# Double integral estimates for Besov type spaces and their applications

Guanlong Bao  Department of Mathematics
Shantou University
Shantou 515063, Guangdong, China<EMAIL_ADDRESS>,  Juntao Du  Department of
Mathematics
Shantou University
Shantou 515063, Guangdong, China<EMAIL_ADDRESS>and  Hasi Wulan  Department
of Mathematics
Shantou University
Shantou 515063, Guangdong, China<EMAIL_ADDRESS>
###### Abstract.

For $0<p<\infty$, we give a complete description of nonnegative radial weight
functions $\omega$ on the open unit disk $\mathbb{D}$ such that

$\int_{\mathbb{D}}|f^{\prime}(z)|^{p}(1-|z|^{2})^{p-2}\omega(z)dA(z)<\infty$

if and only if

$\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f(z)-f(\zeta)|^{p}}{|1-\overline{\zeta}z|^{4+\tau+\sigma}}(1-|z|^{2})^{\tau}(1-|\zeta|^{2})^{\sigma}\omega(\zeta)dA(z)A(\zeta)<\infty$

for all analytic functions $f$ in $\mathbb{D}$, where $\tau$ and $\sigma$ are
some real numbers. As applications, we give some geometric descriptions of
functions in Besove type spaces $B_{p}(\omega)$ with doubling weights, and
characterize the boundedness and compactness of Hankel type operators related
to Besov type spaces with radial Békollé-Bonami weights. Some special cases of
our results are new even for some standard weighted Besov spaces.

###### Key words and phrases:

Besov type space; Dirichlet type space; double integral estimate; Békollé-
Bonami weight; Hankel type operator

###### 2010 Mathematics Subject Classification:

30H25; 30H20; 46E15; 47B35

The work was supported by NNSF of China (No. 11720101003) and Guangdong basic
and applied basic research foundation (No. 2022A1515012117).

##  1\. Introduction

A classical topic in complex analysis and related field is to study double
integral estimates for Dirichlet type spaces or Besove type spaces. Recall
that the Dirichlet integral of a function $f\in L^{2}(\mathbb{T})$ is

$\mathfrak{D}(f)=\int_{\mathbb{T}}\int_{\mathbb{T}}\left|\frac{f(\zeta)-f(\eta)}{\zeta-\eta}\right|^{2}|d\zeta||d\eta|,$

where $\mathbb{T}$ is the boundary of the open unit disk $\mathbb{D}$ in the
complex plane $\mathbb{C}$. By this integral, in 1931 J. Douglas [15] studied
the theory of minimal surfaces, and in 1940 A. Beurling [12] proved that the
Fourier series of a Dirichlet function converges everywhere except on a set of
logarithmic capacity zero. Let $H(\mathbb{D})$ be the space of analytic
functions in $\mathbb{D}$. The Dirichlet space $\mathfrak{D}$ consists of
functions $f\in H(\mathbb{D})$ such that

$\int_{\mathbb{D}}|f^{\prime}(z)|^{2}dA(z)<\infty,$

where $dA(z)=(1/\pi)dxdy$ is the normalized area measure on $\mathbb{D}$. It
is known that every function $f$ in $\mathfrak{D}$ has nontangential limit
denoted by $f(\zeta)$ for almost every $\zeta\in\mathbb{T}$. For $f\in
H(\mathbb{D})$, it is also well known that $f\in\mathfrak{D}$ if and only if
$\mathfrak{D}(f)<\infty$.

For $p>1$, the Besov space $B_{p}$ is the space of functions $f\in
H(\mathbb{D})$ satisfying

$\int_{\mathbb{D}}|f^{\prime}(z)|^{p}(1-|z|^{2})^{p}d\lambda(z)<\infty,$

where

$d\lambda(z)=\frac{dA(z)}{(1-|z|^{2})^{2}}$

is the Möbius invariant measure on $\mathbb{D}$. Clearly, $B_{2}$ is equal to
the Dirichlet space $\mathfrak{D}$. For $\alpha>-1$ and
$p>\max\\{1,2/(2+\alpha)\\}$, in 1988 J. Arazy, S. Fisher and J. Peetre (cf.
[4, Theorem 6.4] and [4, Remark, p. 1026]) showed that a function $f\in B_{p}$
if and only if the following integral

$\int_{\mathbb{D}}\left(\int_{\mathbb{D}}|f\circ\varphi_{a}(z)-f(a)|^{2}(1-|z|^{2})^{\alpha}dA(z)\right)^{p/2}d\lambda(a)$

is finite for $f\in H(\mathbb{D})$, where
$\varphi_{a}(z)=(a-z)/(1-\overline{a}z)$ is a Möbius map interchanging the
points $0$ and $a$. In [4], this characterization was used to investigate the
properties of Hankel operators on weighted Bergman spaces. In 1991 K. Zhu [34]
gave another double integral estimates for Besov spaces; that is, if $f\in
H(\mathbb{D})$, then $f\in B_{p}$ if and only if

$\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f(z)-f(w)|^{p}}{|1-\overline{z}w|^{4}}dA(z)dA(w)<\infty.$

Recall that for $p>0$ and $\alpha<p/2$, the standard weighted Besov space
$B_{p}(\alpha)$ consists of functions $f\in H(\mathbb{D})$ for which

$\int_{\mathbb{D}}|f^{\prime}(z)|^{p}(1-|z|^{2})^{p-1-2\alpha}dA(z)<\infty.$

If $p=2$, then $B_{p}(\alpha)$ is the Dirichlet type space
$\mathfrak{D}_{\alpha}$. In 1993 R. Rochberg and Z. Wu [29] obtained a double
integral characterization of Dirichlet type spaces $\mathfrak{D}_{\alpha}$ and
applied this characterization to study Hankel type operators. In 2008 D. Blasi
and J. Pau [13] generalized this characterization from $\mathfrak{D}_{\alpha}$
to $B_{p}(\alpha)$ by a different method. It is known from [13, Theorem 2.2]
that if $f\in H(\mathbb{D})$, $p>1$, $\sigma,\tau>-1$ and $\alpha\leq 1/2$
such that $\min(\sigma,\tau)+2\alpha>-1$, then $f\in B_{p}(\alpha)$ if and
only if

$\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f(z)-f(w)|^{p}}{|1-\overline{w}z|^{3+\sigma+\tau+2\alpha}}(1-|w|^{2})^{\sigma}(1-|z|^{2})^{\tau}dA(z)dA(w)<\infty.$

This double integral characterization was also used to study Hankel type
operators; see [13, Section 4]. We refer to [7, Section 4] and [31] for some
recent results associated with double integral estimates for some analytic
function spaces. See A. Reijonen [28] for recent results on Besov type spaces
induced by some radial weights.

A function $\omega:\mathbb{D}\rightarrow[0,\infty)$, integrable over
$\mathbb{D}$, is called a weight. If $\omega(z)=\omega(|z|)$ for all
$z\in\mathbb{D}$, then we say that $\omega$ is radial. Suppose $0<p<\infty$
and $(1-|z|^{2})^{p-2}\omega(z)$ is a weight on $\mathbb{D}$. Denote by
$B_{p}(\omega)$ the Besov type space consisting of those functions $f\in
H(\mathbb{D})$ such that

$\|f\|_{B_{p}(\omega)}=|f(0)|+\left(\int_{\mathbb{D}}|f^{\prime}(z)|^{p}(1-|z|^{2})^{p-2}\omega(z)dA(z)\right)^{1/p}<\infty.$

If $\omega(z)=(1-|z|^{2})^{1-2\alpha}$, then $B_{p}(\omega)=B_{p}(\alpha)$.

In this paper, for all $0<p<\infty$, we give a complete description of radial
weights $\omega$ such that an analytic function $f$ belongs to $B_{p}(\omega)$
if and only if $f$ satisfies certain double integral estimates. This result is
new even for $B_{p}(\alpha)$ when $0<p\leq 1$ and $\alpha<p/2$. As
applications, we consider geometric descriptions of functions in Besove type
space with doubling weights. We also apply these double integral estimates to
characterize the boundedness of Hankel type operators from $B_{p}(\omega)$
with radial Békollé-Bonami weights to the corresponding nonanalytic version of
Besov type spaces, which is also new for $B_{p}(\alpha)$ when $1<p<\infty$ and
$1/2<\alpha<p/2$. The compactness of these Hankel type operators is also
investigated.

Throughout this paper, we write $a\lesssim b$ if there exists a positive
constant $C$ such that $a\leq Cb$. If $a\lesssim b\lesssim a$, then we write
$a\thickapprox b$.

##  2\. Double integral estimates for $B_{p}(\omega)$ spaces

This section is devoted to investigate double integral estimates for
$B_{p}(\omega)$ spaces. In particular, for $0<p<\infty$ we characterize
completely radial weights $\omega$ such that

$\displaystyle\int_{\mathbb{D}}|f^{\prime}(z)|^{p}(1-|z|^{2})^{p-2}\omega(z)dA(z)$
$\displaystyle\thickapprox$
$\displaystyle\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f(z)-f(\zeta)|^{p}}{|1-\overline{\zeta}z|^{4+\tau+\sigma}}(1-|z|^{2})^{\tau}(1-|\zeta|^{2})^{\sigma}\omega(\zeta)dA(z)A(\zeta)$

for all $f\in H(\mathbb{D})$, where $\sigma$ and $\tau$ are real numbers in
certain ranges. This result is new even for the space $B_{p}(\alpha)$ when
$0<p\leq 1$ and $\alpha<p/2$. For $p=2$, the double integral estimates for
$B_{p}(\omega)$ spaces also complete corresponding conclusions in [8, p. 1725]
and [17, p. 210], respectively.

