# Volterra type integration operators between weighted Bergman spaces and
Hardy spaces

Yongjiang Duan  Yongjiang DUAN: School of Mathematics and Statistics,
Northeast Normal University, Changchun, Jilin 130024, P.R.China
<EMAIL_ADDRESS>,  Siyu Wang  Siyu WANG: School of Mathematics and
Statistics, Northeast Normal University, Changchun, Jilin 130024, P.R.China
<EMAIL_ADDRESS>and  Zipeng Wang  Zipeng WANG: College of Mathematics
and Statistics, Chongqing University, Chongqing, 401331, P.R.China
<EMAIL_ADDRESS>
###### Abstract.

Let $\mathcal{D}$ be the class of radial weights on the unit disk which
satisfy both forward and reverse doubling conditions. Let $g$ be an analytic
function on the unit disk $\mathbb{D}$. We characterize bounded and compact
Volterra type integration operators

$J_{g}(f)(z)=\int_{0}^{z}f(\lambda)g^{\prime}(\lambda)d\lambda$

between weighted Bergman spaces $L_{a}^{p}(\omega)$ induced by $\mathcal{D}$
weights and Hardy spaces $H^{q}$ for $0<p,q<\infty$.

##  1\. Introduction

A weight $\omega$ in this paper is a positive and integrable function on the
unit disk $\mathbb{D}$. For $0<p<\infty$, the weighted Bergman space
$L^{p}_{a}(\omega)$ is a linear space of analytic functions on $\mathbb{D}$
satisfying

$\|f\|_{L^{p}(\omega)}:=\Big{(}\int_{\mathbb{D}}|f(z)|^{p}\omega(z)dA(z)\Big{)}^{1/p}<\infty,$

where $dA(z)=dxdy/\pi$ is the normalized area measure on $\mathbb{D}$. If
$\omega(z)=\omega_{\alpha}(z):=(1+\alpha)(1-|z|^{2})^{\alpha},-1<\alpha<\infty$,
then $L^{p}_{a}(\omega_{\alpha})$ is known as standard weighted Bergman space
in literatures. Moreover, $L_{a}^{p}(\omega_{0})$ is the classical Bergman
space, and is denoted by $L_{a}^{p}$ [Zhu07]. Recall for $0<p<\infty$, the
Hardy space consists of analytic functions $f$ on $\mathbb{D}$ such that

$\|f\|_{H^{p}}:=\sup_{0\leq
r<1}\Big{(}\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{i\theta})|^{p}d\theta\Big{)}^{1/p}<\infty.$

Let $g$ be an analytic function on $\mathbb{D}$. We consider the Volterra type
integration operator $J_{g}$ with symbol $g$

$J_{g}(f)(z):=\int_{0}^{z}f(\lambda)g^{\prime}(\lambda)d\lambda,\quad
z\in\mathbb{D},$

where $f$ is an analytic function on $\mathbb{D}$.

Pommerenke [Pom77] proved that $J_{g}$ is bounded on Hardy space $H^{2}$ if
and only if $g$ is of bounded mean oscillation. The same characterization of
the boundedness of $J_{g}$ on the general Hardy space $H^{p}$ was obtained by
Aleman and Siskakis [AS95] for $p\geq 1$ and by Aleman and Cima [AC01] for
$0<p<1$. In [AC01], Aleman and Cima further considered any pair of
$(p,q)\in(0,\infty)\times(0,\infty)$ and studied the symbol $g$ such that
$J_{g}:H^{p}\to H^{q}$ is bounded. Analogous characterization of the symbol
$g$ for $J_{g}$ to be bounded on Bergman spaces can be found in [Sis06].
Aleman and Siskakis [AS97] established the result on a large class of weighted
Bergman spaces, which includes the class of standard weights $\omega_{\alpha}$
as special examples. Pau and Peláez [PP10] described the
$L^{p}_{a}(\omega)$-$L^{q}_{a}(\omega)$ boundedness of $J_{g}$ for
$0<p,q<\infty$ when $\omega$ is rapidly decreasing, while Peláez and Rättyä
[PR14] got the corresponding characterizations when $\omega$ is regular or
rapidly increasing weight. Wu [Wu11] established boundedness criterions of
$J_{g}$ from $L_{a}^{p}(\omega_{\alpha})$ to $H^{q}$ for all
$(p,q)\in(0,\infty)\times(0,\infty)$ but $0<q<\min\\{2,p\\}<\infty$. Very
recently, this gap was filled by Miihkinen, Pau, Perälä and Wang in [MPPW20]
using analysis on tent spaces and some other delicate techniques. Moreover, by
modifying the proofs of the boundedness results, one can get the corresponding
compactness results ([Sis06, Wu11, CPW21]).

The goal of the present paper is to study Volterra type integration operators
between weighted Bergman spaces $L_{a}^{p}(\omega)$ and Hardy spaces $H^{q}$
for $0<p,q<\infty$. Here the weight $\omega$ is a $\mathcal{D}$ weight. Recall
that the class of $\mathcal{D}$ weights [PR19] consists of positive integrable
radial functions $\omega$ on $\mathbb{D}$ such that $\omega$ satisfies both
forward and reverse doubling conditions, i.e., there exist constants
$C_{1},C_{2}\in(1,\infty)$ and $C_{3}\in[1,\infty)$,

$C_{1}\widehat{\omega}\bigg{(}1-\frac{1-r}{C_{2}}\bigg{)}\leq\widehat{\omega}(r)\leq
C_{3}\widehat{\omega}\bigg{(}\frac{1+r}{2}\bigg{)},\quad r\in[0,1),$

where $\widehat{\omega}(z)=\int_{|z|}^{1}\omega(r)dr$ for all
$z\in\mathbb{D}$. The class of $\mathcal{D}$ weights can be regarded as the
largest class of radial weights $\omega$ in some sense that most of the
results over $L_{a}^{p}(\omega_{\alpha})$ can still hold on
$L_{a}^{p}(\omega)$. For instance, the Littlewood-Paley estimate holds for
weighted Bergman spaces induced by radial weights if and only if $\omega$ is a
$\mathcal{D}$ weight [PR19].

Let $H(\mathbb{D})$ be the set of analytic functions on $\mathbb{D}$. To
describe our results, let us introduce two function spaces, which can be
regarded as generalized Bloch spaces (little Bloch spaces). For a radial
weight $\omega$ and $\gamma\in\mathbb{R}$, let

$\mathcal{B}^{\gamma,\omega}=\Big{\\{}f\in
H(\mathbb{D}):\sup\limits_{z\in\mathbb{D}}\frac{(1-|z|^{2})^{\gamma}}{\omega(z)}|f^{\prime}(z)|<\infty\Big{\\}}$

and

$\mathcal{B}_{0}^{\gamma,\omega}=\Big{\\{}f\in
H(\mathbb{D}):\lim\limits_{|z|\to
1^{-}}\frac{(1-|z|^{2})^{\gamma}}{\omega(z)}|f^{\prime}(z)|=0\Big{\\}}.$

Let $\mathbb{T}$ be the unit circle. For an interval $I\subset\mathbb{T}$, we
define the Carleson square

$S(I)=\\{z\in\mathbb{D}:z/|z|\in I,~{}1-|I|\leq|z|<1\\},$

where $|\cdot|$ is the normalized arc measure on $\mathbb{T}$. For
$0<s<\infty$, a positive measure $\mu$ on $\mathbb{D}$ is a $s$-Carleson
measure (vanishing $s$-Carleson measure) [Zhu07, Section 9.2] if

$\sup\limits_{I\subseteq\mathbb{T}}\frac{\mu(S(I))}{|I|^{s}}<\infty\quad\Big{(}\lim\limits_{|I|\to
0}\frac{\mu(S(I))}{|I|^{s}}=0\Big{)}.$

The (vanishing) $1$-Carleson measure is called (vanishing) Carleson measure.
Let $\xi\in\mathbb{T}$ and $0<\alpha<\infty$, the Stolz non-tangential
approach region is given by

$\Gamma_{\alpha}(\xi)=\bigg{\\{}z\in\mathbb{D}:|z-\xi|<(\alpha+1)(1-|z|)\bigg{\\}}.$

For convenience, we denote $\Gamma_{1}(\xi)$ by $\Gamma(\xi)$. Let
$0<p<\infty$, $0<q\leq\infty$ and $\nu$ be a positive Borel measure. The tent
space $T_{q,\nu}^{p}$ consists of $\nu$-equivalence classes of
$\nu$-measurable functions $f$ on $\mathbb{D}$ such that

$\|f\|_{T_{q,\nu}^{p}}^{p}:=\int_{\mathbb{T}}\bigg{(}\int_{\Gamma_{\alpha}(\xi)}|f(z)|^{q}d\nu(z)\bigg{)}^{\frac{p}{q}}d\xi<\infty,\quad
q\in(0,\infty)$

and

$\|f\|_{T_{q,\nu}^{p}}^{p}:=\int_{\mathbb{T}}\bigg{(}\textmd{ess}\sup_{z\in\Gamma_{\alpha}(\xi)}|f(z)|\bigg{)}^{p}d\xi<\infty,\quad
q=\infty,$

where the essential supremum is taken with respect to $\nu$. Note that the
tent spaces $T_{q,\nu}^{p}$ does not depend on the choice of $\Gamma_{\alpha}$
[MPPW20]. We denote $T_{q,\nu}^{p}$ by $T_{q,\omega,\beta}^{p}$, where
$d\nu(z)=\Big{(}\frac{(1-|z|^{2})^{2}}{\omega(z)}\Big{)}^{\beta}dA(z)$ and
$0<\beta<\infty$.

Our main results are the followings. The first two results characterize the
boundedness and compactness of Volterra type integration operator $J_{g}$ from
weighted Bergman spaces to Hardy spaces.

######  Theorem 1.1.

Let $0<p,q<\infty$, $\omega\in\mathcal{D}$ and $g\in H(\mathbb{D})$. The
following statements hold:

(A):

If $0<p\leq\min\\{2,q\\}$ or $2<p<q<\infty$, then $J_{g}:L_{a}^{p}(\omega)\to
H^{q}$ is bounded if and only if
$g\in\mathcal{B}^{\gamma,\widehat{\omega}^{\frac{1}{p}}},$ where
$\gamma=1+\frac{1}{q}-\frac{1}{p}$;

(B):

If $2<p=q<\infty$, then $J_{g}:L_{a}^{p}(\omega)\to H^{q}$ is bounded if and
only if the positive measure $\mu_{g}$ is a Carleson measure, where
$d\mu_{g}(z)=|g^{\prime}(z)|^{\frac{2p}{p-2}}\frac{(1-|z|^{2})^{\frac{p+2}{p-2}}}{\widehat{\omega}(z)^{\frac{2}{p-2}}}dA(z)$;

(C):

If $0<q<\infty$ and $p>\max\\{2,q\\}$, then $J_{g}:L_{a}^{p}(\omega)\to H^{q}$
is bounded if and only if $g^{\prime}\in
T_{\frac{2p}{p-2},\widehat{\omega},\frac{2}{p-2}}^{\frac{pq}{p-q}}$;

(D):

If $0<q<p\leq 2$, then $J_{g}:L_{a}^{p}(\omega)\to H^{q}$ is bounded if and
only if $B_{g,\omega}\in L^{\frac{pq}{p-q}}(\mathbb{T})$, where
$B_{g,\omega}(\xi)=\sup\limits_{z\in\Gamma(\xi)}|g^{\prime}(z)|\frac{(1-|z|^{2})}{\widehat{\omega}(z)^{\frac{1}{p}}},~{}\xi\in\mathbb{T}.$

######  Theorem 1.2.

Let $0<p,q<\infty$, $\omega\in\mathcal{D}$ and $g\in H(\mathbb{D})$. The
following statements hold:

(A):

If $0<p\leq\min\\{2,q\\}$ or $2<p<q<\infty$, then $J_{g}:L_{a}^{p}(\omega)\to
H^{q}$ is compact if and only if
$g\in\mathcal{B}_{0}^{\gamma,\widehat{\omega}^{\frac{1}{p}}},$ where
$\gamma=1+\frac{1}{q}-\frac{1}{p}$;

(B):

If $2<p=q<\infty$, then $J_{g}:L_{a}^{p}(\omega)\to H^{q}$ is compact if and
only if the positive measure $\mu_{g}$ is a vanishing Carleson measure, where
$d\mu_{g}(z)=|g^{\prime}(z)|^{\frac{2p}{p-2}}\frac{(1-|z|^{2})^{\frac{p+2}{p-2}}}{\widehat{\omega}(z)^{\frac{2}{p-2}}}dA(z)$;

(C):

If $0<q<\infty$ and $p>\max\\{2,q\\}$, then $J_{g}:L_{a}^{p}(\omega)\to H^{q}$
is compact if and only if $g^{\prime}\in
T_{\frac{2p}{p-2},\widehat{\omega},\frac{2}{p-2}}^{\frac{pq}{p-q}}$;

(D):

If $0<q<p\leq 2$, then $J_{g}:L_{a}^{p}(\omega)\to H^{q}$ is compact if and
only if

$\lim\limits_{r\to 1^{-}}\int_{\mathbb{T}}\sup_{z\in\Gamma(\xi)\backslash
r\mathbb{D}}|g^{\prime}(z)|^{\frac{pq}{p-q}}\frac{(1-|z|^{2})^{\frac{pq}{p-q}}}{\widehat{\omega}(z)^{\frac{q}{p-q}}}d\xi=0.$

The following result gives descriptions for the boundedness and compactness of
$J_{g}$ from Hardy spaces to weighted Bergman spaces. For a positive measure
$\mu$, define

$\widetilde{\mu}(\xi)=\int_{\Gamma(\xi)}\frac{d\mu(\lambda)}{1-|\lambda|^{2}},\quad\xi\in\mathbb{T}.$

######  Theorem 1.3.

Let $0<p,q<\infty$. For $\omega\in\mathcal{D}$ and $g\in H(\mathbb{D})$, let
$d\mu_{g}(z)=|g^{\prime}(z)|^{q}(1-|z|^{2})^{q}\omega(z)dA(z)$. Then

(A):

If $0<p\leq q<\infty$, then $J_{g}:H^{p}\to L_{a}^{q}(\omega)$ is bounded
(compact) if and only if $\mu_{g}$ is a $q/p$-Carleson measure (vanishing
$q/p$-Carleson measure);

(B):

If $0<q<p<\infty$, then $J_{g}:H^{p}\to L_{a}^{q}(\omega)$ is compact if and
only if $J_{g}:H^{p}\to L_{a}^{q}(\omega)$ is bounded if and only if
$\widetilde{\mu_{g}}\in L^{\frac{p}{p-q}}(\mathbb{T})$.

######  Remark 1.1.

Theorem 1.1-Theorem 1.3 generalize the recent work of [MPPW20, CPW21] from
standard weights
$\omega_{\alpha}(z)=(1+\alpha)(1-|z|^{2})^{\alpha}dA(z),-1<\alpha<\infty$ to
$\mathcal{D}$ weights. The main strategies are based on a combination of the
basic ideas in [Wu11, MPPW20, CPW21]. Nevertheless, there are two additional
obstacles we need to overcome. The first one is the lack of explicit
expressions for Bergman kernels of $\mathcal{D}$ weighted Bergman spaces. In
addition, the parameter $\alpha$ plays important roles in [MPPW20, CPW21] for
standard weighted Bergman spaces. We use some delicate analysis of the weights
to capture the suitable information from $\omega$ instead of $\alpha$.

######  Remark 1.2.

Since the proof of boundedness (Theorem 1.1) and compactness (Theorem 1.2) of
$J_{g}$ share several overlaps, we just present detailed proof of Theorem 1.2.

Throughout this paper, for a radial weight $\omega$, we denote
$\omega(E)=\int_{E}\omega(z)dA(z)$, for any measurable set
$E\subseteq\mathbb{D}$. And we will use $a\lesssim b$ to represent that there
exists a constant $C=C(\cdot)>0$ satisfying $a\leq Cb$, $a\gtrsim b$ if there
exists a constant $C=C(\cdot)>0$ satisfying $a\geq Cb$, where the constant
$C(\cdot)$ depends on the parameters indicated in the parenthesis, varying
under different circumstances. Moreover, if $a\lesssim b$ and $b\gtrsim a$,
then we denote by $a\asymp b$. Besides, for a number $p>1$, let $p^{\prime}$
be the conjugate index of $p$ such that
$\frac{1}{p}$+$\frac{1}{p^{\prime}}$=1.

##  2\. From Bergman space into Hardy space

The main goal in this section is to prove Theorem 1.2. The whole proof is
divided into four parts.

###  2.1. Case A

First we are devoted to dealing with the case when $0<p\leq\min\\{2,q\\}$ or
$2<p<q<\infty$.

######  Proposition 2.1.

Assume $0<p\leq\min\\{2,q\\}$ or $2<p<q<\infty$. Let $\omega$ be a
$\mathcal{D}$ weight and $\gamma=1+\frac{1}{q}-\frac{1}{p}$. Then the Volterra
type integration operator $J_{g}:L_{a}^{p}(\omega)\to H^{q}$ is compact if and
only if $g\in\mathcal{B}_{0}^{\gamma,\widehat{\omega}^{1/p}},$ where
$\widehat{\omega}(z)=\int_{|z|}^{1}\omega(r)dr$.

To show Proposition 2.1, we first give some useful lemmas. By a disk version
of Calderón’s theorem, one has the following:

######  Lemma 2.2.

[Pav13, Theorem 1.3] Suppose $f$ is analytic on $\mathbb{D}$ with $f(0)=0$. If
$0<p<\infty$, then

$\|f\|_{H^{p}}^{p}\asymp\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)}|f^{\prime}(z)|^{2}dA(z)\bigg{)}^{\frac{p}{2}}d\xi.$

The following Dirichlet-type embedding theorem can be found in [MPPW20].

