# Difference of weighted composition operators on weighted Bergman spaces over
the unit Ball

Lian Hu, Songxiao Li∗ and Yecheng Shi  Lian Hu
Institute of Fundamental and Frontier Sciences, University of Electronic
Science and Technology of China, 610054, Chengdu, Sichuan, P.R. China.
<EMAIL_ADDRESS>Songxiao Li
Department of Mathematics, Shantou University, Shantou Guangdong 515063, P.R.
China<EMAIL_ADDRESS>Yecheng Shi
School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang
524048, Guangdong, P. R. China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we characterize the boundedness and compactness of differences
of weighted composition operators from weighted Bergman spaces
$A^{p}_{\omega}$ induced by a doubling weight $\omega$ to Lebesgue spaces
$L^{q}_{\mu}$ on the unit ball for full $0<p,q<\infty$, which extend many
results on the unit disk. As a byproduct, a new characterization of
$q$-Carleson the measure for $A^{p}_{\omega}$ in terms of the Bergman metric
ball is also presented. Keywords:Difference, Weighted Bergman space, weighted
composition operator, doubling weight, Carleson measure.

###### 2010 Mathematics Subject Classification:

32A36, 47B38

$\ast$ Corresponding author.

##  1\. Introduction

Denote by $H(\mathbb{B}_{n})$ and $\mathcal{S}(\mathbb{B}_{n})$ the space of
analytic functions and holomorphic self-maps on the unit ball $\mathbb{B}_{n}$
of the complex $n$-space ${\mathbb{C}}^{n}$, respectively. Given
$\varphi\in\mathcal{S}(\mathbb{B}_{n})$ and a Borel function $u$ on
$\mathbb{B}_{n}$, the weighted composition operator $uC_{\varphi}$ with symbol
$\varphi$ and weight $u$ is defined by

$uC_{\varphi}f=f\circ\varphi$

for $f\in H(\mathbb{B}_{n})$. When $u\equiv 1$, the operator $uC_{\varphi}$
becomes the classical composition operator $C_{\varphi}$. The theory of
(weighted) composition operators on various settings has been extensively
studied in recent decades and we refer to the books [5] and [14]. The
difference of composition operators has become an attractive topic in various
areas of function theory and operator theory. Recently, Choe et al. [3, 4]
completely characterized the bounded and compact differences of weighted
composition operators from standard weighted Bergman spaces to Lebesgue spaces
on the unit disk. Subsequently, Choe et al. [2] extend their results to the
unit ball. Chen [1] generalized their results to the doubling weight setting
and characterized the difference of composition operators from weighted
Bergman spaces $A^{p}_{\omega}$ to Lebesgue spaces $L^{q}_{\mu}$ for full
$0<p,q<\infty$. The main purpose of the current paper is to generalize the
main results in [1] to the unit ball setting.

To begin with, we recall some definitions. A positive, measurable and
integrable function $\omega$, satisfying $\omega(z)=\omega(|z|)$ for all
$z\in\mathbb{B}_{n}$, is called a radial weight on $\mathbb{B}_{n}$. Put
$\hat{\omega}(r)=\int_{r}^{1}\omega(s)ds$ for $r\in[0,1)$. A radial weight
$\omega$ is called a doubling weight, denoted by $\omega\in\hat{\mathcal{D}}$,
if there is a constant $C=C(\omega)\geq 1$ such that

$\hat{\omega}(r)<C\hat{\omega}\left({r+1\over
2}\right){\rm~{}for~{}}r\in[0,1).$

A radial weight $\omega$ is called a reverse doubling weight, denoted by
$\omega\in\breve{\mathcal{D}}$, if there exists constants
$C=C(\omega),K=K(\omega)>1$ such that

$\hat{\omega}(r)\geq C\hat{\omega}\left(1-{1-r\over
K}\right){\rm~{}for~{}}r\in[0,1).$

Define $\mathcal{D}=\hat{\mathcal{D}}\cap\breve{\mathcal{D}}$, called the two-
sides doubling weight. See [10, 11] and the references therein for more
results on doubling weights.

For $0<p<\infty$ and a weight function $\omega$, the weighted Bergman space
$A^{p}_{\omega}$ is the space of all functions $f\in H(\mathbb{B}_{n})$ such
that

$\|f\|^{p}_{A^{p}_{\omega}}=\int_{\mathbb{B}_{n}}|f(z)|^{p}\omega(z)dV(z)<\infty,$

where $dV$ is the normalized volume measure on $\mathbb{B}_{n}$. Clearly,
$A^{p}_{\omega}$ is a Banach space when $1\leq p<\infty$ and a complete metric
space with the distance $\rho(f,g)=\|f-g\|^{p}_{A^{p}_{\omega}}$ when $0<p<1$.
Taking
$\omega(z)={\Gamma(n+\alpha+1)\over\Gamma(n+1)\Gamma(\alpha+1)}(1-|z|^{2})^{\alpha},z\in\mathbb{B}_{n}$
with $\alpha>-1$, $A^{p}_{\omega}$ becomes the standard weighted Bergman space
$A^{p}_{\alpha}$.

To present our results, we introduce several notations. For
$\varphi,\psi\in\mathcal{S}(\mathbb{B}_{n})$, put

$\rho(z):=\rho(\varphi(z),\psi(z)),\,\,z\in\mathbb{B}_{n},$

where $\rho(\cdot,\cdot)$ denotes the pseudohyperbolic distance on
$\mathbb{B}_{n}$; see Sect. 2.2. Given a positive Borel measure $\nu$ on
$\mathbb{B}_{n}$, the pullback measure $\nu\circ\varphi^{-1}$ is defined by

$(\nu\circ\varphi^{-1})(E)=\nu[\varphi^{-1}(E)]$

for Borel sets $E\subset\mathbb{B}_{n}$. Let $0<r<1$, $0<\alpha,q<\infty$,
$u,v$ be weights and $\mu$ be a positive Borel measure on $\mathbb{B}_{n}$.
The pullback measures $\eta$ and $\sigma_{r}$ are defined by

$\eta=(|\rho u|^{q}d\mu)\circ\varphi^{-1}+(|\rho v|^{q}d\mu)\circ\psi^{-1},$

and

$\sigma_{r}=(\chi_{G_{r}}|u-v|^{q}d\mu)\varphi^{-1}+(\chi_{G_{r}}|u-v|^{q}d\mu)\psi^{-1},$

respectively, where $\chi_{G_{r}}$ is the characteristic function of the set

$G_{r}:=\\{z\in\mathbb{B}_{n}:\rho(z)<r\\}.$

Recall that for a Banach space $X$ of analytic functions and a positive Borel
measure $\mu$ on $\mathbb{B}_{n}$, $\mu$ is called a (varnishing) $q$-Carleson
measure for $X$ if the identity operator $I_{d}:X\to L^{q}_{\mu}$ is bounded
(compact). In this paper, we completely characterized the boundededness and
compactness of differences of weighted composition operators from weighted
Bergman spaces $A^{p}_{\omega}$ to Lebesgue spaces $L^{q}_{\mu}$ for full
$0<p,q<\infty$ on the unit ball. Our main results are divided into two cases:
$p\leq q$ and $p>q$, which are stated as follows.

######  Theorem 1.1.

Let $0<p\leq q<\infty$, $0<r<1$, $\omega\in\mathcal{D}$,
$\varphi,\psi\in\mathcal{S}(\mathbb{B}_{n})$, $u,v$ be weights and $\mu$ be a
positive Borel measure on $\mathbb{B}_{n}$. Then
$uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$ is bounded (resp.
compact) if and only if $\eta+\sigma_{r}$ is a (resp. varnishing) $q$-Carleson
measure for $A^{p}_{\omega}$.

######  Theorem 1.2.

Let $0<q<p<\infty$, $0<r<1$, $\omega\in\mathcal{D}$,
$\varphi,\psi\in\mathcal{S}(\mathbb{B}_{n})$, $u,v$ be weights and $\mu$ be a
positive Borel measure on $\mathbb{B}_{n}$. Then the following statements are
equivalent:

  1. (i)

$uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$ is bounded;

  2. (ii)

$uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$ is compact;

  3. (iii)

$\eta+\sigma_{r}$ is a $q$-Carleson measure for $A^{p}_{\omega}$.

This paper is organized as follows. We give some preliminaries in Section 2.
In Section 3, some new characterizations of the (varnishing) $q$-Carleson
measure for $A^{p}_{\omega}$ for full $0<p,q<\infty$ are given. In Section 4,
we prove our main results.

Throughout this paper, let $C$ be a positive constant that may change from one
step to the next. For nonnegative quantities $A$ and $B$, we say that
$A\lesssim B$ if there is a constant $C>0$ such that $A\leq CB$. The symbol
$A\approx B$ means that $A\lesssim B\lesssim A$.

##  2\. Preliminaries

In this section, we collect some basic facts that are important to our proof.

###  2.1. Compact operator

Let $X,Y$ be topological vector spaces. A linear map $T:X\to Y$ is said to be
compact if the image of any bounded sequence in $X$ has a subsequence that
converges in $Y$. To study the compactness of $T:A^{p}_{\omega}\to
L^{q}_{\mu}$, we need the following lemma. Its verification is a simplified
modification of the proof for Proposition 3.11 in [5].

######  Lemma 2.1.

Let $0<p,q<\infty$, $\omega\in\mathcal{D}$ and $\mu$ be a positive Borel
measure on $\mathbb{B}_{n}$. Suppose that $T:A^{p}_{\omega}\to L^{q}_{\mu}$ is
linear and bounded. Then $T:A^{p}_{\omega}\to L^{q}_{\mu}$ is compact if and
only if $Tf_{k}\to 0$ in $L^{q}_{\mu}$, where $\\{f_{k}\\}$ is bounded in
$A^{p}_{\omega}$ and converges to $0$ uniformly on any compact subset of
$\mathbb{B}_{n}$.

###  2.2. Pseudohyperbolic distance

Recall that the pseudohyperbolic distance between $a,z\in\mathbb{B}_{n}$ is
defined by

$\rho(a,z)=|\varphi_{a}(z)|,$

where $\varphi_{a}(z)$ is the Möbius transformation in $\mathbb{B}_{n}$. Let
$\beta(\cdot,\cdot)$ denote the Bergman metric on $\mathbb{B}_{n}$, that is,

$\beta(z,w)={1\over 2}\log{1+\rho(z,w)\over
1-\rho(z,w)},\,\,z,w\in\mathbb{B}_{n},$

and $D(z,r)=\\{w\in\mathbb{B}_{n}:\beta(z,w)<r\\}$ for $r\in(0,1)$ be the
Bergman metric ball centered at $z$ with radius $r$. Let $P_{z}$ be the
orthogonal projection from ${\mathbb{C}}^{n}$ onto the one dimensional
subspace $[z]$ generated by $z$ and $Q_{z}$ be the orthogonal projection from
${\mathbb{C}}^{n}$ onto ${\mathbb{C}}^{n}\ominus[z]$. For any
$z\in\mathbb{B}_{n}\backslash\\{0\\}$ and $r>0$, $D(z,r)$ is an ellipsoid
consisting of all points $w\in\mathbb{B}_{n}$ that satisfy

${|P_{z}(w)-c|^{2}\over s^{2}R^{2}}+{|Q_{z}(w)|^{2}\over sR^{2}}<1,$ (2.1)

where

$R=\tanh(r),\,c={(1-R^{2})z\over 1-R^{2}|z|^{2}},\,s={1-|z|^{2}\over
1-R^{2}|z|^{2}}.$

It is clear that

$1-|z|\approx|1-|w|\approx|1-\langle z,w\rangle|$ (2.2)

and

$|1-\langle z,a\rangle|\approx|1-\langle a,w\rangle|$

for any $a\in\mathbb{B}_{n}$ and $z,w\in\mathbb{B}_{n}$ with $\beta(z,w)<r$,
where the constants suppressed depend only on $r$ and $n$.