###  2.1. Littlewood-Paley estimates for weighted Bergman spaces and a
Forelli-Rudin type estimate

In this subsection, we recall some Littlewood-Paley estimates for weighted
Bergman spaces. We also give an elementary proof of a well-known Forelli-Rudin
type estimate. All of these estimates are tools to prove our main theorems in
this section.

Suppose $0<p<\infty$ and $\mu$ is a nonnegative Borel measure on $\mathbb{D}$.
The Lebesgue space $L^{p}(\mathbb{D},d\mu)$ consists of complex-valued
measurable functions $f$ on $\mathbb{D}$ for which

$\|f\|_{L^{p}(\mathbb{D},d\mu)}=\left(\int_{\mathbb{D}}|f(z)|^{p}d\mu(z)\right)^{\frac{1}{p}}<\infty.$

For $0<p<\infty$ and a weight $\omega$ in $\mathbb{D}$, the weighed Bergman
space $A^{p}_{\omega}$ is the space of analytic functions in
$L^{p}(\mathbb{D},\omega dA)$. If $\omega(z)=(\alpha+1)(1-|z|^{2})^{\alpha}$,
$\alpha>-1$, then $A^{p}_{\omega}$ is the standard weighted Bergman space
$A^{p}_{\alpha}$.

For $f\in H(\mathbb{D})$, it is well known that the growth rate of the
following functions are often comparable in some sense:

$f(z),(1-|z|^{2})f^{\prime}(z),(1-|z|^{2})^{2}f^{\prime\prime}(z),(1-|z|^{2})^{3}f^{\prime\prime\prime}(z),\cdots.$

This kind of estimates is usually called Littlewood-Paley estimates. We refer
to [1, 2, 3, 9, 20, 22] for the study of Littlewood-Paley estimates for
$A^{p}_{\omega}$; that is, to characterize weight functions $\omega$ such that

$\int_{\mathbb{D}}|f(z)-f(0)|^{p}\omega(z)dA(z)\thickapprox\int_{\mathbb{D}}|f^{\prime}(z)|^{p}(1-|z|^{2})^{p}\omega(z)dA(z)$

for all $f\in H(\mathbb{D})$. When weight $\omega$ is radial, this question
has been recently solved completely in [22].

The pseudo-hyperbolic metric on $\mathbb{D}$ is defined by

$\rho(z,w)=\left|\frac{z-w}{1-\overline{z}w}\right|,\ \ z,\ w\in\mathbb{D}.$

For $0<r<1$ and $a\in\mathbb{D}$, denote by

$\Delta(a,r)=\\{z\in\mathbb{D}:\rho(a,z)<r\\}$

the pseudo-hyperbolic disk of center $a$ and radius $r$.

The following Littlewood-Paley estimates for $A^{p}_{\omega}$ [9] will be
useful in this paper. For a given weight $\omega$, the conditions appeared in
the following theorems are easy to verify.

######  Theorem A.

Suppose $p>0$ and $\omega$ is a weight. If there exist two constants
$r\in(0,1)$ and $C>0$ such that

$C^{-1}\omega(\zeta)\leq\omega(z)\leq C\omega(\zeta)$

for all $z$ and $\zeta$ satisfying $\rho(z,\zeta)<r$, then there exists
another positive constant $C$ such that

$\int_{\mathbb{D}}(1-|z|^{2})^{p}|f^{\prime}(z)|^{p}\omega(z)\,dA(z)\leq
C\int_{\mathbb{D}}|f(z)-f(0)|^{p}\omega(z)\,dA(z)$

for all $f\in H(\mathbb{D})$.

######  Theorem B.

Suppose $p>0$, $\omega$ is a weight, and there exist $t_{0}\geq 0$ and
$s_{0}\in[-1,0)$ with the following property: for any $t>t_{0}$ and $s>s_{0}$
there is a positive constant $C=C(t,s)$ such that

$\int_{\mathbb{D}}\frac{\omega(\zeta)(1-|\zeta|^{2})^{s}\,dA(\zeta)}{|1-z\overline{\zeta}|^{2+s+t}}\leq\frac{C\omega(z)}{(1-|z|^{2})^{t}}$

for all $z\in\mathbb{D}$. Then there exists another positive constant $C$ such
that

$\int_{\mathbb{D}}|f(z)-f(0)|^{p}\omega(z)\,dA(z)\leq
C\int_{\mathbb{D}}(1-|z|^{2})^{p}|f^{\prime}(z)|^{p}\omega(z)\,dA(z)$

for all $f\in H(\mathbb{D})$.

We recall some well-known estimates as follows (see [33, Lemma 3.10]).

######  Lemma C.

Let $\beta$ be any real number. Then

$\int^{2\pi}_{0}\frac{d\theta}{|1-ze^{-i\theta}|^{1+\beta}}\thickapprox\begin{cases}1&\enspace\text{if}\
\ \beta<0,\\\ \log\frac{2}{1-|z|^{2}}&\enspace\text{if}\ \ \beta=0,\\\
\frac{1}{(1-|z|^{2})^{\beta}}&\enspace\text{if}\ \ \beta>0,\end{cases}$

for all $z\in\mathbb{D}$. Also, suppose $c$ is real and $t>-1$. Then

$\int_{\mathbb{D}}\frac{(1-|w|^{2})^{t}}{|1-\bar{z}w|^{2+t+c}}dA(w)\thickapprox\begin{cases}1&\enspace\text{if}\
\ \ c<0,\\\ \log\frac{2}{1-|z|^{2}}&\enspace\text{if}\ \ \ c=0,\\\
\frac{1}{(1-|z|^{2})^{c}}&\enspace\text{if}\ \ \ c>0,\end{cases}$

for all $z\in\mathbb{D}$.

The following Forelli-Rudin type estimate [19] is very useful in the analysis
of some function spaces. A rather complicated proof of this estimate was given
in [32]. Later, for $s>-1$, $r>0$, $t>0$, and $t+1<s+2<r$, a simple proof was
given in [30, pp. 27-28], where the case of $t<s+2\leq t+1<r$ is missing.

######  Lemma D.

Suppose $s>-1$, $r>0$, $t>0$, and $t<s+2<r$. Then there exists a positive
constant $C$ such that

$\int_{\mathbb{D}}\frac{(1-|w|^{2})^{s}}{|1-\overline{w}z|^{r}|1-\overline{w}\zeta|^{t}}dA(w)\leq
C\,\frac{(1-|z|^{2})^{2+s-r}}{|1-\overline{\zeta}z|^{t}}$

for all $z,\zeta\in\mathbb{D}$.

We present an elementary proof of Lemma D here. For $z$, $\zeta\in\mathbb{D}$,
by a change of variables $w=\varphi_{z}(a)$ and Lemma C, we see that

$\displaystyle\int_{\mathbb{D}}\frac{(1-|w|^{2})^{s}}{|1-\overline{w}z|^{r}|1-\overline{w}\zeta|^{t}}dA(w)$
$\displaystyle=$
$\displaystyle(1-|z|^{2})^{s+2-r}\int_{\mathbb{D}}\frac{(1-|a|^{2})^{s}dA(a)}{|1-\overline{z}a|^{4+2s-r}|1-\overline{\zeta}\varphi_{z}(a)|^{t}}$
$\displaystyle=$
$\displaystyle\frac{(1-|z|^{2})^{s+2-r}}{|1-\overline{z}\zeta|^{t}}\int_{\mathbb{D}}\frac{(1-|a|^{2})^{s}dA(a)}{|1-\overline{z}a|^{4+2s-r-t}|1-a\overline{\varphi_{z}(\zeta)}|^{t}}$
$\displaystyle\leq$
$\displaystyle\frac{(1-|z|^{2})^{s+2-r}}{|1-\overline{z}\zeta|^{t}}\int_{\\{a\in\mathbb{D}:|1-\overline{z}a|\geq|1-a\overline{\varphi_{z}(\zeta)}|\\}}\frac{(1-|a|^{2})^{s}dA(a)}{|1-\overline{z}a|^{4+2s-r-t}|1-a\overline{\varphi_{z}(\zeta)}|^{t}}$
$\displaystyle+\frac{(1-|z|^{2})^{s+2-r}}{|1-\overline{z}\zeta|^{t}}\int_{\\{a\in\mathbb{D}:|1-\overline{z}a|\leq|1-a\overline{\varphi_{z}(\zeta)}|\\}}\frac{(1-|a|^{2})^{s}dA(a)}{|1-\overline{z}a|^{4+2s-r-t}|1-a\overline{\varphi_{z}(\zeta)}|^{t}}$
$\displaystyle\lesssim$
$\displaystyle\frac{(1-|z|^{2})^{s+2-r}}{|1-\overline{z}\zeta|^{t}}\left(\sup_{b\in\mathbb{D}}\int_{\mathbb{D}}\frac{(1-|a|^{2})^{s}dA(a)}{|1-\overline{a}b|^{4+2s-r}}+\sup_{b\in\mathbb{D}}\int_{\mathbb{D}}\frac{(1-|a|^{2})^{s}dA(a)}{|1-\overline{a}b|^{t}}\right)$
$\displaystyle\lesssim$
$\displaystyle\frac{(1-|z|^{2})^{s+2-r}}{|1-\overline{z}\zeta|^{t}},$

which finishes the proof of Lemma D.

