Weighted Bergman spaces induced by doubling weights in the unit ball of $\mathbb{C}^n$

TL;DR

This paper investigates weighted Bergman spaces in the unit ball of complex n-space with doubling weights, characterizing Carleson measures, and analyzing the boundedness and compactness of associated Volterra integral operators.

Contribution

It provides new characterizations of weighted Bergman spaces with doubling weights and studies the boundedness and compactness of Volterra operators on these spaces.

Findings

Characterization of q-Carleson measures via geometric conditions.

Equivalent descriptions of Bergman spaces using radial derivatives.

Boundedness and compactness criteria for Volterra integral operators.

Abstract

This paper is devoted to the study of the weighted Bergman space $A_\omega^p $ in the unit ball $\mathbb{B}$ of $\mathbb{C}^n$ with doubling weight $\omega$ satisfying $$\int_r^1\omega(t)dt <C \int_{\frac{1+r}{2}}^1\omega(t)dt ,\,\, 0\leq r<1.$$ The $q-$Carleson measures for $A_\omega^p$ are characterized in terms of a neat geometric condition involving Carleson block. Some equivalent characterizations for $A_\omega^p$ are obtained by using the radial derivative and admissible approach regions. The boundedness and compactness of Volterra integral operator $T_g:A_\omega^p\to A_\omega^q$ are also investigated in this paper with $0<p\leq q<\infty$, where $$T_gf(z)=\int_0^1 f(tz)\Re g(tz)\frac{dt}{t}, ~~\qquad~~~~f\in H(\mathbb{B}), ~~z\in \mathbb{B}. $$

Equations

$${\int }_r^1\omega (t)dt<C{\int }_{\frac{1+r}{2}}^1\omega (t)dt,0\le r<1.$$

$$T_{g}f(z)={\int }_0^{1}f(tz)\Re g(tz)\frac{dt}{t},f\in H(\mathbb{B}),z\in \mathbb{B}.$$

$$z=(z_1,z_2,\cdots ,z_n)\text{mbox}andw=(w_1,w_2,\cdots ,w_n)$$

$$|z|=\sqrt{⟨z,z⟩}=\sqrt{|z_1{|}^2+\cdots +|z_n{|}^2}.$$

$$\Vert f{\Vert }_{H^p}:=\underset{0<r<1}{\sup }{M}_p(r,f),$$

$$M_p(r,f)={\left({\int }_{\text{SS}}|f(r\xi ){|}^{p}d\sigma (\xi )\right)}^{\frac{1}{p}}<\infty ,\text{mbox}when0<p<\infty ,$$

$$M_{\infty }(r,f)=\underset{|z|=r}{\sup }|f(z)|.$$

$$\Vert f{\Vert }_{A_{\alpha }^p}={\int }_B|f(z){|}^{p}d{V}_{\alpha }(z)=c_{\alpha }{\int }_B|f(z){|}^p(1-|z{|}^2{)}^{\alpha }dV(z)<\infty ,$$

$$\Vert f{\Vert }_{A_{\alpha }^p}^p\asymp |f(0){|}^p+{\int }_{\mathbb{B}}|\Re f(z){|}^p(1-|z{|}^2{)}^{p}d{V}_{\alpha }(z).$$

$$\hat{\omega }(r)<C\hat{\omega }(\frac{1+r}{2}),\text{mbox}when0\le r<1.$$

$$\frac{1}{C}<\frac{\hat{\omega }(r)}{(1-r)\omega (r)}<C,\text{mbox}when0\le r<1.$$

$$\underset{r\to 1}{\lim }\frac{\hat{\omega }(r)}{(1-r)\omega (r)}=\infty .$$

$$\Vert f{\Vert }_{A_{\omega }^p}^p={\int }_{\mathbb{B}}|f(z){|}^p\omega (z)dV(z)<\infty .$$

$$J_{g}f(z)={\int }_0^{z}f(\xi )g^{\prime }(\xi )d\xi ,f\in H(\mathbb{D}),z\in \mathbb{D}.$$

$$T_{g}f(z)={\int }_0^{1}f(tz)\Re g(tz)\frac{dt}{t},f\in H(\mathbb{B}),z\in \mathbb{B}.$$

$$Q(\xi ,r)=\{\eta \in \text{SS}:|1-<\xi ,\eta >|\le r^2\}.$$

$$A_1\le \frac{\sigma (Q(\xi ,r))}{r^{2n}}\le A_2$$

$$S_a=S(Q_a)=\left\{z\in \mathbb{B}:\frac{z}{|z|}\in Q_a,|a|<|z|<1\right\}.$$

$$\frac{{\omega }^{\ast }(r)}{(1-r{)}^a}\text{mbox}isdecreasingon[\delta ,1)\text{mbox}and\underset{r\to 1}{\lim }\frac{{\omega }^{\ast }(r)}{(1-r{)}^a}=0;$$

$$\frac{{\omega }^{\ast }(r)}{(1-r{)}^b}\text{mbox}isincreasingon[\delta ,1)\text{mbox}and\underset{r\to 1}{\lim }\frac{{\omega }^{\ast }(r)}{(1-r{)}^b}=\infty ;$$

$${\int }_{\mathbb{B}}f(z)dV(z)=2n{\int }_0^1{r}^{2n-1}dr{\int }_{\text{SS}}f(r\xi )d\sigma (\xi ).$$

$$\omega (S_a)=2n{\int }_{|a|}^1{r}^{2n-1}\omega (r)dr{\int }_{Q(\frac{a}{|a|},\sqrt{1-|a|})}d\sigma (\xi )\approx (1-|a|{)}^n\hat{\omega }(|a|).$$

$$\sigma \left(Q(\xi ,qr)\backslash Q(\xi ,r)\right)\approx r^{2n}.$$

$$A_1{r}^{2n}\le \sigma (Q(\xi ,r))\le A_2{r}^{2n},\text{mbox}forallr\in (0,\sqrt{2})\text{mbox}and\xi \in {\text{SS}}^n.$$

$$\sigma \left(Q(\xi ,qr)\backslash Q(\xi ,r)\right)\ge (A_1{q}^{2n}-A_2)r^{2n}\gtrsim r^{2n},$$

$$\sigma \left(Q(\xi ,qr)\backslash Q(\xi ,r)\right)\le (A_2{q}^{2n}-A_1)r^{2n}\lesssim r^{2n}.$$

$$\Vert f{\Vert }_{L_{\mu }^q}:={\left({\int }_{\mathbb{B}}|f(z){|}^{q}d\mu (z)\right)}^{\frac{1}{q}}<\infty .$$

$$|f(z){|}^{\alpha }\le C{M}_{\omega }(|f{|}^{\alpha })(z),$$

$$M_{\omega }(\phi )(z)=\underset{z\in S_a}{\sup }\frac{1}{\omega (S_a)}{\int }_{S_a}|\phi (\xi )|\omega (\xi )dV(\xi ).$$

$$Q_k^{\prime }:=\left\{\xi \in \text{SS}:\left|1-⟨\xi ,\frac{z}{|z|}⟩\right|<q^k\left(1-\frac{|z|}{\rho }\right)\right\},$$

Full text

[TABLE]

The $q-$Carleson measures for $A_{\omega}^{p}$ are characterized in terms of a neat geometric condition involving Carleson block. Some equivalent characterizations for $A_{\omega}^{p}$ are obtained by using the radial derivative and admissible approach regions. The boundedness and compactness of Volterra integral operator $T_{g}:A_{\omega}^{p}\to A_{\omega}^{q}$ are also investigated in this paper with $0<p\leq q<\infty$, where

[TABLE]

Keywords: Weighted Bergman space, Carleson measure, Volterra integral operator, doubling weight.

$\dagger$ Corresponding author.

This project was partially supported by the Macao Science and Technology Development Fund (No.186/2017/A3) and NNSF of China (No. 11720101003).