######  Lemma 2.3.

Suppose $f$ is analytic on $\mathbb{D}$ with $f(0)=0$. If $0<q\leq 2$, then

(2.1) $\|f\|_{H^{q}}\lesssim\|f^{\prime}\|_{L_{a}^{q}(\omega_{q-1})}.$

If $0<p<q<\infty$, then

(2.2)
$\|f\|_{H^{q}}\lesssim\|f^{\prime}\|_{L_{a}^{p}(\omega_{p+\frac{p}{q}-2})}.$

Now we recall two classes of weights which have close relations with
$\mathcal{D}$ weights. A weight $\omega$ is called regular [PR14] if $\omega$
is continuous and radial on $\mathbb{D}$ such that
$\psi_{\omega}(r)\asymp(1-r)$ for $0\leq r<1$, where
$\psi_{\omega}(r)=\frac{\widehat{\omega}(r)}{\omega(r)}$ is the distortion
function, which was introduced in [Sis00]. We denote the class of regular
weights by $\mathcal{R}$. And a weight $\omega$ is a $\widehat{\mathcal{D}}$
weight [PRS18] if there exists a positive constant $C$ such that

$\widehat{\omega}(r)\leq
C\widehat{\omega}(\frac{1+r}{2}),\quad\textmd{for~{}all}~{}r\in[0,1).$

Note that $\mathcal{R}\subsetneq\mathcal{D}\subsetneq\widehat{\mathcal{D}}$
[PR14, PRS18, PR19]. Here we list some useful properties of
$\widehat{\mathcal{D}}$, $\mathcal{D}$ and regular weights which are important
in our subsequent work.

######  Lemma 2.4.

[PRS18, Theorem C] Let $1<p<\infty$ and $\omega$ be a $\widehat{\mathcal{D}}$
weight. Then the reproducing kenel of $L_{a}^{2}(\omega)$ at $z$ has the
following estimates,

$\|K_{z}^{\omega}\|_{L_{a}^{p}(\omega)}^{p}\asymp\int_{|z|}^{1}\frac{dt}{\widehat{\omega}(t)^{p-1}(1-t)^{p}},\quad|z|\to
1^{-}.$

From Lemma 2.4, we get

(2.3)
$\|K_{z}^{\omega}\|_{L_{a}^{p}(\omega)}^{p}\asymp\frac{1}{\widehat{\omega}(z)^{p-1}(1-|z|)^{p-1}},\quad
z\in\mathbb{D}.$

By [PRS18, Proposition 5] and its proof, we have

######  Lemma 2.5.

Let $0<p<\infty$ and $\omega$ be a $\mathcal{D}$ weight. Then
$\kappa(z)=(1-|z|)^{-1}\widehat{\omega}(z)$ is regular and
$\|f\|_{L^{p}(\omega)}\asymp\|f\|_{L^{p}(\kappa)}$, for each analytic function
$f$ on $\mathbb{D}$.

Denote by $B(z,r)$ and $D(z,r)$ the Euclidean disk and the pseudo-hyperbolic
disk centered at $z$ with radius $r$ respectively. The next result shows a
nice property of regular weights, which can be found in page 8 of [PR14].

######  Lemma 2.6.

Let $\omega$ be a regular weight. Then for any $z\in\mathbb{D}$ and
$r\in(0,1)$,

$\omega(z)\asymp\omega(u),\quad\textmd{for~{}all}~{}u\in D(z,r).$

Combine Lemma 2.5 with Lemma 2.6, we get the following result.

######  Lemma 2.7.

Let $\omega\in\mathcal{D}$ and $r\in(0,1)$. Then for any $z\in\mathbb{D}$,

$\widehat{\omega}(z)\asymp\widehat{\omega}(u),\quad\textmd{for~{}all}~{}u\in
D(z,r).$

Now we are ready to prove Proposition 2.1.

###### Proof of Proposition 2.1.

Sufficiency. Let $\\{f_{j}\\}_{j=1}^{\infty}$ be any sequence in
$L_{a}^{p}(\omega)$ such that $\|f_{j}\|_{L_{a}^{p}(\omega)}\leq 1$ and
$f_{j}$ uniformly converges to 0 on any compact subset of $\mathbb{D}$. If
$g\in\mathcal{B}_{0}^{\gamma,\widehat{\omega}^{1/p}}$, then we will show

$\lim\limits_{j\to\infty}\|J_{g}(f_{j})\|_{H^{q}}=0.$

For any $\varepsilon>0$, there exists $r\in(0,1)$ such that for any
$z\in\mathbb{D}$ with $r\leq|z|<1$, it holds that

(2.4)
$\frac{(1-|z|^{2})^{\gamma}}{\widehat{\omega}(z)^{1/p}}|g^{\prime}(z)|<\varepsilon.$

Moreover, we can find a constant $k\in\mathbb{N}$ such that for any $j>k$,
$|f_{j}|<\varepsilon$ on $r\mathbb{D}:=\\{z:|z|<r\\}$. By Lemma 2.3, for any
$j\in\mathbb{N}$, we obtain

$\|J_{g}(f_{j})\|_{H^{q}}\lesssim\|(J_{g}f_{j})^{\prime}\|_{L_{a}^{p}(\omega_{p\gamma-1})}.$

Together with Lemma 2.5 and (2.4), for any $j>k$, we get

$\displaystyle\|J_{g}(f_{j})\|_{H^{q}}$
$\displaystyle\lesssim\bigg{(}\int_{r\mathbb{D}}|f_{j}(z)|^{p}\frac{\widehat{\omega}(z)}{1-|z|}|g^{\prime}(z)|^{p}\frac{(1-|z|^{2})^{p\gamma}}{\widehat{\omega}(z)}dA(z)\bigg{)}^{\frac{1}{p}}$
$\displaystyle+\bigg{(}\int_{\mathbb{D}\backslash
r\mathbb{D}}|f_{j}(z)|^{p}\frac{\widehat{\omega}(z)}{1-|z|}|g^{\prime}(z)|^{p}\frac{(1-|z|^{2})^{p\gamma}}{\widehat{\omega}(z)}dA(z)\bigg{)}^{\frac{1}{p}}$
$\displaystyle\lesssim\varepsilon+\varepsilon\|f_{j}\|_{L_{a}^{p}(\omega)}\lesssim\varepsilon.$

Necessity. Assume that $J_{g}:L_{a}^{p}(\omega)\to H^{q}$ is compact. Let
$\alpha>\frac{1}{p}$ and $\lambda\in\mathbb{D}$. Take testing functions

$f_{\lambda}(z)=\frac{(K_{\lambda}^{\omega}(z))^{\alpha}}{\|K_{\lambda}^{\omega}\|_{L_{a}^{\alpha+1-\frac{1}{p}}(\omega)}^{\alpha+1-\frac{1}{p}}},\quad
z\in\mathbb{D}.$

Let
$\omega_{2n+1}=\int_{0}^{1}\omega(r)r^{2n+1}dr,~{}n\in\mathbb{N}\bigcup\\{0\\}$.
It follows from [PR16, formula (20)] that

$|K_{\lambda}^{\omega}(z)|\leq\sum_{n=0}^{\infty}\frac{|z|^{n}}{2\omega_{2n+1}}\asymp\frac{1}{\omega_{1}}+\int_{0}^{|z|}\frac{1}{\widehat{\omega}(t)(1-t)^{2}}dt\lesssim\frac{1}{\widehat{\omega}(z)(1-|z|)},\quad|z|\to
1^{-}.$

Together with (2.3), we obtain

$|f_{\lambda}(z)|\asymp\widehat{\omega}(\lambda)^{\alpha-\frac{1}{p}}(1-|\lambda|)^{\alpha-\frac{1}{p}}|K_{\lambda}^{\omega}(z)|^{\alpha}\lesssim\frac{\widehat{\omega}(\lambda)^{\alpha-\frac{1}{p}}(1-|\lambda|)^{\alpha-\frac{1}{p}}}{\widehat{\omega}(z)^{\alpha}(1-|z|)^{\alpha}},\quad
z\in\mathbb{D}.$

Then for any compact subset $E\subseteq\mathbb{D}$, we have

$\lim\limits_{|\lambda|\to 1^{-}}\sup\limits_{z\in E}|f_{\lambda}(z)|=0.$

Moreover, from (2.3), we know

$\|f_{\lambda}\|_{L_{a}^{p}(\omega)}\asymp\widehat{\omega}(\lambda)^{\alpha-\frac{1}{p}}(1-\lambda)^{\alpha-\frac{1}{p}}\bigg{(}\int_{\mathbb{D}}|K_{\lambda}^{\omega}(z)|^{\alpha
p}\omega(z)dA(z)\bigg{)}^{\frac{1}{p}}\asymp 1.$

Therefore, we deduce

$\lim\limits_{|\lambda|\to 1^{-}}\|J_{g}(f_{\lambda})\|_{H^{q}}=0.$

For any $z,\lambda\in\mathbb{D}$, by using a classical result of the pointwise
estimate for $(J_{g}f_{\lambda})^{\prime}$ (see [Gar07, Page 85-86]), we get

(2.5)
$\displaystyle|f_{\lambda}(z)g^{\prime}(z)|=|(J_{g}f_{\lambda})^{\prime}(z)|\lesssim\frac{1}{(1-|z|^{2})^{1+\frac{1}{q}}}\|J_{g}f_{\lambda}\|_{H^{q}}.$

By choosing $z=\lambda$ in (2.5), together with (2.3), we deduce

$\frac{(1-|\lambda|^{2})^{\gamma}}{\widehat{\omega}(\lambda)^{\frac{1}{p}}}|g^{\prime}(\lambda)|\lesssim\|J_{g}f_{\lambda}\|_{H^{q}},$

from which it follows that

$\lim\limits_{|\lambda|\to
1^{-}}\frac{(1-|\lambda|^{2})^{\gamma}}{\widehat{\omega}(\lambda)^{\frac{1}{p}}}|g^{\prime}(\lambda)|=0.$

Thus, we complete the proof. ∎

###  2.2. Case B

Now we turn to deal with the case $2<p=q<\infty$.

For $k\in\mathbb{N}$, recall the Rademacher functions
$r_{k}(t):=\textmd{sgn}\\{\sin(2^{k}\pi t)\\},~{}t\in[0,1].$ The following
Khinchine’s inequality and Kahane’s inequality [Lue93] are useful in our
subsequent work.

######  Lemma 2.8.

Let $0<p<\infty$. Then for any sequence of complex numbers
$\\{c_{k}\\}_{k=1}^{\infty}$ , it holds that

$\bigg{(}\sum_{k=1}^{\infty}|c_{k}|^{2}\bigg{)}^{\frac{p}{2}}\asymp\int_{0}^{1}\bigg{|}\sum_{k=1}^{\infty}c_{k}r_{k}(t)\bigg{|}^{p}dt.$

######  Lemma 2.9.

Let $0<p,q<\infty$ and $X$ be a quasi-Banach space with quasi-norm
$\|\cdot\|_{X}$. For any sequence $\\{x_{k}\\}_{k=1}^{\infty}\subseteq X$, it
holds that

$\bigg{(}\int_{0}^{1}\big{\|}\sum_{k=1}^{\infty}r_{k}(t)x_{k}\big{\|}_{X}^{q}dt\bigg{)}^{\frac{1}{q}}\asymp\bigg{(}\int_{0}^{1}\big{\|}\sum_{k=1}^{\infty}r_{k}(t)x_{k}\big{\|}_{X}^{p}dt\bigg{)}^{\frac{1}{p}}.$

We first generalize Theorem 1 in [Wu11] to $\mathcal{D}$ weighted Bergman
spaces.

######  Proposition 2.10.

Let $\omega\in\mathcal{D}$, $\kappa(z)=\frac{\widehat{\omega}(z)}{1-|z|}$ and
$\mu$ be a positive measure on $\mathbb{D}$. Let $p\in(0,+\infty)$,
$s\in[1,+\infty)$ and $\iota\in(0,1)$. Then the following statements are
equivalent:

$\big{(}\frac{\mu(D(z,\tilde{\delta}))}{\kappa(D(z,\tilde{\delta}))}\big{)}^{\frac{1}{1-\iota}}\kappa(z)dA(z)$
is a vanishing $s$-Carleson measure, for some $\tilde{\delta}\in(0,1)$;

For any $f\in L_{a}^{p}(\omega)$, $|f(z)|^{p\iota}d\mu(z)$ is a vanishing
$s(1-\iota)$-Carleson measure, that is, for any $f\in L_{a}^{p}(\omega)$ and
any $\varepsilon>0$, there exists $r_{0}\in(0,1)$ such that for any
$I\subseteq\mathbb{T}$ with $0<|I|\leq r_{0}$, it holds that

$\frac{\int_{S(I)}|f(z)|^{p\iota}d\mu(z)}{|I|^{s(1-\iota)}}\lesssim\varepsilon\|f\|_{L_{a}^{p}(\omega)}^{p\iota},$

where the constant of comparison does not depend on the choice of the function
$f$.

Before approaching the proof of Proposition 2.10, we first give some
notations. For $0<\alpha<\infty$ and $\xi\in\mathbb{T}$, we define the arc on
$\mathbb{T}$ as follows,

$I_{\alpha}(z)=\bigg{\\{}\xi\in\mathbb{T}:z\in\Gamma_{\alpha}(\xi)\bigg{\\}},\quad
z\in\mathbb{D}.$

Note that

(2.6) $|I_{\alpha}(z)|\asymp 1-|z|,\quad z\in\mathbb{D},$

see [Wu11, formula (3.3)]. We denote $I_{\alpha}(z)$ by $I(z)$ if $\alpha=1$.
For $I\subseteq\mathbb{T}$, denote by the $\alpha$-tent of $I$

$\wedge_{\alpha}(I)=\bigg{\\{}z\in\mathbb{D}:I_{\alpha}(z)\subseteq
I\bigg{\\}}.$

First observe the relationship between the Carleson square and the
$\alpha$-tent of an arc on $\mathbb{T}$. For any fixed $\alpha>0$, for any
$I\subseteq\mathbb{T}$, there exist $I_{1},I_{2}\subseteq\mathbb{T}$ such that
$S(I_{1})\subseteq\wedge_{\alpha}(I)\subseteq S(I_{2})$ and
$|I_{1}|\asymp|I|\asymp|I_{2}|$. This makes us to choose $S(I)$ or
$\wedge_{\alpha}(I)$ freely in the definition of vanishing Carleson measure.

Next we recall some important observations over geometrical regions by Wu in
[Wu11].

######  Lemma 2.11.

[Wu11, Lemma 2.2 and Lemma 3.1] Let $\alpha>0$, $\xi\in\mathbb{T}$ and
$I\subseteq\mathbb{T}$. Let $\tilde{t}=\tanh t$ and
$\alpha_{t}=(\alpha+2)e^{2t}-2$ for $t\geq 0$. Then

$D(z,\tilde{t})\subseteq\Gamma_{\alpha_{t}}(\xi),\quad\textmd{for~{}any}~{}z\in\Gamma_{\alpha}(\xi),$
$D(z,\tilde{t})\subseteq\wedge_{\alpha}(I),\quad\textmd{for}~{}z\in\wedge_{\alpha_{t}}(I)$

and

$D(z,\tilde{t})\subseteq\wedge_{\alpha}(I),\quad\textmd{for}~{}D(z,\tilde{t})\cap\wedge_{\alpha_{2t}}(I)\neq\emptyset.$

Let $\\{z_{j}\\}_{j=1}^{\infty}$ be a sequence in $\mathbb{D}$, $\delta>0$ and
$r\in(0,1)$. If
$\big{|}\frac{z_{j}-z_{k}}{1-\bar{z_{j}}z_{k}}\big{|}\geq\delta$ for any
$j\neq k$, then we say the sequence $\\{z_{j}\\}_{j=1}^{\infty}$ is
$\delta$-separated. And if $\\{z_{j}\\}_{j=1}^{\infty}$ satisfies
$\mathbb{D}=\bigcup\limits_{j=1}^{\infty}D(z_{j},r)$ and
$\frac{r}{5}$-separated, we say $\\{z_{j}\\}_{j=1}^{\infty}$ is an
$r$-lattice. Notice that every $z\in\mathbb{D}$ belongs to at most $N=N(r)$
pseudohyperbolic disks $D(z_{j},r)$, where $\\{z_{j}\\}_{j=1}^{\infty}$ is a
separated sequence (see Lemma 12 of Chapter 2 in [Dur04] or Lemma 3 in
[Lue93]). Let $\\{z_{k}\\}$ be a separated sequence. For any
$K\in(1,+\infty)$, we re-index the sequence $\\{z_{k}\\}$ as follows: For each
$j\in\mathbb{N}\bigcup\\{0\\}$, we denote the points of the sequence
$\\{z_{k}\\}$ in the annulus $A_{j}=\\{z\in\mathbb{D}:r_{j}\leq|z|<r_{j+1}\\}$
by $\\{z_{j,l}\\}_{l}$, where $r_{j}=1-K^{-j}$. For $p\in(0,\infty)$, note
that a discrete sequence $\\{\lambda_{j,l}\\}_{j,l=1}^{\infty}\in l^{p}$ if

$\bigg{\|}\\{\lambda_{j,l}\\}_{j,l=1}^{\infty}\bigg{\|}_{l^{p}}:=\bigg{(}\sum_{j=1}^{\infty}\sum_{l=1}^{\infty}|\lambda_{j,l}|^{p}\bigg{)}^{\frac{1}{p}}<\infty.$

The following (see [PRS21, Theorem 1]) is the atomic decomposition for Bergman
spaces induced by $\mathcal{D}$ weights. One can consult [PRS21] for more
information.

######  Lemma 2.12.

Let $0<p<\infty$, $1<K<\infty$, $\omega\in\mathcal{D}$ and $\\{z_{k}\\}$ be a
separated sequence in $\mathbb{D}$. Denote by $\\{z_{j,l}\\}_{j,l}$ the
sequence re-indexed from $\\{z_{k}\\}$ using the method above. Then there
exists a positive constant $M$ ($M$ only depends on the choice of $\omega$ and
$p$) such that for any sequence $\\{\lambda_{j,l}\\}_{j,l=1}^{\infty}\in
l^{p}$, the function

$F(z):=\sum_{j,l=1}^{\infty}\lambda_{j,l}\frac{(1-|z_{j,l}|)^{M-\frac{1}{p}}\widehat{\omega}(z_{j,l})^{-\frac{1}{p}}}{(1-\overline{z_{j,l}}z)^{M}}$

is analytic in $\mathbb{D}$ and

$\|F\|_{L_{a}^{p}(\omega)}\lesssim\|\\{\lambda_{j,l}\\}_{j,l=1}^{\infty}\|_{l^{p}}.$

Now we are in a position to prove Proposition 2.10.