###  2.2. Weighted Bergman spaces and doubling weights

In this subsection, we recall characterizations of Carleson measures for
$A^{p}_{\omega}$ induced by doubling weights and also give some estimates
related to these weights.

Let $\hat{\mathcal{D}}$ be the class of radial weights $\omega$ on
$\mathbb{D}$ for which $\hat{\omega}(r)=\int_{r}^{1}\omega(s)ds$ admits the
doubling property $\hat{\omega}(r)\leq C\hat{\omega}(\frac{1+r}{2})$ for all
$r\in[0,1)$, where $C=C(\omega)>1$. A weight in $\hat{\mathcal{D}}$ is usually
called a doubling weight. If there exist $K=K(\omega)>1$ and $C=C(\omega)>1$
such that the radial weight $\omega$ satisfies

$\hat{\omega}(r)\geq C\hat{\omega}\left(1-\frac{1-r}{K}\right),\ \ 0\leq r<1,$

then we say that $\omega\in\check{\mathcal{D}}$. The intersection
$\hat{\mathcal{D}}\cap\check{\mathcal{D}}$ is denoted by $\mathcal{D}$. The
classes of these weights arose naturally in the study of some analytic
function spaces and related operator theory. For instance, by [22], for a
radial weight $\omega$, certain Bergman projection $P_{\omega}$ is bounded
from $L^{\infty}$ to the Bloch space $\mathcal{B}$ if and only if
$\omega\in\hat{\mathcal{D}}$; $P_{\omega}:\ L^{\infty}\rightarrow\mathcal{B}$
is bonded and onto if and only if $\omega\in\mathcal{D}$; the classes
$\hat{\mathcal{D}}$ and $\mathcal{D}$ also characterize completely Littlewood-
Paley estimates for weighted Bergman spaces with radial weights. See [21, 23]
for properties of these weights. Our investigation in this section will guide
us to find new and more significance of the class $\hat{\mathcal{D}}$; that
is, the weight in $\hat{\mathcal{D}}$ describes precisely certain double
integral estimate for Besov type spaces.

For a space $X$ of analytic functions on $\mathbb{D}$ and $0<p<\infty$, a
nonnegative Borel measures $\mu$ on $\mathbb{D}$ is said to be a $p$-Carleson
measure for $X$ if the identity operator $I_{d}:X\to L^{p}(\mathbb{D},d\mu)$
is bounded; that is,

$\left(\int_{\mathbb{D}}|f(z)|^{p}d\mu(z)\right)^{\frac{1}{p}}\lesssim\|f\|_{X}$

for all $f\in X$. If $I_{d}:X\to L^{p}(\mathbb{D},d\mu)$ is compact, then we
say that $\mu$ is a vanishing $p$-Carleson measure for $X$.

For a doubling weight $\omega$ and $0<p,q<\infty$, J. Peláez and J. Rättyä
[25] characterized nonnegative Borel measures $\mu$ on $\mathbb{D}$ such that
the differentiation operator of order $n\in\mathbb{N}\cup\\{0\\}$ is bounded
from $A_{\omega}^{p}$ into $L^{q}(\mathbb{D},d\mu)$. In particular, they gave
the following result.

######  Theorem E.

Suppose $0<p<\infty$, $\omega\in\hat{\mathcal{D}}$ and $\mu$ is a nonnegative
Borel measure on $\mathbb{D}$. Then $\mu$ is a $p$-Carleson measure for
$A^{p}_{\omega}$ if and only if the function

$a\mapsto\frac{\mu(S(a))}{\int_{S(a)}wdA},\ \ a\in\mathbb{D}\setminus\\{0\\},$

is essentially bounded, where

$S(a)=\left\\{z\in\mathbb{D}:\left|\frac{\arg z-\arg
a}{2\pi}\right|<\frac{1-|a|}{2},\ |z|\geq|a|\right\\}$

is the Carleson box with vertex at $a$.

The following characterizations of doubling weights can be found in [21].

######  Lemma F.

Suppose $\omega$ is a radial weight. Then the following conditions are
equivalent:

  1. (i)

$\omega\in\hat{\mathcal{D}}$;

  2. (ii)

there exists a positive constant $\beta$ depending only on $\omega$ such that

$\displaystyle\frac{\hat{\omega}(r)}{(1-r)^{\beta}}\lesssim\frac{\hat{\omega}(t)}{(1-t)^{\beta}}$
(2.1)

for all $0\leq r\leq t<1$;

  3. (iii)

there exists a positive constant $\gamma$ depending only on $\omega$ such that

$\displaystyle\int_{0}^{t}\frac{\omega(s)}{(1-s)^{\gamma}}ds\lesssim\frac{\hat{\omega}(t)}{(1-t)^{\gamma}}$
(2.2)

for all $0\leq t<1$;

  4. (iv)

there exists a nonnegative constant $\lambda$ depending only on $\omega$ such
that

$\displaystyle\int_{\mathbb{D}}\frac{\omega(z)dA(z)}{|1-\overline{\xi}z|^{\lambda+1}}\lesssim\frac{\hat{\omega}(\xi)}{(1-|\xi|)^{\lambda}}$
(2.3)

for all $\xi\in\mathbb{D}$;

  5. (v)

there exists a positive constant $\eta$ depending only on $\omega$ such that

$\displaystyle\int_{0}^{1}s^{x}\omega(s)ds\lesssim\left(\frac{y}{x}\right)^{\eta}\int_{0}^{1}s^{y}\omega(s)ds$
(2.4)

for all $0<x\leq y<\infty$.

Of course, constants $\beta$, $\gamma$, $\lambda$ and $\eta$ in Lemma F are
not unique. In fact, if $\beta$ satisfies (ii) in Lemma F, then any constant
bigger than $\beta$ also satisfies (ii). The same phenomenon occurs for
$\gamma$, $\lambda$ and $\eta$. Now we give the following relation among these
parameters. The infimum below is useful in our results.

######  Lemma 2.1.

Suppose $\omega\in\hat{\mathcal{D}}$. Then

$\displaystyle\inf\\{\beta:\ \beta\ \text{satisfies (ii) in Lemma
\ref{property of doub w}}\\}$ $\displaystyle=$ $\displaystyle\inf\\{\gamma:\
\gamma\ \text{satisfies (iii) in Lemma \ref{property of doub w}}\\}$
$\displaystyle=$ $\displaystyle\inf\\{\lambda:\ \lambda\ \text{satisfies (iv)
in Lemma \ref{property of doub w}}\\}$ $\displaystyle=$
$\displaystyle\inf\\{\eta:\ \eta\ \text{satisfies (v) in Lemma \ref{property
of doub w}}\\}.$

###### Proof.

Checking that proof of Lemma F [21], we get

$\\{\gamma:\ \gamma\ \text{satisfies (iii) in Lemma \ref{property of doub
w}}\\}\subseteq\\{\lambda:\ \lambda\ \text{satisfies (iv) in Lemma
\ref{property of doub w}}\\},$

which yields

$\displaystyle\inf\\{\lambda:\ \lambda\ \text{satisfies (iv) in Lemma
\ref{property of doub w}}\\}$ $\displaystyle\leq$
$\displaystyle\inf\\{\gamma:\ \gamma\ \text{satisfies (iii) in Lemma
\ref{property of doub w}}\\}.$ (2.5)

Now suppose $\beta$ satisfies (ii) in Lemma F. Then for any $\epsilon>0$,
$\beta+\epsilon$ also satisfies the same property. Hence

$\hat{\omega}(0)\lesssim\frac{\hat{\omega}(t)}{(1-t)^{\beta+\epsilon}}$

for all $0\leq t<1$. Consequently,

$\displaystyle\int_{0}^{t}\frac{\omega(s)}{(1-s)^{\beta+\epsilon}}ds=$
$\displaystyle\hat{\omega}(0)-\frac{\hat{\omega}(t)}{(1-t)^{\beta+\epsilon}}+(\beta+\epsilon)\int_{0}^{t}\frac{\hat{\omega}(s)}{(1-s)^{\beta+\epsilon+1}}ds$
$\displaystyle\lesssim$
$\displaystyle\frac{\hat{\omega}(t)}{(1-t)^{\beta+\epsilon}}+(\beta+\epsilon)\frac{\hat{\omega}(t)}{(1-t)^{\beta}}\int_{0}^{t}\frac{1}{(1-s)^{\epsilon+1}}ds$
$\displaystyle\lesssim$
$\displaystyle\left(1+\frac{\beta+\epsilon}{\epsilon}\right)\frac{\hat{\omega}(t)}{(1-t)^{\beta+\epsilon}},$
(2.6)

which means that when $\beta$ satisfies (ii) in Lemma F,

$\beta+\epsilon\in\\{\gamma:\ \gamma\ \text{satisfies (iii) in Lemma
\ref{property of doub w}}\\}$

for any $\epsilon>0$. Thus

$\displaystyle\inf\\{\gamma:\ \gamma\ \text{satisfies (iii) in Lemma
\ref{property of doub w}}\\}$ $\displaystyle\leq$ $\displaystyle\inf\\{\beta:\
\beta\ \text{satisfies (ii) in Lemma \ref{property of doub w}}\\}.$ (2.7)