1. Introduction

Let $\mathbb{B}$ be the open unit ball of $\mathbb{C}^{n}$ and $\SS$ the boundary of $\mathbb{B}$. When $n=1$, then $\mathbb{B}$ is the open unit disk in complex plane $\mathbb{C}$ and always denoted by $\mathbb{D}$. Let $H(\mathbb{B})$ denote the space of all holomorphic functions on $\mathbb{B}$. For any two points

[TABLE]

in $\mathbb{C}^{n}$, we define $\langle z,w\rangle=z_{1}\overline{w_{1}}+\cdots+z_{n}\overline{w_{n}}$ and

[TABLE]

Let $d\sigma$ and $dV$ be the normalized surface and volume measures on $\SS$ and $\mathbb{B}$, respectively. For $0<p\leq\infty$, the Hardy space $H^{p}(\mathbb{B})$(or $H^{p}$) is the space consisting of all functions $f\in H(\mathbb{B})$ such that

[TABLE]

where

[TABLE]

and

[TABLE]

For $-1<\alpha<\infty$ and $0<p<\infty$, the weighted Bergman space $A_{\alpha}^{p}(\mathbb{B})$( or $A_{\alpha}^{p}$) consists of all $f\in H(\mathbb{B})$ such that

[TABLE]

where $c_{\alpha}=\Gamma(n+\alpha+1)/(\Gamma(n+1)\Gamma(\alpha+1))$. When $\alpha=0$, $A^{p}_{0}(\mathbb{B})=A^{p}(\mathbb{B})$ is the standard Bergman space. It is known that $f\in A^{p}_{\alpha}$ if and only if $\Re f(z)\in A_{\alpha+p}^{p}$. Moreover

[TABLE]

Here $\Re f$ is the radial derivative of $f,$ i.e., $\Re f(z)=\sum^{n}_{j=1}z_{j}\frac{\partial f}{\partial z_{j}}(z).$ See [17, 18] for the theory of $H^{p}$ and $A_{\alpha}^{p}$ in the unit ball.

Suppose $\omega$ is a radial weight ($i.e.$, a positive and integrable function in $\mathbb{B}$ such that $\omega(z)=\omega(|z|)$). Let $\hat{\omega}(r)=\int_{r}^{1}\omega(t)dt$. $\omega$ is called a doubling weight, denoted by $\omega\in\hat{\mathcal{D}}$, if there is a constant $C>0$ such that

[TABLE]

$\omega$ is called a regular weight, denote by $\omega\in\mathcal{R}$, if there is a constant $C>0$ determined by $\omega$, such that

[TABLE]

$\omega$ is called a rapidly increasing weight, denote by $\omega\in\mathcal{I}$, if

[TABLE]

After a calculation, we see that $\mathcal{I}\cup\mathcal{R}\subset\hat{\mathcal{D}}.$ See [9, 10] for more details about $\mathcal{I},\mathcal{R}$, $\hat{\mathcal{D}}$.

In [10], J. Peláez and J. Rättyä introduced a new class function space $A_{\omega}^{p}(\mathbb{D})$, the weighted Bergman space induced by rapidly increasing weights in $\mathbb{D}$. In [10], they investigated some basic properties of $\omega$ with $\omega\in\mathcal{R}\cup\mathcal{I}$, described the $q-$Carleson measure for $A_{\omega}^{p}(\mathbb{D})$, gave equivalent characterizations of $A_{\omega}^{p}(\mathbb{D})$, characterized the boundedness, compactness and Schatten classes of Volterra integral operator $J_{g}$ on $A_{\omega}^{p}(\mathbb{D})$. In [9], J. Peláez extended many results from $\omega\in\mathcal{R}\cup\mathcal{I}$ to $\omega\in\hat{\mathcal{D}}$. See [9, 10, 11, 12, 13, 14, 15] for many results on $A_{\omega}^{p}(\mathbb{D})$ with $\omega\in\hat{\mathcal{D}}$.

Motivated by [10], we extend the Bergman space $A_{\omega}^{p}(\mathbb{D})$ with $\omega\in\hat{\mathcal{D}}$ to the unit ball. Let $\omega\in\hat{\mathcal{D}}$ and $0<p<\infty$. The weighted Bergman space $A_{\omega}^{p}=A_{\omega}^{p}(\mathbb{B})$ is the space of all $f\in H(\mathbb{B})$ for which

[TABLE]

It is easy to check that $A_{\omega}^{p}$ is a Banach space when $p\geq 1$ and a complete metric space with the distance $\rho(f,g)=\|f-g\|_{A_{\omega}^{p}}^{p}$ when $0<p<1$. When $\omega(z)=c_{\alpha}(1-|z|^{2})^{\alpha}(\alpha>-1)$, the space $A_{\omega}^{p}$ becomes the classical weighted Bergman space $A_{\alpha}^{p}$.

Suppose that $g\in H(\mathbb{D})$. The integral operator $J_{g}$, called the Volterra integral operator, is defined by

[TABLE]

The operator $J_{g}$ was first introduced by Pommerenke in [16]. He showed that $J_{g}$ is a bounded operator on the Hardy space $H^{2}(\mathbb{D})$ if and only if $g\in BMOA(\mathbb{D})$. See [1, 2, 3, 4] for the study of the boundedness, compactness and the spectrum of $J_{g}$ in $H^{p}(\mathbb{D})$ and $A_{\alpha}^{p}(\mathbb{D})$.

Let $g\in H(\mathbb{B})$. Define

[TABLE]

This operator is also called the Volterra type integral operator (or the Riemann-Stieltjes operator, or the Extended Cesàro operator). The operator $T_{g}$ was introduced by Z. Hu in [5] and studied, for example in [5, 6, 7, 8]. In particular, J. Pau completely described the boundedness and compactness of $T_{g}$ between different Hardy spaces in the unit ball of $\mathbb{C}^{n}$ in [8].

In this paper, we will investigate some properties of $A_{\omega}^{p}$ in the unit ball of $\mathbb{C}^{n}$ and study the boundedness and compactness of $T_{g}:A_{\omega}^{p}\to A_{\omega}^{q}$ with $\omega\in\hat{\mathcal{D}}$ and $0<p\leq q<\infty$, The paper is organized as follows. In section 2, we recall some well-known results and notations, define (standard) Carleson block $S_{a}$ for $a\in\mathbb{B}$ and estimate the volume of $S_{a}$. In section 3, we characterize the $q-$Carleson measure for $A_{\omega}^{p}$ with standard Carleson block $S_{a}$ for $a\in\mathbb{B}$. In section 4, we extend the admissible approach region $\Gamma_{u}$ from $u\in\SS$ to $u\in\overline{\mathbb{B}}$ by dilation transformation, get some equivalent characterizations for $A_{\omega}^{p}$ by using the radial derivative and admissible approach regions. In section 5, we define a new class of holomorphic functions $\mathcal{C}^{\kappa}(\omega^{*})(\kappa\geq 1)$ and the little-oh subspace of it, and then we investigate the boundedness and compactness of $T_{g}:A_{\omega}^{p}\to A_{\omega}^{q}$ with $0<p\leq q<\infty$ and $\omega\in\hat{\mathcal{D}}$. In section 6, we discuss the inclusion relationship between $\mathcal{C}^{1}(\omega^{*})(\mathcal{C}_{0}^{1}(\omega^{*}))$ and some other function spaces, such as the Bloch space $\mathcal{B}$, the little Bloch space $\mathcal{B}_{0}$, the $BMOA$ space and the $VMOA$ space.

Throughout this paper, the letter $C$ will denote constants and may differ from one occurrence to the other. The notation $A\lesssim B$ means that there is a positive constant C such that $A\leq CB$. The notation $A\approx B$ means $A\lesssim B$ and $B\lesssim A$.

2. Preliminary results

For any $\xi,\tau\in\overline{\mathbb{B}}$, let $d(\xi,\tau)=|1-\langle\xi,\tau\rangle|^{\frac{1}{2}}$. Then $d(\cdot,\cdot)$ is the nonisotropic metric. For $r>0$ and $\xi\in\SS$, let

[TABLE]

$Q(\xi,r)$ is a ball in $\SS$ for all $\xi\in\SS^{n}$ and $r\in(0,\sqrt{2})$. More information about $d(\cdot,\cdot)$ and $Q(\xi,r)$ can be found in [17, 18]. Lemma 4.6 in [18] is very useful in this paper, and we express it as follows.

Lemma 1**.**

There exist positive constants $A_{1}$ and $A_{2}$ (depending on $n$ only) such that

[TABLE]

for all $\xi\in\SS^{n}$ and $r\in(0,\sqrt{2})$.

For any $a\in\mathbb{B}\backslash\{0\}$, let $Q_{a}=Q({a}/{|a|},\sqrt{1-|a|}),$ and define

[TABLE]

For convince, if $a=0$, let $Q_{a}=\SS$ and $S_{a}=\mathbb{B}$. We call $S_{a}$ the Carleson block. Now we give a estimate for the volume of $S_{a}$. As usual, for a measurable set $E\subset\mathbb{B}$, $\omega(E)=\int_{E}\omega(z)dV(z)$.

Lemma 2**.**

Assume that $\omega\in\hat{\mathcal{D}}$, $r\in[0,1]$ and $\omega^{*}(r)=\int_{r}^{1}s\omega(s)\log\frac{s}{r}ds$. Then the following statements hold.

1. (i)

$\omega^{*}\in\mathcal{R}$* and $\omega^{*}(r)\approx(1-r)\int_{r}^{1}\omega(t)dt$ when $r\in(\frac{1}{2},1)$;* 2. (ii)

There are $0<a<b<+\infty$ and $\delta\in[0,1)$, such that

[TABLE]

[TABLE] 3. (iii)

$\omega^{*}(r)$* is decreasing on $[\delta,1)$ and $\lim\limits_{r\to 1}\omega^{*}(r)=0.$* 4. (iv)

$\omega(S_{a})\approx(1-|a|)^{n}\int_{|a|}^{1}\omega(r)dr$.