For $\xi\in\partial\mathbb{B}_{n}$ and $\delta>0$, let

$S(\xi,\delta):=\\{z\in\mathbb{B}_{n}|~{}|1-\langle z,\xi\rangle|<\delta\\}$

be the Carleson tube at $\xi$. From (2.13) in [2], we see that

$D(te_{1},r)\subset S\left(e_{1},{2(1-t)\over 1-\tanh^{-1}r}\right)$ (2.3)

for all $r,t\in(0,1)$.

###  2.3. Separated sequences and lattices

A sequence of points $\\{a_{j}\\}\subset\mathbb{B}_{n}$ is said to be
separated if there is a constant $\delta>0$ such that the Bergman metric
$\beta(a_{i},a_{j})\geq\delta$ for all $i$ and $j$ with $i\neq j$. This
implies that there exists $r>0$ such that the Bergman metric balls
$D(a_{j},r)$ are pairwise disjoint.

######  Lemma 2.2.

[15, Theorem 2.23] There exists a positive integer $N$ such that, for any
$0<r\leq 1$ we can find a sequence $\\{a_{k}\\}\in\mathbb{B}_{n}$ with the
following properties:

  1. (i)

$\mathbb{B}_{n}=\cup_{k}D(a_{k},r)$;

  2. (ii)

The sets $D(a_{k},{r\over 4})$ are mutually disjoint;

  3. (iii)

Each point $z\in\mathbb{B}_{n}$ belongs to at most $N$ of the sets
$D(a_{k},4r)$.

Any sequence $\\{a_{k}\\}$ satisfying the above conditions is called an
$r$-lattice in the Bergman metric. Clearly, any $r$-lattice sequence is
separated.

###  2.4. Function property

From [6, Lemma 2.2], we know that the weight $\omega$ has the following
properties.

######  Lemma 2.3.

Let $\omega\in\mathcal{D}$. For $a,z\in\mathbb{B}_{n}$, then the following
statements hold:

  1. (i)

$\omega(S_{a})\approx(1-|a|)^{n}\int_{|a|}^{1}\omega(s)ds$, where $S_{a}$ is
the Carleson block;

  2. (ii)

$\hat{\omega}(a)\approx\hat{\omega}(z)$, if $1-|a|\approx 1-|z|$;

  3. (iii)

There are $0<\alpha:=\alpha(\omega)\leq\beta:=\beta(\omega)$ and
$C=C(\omega)>1$ such that

${1\over C}\left({1-s\over
1-t}\right)^{\lambda}\leq{\hat{\omega}(s)\over\hat{\omega}(t)}\leq
C\left({1-s\over 1-t}\right)^{\beta},0\leq s\leq t<1.$

For $\omega\in\mathcal{D}$, the twisted weight $W$ is defined as follows.

$W(z)=W_{\omega}(r):={\hat{\omega}(z)\over 1-|z|},\,\,z\in\mathbb{B}_{n}.$

The following result may have appeared in some literature, but for the benefit
of the reader, we provide a brief proof here.

######  Lemma 2.4.

Let $0<p<\infty,\omega\in\mathcal{D}$. Then
$\|f\|_{A^{p}_{\omega}}\approx\|f\|_{A^{p}_{W}}$ for all $f\in
H(\mathbb{B}_{n})$.

###### Proof.

Let $r_{k}=1-2^{-k}$ for all $k\in{\mathbb{N}}\cup\\{0\\}$. By
$\omega\in\mathcal{D}$, there is $C=C(\omega)\geq 1$ such that

$\displaystyle\widehat{W}(r_{k})=$
$\displaystyle\int_{r_{k}}^{r_{k+1}}{\hat{\omega}(s)\over
1-s}ds+\int_{r_{k+1}}^{1}W(s)ds$ $\displaystyle\leq$
$\displaystyle\hat{\omega}(r_{k})\int_{r_{k}}^{r_{k+1}}{1\over
1-s}ds+\int_{r_{k+1}}^{1}W(s)ds$ $\displaystyle\leq$ $\displaystyle C\log
2\cdot\hat{\omega}(r_{k+1})+\int_{r_{k+1}}^{1}W(s)ds\leq
C_{1}\widehat{W}(r_{k+1})$

for some $C_{1}=C_{1}(\omega)\geq 1$. Then similar to the proof of Lemma
2.1(${\rm i}\Rightarrow{\rm iii}$) in [10], there are
$\gamma:=\gamma(\omega)>0$ and $C:=C(\omega)>0$ such that

$\int_{0}^{t}\left({1-t\over 1-s}\right)^{\gamma}W(s)ds\leq
C\widehat{W}(t),\,\,0\leq t<1,$

which gives that $W\in\hat{\mathcal{D}}$ by Lemma A(iii) in [13]. Applying
Theorem 1 in [7], $\|f\|_{A^{p}_{\omega}}\approx\|f\|_{A^{p}_{W}}$ for all
$f\in H(\mathbb{B}_{n})$ if $W(S_{a})\approx\omega(S_{a})$ for
$a\in\mathbb{B}_{n}$. Since $W$ and $\omega$ are radial weights, we get the
desired result if $\widehat{W}(r)\approx\hat{\omega}(r)$ for $0\leq r<1$.
Using Lemma B(iii) in [13], for $0\leq r<1$, there is a $C:=C(\omega)>0$ such
that

$\widehat{W}(r)=\int_{r}^{1}{\hat{\omega}(s)\over 1-s}ds\leq C\hat{\omega}(r)$

and by Lemma A(ii) in [13], there are constants $\beta:=\beta(\omega)>0$ and
$C:=C(\omega)>0$ such that

$\widehat{W}(r)=\int_{r}^{1}{\hat{\omega}(s)\over 1-s}ds\geq
C\int_{r}^{1}{\hat{\omega}(r)\over 1-s}\left(1-s\over 1-r\right)^{\beta}ds\geq
C\hat{\omega}(r).$

Therefore, $\widehat{W}(r)\approx\hat{\omega}(r)$ for $0\leq r<1$. The proof
is complete. ∎

The following Lemma deals with the integrability of doubling weights.

######  Lemma 2.5.

If $\omega\in{\mathcal{D}}$, then there is
$\lambda_{0}=\lambda_{0}(\omega)\geq 0$ such that

$\int_{\mathbb{B}_{n}}{\omega(z)\over|1-\langle z,a\rangle|^{\lambda
n+n}}dV(z)\approx{\hat{\omega}(a)\over(1-|a|)^{\lambda
n}},a\in\mathbb{B}_{n},$

for all $\lambda>\lambda_{0}$.

###### Proof.

Using Theorem 1.12 in [15] and Lemma 2.1 (iii) in [10], we have

$\displaystyle\int_{\mathbb{B}_{n}}{\omega(z)\over|1-\langle
z,a\rangle|^{\lambda n+n}}dV(z)=$ $\displaystyle
2n\int_{0}^{1}\omega(r)r^{2n-1}\int_{\SS_{n}}{1\over|1-\langle
r\xi,a\rangle|^{\lambda n+n}}d\sigma(\xi)dr$ $\displaystyle\approx$
$\displaystyle\int_{0}^{1}{\omega(r)r^{2n-1}\over(1-|a|r)^{\lambda n}}dr$
$\displaystyle\leq$ $\displaystyle\int_{0}^{|a|}{\omega(r)\over(1-r)^{\lambda
n}}dr+\int_{|a|}^{1}{\omega(r)\over(1-|a|)^{\lambda n}}dr$
$\displaystyle\lesssim$
$\displaystyle{\int_{|a|}^{1}\omega(r)dr\over(1-|a|)^{\lambda
n}}={\hat{\omega}(a)\over(1-|a|)^{\lambda n}}$

and

$\displaystyle\int_{\mathbb{B}_{n}}{\omega(z)\over|1-\langle
z,a\rangle|^{\lambda n+n}}dV(z)=$ $\displaystyle
2n\int_{0}^{1}\omega(r)r^{2n-1}\int_{\SS_{n}}{1\over|1-\langle
r\xi,a\rangle|^{\lambda n+n}}d\sigma(\xi)dr$ $\displaystyle\gtrsim$
$\displaystyle\int_{|a|}^{1}{\omega(r)r^{2n-1}\over(1-|a|)^{\lambda
n}}dr\approx{\int_{|a|}^{1}\omega(r)dr\over(1-|a|)^{\lambda
n}}={\hat{\omega}(a)\over(1-|a|)^{\lambda n}}.$

The proof is complete. ∎

To study the boundedness and compactness of
$uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$, we choose
$K_{a}(z)={1\over 1-\langle a,z\rangle},z,a\in\mathbb{B}_{n}$ as the test
function in the case $p\leq q$. From the above lemma, we see that

$\|K_{a}^{s}\|_{A^{p}_{\omega}}^{p}\approx{\hat{\omega}(a)\over(1-|a|)^{ps-n}}$
(2.4)

for any $s>{\lambda n+n\over p}$. In the case $p>q$, we need to consider
another test function, given by the following Lemma 2.7. Before presenting
Lemma 2.7, we first introduce a necessary lemma, whose proof is similar to
that of Lemma 2, and we only give the key steps.

######  Lemma 2.6.

If $\omega\in{\mathcal{D}}$, then there is $\beta=\beta(\omega)>0$ and
$\lambda_{0}=\lambda_{0}(\omega)\geq 0$ such that

$\int_{\mathbb{B}_{n}}{\omega(z)\over(1-|z|^{2})^{t}|1-\langle
z,a\rangle|^{\lambda n+n}}dV(z)\lesssim{\hat{\omega}(a)\over(1-|a|)^{\lambda
n+t}},a\in\mathbb{B}_{n},$

for all $t>\beta,\lambda>\lambda_{0}$.

###### Proof.

Let $\beta=\beta(\omega)$ be the constant in Lemma 2.3(iii). Applying Theorem
1.12 in [15] and Lemma 2.1 (iii) in [10], we have

$\displaystyle\int_{\mathbb{B}_{n}}{\omega(z)\over(1-|z|^{2})^{t}|1-\langle
z,a\rangle|^{\lambda n+n}}dV(z)\approx$
$\displaystyle\int_{0}^{1}{\omega(r)r^{2n-1}\over(1-r)^{t}(1-|a|r)^{\lambda
n}}dr$ $\displaystyle\leq$
$\displaystyle\int_{0}^{|a|}{\omega(r)\over(1-r)^{\lambda
n+t}}dr+\int_{|a|}^{1}{\omega(r)\over(1-r)^{\lambda n+t}}dr.$

Note that Lemma 2.3(iii) gives that

$\displaystyle\int_{|a|}^{1}{\omega(r)\over(1-r)^{\lambda n+t}}dr\leq$
$\displaystyle\int_{|a|}^{1}{\omega(r)\over(1-|a|)^{\lambda
n+t}}{\hat{\omega}(a)\over\hat{\omega}(r)}dr\leq{\hat{\omega}(a)\over(1-|a|)^{\lambda
n+t}},$

which deduces the desired result. ∎

The unit disk case of the following lemma is given by Peláez and Rättyä in
[12, Theorem 1]. Here, we prove the high-dimensional case by using a different
method from theirs.

