^{\lambda -(\alpha +\beta )}}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{n+2\lambda +1-(\alpha +\beta )}}d\mu (\mathbf{z})d\nu (\mathbf{w})\\ +C{\int }_{{\mathbb{B}}_n\mathrm{\setminus }\delta {\mathbb{B}}_n}|f(\mathbf{w})|{(1-{|\mathbf{w}|}^2)}^{\lambda }{\int }_{\eta {\mathbb{B}}_n}\frac{{(1-{|\mathbf{z}|}^2)}^{\lambda -(\alpha +\beta )}}{{|1-〈\mathbf{z},\mathbf{w}〉|}^{n+2\lambda +1-(\alpha +\beta )}}d\mu (\mathbf{z})d\nu (\mathbf{w})\\ \le & C\epsilon {\int }_{{\mathbb{B}}_n\mathrm{\setminus }\delta {\mathbb{B}}_n}|f(\mathbf{w})|d\nu (\mathbf{w})+C{\int }_{{\mathbb{B}}_n\mathrm{\setminus }\delta {\mathbb{B}}_n}|f(\mathbf{w})|\frac{{(1-\delta )}^{\lambda }}{{(1-\eta )}^{n+1+\lambda }}d\nu (\mathbf{w})\\ \le & C\epsilon {\parallel f\parallel }_{A^1({\mathbb{B}}_n)}.\end{array}$$

Hence $|J_1|<C\epsilon$, where C does not depend on $f(\mathbf{z})$, and so

$$\underset{j\to \mathrm{\infty }}{lim}\underset{{\parallel f\parallel }_{A^1({\mathbb{B}}_n)}\le 1}{sup}|J_1|=0.$$

Thus, $T_{\mu }^{\alpha ,\beta ;\omega }$ is compact on ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$ if and only if

$$\underset{j\to \mathrm{\infty }}{lim}\underset{{\parallel f\parallel }_{A^1({\mathbb{B}}_n)}\le 1}{sup}|J_2|=0.$$

Again, as in the proof of Theorem 3.1, we have

$$|J_2|\le C{\int }_{{\mathbb{B}}_n}{\parallel f\parallel }_{A^1({\mathbb{B}}_n)}{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta }|g(\mathbf{w})|Q_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})d\nu (\mathbf{w}).$$

From (8) it is easy to see that

(1) if $\alpha +\beta =1$ and $P_{\alpha +\alpha -1;\omega }(\mu )\in {\mathcal{LB}}_{\omega ,0}^{\alpha ,\beta }({\mathbb{B}}_n)$, then

$$\underset{|\mathbf{w}|\to 1}{lim}(1-{|\mathbf{w}|}^2)Q_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})(ln\frac{2}{1-{|\mathbf{w}|}^2})=0;$$

(2) if $\alpha =\beta =1$, $P_{\alpha +\alpha -1;\omega }(\mu )\in {\mathcal{B}}_{\omega ;0}({\mathbb{B}}_n)\cap {\mathcal{LB}}_{\omega ;0}^2({\mathbb{B}}_n)$, then

$$\underset{|\mathbf{w}|\to 1}{lim}(1-{|\mathbf{w}|}^2)Q_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})=0$$

and

$$\underset{|\mathbf{w}|\to 1}{lim}{(1-{|\mathbf{w}|}^2)}^2{Q}_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})(ln\frac{2}{1-{|\mathbf{w}|}^2})=0;$$

(3) if $\alpha >1$, $\beta >1$, and $P_{\alpha +\alpha -1;\omega }(\mu )\in {\mathcal{B}}_{\omega ;0}^{\alpha ,\beta }({\mathbb{B}}_n)$, then

$$\underset{|\mathbf{w}|\to 1}{lim}{(1-{|\mathbf{w}|}^2)}^{\alpha +\beta }{Q}_{\mu }^{\alpha ,\beta ;\omega }(\mathbf{w})=0.$$

Combined with $g_j\to 0$ as $j\to \mathrm{\infty }$ on compact subsets of ${\mathbb{B}}_n$, we have

$$\underset{j\to \mathrm{\infty }}{lim}\underset{{\parallel f\parallel }_{A^1({\mathbb{B}}_n)}\le 1}{sup}|J_2|=0.$$

Therefore,

$$\underset{j\to \mathrm{\infty }}{lim}{\parallel T_{\mu }^{\alpha ,\beta ;\omega }{g}_j\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}=0,$$

which implies that $T_{\mu }^{\alpha ,\beta ;\omega }$ is a compact operator.

Next assume that $T_{\mu }^{\alpha ,\beta ;\omega }$ is a compact operator on ${\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)$. Again, as in the proof of Theorem 3.1, we take

$$f_{\mathbf{w}}(\mathbf{z})=\frac{{(1-{|\mathbf{w}|}^2)}^t}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+t+1}}\text{for }t>0.$$

We know that ${\parallel f_{\mathbf{w}}\parallel }_{A^1({\mathbb{B}}_n)}\le C$. On the other hand, take

$$g_{\mathbf{w}}(\mathbf{z})=\frac{{(1-{|\mathbf{w}|}^2)}^{n+2+t-(\alpha +\beta )}}{{(1-〈\mathbf{z},\mathbf{w}〉)}^{n+t+1}};{\phi }_{\mathbf{w}}(\mathbf{z})\equiv 1\text{and}\omega (1-|\mathbf{z}|)\equiv 1\text{for }t>0.$$

Then ${\parallel g_{\mathbf{w}}\parallel }_{{\mathcal{B}}_{\omega }^{\alpha ,\beta }({\mathbb{B}}_n)}\le C$ and $g_{\mathbf{w}}\to 0$ uniformly on compact subsets of ${\mathbb{B}}_n$, as $|\mathbf{w}|\to 1$,

$$\begin{array}{c}|{〈f,{}_{}^{}}_{}\hfill \end{array}$$