###### Proof of Proposition 2.10.

First we prove that $\textmd{(i)}\Rightarrow\textmd{(ii)}$. Choose
$\delta\in(0,\infty)$ such that $\tilde{\delta}:=\tanh\delta$ satisfies the
condition in (i), it follows from Lemma 2.5 that
$\kappa(z)=\frac{\widehat{\omega}(z)}{1-|z|}$ is a regular weight. By Lemma
2.6, we know that

(2.7) $\kappa(z)\asymp\kappa(\zeta),\quad\zeta\in D(z,\tilde{\delta}).$

Let $0<p<\infty$, $1\leq s<\infty$, $0<\iota<1$ and $\alpha>0$. Let
$\alpha_{\delta}=(\alpha+2)e^{2\delta}-2$. For any $f\in L_{a}^{p}(\omega)$
and $I\subseteq\mathbb{T}$, using the subharmonicity of $|f|^{p}$, Lemma 2.11
and (2.7), we obtain

$\displaystyle|f(z)|^{p\iota}$
$\displaystyle\lesssim\frac{1}{(1-|z|)^{2}}\int_{B(z,\tilde{\delta}(1-|z|))}|f(\zeta)|^{p\iota}dA(\zeta)$
$\displaystyle\lesssim\frac{1}{\kappa(z)(1-|z|)^{2}}\int_{D(z,\tilde{\delta})}|f(\zeta)|^{p\iota}\kappa(\zeta)dA(\zeta)$
(2.8)
$\displaystyle=\frac{1}{\kappa(z)(1-|z|)^{2}}\int_{\wedge_{\alpha}(I)\cap
D(z,\tilde{\delta})}|f(\zeta)|^{p\iota}\kappa(\zeta)dA(\zeta),\quad\textmd{for~{}any}~{}z\in\wedge_{\alpha_{\delta}}(I).$

Combining Lemma 2.5, Fubini’s theorem,
$\textmd{H}\ddot{\textmd{o}}\textmd{lder's}$ inequality, (2.7) with (2.2), for
any $I\subseteq\mathbb{T}$, we get

$\displaystyle\int_{\wedge_{\alpha_{\delta}}(I)}|f(z)|^{p\iota}d\mu(z)$
$\displaystyle\lesssim\int_{\wedge_{\alpha_{\delta}}(I)}\frac{1}{\kappa(z)(1-|z|)^{2}}\int_{\wedge_{\alpha}(I)\cap
D(z,\tilde{\delta})}|f(\zeta)|^{p\iota}\kappa(\zeta)dA(\zeta)d\mu(z)$
$\displaystyle\asymp\int_{\wedge_{\alpha}(I)}\frac{1}{\kappa(\zeta)(1-|\zeta|)^{2}}\int_{\wedge_{\alpha_{\delta}}(I)\cap
D(\zeta,\tilde{\delta})}d\mu(z)|f(\zeta)|^{p\iota}\kappa(\zeta)dA(\zeta)$
$\displaystyle\lesssim\int_{\wedge_{\alpha}(I)}\frac{\mu(D(\zeta,\tilde{\delta}))}{\kappa(D(\zeta,\tilde{\delta}))}|f(\zeta)|^{p\iota}\kappa(\zeta)dA(\zeta)$
(2.9)
$\displaystyle\lesssim\|f\|_{L_{a}^{p}(\omega)}^{p\iota}\bigg{(}\int_{\wedge_{\alpha}(I)}\bigg{(}\frac{\mu(D(\zeta,\tilde{\delta}))}{\kappa(D(\zeta,\tilde{\delta}))}\bigg{)}^{\frac{1}{1-\iota}}\kappa(\zeta)dA(\zeta)\bigg{)}^{1-\iota}.$

Since
$\big{(}\frac{\mu(D(\zeta,\tilde{\delta}))}{\kappa(D(\zeta,\tilde{\delta}))}\big{)}^{\frac{1}{1-\iota}}\kappa(\zeta)dA(\zeta)$
is a vanishing $s$-Carleson measure, then for any $\varepsilon>0$, there
exists $r_{1}\in(0,1)$ such that for any $I\subseteq\mathbb{T}$ with
$0<|I|\leq r_{1}$, it holds that

$\frac{\int_{\wedge_{\alpha}(I)}\big{(}\frac{\mu(D(\zeta,\tilde{\delta}))}{\kappa(D(\zeta,\tilde{\delta}))}\big{)}^{\frac{1}{1-\iota}}\kappa(\zeta)dA(\zeta)}{|I|^{s}}<\varepsilon^{\frac{1}{1-\iota}}.$

Combining with (2.2), there exists a constant $C$ ($C$ does not depend on the
choice of $f$) such that for any $I\subseteq\mathbb{T}$ with $0<|I|\leq
r_{1}$, for any $f\in L_{a}^{p}(\omega)$, we have

$\frac{\int_{\wedge_{\alpha_{\delta}}(I)}|f(z)|^{p\iota}d\mu(z)}{|I|^{s(1-\iota)}}\leq\varepsilon
C\|f\|_{L_{a}^{p}(\omega)}^{p\iota},$

as desired. Next we show $\textmd{(ii)}\Rightarrow\textmd{(i)}$. By the
hypothesis, for any $f\in L_{a}^{p}(\omega)$ and $\varepsilon>0$, there exists
$r_{2}\in(0,1)$ such that for any $I\subseteq\mathbb{T}$ with $0<|I|\leq
r_{2}$, it holds that

$\frac{\int_{\wedge_{\alpha}(I)}|f(z)|^{p\iota}d\mu(z)}{|I|^{s(1-\iota)}}\lesssim\varepsilon\|f\|_{L_{a}^{p}(\omega)}^{p\iota}.$

Take $\delta>0$ such that $\tilde{\delta}=\tanh\delta\in(0,\frac{1}{2})$. Let
$\\{z_{k}\\}_{k=1}^{\infty}$ be a $\tilde{\delta}$-lattice. Take the constant
$M$ and denote by $\\{z_{j,l}\\}_{j,l=1}^{\infty}$ the sequence re-indexed
from $\\{z_{k}\\}_{k=1}^{\infty}$ as in Lemma 2.12. Hence, by Lemma 2.12, for
any $\\{\lambda_{j,l}\\}_{j,l=1}^{\infty}\in l^{p}$, for any
$I\subseteq\mathbb{T}$ with $0<|I|\leq r_{2}$, it holds that

$\frac{\int_{\wedge_{\alpha}(I)}\big{|}\sum_{j,l=1}^{\infty}\lambda_{j,l}f_{j,l}(z)\big{|}^{p\iota}d\mu(z)}{|I|^{s(1-\iota)}}\lesssim\varepsilon\bigg{\|}\sum_{j,l=1}^{\infty}\lambda_{j,l}f_{j,l}(z)\bigg{\|}_{L_{a}^{p}(\omega)}^{p\iota}\lesssim\varepsilon\|\\{\lambda_{j,l}\\}_{j,l=1}^{\infty}\|_{l^{p}}^{p\iota},$

where

$f_{j,l}(z)=\frac{(1-|z_{j,l}|)^{M-\frac{1}{p}}\widehat{\omega}(z_{j,l})^{-\frac{1}{p}}}{(1-\overline{z_{j,l}}z)^{M}},\quad
z\in\mathbb{D}.$

For $t\in(0,1)$, replace $\lambda_{j,l}$ by $\lambda_{j,l}r_{j,l}(t)$, where
$r_{j,l}(t)$ is the Rademacher function. Then for any $I\subseteq\mathbb{T}$
with $0<|I|\leq r_{2}$, we have

$\frac{\int_{\wedge_{\alpha}(I)}\big{|}\sum_{j,l=1}^{\infty}\lambda_{j,l}r_{j,l}(t)f_{j,l}(z)\big{|}^{p\iota}d\mu(z)}{|I|^{s(1-\iota)}}\lesssim\varepsilon\|\\{\lambda_{j,l}\\}_{j,l=1}^{\infty}\|_{l^{p}}^{p\iota}.$

Integrate both sides of the above inequality over $(0,1)$ with respect to $t$,
we obtain

$\frac{\int_{\wedge_{\alpha}(I)}\int_{0}^{1}\big{|}\sum_{j,l=1}^{\infty}\lambda_{j,l}r_{j,l}(t)f_{j,l}(z)\big{|}^{p\iota}dtd\mu(z)}{|I|^{s(1-\iota)}}\lesssim\varepsilon\|\\{\lambda_{j,l}\\}_{j,l=1}^{\infty}\|_{l^{p}}^{p\iota}.$

Combining with Lemma 2.8, we get

(2.10)
$\int_{\wedge_{\alpha}(I)}\bigg{(}\sum_{j,l=1}^{\infty}|\lambda_{j,l}|^{2}|f_{j,l}(z)|^{2}\bigg{)}^{\frac{p\iota}{2}}d\mu(z)\lesssim\varepsilon|I|^{s(1-\iota)}\|\\{\lambda_{j,l}\\}_{j,l=1}^{\infty}\|_{l^{p}}^{p\iota},$

for $I\subseteq\mathbb{T}~{}\textmd{with}~{}0<|I|\leq r_{2}$. It follows from
(2.10) that there exists a constant $N=N(\delta)$ such that

$\displaystyle\sum_{\\{j,l:D(z_{j,l},2\tilde{\delta})\subseteq\wedge_{\alpha}(I)\\}}|\lambda_{j,l}|^{p\iota}\frac{\mu(D(z_{j,l},2\tilde{\delta}))}{\widehat{\omega}(z_{j,l})^{\iota}(1-|z_{j,l}|)^{\iota}}$
$\displaystyle\asymp\sum_{\\{j,l:D(z_{j,l},2\tilde{\delta})\subseteq\wedge_{\alpha}(I)\\}}\int_{D(z_{j,l},2\tilde{\delta})}\big{(}|\lambda_{j,l}|\cdot|f_{j,l}(z)|\big{)}^{p\iota}d\mu(z)$
$\displaystyle\leq\sum_{\\{j,l:D(z_{j,l},2\tilde{\delta})\subseteq\wedge_{\alpha}(I)\\}}\int_{D(z_{j,l},2\tilde{\delta})}\big{(}\sum_{j,l=1}^{\infty}|\lambda_{j,l}|^{2}\cdot|f_{j,l}(z)|^{2}\big{)}^{\frac{p\iota}{2}}d\mu(z)$
$\displaystyle\leq
N\int_{\wedge_{\alpha}(I)}\big{(}\sum_{j,l=1}^{\infty}|\lambda_{j,l}|^{2}\cdot|f_{j,l}(z)|^{2}\big{)}^{\frac{p\iota}{2}}d\mu(z)$
(2.11) $\displaystyle\lesssim\varepsilon
N|I|^{s(1-\iota)}\|\\{\lambda_{j,l}\\}_{j,l=1}^{\infty}\|_{l^{p}}^{p\iota},\quad\textmd{for}~{}I\subseteq\mathbb{T}~{}\textmd{with}~{}0<|I|\leq
r_{2}.$

Since the dual space of $l^{\frac{1}{\iota}}$ can be identified with
$l^{\frac{1}{1-\iota}}$, together with (2.2), we deduce

(2.12)
$\bigg{[}\sum_{\\{j,l:D(z_{j,l},2\tilde{\delta})\subseteq\wedge_{\alpha}(I)\\}}\bigg{(}\frac{\mu(D(z_{j,l},2\tilde{\delta}))}{\widehat{\omega}(z_{j,l})^{\iota}(1-|z_{j,l}|)^{\iota}}\bigg{)}^{\frac{1}{1-\iota}}\bigg{]}^{1-\iota}\lesssim\varepsilon|I|^{s(1-\iota)},\quad\textmd{for}~{}I\subseteq\mathbb{T}~{}\textmd{with}~{}0<|I|\leq
r_{2}.$

Take $t=2\tanh^{-1}(2\tilde{\delta})$. Let $\alpha_{t}=(\alpha+2)e^{2t}-2$. By
Lemma 2.7, Lemma 2.11 and (2.12), for any $I\subseteq\mathbb{T}$ with
$0<|I|\leq r_{2}$, we get

$\displaystyle\int_{\wedge_{\alpha_{t}}(I)}\bigg{(}\frac{\mu(D(z,\tilde{\delta}))}{\kappa(D(z,\tilde{\delta}))}\bigg{)}^{\frac{1}{1-\iota}}\kappa(z)dA(z)$
$\displaystyle\leq\sum_{\\{j,l:D(z_{j,l},\tilde{\delta})\bigcap\wedge_{\alpha_{t}(I)\neq\varnothing}\\}}\int_{D(z_{j,l},\tilde{\delta})}\bigg{(}\frac{\mu(D(z,\tilde{\delta}))}{\kappa(D(z,\tilde{\delta}))}\bigg{)}^{\frac{1}{1-\iota}}\kappa(z)dA(z)$
$\displaystyle\leq\sum_{\\{j,l:D(z_{j,l},2\tilde{\delta})\subseteq\wedge_{\alpha}(I)\\}}\int_{D(z_{j,l},\tilde{\delta})}\bigg{(}\frac{\mu(D(z,\tilde{\delta}))}{\kappa(D(z,\tilde{\delta}))}\bigg{)}^{\frac{1}{1-\iota}}\kappa(z)dA(z)$
$\displaystyle\lesssim\varepsilon^{\frac{1}{1-\iota}}|I|^{s},$

which yields that
$\big{(}\frac{\mu(D(z,\tilde{\delta}))}{\kappa(D(z,\tilde{\delta}))}\big{)}^{\frac{1}{1-\iota}}\kappa(z)dA(z)$
is a vanishing $s$-Carleson measure, as desired. ∎

For a positive measure $\mu$ on $\mathbb{D}$ and $\alpha>0$, define the area
integral operator $A_{\mu,\alpha}^{2}$ acting on $f\in H(\mathbb{D})$ by

$A_{\mu,\alpha}^{2}(f)(\xi)=\bigg{(}\int_{\Gamma_{\alpha}(\xi)}|f(z)|^{2}\frac{d\mu(z)}{1-|z|}\bigg{)}^{\frac{1}{2}},\quad\xi\in\mathbb{T}.$

For a $\mathcal{D}$ weight $\omega$, $0<q,\alpha<\infty$ and $g\in
H(\mathbb{D})$, it follows from Lemma 2.2 that the Volterra type integration
operator is intimately related to area integral operator as follows,

(2.13)
$\|J_{g}(f)\|_{H^{q}}\asymp\|A_{\mu_{g},\alpha}^{2}(f)\|_{L^{q}(\mathbb{T})},\quad\textmd{for~{}any}~{}f\in
L_{a}^{p}(\omega),$

where $d\mu_{g}(z)=|g^{\prime}(z)|^{2}(1-|z|)dA(z)$.

######  Proposition 2.13.

Let $2<p<\infty$, $\omega\in\mathcal{D}$, $g\in H(\mathbb{D})$ and

$d\mu_{g}(z)=|g^{\prime}(z)|^{\frac{2p}{p-2}}\frac{(1-|z|^{2})^{\frac{p+2}{p-2}}}{\widehat{\omega}(z)^{\frac{2}{p-2}}}dA(z).$

Then the Volterra type integration operator $J_{g}:L_{a}^{p}(\omega)\to H^{p}$
is compact if and only if $\mu_{g}$ is a vanishing Carleson measure.

###### Proof.

Sufficiency. Let $\\{f_{j}\\}_{j=1}^{\infty}$ be any sequence in
$L_{a}^{p}(\omega)$ with $\|f_{j}\|_{L_{a}^{p}(\omega)}\leq 1$ and $f_{j}$
converges to 0 uniformly on any compact subset of $\mathbb{D}$. Since
$\mu_{g}$ is a vanishing Carleson measure, for any $\varepsilon>0$, there
exists $r_{0}\in(0,1)$ such that for any $I\subseteq\mathbb{T}$ with $|I|\leq
1-r_{0}$, it holds that

(2.14) $\frac{\mu_{g}(S(I))}{|I|}<\varepsilon^{\frac{2p}{p-2}}.$

Moreover, there also exists $k_{0}\in\mathbb{N}$ such that $j>k_{0}$,
$|f_{j}|<\varepsilon$ on $r_{0}\mathbb{D}$. Let

$d\mu_{f_{j},g}(z)=|f_{j}(z)|^{2}|g^{\prime}(z)|^{2}(1-|z|^{2})dA(z).$

By virtue of (2.14), we obtain

(2.15) $\frac{\mu_{g}(S(I)\backslash
r_{0}\mathbb{D})}{|I|}<\varepsilon^{\frac{2p}{p-2}},\quad\textmd{for~{}any}~{}I\subseteq\mathbb{T}~{}\textmd{with}~{}|I|\leq
1-r_{0}.$

In fact, we have

(2.16) $\frac{\mu_{g}(S(I)\backslash
r_{0}\mathbb{D})}{|I|}<2\varepsilon^{\frac{2p}{p-2}},\quad\textmd{for~{}any}~{}I\subseteq\mathbb{T}.$

For any $|I|>1-r_{0}$, there exists $k\in\mathbb{N}$ such that
$(k-1)(1-r_{0})<|I|\leq k(1-r_{0})$. There also exist intervals $I_{l}$ on
$\mathbb{T}$, $l=1,2,\cdots,k$ such that $|I_{l}|=1-r_{0}$, $l=1,2,\cdots,k$
and $I\subseteq\bigcup\limits_{l=1}^{k}I_{l}$. Notice that $k(1-r_{0})<2|I|$.
Combining with (2.15), we deduce

$\mu_{g}(S(I)\backslash
r_{0}\mathbb{D})\leq\sum_{l=1}^{k}\mu_{g}(S(I_{l})\backslash
r_{0}\mathbb{D})<\varepsilon^{\frac{2p}{p-2}}k(1-r_{0})<\varepsilon^{\frac{2p}{p-2}}2|I|,$

which yields that (2.16) holds.