######  Lemma 2.7.

Let $0<p<\infty,\omega\in\mathcal{D}$ and $\\{a_{k}\\}$ be a separated
sequence in $\mathbb{B}_{n}$. If $t>n+{\beta(\omega)+\lambda(\omega)n+n\over
p}$ for some $\beta(\omega),\lambda(\omega)>0$ and
$\lambda=\\{\lambda_{k}\\}\in l^{p}$, then the function $F$ defined by

$F(z):=\sum_{k=1}^{\infty}\lambda_{k}{(1-|a_{k}|^{2})^{t-{n\over
p}}\over\hat{\omega}(a_{k})^{1\over p}(1-\langle z,a_{k}\rangle)^{t}}$

belongs to $A^{p}_{\omega}(\mathbb{B}_{n})$ and there is a constant
$C:=C(t,p,\omega)>0$ such that

$\|F\|_{A^{p}_{\omega}}\leq
C\sum_{k=1}^{n}|\lambda_{k}|^{p}=C\|\lambda\|_{l^{p}}.$

###### Proof.

Let

$g_{k}(z):={(1-|a_{k}|^{2})^{t-{n\over p}}\over\hat{\omega}(a_{k})^{1\over
p}(1-\langle z,a_{k}\rangle)^{t}},\,\,z\in\mathbb{B}_{n}.$

Then $g_{k}\in A^{p}_{\omega}$ with $\|g_{k}\|_{A^{p}_{\omega}}\approx 1$ by
(2.4) for $t>{n\over p}$. Therefore, for $0<p\leq 1$, there exists
$C:=C(p,\omega)>0$ such that

$\|F\|_{A^{p}_{\omega}}\leq\sum_{k=1}^{\infty}|\lambda_{k}|^{p}\|g_{k}\|_{A^{p}_{\omega}}^{p}\leq
C\sum_{k=1}^{\infty}|\lambda_{k}|^{p}.$

Next consider the case $1<p<\infty$. For $g\in H(\mathbb{B}_{n})$, let

$\Lambda g(z):=\int_{\mathbb{B}_{n}}{(1-|w|^{2})^{t-n-1}\over|1-\langle
z,w\rangle|^{t}}g(w)dV(w),\,\,z\in\mathbb{B}_{n}.$

Consider the function

$f(z):=\sum_{k=1}^{\infty}{|\lambda_{k}|\chi_{D(a_{k},r)}(z)\over\omega(a_{k})^{1\over
p}V\left(D(a_{k},r)\right)^{1\over p}},$

where $\chi$ is the characteristic function. We get

$\displaystyle\Lambda f(z)=$
$\displaystyle\sum_{k=1}^{\infty}{|\lambda_{k}|\over\omega(a_{k})^{1\over
p}V\left(D(a_{k},r)\right)^{1\over
p}}\int_{D(a_{k},r)}{(1-|w|^{2})^{t-n-1}\over|1-\langle z,w\rangle|^{t}}dV(w)$
$\displaystyle\approx$ $\displaystyle\sum_{k=1}^{\infty}{|\lambda_{k}|\over
V\left(D(a_{k},r)\right)^{1\over p}}\cdot{(1-|a_{k}|^{2})^{t+{1\over
p}-n-1}\over\hat{\omega}(a_{k})^{1\over p}|1-\langle
z,a_{k}\rangle|^{t}}\int_{D(a_{k},r)}{\hat{\omega}(|w|)^{1\over
p}\over\omega(w)^{1\over p}(1-|w|)^{1\over p}}dV(w)$ $\displaystyle\gtrsim$
$\displaystyle\sum_{k=1}^{\infty}{|\lambda_{k}|\over\hat{\omega}(a_{k})^{1\over
p}V\left(D(a_{k},r)\right)^{1\over p}}\cdot{(1-|a_{k}|^{2})^{t+{1\over
p}-n-1}\over|1-\langle z,a_{k}\rangle|^{t}}V(D(a_{k},r))$
$\displaystyle\approx$
$\displaystyle\sum_{k=1}^{\infty}|\lambda_{k}|{(1-|a_{k}|^{2})^{t-{n\over
p}}\over\hat{\omega}(a_{k})^{1\over p}|1-\langle
z,a_{k}\rangle|^{t}}\geq|F(z)|$

for all $z\in\mathbb{B}_{n}$. We claim that $\Lambda$ is bounded on
$L^{p}_{\omega}$ at this moment. Then there is a constant $C:=C(t,p,\omega)>0$
such that

$\|F\|_{A^{p}_{\omega}}\leq C\|f\|_{L^{p}_{\omega}}.$ (2.5)

Noting that each point $z\in\mathbb{B}_{n}$ belongs to at most $N$ of the sets
$D(a_{k},4r)$ by Lemma 2.2, we have

$|f(z)|^{p}\leq
N^{p-1}\sum_{k=1}^{\infty}{|\lambda_{k}|^{p}\chi_{D(a_{k},4r)}(z)\over\omega(a_{k})V\left(D(a_{k},r)\right)},\,z\in\mathbb{B}_{n}.$

Therefore, integrating term by term, we get

$\int_{\mathbb{B}_{n}}|f(z)|^{p}\omega(z)dV(z)\leq
N^{p-1}\sum_{k=1}^{\infty}|\lambda_{k}|^{p},$

which combined with (2.5) deduces the desired result.

Finally, we only need to verify our claim, that is, $\Lambda$ is bounded on
$L^{p}_{\omega}$. Let

$h(z)=(1-|z|^{2})^{-{t-n-1\over 2}},\,\,z\in\mathbb{B}_{n}.$

By Lemmas 2.5 and 2.6, we obtain

$\displaystyle\int_{\mathbb{B}_{n}}{(1-|w|^{2})^{t-n-1}\over|1-\langle
z,w\rangle|^{t}}h^{2}(w)\omega(w)dV(w)=$
$\displaystyle\int_{\mathbb{B}_{n}}{\omega(w)\over|1-\langle
z,w\rangle|^{t}}dV(w)$ $\displaystyle\lesssim$
$\displaystyle{\hat{\omega}(z)\over(1-|z|^{2})^{t-n}}\lesssim{1\over(1-|z|^{2})^{t-n-1}}$

and

$\displaystyle\int_{\mathbb{B}_{n}}{(1-|w|^{2})^{t-n-1}\over|1-\langle
z,w\rangle|^{t}}h^{2}(z)\omega(z)dV(z)=$
$\displaystyle\int_{\mathbb{B}_{n}}{(1-|w|^{2})^{t-n-1}\omega(z)\over(1-|z|^{2})^{t-n-1}|1-\langle
z,w\rangle|^{t}}dV(z)$ $\displaystyle\lesssim$
$\displaystyle{\hat{\omega}(w)\over(1-|w|^{2})^{t-n}}\lesssim{1\over(1-|w|^{2})^{t-n-1}},$

which implies that $\Lambda$ is bounded on $L^{p}_{\omega}$ by Schur’s test.
The proof is complete. ∎

The following lemma extends the one dimensional result [9, Lemma 1] to higher
dimensions and classical weighted Bergman spaces [8, Lemma 2.2] to weighted
Bergman spaces induced by doubling weights.

######  Lemma 2.8.

Let $0<p\leq q<\infty$, $\omega\in\mathcal{D}$ and $0<r_{2}<r_{1}<1$. Then
there is a constant $C:=C(\omega,p,q,r_{1},r_{2})>0$ such that

$|f(a)-f(b)|^{q}\leq
C{\rho(a,b)^{q}\over\left((1-|a|)^{n}\hat{\omega}(a)\right)^{q\over
p}}\int_{D(a,r_{1})}|f(z)|^{p}W(z)dV(z)$

for any $a\in\mathbb{B}_{n}$, $b\in D(a,r_{2})$ and $f\in A^{p}_{\omega}$.

###### Proof.

Note that for any $z\in\mathbb{B}_{n}\backslash\\{0\\}$ and $r>0$,
$\Delta(z,r)$ is an ellipsoid consisting of all $w\in\mathbb{B}_{n}$ such that

${|P_{z}(w)-c|^{2}\over r^{2}s^{2}}+{|Q_{z}(w)|^{2}\over r^{2}s}<1,$ (2.6)

where

$\displaystyle c={(1-r^{2})z\over
1-r^{2}|z|^{2}}{\rm~{}and~{}}s={1-|z|^{2}\over 1-r^{2}|z|^{2}}.$

This combined with (2.1) and (2.7) in [8] deduces that

$|f(0)-f(b)|^{p}\leq C|b|^{p}\int_{\tanh(r_{1})\mathbb{B}_{n}}|f(z)|^{p}dV(z)$

for $b\in D(a,\tanh(r_{2}))$ and some $C:=C(r_{1},r_{2})>0$, where
$a\in\mathbb{B}_{n}$ and $0<r_{2}<r_{1}<1$. Write $R:=R(r_{1})=\tanh r_{1}$.
By Lemma 2.3, we obtain

$\displaystyle|f(a)-f(b)|^{p}=$
$\displaystyle|f\circ\varphi_{a}(0)-f\circ\varphi_{a}(\varphi_{a}(b))|^{p}$
$\displaystyle\leq$ $\displaystyle
C|\varphi_{a}(b)|^{p}\int_{R\mathbb{B}_{n}}|f\circ\varphi_{a}(z)|^{p}dV(z)$
$\displaystyle\leq$ $\displaystyle
C\rho(a,b)^{p}\int_{D(a,R)}|f(z)|^{p}\left({1-|a|^{2}\over|1-\langle
z,a\rangle|^{2}}\right)^{n+1}dV(z)$ $\displaystyle\leq$ $\displaystyle
C_{1}{\rho(a,b)^{p}\over(1-|a|)^{n}\hat{\omega}(a)}\int_{D(a,R)}|f(z)|^{p}{\hat{\omega}(z)\over
1-|z|}dV(z)$ $\displaystyle=$ $\displaystyle
C_{1}{\rho(a,b)^{p}\over(1-|a|)^{n}\hat{\omega}(a)}\int_{D(a,R)}|f(z)|^{p}W(z)dV(z)$

for some $C_{1}:=C_{1}(\omega,r_{1},r_{2})>0$.

Next, we consider the case $p<q$. Using the result of $p=q$ above and Lemma
2.4, we get

$\displaystyle|f(a)-f(b)|^{q}\leq$ $\displaystyle C^{q\over
p}{\rho(a,b)^{q}\over\left((1-|a|)^{n}\hat{\omega}(a)\right)^{q\over
p}}\left(\int_{D(a,R)}|f(z)|^{p}W(z)dV(z)\right)^{q\over p}$
$\displaystyle\leq$ $\displaystyle C^{q\over
p}{\rho(a,b)^{q}\|f\|^{q-p}_{A^{p}_{W}}\over\left((1-|a|)^{n}\hat{\omega}(a)\right)^{q\over
p}}\int_{D(a,R)}|f(z)|^{p}W(z)dV(z)$ $\displaystyle\leq$ $\displaystyle
C_{2}{\rho(a,b)^{q}\over\left((1-|a|)^{n}\hat{\omega}(a)\right)^{q\over
p}}\int_{D(a,R)}|f(z)|^{p}W(z)dV(z)$

for some $C_{2}:=C_{2}(\omega,p,q,r_{1},r_{2})>0$. The proof is complete. ∎

##  3\. Carleson measure

In this section, we characterize the (vanishing) $q$-Carleson measure for
$A^{p}_{\omega}$ for full $0<p,q<\infty$. To present our results, we introduce
a notation first. Given $\omega\in\mathcal{D},0<r<1$, $0<s<\infty$ and a
positive Borel measurev $\mu$ on $\mathbb{B}_{n}$, the weighted mean function
$\hat{\mu}_{\omega,r,s}$ is defined by

$\hat{\mu}_{\omega,r,s}(z):={\mu(D(z,r))\over\omega(D(z,r))^{s}},\,\,z\in\mathbb{B}_{n}.$

Write $\hat{\mu}_{\omega,r}:=\hat{\mu}_{\omega,r,1}$ for simplicity.