Proof.

$(i)-(iii)$ can be found in [10, 11]. From [18, Lemma 1.8], we see that

[TABLE]

Then by Lemma 1, we have

[TABLE]

The proof is complete. ∎

Lemma 3**.**

There exists $q=q(n)>1$, such that for all $r\in\left(0,\frac{\sqrt{2}}{q}\right)$ and $\xi\in\SS^{n}$,

[TABLE]

Proof.

By Lemma 1, there exist $A_{2}>A_{1}>0$, such that

[TABLE]

Fix a $q>1$ such that $A_{1}q^{2n}>A_{2}$. Then we have

[TABLE]

and

[TABLE]

The proof is complete. ∎

For $q>0$, if $\mu$ is a positive Borel measure on $\mathbb{B}$, $L_{\mu}^{q}$ consists of the Lebesgue measurable functions on $\mathbb{B}$ such that

[TABLE]

To study the compactness of a linear operator $T$ from $A_{\omega}^{p}$ to $L_{\mu}^{q}$, we need the following lemma which can be obtained in a standard way.

Lemma 4**.**

Suppose $0<p,q<\infty,\omega\in\hat{\mathcal{D}}$ and $\mu$ is a positive Borel measure on $\mathbb{B}$. If $T:A_{\omega}^{p}\to L_{\mu}^{q}$ is linear and bounded, then $T$ is compact if and only if whenever $\{f_{k}\}$ is bounded in $A_{\omega}^{p}$ and $f_{k}\to 0$ uniformly on compact subsets of $\mathbb{B}$, $\lim\limits_{k\to\infty}\|Tf_{k}\|_{L_{\mu}}^{q}=0.$

Lemma 5**.**

Suppose $\omega\in\hat{\mathcal{D}},0<\alpha<\infty$. Then there exists a constant $C=C(\alpha,\omega,n)$ such that

[TABLE]

for all $f\in H(\mathbb{B})$. Here and henceforth,

[TABLE]

Proof.

Fix $q=q(n)$ such that Lemma 3 holds. Let $r_{0}=\max\left(\frac{1}{2},1-\frac{1}{q}\right)$.

For any $z\in\mathbb{B}$ such that $r_{0}<|z|<1$, if $\frac{1+|z|}{2}<\rho<1$, let $N$ be the largest nature number such that $q^{N}\left(1-|z|\right)<1$ and

[TABLE]

for $k=0,1,2,\cdots,N$. For convenience, let $Q_{-1}^{\prime}=\O$. Then,

[TABLE]

When $\xi\in Q_{k+1}^{\prime}\backslash Q_{k}^{\prime}$ with $k=1,2,\cdots,N$, we have

[TABLE]

Then for all $\xi\in Q_{k+1}^{\prime}\backslash Q_{k}^{\prime}$ wiht $k=-1,0,1,\cdots,N$, we have

[TABLE]

Since $\omega\in\hat{\mathcal{D}}$, by Lemma 2.1(ii) in [9], there exist $C_{0}=C_{0}(\omega)\geq 1$ and $\beta=\beta(\omega)>0$, such that

[TABLE]

Write $\alpha=s\gamma$, where $\gamma>1+\frac{\beta}{n}>1$. Suppose $\frac{1}{\gamma}+\frac{1}{\gamma^{\prime}}=1$. Then by Corollary 4.5 in [18], Hölder’s inequality and Theorem 1.12 in [18], we have

[TABLE]

Therefore,

[TABLE]

When $k=0,1,2,\cdots,N$, let $t_{k}=1-q^{k}(1-|z|)$ and $a_{k}=t_{k}z$. When $k=N+1$, let $a_{k}=0$, $Q_{a_{k}}=\SS$ and $S_{a_{k}}=\mathbb{B}$. For $0\leq k\leq N+1$, we have

[TABLE]

For all $|z|\geq r_{0}$, $\int_{|z|}^{1}r^{2n-1}\omega(r)dr\approx\int_{\frac{|z|+1}{2}}^{1}r^{2n-1}\omega(r)dr$. So, we have

[TABLE]

By Lemma 2, we have

[TABLE]

Then,

[TABLE]

Next we suppose that $|z|\leq r_{0}$. For all $a\in\mathbb{B}$ such that $z\in S_{a}$, we have $|a|<|z|\leq r_{0}$. By Lemma 2, $\omega(S_{a})\approx 1$. Then,

[TABLE]

Using Cauchy’s formula, we have

[TABLE]

The proof is complete. ∎

Here and henceforth, for all $a\in\mathbb{B}$ and $0<p<\infty$, set

[TABLE]

We obtain the following lemma.

Lemma 6**.**

Suppose $\omega\in\hat{\mathcal{D}},0<p<\infty$ and $\gamma$ is large enough. For all $a\in\mathbb{B}$,

[TABLE]

and

[TABLE]

Proof.

For all $z\in S_{a}$, we have

[TABLE]

and

[TABLE]

It follows that (3) holds.

By Lemmas 1 and 2, we have

[TABLE]

By Theorem 1.12 in [18] and Lemma 2.1(iii) in [9], if $\gamma$ is large enough, we have

[TABLE]

By (3), $\|F_{a,p}\|_{A_{\omega}^{p}}^{p}\gtrsim\omega(S_{a})$ is obvious. The proof is complete. ∎

In the rest this paper, we always assume $F_{a,p}$ satisfies the condition of Lemma 6.

In the last of this section, we define a $\alpha-$Carleson block $S_{a,\alpha}$ for all $a\in\mathbb{B}\backslash\{0\}$ and any fixed $\alpha>-1$. That is

[TABLE]

When $a=0$, we define $S_{a,\alpha}=\mathbb{B}$. Obviously, for all $a\in\mathbb{B}$, we have $S_{a,0}=S_{a}$ and $S_{a}\subset S_{a,\alpha}(\alpha>0)$. The following proposition is useful in this paper.

Proposition 1**.**

For any fixed $\alpha\geq 0$, there exist $N\in\mathbb{N}$, such that, for all $a\in\mathbb{B}$, there are $a_{1},a_{2},\cdots,a_{k}$ satisfy the following condition:

1. (i)

$k\leq N$* and $|a_{1}|=|a_{2}|=\cdots=|a_{k}|=|a|$;* 2. (ii)

$S_{a,\alpha}\subset\cup_{i=1}^{k}S_{a_{i}}$.

Proof.

Suppose $a\in\mathbb{B}\backslash\{0\}$ is fixed. For any $\tau\in\SS$, define

[TABLE]

Since $\frac{\sigma(E_{a}^{\prime})}{\sigma(E_{\tau})}<\infty$, there are at most $M:=M(a)$ elements $\xi_{1},\xi_{2},\cdots,\xi_{M}$ in $\SS$ such that

1. (a)

$E_{\xi_{i}}\cap E_{\xi_{j}}=\O$ for all $1\leq i<j\leq M$; 2. (b)

$E_{\xi_{i}}\subset E_{a}^{\prime}$ for all $1\leq i\leq M$.

Then we have

[TABLE]

Otherwise, there is a $\xi\in Q(\frac{a}{|a|},\sqrt{(1+\alpha)(1-|a|)})$ but $\xi\not\in\cup_{i=1}^{M}Q(\xi_{i},\sqrt{1-|a|})$. Then for any $\eta\in E_{\xi}$, we have

[TABLE]

and

[TABLE]

That is a contraction with $M$ is the maximum number. By Lemma 1, we have

[TABLE]

Then by letting $a_{i}=|a|\xi_{i}$, we finish the proof. ∎

Remark 1**.**

By Lemma 2, for any fixed $\alpha>0$, $\omega(S_{a})\approx\omega(S_{a,\alpha})$. Hence, many results described by Carleson block also hold for $\alpha$-Carleson block.

For $\xi\in\SS$ and $r>0$, a Carleson tube $S^{*}(\xi,r)$ can be define as

[TABLE]

As we know, Carleson tube is very useful in the study of the function space on the unit ball of $\mathbb{C}^{n}$. For the convenience, we often restrict $0<r<\delta$ for some $\delta>0$. Here, we will compare Carleson tube with Carleson block.

Proposition 2**.**

The following assertions hold.

1. (i)

For any $\xi\in\SS$ and $0<r<1$, there exists $a\in\mathbb{B}$ such that $S^{*}(\xi,r)\subset S_{a,2}$. 2. (ii)

For any $a\in\mathbb{B}$ with $|a|>\frac{1}{2}$, there exist $\xi\in\SS$ and $0<r<1$ such that $S_{a}\subset S^{*}(\xi,r)$.