Let

$\kappa(z)=\frac{\widehat{\omega}(z)}{1-|z|},\quad z\in\mathbb{D}.$

Since $\chi_{\mathbb{D}\backslash r_{0}\mathbb{D}}\mu_{g}$ is a Carleson
measure, by [Dur70, Theorem 9.4], Lemma 2.5,
$\textmd{H}\ddot{\textmd{o}}\textmd{lder's}$ inequality and (2.16), for any
$h\in H^{\frac{p}{p-2}}$, we have

$\displaystyle\int_{\mathbb{D}\backslash
r_{0}\mathbb{D}}|h(z)|d\mu_{f_{j},g}(z)$
$\displaystyle\leq\bigg{(}\int_{\mathbb{D}\backslash
r_{0}\mathbb{D}}|h(z)|^{\frac{p}{p-2}}d\mu_{g}(z)\bigg{)}^{\frac{p-2}{p}}\|f_{j}\|_{L_{a}^{p}(\kappa)}^{2}$
$\displaystyle\lesssim\|f_{j}\|_{L_{a}^{p}(\omega)}^{2}\|h\|_{H^{\frac{p}{p-2}}}\cdot\bigg{(}\sup\limits_{I\subseteq\mathbb{T}}\frac{\mu_{g}(S(I)\backslash
r_{0}\mathbb{D})}{|I|}\bigg{)}^{\frac{p-2}{p}}$ (2.17)
$\displaystyle\lesssim\varepsilon^{2}\|h\|_{H^{\frac{p}{p-2}}}.$

Furthermore, it follows from Lemma 2.2 that

(2.18)
$\|J_{g}(f_{j})\|_{H^{p}}^{p}\asymp\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)}|f_{j}(z)|^{2}|g^{\prime}(z)|^{2}dA(z)\bigg{)}^{\frac{p}{2}}d\xi.$

Recall

(2.19)
$\widetilde{\mu_{f_{j},g}}(\xi)=\int_{\Gamma(\xi)}\frac{d\mu_{f_{j},g}(z)}{1-|z|^{2}},\quad\xi\in\mathbb{T}.$

By [Pau16, Theorem E], for any $j\in\mathbb{N}$, we have

$\|Id\|_{H^{\frac{p}{p-2}}\to L^{1}(\mu_{f_{j},g}\chi_{\mathbb{D}\backslash
r\mathbb{D}})}\asymp\bigg{[}\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)\backslash
r_{0}\mathbb{D}}\frac{d\mu_{f_{j},g}(z)}{1-|z|^{2}}\bigg{)}^{\frac{p}{2}}d\xi\bigg{]}^{\frac{2}{p}}.$

Together with (2.2), (2.18) and (2.19), for $j>k_{0}$, we deduce

$\|J_{g}(f_{j})\|_{H^{p}}\asymp\|\widetilde{\mu_{f_{j},g}}\|_{L^{\frac{p}{2}}(\mathbb{T})}^{\frac{1}{2}}\lesssim\varepsilon.$

Hence, we know that $J_{g}$ is a compact operator as desired. Necessity.
Assume that $J_{g}:L_{a}^{p}(\omega)\to H^{p}$ is compact. Take a positive
constant $\alpha$. Let $d\nu_{g}(z)=|g^{\prime}(z)|^{2}(1-|z|)dA(z)$ and
$A_{\nu_{g},\alpha}^{2}$ be the area integral operator from
$L_{a}^{p}(\omega)$ to $L^{p}(\mathbb{T})$. Since the dual space of
$L^{\frac{p}{2}}(\mathbb{T})$ can be identified with
$L^{\frac{p}{p-2}}(\mathbb{T})$, together with Fubini’s theorem, for any $h\in
L^{\frac{p}{p-2}}(\mathbb{T})$ with $h\geq 0$ and $f\in L_{a}^{p}(\omega)$, we
obtain

(2.20)
$\displaystyle\int_{\mathbb{T}}h(\xi)\big{(}A_{\nu_{g},\alpha}^{2}(f)(\xi)\big{)}^{2}d\xi=\int_{\mathbb{D}}\frac{1}{1-|z|}\int_{I_{\alpha}(z)}h(\xi)d\xi|f(z)|^{2}d\nu_{g}(z).$

Notice that for any $I\subseteq\mathbb{T}$, there exists $a\in\mathbb{D}$ such
that $I=I_{\alpha}(a)$. For any $a\in\mathbb{D}$, take testing functions

$h_{a}(\xi)=\chi_{I_{\alpha}(a)}(\xi),\quad\xi\in\mathbb{T}.$

Notice that

(2.21)
$\|h_{a}\|_{L^{\frac{p}{p-2}}(\mathbb{T})}\asymp(1-|a|)^{\frac{p-2}{p}}$

and

(2.22) $\int_{I_{\alpha}(z)}h_{a}(\xi)d\xi\asymp 1-|z|,\quad
z\in\wedge_{\alpha}(I_{\alpha}(a)).$

Combining (2.20), (2.21), (2.22) with
$\textmd{H}\ddot{\textmd{o}}\textmd{lder's}$ inequality, for any
$a\in\mathbb{D}$, we deduce

$\displaystyle\int_{\wedge_{\alpha}(I_{\alpha}(a))}|f(z)|^{2}d\nu_{g}(z)$
$\displaystyle\lesssim\int_{\mathbb{D}}\frac{1}{1-|z|}\int_{I_{\alpha}(z)}h_{a}(\xi)d\xi|f(z)|^{2}d\nu_{g}(z)$
$\displaystyle=\int_{\mathbb{T}}h_{a}(\xi)\big{(}A_{\nu_{g},\alpha}^{2}(f)(\xi)\big{)}^{2}d\xi$
$\displaystyle\leq\|h_{a}\|_{L^{\frac{p}{p-2}}(\mathbb{T})}\|A_{\nu_{g},\alpha}^{2}(f)\|_{L^{p}(\mathbb{T})}^{2}$
$\displaystyle\asymp(1-|a|)^{\frac{p-2}{p}}\|A_{\nu_{g},\alpha}^{2}(f)\|_{L^{p}(\mathbb{T})}^{2},$

from which it follows that

(2.23)
$\frac{\int_{\wedge_{\alpha}(I_{\alpha}(a))}|f(z)|^{2}d\nu_{g}(z)}{(1-|a|)^{\frac{p-2}{p}}}\lesssim\|A_{\nu_{g},\alpha}^{2}(f)\|_{L^{p}(\mathbb{T})}^{2}.$

For any $\zeta\in\mathbb{D}$ and $\beta>0$, consider test functions

$f_{\zeta,\beta}(z):=\frac{(1-|\zeta|)^{\frac{\beta+1}{p}}}{(1-\bar{\zeta}z)^{\frac{\beta+1}{p}}\widehat{\omega}(\zeta)^{\frac{1}{p}}(1-|\zeta|)^{\frac{1}{p}}},\quad
z\in\mathbb{D}.$

Recall Lemma A(vi) in [PRS18] states that there exists
$\gamma_{0}=\gamma_{0}(\omega)>0$ such that for any $\gamma\geq\gamma_{0}$, it
holds that

$\int_{\mathbb{D}}\frac{\omega(z)}{|1-\bar{\zeta}z|^{\gamma+1}}dA(z)\asymp\frac{\widehat{\omega}(\zeta)}{(1-|\zeta|)^{\gamma}},\quad\zeta\in\mathbb{D}.$

Hence, we can choose $\gamma$ large enough such that
$\|f_{\zeta,\gamma}\|_{L_{a}^{p}(\omega)}\asymp 1$ and $f_{\zeta,\gamma}$
converges uniformly on any compact subset of $\mathbb{D}$ as $|\zeta|\to
1^{-}$. Combining with (2.23), we obtain that for any $\zeta\in\mathbb{D}$ and
$a\in\mathbb{D}$, it holds that

(2.24)
$\displaystyle\frac{\int_{\wedge_{\alpha}(I_{\alpha}(a))}|f_{\zeta,\gamma}(z)|^{2}d\nu_{g}(z)}{(1-|a|)^{\frac{p-2}{p}}}\lesssim\|A_{\nu_{g},\alpha}^{2}(f_{\zeta,\gamma})\|_{L^{p}(\mathbb{T})}^{2}.$

In view of the compactness of $J_{g}$ and (2.13), for any $\varepsilon>0$,
there exists $r_{1}\in(0,1)$ such that for any $\zeta\in\mathbb{D}$ with
$r_{1}\leq|\zeta|<1$, it holds that
$\|A_{\nu_{g},\alpha}^{2}(f_{\zeta,\gamma})\|_{L^{p}(\mathbb{T})}^{2}<\varepsilon$.
Notice that there exists $r_{2}\in(r_{1},1)$ such that for any
$a\in\mathbb{D}$ with $r_{2}\leq|a|<1$,

$D(z,r)\subseteq\mathbb{D}\backslash
r_{1}\mathbb{D},\quad\textmd{for~{}any}~{}z\in\wedge_{\alpha}(I_{\alpha}(a)).$

For $a\in\mathbb{D}$ with $r_{2}\leq|a|<1$, take $\zeta\in\mathbb{D}$ with

$\zeta\in\bigcup_{z\in\wedge_{\alpha}(I_{\alpha}(a))}D(z,r).$

Together with (2.24), for any $a\in\mathbb{D}$ with $r_{2}\leq|a|<1$, we
deduce

(2.25)
$\frac{\int_{\wedge_{\alpha}(I_{\alpha}(a))}\frac{1}{\widehat{\omega}(z)^{\frac{2}{p}}(1-|z|)^{\frac{2}{p}}}d\nu_{g}(z)}{(1-|a|)^{\frac{p-2}{p}}}\lesssim\varepsilon.$

Next we show that for any $f\in L_{a}^{p}(\omega)$, $|f(z)|^{2}d\nu_{g}(z)$ is
a vanishing $\frac{p-2}{p}$-Carleson measure. By the subharmonicity of
$|f|^{p}$, Lemma 2.5, Lemma 2.7 and (2.25), for any $a\in\mathbb{D}$ with
$r_{2}\leq|a|<1$, we obtain

$\displaystyle\frac{\int_{\wedge_{\alpha}(I_{\alpha}(a))}|f(z)|^{2}d\nu_{g}(z)}{(1-|a|)^{\frac{p-2}{p}}}$
$\displaystyle\lesssim\frac{\int_{\wedge_{\alpha}(I_{\alpha}(a))}\bigg{(}\frac{1}{\widehat{\omega}(z)(1-|z|)}\int_{D(z,r)}|f(u)|^{p}\frac{\widehat{\omega}(u)}{1-|u|}dA(u)\bigg{)}^{\frac{2}{p}}d\nu_{g}(z)}{(1-|a|)^{\frac{p-2}{p}}}$
$\displaystyle\leq\|f\|_{L_{a}^{p}(\omega)}^{2}\frac{\int_{\wedge_{\alpha}(I_{\alpha}(a))}\frac{1}{\widehat{\omega}(z)^{\frac{2}{p}}(1-|z|)^{\frac{2}{p}}}d\nu_{g}(z)}{(1-|a|)^{\frac{p-2}{p}}}$
(2.26) $\displaystyle\lesssim\varepsilon\|f\|_{L_{a}^{p}(\omega)}^{2}.$

It follows from Proposition 2.10 and (2.2) that there exits $t\in(0,1)$ such
that

$\bigg{(}\frac{\nu_{g}(D(z,t))}{\widehat{\omega}(z)(1-|z|)}\bigg{)}^{\frac{p}{p-2}}\frac{\widehat{\omega}(z)}{1-|z|}dA(z)$

is a vanishing Carleson measure. By the subharmonicity of $|g^{\prime}|^{2}$,
we deduce

$|g^{\prime}(z)|^{2}\frac{(1-|z|)^{2}}{\widehat{\omega}(z)}\lesssim\frac{1}{\widehat{\omega}(z)(1-|z|)}\int_{D(z,t)}|g^{\prime}(\zeta)|^{2}(1-|\zeta|)dA(\zeta)=\frac{\nu_{g}(D(z,t))}{\widehat{\omega}(z)(1-|z|)},$

which yields that

$\bigg{(}|g^{\prime}(z)|^{2}\frac{(1-|z|)^{2}}{\widehat{\omega}(z)}\bigg{)}^{\frac{p}{p-2}}\frac{\widehat{\omega}(z)}{1-|z|}dA(z)$

is a vanishing Carleson measure. Thus, we conclude that $\mu_{g}$ is a
vanishing Carleson measure. ∎

###  2.3. Case C

Now we are devoted to dealing with the case $0<q<\infty$ and
$p>\max\\{2,q\\}$.

For a separated sequence $\\{a_{k}\\}_{k=1}^{\infty}\subset\mathbb{D}$, let
$\nu=\sum\limits_{k=1}^{\infty}\delta_{a_{k}}$ where $\delta_{a_{k}}$ is Dirac
measure at point $a_{k}$, then we denote $T_{q,\nu}^{p}$ by
$T_{q}^{p}(\\{a_{k}\\})$. For $0<p<\infty$ and $0<q\leq\infty$, if

$\|\\{b_{k}\\}\|_{T_{q}^{p}(\\{a_{k}\\})}^{p}:=\int_{\mathbb{T}}\bigg{(}\sum_{a_{k}\in\Gamma(\xi)}|b_{k}|^{q}\bigg{)}^{\frac{p}{q}}d\xi<\infty,\quad
q\in(0,\infty)$

then we say the sequence $\\{b_{k}\\}_{k=1}^{\infty}\in
T_{q}^{p}(\\{a_{k}\\})$, and if

$\|\\{b_{k}\\}\|_{T_{q}^{p}(\\{a_{k}\\})}^{p}:=\int_{\mathbb{T}}\bigg{(}\sup_{a_{k}\in\Gamma(\xi)}|b_{k}|\bigg{)}^{p}d\xi<\infty,\quad
q=\infty.$

we say the sequence $\\{b_{k}\\}_{k=1}^{\infty}\in
T_{\infty}^{p}(\\{a_{k}\\})$. The following two results characterize the dual
space and the factorization of tent spaces of sequences respectively (see
[Ars99] and [MPPW20]).

######  Lemma 2.14.

Let $1<p<\infty$ and $\\{a_{k}\\}_{k=1}^{\infty}$ be a separated sequence.

  * •

If $1<q<\infty$, then the dual space of $T_{q}^{p}(\\{a_{k}\\})$ can be
identified with $T_{q^{\prime}}^{p^{\prime}}(\\{a_{k}\\})$ via the pairing,

$\langle\\{b_{k}\\},\\{d_{k}\\}\rangle_{T_{2}^{2}(\\{a_{k}\\})}=\sum\limits_{k=1}^{\infty}b_{k}\bar{d_{k}}(1-|a_{k}|^{2}),\quad\\{b_{k}\\}\in
T_{q}^{p}(\\{a_{k}\\}),~{}\\{d_{k}\\}\in
T_{q^{\prime}}^{p^{\prime}}(\\{a_{k}\\}).$

  * •

If $0<q\leq 1$, the the dual space of $T_{q}^{p}(\\{a_{k}\\})$ can be
identified with $T_{\infty}^{p^{\prime}}(\\{a_{k}\\})$ via the pairing,

$\langle\\{b_{k}\\},\\{d_{k}\\}\rangle_{T_{2}^{2}(\\{a_{k}\\})}=\sum\limits_{k=1}^{\infty}b_{k}\bar{d_{k}}(1-|a_{k}|^{2}),\quad\\{b_{k}\\}\in
T_{q}^{p}(\\{a_{k}\\}),~{}\\{d_{k}\\}\in
T_{\infty}^{p^{\prime}}(\\{a_{k}\\}).$

######  Lemma 2.15.

Let $0<p,q<\infty$ and $\\{a_{k}\\}_{k=1}^{\infty}$ be an $r$-lattice. If
$p<p_{1},p_{2}<\infty$, $q<q_{1},q_{2}<\infty$,
$\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p}$ and
$\frac{1}{q_{1}}+\frac{1}{q_{2}}=\frac{1}{q}$. Then

$T_{q}^{p}(\\{a_{k}\\})=T_{q_{1}}^{p_{1}}(\\{a_{k}\\})\cdot
T_{q_{2}}^{p_{2}}(\\{a_{k}\\}),$

where $T_{q_{1}}^{p_{1}}(\\{a_{k}\\})\cdot
T_{q_{2}}^{p_{2}}(\\{a_{k}\\})=\\{\\{b_{k}\\}:~{}b_{k}=c_{k}d_{k},~{}\\{c_{k}\\}\in
T_{q_{1}}^{p_{1}}(\\{a_{k}\\}),\\{d_{k}\\}\in
T_{q_{2}}^{p_{2}}(\\{a_{k}\\})\\}$. Moreover, for any $\\{b_{k}\\}\in
T_{q}^{p}(\\{a_{k}\\})$, it holds that

$\inf\|\\{c_{k}\\}\|_{T_{q_{1}}^{p_{1}}(\\{a_{k}\\})}\cdot\|\\{d_{k}\\}\|_{T_{q_{2}}^{p_{2}}(\\{a_{k}\\})}\lesssim\|\\{b_{k}\\}\|_{T_{q}^{p}(\\{a_{k}\\})},$

where the infimum above is taken over all possible factorizations of
$\\{b_{k}\\}$.

We also need the following useful estimates.

######  Lemma 2.16.

[Ars99, Lemma 4] Let $0<s<\infty$, $\iota>\max\\{1,\frac{1}{s}\\}$ and $\mu$
be a positive measure on $\mathbb{D}$. Then

$\int_{\mathbb{T}}\bigg{(}\int_{\mathbb{D}}\bigg{(}\frac{1-|z|^{2}}{|1-z\bar{\xi}|}\bigg{)}^{\iota}d\mu(z)\bigg{)}^{s}d\xi\asymp\int_{\mathbb{T}}\mu(\Gamma(\xi))^{s}d\xi.$

Using similar analysis to that in [PRS18, Proposition 14], we get the
following result.