To characterize the $q$-Carleson measure for $A^{p}_{\omega}$ in the case
$p\leq q$, we need to prove the following lemma.

######  Lemma 3.1.

Let $0<r<1$, $0<p\leq q<\infty$, $\omega\in\mathcal{D}$ and $\mu$ be a
positive Borel measure on $\mathbb{B}_{n}$. Then for any $f\in
A^{p}_{\omega}$, one has

$\int_{\mathbb{B}_{n}}|f(z)|^{q}d\mu(z)\lesssim\|f\|^{q-p}_{A^{p}_{\omega}}\int_{\mathbb{B}_{n}}|f(z)|^{p}\hat{\mu}_{\omega,r,{q\over
p}}(z)W(z)dV(z).$

###### Proof.

For $f\in A^{p}_{\omega}$, using the subharmonic property of $|f|^{p}$ (see
Lemma 2.24 in [15]), (2.2), Lemmas 2.3 and 2.4, we have

$\displaystyle|f(z)|^{q}\lesssim$
$\displaystyle\left({1\over(1-|z|^{2})^{n+1}}\int_{D(z,r)}|f(w)|^{p}dV(w)\right)^{q\over
p}$ $\displaystyle\lesssim$
$\displaystyle\left({1\over\hat{\omega}(z)(1-|z|^{2})^{n}}\int_{D(z,r)}|f(w)|^{p}W(w)dV(w)\right)^{q\over
p}$ $\displaystyle\lesssim$
$\displaystyle{1\over\left(\hat{\omega}(z)(1-|z|^{2})^{n}\right)^{q\over
p}}\|f\|^{q-p}_{A^{p}_{\omega}}\int_{D(z,r)}|f(w)|^{p}W(w)dV(w).$

Combined with (2.1), (2.6), Proposition 1 in [6] and Lemma 2.3, we see that

$\omega(D(a,r))\approx\omega(S_{a})\approx\hat{\omega}(a)(1-|a|^{2})^{n}.$
(3.1)

Therefore, by Fubini’s theorem and Lemma 2.4,

$\displaystyle\int_{\mathbb{B}_{n}}|f(z)|^{q}d\mu(z)\lesssim$
$\displaystyle\|f\|^{q-p}_{A^{p}_{\omega}}\int_{\mathbb{B}_{n}}{1\over\left(\hat{\omega}(z)(1-|z|^{2})^{n}\right)^{q\over
p}}\int_{D(z,r)}|f(w)|^{p}W(w)dV(w)d\mu(z)$ $\displaystyle\lesssim$
$\displaystyle\|f\|^{q-p}_{A^{p}_{\omega}}\int_{\mathbb{B}_{n}}|f(w)|^{p}W(w){\mu(D(w,r))\over\left(\hat{\omega}(w)(1-|w|^{2})^{n}\right)^{q\over
p}}dV(w)$ $\displaystyle\lesssim$
$\displaystyle\|f\|^{q-p}_{A^{p}_{\omega}}\int_{\mathbb{B}_{n}}|f(w)|^{p}\hat{\mu}_{\omega,r,{q\over
p}}(w)W(w)dV(w).$

The proof is complete. ∎

The following result gives the characterization of the (vanishing)
$q$-Carleson measure in the case $A^{p}_{\omega}$ for $p\leq q$.

######  Theorem 3.2.

Let $0<p\leq q<\infty$, $\omega\in\mathcal{D}$ and $\mu$ be a positive Borel
measure on $\mathbb{B}_{n}$. Then there is $r_{0}:=r_{0}(\omega)\in(0,1)$ such
that the following statements hold:

  1. (i)

$\mu$ is a $q$-Carleson measure for $A^{p}_{\omega}$ if and only if for some
(or all) $r\in[r_{0},1)$ such that $\hat{\mu}_{\omega,r,{q\over p}}\in
L^{\infty}(\mathbb{B}_{n})$;

  2. (ii)

$\mu$ is a vanishing $q$-Carleson measure for $A^{p}_{\omega}$ if and only if
for some (or all) $r\in[r_{0},1)$ such that $\hat{\mu}_{\omega,r,{q\over
p}}(a)\to 0$ as $|a|\to 1$.

###### Proof.

(i). If $\hat{\mu}_{\omega,r,{q\over p}}\in L^{\infty}(\mathbb{B}_{n})$, then
$\mu$ is a $q$-Carleson measure for $A^{p}_{\omega}$ by Lemmas 3.1 and 2.4. We
next verify the necessity. Suppose that $\mu$ is a $q$-Carleson measure for
$A^{p}_{\omega}$. Consider the test function

$f_{a}(z)={1\over\hat{\omega}(a)^{1\over p}(1-|a|^{2})^{n\over
p}}\left({1-|a|^{2}\over|1-\langle z,a\rangle|}\right)^{\gamma+n\over
p},\,\,a,z\in\mathbb{B}_{n},$ (3.2)

where $\gamma$ is large enough. From Lemma 2.3 and Lemma 6 in [7], $f_{a}\in
A^{p}_{\omega}$. Applying (2.2), (3.1) and the fact
$W(S_{a})\approx\omega(S_{a}),a\in\mathbb{B}_{n}$ ( see the proof of Lemma
2.4), we have

$\displaystyle(1-|a|^{2})^{n}\hat{\omega}(a)\approx$
$\displaystyle{\hat{\omega}(a)\over
1-|a|^{2}}\int_{D(a,r)}dV(z)\approx\int_{D(a,r)}{\hat{\omega}(z)\over
1-|z|^{2}}dV(z)$ (3.3) $\displaystyle=$ $\displaystyle W(D(a,r))\approx
W(S_{a})\approx\omega(S_{a})\approx\omega(D(a,r)).$

Therefore, for $a\in\mathbb{B}_{n}$, by the assumption,

$\hat{\mu}_{\omega,r,{q\over
p}}(a)\approx{\mu(D(a,r))\over\left(\hat{\omega}(a)(1-|a|^{2})^{n}\right)^{q\over
p}}\lesssim\|f_{a}\|^{q}_{L^{q}_{\mu}}\lesssim\|f\|^{q}_{A^{p}_{\omega}}\approx
1,$

which means that

$\mu(D(a,r))\lesssim\left(\hat{\omega}(a)(1-|a|^{2})^{n}\right)^{q\over
p}\lesssim(\omega(D(a,r)))^{q\over p}.$

This completes the proof of (i).

(ii) Assume that $\mu$ is a vanishing $q$-Carleson measure for
$A^{p}_{\omega}$. Consider the test function $f_{a}$ defined as (3.2). Then
$f_{a}\in A^{p}_{\omega}$ and converges to $0$ uniformly on any compact subset
of $\mathbb{B}_{n}$ as $|a|\to 1$. Therefore, by Lemma 2.1,

$0=\lim_{|a|\to 1^{-}}\|f_{a}\|^{q}_{L^{q}_{\mu}}\geq\lim_{|a|\to
1^{-}}\int_{D(a,r)}|f_{a}(z)|^{q}d\mu(z)\gtrsim\lim_{|a|\to
1^{-}}{\mu(D(a,r))\over\left(\hat{\omega}(a)(1-|a|^{2})^{n}\right)^{q\over
p}},$

which means that $\hat{\mu}_{\omega,r,{q\over p}}(a)\to 0$ as $|a|\to 1$.

Conversely, assume that $\hat{\mu}_{\omega,r,{q\over p}}(a)\to 0$ as $|a|\to
1$. Then for any $\varepsilon>0$, there is $r:=r(\omega)\in(0,1)$ such that
${\mu(D(a,r))\over\left(\hat{\omega}(a)(1-|a|^{2})^{n}\right)^{q\over
p}}<\varepsilon$ when $|a|>r$. Put $d\mu_{r}(z):=\chi_{r\leq|z|<1}d\mu(z)$. If
$|a|\geq r$, then $\mu_{r}(D(a,r))\leq\mu(D(a,r))$. If $0<|a|<r$, by the fact
$D(a,r)=\varphi_{a}(D(0,r))$, then

$\displaystyle\mu_{r}(D(a,r))=$
$\displaystyle(1-|a|^{2})^{n+1}\int_{D(0,r)\backslash
D(0,\tanh^{-1}r)}{d\mu(z)\over|1-\langle z,a\rangle|^{2(n+1)}}$
$\displaystyle\lesssim$ $\displaystyle\mu(D(0,r)\backslash D(0,\tanh^{-1}r)).$

From [15, page 64], we see that there is a finite sequence
$\\{a^{(1)},\cdots,a^{(N)}\\}$ such that

$D(0,r)\backslash D(0,\tanh^{-1}r)\subset\cup_{j=1}^{N}D(a^{(j)},\delta)$

for some $\delta\in(0,1)$, where $a^{(j)}\in D(0,r)\backslash
D(0,\tanh^{-1}r)$, $j=1,\cdots,n$. Therefore,

$\mu_{r}(D(a,r))\lesssim\cup_{j=1}^{N}D(a^{(j)},\delta)\lesssim\varepsilon,$

which implies that $\hat{\mu}_{\omega,r,{q\over p}}(a)\lesssim\varepsilon$.
Thus, $\|I_{d}\|^{q}_{A^{p}_{\omega}\to L^{q}_{\mu_{r}}}\lesssim\varepsilon$.
Let $\\{f_{k}\\}\in A^{p}_{\omega}$ and converge to $0$ uniformly on any
compact subset of $\mathbb{B}_{n}$. Then

$\displaystyle\lim\sup_{k\to\infty}\|f_{k}\|^{q}_{L^{q}_{\mu}}=$
$\displaystyle\lim\sup_{k\to\infty}\left(\int_{\mathbb{B}_{n}}|f_{k}(z)|^{q}d\mu_{r}(z)+\int_{r\mathbb{B}_{n}}|f_{k}(z)|^{q}d\mu(z)\right)$
$\displaystyle\lesssim$
$\displaystyle\varepsilon\lim\sup_{k\to\infty}\|f_{k}\|^{q}_{A^{p}_{\omega}}\lesssim\varepsilon,$

which gives that $\mu$ is a vanishing $q$-Carleson measure for
$A^{p}_{\omega}$ by the arbitrariness of $\varepsilon$. ∎

Next, we consider the case $q>p$.