Proof.

(i). For any $z\in S^{*}(\xi,r)$ with $0<r<1$, by letting $a=(1-r)\xi$, we have $|z|>|a|$ and

[TABLE]

Then $S^{*}(\xi,r)\subset S_{a,2}$.

(ii). Suppose $a\neq 0$. Let $\xi=\frac{a}{|a|}$ and $2(1-|a|)<r<1$. For any $z\in S_{a}$, we have

[TABLE]

Then $S_{a}\subset S^{*}(\xi,r)$. The proof is complete. ∎

3. The $q$-Carleson Measure for $A_{\omega}^{p}$

In this section, we give some descriptions of $q$-Carleson measure for $A_{\omega}^{p}$ when $0<p\leq q<\infty$. For a given Banach space (or a complete metric space) $X$ of analytic functions on $\mathbb{B}$, a positive Borel measure $\mu$ on $\mathbb{B}$ is called a $q$-Carleson measure for $X$ if the identity operator $Id:X\to L^{q}_{\mu}$ is bounded. Moveover, if $Id:X\to L^{q}_{\mu}$ is compact, then we say that $\mu$ is a vanishing $q$-Carleson measure for $X$.

Theorem 1**.**

Let $0<p\leq q<\infty$, $\omega\in\hat{\mathcal{D}}$, and $\mu$ be a positive Borel measure on $\mathbb{B}$. The the following statements hold:

1. (i)

$\mu$* is a $q$-Carleson measure for $A_{\omega}^{p}$ if and only if*

[TABLE]

Moreover, if $\mu$ is a $q$-Carleson measure for $A_{\omega}^{p}$, then the identity operator $Id:A_{\omega}^{p}\to L_{\mu}^{q}$ satisfies

[TABLE] 2. (ii)

$\mu$* is a vanishing $q$-Carleson measure for $A_{\omega}^{p}$ if and only if*

[TABLE]

Proof.

First assume that $\mu$ is a $q$-Carleson measure for $A_{\omega}^{p}$, By Lemma 6, we have

[TABLE]

So,

[TABLE]

Conversely, suppose $M:=\sup\limits_{a\in\mathbb{B}}\frac{\mu(S_{a})}{(\omega(S_{a}))^{\frac{q}{p}}}<\infty.$ We begin with proving that there exists a constant $K=K(p,q,\omega)$ sucht that

[TABLE]

is valid for all $\varphi\in L_{\omega}^{1}$ and $0<s<\infty.$ Here $E_{s}=\{z\in\mathbb{B}:M_{\omega}(\varphi)(z)>s\}$.

If $E_{s}=\O$, (6) holds. If $E_{s}\neq\O$, define $A_{s}^{\varepsilon}$ and $B_{s}^{\varepsilon}$ for each $\varepsilon>0$ as follows.

[TABLE]

and

[TABLE]

Then we have

[TABLE]

Let $E\subset A_{s}^{\varepsilon}$ such that for all $\xi,\eta\in E$ we have either $\xi=\eta$ or $Q_{\xi}\cap Q_{\eta}=\O$. Since

[TABLE]

we get that there are only finite elements in $E$. By Lemma 5.6 in [18], there are $\{z_{1},z_{2},\cdots,z_{m}\}\subset A_{s}^{\varepsilon}$ such that $Q_{z_{j}}(1\leq j\leq m)$ are disjoint and

[TABLE]

where

[TABLE]

For any $z\in B_{s}^{\varepsilon}$, there is a $u\in A_{s}^{\varepsilon}$ such that $S_{z}\subset S_{u}$. So, $Q_{z}\subset Q_{u}$. By (8), we have

[TABLE]

Let $r_{k}=1-25(1-|z_{k}|)$. If $r_{k}>0$, let $z_{k}^{\prime}=\frac{r_{k}z_{k}}{|z_{k}|}$, and otherwise, let $z_{k}^{\prime}=0$. Then we have $Q_{z_{k}}^{\prime}\subset Q_{z_{k}^{\prime}}$. Therefore,

[TABLE]

Here, the last equivalent relation can be get by Lemma 2 and $\omega\in\hat{\mathcal{D}}$. Then, by (7) and (9), we have

[TABLE]

Let $\varepsilon\to 0^{+}$, we have $K=K(p,q,\omega)$ such that

[TABLE]

Then we obtain (6).

Next, we will show that $\mu$ is a $q-$Carleson measure for $A_{\omega}^{p}$. The proof is similar to the proof of [10, Theorem 2.1], but for the benefits of the readers and the completeness of the paper, we give the details of the proof.

Fix $\alpha>\frac{1}{p}$ and let $f\in A_{\omega}^{p}$. For $s>0$, let

[TABLE]

where

[TABLE]

and $K$ is the constant in (6) such that $K\geq 1$. Since $p>\frac{1}{\alpha}$, the function $\psi_{\frac{1}{\alpha},s}$ belongs to $L_{\omega}^{1}$ for all $s>0$, and

[TABLE]

Then,

[TABLE]

Using Lemma 5, (10), (6) and Minkowski’s inequality (Fubini’s Theorem in the case $p=q$) in order, we have

[TABLE]

So, we get $\|Id\|_{A_{\omega}^{p}\to A_{\omega}^{q}}^{q}\lesssim M$. The proof of $(i)$ is complete.

Next we prove $(ii)$. First we suppose that $\mu$ is a vanishing $q$-Carleson measure for $A_{\omega}^{p}$. Let

[TABLE]

for some $\gamma$ is large enough. By Lemmas 2 and 6 $f_{a,p}$ is bounded in $A_{\omega}^{p}$ and converges to 0 uniformly on compact subset of $\mathbb{B}$ as $|a|\to 1$. By Lemma 4, we have $\lim\limits_{|a|\to 1}\|f_{a,p}\|_{L_{\mu}^{q}}=0.$ Since

[TABLE]

we have $\lim\limits_{|a|\to 1}\frac{\mu(S_{a})}{(\omega(S_{a}))^{\frac{q}{p}}}=0.$

Conversely we suppose that $\lim\limits_{|a|\to 1}\frac{\mu(S_{a})}{(\omega(S_{a}))^{\frac{q}{p}}}=0.$ For all $\varepsilon>0$, there exists $r=r(\varepsilon)\in(0,1)$ such that when $|a|>r$, $\frac{\mu(S_{a})}{(\omega(S_{a}))^{\frac{q}{p}}}<\varepsilon$. Let $d\mu_{r}(z)=\chi_{r\leq|z|<1}d\mu(z)$.

If $|a|\geq r$, $\mu_{r}(S_{a})=\mu(S_{a})$. Then suppose $0<|a|<r$. Since $\sigma(\SS)<\infty$ and

[TABLE]

by the proof of Proposition 1, for all $\xi\in\SS$, there are at most $N$ elements $\xi_{1},\xi_{2},\cdots,\xi_{N}$ in $Q_{a}$ such that,

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

Since

[TABLE]

by Lemma 2, we have

[TABLE]

and

[TABLE]

Then, $\|Id\|_{A_{\omega}^{p}\to L_{\mu_{r}}^{q}}^{q}\lesssim\varepsilon$.

So, if $\{f_{k}\}$ is bounded in $A_{\omega}^{p}$ and converges to 0 uniformly on compact subset of $\mathbb{B}$, then we have

[TABLE]

Since $\varepsilon$ is arbitrary and $\sup_{k\to\infty}\|f_{k}\|_{A_{\omega}^{p}}<\infty$, $\lim\limits_{k\to\infty}\|f_{k}\|_{L_{\mu}^{q}}=0$. So, $\mu$ is a vanishing $q$-Carleson measure for $A_{\omega}^{p}$. The proof is complete. ∎

As a by-product of the proof of Theorem 1, we have the following result which is of independent interest.

Corollary 1**.**

Let $0<p\leq q<\infty$ and $0<\alpha<\infty$ such that $p\alpha>1$. Let $\mu$ be a positive Borel measure on $\mathbb{B}$ and $\omega\in\hat{\mathcal{D}}$. Then $[M_{\omega}((\cdot)^{\frac{1}{\alpha}})]^{\alpha}:L_{\omega}^{p}\to L_{\mu}^{q}$ is bounded if and only if (5) holds. Moreover,

[TABLE]

Proof.

By (11), we obtain

[TABLE]

By Theorem 1 and Lemma 5, we have

[TABLE]

The proof is complete. ∎

4. Equivalent norms for $A_{\omega}^{p}$ space

In this section, we give some equivalent norms for the space $A_{\omega}^{p}$ on the unit ball. These norms are inherited from different equivalent $H^{p}$ norms. First, we give some notations.