Next let $\lambda$ satisfy (iv) in Lemma F. Without loss of generality, we can
assume $\lambda>0$. In fact, if $\lambda=0$, it is enough to consider the case
of $\lambda+\epsilon$ for any $\epsilon>0$. For $0\leq r\leq t<1$, take
$\xi\in\mathbb{D}$ such that $|\xi|=t$. It follows from (2.3) and Lemma C that

$\displaystyle\frac{\hat{\omega}(t)}{(1-t)^{\lambda}}\gtrsim\int_{\mathbb{D}}\frac{\omega(z)dA(z)}{|1-\overline{\xi}z|^{\lambda+1}}\gtrsim\int_{r}^{1}\frac{\omega(s)}{(1-st)^{\lambda}}ds\gtrsim\frac{\hat{\omega}(r)}{(1-r^{2})^{\lambda}},$

which gives that $\lambda\in\\{\beta:\ \beta\ \text{satisfies (ii) in Lemma
\ref{property of doub w}}\\}$. Hence

$\displaystyle\inf\\{\beta:\ \beta\ \text{satisfies (ii) in Lemma
\ref{property of doub w}}\\}$ $\displaystyle\leq$
$\displaystyle\inf\\{\lambda:\ \lambda\ \text{satisfies (iv) in Lemma
\ref{property of doub w}}\\}.$ (2.8)

By (2.2), (2.2) and (2.2), we get

$\displaystyle\inf\\{\beta:\ \beta\ \text{satisfies (ii) in Lemma
\ref{property of doub w}}\\}$ $\displaystyle=$ $\displaystyle\inf\\{\gamma:\
\gamma\ \text{satisfies (iii) in Lemma \ref{property of doub w}}\\}$ (2.9)
$\displaystyle=$ $\displaystyle\inf\\{\lambda:\ \lambda\ \text{satisfies (iv)
in Lemma \ref{property of doub w}}\\}.$

Let $\beta$ satisfy (ii) in Lemma F, so is $\beta+\varepsilon$ for any
$\varepsilon>0$. Consider $2<x\leq y<\infty$ first. For $0<s<1$ the function
$f_{x,\beta+\varepsilon}(s)=x^{\beta+\varepsilon}s^{x}(1-s)^{\beta+\varepsilon}$
takes its maximum at $s=x/(x+\beta+\varepsilon)$. Thus when $s\in(0,1)$,

$x^{\beta+\varepsilon}s^{x}(1-s)^{\beta+\varepsilon}\leq\left(\frac{x}{x+\beta+\varepsilon}\right)^{x}\left(\frac{x(\beta+\varepsilon)}{x+\beta+\varepsilon}\right)^{\beta+\varepsilon}\leq(\beta+\varepsilon)^{\beta+\varepsilon}.$

Combining this with (2.6) and (2.1), we deduce

$\displaystyle x^{\beta+\varepsilon}\int_{0}^{1}s^{x}\omega(s)ds$
$\displaystyle=$ $\displaystyle
x^{\beta+\varepsilon}\int_{1-\frac{1}{x}}^{1}s^{x}\omega(s)ds+x^{\beta+\varepsilon}\int_{0}^{1-\frac{1}{x}}s^{x}\omega(s)ds$
$\displaystyle\lesssim$
$\displaystyle\frac{\hat{\omega}(1-\frac{1}{x})}{(1-(1-\frac{1}{x}))^{\beta+\varepsilon}}+\int_{0}^{1-\frac{1}{x}}x^{\beta+\varepsilon}s^{x}(1-s)^{\beta+\varepsilon}\frac{\omega(s)}{(1-s)^{\beta+\varepsilon}}ds$
$\displaystyle\lesssim$
$\displaystyle\frac{\hat{\omega}(1-\frac{1}{x})}{(1-(1-\frac{1}{x}))^{\beta+\varepsilon}}+\int_{0}^{1-\frac{1}{x}}\frac{\omega(s)}{(1-s)^{\beta+\varepsilon}}ds$
$\displaystyle\lesssim$
$\displaystyle\frac{\hat{\omega}(1-\frac{1}{x})}{(1-(1-\frac{1}{x}))^{\beta+\varepsilon}}\lesssim\frac{\hat{\omega}(1-\frac{1}{y})}{(1-(1-\frac{1}{y}))^{\beta+\varepsilon}}.$

Also,

$\displaystyle\frac{\hat{\omega}(1-\frac{1}{y})}{(1-(1-\frac{1}{y}))^{\beta+\varepsilon}}\thickapprox
y^{\beta+\varepsilon}\int_{1-\frac{1}{y}}^{1}s^{y}\omega(s)ds\lesssim
y^{\beta+\varepsilon}\int_{0}^{1}s^{y}\omega(s)ds.$

Thus for $2<x\leq y<\infty$, one gets

$\int_{0}^{1}s^{x}\omega(s)ds\lesssim\left(\frac{y}{x}\right)^{\beta+\varepsilon}\int_{0}^{1}s^{y}\omega(s)ds.$
(2.10)

When $0<x\leq y\leq 2$, (2.10) also holds because

$\int_{0}^{1}s^{x}\omega(s)ds\thickapprox\int_{0}^{1}s^{y}\omega(s)ds\thickapprox\int_{0}^{1}\omega(s)ds.$

When $0<x\leq 2<y<\infty$, there is a small enough positive constant $c$ with
$2<2+c<y$. Then

$\displaystyle\int_{0}^{1}s^{x}\omega(s)ds\thickapprox\int_{0}^{1}s^{2+c}\omega(s)ds$
$\displaystyle\lesssim\left(\frac{y}{2+c}\right)^{\beta+\varepsilon}\int_{0}^{1}s^{y}\omega(s)ds$
$\displaystyle\lesssim\left(\frac{y}{x}\right)^{\beta+\varepsilon}\int_{0}^{1}s^{y}\omega(s)ds.$

Consequently, $\beta+\varepsilon\in\\{\eta:\ \eta\ \text{satisfies (v) in
Lemma \ref{property of doub w}}\\}$. Then

$\displaystyle\inf\\{\eta:\ \eta\ \text{satisfies (v) in Lemma \ref{property
of doub w}}\\}$ $\displaystyle\leq$ $\displaystyle\inf\\{\beta:\ \beta\
\text{satisfies (ii) in Lemma \ref{property of doub w}}\\}.$ (2.11)

Conversely, suppose (v) in Lemma F holds for some positive constant $\eta$.
Then $\omega\in\hat{\mathcal{D}}$ and hence there exits a positive constant
$C=C(\omega)>1$ such that

$\hat{\omega}(r)\leq C\hat{\omega}\left(\frac{r+1}{2}\right),$

for all $r\in(0,1)$. Let $t\in(\frac{3}{4},1)$ and $t_{n}=1-2^{n}(1-t)$,
$n=0,1,\cdots,N-1$. Here $N=N(t)$ is the largest positive integer such that
$t_{N-1}>0$. We set $t_{N}=0$. Then

$\displaystyle\int_{0}^{t}s^{\frac{1}{1-t}}\omega(s)ds$
$\displaystyle\thickapprox\sum_{n=0}^{N-1}\int_{t_{n+1}}^{t_{n}}s^{\frac{1}{1-t}}\omega(s)ds$
$\displaystyle\lesssim\sum_{n=0}^{N-1}\left(t_{n}^{\frac{1}{1-t}}C^{n}\hat{\omega}(t)\right)$
$\displaystyle\lesssim\left(\sum_{n=0}^{N-1}2^{-2^{n}}C^{n}\right)\hat{\omega}(t)\lesssim\hat{\omega}(t),$

where we use

$\displaystyle t_{n}^{\frac{1}{1-t}}$
$\displaystyle=\left(\big{(}1-2^{n}(1-t)\big{)}^{\frac{1}{2^{n}(1-t)}}\right)^{2^{n}}\lesssim
2^{-2^{n}}$

for all $n=0,1,\cdots,N$ and $t\in(\frac{3}{4},1)$. Hence,

$\displaystyle\int_{0}^{1}s^{\frac{1}{1-t}}\omega(s)ds=\int_{0}^{t}s^{\frac{1}{1-t}}\omega(s)ds+\int_{t}^{1}s^{\frac{1}{1-t}}\omega(s)ds\lesssim\hat{\omega}(t).$

Hence, for $\frac{4}{3}\leq x\leq y<\infty$, we get

$\displaystyle\frac{\hat{\omega}(1-\frac{1}{x})}{(1-(1-\frac{1}{x}))^{\eta}}$
$\displaystyle\lesssim x^{\eta}\int_{0}^{1}s^{x}\omega(s)ds\lesssim
y^{\eta}\int_{0}^{1}s^{y}\omega(s)ds$
$\displaystyle=\frac{\int_{0}^{1}s^{\frac{1}{1-(1-\frac{1}{y})}}\omega(s)ds}{(1-(1-\frac{1}{y}))^{\eta}}\lesssim\frac{\hat{\omega}(1-\frac{1}{y})}{(1-(1-\frac{1}{y}))^{\eta}},$

which gives

$\frac{\hat{\omega}(r)}{(1-r)^{\eta}}\lesssim\frac{\hat{\omega}(t)}{(1-t)^{\eta}}$

for all $1/4\leq r\leq t<1$. Joining this with some elementary estimates, we
see that $\eta$ satisfies (2.1) for all $0\leq r\leq t<1$. This yields

$\displaystyle\inf\\{\beta:\ \beta\ \text{satisfies (ii) in Lemma
\ref{property of doub w}}\\}$ $\displaystyle\leq$ $\displaystyle\inf\\{\eta:\
\eta\ \text{satisfies (v) in Lemma \ref{property of doub w}}\\}.$ (2.12)

By (2.9), (2.2) and (2.2), we get the desired result. ∎

For $\omega\in\hat{\mathcal{D}}$, denoted by $U(\omega)$ the infimum in Lemma
2.1. Generally speaking, (2.1), (2.2), (2.3) and (2.4) are not true when
$\beta=\gamma=\lambda=\eta=U(\omega)$. For example, when $\omega(z)=1$, (2.2)
and (2.3) do not hold when $\gamma=\lambda=U(\omega)=1$.