######  Theorem 3.3.

Let $0<q<p<\infty$, $\omega\in\mathcal{D}$ and $\mu$ be a positive Borel
measure on $\mathbb{B}_{n}$. Then there is $r_{0}:=r_{0}(\omega)\in(0,1)$ such
that the following statements are equivalent:

  1. (i)

$\mu$ is a $q$-Carleson measure for $A^{p}_{\omega}$;

  2. (ii)

$\mu$ is a vanishing $q$-Carleson measure for $A^{p}_{\omega}$;

  3. (iii)

For some (or all) $r\in[r_{0},1)$, $\hat{\mu}_{\omega,r}\in L^{p\over
p-q}_{W}$;

  4. (iv)

For some (or all) $r\in[r_{0},1)$ and any $\delta$-lattice $\\{a_{k}\\}$,

$\hat{\mu}_{\omega,r,{q\over p}}(a_{k})\in l^{p\over p-q}.$

###### Proof.

$(i)\Rightarrow(iv)$. Assume that $\mu$ is a $q$-Carleson measure for
$A^{p}_{\omega}$. Consider the function

$F_{t}(z)=\sum_{k=1}^{\infty}\lambda_{k}r_{k}(t)f_{k}(z),\,t\in[0,1],\,z\in\mathbb{B}_{n},$

where $\lambda=\\{\lambda_{k}\\}\in l^{p}$, $r_{k}$ are the Rademacher
functions and

$f_{k}(z)={(1-|a_{k}|^{2})^{t-{n\over p}}\over\hat{\omega}(a_{k})^{1\over
p}(1-\langle z,a_{k}\rangle)^{t}},\,z\in\mathbb{B}_{n},$

for some sufficiently large $t>0$. Then $F_{t}\in A^{p}_{\omega}$ with
$\|F_{t}\|_{A^{p}_{\omega}}\lesssim\|\lambda\|_{l^{p}}$ by Lemma 2.7. Thus, by
the assumption, we obtain that

$\displaystyle\int_{\mathbb{B}_{n}}\left|\sum_{k=1}^{\infty}\lambda_{k}r_{k}(t)f_{k}(z)\right|^{q}d\mu(z)\leq\|I_{d}\|^{q}_{A^{p}_{\omega}\to
L^{q}_{\mu}}\|F_{t}\|^{q}_{A^{p}_{\omega}}\lesssim\|I_{d}\|^{q}_{A^{p}_{\omega}\to
L^{q}_{\mu}}\|\lambda\|^{q}_{l^{q}}.$

Then integrating with respect to $t$ on $[0,1]$, and employing Fubini’s
theorem and Khinchine’s inequality,

$\int_{\mathbb{B}_{n}}\left(\sum_{k=1}^{\infty}|\lambda_{k}|^{2}|f_{k}(z)|^{2}\right)^{q\over
2}d\mu(z)\lesssim\|I_{d}\|^{q}_{A^{p}_{\omega}\to
L^{q}_{\mu}}\|\lambda\|^{q}_{l^{q}}.$

By Lemma 2.2, we get

$\displaystyle\left(\sum_{k=1}^{\infty}|\lambda_{k}|^{2}|f_{k}(z)|^{2}\right)^{q\over
2}$
$\displaystyle\gtrsim\sum_{k=1}^{\infty}|\lambda_{k}|^{q}|f_{k}(z)|^{q}\chi_{D(a_{k},4r)}(z)$
$\displaystyle\approx\sum_{k=1}^{\infty}|\lambda_{k}|^{q}{\chi_{D(a_{k},4r)}(z)\over\hat{\omega}(a_{k})^{q\over
p}(1-|a_{k}|^{2})^{{qn\over p}}}.$

Therefore,

$\sum_{k=1}^{\infty}|\lambda_{k}|^{q}{\mu(D(a_{k},4r))\over\left(\hat{\omega}(a_{k})(1-|a_{k}|^{2})^{n}\right)^{q\over
p}}\lesssim\|I_{d}\|^{q}_{A^{p}_{\omega}\to
L^{q}_{\mu}}\|\lambda\|^{q}_{l^{q}},$

which combined with the duality

$\left(l^{p\over q}\right)^{\ast}=l^{p\over p-q}$

yields that (iv) holds.

$(iv)\Rightarrow(iii)$. Suppose that $(iv)$ holds. By Lemma 2.2, we can choose
$\tilde{r}\in(0,1-r)$ and $N\geq 1$ such that
$\mathbb{B}_{n}\backslash\varepsilon\mathbb{B}_{n}\subset\cup_{k\geq
N}D(a_{k},\tilde{r})$ for some $0<\varepsilon_{0}<\varepsilon<1$. Note that
$\hat{\mu}_{\omega,r}$ is continuous on $\mathbb{B}_{n}$. Therefore, by (3.3),

$\displaystyle\int_{\mathbb{B}_{n}}\hat{\mu}_{\omega,r}(z)^{p\over
p-q}W(z)dV(z)$ $\displaystyle=$
$\displaystyle\left(\int_{\varepsilon\mathbb{B}_{n}}+\int_{\mathbb{B}_{n}\backslash\varepsilon\mathbb{B}_{n}}\right)\hat{\mu}_{\omega,r}(z)^{p\over
p-q}W(z)dV(z)$ $\displaystyle\leq$
$\displaystyle\int_{\varepsilon\mathbb{B}_{n}}{\hat{\omega}(z)\over
1-|z|}dV(z)+\sum_{k\geq
N}\int_{D(a_{k},\tilde{r})}\hat{\mu}_{\omega,r}(z)^{p\over
p-q}{\hat{\omega}(z)\over 1-|z|}dV(z)$ $\displaystyle\lesssim$
$\displaystyle\hat{\omega}(0)+\sum_{k\geq N}{\hat{\omega}(a_{k})\over
1-|a_{k}|}\int_{D(a_{k},\tilde{r})}\left({\mu(D(z,r))\over\hat{\omega}(z)(1-|z|)^{n}}\right)^{p\over
p-q}dV(z)$ $\displaystyle\lesssim$ $\displaystyle\hat{\omega}(0)+\sum_{k\geq
N}\left({\mu(D(a_{k},r+\tilde{r}))\over\left(\hat{\omega}(a_{k})(1-|a_{k}|)^{n}\right)^{q\over
p}}\right)^{p\over p-q}<\infty,$

which means that $(iii)$ holds.

$(iii)\Rightarrow(ii)$. Suppose that $(iii)$ holds. Let $\\{f_{k}\\}$ be a
bounded sequence in $A^{p}_{\omega}$ and converge to $0$ uniformly on any
compact subset of $\mathbb{B}_{n}$. By the assumption, for any
$\varepsilon>0$, there is $\varepsilon_{0}\in(0,1)$ such that

$\int_{\mathbb{B}_{n}\backslash\varepsilon_{0}\mathbb{B}_{n}}\hat{\mu}_{\omega,r}(z)^{p\over
p-q}W(z)dV(z)<\varepsilon^{p\over p-q}.$

Therefore, using the subharmonic property of $|f_{k}|^{q}$, Fubini’s theorem,
Hölder’s inequality and Lemma 2.4, we have

$\displaystyle\|f_{k}\|^{q}_{L^{q}_{\mu}}=$
$\displaystyle\int_{\mathbb{B}_{n}}|f_{k}(z)|^{q}d\mu(z)\lesssim\int_{\mathbb{B}_{n}}{1\over(1-|z|)^{n+1}}\int_{D(z,r)}|f_{k}(w)|^{q}dV(w)d\mu(z)$
$\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{B}_{n}}|f_{k}(w)|^{q}{\mu(D(w,r))\over(1-|w|)^{n+1}}dV(w)$
$\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{B}_{n}}|f_{k}(w)|^{q}\hat{\mu}_{\omega,r}(w)W(w)dV(w)$
$\displaystyle=$
$\displaystyle\left(\int_{\varepsilon_{0}\mathbb{B}_{n}}+\int_{\mathbb{B}_{n}\backslash\varepsilon_{0}\mathbb{B}_{n}}\right)|f_{k}(w)|^{q}\hat{\mu}_{\omega,r}(w)W(w)dV(w)$
$\displaystyle\leq$
$\displaystyle\left(\int_{\varepsilon_{0}\mathbb{B}_{n}}|f_{k}(w)|^{q}W(w)dV(w)\right)^{q\over
p}\left(\int_{\mathbb{B}_{n}}\hat{\mu}_{\omega,r}(w)^{p\over
p-q}W(w)dV(w)\right)^{p-q\over p}$
$\displaystyle+\left(\int_{\mathbb{B}_{n}}|f_{k}(w)|^{q}W(w)dV(w)\right)^{q\over
p}\left(\int_{\mathbb{B}_{n}\backslash\varepsilon_{0}\mathbb{B}_{n}}\hat{\mu}_{\omega,r}(w)^{p\over
p-q}W(w)dV(w)\right)^{p-q\over p}$ $\displaystyle\lesssim$
$\displaystyle\varepsilon\|f_{k}\|^{q}_{A^{p}_{\omega}},$

which implies that $(ii)$ holds.

$(ii)\Rightarrow(i)$. It is obvious. The proof is complete. ∎

##  4\. Main results

In this section, we prove Theorems 1.1 and 1.2. To present our results, we
introduce some notations first. The measures $\eta$ and $\sigma_{r}$ are
defined by

$\eta=\eta_{\varphi,u}+\eta_{\psi,u},$
$\sigma_{r}=\sigma_{\varphi,r}+\sigma_{\psi,r},$

where the measures $\eta_{\varphi,u}$ and $\sigma_{\varphi,r}$ are defined by

$\eta_{\varphi,u}=\left(|\rho u|^{q}d\mu\right)\circ\varphi^{-1},$
$\sigma_{\varphi,r}=\left(\chi_{G_{r}}|u-v|^{q}d\mu\right)\circ\varphi^{-1},$

and the measures $\eta_{\psi,u},\sigma^{\alpha}_{\psi}$ and $\sigma_{\psi,r}$
are defined similarly.

The following theorem, encompassing the content of Theorem 1.1, is the main
result of this subsection.

###  4.1. The case $p\leq q$

We first investigate some sufficient conditions for
$uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$ to be bounded and
compact.

######  Theorem 4.1.

Let $0<r<1$, $0<p\leq q<\infty$ and $\omega\in\mathcal{D}$,
$\varphi,\psi\in\mathcal{S}(\mathbb{B}_{n})$, $u,v$ be weights and $\mu$ be a
positive Borel measure on $\mathbb{B}_{n}$. If $\eta+\sigma_{\varphi,r}$ or
$\eta+\sigma_{\psi,r}$ is a (resp. vanishing) $q$-Carleson measure for
$A^{p}_{\omega}$, then $uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$
is bounded (resp. compact).