Let $\alpha>2$. The admissible approach region $\Gamma_{\zeta,\alpha}$ for some $\zeta\in\overline{\mathbb{B}}\backslash\{0\}$ can be defined as

[TABLE]

When $\zeta=0$, let $\Gamma_{\zeta,\alpha}=\{0\}$. Obviously, if $r>0$ and $r\zeta,\zeta\in\overline{\mathbb{B}}$, $z\in\Gamma_{\zeta,\alpha}$ if and only if $rz\in\Gamma_{r\zeta,\alpha}.$ Define

[TABLE]

It follows from Fubini’s Theorem, for a positive function $\varphi$ and a finite positive measure $\mu$, one has

[TABLE]

See [8], for example. This fact will be used frequently in this paper.

Proposition 3**.**

Suppose $\alpha>2$ is fixed and $\omega\in\hat{\mathcal{D}}$. Then we have the following statements.

1. (i)

$T_{z,\alpha}\subset S_{z,\alpha}$. 2. (ii)

There exist $r=r(\alpha)$ and $\beta>-1$, such that $S_{\frac{1+|z|}{2|z|}z,\beta}\subset T_{z,\alpha}$ when $|z|>r$. 3. (iii)

$\omega(T_{z,\alpha})\approx\omega(S_{z,0}).$**

Proof.

**(i). **Suppose $\zeta\in T_{z}$, that is,

[TABLE]

So, we have $|\zeta|>|z|$ and

[TABLE]

Therefore, $\zeta\in S_{z,\alpha}$, i.e. $T_{z}\subset S_{z,\alpha}$.

**(ii). ** Suppose $\zeta\in S_{\frac{1+|z|}{2|z|}z,\beta}$. Then we have $|\zeta|>\frac{1+|z|}{2}$ and

[TABLE]

Since $1-\frac{|z|^{2}}{|\zeta|^{2}}>\frac{(1-|z|)(3|z|+1)}{(1+|z|)^{2}}$,

[TABLE]

Let $\beta=\frac{2\alpha-7}{3}$ and $r(\alpha)\in(0,1)$ such that

[TABLE]

So, if $|z|>r(\alpha)$, we have

[TABLE]

That is, (ii) holds.

(iii). By (i), (1) and Lemma 1, we have

[TABLE]

Let $\beta$ and $r(\alpha)$ be fixed as in the proof of (ii). When $|z|>r$, similarly to the proof of (13), we have

[TABLE]

When $0<|z|\leq r$, let

[TABLE]

For any $\zeta\in E_{z}$, we have

[TABLE]

that is, $E_{z}\subset T_{z,\alpha}$. Therefore, for all $0<|z|\leq r$, we have

[TABLE]

So, (iv) holds. The proof is complete. ∎

In the rest of this paper, for simplicity, we write $\Gamma_{\eta,\alpha}$ and $T_{z,\alpha}$ by $\Gamma_{\eta}$ and $T_{z}$, respectively. Moreover, if $\omega\in\hat{\mathcal{D}}$ and $z\in\mathbb{B}\backslash\{0\}$, let

[TABLE]

The main result in this section is the following theorem.

Theorem 2**.**

Let $0<p<\infty$ and $\omega$ be a radial weight. Then

[TABLE]

Moreover, if $p\geq 2$ and $\omega\in\hat{\mathcal{D}}$,

[TABLE]

Proof.

For $0<r<1$, let $f_{r}(z)=f(rz)$. By Theorem 4.22 in [18], we have

[TABLE]

In the following, we always suppose $f(0)=0$. Then Fubini’s Theorem yields

[TABLE]

Hence (14) holds.

Suppose $p\geq 2$. By Theorem 4.17 in [18], we have

[TABLE]

So, for all $|z|<\frac{3}{4}$, we have

[TABLE]

Let $z=(z_{1},z_{2},\cdots,z_{n})\in\mathbb{B}$. By Cauchy’s Formula, we obtain

[TABLE]

when $|z|<\frac{1}{2}$ and $i=1,2,\cdots,n$. So, we have

[TABLE]

Here \nabla f(z)=\Big{(}\frac{\partial f}{\partial z_{1}}(z),\cdot\cdot\cdot,\frac{\partial f}{\partial z_{n}}(z)\Big{)}. For all $\tau<\frac{1}{2}$, we have

[TABLE]

Since $\lim\limits_{\tau\to 0}\int_{0}^{\tau}s\log\frac{1}{s}ds=0$, we can choose a fixed $\tau\in(0,\frac{1}{2})$ such that

[TABLE]

By Fubini’s Theorem, we have

[TABLE]

So, we get (16).

Since $\tau\in(0,1)$ is fixed, after a calculation, we have

• •

For all $z\in\mathbb{B}$, $\omega^{*}(z)\geq\omega^{n*}(z)\gtrsim(1-|z|)\hat{\omega}(z)$.

• •

For all $|z|>\frac{\tau^{2}}{2}$, $\omega^{*}(z)\approx\omega^{n*}(z)\approx(1-|z|)\hat{\omega}(z)$.

So, using (18) and (19), we obtain (17).

By Theorem B in [8], if $f(0)=0$, we have

[TABLE]

By Fubini’s Theorem, we have

[TABLE]

Similarly, by using the monotonicity of $\|f_{r}\|_{H^{p}}$, we have

[TABLE]

Then, (15) holds. The proof is complete. ∎

For any $f\in H(\mathbb{B})$ and $u\in\overline{\mathbb{B}}\backslash\{0\}$, let

[TABLE]

Then we have the following theorem.

Theorem 3**.**

Let $0<p<\infty$ and $\omega$ be a radial weight. Then for all $f\in H(\mathbb{B})$,

[TABLE]

Proof.

For any $u\in\mathbb{B}\backslash\{0\}$, let $r=|u|$ and $\xi=\frac{u}{|u|}$. Then

[TABLE]

and

[TABLE]

By Theorem A in [8], we have $\|N(f)\|_{L^{p}(\SS)}^{p}\lesssim\|f\|_{H^{p}}^{p}.$ Therefore,

[TABLE]

The fact that $\|f\|_{A_{\omega}^{p}}^{p}\leq\|N(f)\|_{L_{\omega}^{p}}^{p}$ is obvious. The proof is complete. ∎

5. Volterra integral operator from $A_{\omega}^{p}$ to $A_{\omega}^{q}$

In this section, we will describe the boundedness and compactness of $T_{g}:A_{\omega}^{p}\to A_{\omega}^{q}$. For this purpose, we first introduce some new function spaces.

Let $0<p\leq q<\infty$, $g\in H(\mathbb{B})$ and $\omega\in\hat{\mathcal{D}}$. We say that $g$ belongs to $\mathcal{C}^{q,p}(\omega^{*})$, if the measure $|\Re g(z)|^{2}\omega^{*}(z)dV(z)$ is a $q-$Carleson measure for $A_{\omega}^{p}$. $g\in\mathcal{C}_{0}^{q,p}(\omega^{*})$ if $|\Re g(z)|^{2}\omega^{*}(z)dV(z)$ is a vanishing $q-$Carleson measure for $A_{\omega}^{p}$. If $0<p\leq q<\infty$, Theorem 1 shows that $\mathcal{C}^{q,p}(\omega^{*})$ depends only on $\frac{q}{p}$. Consequently, for $0<p\leq q<\infty$, we will write $\mathcal{C}^{\kappa}(\omega^{*})$ instead of $\mathcal{C}^{q,p}(\omega^{*})$ where $\kappa=\frac{q}{p}$. Similarly, we can define $\mathcal{C}_{0}^{\kappa}(\omega^{*})$. Thus, if $\kappa\geq 1$, $\mathcal{C}^{\kappa}(\omega^{*})$ consists of those $g\in H(\mathbb{B})$ such that

[TABLE]

Before state and prove the main results in this section, we state some lemmas which will be used. For brief, if $r\in(0,1)$, let $S_{r}$ denote any Carleson block $S_{a}$ with $|a|=r$.

Lemma 7**.**

Let $0<p,q<\infty$, $g\in H(\mathbb{B})$ and $\omega\in\hat{\mathcal{D}}$.

1. (i)

If $T_{g}:A_{\omega}^{p}\to A_{\omega}^{q}$ is bounded, then

[TABLE] 2. (ii)

If $T_{g}:A_{\omega}^{p}\to A_{\omega}^{q}$ is compact, then

[TABLE]

Proof.

Assume $T_{g}:A_{\omega}^{p}\to A_{\omega}^{q}$ is bounded. Let

[TABLE]

for some $\gamma$ which is large enough such that Lemma 6 holds. For all $\frac{1}{2}<r<1$ and $h\in A_{\omega}^{q}$, we have

[TABLE]

Then, when $\frac{1}{2}<r<1$, for all $a\in\mathbb{B}$, by Lemma 6, we have

[TABLE]

The following facts are well know, that are

[TABLE]

Letting $f_{r}(z)=f(rz)$ for all $0<r<1$, when $|a|>\frac{1}{2}$, by (22), we have

[TABLE]

By Lemma 2, we have

[TABLE]

which implies the desired result.