######  Proposition 2.17.

Let $1<p<\infty$ and $\omega\in\mathcal{D}$. Assume that
$\\{a_{k}\\}_{k=1}^{\infty}\subseteq\mathbb{D}\backslash\\{0\\}$ is a
separated sequence. Then

$F:=\sum_{k=1}^{\infty}c_{k}\frac{K_{a_{k}}^{\omega}}{\|K_{a_{k}}^{\omega}\|_{L_{a}^{2-\frac{1}{p}}(\omega)}^{2-\frac{1}{p}}}\in
H(\mathbb{D}).$

Moreover,
$\|F\|_{L_{a}^{p}(\omega)}\lesssim\|\\{c_{k}\\}_{k=1}^{\infty}\|_{l^{p}}$ for
all $\\{c_{k}\\}_{k=1}^{\infty}\in l^{p}$.

###### Proof.

Let $\kappa(z)=\frac{\widehat{\omega}(z)}{1-|z|},~{}z\in\mathbb{D}$ and
$\\{c_{k}\\}_{k=1}^{\infty}\in l^{p}$. From Lemma 2.5, we know that $\kappa$
is regular. Assume that $\rho\in(0,1)$. For any $z\in\mathbb{D}$ with
$0<|z|\leq\rho$, it follows from Lemma 2.6,
$\textmd{H}\ddot{\textmd{o}}\textmd{lder's}$ inequality and (2.3) that

$\displaystyle|F(z)|\leq$
$\displaystyle\bigg{(}\sum_{k=1}^{\infty}|c_{k}|^{p}\bigg{)}^{\frac{1}{p}}\bigg{[}\sum_{k=1}^{\infty}\bigg{(}\frac{|K_{a_{k}}^{\omega}(z)|}{\|K_{a_{k}}^{\omega}\|_{L_{a}^{2-\frac{1}{p}}(\omega)}^{2-\frac{1}{p}}}\bigg{)}^{p^{\prime}}\bigg{]}^{\frac{1}{p^{\prime}}}$
$\displaystyle\lesssim$
$\displaystyle\bigg{(}\sum_{k=1}^{\infty}|c_{k}|^{p}\bigg{)}^{\frac{1}{p}}\sup\limits_{k\geq
1}|K_{a_{k}}^{\omega}(z)|\bigg{(}\sum_{k=1}^{\infty}\widehat{\omega}(a_{k})(1-|a_{k}|)\bigg{)}^{\frac{1}{p^{\prime}}}$
$\displaystyle\asymp$
$\displaystyle\bigg{(}\sum_{k=1}^{\infty}|c_{k}|^{p}\bigg{)}^{\frac{1}{p}}\sup\limits_{k\geq
1}|K_{a_{k}}^{\omega}(z)|\bigg{(}\sum_{k=1}^{\infty}\kappa(D(a_{k},r))\bigg{)}^{\frac{1}{p^{\prime}}}$
$\displaystyle\leq$ $\displaystyle
C\bigg{(}\sum_{k=1}^{\infty}|c_{k}|^{p}\bigg{)}^{\frac{1}{p}},$

where $r\in(0,1)$ and $C$ is a constant which only depends on $p,\rho,r$ and
$\omega$. Thus, we know that $F$ is analytic on $\mathbb{D}$. Take Lemma 2.5
into account, we deduce
$\|F\|_{L_{a}^{p}(\omega)}\asymp\|F\|_{L_{a}^{p}(\kappa)}.$ Since
$(L_{a}^{p}(\kappa))^{*}\simeq L_{a}^{p^{\prime}}(\kappa)$ (see [PRS18,
Theorem D]), together with the property of subharmonicity, similar analysis as
above gives the desired result. ∎

Now we give the main result of this subsection.

######  Proposition 2.18.

Let $0<q<\infty$, $p>\max\\{2,q\\}$, $\omega\in\mathcal{D}$ and $g\in
H(\mathbb{D})$. Then the Volterra type integration operator
$J_{g}:L_{a}^{p}(\omega)\to H^{q}$ is compact if and only if $J_{g}$ is
bounded if and only if $g^{\prime}\in
T_{\frac{2p}{p-2},\widehat{\omega},\frac{2}{p-2}}^{\frac{pq}{p-q}}$. Moreover,
$\|J_{g}\|_{L_{a}^{p}(\omega)\to
H^{q}}\asymp\|g^{\prime}\|_{T_{\frac{2p}{p-2},\widehat{\omega},\frac{2}{p-2}}^{\frac{pq}{p-q}}}$.

###### Proof.

We divide the proof into two steps.

Step 1. Let $g$ be an analytic function on $\mathbb{D}$ with satisfying

$g^{\prime}\in
T_{\frac{2p}{p-2},\widehat{\omega},\frac{2}{p-2}}^{\frac{pq}{p-q}},$

we will show that $J_{g}:L_{a}^{p}(\omega)\to H^{q}$ is a compact operator.
Let $\\{f_{j}\\}_{j=1}^{\infty}$ be any sequence in $L_{a}^{p}(\omega)$ with
$\|f_{j}\|_{L_{a}^{p}(\omega)}\leq 1$ and $f_{j}$ converges to 0 uniformly on
any compact subset of $\mathbb{D}$. We are devoted to proving

(2.27) $\lim\limits_{j\to\infty}\|J_{g}(f_{j})\|_{H^{q}}=0.$

Since $g^{\prime}\in
T_{\frac{2p}{p-2},\widehat{\omega},\frac{2}{p-2}}^{\frac{pq}{p-q}}$, then for
any $\varepsilon>0$, there exists $r_{0}\in(0,1)$ such that

(2.28) $\bigg{[}\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)\backslash
r_{0}\mathbb{D}}|g^{\prime}(z)|^{\frac{2p}{p-2}}\bigg{(}\frac{(1-|z|^{2})^{2}}{\widehat{\omega}(z)}\bigg{)}^{\frac{2}{p-2}}dA(z)\bigg{)}^{\frac{q(p-2)}{2(p-q)}}d\xi\bigg{]}^{\frac{p-q}{p}}<\varepsilon.$

Moreover, there exists $k_{0}\in\mathbb{N}$ such that for any $j>k_{0}$,
$|f_{j}|^{q}<\varepsilon$ on $r_{0}\mathbb{D}$. Combining Lemma 2.5, (2.6)
with Fubini’s theorem, for any $j\in\mathbb{N}$, we deduce

(2.29) $\int_{\mathbb{T}}\int_{\Gamma(\xi)\backslash
r_{0}\mathbb{D}}|f_{j}(z)|^{p}\frac{\widehat{\omega}(z)}{(1-|z|^{2})^{2}}dA(z)d\xi\lesssim\int_{\mathbb{D}}|f_{j}(z)|^{p}\frac{\widehat{\omega}(z)}{(1-|z|^{2})^{2}}\int_{I(z)}d\xi
dA(z)\asymp\|f_{j}\|_{L_{a}^{p}(\omega)}^{p}.$

By Lemma 2.2, (2.28), (2.29) and $\textmd{H}\ddot{\textmd{o}}\textmd{lder's}$
inequality, we obtain

$\displaystyle\|J_{g}(f_{j})\|_{H^{q}}^{q}$
$\displaystyle\lesssim\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)\bigcap
r_{0}\mathbb{D}}|g^{\prime}(z)|^{2}|f_{j}(z)|^{2}dA(z)\bigg{)}^{\frac{q}{2}}d\xi+\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)\backslash
r_{0}\mathbb{D}}|g^{\prime}(z)|^{2}|f_{j}(z)|^{2}dA(z)\bigg{)}^{\frac{q}{2}}d\xi$
$\displaystyle\lesssim\varepsilon+\int_{\mathbb{T}}\bigg{[}\bigg{(}\int_{\Gamma(\xi)\backslash
r_{0}\mathbb{D}}|g^{\prime}(z)|^{\frac{2p}{p-2}}\bigg{(}\frac{(1-|z|^{2})^{2}}{\widehat{\omega}(z)}\bigg{)}^{\frac{2}{p-2}}dA(z)\bigg{)}^{\frac{q(p-2)}{2p}}$
$\displaystyle\cdot\bigg{(}\int_{\Gamma(\xi)\backslash
r_{0}\mathbb{D}}|f_{j}(z)|^{p}\frac{\widehat{\omega}(z)}{(1-|z|^{2})^{2}}dA(z)\bigg{)}^{\frac{q}{p}}\bigg{]}d\xi$
$\displaystyle\leq\varepsilon+\bigg{[}\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)\backslash
r_{0}\mathbb{D}}|g^{\prime}(z)|^{\frac{2p}{p-2}}\bigg{(}\frac{(1-|z|^{2})^{2}}{\widehat{\omega}(z)}\bigg{)}^{\frac{2}{p-2}}dA(z)\bigg{)}^{\frac{q(p-2)}{2(p-q)}}d\xi\bigg{]}^{\frac{p-q}{p}}$
$\displaystyle\cdot\bigg{[}\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)\backslash
r_{0}\mathbb{D}}|f_{j}(z)|^{p}\frac{\widehat{\omega}(z)}{(1-|z|^{2})^{2}}dA(z)\bigg{)}d\xi\bigg{]}^{\frac{q}{p}}$
$\displaystyle\lesssim\varepsilon+\varepsilon\|f_{j}\|_{L_{a}^{p}(\omega)}^{q}\lesssim\varepsilon,\quad\textmd{for~{}any}~{}j>k_{0}.$

Step 2. Assume that $J_{g}$ is bounded. For $r\in(0,1)$, let $\tilde{r}=\tanh
r$ and $\\{a_{k}\\}_{k=1}^{\infty}$ be an $\tilde{r}$-lattice. It follows from
Lemma 2.7 that

(2.30) $\widehat{\omega}(z)\asymp\widehat{\omega}(a_{k}),\quad z\in
D(a_{k},\tilde{r}).$

By the subharmonicity of $|g^{\prime}|^{2}$ and (2.30), for any
$\xi\in\mathbb{T}$, we get

$\displaystyle\int_{\Gamma(\xi)}|g^{\prime}(z)|^{\frac{2p}{p-2}}\frac{(1-|z|^{2})^{\frac{4}{p-2}}}{\widehat{\omega}(z)^{\frac{2}{p-2}}}dA(z)$
$\displaystyle\lesssim\int_{\Gamma(\xi)}\bigg{(}\int_{D(z,\tilde{r})}|g^{\prime}(\zeta)|^{2}dA(\zeta)\bigg{)}^{\frac{p}{p-2}}\frac{1}{\widehat{\omega}(z)^{\frac{2}{p-2}}(1-|z|^{2})^{2}}dA(z)$
$\displaystyle\leq\sum_{k}\int_{D(a_{k},\tilde{r})}\Big{(}\int_{D(z,\tilde{r})}|g^{\prime}(\zeta)|^{2}dA(\zeta)\Big{)}^{\frac{p}{p-2}}\frac{1}{\widehat{\omega}(z)^{\frac{2}{p-2}}(1-|z|^{2})^{2}}dA(z)$
$\displaystyle\lesssim\sum_{k}\Big{(}\sup_{\zeta\in
D(a_{k},2\tilde{r})}|g^{\prime}(\zeta)|^{\frac{2p}{p-2}}\Big{)}\frac{(1-|a_{k}|^{2})^{\frac{2p}{p-2}}}{\widehat{\omega}(a_{k})^{\frac{2}{p-2}}},$

where the summation runs
$\\{k:D(a_{k},\tilde{r})\bigcap\Gamma(\xi)\neq\emptyset\\}$. Let
$\delta=3e^{2r}-2$. By Lemma 2.11, for any $\xi\in\mathbb{T}$, we have

$D(\zeta,\tilde{r})\subseteq\Gamma_{\delta}(\xi),\quad\forall\zeta\in\Gamma(\xi).$

Hence, for any $\xi\in\mathbb{T}$, we deduce

$\displaystyle\int_{\Gamma(\xi)}|g^{\prime}(z)|^{\frac{2p}{p-2}}\frac{(1-|z|^{2})^{\frac{4}{p-2}}}{\widehat{\omega}(z)^{\frac{2}{p-2}}}dA(z)\lesssim\sum_{a_{k}\in\Gamma_{\delta}(\xi)}\bigg{(}\sup_{\zeta\in
D(a_{k},2\tilde{r})}|g^{\prime}(\zeta)|^{\frac{2p}{p-2}}\bigg{)}\frac{(1-|a_{k}|^{2})^{\frac{2p}{p-2}}}{\widehat{\omega}(a_{k})^{\frac{2}{p-2}}}.$

Thus, we obtain

$\displaystyle\begin{split}&~{}~{}~{}~{}\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)}|g^{\prime}(z)|^{\frac{2p}{p-2}}\frac{(1-|z|^{2})^{\frac{4}{p-2}}}{\widehat{\omega}(z)^{\frac{2}{p-2}}}dA(z)\bigg{)}^{\frac{q(p-2)}{2(p-q)}}d\xi\\\
&\lesssim\int_{\mathbb{T}}\bigg{(}\sum_{a_{k}\in\Gamma(\xi)}\bigg{(}\sup_{\zeta\in
D(a_{k},2\tilde{r})}|g^{\prime}(\zeta)|^{\frac{2p}{p-2}}\bigg{)}\frac{(1-|a_{k}|^{2})^{\frac{2p}{p-2}}}{\widehat{\omega}(a_{k})^{\frac{2}{p-2}}}\bigg{)}^{\frac{q(p-2)}{2(p-q)}}d\xi.\end{split}$

In what follows, we only need to prove

(2.31)
$\int_{\mathbb{T}}\bigg{(}\sum\limits_{a_{k}\in\Gamma(\xi)}\bigg{(}\sup_{\zeta\in
D(a_{k},2\tilde{r})}|g^{\prime}(\zeta)|^{\frac{2p}{p-2}}\bigg{)}\frac{(1-|a_{k}|^{2})^{\frac{2p}{p-2}}}{\widehat{\omega}(a_{k})^{\frac{2}{p-2}}}\bigg{)}^{\frac{q(p-2)}{2(p-q)}}d\xi\lesssim\|J_{g}\|_{L_{a}^{p}(\omega)\to
H^{q}}^{\frac{pq}{p-q}}.$

Define a sequence $\\{b_{k}\\}_{k=1}^{\infty}$ by

$b_{k}=\bigg{(}\sup_{\zeta\in
D(a_{k},2\tilde{r})}|g^{\prime}(\zeta)|^{q}\bigg{)}\frac{(1-|a_{k}|^{2})^{q}}{\widehat{\omega}(a_{k})^{\frac{q}{p}}}.$

Let $\eta=\frac{2p}{q(p-2)}$. We turn to prove that there exists a positive
constant $s$ such that

(2.32) $\|\\{b_{k}^{\frac{1}{s}}\\}\|_{T_{\eta
s}^{\frac{ps}{p-q}}(\\{a_{k}\\})}\lesssim\|J_{g}\|_{L_{a}^{p}(\omega)\to
H^{q}}^{\frac{q}{s}}.$

Choose a constant $s$ with $s>\max\\{1,\frac{q}{2}\\}$. From Lemma 2.14, the
dual space of $T_{(\eta
s)^{\prime}}^{\big{(}\frac{ps}{p-q}\big{)}^{\prime}}(\\{a_{k}\\})$ can be
identified with $T_{\eta s}^{\frac{ps}{p-q}}(\\{a_{k}\\})$. Thus, by (2.6),
for any sequence $\\{e_{k}\\}\in T_{(\eta
s)^{\prime}}^{\big{(}\frac{ps}{p-q}\big{)}^{\prime}}(\\{a_{k}\\})$, it holds
that

(2.33) $|\langle
e_{k},b_{k}^{\frac{1}{s}}\rangle_{T_{2}^{2}(\\{a_{k}\\})}|\leq\sum\limits_{k=1}^{\infty}|e_{k}b_{k}^{\frac{1}{s}}|(1-|a_{k}|^{2})\asymp\sum\limits_{k=1}^{\infty}|e_{k}b_{k}^{\frac{1}{s}}|\int_{I(a_{k})}d\xi=\int_{\mathbb{T}}\sum\limits_{a_{k}\in\Gamma(\xi)}|e_{k}b_{k}^{\frac{1}{s}}|d\xi.$

By Lemma 2.15, we obtain

(2.34) $T_{(\eta
s)^{\prime}}^{\big{(}\frac{ps}{p-q}\big{)}^{\prime}}(\\{a_{k}\\})=T_{\frac{2s}{2s-q}}^{s^{\prime}}(\\{a_{k}\\})\cdot
T_{\frac{ps}{q}}^{\frac{ps}{q}}(\\{a_{k}\\}),$

from which there exist two sequences $\\{x_{k}\\}\in
T_{\frac{2s}{2s-q}}^{s^{\prime}}(\\{a_{k}\\})$ and $\\{y_{k}\\}\in
T_{p}^{p}(\\{a_{k}\\})$ such that

$e_{k}=x_{k}y_{k}^{\frac{q}{s}}.$

Together with (2.33) and $\textmd{H}\ddot{\textmd{o}}\textmd{lder's}$
inequality, we get

$\displaystyle|\langle
e_{k},b_{k}^{\frac{1}{s}}\rangle_{T_{2}^{2}(\\{a_{k}\\})}|$
$\displaystyle\lesssim\int_{\mathbb{T}}\bigg{(}\sum\limits_{a_{k}\in\Gamma(\xi)}|x_{k}|^{\frac{2s}{2s-q}}\bigg{)}^{\frac{2s-q}{2s}}\bigg{(}\sum\limits_{a_{k}\in\Gamma(\xi)}|y_{k}^{2}b_{k}^{\frac{2}{q}}|\bigg{)}^{\frac{q}{2s}}d\xi$
(2.35)
$\displaystyle\leq\|\\{x_{k}\\}\|_{T_{\frac{2s}{2s-q}}^{s^{\prime}}(\\{a_{k}\\})}\bigg{[}\int_{\mathbb{T}}\bigg{(}\sum\limits_{a_{k}\in\Gamma(\xi)}|y_{k}^{2}b_{k}^{\frac{2}{q}}|\bigg{)}^{\frac{q}{2}}d\xi\bigg{]}^{\frac{1}{s}}.$