For a radial weight $\omega$, by [26, Lemma B], $\omega\in\check{\mathcal{D}}$
if and only if there exists $\alpha=\alpha(\omega)>0$ such that

$\frac{\hat{\omega}(t)}{(1-t)^{\alpha}}\lesssim\frac{\hat{\omega}(r)}{(1-r)^{\alpha}}$
(2.13)

for all $0\leq r\leq t<1$. Let $L(\omega)$ be the supremum of the set of these
parameters $\alpha$. There also exists $\omega\in\check{\mathcal{D}}$ such
that (2.13) does not hold when $\alpha=L(\omega)$. For example, set
$\omega(t)=(1-t)\left(2\log\frac{e}{1-t}-1\right)$. Then
$\hat{\omega}(t)=(1-t)^{2}\log\frac{e}{1-t}$ and $L(\omega)=2$ , but (2.13)
does not hold for $\alpha=2$ and this $\hat{\omega}$.

Note that $\mathcal{D}=\hat{\mathcal{D}}\cap\check{\mathcal{D}}$. Then for any
$\omega\in\mathcal{D}$, both $U(\omega)$ and $L(\omega)$ are well defined. It
is also clear that $0<L(\omega)\leq U(\omega)<\infty.$

###  2.3. Double integral estimates for $B_{p}(\omega)$ spaces

In this subsection, we give a complete description of radial weights $\omega$
such that double integral estimates for $B_{p}(\omega)$ spaces hold.

For real parameters $p$ and nonnegative functions $\omega$ on $\mathbb{D}$, we
will write $\omega_{[p]}(z)=(1-|z|^{2})^{p}\omega(z)$ and
$dA_{p}(z)=(1-|z|^{2})^{p}dA(z)$ for convenience. Now we give Theorem 2.2, one
side of double integral estimates for $B_{p}(\omega)$ which always holds. It
is worth mentioning that the weight function $\omega$ in the following theorem
is not necessarily radial.

######  Theorem 2.2.

Suppose $p>0$, $\tau>-1$ and $\omega$ is a nonnegative function on
$\mathbb{D}$. Let $\sigma$ be a real number such that $\omega_{[\sigma]}$ is a
weight. Then

$\displaystyle\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f(z)-f(\zeta)|^{p}}{|1-\overline{\zeta}z|^{4+\tau+\sigma}}\omega(\zeta)dA_{\tau}(z)dA_{\sigma}(\zeta)\gtrsim\int_{\mathbb{D}}|f^{\prime}(z)|^{p}\omega(z)dA_{p-2}(z)$
(2.14)

for all $f\in H(\mathbb{D})$.

###### Proof.

For $f\in H(\mathbb{D})$ and $\zeta\in\mathbb{D}$, by a change of variable
$z=\varphi_{\zeta}(u)$, we get

$\displaystyle J_{1}(\zeta):$
$\displaystyle=(1-|\zeta|^{2})^{\sigma}\int_{\mathbb{D}}\frac{|f(z)-f(\zeta)|^{p}}{|1-\overline{\zeta}z|^{4+\tau+\sigma}}dA_{\tau}(z)$
$\displaystyle=\frac{1}{(1-|\zeta|^{2})^{2}}\int_{\mathbb{D}}\frac{|f\circ\varphi_{\zeta}(u)-f\circ\varphi_{\zeta}(0)|^{p}}{|1-\overline{\zeta}u|^{\tau-\sigma}}dA_{\tau}(u).$
(2.15)

From Proposition 4.5 and Lemma 4.30 in [33],

$\displaystyle|1-\overline{\zeta}u|\thickapprox|1-\overline{\zeta}\eta|,1-|u|^{2}\thickapprox
1-|\eta|^{2}\thickapprox|1-\overline{u}\eta|,$ (2.16)

for all $\zeta\in\mathbb{D}$, and all $u$ and $\eta$ in $\mathbb{D}$
satisfying $\rho(u,\eta)<1/2$. The comparison constants in (2.16) are
independent of $\zeta$, $u$ and $\eta$. These facts allow us to apply a
Littlewood-Paley estimate for Bergman spaces with nonradial weights
$\omega_{\zeta}(u)=(1-|u|^{2})^{\tau}/{|1-\overline{\zeta}u|^{\tau-\sigma}}$
(see Theorem A and its proof in [9]). We get

$\displaystyle\int_{\mathbb{D}}\frac{|f\circ\varphi_{\zeta}(u)-f\circ\varphi_{\zeta}(0)|^{p}}{|1-\overline{\zeta}u|^{\tau-\sigma}}dA_{\tau}(u)$
$\displaystyle\geq$ $\displaystyle
C\int_{\mathbb{D}}\frac{|(f\circ\varphi_{\zeta})^{\prime}(u)|^{p}}{|1-\overline{\zeta}u|^{\tau-\sigma}}dA_{\tau+p}(u),$
(2.17)

where $C$ is a positive constant independent of $\zeta$ and $f$. Bearing in
mind (2.15), (2.16), (2.3), the change of variable with $u=\varphi_{\zeta}(z)$
and the sub-mean value property for $|f|^{p}$, we deduce

$\displaystyle J_{1}(\zeta)$
$\displaystyle\gtrsim\frac{1}{(1-|\zeta|^{2})^{2}}\int_{\mathbb{D}}\frac{|(f\circ\varphi_{\zeta})^{\prime}(u)|^{p}}{|1-\overline{\zeta}u|^{\tau-\sigma}}dA_{\tau+p}(u)$
$\displaystyle\thickapprox\int_{\mathbb{D}}|f^{\prime}(z)|^{p}\frac{(1-|\zeta|^{2})^{\sigma}}{|1-\overline{\zeta}z|^{\sigma+\tau+4}}dA_{\tau+p}(z)$
$\displaystyle\gtrsim(1-|\zeta|^{2})^{p-4}\int_{\Delta(\zeta,1/2)}|f^{\prime}(z)|^{p}dA(z)$
$\displaystyle\gtrsim(1-|\zeta|^{2})^{p-2}|f^{\prime}(\zeta)|^{p}$

for all $\zeta\in\mathbb{D}$. This implies (2.14). The proof is complete. ∎

The conditions of $\tau$ and $\sigma$ in Theorem 2.2 are only used to ensure
the convergence of the integrals in the left-hand of (2.14).

Now we give the other side of double integral estimates for $B_{p}(\omega)$
spaces as follows.

######  Theorem 2.3.

Suppose $p>0$ and ${\omega_{[p-2]}}$ is a radial weight. Then the following
conditions are equivalent:

  1. (i)

there exist real numbers $\sigma$ and $\tau$ such that

$\displaystyle\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f(z)-f(\zeta)|^{p}}{|1-\overline{\zeta}z|^{4+\tau+\sigma}}\omega(\zeta)dA_{\tau}(z)A_{\sigma}(\zeta)\lesssim\int_{\mathbb{D}}|f^{\prime}(z)|^{p}\omega(z)dA_{p-2}(z)$
(2.18)

for all $f\in H(\mathbb{D})$;

  2. (ii)

${\omega_{[p-2]}}\in\hat{\mathcal{D}}$.

To understand well the existence of parameters $\sigma$ and $\tau$ in Theorem
2.3, we prove the following result which implies Theorem 2.3.