$(ii)$ Assume that $T_{g}:A_{\omega}^{p}\to A_{\omega}^{q}$ is compact. By Lemma 6, $\{f_{a,p}\}$ is bounded and converges to 0 uniformly on compact subset of $\mathbb{B}$ as $|a|\to 1$. By Lemma 4,

[TABLE]

By (22), for any given $\varepsilon>0$, there exists a $r_{0}\in(0,1)$, such that when $|a|>r_{0}$,

[TABLE]

Then by repeating the proof of $(i)$, we can prove $(ii)$. The proof is complete. ∎

Lemma 8**.**

Let $0<\kappa<\infty$, $\omega\in\hat{\mathcal{D}}$ and $g\in H(\mathbb{B})$. Then the following statements hold.

1. (i)

$g\in\mathcal{C}^{2\kappa+1}(\omega^{*})$* if and only if*

[TABLE] 2. (ii)

$g\in\mathcal{C}_{0}^{2\kappa+1}(\omega^{*})$* if and only if*

[TABLE]

Proof.

Let $r_{0}>0$ be fixed and $D(a,r_{0})$ be the Bergman metric ball at $a$ with radius $r_{0}$. By Lemma 2.20 in [18], there exists $B=B(r_{0})>1$ such that, for all $z\in D(a,r_{0})$,

[TABLE]

When $|a|>\max\{\frac{B-1}{B},\frac{2B}{2B+1}\}=\frac{2B}{2B+1}$, let

[TABLE]

Then, $1-|\beta_{3}(a)|=(2B+1)(1-|a|)$. For all $z\in D(a,r_{0})$, we have

[TABLE]

and

[TABLE]

Therefore, $D(a,r_{0})\subset S_{\beta_{3}(a)}$ for all $|a|>\max\{\frac{B-1}{B},\frac{2B}{2B+1}\}$.

Assume that $g\in\mathcal{C}^{2\kappa+1}(\omega^{*})$. It is enough to prove (23) holds for $|a|>\frac{2B}{2B+1}$. By Lemma 2, we have $\omega^{*}\in\mathcal{R}$ and

[TABLE]

here $\varepsilon$ is any fixed positive number in (0,1).

When $|a|>\max\{\frac{B-1}{B},\frac{2B}{2B+1}\}$, by Lemma 2.24 in [18], Lemma 2 and Theorem 1, there is a $C=C(\kappa,r_{0},\omega)$ such that

[TABLE]

Hence, there exists $C=C(k,r_{0},\omega)$, such that

[TABLE]

Therefore, (23) holds.

Conversely, suppose that

[TABLE]

By Lemmas 1 and 2, we have

[TABLE]

It follows that $g\in\mathcal{C}^{2\kappa+1}(\omega^{*})$.

The assertion $(ii)$ can be proved by modifying the above proof in a standard way and we omit the detail. The proof is complete. ∎

Theorem 4**.**

Let $0<p\leq q<\infty$, $\omega\in\hat{\mathcal{D}}$, $\kappa=\frac{1}{p}-\frac{1}{q}$ and $g\in H(\mathbb{B})$.

1. (i)

If $n\kappa\geq 1$, then $T_{g}:A_{\omega}^{p}\to A_{\omega}^{q}$ is bounded if and only if $g$ is constant. 2. (ii)

If $0<n\kappa<1$, then the following conditions are equivalent:

1. (iia)

$T_{g}:A_{\omega}^{p}\to A_{\omega}^{q}$* is bounded;* 2. (iib)

$M_{\infty}(r,\Re g)\lesssim\frac{\omega^{\kappa}(S_{r})}{1-r}$** 3. (iic)

$g\in\mathcal{C}^{2\kappa+1}(\omega^{*})$. 3. (iii)

The following conditions are equivalent.

1. (iiia)

$T_{g}:A_{\omega}^{p}\to A_{\omega}^{p}$* is bounded;* 2. (iiib)

$g\in\mathcal{C}^{1}(\omega^{*})$.

Proof.

By Lemmas 2, 7 and 8, we see that $(i)$ holds, and $(iia)\Rightarrow(iib)\Leftrightarrow(iic)$. Let $d\mu_{g}(z)=|\Re g(z)|^{2}\omega^{*}(z)dV(z)$. First, we prove the statement $(ii)$.

Suppose that $(iic)$ holds and $q=2$. Then $d\mu_{g}$ is a 2-Carleson measure for $A_{\omega}^{p}$. By using (16) and Theorem 1, we have

[TABLE]

So, $T_{g}:A_{\omega}^{p}\to A_{\omega}^{2}$ is bounded.

Assume $f\in H^{\infty}$. By Lemmas 2 and 7, we get

[TABLE]

and

[TABLE]

That is to say $T_{g}f\in H^{\infty}$.

Suppose that $(iic)$ holds and $q>2$. Let $\beta=\frac{(2\kappa+1)q}{2kq+2}$ and $\beta^{\prime}=\frac{(2\kappa+1)q}{q-2}$. By (16) and Hölder’s inequality, we have

[TABLE]

Therefore, when $q>2$,

[TABLE]

For all $f\in A_{\omega}^{p}$, we have $\lim\limits_{r\to 1}\|f-f_{r}\|_{A_{\omega}^{p}}=0$. Then $\{f_{r}\}$ is a Cauchy’s sequence in $A_{\omega}^{p}$. By (24), there is a $F\in A_{\omega}^{q}$, such that $\lim\limits_{r\to 1}\|T_{g}f_{r}-F\|_{A_{\omega}^{q}}=0$. So, for all $z\in\mathbb{B}$, we have $\lim\limits_{r\to 1}|T_{g}f_{r}(z)-F(z)|=0$.

Meanwhile, for any fixed $z\in\mathbb{B}$, we have

[TABLE]

Therefore, $F=T_{g}f$ and

[TABLE]

So, $(iic)$ deduce $(iia)$ when $q>2$.

Suppose that $(iic)$ holds and $q<2$. Let $\tau=\frac{(2-q)p}{q}$. By (15), Hölder’s inequality and Theorem 3, we have

[TABLE]

where

[TABLE]

By Fubini’s Theorem, we have

[TABLE]

Here, $E_{z}=\{\eta\in\SS:\left|1-\langle z,\eta\rangle\right|<\frac{\alpha}{2}(1-|z|^{2})\}$. By the definition of $\Gamma_{\eta}$, we have $\alpha>2$. If $z\neq 0$, we have

[TABLE]

by the facts of

[TABLE]

and

[TABLE]

when $\zeta\in Q\left(\frac{z}{|z|},\sqrt{\frac{(\alpha-2)(1-|z|)}{2}}\right)$ and $\eta\in E_{z}$, respectively. By Lemma 1, we have $\sigma(E_{z})\approx(1-|z|)^{n}$.

Since $\int_{t}^{1}r^{2n-1}\omega(r)(1-\frac{t}{r})dr\approx(1-t)\hat{\omega}(t)$, by Fubini’s Theorem, (16) and (17), we have

[TABLE]

So, $\|T_{g}f\|_{A_{\omega}^{q}}\lesssim\|f\|_{A_{\omega}^{p}}$. Then we finish the proof of assertion $(ii)$.

Next we prove the assertion $(iii)$. When $p=2$, (iiia)$$\Leftrightarrow$$(iiib) is obvious. When $p<2$, by the the proof of (iic)$$\Rightarrow$$(iia) when $q<2$, we obtain (iiib)$$\Rightarrow$$(iiia).

Suppose $2<p\leq 4$ and $(iiib)$ holds. For all $f\in H^{\infty}$, by (16), we have

[TABLE]

By $g\in\mathcal{C}^{1}(\omega^{*})$, we see that $T_{g}$ is bounded on $A_{\omega}^{p-2}$. Since $f\in H^{\infty}\subset A_{\omega}^{p-2}$, $T_{g}f\in A_{\omega}^{p-2}$. By Theorem 1, we have

[TABLE]

Similar to the proof of $(iic)\Rightarrow(iia)$, we obtain (iiib)$$\Rightarrow$$(iiia) when $2<p\leq 4$. Using mathematical induction, we have (iiib)$$\Rightarrow$$(iiia) when $p>2$.

So, it remains to show that (iiia)$$\Rightarrow$$(iiib) when $p\neq 2$.

Suppose $p>2$ and $(iiia)$ holds. By the calculations from (26) to (28), Hölder’s inequality, Theorem 3 and (15), we have

[TABLE]

So, $|\Re g(z)|^{2}\omega^{*}(z)dV(z)$ is a $p-$Carleson measure for $A_{\omega}^{p}$, and thus $g\in\mathcal{C}^{1}(\omega^{*})$.