###### Proof.

By symmetry, we only need to consider the measure $\nu:=\eta+\sigma_{\psi,r}$.
Suppose that $\nu$ is a $q$-Carleson measure for $A^{p}_{\omega}$. Then
Theorem 3.2 gives that $\hat{\nu}_{\omega,\tilde{r},{q\over p}}\in
L^{\infty}(\mathbb{B}_{n})$ for some $\tilde{r}\in(r_{0},1)$ with
$r_{0}=r_{0}(\omega)>0$. For any function $f\in A^{p}_{\omega}$, we have

$\displaystyle\|(uC_{\varphi}-vC_{\psi})f\|^{q}_{L^{q}_{\mu}}=$
$\displaystyle\left(\int_{G_{r}}+\int_{\mathbb{B}_{n}\backslash
G_{r}}\right)|u(z)f\circ\varphi(z)-v(z)f\circ\psi(z)|^{q}d\mu(z)$
$\displaystyle=:$ $\displaystyle I+II.$

Using Lemma 2.8, Fubini’s theorem, Lemma 2.3 and (3.3),

$\displaystyle A:=$
$\displaystyle\int_{G_{r}}|v(z)(f\circ\varphi(z)-f\circ\psi(z))|^{q}d\mu(z)$
$\displaystyle\lesssim$
$\displaystyle\int_{G_{r}}{\rho(z)^{q}|v(z)|^{q}\over\left((1-|\psi(z)|)^{n}\hat{\omega}(\psi(z))\right)^{q\over
p}}\int_{D(\psi(z),\tilde{r})}|f(w)|^{p}W(w)dV(w)d\mu(z)$ $\displaystyle\leq$
$\displaystyle\int_{\mathbb{B}_{n}}|f(w)|^{p}\int_{\psi^{-1}(D(w,\tilde{r}))}{\rho(z)^{q}|v(z)|^{q}\over\left((1-|\psi(z)|)^{n}\hat{\omega}(\psi(z))\right)^{q\over
p}}d\mu(z)W(w)dV(w)$ $\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{B}_{n}}|f(w)|^{p}{\eta(D(w,\tilde{r}))\over\left((1-|w|)^{n}\hat{\omega}(w)\right)^{q\over
p}}W(w)dV(w)$ $\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{B}_{n}}|f(w)|^{p}{\nu(D(w,\tilde{r}))\over\left((1-|w|)^{n}\hat{\omega}(w)\right)^{q\over
p}}W(w)dV(w)$ $\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{B}_{n}}|f(w)|^{p}\hat{\nu}_{\omega,\tilde{r},{q\over
p}}(w)W(w)dV(w).$

Therefore, by Lemma 3.1, we get

$\displaystyle I\lesssim$
$\displaystyle\int_{G_{r}}\left(|(u(z)-v(z))f\circ\varphi(z)|^{q}+|v(z)(f\circ\varphi(z)-f\circ\psi(z))|^{q}\right)d\mu(z)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{B}_{n}}|f(z)|^{q}d\sigma_{\varphi,r}(z)+A$
$\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{B}_{n}}|f(z)|^{p}\hat{\nu}_{\omega,\tilde{r},{q\over
p}}(w)W(w)dV(w).$

For $II$, applying Lemma 3.1 again, we obtain

$\displaystyle II\lesssim$ $\displaystyle\int_{\mathbb{B}_{n}\backslash
G_{r}}(|u(z)f\circ\varphi(z)|^{q}+|v(z)f\circ\psi(z)|^{q})d\mu(z)$
$\displaystyle\leq$ $\displaystyle{1\over r^{q}}\int_{\mathbb{B}_{n}\backslash
G_{r}}\rho(z)(|u(z)f\circ\varphi(z)|^{q}+|v(z)f\circ\psi(z)|^{q})d\mu(z)$
$\displaystyle\leq$ $\displaystyle{1\over
r^{q}}\int_{\mathbb{B}_{n}}|f(z)|^{q}d\eta(z)$ $\displaystyle\leq$
$\displaystyle{1\over r^{q}}\int_{\mathbb{B}_{n}}|f(z)|^{q}d\nu(z)$
$\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{B}_{n}}|f(z)|^{p}\hat{\nu}_{\omega,\tilde{r},{q\over
p}}(w)W(w)dV(w).$

Thus, by Lemma 2.4,

$\|(uC_{\varphi}-vC_{\psi})f\|^{q}_{L^{q}_{\mu}}\lesssim\int_{\mathbb{B}_{n}}|f(z)|^{p}\hat{\nu}_{\omega,\tilde{r},{q\over
p}}(z)W(z)dV(z)\lesssim\|\hat{\nu}_{\omega,\tilde{r},{q\over
p}}\|_{L^{\infty}(\mathbb{B}_{n})}<\infty,$ (4.1)

which means that $uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$ is
bounded.

Suppose that $\nu$ is a vanishing $q$-Carleson measure for $A^{p}_{\omega}$.
Using Theorem 3.2, for any $\varepsilon>0$, there is a constant $R\in(0,1)$
such that $\hat{\nu}_{\omega,\tilde{r},{q\over p}}(z)<\varepsilon$ for
$|z|>R$. Let $\\{f_{k}\\}$ be a bounded sequence in $A^{p}_{\omega}$ and
converge to $0$ uniformly on any compact subset of $\mathbb{B}_{n}$. Then by
(4.1) and Lemma 2.4, we have

$\displaystyle\lim\sup_{k\to\infty}\|(uC_{\varphi}-vC_{\psi})f_{k}\|^{q}_{L^{q}_{\mu}}$
$\displaystyle\lesssim$
$\displaystyle\lim\sup_{k\to\infty}\|f_{k}\|^{q-p}_{A^{p}_{\omega}}\left(\int_{R\mathbb{B}_{n}}+\int_{\mathbb{B}_{n}\backslash
R\mathbb{B}_{n}}\right)|f_{k}(z)|^{p}\hat{\nu}_{\omega,\tilde{r},{q\over
p}}(z)W(z)dV(z)\lesssim\varepsilon,$

which combined with Lemma 2.1 gives that
$uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$ is compact. ∎

Before studying the necessary conditions, we introduce some notations. For
$\varphi,\psi\in\mathcal{S}(\mathbb{B}_{n})$ and $b\in\mathbb{B}_{n}$, let

$Q_{b}(z):={1-\langle b,\varphi(z)\rangle\over 1-\langle
b,\psi(z)\rangle},z\in\mathbb{B}_{n}.$

Given $r\in(0,1)$, $0<q,s<\infty$ and a positive Borel measure $\mu$ on
$\mathbb{B}_{n}$, put

$R_{s,r,q}(a,b):=\int_{\varphi^{-1}(D(a,r))}|u-vQ_{b}^{s}|^{q}d\mu$

for $a,b\in\mathbb{B}_{n}$. For $a\in\mathbb{B}_{n}\backslash\\{0\\}$, set
$w_{1}(a):={a\over|a|}$ and extend it to any orthonormal basis
$\\{w_{j}(a)\\}_{j=1}^{n}$ for ${\mathbb{C}}^{n}$. For $N>0$ with
$N^{2}(1-|a|)<1$, let

$a^{1,N}=\left(1-N^{2}\sqrt{1-|a|}\right)w_{1}(a)$

and

$a^{j,N}=a^{1,N}+N\sqrt{1-|a|}w_{j}(a),\,j=2,\cdots,n.$

Since

$1-|a^{j,N}|^{2}=N^{2}(1-|a|)\times\begin{cases}\left(2-N^{2}(1-|a|)\right){~{}\rm
if~{}}j=1\\\ \left(1-N^{2}(1-|a|)\right){~{}\rm if~{}}j\neq 1\end{cases},$
(4.2)

we have $a^{j,N}\in\mathbb{B}_{n},j=1,\cdots,n$. In case $a=te_{1}$, take
$w_{j}(te_{1}):=e_{j},j=1,\cdots,n$. For $t\in(0,1)$, $a=te_{1}$ and $N\geq 1$
with $N^{2}(1-|a|)<1$, set

$J_{N}(a):=\left\\{a,a^{j,N^{2}},a^{j,N^{3}}\right\\}_{j=1}^{n}.$

From Lemma 3.14 in [2], we see the following result hold.

######  Lemma 4.2.

Let $0<q<\infty$, $0<r,R<1$, $s>0$,
$\varphi,\psi\in\mathcal{S}(\mathbb{B}_{n})$ and $u,v$ be weights. Put
$\mu:=\eta_{\varphi,u}+\sigma_{\varphi,r}$. Then there exist constants
$N:=N(s,R)\geq 1$, $t_{0}:=t_{0}(s,R,N)\in(0,1)$ and $C:=C(q,s,R,N)>0$ such
that

$\mu(D(te_{1},R))\leq C\sum_{b\in J_{N}(te_{1})}R_{s,r,q}(te_{1},b)$

for all $t\in(t_{0},1)$.

Next, we give some necessary conditions for
$uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$ to be bounded and
compact.

######  Theorem 4.3.

Let $0<p\leq q<\infty$ and $\omega\in\mathcal{D}$. Let
$\varphi,\psi\in\mathcal{S}(\mathbb{B}_{n})$, $u,v$ be weights and $\mu$ be a
positive Borel measure on $\mathbb{B}_{n}$. If
$uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$ is bounded (resp.
compact), then the measures $\eta_{\varphi,u}+\sigma_{\varphi,r}$ and
$\eta_{\psi,u}+\sigma_{\psi,r}$ are (resp. vanishing) $q$-Carleson measure for
$A^{p}_{\omega}$.

###### Proof.

Write $\nu:=\eta_{\varphi,u}+\sigma_{\varphi,r}$. Let $0<\delta<\infty$. Fix
numbers $N:=N(s,R)$ and $t_{0}:=t_{0}(s,R,N)$ provided by Lemma 4.2, and
choosing $\delta_{1}=\delta_{1}(t_{0})\in(0,1)$ such that

$\bigcup_{z\in
t_{0}\overline{\mathbb{B}_{n}}}D(z,\delta)\subset\delta_{1}\overline{\mathbb{B}_{n}}.$

Since

$\begin{split}\rho|u|\leq&\rho(|u|+|v|)\leq\frac{(|\varphi-\psi|(|u|+|v|)}{1-\delta_{1}}\\\
&\leq\frac{2|u\varphi-v\psi|+|u-v|(|\varphi|+|\psi|)}{1-\delta_{1}}\\\
&\leq\frac{2|(uC_{\varphi}-vC_{\psi})(id)|+2|(uC_{\varphi}-vC_{\psi})(1)|}{1-\delta_{1}},\end{split}$

so that

$\nu(\delta_{1}\overline{\mathbb{B}_{n}})\lesssim\|uC_{\varphi}-vC_{\psi})(id)\|_{L^{q}_{\mu}}^{q}+\|uC_{\varphi}-vC_{\psi})(1)\|_{L^{q}_{\mu}}^{q}.$

By Lemma 2.5, there exists a constant $r_{1}>0$ such that

$\inf_{z\in t_{0}\overline{\mathbb{B}_{n}}}\hat{\omega}(z)>r_{1}.$

This, together with (3.1), yields

$\begin{split}\sup_{z\in
t_{0}\overline{\mathbb{B}_{n}}}\hat{\nu}_{\omega,R,{q\over
p}}(z)&\leq\frac{\nu(\delta_{1}\overline{\mathbb{B}_{n}})}{r_{1}^{q\over
p}(1-t_{0}^{2})^{qn\over p}}\\\
&\lesssim\|uC_{\varphi}-vC_{\psi})(id)\|_{L^{q}_{\mu}}^{q}+\|uC_{\varphi}-vC_{\psi})(1)\|_{L^{q}_{\mu}}^{q}.\end{split}$
(4.3)

We now turn to the estimate $\hat{\nu}_{\omega,R,{q\over p}}$ for
$\mathbb{B}_{n}\backslash t_{0}\overline{\mathbb{B}_{n}}$. Consider the test
function

$f_{b,s}={K_{b}^{s}\over\|K_{b}^{s}\|_{A^{p}_{\omega}}},b\in J_{N}(te_{1}),$

where $s>{\lambda(\omega)+n\over p}$ for some $\lambda(\omega)>0$. Then by
(2.4) and (2.2),