######  Theorem 2.4.

Suppose $p>0$ and $\omega$ is a radial nonnegative function on $\mathbb{D}$.
Then the following statements hold:

  * (i)

if ${\omega_{[p-2]}}$ is a weight and there exist real numbers $\sigma$ and
$\tau$ such that (2.18) holds for all $f\in H(\mathbb{D})$, then
${\omega_{[p-2]}}\in\hat{\mathcal{D}}$;

  * (ii)

if ${\omega_{[p-2]}}\in\hat{\mathcal{D}}$, then (2.18) holds for all $f\in
H(\mathbb{D})$ when

$\min\\{\sigma,\tau\\}>p-2,\,\,\,\tau>\max\\{U({\omega_{[p-2]}})-p-1,-1\\}.$

###### Proof.

(i) Let ${\omega_{[p-2]}}$ be a weight and there exist real numbers $\sigma$
and $\tau$ such that (2.18) holds for all $f\in H(\mathbb{D})$. This forces
$\tau>-1$ and $\omega_{[\sigma]}$ is a weight. If $\sigma>\tau$, then

$\displaystyle\frac{(1-|z|)^{\sigma}(1-|\zeta|)^{\sigma}\omega(\zeta)}{|1-\overline{\zeta}z|^{4+2\sigma}}$
$\displaystyle\leq$
$\displaystyle\frac{(1-|z|)^{\tau}(1-|\zeta|)^{\sigma}\omega(\zeta)}{|1-\overline{\zeta}z|^{4+\tau+\sigma}}$
$\displaystyle\leq$
$\displaystyle\frac{(1-|z|)^{\tau}(1-|\zeta|)^{\tau}\omega(\zeta)}{|1-\overline{\zeta}z|^{4+2\tau}}$

for all $z$, $\zeta\in\mathbb{D}$. Thus, without loss of generality, we can
assume that $\sigma=\tau$. Checking the proof of Theorem 2.2, we get that

$\displaystyle J_{1}(\zeta)$
$\displaystyle=(1-|\zeta|^{2})^{\sigma}\int_{\mathbb{D}}\frac{|f(z)-f(\zeta)|^{p}}{|1-\overline{\zeta}z|^{4+\tau+\sigma}}dA_{\tau}(z)$
$\displaystyle\gtrsim\int_{\mathbb{D}}|f^{\prime}(z)|^{p}\frac{(1-|\zeta|^{2})^{\tau}dA_{\tau+p}(z)}{|1-\overline{\zeta}z|^{2\tau+4}}$
(2.19)

for all $\zeta\in\mathbb{D}$. It follows from (2.18), (2.19) and the Fubini
theorem that

$\displaystyle\int_{\mathbb{D}}|f^{\prime}(z)|^{p}\int_{\mathbb{D}}\frac{\omega(\zeta)}{|1-\overline{\zeta}z|^{2\tau+4}}dA_{\tau}(\zeta)dA_{p+\tau}(z)$
$\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{D}}|f^{\prime}(z)|^{p}\omega(z)dA_{p-2}(z)$ (2.20)

for all $f\in H(\mathbb{D})$. Take $f^{\prime}(z)=z^{n}$ in (2.20), where
$n=1,2,\cdots$. Bear in mind $2\tau+3>0$ and Lemma C. Then

$\displaystyle\int_{0}^{1}r^{np+1}(1-r)^{p-2}\omega(r)dr$ (2.21)
$\displaystyle\gtrsim$
$\displaystyle\int_{0}^{1}t^{np+1}(1-t)^{p+\tau}\int_{0}^{1}\frac{\omega(r)(1-r)^{\tau}r}{(1-tr)^{2\tau+3}}drdt$
$\displaystyle\thickapprox$
$\displaystyle\int_{0}^{1}\omega(r)(1-r)^{\tau}\left(\int_{0}^{1}\frac{t^{np+1}(1-t)^{p+\tau}}{(1-tr)^{2\tau+3}}dt\right)dr.$

Also,

$\displaystyle\int_{0}^{1}\frac{t^{np+1}(1-t)^{p+\tau}}{(1-tr)^{2\tau+3}}dt$
$\displaystyle\geq$
$\displaystyle\int_{\sqrt{r}}^{\frac{1+\sqrt{r}}{2}}\frac{t^{np+1}(1-t)^{p+\tau}}{(1-tr)^{2\tau+3}}dt$
$\displaystyle\gtrsim$ $\displaystyle r^{\frac{np+1}{2}}(1-r)^{p-\tau-2}$

for all $r\in[0,1)$ and for all positive integers $n$. By this and (2.21),
there exists a constant $M\in(1,\infty)$ such that

$\int_{0}^{1}r^{\frac{np+1}{2}}{\omega_{[p-2]}}(r)dr\leq
M\int_{0}^{1}r^{np+1}{\omega_{[p-2]}}(r)dr$ (2.22)

for all positive integers $n$. For $b\in\mathbb{R}$, let $E(b)$ be the integer
with $E(b)\leq b<E(b)+1$. Next we use (v) in Lemma F to show
${\omega_{[p-2]}}\in\hat{\mathcal{D}}$.

Let $\frac{p+1}{2}\leq x\leq y<\infty$. Write $n_{0}=E(\frac{2x-1}{p})$ and
$k=E(\log_{2}\frac{2y}{n_{0}p})+1$ for convenience. Then

$\frac{n_{0}p+1}{2}\leq x<\frac{(n_{0}+1)p+1}{2},\ \text{and}\ \
y<2^{k-1}n_{0}p.$

Combining these with (2.22), we deduce

$\displaystyle\int_{0}^{1}r^{x}{\omega_{[p-2]}}(r)dr$ $\displaystyle\leq$
$\displaystyle\int_{0}^{1}r^{\frac{n_{0}p+1}{2}}{\omega_{[p-2]}}(r)dr$ (2.23)
$\displaystyle\leq$ $\displaystyle
M^{k}\int_{0}^{1}r^{2^{k-1}n_{0}p+1}{\omega_{[p-2]}}(r)dr$ $\displaystyle\leq$
$\displaystyle
M\left(\frac{2y}{n_{0}p}\right)^{\log_{2}M}\int_{0}^{1}r^{y}{\omega_{[p-2]}}(r)dr$
$\displaystyle\leq$ $\displaystyle
C_{M}\left(\frac{y}{x}\right)^{\log_{2}M}\int_{0}^{1}r^{y}{\omega_{[p-2]}}(r)dr,$

where $C_{M}=M\sup\limits_{n_{0}\geq
1}(\frac{n_{0}p+p+1}{n_{0}p})^{\log_{2}M}$ is independent of $x$ and $y$.

Let $0<x\leq y\leq\frac{p+1}{2}$. Clearly,

$\displaystyle\int_{0}^{1}r^{x}{\omega_{[p-2]}}(r)dr$
$\displaystyle\thickapprox\int_{0}^{1}r^{y}{\omega_{[p-2]}}(r)dr$
$\displaystyle\lesssim\left(\frac{y}{x}\right)^{\log_{2}M}\int_{0}^{1}r^{y}{\omega_{[p-2]}}(r)dr.$

Let $0<x<\frac{p+1}{2}<y<\infty$. Using (2.23), we get

$\displaystyle\int_{0}^{1}r^{x}{\omega_{[p-2]}}(r)dr$
$\displaystyle\thickapprox\int_{0}^{1}r^{\frac{p+1}{2}}{\omega_{[p-2]}}(r)dr$
$\displaystyle\lesssim
C_{M}\left(\frac{y}{x}\right)^{\log_{2}M}\int_{0}^{1}r^{y}{\omega_{[p-2]}}(r)dr.$

Consequently,

$\int_{0}^{1}r^{x}{\omega_{[p-2]}}(r)dr\lesssim\left(\frac{y}{x}\right)^{\log_{2}M}\int_{0}^{1}r^{y}{\omega_{[p-2]}}(r)dr$

for all $0<x\leq y<\infty$. Hence Lemma F yields
${\omega_{[p-2]}}\in\hat{\mathcal{D}}$.

(ii) Suppose ${\omega_{[p-2]}}\in\hat{\mathcal{D}}$,

$\min\\{\sigma,\tau\\}>p-2,\ \text{and}\ \
\tau>\max\\{U({\omega_{[p-2]}})-p-1,-1\\}.$ (2.24)

Of course, ${\omega_{[p-2]}}$ is a weight. Then $\omega_{[\sigma]}$ is also a
weight. Next we show that (2.18) holds for all $f\in H(\mathbb{D})$.

Let $\sigma\geq\tau$. Following the proof of Theorem 2.2 and using Theorem B,
we obtain

$\displaystyle\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f(z)-f(\zeta)|^{p}}{|1-\overline{\zeta}z|^{4+\tau+\sigma}}\omega(\zeta)dA_{\tau}(z)A_{\sigma}(\zeta)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{D}}\frac{\omega(\zeta)}{(1-|\zeta|^{2})^{2}}dA(\zeta)\int_{\mathbb{D}}\frac{|f\circ\varphi_{\zeta}(u)-f\circ\varphi_{\zeta}(0)|^{p}}{|1-\overline{\zeta}u|^{\tau-\sigma}}dA_{\tau}(u)$
$\displaystyle\leq$ $\displaystyle
2^{\sigma-\tau}\int_{\mathbb{D}}\frac{\omega(\zeta)}{(1-|\zeta|^{2})^{2}}dA(\zeta)\int_{\mathbb{D}}|f\circ\varphi_{\zeta}(u)-f\circ\varphi_{\zeta}(0)|^{p}dA_{\tau}(u)$
$\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{D}}\frac{\omega(\zeta)}{(1-|\zeta|^{2})^{2}}dA(\zeta)\int_{\mathbb{D}}|(f\circ\varphi_{\zeta}(u))^{\prime}|^{p}dA_{\tau+p}(u)$
$\displaystyle\thickapprox$
$\displaystyle\int_{\mathbb{D}}|f^{\prime}(z)|^{p}\left(\int_{\mathbb{D}}\frac{(1-|\zeta|^{2})^{\tau}\omega(\zeta)dA(\zeta)}{|1-\overline{\zeta}z|^{2\tau+4}}\right)dA_{\tau+p}(z).$
(2.25)