Suppose $0<p<2$ and $(iiia)$ holds. Recall that $d\mu_{g}(z)=|\Re g(z)|^{2}\omega^{*}(z)dV(z)$. Then by Lemma 7 and its proof we get

[TABLE]

Here, $\mathcal{B}$ is the Bloch space on the unit ball and $\|T_{g}\|$ is $\|T_{g}\|_{A_{\omega}^{p}\to A_{\omega}^{p}}$. Let $F_{a,p}$ be defined as (2) for some $\gamma$ large enough. Let $1<\tau_{1},\tau_{2}<\infty$ such that $\frac{\tau_{2}}{\tau_{1}}=\frac{p}{2}<1$, and let $\tau_{1}^{\prime},\tau_{2}^{\prime}$ be the conjugate indexes of $\tau_{1},\tau_{2}$ respectively.

By Lemma 2, Proposition 3, Hölder’s inequality, Fubini’s Theorem and (15), for any $a\in\mathbb{B}$ with $|a|\geq\frac{1}{2}$, we have

[TABLE]

[TABLE]

where

[TABLE]

Since $\frac{\tau_{2}^{\prime}}{\tau_{1}^{\prime}}>1$, we have $\left(\frac{\tau_{2}^{\prime}}{\tau_{1}^{\prime}}\right)^{\prime}=\frac{\tau_{2}(\tau_{1}-1)}{\tau_{1}-\tau_{2}}>1$. Let $\tau_{3}=\frac{\tau_{2}(\tau_{1}-1)}{\tau_{1}-\tau_{2}}$. We have

[TABLE]

By using Fubini’s Theorem, Proposition 3, Lemma 6, Lemma 5, Propositions 1, Remark 1, Lemma 2 and Corollary 1 in order, we have

[TABLE]

Then we have

[TABLE]

By the process of obtaining (31), if we replace $g(z)$ by $g_{r}(z)=g(rz)$, we have

[TABLE]

We now claim that there exists a constant $C=C(\omega)>0$ such that

[TABLE]

Taking this for granted for a moment, by (32) and (33) we have

[TABLE]

By Fatou’s Lemma,

[TABLE]

So $g\in\mathcal{C}^{1}(\omega^{*})$.

It remains to prove (33). For any fixed $r\in(\frac{1}{2},1)$, when $\frac{1}{2}<|a|\leq\frac{1}{2-r}$, by triangle inequality, we have

[TABLE]

When $\frac{1}{2}<|a|\leq\frac{1}{2-r}$, by (15) and (21), we have

[TABLE]

By (20) and the similar calculation, when $t\geq\frac{1}{4}$, we have

[TABLE]

Then

[TABLE]

When $|a|\geq\frac{1}{2-r}$, by (15), Lemma 7, (30) and $\gamma$ is large enough, we have

[TABLE]

Let $\Re^{-1}F_{a,p}(z)=\frac{1}{\frac{\gamma+n}{p}-1}(1-|a|^{2})\left(\frac{1-|a|^{2}}{1-\langle z,a\rangle}\right)^{\frac{\gamma+n}{p}-1}$,

[TABLE]

and

[TABLE]

Then, by using (15) and Lemma 6, we have

[TABLE]

[TABLE]

[TABLE]

For any $u\in\mathbb{B}\backslash\{0\}$ and $\xi\in\Gamma_{u}\backslash\{0\}$, let $r=|\xi|$ and $\eta=\frac{\xi}{|\xi|}$. Then we have

[TABLE]

By (27), we have

[TABLE]

By Lemma 2, when $\gamma$ is large enough, we have $J_{4}\lesssim(1-|a|)^{p}\omega(S_{a})$. So, we have $\|T_{g_{r}}F_{a,p}\|_{A_{\omega}^{p}}^{p}\lesssim\|T_{g}\|^{p}\omega(S_{a})$, that is, (33) holds. The proof is complete. ∎

Theorem 5**.**

Let $0<p\leq q<\infty$, $\omega\in\hat{\mathcal{D}}$, $\kappa=\frac{1}{p}-\frac{1}{q}$ and $g\in H(\mathbb{B})$.

1. (i)

If $0<n\kappa<1$, then the following conditions are equivalent:

1. (ia)

$T_{g}:A_{\omega}^{p}\to A_{\omega}^{q}$* is compact;* 2. (ib)

$M_{\infty}(r,\Re g)=o\left(\frac{\omega^{\kappa}(S_{a})}{1-r}\right)$; 3. (ic)

$g\in\mathcal{C}_{0}^{2\kappa+1}(\omega^{*})$. 2. (ii)

The following conditions are equivalent.

1. (iia)

$T_{g}:A_{\omega}^{p}\to A_{\omega}^{p}$* is compact;* 2. (iib)

$g\in\mathcal{C}_{0}^{1}(\omega^{*})$.

Proof.

By Lemmas 7 and 8, we have (ia)$$\Rightarrow$$(ib)$$\Leftrightarrow$$(ic). Let $d\mu_{g}(z)=|\Re g(z)|^{2}\omega^{*}(z)dV(z)$. First, we prove $(i)$.

Suppose that $(ic)$ holds and $q=2$. Then $\mu_{g}$ is a vanishing 2-Carleson measure for $A_{\omega}^{p}$. Using (16), we have

[TABLE]

So, $T_{g}:A_{\omega}^{p}\to A_{\omega}^{2}$ is compact by Theorem 1.

Suppose that $(ic)$ holds. By Theorem 4, $T_{g}:A_{\omega}^{p}\to A_{\omega}^{q}$ is bounded. For every $\varepsilon>0$, there is a $r\in(0,1)$, such that

[TABLE]

For any measurable subset $E$ of $\mathbb{B}$, define $\mu_{g,r}(E)=\mu_{g}(E\cap(\mathbb{B}\backslash r\mathbb{B}))$. By (12),

[TABLE]

When $q>2$, let $\beta=\frac{(2\kappa+1)q}{2\kappa q+2}$ and $\beta^{\prime}=\frac{(2\kappa+1)q}{q-2}$. For any $0<r<1$, by (16), Hölder’s inequality and Theorem 1, we have

[TABLE]

Then by Lemma 4, $(ia)$ holds.

When $0<q<2$, by (25) and (28), we have

[TABLE]

Here $\tau=\frac{(2-q)p}{q}$. By Lemma 4, $T_{g}:A_{\omega}^{p}\to A_{\omega}^{q}$ is compact. So, we finish the proof of (ic)$$\Rightarrow$$(ia).

When $p=2$, (iia)$$\Leftrightarrow$$(iib) is obvious.

When $p\neq 2$, by the proof of (ic)$$\Rightarrow$$(ia), we get (iib)$$\Rightarrow$$(iia).

Suppose $p>2$ and $(iia)$ holds. By (29), we have

[TABLE]

Let $f_{a,p}(z)=\frac{F_{a,p}(z)}{\|F_{a,p}\|_{A_{\omega}^{p}}}$ for some $\gamma$ is large enough. Then we have

[TABLE]

By Lemma 4, $(iib)$ holds.

Suppose $0<p<2$ and $(iia)$ holds. Let $f_{a,p}(z)=\frac{F_{a,p}(z)}{\|F_{a,p}\|_{A_{\omega}^{p}}}$ for some $\gamma$ is large enough. Then $\sup_{a\in\mathbb{B}}\frac{\mu_{g}(S_{a})}{\omega(S_{a})}<\infty$. By (31), we have

[TABLE]

By Lemma 4, $(iib)$ holds. The proof is complete. ∎

6. inclusion relations about $\mathcal{C}^{1}(\omega^{*})(\mathcal{C}_{0}^{1}(\omega^{*}))$

In this section, we discuss the inclusion relationship between $\mathcal{C}^{1}(\omega^{*})(\mathcal{C}_{0}^{1}(\omega^{*}))$ and some other function spaces, such as $\mathcal{B}(\mathcal{B}_{0})$ and $BMOA(VMOA)$.

Recall that a function $f\in H(\mathbb{B})$ is said to belong to the Bloch space, denoted by ${\mathcal{B}}={\mathcal{B}}(\mathbb{B})$, if

[TABLE]

It is well known that ${\mathcal{B}}$ is a Banach space with the above norm. Let $\mathcal{B}_{0}$, called the little Bloch space, denote the subspace of $\mathcal{B}$ consisting of those $f\in\mathcal{B}$ for which

[TABLE]

Recall that

[TABLE]

A function $f\in H(\mathbb{B})$ is said to belong to the space $BMOA$ if and only if

[TABLE]

for some (equivalently, for any) $\delta>0$. Let $VMOA$ denote the subspace of $BMOA$ for which

[TABLE]

More information about $\mathcal{B}$, BMOA and VMOA can be found in [18].