Let

(2.36)
$f_{k}(z)=\frac{K_{a_{k}}^{\omega}(z)}{\|K_{a_{k}}^{\omega}\|_{L_{a}^{2-\frac{1}{p}}(\omega)}^{2-\frac{1}{p}}},\quad
z\in\mathbb{D}$

and $\\{r_{k}:k\in\mathbb{N}\\}$ be the set of the Rademacher functions. For
$t\in[0,1]$, set

$F_{t}(z)=\sum_{k=1}^{\infty}(1-|a_{k}|^{2})^{\frac{1}{p}}y_{k}r_{k}(t)f_{k}(z),\quad
z\in\mathbb{D}.$

Then

(2.37)
$\|\\{y_{k}\\}\|_{T_{p}^{p}(\\{a_{k}\\})}^{p}=\int_{\mathbb{T}}\sum_{a_{k}\in\Gamma(\xi)}|y_{k}|^{p}d\xi=\sum\limits_{k=1}^{\infty}\int_{I(a_{k})}|y_{k}|^{p}d\xi\asymp\sum_{k=1}^{\infty}|y_{k}|^{p}(1-|a_{k}|^{2}).$

By using Proposition 2.17 and (2.37), we obtain

$\|J_{g}(F_{t})\|_{H^{q}}^{q}\lesssim\|J_{g}\|_{L_{a}^{p}(\omega)\to
H^{q}}^{q}\big{(}\sum_{k=1}^{\infty}|y_{k}|^{p}(1-|a_{k}|^{2})\big{)}^{\frac{q}{p}}\asymp\|J_{g}\|_{L_{a}^{p}(\omega)\to
H^{q}}^{q}\|\\{y_{k}\\}\|_{T_{p}^{p}(\\{a_{k}\\})}^{q}.$

Together with Lemma 2.2, we deduce

(2.38)
$\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)}|g^{\prime}(z)|^{2}\bigg{|}\sum_{k=1}^{\infty}(1-|a_{k}|^{2})^{\frac{1}{p}}y_{k}r_{k}(t)f_{k}(z)\bigg{|}^{2}dA(z)\bigg{)}^{\frac{q}{2}}d\xi\lesssim\|J_{g}\|_{L_{a}^{p}(\omega)\to
H^{q}}^{q}\cdot\|\\{y_{k}\\}\|_{T_{p}^{p}(\\{a_{k}\\})}^{q}.$

Integrate both sides of the inequality (2.38) from 0 to 1 with respect to $t$,
by Fubini’s theorem, we have

(2.39)
$\int_{\mathbb{T}}\int_{0}^{1}\bigg{(}\int_{\Gamma(\xi)}|g^{\prime}(z)|^{2}\bigg{|}\sum_{k=1}^{\infty}(1-|a_{k}|^{2})^{\frac{1}{p}}y_{k}r_{k}(t)f_{k}(z)\bigg{|}^{2}dA(z)\bigg{)}^{\frac{q}{2}}dtd\xi\lesssim\|J_{g}\|_{L_{a}^{p}(\omega)\to
H^{q}}^{q}\cdot\|\\{y_{k}\\}\|_{T_{p}^{p}(\\{a_{k}\\})}^{q}.$

By the subharmonicity of $|g^{\prime 2}|$ and Lemma 2.7, for any
$k\in\mathbb{N}$, we obtain

$\displaystyle\sup\limits_{\zeta\in
D(a_{k},2\tilde{r})}|g^{\prime}(\zeta)|^{2}$
$\displaystyle\lesssim\sup\limits_{\zeta\in
D(a_{k},2\tilde{r})}\frac{1}{(1-|\zeta|)^{2}}\int_{B(\zeta,\tilde{r}(1-|\zeta|))}|g^{\prime}(z)|^{2}dA(z)$
$\displaystyle\leq\sup\limits_{\zeta\in
D(a_{k},2\tilde{r})}\frac{1}{(1-|\zeta|)^{2}}\int_{D(\zeta,\tilde{r})}|g^{\prime}(z)|^{2}dA(z)$
(2.40)
$\displaystyle\leq\frac{1}{\widehat{\omega}(a_{k})(1-|a_{k}|)}\int_{D(a_{k},3\tilde{r})}|g^{\prime}(z)|^{2}\frac{\widehat{\omega}(z)}{(1-|z|)}dA(z).$

For $z\in D(a_{k},3\tilde{r})$, it follows from [PRS18, Lemma 8] and (2.3)
that for $r$ small enough, we have

(2.41)
$|f_{k}(z)|\asymp\big{[}\widehat{\omega}(a_{k})(1-|a_{k}|)\big{]}^{1-\frac{1}{p}}|K_{a_{k}}^{\omega}(z)|\asymp\frac{1}{\widehat{\omega}(a_{k})^{\frac{1}{p}}(1-|a_{k}|^{2})^{\frac{1}{p}}}.$

Take $\iota>\max\\{1,\frac{2}{q}\\}$. Combining Lemma 2.7, (2.3) with (2.41),
for $\xi\in\mathbb{T}$, we deduce

$\displaystyle\sum\limits_{a_{k}\in\Gamma(\xi)}|y_{k}^{2}b_{k}^{\frac{2}{q}}|$
$\displaystyle\asymp\sum\limits_{a_{k}\in\Gamma(\xi)}|y_{k}|^{2}\sup\limits_{\zeta\in
D(a_{k},2\tilde{r})}|g^{\prime}(\zeta)|^{2}\bigg{(}\frac{1-|a_{k}|^{2}}{|1-\bar{a_{k}}\xi|}\bigg{)}^{\iota}\frac{(1-|a_{k}|^{2})^{2}}{\widehat{\omega}(a_{k})^{\frac{2}{p}}}$
$\displaystyle\lesssim\sum\limits_{k=1}^{\infty}|y_{k}|^{2}\frac{1}{\widehat{\omega}(a_{k})(1-|a_{k}|)}\int_{D(a_{k},3\tilde{r})}|g^{\prime}(z)|^{2}\frac{\widehat{\omega}(z)}{1-|z|}\bigg{(}\frac{1-|z|^{2}}{|1-\bar{z}\xi|}\bigg{)}^{\iota}\frac{(1-|z|^{2})^{2}}{\widehat{\omega}(z)^{\frac{2}{p}}}dA(z)$
$\displaystyle\asymp\sum\limits_{k=1}^{\infty}|y_{k}|^{2}(1-|a_{k}|^{2})^{\frac{2}{p}}\int_{D(a_{k},3\tilde{r})}|g^{\prime}(z)|^{2}\bigg{(}\frac{1-|z|^{2}}{|1-\bar{z}\xi|}\bigg{)}^{\iota}|f_{k}(z)|^{2}dA(z)$
$\displaystyle\leq\int_{\mathbb{D}}\bigg{(}\frac{1-|z|^{2}}{|1-\bar{z}\xi|}\bigg{)}^{\iota}\sum\limits_{k=1}^{\infty}|y_{k}|^{2}(1-|a_{k}|^{2})^{\frac{2}{p}}|f_{k}(z)|^{2}|g^{\prime}(z)|^{2}dA(z).$

Together with Lemma 2.16, we know

$\displaystyle\int_{\mathbb{T}}\bigg{(}\sum\limits_{a_{k}\in\Gamma(\xi)}|y_{k}^{2}b_{k}^{\frac{2}{q}}|\bigg{)}^{\frac{q}{2}}d\xi$
$\displaystyle\lesssim\int_{\mathbb{T}}\bigg{(}\int_{\mathbb{D}}\bigg{(}\frac{1-|z|^{2}}{|1-\bar{z}\xi|}\bigg{)}^{\iota}\sum\limits_{k=1}^{\infty}|y_{k}|^{2}(1-|a_{k}|^{2})^{\frac{2}{p}}|f_{k}(z)|^{2}|g^{\prime}(z)|^{2}dA(z)\bigg{)}^{\frac{q}{2}}d\xi$
$\displaystyle\asymp\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)}\sum\limits_{k=1}^{\infty}|y_{k}|^{2}(1-|a_{k}|^{2})^{\frac{2}{p}}|f_{k}(z)|^{2}|g^{\prime}(z)|^{2}dA(z)\bigg{)}^{\frac{q}{2}}d\xi.$

Moreover, using Lemma 2.8, Lemma 2.9 and Fubini’s theorem, we deduce

$\displaystyle\begin{split}&~{}~{}~{}~{}\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)}\sum\limits_{k=1}^{\infty}|y_{k}|^{2}(1-|a_{k}|^{2})^{\frac{2}{p}}|f_{k}(z)|^{2}|g^{\prime}(z)|^{2}dA(z)\bigg{)}^{\frac{q}{2}}d\xi\\\
&\asymp\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)}|g^{\prime}(z)|^{2}\int_{0}^{1}\bigg{|}\sum\limits_{k=1}^{\infty}y_{k}r_{k}(t)(1-|a_{k}|^{2})^{\frac{1}{p}}f_{k}(z)\bigg{|}^{2}dtdA(z)\bigg{)}^{\frac{q}{2}}d\xi\\\
&=\int_{\mathbb{T}}\bigg{(}\int_{0}^{1}\int_{\Gamma(\xi)}\bigg{|}g^{\prime}(z)\sum\limits_{k=1}^{\infty}y_{k}r_{k}(t)(1-|a_{k}|^{2})^{\frac{1}{p}}f_{k}(z)\bigg{|}^{2}dA(z)dt\bigg{)}^{\frac{q}{2}}d\xi\\\
&\asymp\int_{\mathbb{T}}\int_{0}^{1}\bigg{(}\int_{\Gamma(\xi)}\bigg{|}g^{\prime}(z)\sum\limits_{k=1}^{\infty}y_{k}r_{k}(t)(1-|a_{k}|^{2})^{\frac{1}{p}}f_{k}(z)\bigg{|}^{2}dA(z)\bigg{)}^{\frac{q}{2}}dtd\xi.\end{split}$

Combining with (2.39), we conclude that

(2.42)
$\int_{\mathbb{T}}\bigg{(}\sum\limits_{a_{k}\in\Gamma(\xi)}|y_{k}^{2}b_{k}^{\frac{2}{q}}|\bigg{)}^{\frac{q}{2}}d\xi\lesssim\|J_{g}\|_{L_{a}^{p}(\omega)\to
H^{q}}^{q}\cdot\|\\{y_{k}\\}\|_{T_{p}^{p}(\\{a_{k}\\})}^{q}.$

According to (2.3) and (2.42), we deduce

(2.43) $|\langle
e_{k},b_{k}^{\frac{1}{s}}\rangle_{T_{2}^{2}(\\{a_{k}\\})}|\lesssim\|\\{x_{k}\\}\|_{T_{\frac{2s}{2s-q}}^{s^{\prime}}(\\{a_{k}\\})}\cdot\|J_{g}\|_{L_{a}^{p}(\omega)\to
H^{q}}^{\frac{q}{s}}\cdot\|\\{y_{k}\\}\|_{T_{p}^{p}(\\{a_{k}\\})}^{\frac{q}{s}}.$

Take infimum over all possible factorizations of $\\{e_{k}\\}$ as in (2.34),
we deduce

$\inf\|\\{x_{k}\\}\|_{T_{\frac{2s}{2s-q}}^{s^{\prime}}(\\{a_{k}\\})}\cdot\|\\{y_{k}\\}\|_{T_{p}^{p}(\\{a_{k}\\})}^{\frac{q}{s}}\lesssim\|\\{e_{k}\\}\|_{T_{(\eta
s)^{\prime}}^{\big{(}\frac{ps}{p-q}\big{)}^{\prime}}(\\{a_{k}\\})}.$

Combining with (2.43), we get

$|\langle
e_{k},b_{k}^{\frac{1}{s}}\rangle_{T_{2}^{2}(\\{a_{k}\\})}|\lesssim\|J_{g}\|_{L_{a}^{p}(\omega)\to
H^{q}}^{\frac{q}{s}}\cdot\|\\{e_{k}\\}\|_{T_{(\eta
s)^{\prime}}^{\big{(}\frac{ps}{p-q}\big{)}^{\prime}}(\\{a_{k}\\})}.$

Finally, it follows from Lemma 2.14 that (2.32) holds. Thus, we get the result
as desired. ∎

###  2.4. Case D

Last, we will deal with the remaining case $0<q<p\leq 2$, with the main result
as follows.

######  Proposition 2.19.

Let $0<q<p\leq 2$, $\omega\in\mathcal{D}$ and $g\in H(\mathbb{D})$. Then the
Volterra type integration operator $J_{g}:L_{a}^{p}(\omega)\to H^{q}$ is
compact if and only if

$\lim\limits_{r\to 1^{-}}\int_{\mathbb{T}}\sup_{z\in\Gamma(\xi)\backslash
r\mathbb{D}}|g^{\prime}(z)|^{\frac{pq}{p-q}}\frac{(1-|z|^{2})^{\frac{pq}{p-q}}}{\widehat{\omega}(z)^{\frac{q}{p-q}}}d\xi=0.$

###### Proof.

Sufficiency. Let $\\{f_{j}\\}_{j=1}^{\infty}$ be any sequence in
$L_{a}^{p}(\omega)$ with $\|f_{j}\|_{L_{a}^{p}(\omega)}\leq 1$ and $f_{j}$
converges to 0 uniformly on any compact subset of $\mathbb{D}$. Then we will
prove $\|J_{g}(f_{j})\|_{H^{q}}\to 0$ as $j\to\infty$. By the assumption, it
follows that for any $\varepsilon>0$, there exists $R_{0}\in(0,1)$ such that
for any $R_{1}\in[R_{0},1)$, it holds that

(2.44) $\int_{\mathbb{T}}\sup_{z\in\Gamma(\xi)\backslash
R_{1}\mathbb{D}}|g^{\prime}(z)|^{\frac{pq}{p-q}}\frac{(1-|z|^{2})^{\frac{pq}{p-q}}}{\widehat{\omega}(z)^{\frac{q}{p-q}}}d\xi<\varepsilon^{\frac{p}{p-q}}.$

Note that there also exists $k_{0}\in\mathbb{N}$ such that

(2.45)
$|f_{j}|<\varepsilon^{\frac{1}{q}}~{}\textmd{on}~{}R_{0}\mathbb{D},\quad\textmd{for~{}any}~{}j>k_{0}.$

Choose $\gamma\in(0,\infty)$ with
$\tilde{\gamma}:=\tanh\gamma\in(0,\frac{1}{2})$ and
$\\{a_{k}\\}_{k=1}^{\infty}$ be a $\tilde{\gamma}$-lattice. Then we know

$\Gamma(\xi)\subseteq\bigcup_{\\{a_{k}:D(a_{k},\tilde{\gamma})\bigcap\Gamma(\xi)\neq\emptyset\\}}D(a_{k},\tilde{\gamma}).$

Take $\alpha=3e^{2\gamma}-2$. By Lemma 2.11, we get

$D(\zeta,\tilde{\gamma})\subseteq\Gamma_{\alpha}(\xi),\quad\zeta\in\Gamma(\xi).$

Hence, we obtain

(2.46)
$\Gamma(\xi)\subseteq\bigcup_{a_{k}\in\Gamma_{\alpha}(\xi)}D(a_{k},\tilde{\gamma}).$

Observe ([MPPW20]) that there exists a constant $\beta>\alpha$ such that

(2.47)
$\bigcup_{a_{k}\in\Gamma_{\alpha}(\xi)}D(a_{k},\tilde{\gamma})\subseteq\Gamma_{\beta}(\xi),\quad\xi\in\mathbb{T}.$

For any $j\in\mathbb{N}$, by the subharmonicity of $|f_{j}|^{2}$, we deduce

(2.48) $\displaystyle|f_{j}(z)|^{2}$
$\displaystyle\lesssim\bigg{(}\frac{1}{(1-|a_{k}|)^{2}}\int_{D(a_{k},2\tilde{\gamma})}|f_{j}(u)|^{p}dA(u)\bigg{)}^{\frac{2}{p}},\quad
z\in D(a_{k},\tilde{\gamma}).$

It follows from Lemma 2.7 that

(2.49) $\widehat{\omega}(z)\asymp\widehat{\omega}(a_{k}),\quad z\in
D(a_{k},2\tilde{\gamma}).$

According to [Zhu07, Theorem 3.3], if $p\in(0,1]$, then for any sequence
$\\{z_{k}\\}_{k=1}^{\infty}\in l^{p}$,

(2.50)
$\bigg{|}\sum_{k=1}^{\infty}z_{k}\bigg{|}\leq\bigg{(}\sum_{k=1}^{\infty}|z_{k}|^{p}\bigg{)}^{\frac{1}{p}}.$

Combining (2.46), (2.47), (2.48), (2.49) with (2.50), for any
$j\in\mathbb{N}$, we obtain

(2.51)
$\bigg{(}\int_{\Gamma(\xi)}|f_{j}(z)|^{2}(1-|z|^{2})^{-2}\widehat{\omega}(z)^{\frac{2}{p}}dA(z)\bigg{)}^{\frac{p}{2}}\lesssim\int_{\Gamma_{\beta}(\xi)}|f_{j}(u)|^{p}\frac{\widehat{\omega}(u)}{(1-|u|)^{2}}dA(u).$

By Lemma 2.5, (2.6) and Fubini’s theorem, for any $j\in\mathbb{N}$, we have

$\int_{\mathbb{T}}\int_{\Gamma_{\beta}(\xi)}|f_{j}(u)|^{p}\frac{\widehat{\omega}(u)}{(1-|u|)^{2}}dA(u)d\xi=\int_{\mathbb{D}}|f_{j}(u)|^{p}\frac{\widehat{\omega}(u)}{(1-|u|)^{2}}\int_{I_{\beta}(u)}d\xi
dA(u)\asymp\|f_{j}\|_{L_{a}^{p}(\omega)}^{p}.$

Together with Lemma 2.2, (2.44), (2.45), (2.51) and
$\textmd{H}\ddot{\textmd{o}}\textmd{lder's}$ inequality, for any $j>k_{0}$, we
get

$\displaystyle\|J_{g}(f_{j})\|_{H^{q}}^{q}$
$\displaystyle\lesssim\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)\bigcap
R_{0}\mathbb{D}}|g^{\prime}(z)|^{2}|f_{j}(z)|^{2}dA(z)\bigg{)}^{\frac{q}{2}}d\xi+\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)\backslash
R_{0}\mathbb{D}}|g^{\prime}(z)|^{2}|f_{j}(z)|^{2}dA(z)\bigg{)}^{\frac{q}{2}}d\xi$
$\displaystyle\lesssim\varepsilon+\int_{\mathbb{T}}\bigg{(}\sup\limits_{z\in\Gamma(\xi)\backslash
R_{0}\mathbb{D}}|g^{\prime}(z)|^{q}\frac{(1-|z|^{2})^{q}}{\widehat{\omega}(z)^{\frac{q}{p}}}\bigg{)}\cdot\bigg{(}\int_{\Gamma(\xi)}|f_{j}(z)|^{2}\frac{\widehat{\omega}(z)^{\frac{2}{p}}}{(1-|z|^{2})^{2}}dA(z)\bigg{)}^{\frac{q}{2}}d\xi$
$\displaystyle\leq\varepsilon+\bigg{[}\int_{\mathbb{T}}\bigg{(}\sup_{z\in\Gamma(\xi)\backslash
R_{0}\mathbb{D}}|g^{\prime}(z)|^{q}\frac{(1-|z|^{2})^{q}}{\widehat{\omega}(z)^{\frac{q}{p}}}\bigg{)}^{\frac{p}{p-q}}d\xi\bigg{]}^{\frac{p-q}{p}}$
$\displaystyle\cdot\bigg{[}\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)}|f_{j}(z)|^{2}\frac{\widehat{\omega}(z)^{\frac{2}{p}}}{(1-|z|^{2})^{2}}dA(z)\bigg{)}^{\frac{p}{2}}d\xi\bigg{]}^{\frac{q}{p}}$
$\displaystyle\lesssim\varepsilon+\varepsilon\|f_{j}\|_{L_{a}^{p}(\omega)}^{q}\lesssim\varepsilon,$

from which it follows that $J_{g}:L_{a}^{p}(\omega)\to H^{q}$ is compact.
Necessity. We divide the proof into two steps.