$\displaystyle\|(uC_{\varphi}-vC_{\psi})f_{b,s}\|^{q}_{L^{q}_{\mu}}=$
$\displaystyle{1\over\|K_{b}^{s}\|_{A^{p}_{\omega}}^{q}}\int_{\mathbb{B}_{n}}\left|{u\over(1-\langle
b,\varphi\rangle)^{s}}-{v\over(1-\langle b,\psi\rangle)^{s}}\right|^{q}d\mu$
$\displaystyle\gtrsim$ $\displaystyle{1\over(1-t)^{-qs+{qn\over
p}}\hat{\omega}(t)^{q\over
p}}\int_{\varphi^{-1}(D(te_{1},R))}{|u-vQ_{b}^{s}|^{q}\over|1-\langle
b,\varphi\rangle|^{qs}}d\mu$ $\displaystyle\approx$
$\displaystyle{1\over\left((1-t)^{n}\hat{\omega}(t)\right)^{q\over
p}}\int_{\varphi^{-1}(D(te_{1},R))}|u-vQ_{b}^{s}|^{q}d\mu,$

for some $R\in(0,1)$. Therefore, using Lemma 4.2,

$\displaystyle{\nu(D(te_{1},R))\over\left((1-t)^{n}\hat{\omega}(t)\right)^{q\over
p}}\lesssim$ $\displaystyle\sum_{b\in
J_{N}(te_{1})}\|(uC_{\varphi}-vC_{\psi})f_{b,s}\|^{q}_{L^{q}_{\mu}}$

for all $t\in(t_{0},1)$, which combined with Lemma 2.3 and (4.2), we can find
a constant $C>0$ such that

$\hat{\nu}_{\omega,R,{q\over
p}}(te_{1})\lesssim{\nu(D(te_{1},R))\over\left((1-t)^{n}\hat{\omega}(t)\right)^{q\over
p}}\lesssim\sup_{1-|a|\leq
C(1-|t|)}\|(uC_{\varphi}-vC_{\psi})f_{a,s}\|_{L^{q}_{\mu}}^{q}$ (4.4)

for all $t\in(t_{0},1)$. For any $z\in\mathbb{B}_{n}\backslash
t_{0}\overline{\mathbb{B}_{n}}$, we have $|z|\in(t_{0},1)$. Suppose
$\mathcal{U}$ is the unitary operator on ${\mathbb{C}}^{n}$, which maps $z$ to
$|z|e_{1}$. Denote

$\nu_{\mathcal{U}}:=\eta_{\mathcal{U}\circ\varphi,u}+\sigma_{\mathcal{U}\circ\varphi,r}.$

Noting

$\widehat{\nu_{\mathcal{U}}}_{\omega,R,{q\over
p}}(z)=\hat{\nu}_{\omega,R,{q\over p}}(\mathcal{U}^{*}z)$

for all $z\in\mathbb{B}_{n}$, and

$(uC_{\mathcal{U}\circ\varphi}-vC_{\mathcal{U}\circ\psi})f_{a,s}=(uC_{\varphi}-vC_{\psi})f_{\mathcal{U}^{*}a,s}$

for all $a\in\mathbb{B}_{n}$, where $\mathcal{U}^{*}=\mathcal{U}^{-1}.$ So,
applying (4.4) with $\mathcal{U}\circ\varphi$ and $\mathcal{U}\circ\psi$ in
place of $\varphi$ and $\psi$, we obtain

$\hat{\nu}_{\omega,R,{q\over
p}}(z)=\widehat{\nu_{\mathcal{U}}}_{\omega,R,{q\over
p}}(|z|e_{1})\lesssim\sup_{1-|a|\leq
C(1-|z|)}\|(uC_{\varphi}-vC_{\psi})f_{a,s}\|_{L^{q}_{\mu}}^{q}.$ (4.5)

for all $z\in\mathbb{B}_{n}\backslash t_{0}\overline{\mathbb{B}_{n}}$. Using
this and (4.3), we obtain that $\hat{\nu}_{\omega,R,{q\over p}}\in
L^{\infty}(\mathbb{B}_{n})$. Then Theorem 3.2 yields that $\nu$ is a
$q$-Carleson measure for $A^{p}_{\omega}$.

If $uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$ is compact, then
$\|(uC_{\varphi}-vC_{\psi})f_{b,s}\|_{L^{q}_{\mu}}\to 0$ as $|b|\to 1$, where
$s>{\lambda(\omega)+n\over p}$ for some $\lambda(\omega)>0$. By Lemma 2.3 and
(4.5), we have $\hat{\nu}_{\omega,R,{q\over p}}(z)\to 0$ as $|z|\to 1$, which
deduces that $\nu$ is vanishing $q$-Carleson measure for $A^{p}_{\omega}$ by
Theorem 3.2.

By symmetry, we obtain that $\eta_{\psi,v}+\sigma_{\psi,r}$ is also a
(vanishing) $q$-Carleson measure for $A^{p}_{\omega}$. ∎

From Theorems 4.1 and 4.3, we obtain the following corollary, which contains
Theorem 1.1.

######  Corollary 4.4.

Let $0<p\leq q<\infty$, $0<r<1$, $\omega\in\mathcal{D}$,
$\varphi,\psi\in\mathcal{S}(\mathbb{B}_{n})$, $u,v$ be weights and $\mu$ be a
positive Borel measure on $\mathbb{B}_{n}$. Then the following statements are
equivalent:

  1. (i)

$uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$ is bounded (resp.
compact);

  2. (ii)

$\eta+\sigma_{\varphi,r}$ or $\eta+\sigma_{\psi,r}$ is (resp. vanishing)
$q$-Carleson measure for $A^{p}_{\omega}$;

  3. (iii)

$\eta+\sigma_{\varphi,r}$ and $\eta+\sigma_{\psi,r}$ are (resp. vanishing)
$q$-Carleson measure for $A^{p}_{\omega}$.

###  4.2. The case $q<p$

We first explore the sufficient condition for
$uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$ to be compact.

######  Theorem 4.5.

Let $0<q<p<\infty$, $0<r<1$, $\omega\in\mathcal{D}$,
$\varphi,\psi\in\mathcal{S}(\mathbb{B}_{n})$, $u,v$ be weights and $\mu$ be a
positive Borel measure on $\mathbb{B}_{n}$. If $\eta+\sigma_{\varphi,r}$ or
$\eta+\sigma_{\psi,r}$ is $q$-Carleson measure for $A^{p}_{\omega}$, then
$uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$ is compact.

###### Proof.

Suppose that $\nu:=\eta+\sigma_{\varphi,r}$ is $q$-Carleson measure for
$A^{p}_{\omega}$. By Theorem 3.3, there is $r_{0}=r_{0}(\omega)\in(0,1)$ such
that $\hat{\nu}_{\omega,r}\in L^{p\over p-q}_{W}$ for some $r\in(r_{0},1)$.
For $f\in A^{p}_{\omega}$, employing (4.1) with $p=q$, Hölder’s inequality and
Lemma 2.4, we have

$\displaystyle\|(uC_{\varphi}-vC_{\psi})f\|^{q}_{L^{q}_{\mu}}\lesssim$
$\displaystyle\int_{\mathbb{B}_{n}}|f(z)|^{q}\hat{\nu}_{\omega,r}(z)W(z)dV(z)$
$\displaystyle\leq$
$\displaystyle\left(\int_{\mathbb{B}_{n}}|f(z)|^{p}W(z)dV(z)\right)^{q\over
p}\left(\int_{\mathbb{B}_{n}}\hat{\nu}_{\omega,r}(z)^{p\over
p-q}W(z)dV(z)\right)^{p-q\over p}$ $\displaystyle\lesssim$
$\displaystyle\|f\|^{q}_{A^{p}_{\omega}}\|\hat{\nu}_{\omega,r}\|_{L_{W}^{p\over
p-q}},$

which gives that $uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$ is
bounded. Let $\\{f_{k}\\}$ be a bounded sequence in $A^{p}_{\omega}$ and
converge to $0$ uniformly on any compact subset of $\mathbb{B}_{n}$. For any
$\varepsilon>0$, there is $R\in(0,1)$ such that

$\int_{\mathbb{B}_{n}\backslash
R\mathbb{B}_{n}}\hat{\nu}_{\omega,R}(z)^{p\over
p-q}W(z)dV(z)<\varepsilon^{p\over p-q}.$

Similar to the above,

$\displaystyle\lim\sup_{k\to\infty}\|(uC_{\varphi}-vC_{\psi})f_{k}\|^{q}_{L^{q}_{\mu}}$
$\displaystyle\lesssim$
$\displaystyle\lim\sup_{k\to\infty}\left(\int_{\mathbb{B}_{n}}+\int_{\mathbb{B}_{n}\backslash
R\mathbb{B}_{n}}\right)|f_{k}(z)|^{q}\hat{\nu}_{\omega,r}(z)W(z)dV(z)$
$\displaystyle\leq$
$\displaystyle\lim\sup_{k\to\infty}\left(\int_{R\mathbb{B}_{n}}|f_{k}(z)|^{p}W(z)dV(z)\right)^{q\over
p}\|\hat{\nu}_{\omega,R}\|_{L_{W}^{p\over p-q}}$ $\displaystyle+\sup_{k\geq
1}\|f_{k}\|^{q}_{A^{p}_{\omega}}\left(\int_{\mathbb{B}_{n}\backslash
R\mathbb{B}_{n}}\hat{\nu}_{\omega,r}(z)W(z)dV(z)\right)^{p-q\over
p}\lesssim\varepsilon,$

which implies that $uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$ is
compact by the arbitrariness of $\varepsilon$ and Lemma 2.1. Similarly, if
$\eta+\sigma_{\psi,r}$ is $q$-Carleson measure for $A^{p}_{\omega}$, we can
obtain the same result. ∎

The following decomposition lemma is crucial for studying the necessary
condition for $uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$ to be
bounded.

######  Lemma 4.6.

Let $0<\delta<1$, $N>0$, $r=r(N)\in\left(0,\tanh^{-1}\sqrt{1-{N^{2}\over
2(N^{2}+1)^{2}}}\right)$, $M=M(n,\delta,r,N)$ be a positive integer and
$\\{a_{k}\\}$ be a $\delta$-separated sequence in $\mathbb{B}_{n}$ with
$N^{2}(1-|a_{k}|)<{1\over 2}$ for all $k$. If any collection of more than $M$
of the pseudohyperbolic balls
$\left\\{D(a_{k}^{1,N},r),\cdots,D(a_{k}^{n,N},r)\right\\}_{k=1}^{\infty}$
contains no point in common, then $\\{a_{k}\\}$ is a union of $M+1$ separated
sequences.