Let $\sigma<\tau$. For $\zeta\in\mathbb{D}$, write
$\eta_{\zeta}(u)=|1-\overline{\zeta}u|^{\sigma-\tau}(1-|u|^{2})^{\tau}$. Note
that $\sigma>p-2>-2$. Clearly, there exists $s_{0}\in[-1,0)$ and $t_{0}>0$
such that if $s>s_{0}$ and $t>t_{0}$, then

$s+\tau>-1,\ 2+s+t>0,\ t>\tau,\ \sigma+s>-2.$

It follows from Lemma D that

$\displaystyle\int_{\mathbb{D}}\frac{\eta_{\zeta}(u)(1-|u|^{2})^{s}dA(u)}{|1-z\overline{u}|^{2+s+t}}$
$\displaystyle=\int_{\mathbb{D}}\frac{(1-|u|^{2})^{s+\tau}dA(u)}{|1-z\overline{u}|^{2+s+t}|1-\overline{\zeta}u|^{\tau-\sigma}}$
$\displaystyle\leq C\frac{\eta_{\zeta}(z)}{(1-|z|^{2})^{t}},$

where $C$ is a positive constant independent of $\zeta$. Using Theorem B and
checking its proof in [9], we get that there exits another positive constant
$C$ independent of $\zeta$ such that

$\displaystyle\
\int_{\mathbb{D}}\frac{|f\circ\varphi_{\zeta}(u)-f\circ\varphi_{\zeta}(0)|^{p}}{|1-\overline{\zeta}u|^{\tau-\sigma}}dA_{\tau}(u)$
$\displaystyle\leq
C\int_{\mathbb{D}}\frac{|(f\circ\varphi_{\zeta}(u))^{\prime}|^{p}(1-|u|^{2})^{p}}{|1-\overline{\zeta}u|^{\tau-\sigma}}dA_{\tau}(u)$
$\displaystyle=C\int_{\mathbb{D}}|f^{\prime}(z)|^{p}\frac{(1-|\zeta|^{2})^{\sigma+2}}{|1-\overline{\zeta}z|^{\sigma+\tau+4}}dA_{\tau+p}(z).$

This yields

$\displaystyle\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f(z)-f(\zeta)|^{p}}{|1-\overline{\zeta}z|^{4+\tau+\sigma}}\omega(\zeta)dA_{\tau}(z)A_{\sigma}(\zeta)$
$\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{D}}|f^{\prime}(z)|^{p}\left(\int_{\mathbb{D}}\frac{(1-|\zeta|^{2})^{\sigma}\omega(\zeta)dA(\zeta)}{|1-\overline{\zeta}z|^{\sigma+\tau+4}}\right)dA_{\tau+p}(z).$
(2.26)

Write $x=\min\\{\sigma,\tau\\}$. It follows from (2.25) and (2.26) that

$\displaystyle\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f(z)-f(\zeta)|^{p}}{|1-\overline{\zeta}z|^{4+\tau+\sigma}}\omega(\zeta)dA_{\tau}(z)A_{\sigma}(\zeta)$
$\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{D}}|f^{\prime}(z)|^{p}\left(\int_{\mathbb{D}}\frac{(1-|\zeta|^{2})^{x}\omega(\zeta)dA(\zeta)}{|1-\overline{\zeta}z|^{\tau+x+4}}\right)dA_{\tau+p}(z)$
$\displaystyle:=$ $\displaystyle\int_{\mathbb{D}}|f^{\prime}(z)|^{p}d\mu(z).$
(2.27)

By (2.24), $\tau+x+3>p>0$. For any $a\in\mathbb{D}\backslash\\{0\\}$, Lemma C
yields

$\displaystyle\mu(S(a))=$
$\displaystyle\int_{S(a)}\left(\int_{\mathbb{D}}\frac{(1-|\zeta|^{2})^{x}\omega(\zeta)dA(\zeta)}{|1-\overline{\zeta}z|^{\tau+x+4}}\right)dA_{\tau+p}(z)$
$\displaystyle\thickapprox$
$\displaystyle(1-|a|)\int_{|a|}^{1}(1-r)^{\tau+p}\int_{0}^{|a|}\frac{(1-t)^{x}\omega(t)dt}{(1-rt)^{\tau+x+3}}dr$
$\displaystyle+(1-|a|)\int_{|a|}^{1}(1-r)^{\tau+p}\int_{|a|}^{1}\frac{(1-t)^{x}\omega(t)dt}{(1-rt)^{\tau+x+3}}dr$
$\displaystyle:=$ $\displaystyle(1-|a|)(I_{1}(a)+I_{2}(a)).$ (2.28)

If $|a|<r<1$ and $0<t<|a|$, then

$1>\frac{1-t}{1-rt}>\frac{1-|a|}{1-r|a|}>\frac{1-|a|}{1-|a|^{2}}>\frac{1}{2}.$

Note that $\tau>U({\omega_{[p-2]}})-p-1$. By Lemma F, we have

$\displaystyle I_{1}(a)$ $\displaystyle\thickapprox$
$\displaystyle\int_{|a|}^{1}(1-r)^{p+\tau}\int_{0}^{|a|}\frac{\omega(t)(1-t)^{p-2}}{(1-t)^{\tau+p+1}}dtdr$
(2.29) $\displaystyle\lesssim$
$\displaystyle\frac{\widehat{{\omega_{[p-2]}}}(a)}{(1-|a|)^{\tau+p+1}}\int_{|a|}^{1}(1-r)^{p+\tau}dr$
$\displaystyle\thickapprox$ $\displaystyle\widehat{{\omega_{[p-2]}}}(a).$

Due to $x>p-2$ and $\tau+p>-1$,

$\int_{0}^{1}\frac{(1-r)^{\tau+p}}{(1-rt)^{\tau+x+3}}dr\thickapprox\frac{1}{(1-t)^{x+2-p}}$

for all $t\in(0,1)$. By this and the Fubini theorem, one gets

$\displaystyle I_{2}(a)=$
$\displaystyle\int_{|a|}^{1}\omega(t)(1-t)^{x}\int_{|a|}^{1}\frac{(1-r)^{\tau+p}}{(1-rt)^{\tau+x+3}}drdt$
$\displaystyle\lesssim$
$\displaystyle\int_{|a|}^{1}\omega(t)(1-t)^{p-2}\int_{0}^{1}\frac{(1-r)^{\tau+p}(1-t)^{x+2-p}}{(1-rt)^{\tau+x+3}}drdt$
$\displaystyle\thickapprox$ $\displaystyle\widehat{{\omega_{[p-2]}}}(a).$

Note that

$\int_{S(a)}\omega_{[p-2]}(z)dA(z)\thickapprox(1-|a|)\widehat{{\omega_{[p-2]}}}(a)$

for all $a\in\mathbb{D}\backslash\\{0\\}$. Consequently,

$\sup_{a\in\mathbb{D}\backslash\\{0\\}}\frac{\mu(S(a))}{\int_{S(a)}\omega_{[p-2]}(z)dA(z)}<\infty.$

It follows from (2.27) and Theorem E that

$\displaystyle\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f(z)-f(\zeta)|^{p}}{|1-\overline{\zeta}z|^{4+\tau+\sigma}}\omega(\zeta)dA_{\tau}(z)A_{\sigma}(\zeta)$
$\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{D}}|f^{\prime}(z)|^{p}\left(\int_{\mathbb{D}}\frac{(1-|\zeta|^{2})^{x}\omega(\zeta)dA(\zeta)}{|1-\overline{\zeta}z|^{\tau+x+4}}\right)dA_{\tau+p}(z)$
$\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{D}}|f^{\prime}(z)|^{p}{\omega_{[p-2]}}(z)dA(z)$

for all $f\in H(\mathbb{D})$. The proof is complete. ∎

For a nonnegative function $\eta$ on [0, 1), we say that $\eta$ is essentially
decreasing on [0, 1) if $\eta(t_{1})\gtrsim\eta(t_{2})$ for all $0\leq
t_{1}<t_{2}<1$. Also, $\eta$ is said to be essentially increasing on [0, 1) if
$\eta(t_{1})\lesssim\eta(t_{2})$ for all $0\leq t_{1}<t_{2}<1$.

Note that $\mathcal{D}\subsetneqq\hat{\mathcal{D}}$. For $0<p<\infty$ and a
radial nonnegative function $\omega$ on $\mathbb{D}$, if we assume
${\omega_{[p-2]}}\in\mathcal{\mathcal{D}}$, then the range of parameters
$\sigma$ or $\tau$ in (ii) of Theorem 2.4 should be larger. Because of this
observation, we give the following result.

######  Proposition 2.5.

Suppose $p>0$ and $\omega$ is a radial nonnegative function on $\mathbb{D}$.
If ${\omega_{[p-2]}}\in\mathcal{D}$, then (2.18) holds for all $f\in
H(\mathbb{D})$ when

$\tau>-1,\sigma>-2,\min\\{\sigma,\tau\\}>p-2-L({\omega_{[p-2]}}),\tau>U({\omega_{[p-2]}})-p-1,$
(2.30)

and $\omega_{[\sigma]}$ is a weight.