Step 1. Let $\delta$ be a positive constant such that
$\tilde{\delta}:=\tanh\delta\in(0,\frac{1}{3})$. Take
$\\{a_{k}\\}_{k=1}^{\infty}$ be a $\tilde{\delta}$-lattice. Rearrange the
sequence $\\{a_{k}\\}_{k=1}^{\infty}$ such that
$|a_{1}|\leq|a_{2}|\leq\cdots\to 1^{-}$. For any $\rho_{0}\in(0,1)$ and
$\xi\in\mathbb{T}$, we know that

$\Gamma(\xi)\backslash\rho_{0}\mathbb{D}\subseteq\bigcup_{\\{a_{k}:D(a_{k},\tilde{\delta})\cap(\Gamma(\xi)\backslash\rho_{0}\mathbb{D})\neq\emptyset\\}}D(a_{k},\tilde{\delta}).$

Let $\lambda=3e^{2\delta}-2$. By Lemma 2.11, for any $\xi\in\mathbb{T}$, we
have

$D(\zeta,\tilde{\delta})\subseteq\Gamma_{\lambda}(\xi),\quad\textmd{for~{}any}~{}\zeta\in\Gamma(\xi).$

Hence, for any $\zeta\in
D(a_{k},\tilde{\delta})\cap(\Gamma(\xi)\backslash\rho_{0}\mathbb{D})$, there
exists $\rho_{1}\in(0,\rho_{0}]$ such that

(2.52)
$\Gamma(\xi)\backslash\rho_{0}\mathbb{D}\subseteq\bigcup_{a_{k}\in\Gamma_{\lambda}(\xi)\backslash\rho_{1}\mathbb{D}}D(a_{k},\tilde{\delta})$

and $\rho_{1}$ tends to 1 as $\rho_{0}$ tends to $1$. Recall [Zhu07,
Proposition 4.5], for any $z\in D(a_{k},\tilde{\delta})$, we know that

$\frac{1-\tilde{\delta}}{1+\tilde{\delta}}(1-|z|^{2})\leq
1-|a_{k}|^{2}\leq\frac{1+\tilde{\delta}}{1-\tilde{\delta}}(1-|z|^{2})$

and

(2.53)
$A(D(a_{k},\tilde{\delta}))=\frac{\tilde{\delta}^{2}(1-|a_{k}|^{2})^{2}}{(1-|a_{k}|^{2}\tilde{\delta}^{2})^{2}}.$

Recall (see [PRS18]) there exist positive constants $\varrho_{1}$ and
$\varrho_{2}$ such that

(2.54)
$\big{(}\frac{1-t_{1}}{1-t_{2}}\big{)}^{\varrho_{1}}\widehat{\omega}(t_{2})\lesssim\widehat{\omega}(t_{1})\lesssim\big{(}\frac{1-t_{1}}{1-t_{2}}\big{)}^{\varrho_{2}}\widehat{\omega}(t_{2}),\quad
0\leq t_{1}\leq t_{2}<1.$

By the subharmonicity of $|g^{\prime}|$, for any $z\in
D(a_{k},\tilde{\delta})$, we have

$\displaystyle|g^{\prime}(z)|$
$\displaystyle\leq\frac{1}{\tilde{\delta}^{2}(1-|z|)^{2}}\int_{B(z,\tilde{\delta}(1-|z|))}|g^{\prime}(u)|dA(u)$
$\displaystyle\leq\frac{1}{\tilde{\delta}^{2}(1-|z|)^{2}}\int_{D(z,\tilde{\delta})}|g^{\prime}(u)|dA(u)$
(2.55)
$\displaystyle\lesssim\frac{1}{\tilde{\delta}^{2}(1-|a_{k}|)^{2}}\int_{D(a_{k},2\tilde{\delta})}|g^{\prime}(u)|dA(u).$

Combining (2.52), (2.53), (2.54) with (2.4), we obtain

$\displaystyle\sup\limits_{z\in\Gamma(\xi)\backslash\rho_{0}\mathbb{D}}\frac{(1-|z|^{2})}{\widehat{\omega}(z)^{\frac{1}{p}}}|g^{\prime}(z)|$
$\displaystyle\lesssim\sup\limits_{a_{k}\in\Gamma_{\lambda}(\xi)\backslash\rho_{1}\mathbb{D}}\frac{1}{\tilde{\delta}^{2}(1-|a_{k}|^{2})\widehat{\omega}(a_{k})^{\frac{1}{p}}}\int_{D(a_{k},2\tilde{\delta})}|g^{\prime}(u)|dA(u)$
$\displaystyle\lesssim\sup\limits_{a_{k}\in\Gamma_{\lambda}(\xi)\backslash\rho_{1}\mathbb{D}}\frac{(1-|a_{k}|^{2})}{\widehat{\omega}(a_{k})^{\frac{1}{p}}}\bigg{(}\sup\limits_{u\in
D(a_{k},2\tilde{\delta})}|g^{\prime}(u)|\bigg{)}.$

Notice that the constants of comparison in the above inequality do not depend
on $\tilde{\delta},\rho_{0}$ and $\rho_{1}$. Then we deduce there exists a
constant $C>0$ ($C$ does not depend on the choices of
$\tilde{\delta},\rho_{0}$ and $\rho_{1}$) such that

$\displaystyle\bigg{(}\sup\limits_{z\in\Gamma(\xi)\backslash\rho_{0}\mathbb{D}}\frac{(1-|z|^{2})}{\widehat{\omega}(z)^{\frac{1}{p}}}|g^{\prime}(z)|\bigg{)}^{\frac{pq}{p-q}}$
$\displaystyle\leq
C\bigg{[}\sup\limits_{a_{k}\in\Gamma_{\lambda}(\xi)\backslash\rho_{1}\mathbb{D}}\frac{(1-|a_{k}|^{2})}{\widehat{\omega}(a_{k})^{\frac{1}{p}}}\bigg{(}\sup\limits_{u\in
D(a_{k},2\tilde{\delta})}|g^{\prime}(u)|\bigg{)}\bigg{]}^{\frac{pq}{p-q}}.$

Thus, by integrating over $\mathbb{T}$ with respect to $\xi$, we conclude that

(2.56)
$\int_{\mathbb{T}}\bigg{(}\sup\limits_{z\in\Gamma(\xi)\backslash\rho_{0}\mathbb{D}}\frac{(1-|z|^{2})}{\widehat{\omega}(z)^{\frac{1}{p}}}|g^{\prime}(z)|\bigg{)}^{\frac{pq}{p-q}}d\xi\leq
C\int_{\mathbb{T}}\bigg{[}\sup\limits_{a_{k}\in\Gamma_{\lambda}(\xi)\backslash\rho_{1}\mathbb{D}}\frac{(1-|a_{k}|^{2})}{\widehat{\omega}(a_{k})^{\frac{1}{p}}}\bigg{(}\sup\limits_{u\in
D(a_{k},2\tilde{\delta})}|g^{\prime}(u)|\bigg{)}\bigg{]}^{\frac{pq}{p-q}}d\xi.$

Step 2. We will prove that

(2.57) $\lim\limits_{r\to
1^{-}}\int_{\mathbb{T}}\bigg{[}\sup\limits_{a_{k}\in\Gamma(\xi)\backslash
r\mathbb{D}}\frac{(1-|a_{k}|^{2})}{\widehat{\omega}(a_{k})^{\frac{1}{p}}}\bigg{(}\sup\limits_{u\in
D(a_{k},2\tilde{\delta})}|g^{\prime}(u)|\bigg{)}\bigg{]}^{\frac{pq}{p-q}}d\xi=0.$

Note that for the any $\varepsilon>0$, there exists $R_{2}\in(0,1)$ such that
for any $R_{3}\in(R_{2},1)$, it holds that

(2.58) $\widehat{\omega}(R_{3})(1-R_{3})<\varepsilon.$

Moreover there exists $k_{1}\in\mathbb{N}$ such that for any $k>k_{1}$, it
holds that $|a_{k}|\in(R_{2},1)$. Combining with (2.58), we have

(2.59)
$\widehat{\omega}(a_{k})(1-|a_{k}|)<\varepsilon,\quad\textmd{for~{}any}~{}k>k_{1}.$

Since every $z\in\mathbb{D}$ belongs to at most $N$ ($N$ is a constant only
depends on $\delta$) $D(a_{k},\tilde{\delta})$, then we get

$\sum_{k=1}^{\infty}\kappa(D(a_{k},\tilde{\delta}))\leq N\kappa(\mathbb{D}),$

where

$\kappa(z)=\frac{\widehat{\omega}(z)}{1-|z|},\quad z\in\mathbb{D}.$

Hence, there exists $k_{2}\in\mathbb{N}$ such that for any $k_{3}>k_{2}$,

(2.60) $\sum_{k=k_{3}}^{\infty}\kappa(D(a_{k},\tilde{\delta}))<\varepsilon.$

Let $r_{k}$ be the Rademacher functions. For $k\in\mathbb{N}$ and $p\in(0,2]$,
consider the functions

$f_{k,p}(z)=\frac{K_{a_{k}}^{\omega}(z)^{\frac{1}{q}}}{\|K_{a_{k}}^{\omega}\|_{L_{a}^{\frac{p}{q}}(\omega)}^{\frac{1}{q}}}\chi_{\\{p:0<p\leq
1\\}}+\frac{K_{a_{k}}^{\omega}(z)}{\|K_{a_{k}}^{\omega}\|_{L_{a}^{2-\frac{1}{p}}(\omega)}^{2-\frac{1}{p}}}\chi_{\\{p:1<p\leq
2\\}},\quad z\in\mathbb{D}.$

For $t\in(0,1)$, $k_{4}\in\mathbb{N}$ and $\\{y_{k}\\}_{k=1}^{\infty}\in
T_{p}^{p}(\\{a_{k}\\})$, choose test functions

$F_{t,p,k_{4},\\{y_{k}\\}}(z)=\frac{\sum\limits_{k=k_{4}}^{\infty}(1-|a_{k}|^{2})^{\frac{1}{p}}y_{k}r_{k}(t)f_{k,p}(z)}{\|\\{y_{k}\\}\|_{T_{p}^{p}(\\{a_{k}\\})}},\quad
z\in\mathbb{D}.$

Combining Lemma 2.6, (2.3), (2.37), (2.50), (2.59) with (2.60), for any
$z\in\overline{D(0,\varrho)}$ with $\varrho\in(0,1)$, for any
$k_{4}>\max\\{k_{1},k_{2}\\}$, we deduce

$\displaystyle\big{|}F_{t,p,k_{4},\\{y_{k}\\}}(z)\big{|}\leq$
$\displaystyle\frac{1}{\|\\{y_{k}\\}\|_{T_{p}^{p}(\\{a_{k}\\})}}\bigg{(}\sum_{k=k_{4}}^{\infty}(1-|a_{k}|^{2})|y_{k}|^{p}\frac{|K_{a_{k}}^{\omega}(z)|^{\frac{p}{q}}}{\|K_{a_{k}}^{\omega}\|_{L_{a}^{\frac{p}{q}}(\omega)}^{\frac{p}{q}}}\bigg{)}^{\frac{1}{p}}$
$\displaystyle\lesssim$ $\displaystyle\frac{\sup\limits_{k\geq
k_{4}}|K_{a_{k}}^{\omega}(z)|^{\frac{1}{q}}}{\|\\{y_{k}\\}\|_{T_{p}^{p}(\\{a_{k}\\})}}\bigg{[}\sum_{k=k_{4}}^{\infty}(1-|a_{k}|^{2})|y_{k}|^{p}\bigg{(}\widehat{\omega}(a_{k})(1-|a_{k}|)\bigg{)}^{\frac{p}{q}-1}\bigg{]}^{\frac{1}{p}}$
$\displaystyle\leq$ $\displaystyle\sup\limits_{k\geq
k_{4}}|K_{a_{k}}^{\omega}(z)|^{\frac{1}{q}}\sup\limits_{k\geq
k_{4}}\big{(}\widehat{\omega}(a_{k})(1-|a_{k}|)\big{)}^{\frac{p-q}{pq}}\frac{\bigg{(}\sum\limits_{k=k_{4}}^{\infty}(1-|a_{k}|^{2})|y_{k}|^{p}\bigg{)}^{\frac{1}{p}}}{\|\\{y_{k}\\}\|_{T_{p}^{p}(\\{a_{k}\\})}}$
$\displaystyle\lesssim$
$\displaystyle\varepsilon^{\frac{p-q}{pq}}\sup\limits_{k\geq
k_{4}}|K_{a_{k}}^{\omega}(z)|^{\frac{1}{q}},\quad p\in(0,1]$

and

$\displaystyle\big{|}F_{t,p,k_{4},\\{y_{k}\\}}(z)\big{|}\leq$
$\displaystyle\frac{1}{\|\\{y_{k}\\}\|_{T_{p}^{p}(\\{a_{k}\\})}}\bigg{(}\sum_{k=k_{4}}^{\infty}(1-|a_{k}|^{2})|y_{k}|^{p}\bigg{)}^{\frac{1}{p}}\bigg{[}\sum_{k=k_{4}}^{\infty}\bigg{(}\frac{|K_{a_{k}}^{\omega}(z)|}{\|K_{a_{k}}^{\omega}\|_{L_{a}^{2-\frac{1}{p}}(\omega)}^{{}^{2-\frac{1}{p}}}}\bigg{)}^{\frac{p}{p-1}}\bigg{]}^{\frac{p-1}{p}}$
$\displaystyle\lesssim$ $\displaystyle\sup\limits_{k\geq
k_{4}}|K_{a_{k}}^{\omega}(z)|\bigg{(}\sum_{k=k_{4}}^{\infty}\widehat{\omega}(a_{k})(1-|a_{k}|)\bigg{)}^{\frac{p-1}{p}}$
$\displaystyle\asymp$ $\displaystyle\sup\limits_{k\geq
k_{4}}|K_{a_{k}}^{\omega}(z)|\bigg{(}\sum_{k=k_{4}}^{\infty}\kappa(D(a_{k},\tilde{\delta}))\bigg{)}^{\frac{p-1}{p}}$
$\displaystyle\leq$
$\displaystyle\varepsilon^{\frac{p-1}{p}}\sup\limits_{k\geq
k_{4}}|K_{a_{k}}^{\omega}(z)|,\quad p\in(1,2],$

which yields that for $p\in(0,2]$, $F_{t,p,k_{4},\\{y_{k}\\}}$ uniformly
converges to 0 on any compact subset of $\mathbb{D}$ as $k_{4}\to\infty$.
Furthermore, by using Proposition 2.17, (2.37) and (2.50), for any
$k_{4}\in\mathbb{N}$, we obtain

$\|F_{t,p,k_{4},\\{y_{k}\\}}\|_{L_{a}^{p}(\omega)}^{p}\leq\frac{1}{\|\\{y_{k}\\}\|_{T_{p}^{p}(\\{a_{k}\\})}^{p}}\int_{\mathbb{D}}\sum_{k=k_{4}}^{\infty}(1-|a_{k}|^{2})|y_{k}|^{p}\frac{|K_{a_{k}}^{\omega}(z)|^{\frac{p}{q}}}{\|K_{a_{k}}^{\omega}\|_{L_{a}^{\frac{p}{q}}(\omega)}^{\frac{p}{q}}}\omega(z)dA(z)\lesssim
1,\quad p\in(0,1]$

and

$\|F_{t,p,k_{4},\\{y_{k}\\}}\|_{L_{a}^{p}(\omega)}^{p}\lesssim\frac{\sum\limits_{k=k_{4}}^{\infty}(1-|a_{k}|^{2})|y_{k}|^{p}}{\|\\{y_{k}\\}\|_{T_{p}^{p}(\\{a_{k}\\})}^{p}}\lesssim
1,\quad p\in(1,2].$

Thus, we know that for $p\in(0,2]$, $\\{F_{t,p,k_{4},\\{y_{k}\\}}\\}$ is a
sequence in $L_{a}^{p}(\omega)$ with

$\sup\limits_{k_{4}\in\mathbb{N}}\|F_{t,p,k_{4},\\{y_{k}\\}}\|_{L_{a}^{p}(\omega)}\lesssim
1.$

It follows from the compactness of $J_{g}$ that

$\lim\limits_{k_{4}\to\infty}\|J_{g}(F_{t,p,k_{4},\\{y_{k}\\}})\|_{H^{q}}^{q}=0.$