###### Proof.

Write $a_{1,1}=a_{1},a_{2,1}=a_{2}\cdots,a_{M+1,1}=a_{M+1}$. For $a_{L+2}$, by
the assumption, there is $m\in\\{1,2,\cdots,M+1\\}$ such that
$a_{M+2}\notin\left\\{D(a_{m}^{1,N},r),\cdots,D(a_{m}^{n,N},r)\right\\}$, that
is, $\beta(a_{M+2},a_{m}^{j,N})\geq r$ for all $j\in\\{1,2,\cdots,n\\}$. Hence
by the fact $\rho(z,w)=\tanh\beta(z,w)$ for all $z,w\in\mathbb{B}_{n}$ and
(4.5) in [2],

$\displaystyle\beta(a_{M+2},a_{m,1})=$
$\displaystyle\beta(a_{M+2},a_{m})\geq\beta(a_{M+2},a_{m}^{j,N})-\beta(a_{m},a_{m}^{j,N})$
$\displaystyle\geq$ $\displaystyle r-\tanh^{-1}{\sqrt{1-{N^{2}\over
2(N^{2}+1)^{2}}}}:=R.$

Put $a_{m,2}=a_{M+2}$. Inductively, suppose that $L$ is a positive integer and

$\\{a_{k}\\}_{k=1}^{L}=\cup_{m=1}^{M+1}\\{a_{m,k}\\}_{k=1}^{N_{m}}$

such that for any $m\in\\{1,2,\cdots,M+1\\}$ and $1\leq i,j\leq N_{m}$, $i\neq
j$,

$\beta(a_{m,i},a_{m,j})\geq R.$

For $a_{L+1}$, we claim that there is $m_{0}\in\\{1,2,\cdots,M+1\\}$ such that

$a_{L+1}\notin\cup_{k=1}^{N_{m_{0}}}\left\\{D(a_{m_{0},k}^{1,N},R),\cdots,D(a_{m_{0},k}^{n,N},R)\right\\}.$

If not, suppose that

$a_{L+1}\in\cap_{m=1}^{M+1}\cup_{k=1}^{N_{m}}\left\\{D(a_{m,k}^{1,N},R),\cdots,D(a_{m,k}^{n,N},R)\right\\}.$

Then for any $m\in\\{1,2,\cdots,M+1\\}$, there is
$k_{m}\in\\{1,2,\cdots,N_{m}\\}$ such that

$a_{L+1}\in\left\\{D(a_{m,k_{m}}^{1,N},R),\cdots,D(a_{m,k_{m}}^{n,N},R)\right\\},$

Thus,

$a_{L+1}\in\cap_{m=1}^{M+1}\left\\{D(a_{m,k_{m}}^{1,N},R),\cdots,D(a_{m,k_{m}}^{n,N},R)\right\\},$

which gives a contradiction, and our claim is correct. Then putting
$a_{m_{0},N_{m_{0}+1}}=a_{L+1}$, we obtain

$\\{a_{k}\\}_{k=1}^{\infty}=\cup_{m=1}^{M+1}\\{a_{m,k}\\}_{k=1}^{\infty}$

and each $\\{a_{m,k}\\}_{k=1}^{\infty}$ is a $R$-separated sequence. ∎

We next give some necessary conditions for
$uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$ to be bounded.

######  Theorem 4.7.

Let $0<q<p<\infty$ and $\omega\in\mathcal{D}$,
$\varphi,\psi\in\mathcal{S}(\mathbb{B}_{n})$, $u,v$ be weights and $\mu$ be a
positive Borel measure on $\mathbb{B}_{n}$. If
$uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$ is bounded, then
$\eta_{\varphi,u}+\sigma_{\varphi,r}$ and $\eta_{\psi,v}+\sigma_{\psi,r}$ are
$q$-Carleson measure for $A^{p}_{\omega}$.

###### Proof.

For $N\geq 36$ and $\delta\in(0,1)$, let $\\{a_{k}\\}$ be a $\delta$-separated
sequence in $\mathbb{B}_{n}$ such that $N^{4}(1-|a_{k}|)<{1\over 8}$ for each
$k$ and $\\{b_{k}\\}$ be a sequence given by one of
$\\{a_{k}\\},\\{a_{k}^{1,N}\\},\cdots,$ $\\{a_{k}^{n,N}\\}$. Consider the
function

$F_{t}(z)=\sum_{k=1}^{\infty}\lambda_{k}r_{k}(t)f_{k}(z),\,t\in[0,1],\,z\in\mathbb{B}_{n},$

where $\lambda=\\{\lambda_{k}\\}\in l^{p}$, $r_{k}$ are the Rademacher
functions and

$f_{k}(z)={(1-|a_{k}|^{2})^{s-{n\over p}}\over\hat{\omega}(a_{k})^{1\over
p}(1-\langle z,b_{k}\rangle)^{s}}$

for large $s$. Employing Lemma 4.4 in [2] and Lemma 4.6, there is a positive
integer $M$ such that $\\{b_{k}\\}$ is a union of $M$ separated sequences.
Thus, Lemma 2.7 yields that $F_{t}\in A^{p}_{\omega}$ with
$\|F_{t}\|_{A^{p}_{\omega}}\lesssim\|\lambda\|_{l^{p}}$ for $t\in[0,1]$. Since
$uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$ is bounded, we have

$\displaystyle\|(uC_{\varphi}-vC_{\psi})F_{t}\|^{q}_{L^{q}_{\mu}}=$
$\displaystyle\int_{\mathbb{B}_{n}}\left|\sum_{k=1}^{\infty}\lambda_{k}r_{k}(t){(1-|a_{k}|^{2})^{s-{n\over
p}}\over\hat{\omega}(a_{k})^{1\over
p}(1-\langle\varphi,b_{k}\rangle)^{s}}(u-vQ_{b_{k}}^{s})\right|^{q}d\mu$
$\displaystyle\lesssim$
$\displaystyle\|uC_{\varphi}-vC_{\psi}\|^{q}_{A^{p}_{\omega}\to
L^{q}_{\mu}}\|\lambda\|^{q}_{l^{p}}.$

Fix $0<r<1$ with ${2\over 1-\tanh r}\geq 36$. Integrating with respect to $t$
on $[0,1]$, and applying Fubini’s theorem, Khinchines’s inequality and Lemma
2.2, we get

$\displaystyle\int_{\mathbb{B}_{n}}\sum_{k=1}^{\infty}|\lambda_{k}|^{q}\left|{(1-|a_{k}|^{2})^{s-{n\over
p}}\over\hat{\omega}(a_{k})^{1\over
p}(1-\langle\varphi,b_{k}\rangle)^{s}}(u-vQ_{b_{k}}^{s})\right|^{q}\chi_{D(b_{k},r)\circ\varphi}d\mu$
$\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{B}_{n}}\left(\sum_{k=1}^{\infty}\left|\lambda_{k}{(1-|a_{k}|^{2})^{s-{n\over
p}}\over\hat{\omega}(a_{k})^{1\over
p}(1-\langle\varphi,b_{k}\rangle)^{s}}(u-vQ_{b_{k}}^{s})\right|^{2}\right)^{q\over
2}d\mu.$

Since $N\geq 36\geq{2\over 1-\tanh r}$, by (2.3), $D(b_{k},r)\subset
S(w_{1}(b_{k}),N(1-|b_{k}|))$ for all $k$. Using Lemma 3.6(a) in [2],

${\chi_{D(b_{k},r)\circ\varphi}\over|1-\langle\varphi,b_{k}\rangle|}\approx{\chi_{D(b_{k},r)\circ\varphi}\over
N^{2}(1-|b_{k}|)}\approx{\chi_{D(b_{k},r)\circ\varphi}\over N^{2}(1-|a_{k}|)}$

for all $k$. Therefore,

$\displaystyle\sum_{k=1}^{\infty}{|\lambda_{k}|^{q}R_{s,r,q}(a_{k},b_{k})\over\left((1-|a_{k}|)^{n}\hat{\omega}(a_{k})\right)^{q\over
p}}\lesssim$
$\displaystyle\int_{\mathbb{B}_{n}}\left(\sum_{k=1}^{\infty}\left|\lambda_{k}{(1-|a_{k}|^{2})^{s-{n\over
p}}\over\hat{\omega}(a_{k})^{1\over p}(1-\langle
z,b_{k}\rangle)^{s}}(u-vQ_{b_{k}}^{s})\right|^{2}\right)^{q\over 2}d\mu$
$\displaystyle\lesssim$
$\displaystyle\|uC_{\varphi}-vC_{\psi}\|^{q}_{A^{p}_{\omega}\to
L^{q}_{\mu}}\|\lambda\|^{q}_{l^{p}},$

which combined with the duality $\left(l^{p\over q}\right)^{\ast}=l^{p\over
p-q}$ deduces that

$\left\|\left\\{{R_{s,r,q}(a_{k},b_{k})\over\left((1-|a_{k}|)^{n}\hat{\omega}(a_{k})\right)^{q\over
p}}\right\\}\right\|_{l^{p\over
p-q}}\lesssim\|uC_{\varphi}-vC_{\psi}\|^{q}_{A^{p}_{\omega}\to L^{q}_{\mu}}.$

Write $\nu:=\eta_{\varphi,u}+\sigma_{\varphi,r}$. Then by Lemmas 2.3 and 4.2,

$\displaystyle\hat{\nu}_{\omega,r,{q\over p}}(a_{k})\lesssim\sum_{b\in
J_{N}(a_{k})}{R_{s,r,q}(a_{k},b_{k})\over\left((1-|a_{k}|)^{n}\hat{\omega}(a_{k})\right)^{q\over
p}}$

for all large $k$, since $|a_{k}|\to 1$. If $a_{k}\in
t_{0}\overline{\mathbb{B}}_{n}$ for some $t_{0}\in(0,1)$, then using (4.3),

$\sup_{a_{k}\in t_{0}\overline{\mathbb{B}}_{n}}\hat{\nu}_{\omega,r,{q\over
p}}(a_{k})\lesssim\|uC_{\varphi}-vC_{\psi})(id)\|_{L^{q}_{\mu}}^{q}+\|uC_{\varphi}-vC_{\psi})(1)\|_{L^{q}_{\mu}}^{q}<\infty.$

Therefore,

$\\{\hat{\nu}_{\omega,r,{q\over p}}(a_{k})\\}_{k=1}^{\infty}\in l^{p\over
p-q},$

which implies that $\nu$ is a $q$-Carleson measure for $A^{p}_{\omega}$ by
Theorem 3.3. By symmetry, we obtain that $\eta_{\psi,v}+\sigma_{\psi,r}$ is
also a $q$-Carleson measure for $A^{p}_{\omega}$. ∎

From Theorems 4.5 and 4.7, we obtain the following corollary, which contains
Theorem 1.2.

######  Corollary 4.8.

Let $0<q<p<\infty$, $0<r<1$, $\omega\in\mathcal{D}$,
$\varphi,\psi\in\mathcal{S}(\mathbb{B}_{n})$, $u,v$ be weights and $\mu$ be a
positive Borel measure on $\mathbb{B}_{n}$. Then the following statements are
equivalent:

  1. (i)

$uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$ is bounded;

  2. (ii)

$uC_{\varphi}-vC_{\psi}:A^{p}_{\omega}\to L^{q}_{\mu}$ is compact;

  3. (iii)

$\eta+\sigma_{\varphi,r}$ or $\eta+\sigma_{\psi,r}$ are $q$-Carleson measure
for $A^{p}_{\omega}$;

  4. (iv)

$\eta+\sigma_{\varphi,r}$ and $\eta+\sigma_{\psi,r}$ are $q$-Carleson measure
for $A^{p}_{\omega}$.

Data Availability No data was used to support this study.

Conflicts of Interest The authors declare that they have no conflicts of
interest.

Acknowledgements The corresponding author was supported by NNSF of China (No.
12371131), STU Scientific Research Initiation Grant(No. NTF23004), LKSF STU-
GTIIT Joint-research Grant (No. 2024 LKSFG06), Guangdong Basic and Applied
Basic Research Foundation (No. 2023A1515010614).

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