Proposition 4**.**

1. (i)

If $\omega\in\hat{\mathcal{D}}$, then $\mathcal{C}^{1}(\omega^{*})\subset A_{\omega}^{p}$ for all $p>0$. 2. (ii)

If $\omega\in\hat{\mathcal{D}}$, then $BMOA\subset\mathcal{C}^{1}(\omega^{*})\subset\mathcal{B}$ and $VMOA\subset\mathcal{C}_{0}^{1}(\omega^{*})\subset\mathcal{B}_{0}$. 3. (iii)

If $\omega\in\hat{\mathcal{D}}$, with the norm $\|\cdot\|_{\mathcal{C}^{1}(\omega^{*})}$, ${C}^{1}(\omega^{*})$ is a Banach space and $\mathcal{C}_{0}^{1}(\omega^{*})$ is a closed subspace of ${C}^{1}(\omega^{*})$. 4. (iv)

If $\omega\in\mathcal{R}$, then $\mathcal{C}^{1}(\omega^{*})=\mathcal{B}$ and $\mathcal{C}_{0}^{1}(\omega^{*})=\mathcal{B}_{0}$. 5. (v)

If $\omega\in\mathcal{I}$, then $\mathcal{C}^{1}(\omega^{*})\subsetneq\mathcal{B}$ and $\mathcal{C}_{0}^{1}(\omega^{*})\subsetneq\mathcal{B}_{0}$. 6. (vi)

If $\omega\in\mathcal{I}$ and both $\omega$ and $\frac{\hat{\omega}(r)}{(1-r)\omega(r)}$ are increasing on $[0,1)$, then $VMOA\subsetneq\mathcal{C}_{o}^{1}(\omega^{*})$ and $BMOA\subsetneq\mathcal{C}^{1}(\omega^{*})$.

Proof.

$(i)$. Suppose $g\in\mathcal{C}^{1}(\omega^{*})$. By Theorem 2,

[TABLE]

So, $g\in A_{\omega}^{2}$. Similarly, we have

[TABLE]

By mathematical induction, for all $k\in\mathbb{N}$, $g-g(0)\in A_{\omega}^{2k}$. So, $\mathcal{C}^{1}(\omega^{*})\subset A_{\omega}^{p}$ for all $p>0$.

$(ii)$. By Theorems 4, 5 and Lemma 7, we have $\mathcal{C}^{1}(\omega^{*})\subset\mathcal{B}$ and $\mathcal{C}_{0}^{1}(\omega^{*})\subset\mathcal{B}_{0}$. Suppose $g\in BMOA$, let $d\mu_{g}^{*}(z)=(1-|z|^{2})|\Re g(z)|^{2}dV(z).$ Then

[TABLE]

where $\delta$ is any fixed positive constant. By Proposition 2, for any $a\in\mathbb{B}$ with $|a|>\frac{1}{2}$, there are $\xi\in\SS$ and $r=2(1-|a|)$, such that $S_{a}\subset S^{*}(\xi,r)$. By Lemma 2, we have

[TABLE]

So, $g\in\mathcal{C}^{1}(\omega^{*})$. That is $BMOA\subset\mathcal{C}^{1}(\omega^{*})$. Similarly, by Theorem 5.19 in [18], we have $VMOA\subset\mathcal{C}_{0}^{1}(\omega^{*})$.

$(iii)$. Suppose that $\{g_{k}\}$ is a Cauchy’s sequence in $\mathcal{C}^{1}(\omega^{*})$. By (35) and Theorem 1, $\{g_{k}\}$ is a Cauchy’s sequence in $A_{\omega}^{2}$. Then we have $g\in A_{\omega}^{2}$ such that $\lim\limits_{k\to\infty}\|g_{k}-g\|_{A_{\omega}^{2}}=0.$ By Theorem 2, Fatou’s Lemma and Theorem 1, for any $f\in A_{\omega}^{2}$, we have

[TABLE]

So, $T_{g}:A_{\omega}^{2}\to A_{\omega}^{2}$ is bounded. Then $g\in\mathcal{C}^{1}(\omega^{*})$. Similarly, for all $f\in A_{\omega}^{2}$, we have

[TABLE]

By Theorem 1, $\lim\limits_{j\to\infty}\|g-g_{j}\|_{\mathcal{C}^{1}(\omega^{*})}=0$. So, $\mathcal{C}^{1}(\omega^{*})$ is a Banach space.

Suppose $\{g_{k}\}$ is a Cauchy’s sequence in $\mathcal{C}_{0}^{1}(\omega^{*})$. Then there exists $g\in\mathcal{C}^{1}(\omega^{*})$ such that $\lim_{k\to\infty}\|g_{k}-g\|_{\mathcal{C}^{1}(\omega^{*})}=0$. Let $\{f_{j}\}$ be a bounded sequence in $A_{\omega}^{2}$ such that $\{f_{j}\}$ converges to 0 uniformly on compact subsets of $\mathbb{B}$. By Theorems 1 and 2, we have

[TABLE]

For any given $\varepsilon>0$, we can choose a $k\in\mathbb{N}$ such that $\|g-g_{k}\|_{\mathcal{C}^{1}(\omega^{*})}<\varepsilon^{2}$. By Lemma 4,

[TABLE]

Then $T_{g}:A_{\omega}^{2}\to A_{\omega}^{2}$ is compact. So, $g\in\mathcal{C}_{0}^{1}(\omega^{*})$. That is, $\mathcal{C}_{0}^{1}(\omega^{*})$ is a closed subspace of ${C}^{1}(\omega^{*})$.

$(iv)$. Suppose $\omega\in\mathcal{R}$. By observation (v) after Lemma 1.1 in [10], there exists $\beta>-1$ and $\delta\in(0,1)$, such that $\frac{\omega(r)}{(1-r)^{\beta}}$ is decreasing on $[\delta,1)$. Without loss of generality, let $\delta=0$. Then for all $g\in\mathcal{B}$ and $a\in\mathbb{B}$ such that $|a|>\frac{1}{2}$, by Lemmas 1 and 2, we have

[TABLE]

Then $\mathcal{B}\subset\mathcal{C}^{1}(\omega^{*})$. So, $\mathcal{B}=\mathcal{C}^{1}(\omega^{*})$. Similarly, $\mathcal{B}_{0}=\mathcal{C}_{0}^{1}(\omega^{*})$.

$(v)$ and $(vi)$ have been proved in [10] when $n=1$, so they also hold for $n>1$. The proof is complete. ∎

Proposition 5**.**

Let $\omega\in\hat{\mathcal{D}}$ and $g\in\mathcal{C}^{1}(\omega^{*})$. Then the following statements are equivalent.

1. (i)

$g\in C_{0}^{1}(\omega^{*})$; 2. (ii)

$\lim\limits_{r\to 1}\|g-g_{r}\|_{\mathcal{C}^{1}(\omega^{*})}=0$, here $g_{r}(z)=g(rz)$; 3. (iii)

There is a sequence of polynomials $\{p_{k}\}$ such that $\lim\limits_{r\to 1}\|g-p_{k}\|_{\mathcal{C}^{1}(\omega^{*})}=0$.

Proof.

(i)$$\Rightarrow$$(ii). Suppose $g\in C_{0}^{1}(\omega^{*})$. Let $\gamma$ be large enough and

[TABLE]

Then Lemma 4, Theorems 1 and 2 yield

[TABLE]

and for any $r\in(0,1)$,

[TABLE]

By (36) and $(ii)$ in Proposition 4, for any $\varepsilon>0$, there is a $r_{0}\in(\frac{1}{2},1)$ such that

[TABLE]

By (34), if $r_{0}<|a|<\frac{1}{2-r}$, we have

[TABLE]

If $|a|>\max\left\{r_{0},\frac{1}{2-r}\right\}$, by (16), we have

[TABLE]

By Theorem 1.12 in [18] and Lemma 2, if $\gamma$ is large enough, we have

[TABLE]

So, when $|a|>\max\left\{r_{0},\frac{1}{2-r}\right\}$, by (38),

[TABLE]

By (37), (39) and (40), we obtain

[TABLE]

When $|a|\leq r_{0}$, we have

[TABLE]

So, there is a $r_{1}\geq r_{0}$, such that

[TABLE]

Therefore,

[TABLE]

Then $(ii)$ holds.

(ii)$$\Rightarrow$$(iii). For any $n\in\mathbb{N}$, there is a polynomial $p_{n}$ such that

[TABLE]

Since

[TABLE]

we obtain $(iii)$.

(iii)$$\Rightarrow$$(i). For any polynomial $p_{n}$, we have $\|\Re p_{n}\|_{H^{\infty}}<\infty$. Then by Lemmas 1 and 2, for $|a|>\frac{1}{2}$ we have

[TABLE]

So, $p_{n}\in\mathcal{C}_{0}^{1}(\omega^{*})$. Then by $(iii)$ of Proposition 4, $(i)$ holds. The proof is complete. ∎