Then there exists $k_{5}\in\mathbb{N}$ such that for any
$k_{4}\in[k_{5},\infty)$, it holds that

(2.61) $\|J_{g}(F_{t,p,k_{4},\\{y_{k}\\}})\|_{H^{q}}^{q}<\varepsilon.$

Note that $k_{5}$ does not depend on the choice of
$\\{y_{k}\\}_{k=1}^{\infty}$ in $T_{p}^{p}(\\{a_{k}\\})$. Combine with Lemma
2.2, we obtain

$\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)}|g^{\prime}(z)|^{2}\bigg{|}\sum_{k=k_{4}}^{\infty}(1-|a_{k}|^{2})^{\frac{1}{p}}y_{k}r_{k}(t)f_{k,p}(z)\bigg{|}^{2}dA(z)\bigg{)}^{\frac{q}{2}}d\xi\asymp\|J_{g}(F_{t,p,k_{4},\\{y_{k}\\}})\|_{H^{q}}^{q}\|\\{y_{k}\\}\|_{T_{p}^{p}(\\{a_{k}\\})}^{q}.$

Integrate both sides of the inequality above from 0 to 1 with respect to $t$,
by Fubini’s theorem and (2.61), for any $k_{4}\in[k_{5},\infty)$, we have

(2.62)
$\int_{\mathbb{T}}\int_{0}^{1}\bigg{(}\int_{\Gamma(\xi)}|g^{\prime}(z)|^{2}\bigg{|}\sum_{k=k_{4}}^{\infty}(1-|a_{k}|^{2})^{\frac{1}{p}}y_{k}r_{k}(t)f_{k,p}(z)\bigg{|}^{2}dA(z)\bigg{)}^{\frac{q}{2}}dtd\xi\lesssim\varepsilon\|\\{y_{k}\\}\|_{T_{p}^{p}(\\{a_{k}\\})}^{q}.$

Notice that there exists $R_{4}\in(0,1)$ such that for all
$R_{5}\in[R_{4},1)$, we have

$k\geq k_{5},\quad\textmd{for~{}any}~{}a_{k}\in\mathbb{D}\backslash
R_{5}\mathbb{D}.$

Take a sequence $\\{b_{k}\\}_{k=1}^{\infty}$, where

$b_{k}=\frac{(1-|a_{k}|^{2})^{q}}{\widehat{\omega}(a_{k})^{\frac{q}{p}}}\bigg{(}\sup\limits_{u\in
D(a_{k},2\tilde{\delta})}|g^{\prime}(u)|\bigg{)}^{q}\chi_{\\{a_{k}:R_{4}\leq|a_{k}|<1\\}}.$

Choose $s>1$. Let $\iota=\frac{2ps}{2ps-pq+2q}$. Then $\iota\leq 1$. Due to
Lemma 2.14, the dual space of

$T_{\iota}^{\big{(}\frac{ps}{p-q}\big{)}^{\prime}}(\\{a_{k}\\})$

can be identified with

$T_{\infty}^{\frac{ps}{p-q}}(\\{a_{k}\\}).$

Take

$\\{e_{k}\\}_{k=1}^{\infty}\in
T_{\iota}^{\big{(}\frac{ps}{p-q}\big{)}^{\prime}}(\\{a_{k}\\}).$

The factorization in Lemma 2.15 shows that there exist sequences

$\\{x_{k}\\}_{k=1}^{\infty}\in
T_{\frac{2s}{2s-q}}^{s^{\prime}}(\\{a_{k}\\})\text{ and
}\\{y_{k}\\}_{k=1}^{\infty}\in T_{p}^{p}(\\{a_{k}\\})$

such that $e_{k}=x_{k}y_{k}^{\frac{q}{s}}.$ Using a similar estimation as in
(2.3), we get

(2.63) $|\langle
e_{k},b_{k}^{\frac{1}{s}}\rangle_{T_{2}^{2}(\\{a_{k}\\})}|\lesssim\|\\{x_{k}\\}\|_{T_{\frac{2s}{2s-q}}^{s^{\prime}}(\\{a_{k}\\})}\bigg{[}\int_{\mathbb{T}}\bigg{(}\sum\limits_{a_{k}\in\Gamma(\xi)\backslash
R_{4}\mathbb{D}}|y_{k}^{2}b_{k}^{\frac{2}{q}}|\bigg{)}^{\frac{q}{2}}d\xi\bigg{]}^{\frac{1}{s}}.$

Take $\tau>\max\\{1,\frac{2}{q}\\}$. By the subharmonicity of $|g^{\prime
2}|$, [PRS18, Lemma 8], Lemma 2.7 and (2.3), we deduce for $\delta$ small
enough,

$\displaystyle\sum\limits_{a_{k}\in\Gamma(\xi)\backslash
R_{4}\mathbb{D}}|y_{k}^{2}b_{k}^{\frac{2}{q}}|\asymp$
$\displaystyle\sum\limits_{a_{k}\in\Gamma(\xi)\backslash
R_{4}\mathbb{D}}|y_{k}|^{2}\bigg{(}\sup\limits_{u\in
D(a_{k},2\tilde{\delta})}|g^{\prime}(u)|\bigg{)}^{2}\bigg{(}\frac{1-|a_{k}|^{2}}{|1-\bar{a_{k}}\xi|}\bigg{)}^{\tau}\frac{(1-|a_{k}|^{2})^{2}}{\widehat{\omega}(a_{k})^{\frac{2}{p}}}$
$\displaystyle\lesssim$
$\displaystyle\sum\limits_{k=k_{5}}^{\infty}|y_{k}|^{2}\frac{1}{\widehat{\omega}(a_{k})(1-|a_{k}|)}\int_{D(a_{k},3\tilde{\delta})}|g^{\prime}(z)|^{2}\frac{\widehat{\omega}(z)}{1-|z|}\bigg{(}\frac{1-|z|^{2}}{|1-\bar{z}\xi|}\bigg{)}^{\tau}\frac{(1-|z|^{2})^{2}}{\widehat{\omega}(z)^{\frac{2}{p}}}dA(z)$
$\displaystyle\asymp$
$\displaystyle\sum\limits_{k=k_{5}}^{\infty}|y_{k}|^{2}(1-|a_{k}|^{2})^{\frac{2}{p}}\int_{D(a_{k},3\tilde{\delta})}|g^{\prime}(z)|^{2}\bigg{(}\frac{1-|z|^{2}}{|1-\bar{z}\xi|}\bigg{)}^{\tau}|f_{k,p}(z)|^{2}dA(z)$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{D}}\bigg{(}\frac{1-|z|^{2}}{|1-\bar{z}\xi|}\bigg{)}^{\tau}\sum\limits_{k=k_{5}}^{\infty}|y_{k}|^{2}(1-|a_{k}|^{2})^{\frac{2}{p}}|g^{\prime}(z)|^{2}|f_{k,p}(z)|^{2}dA(z).$

Together with Lemma 2.16, we know

$\displaystyle\int_{\mathbb{T}}\bigg{(}\sum\limits_{a_{k}\in\Gamma(\xi)\backslash
R_{4}\mathbb{D}}|y_{k}^{2}b_{k}^{\frac{2}{q}}|\bigg{)}^{\frac{q}{2}}d\xi$
$\displaystyle\lesssim\int_{\mathbb{T}}\bigg{(}\int_{\mathbb{D}}\bigg{(}\frac{1-|z|^{2}}{|1-\bar{z}\xi|}\bigg{)}^{\tau}\sum\limits_{k=k_{5}}^{\infty}|y_{k}|^{2}(1-|a_{k}|^{2})^{\frac{2}{p}}|g^{\prime}(z)|^{2}|f_{k,p}(z)|^{2}dA(z)\bigg{)}^{\frac{q}{2}}d\xi$
(2.64)
$\displaystyle\asymp\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)}\sum\limits_{k=k_{5}}^{\infty}|y_{k}|^{2}(1-|a_{k}|^{2})^{\frac{2}{p}}|f_{k,p}(z)|^{2}|g^{\prime}(z)|^{2}dA(z)\bigg{)}^{\frac{q}{2}}d\xi.$

Moreover, according to Lemma 2.8, Lemma 2.9 and Fubini’s theorem, we obtain

$\displaystyle\begin{split}&~{}~{}~{}~{}\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)}\sum\limits_{k=k_{5}}^{\infty}|y_{k}|^{2}(1-|a_{k}|^{2})^{\frac{2}{p}}|f_{k,p}(z)|^{2}|g^{\prime}(z)|^{2}dA(z)\bigg{)}^{\frac{q}{2}}d\xi\\\
&\asymp\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)}|g^{\prime}(z)|^{2}\int_{0}^{1}\bigg{|}\sum\limits_{k=k_{5}}^{\infty}y_{k}r_{k}(t)(1-|a_{k}|^{2})^{\frac{1}{p}}f_{k,p}(z)\bigg{|}^{2}dtdA(z)\bigg{)}^{\frac{q}{2}}d\xi\\\
&=\int_{\mathbb{T}}\bigg{(}\int_{0}^{1}\int_{\Gamma(\xi)}\bigg{|}g^{\prime}(z)\sum\limits_{k=k_{5}}^{\infty}y_{k}r_{k}(t)(1-|a_{k}|^{2})^{\frac{1}{p}}f_{k,p}(z)\bigg{|}^{2}dA(z)dt\bigg{)}^{\frac{q}{2}}d\xi\\\
&\asymp\int_{\mathbb{T}}\int_{0}^{1}\bigg{(}\int_{\Gamma(\xi)}\bigg{|}g^{\prime}(z)\sum\limits_{k=k_{5}}^{\infty}y_{k}r_{k}(t)(1-|a_{k}|^{2})^{\frac{1}{p}}f_{k,p}(z)\bigg{|}^{2}dA(z)\bigg{)}^{\frac{q}{2}}dtd\xi.\end{split}$

Together with (2.62) and (2.4), we get

(2.65) $\int_{\mathbb{T}}\bigg{(}\sum\limits_{a_{k}\in\Gamma(\xi)\backslash
R_{4}\mathbb{D}}|y_{k}^{2}b_{k}^{\frac{2}{q}}|\bigg{)}^{\frac{q}{2}}d\xi\lesssim\varepsilon\|\\{y_{k}\\}\|_{T_{p}^{p}(\\{a_{k}\\})}^{q}.$

According to (2.63) and (2.65), it holds that

$|\langle
e_{k},b_{k}^{\frac{1}{s}}\rangle_{T_{2}^{2}(\\{a_{k}\\})}|\lesssim\|\\{x_{k}\\}\|_{T_{\frac{2s}{2s-q}}^{s^{\prime}}(\\{a_{k}\\})}\varepsilon^{\frac{1}{s}}\|\\{y_{k}\\}\|_{T_{p}^{p}(\\{a_{k}\\})}^{\frac{q}{s}}.$

By taking infimum over all possible factorizations of $\\{e_{k}\\}$, we deduce

$|\langle e_{k},b_{k}^{\frac{1}{s}}\rangle_{T_{2}^{2}(\\{a_{k}\\})}|\lesssim\
\varepsilon^{\frac{1}{s}}\|\\{e_{k}\\}\|_{T_{\iota}^{\big{(}\frac{ps}{p-q}\big{)}^{\prime}}(\\{a_{k}\\})}.$

Therefore, using Lemma 2.14, we know that for any $\rho_{1}\geq R_{4}$,

(2.66)
$\int_{\mathbb{T}}\bigg{(}\sup\limits_{a_{k}\in\Gamma_{\lambda}(\xi)\backslash\rho_{1}\mathbb{D}}\frac{(1-|a_{k}|^{2})}{\widehat{\omega}(a_{k})^{\frac{1}{p}}}\bigg{(}\sup\limits_{u\in
D(a_{k},2\tilde{\delta})}|g^{\prime}(u)|\bigg{)}\bigg{)}^{\frac{pq}{p-q}}d\xi\lesssim\varepsilon^{\frac{p}{p-q}}.$

Hence, (2.57) holds.

Recall in Step 1, we can choose $\rho_{1}\in(0,\rho_{0}]$ such that $\rho_{1}$
tends to 1 as $\rho_{0}$ tends to 1. Then we can choose $\rho_{0}\in(0,1)$
such that $\rho_{1}\geq R_{4}$. Combining (2.56) with (2.66), we deduce

$\int_{\mathbb{T}}\bigg{(}\sup\limits_{z\in\Gamma(\xi)\backslash\rho_{0}\mathbb{D}}\frac{(1-|z|^{2})}{\widehat{\omega}(z)^{\frac{1}{p}}}|g^{\prime}(z)|\bigg{)}^{\frac{pq}{p-q}}d\xi\lesssim\varepsilon^{\frac{p}{p-q}}.$

Thus, we complete the whole proof of the necessity part. ∎

Combining Proposition 2.1, Proposition 2.13, Proposition 2.18 with Proposition
2.19, we get Theorem 1.2. Moreover, one can get Theorem 1.1 by modifying the
corresponding proofs of the above four propositions slightly.

##  3\. From Hardy space into Bergman space

In this section, we are devoted to proving Theorem 1.3. To begin with, we
state a result (cf. [PR19, Theorem 5]) which is vital to the proof of our main
result in this section. It describes that the class of $\mathcal{D}$ weights
is the largest class of radial weights such that the Littlewood-Paley estimate
holds.

######  Lemma 3.1.

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

$\|f\|_{L_{a}^{p}(\omega)}^{p}\asymp\int_{\mathbb{D}}|f^{\prime}(z)|^{p}(1-|z|^{2})^{p}\omega(z)dA(z)+|f(0)|^{p},\quad
f\in H(\mathbb{D})$

holds if and only if $\omega\in\mathcal{D}$.

###### Proof of Theorem 1.3.

Let

$d\mu_{g}(z)=|g^{\prime}(z)|^{q}(1-|z|^{2})^{q}\omega(z)dA(z).$

It follows from Lemma 3.1 that

(3.67)
$\|J_{g}(f)\|_{L_{a}^{q}(\omega)}^{q}\asymp\int_{\mathbb{D}}|f(z)|^{q}d\mu_{g}(z).$

By (3.67), the boundedness (compactness) of $J_{g}:H^{p}\to L^{q}_{a}(\omega)$
is equivalent to that of $Id:H^{p}\to L^{q}(\mu_{g})$. For the case when
$0<p\leq q<\infty$, the identity operator $Id:H^{p}\to L^{q}(\mu_{g})$ is
bounded (compact) if and only if $\mu_{g}$ is a $\frac{q}{p}$-Carleson measure
(vanishing $\frac{q}{p}$-Carleson measure) (cf. [Zhu07, Section 9.2]).

Now we are in a position to deal with the case when $0<q<p<\infty$. For
$\xi\in\mathbb{T}$, recall

$\widetilde{\mu_{g}}(\xi)=\int_{\Gamma(\xi)}\frac{d\mu_{g}(\lambda)}{1-|\lambda|^{2}}.$

By [Pau16, Theorem E] and (3.67), we know that $J_{g}:H^{p}\to
L_{a}^{q}(\omega)$ is bounded if and only if $\widetilde{\mu_{g}}\in
L^{\frac{p}{p-q}}(\mathbb{T})$.

It suffices to prove that $J_{g}:H^{p}\to L_{a}^{q}(\omega)$ is compact if
$\widetilde{\mu_{g}}\in L^{\frac{p}{p-q}}(\mathbb{T})$. Let
$\\{f_{j}\\}_{j=1}^{\infty}$ be any sequence in $H^{p}$ with
$\|f_{j}\|_{H^{p}}\leq 1$ and $f_{j}$ converges to 0 uniformly on any compact
subset of $\mathbb{D}$. By Lemma 3.1, we obtain

(3.68)
$\|J_{g}(f_{j})\|_{L_{a}^{q}(\omega)}^{q}\asymp\int_{\mathbb{D}}|f_{j}(z)|^{q}d\mu_{g}(z).$

Then we only need to show

(3.69) $\lim\limits_{j\to\infty}\int_{\mathbb{D}}|f_{j}(z)|^{q}d\mu_{g}(z)=0.$

Since $\widetilde{\mu_{g}}\in L^{\frac{p}{p-q}}(\mathbb{T})$, then for any
$\varepsilon>0$, there exists $r_{0}\in(0,1)$ such that

(3.70) $\bigg{[}\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)\backslash
r_{0}\mathbb{D}}\frac{d\mu_{g}(z)}{1-|z|^{2}}\bigg{)}^{\frac{p}{p-q}}d\xi\bigg{]}^{\frac{p-q}{p}}<\varepsilon.$

Moreover, there exists $k_{0}\in\mathbb{N}$ such that for any $j>k_{0}$,
$|f_{j}|^{q}<\varepsilon$ on $r_{0}\mathbb{D}$. It follows from [Gar07,
Theorem 3.1 in Chapter II], $\textmd{H}\ddot{\textmd{o}}\textmd{lder's}$
inequality, Fubini’s theorem and (2.6), that

$\displaystyle\int_{\mathbb{D}}|f_{j}(z)|^{q}d\mu_{g}(z)$
$\displaystyle\lesssim\varepsilon+\int_{\mathbb{D}\backslash
r_{0}\mathbb{D}}\frac{|f_{j}(z)|^{q}}{1-|z|^{2}}\int_{I(z)}d\xi d\mu_{g}(z)$
$\displaystyle\leq\varepsilon+\bigg{(}\int_{\mathbb{T}}\sup_{z\in\Gamma(\xi)}|f_{j}(z)|^{p}d\xi\bigg{)}^{\frac{q}{p}}\bigg{[}\int_{\mathbb{T}}\bigg{(}\int_{\Gamma(\xi)\backslash
r_{0}\mathbb{D}}\frac{d\mu_{g}(z)}{1-|z|^{2}}\bigg{)}^{\frac{p}{p-q}}d\xi\bigg{]}^{\frac{p-q}{p}}$
$\displaystyle\lesssim\varepsilon+\varepsilon\|f_{j}\|_{H^{p}}^{q}\lesssim\varepsilon.$

Thus, we get the desired conclusion and complete the proof. ∎

## Acknowledgements

This work is partially supported by National Natural Science Foundation of
China.

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