###### Proof.

Checking the proof of (ii) of Theorem 2.4 step by step, we see that (2.28) and
(2.29) hold. Also, $I_{2}(a)\lesssim\widehat{{\omega_{[p-2]}}}(a)$ for all
$a\in\mathbb{D}\backslash\\{0\\}$ when $\min\\{\sigma,\tau\\}>p-2$. Hence, it
suffices to prove $I_{2}(a)\lesssim\widehat{{\omega_{[p-2]}}}(a)$ for all
$a\in\mathbb{D}\backslash\\{0\\}$ when
$p-2-L({\omega_{[p-2]}})<\min\\{\sigma,\tau\\}\leq p-2$. Recall that
$x=\min\\{\sigma,\tau\\}$. Write

$\displaystyle I_{2}(a)$
$\displaystyle=\int_{|a|}^{1}\omega(t)\int_{|a|}^{t}\frac{(1-r)^{\tau+p}(1-t)^{x}}{(1-rt)^{\tau+x+3}}drdt$
$\displaystyle\ \ \ \
+\int_{|a|}^{1}\omega(t)\int_{t}^{1}\frac{(1-r)^{\tau+p}(1-t)^{x}}{(1-rt)^{\tau+x+3}}drdt$
$\displaystyle:=I_{21}(a)+I_{22}(a).$ (2.31)

Clearly,

$I_{22}(a)\lesssim\int_{|a|}^{1}\omega(t)(1-t)^{-\tau-3}\int_{t}^{1}(1-r)^{\tau+p}drdt\thickapprox\widehat{{\omega_{[p-2]}}}(a).$
(2.32)

The Fubini theorem yields

$\displaystyle I_{21}(a)$
$\displaystyle=\int_{|a|}^{1}\int_{r}^{1}\frac{(1-r)^{\tau+p}(1-t)^{x}\omega(t)}{(1-rt)^{\tau+x+3}}dtdr$
$\displaystyle\lesssim\int_{|a|}^{1}\int_{r}^{1}(1-r)^{p-x-3}(1-t)^{x}\omega(t)dtdr.$

If $x=p-2$, due to ${\omega_{[p-2]}}\in\check{\mathcal{D}}$, it follows from
(2.13) that

$\displaystyle
I_{21}(a)\lesssim\int_{|a|}^{1}(1-r)^{-1}\widehat{{\omega_{[p-2]}}}(r)dr\lesssim\widehat{{\omega_{[p-2]}}}(a).$
(2.33)

If $p-2-L({\omega_{[p-2]}})<x<p-2$, we can choose a small enough positive
number $\varepsilon$ such that $p-2-L({\omega_{[p-2]}})+\varepsilon<x$ and
$\frac{\widehat{{\omega_{[p-2]}}}(t)}{(1-t)^{L({\omega_{[p-2]}})-\varepsilon}}$
is essentially decreasing. An integration by parts gives that

$\displaystyle\int_{|a|}^{1}\omega(t)(1-t)^{x}dt$ $\displaystyle=$
$\displaystyle(1-|a|)^{x-p+2}\widehat{{\omega_{[p-2]}}}(a)-(x-p+2)\int_{|a|}^{1}\widehat{{\omega_{[p-2]}}}(t)(1-t)^{x-p+1}dt.$

Consequently,

$\displaystyle I_{21}(a)$
$\displaystyle\lesssim\int_{|a|}^{1}\int_{r}^{1}(1-r)^{p-x-3}(1-t)^{x}\omega(t)dtdr$
$\displaystyle\thickapprox\int_{|a|}^{1}\omega(t)(1-t)^{x}dt\int_{|a|}^{t}(1-r)^{p-x-3}dr$
$\displaystyle\lesssim(1-|a|)^{p-x-2}\int_{|a|}^{1}\omega(t)(1-t)^{x}dt$
$\displaystyle\lesssim\widehat{{\omega_{[p-2]}}}(a)+(1-|a|)^{p-x-2}\int_{|a|}^{1}\frac{\widehat{{\omega_{[p-2]}}}(t)(1-t)^{L({\omega_{[p-2]}})-\varepsilon}}{(1-t)^{L({\omega_{[p-2]}})-\varepsilon}(1-t)^{p-x-1}}dt$
$\displaystyle\lesssim\widehat{{\omega_{[p-2]}}}(a).$ (2.34)

Joining (2.31), (2.32), (2.33) and (2.34), we see that
$I_{2}(a)\lesssim\widehat{{\omega_{[p-2]}}}(a)$ for all
$a\in\mathbb{D}\backslash\\{0\\}$ when
$p-2-L({\omega_{[p-2]}})<\min\\{\sigma,\tau\\}\leq p-2$. The proof is
finished. ∎

Theorem 2.2 and Theorem 2.3 give complete descriptions of nonnegative radial
weight functions $\omega$ such that the double integral estimates for
$B_{p}(\omega)$ hold. Theorem 2.4 and Proposition 2.5 explain further
parameters $\sigma$ and $\tau$ in (2.18).

As stated in Section 1 (cf. [13, Theorem 2.2]), when $p>1$ and $\alpha\leq
1/2$, there is a double integral characterization for $B_{p}(\alpha)$. Theorem
2.6 below covers this case exactly. In fact, Theorem 2.6 gives that this
double integral characterization for $B_{p}(\alpha)$ exists for all possible
$p$ and $\alpha$.

######  Theorem 2.6.

Suppose $p>\max\\{0,2\alpha\\}$, $\beta>\max\\{-1,-2\alpha-1\\}$ and

$\tau>\max\\{-1,-2+2\alpha,-1-2\alpha\\}.$

Then

$\displaystyle\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f(z)-f(\zeta)|^{p}}{|1-\overline{\zeta}z|^{3+\beta+\tau+2\alpha}}(1-|\zeta|^{2})^{\beta}(1-|z|^{2})^{\tau}dA(z)dA(\zeta)$
$\displaystyle\thickapprox$
$\displaystyle\int_{\mathbb{D}}|f^{\prime}(z)|^{p}(1-|z|^{2})^{p-1-2\alpha}dA(z)$

for all $f\in H(\mathbb{D})$.

###### Proof.

Take $\omega(z)=(1-|z|^{2})^{1-2\alpha}$ and $\sigma=\beta-1+2\alpha$. Then
$U({\omega_{[p-2]}})=L({\omega_{[p-2]}})=p-2\alpha$. Applying Theorem 2.2 and
Proposition 2.5 to this $\omega$, we obtain

$\displaystyle\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f(z)-f(\zeta)|^{p}}{|1-\overline{\zeta}z|^{4+\tau+\sigma}}(1-|\zeta|^{2})^{1-2\alpha+\sigma}(1-|z|^{2})^{\tau}dA(z)dA(\zeta)$
$\displaystyle\thickapprox$
$\displaystyle\int_{\mathbb{D}}|f^{\prime}(z)|^{p}(1-|z|^{2})^{p-1-2\alpha}dA(z)$

for all $f\in H(\mathbb{D})$. The desired result follows. ∎

##  3\. Geometric characterizations of functions in $B_{p}(\omega)$ spaces

In this section, based on double integral estimates for $B_{p}(\omega)$
spaces, we give some geometric descriptions of functions in $B_{p}(\omega)$
spaces when $0<p<\infty$ and ${\omega_{[p-2]}}\in\hat{\mathcal{D}}$.

For any $z$, $\zeta$ in $\mathbb{D}$, denote by $\beta(z,\zeta)$ the distance
between $z$ and $\zeta$ in the Bergman metric. It is well known that

$\beta(z,\zeta)=\frac{1}{2}\log\frac{1+\rho(z,\zeta)}{1-\rho(z,\zeta)}.$

Suppose $z\in\mathbb{D}$ and $r>0$. Let
$D(z,r)=\\{\zeta\in\mathbb{D}:\beta(z,\zeta)<r\\}$ and let $|D(z,r)|$ be the
area of $D(z,r)$ with respect to the measure $dA$. For $f\in H(\mathbb{D})$,
denote by $|f(D(z,r))|$ the area of the image of $D(z,r)$ under $f$. Recall
that

$O_{r}(f)(z)=\sup\\{|f(z)-f(\zeta)|:\ \ \zeta\in D(z,r)\\}$

is the oscillation of $f$ at $z$ in the Bergman metric. The mean oscillation
of $f$ at $z$ in the Bergman metric is

$MO_{r}(f)(z)=\frac{1}{|D(z,r)|}\int_{D(z,r)}|f(\zeta)-\widehat{f_{r}}(z)|dA(\zeta),$

where $\widehat{f_{r}}$ is the averaging function given by

$\widehat{f_{r}}(z)=\frac{1}{|D(z,r)|}\int_{D(z,r)}f(u)dA(u).$

Applying Theorem 2.4, we obtain the following conclusion which generalizes
Theorem B in [34] from the Besov space $B_{p}$, $p>1$, to the Besov type space
$B_{p}(\omega)$ for $0<p<\infty$ and ${\omega_{[p-2]}}\in\hat{\mathcal{D}}$.
The proof of Theorem B in [34] used the Hölder inequality. Theorem 3.1 below
includes the case of $0<p<1$ for which the Hölder inequality is invalid.

######  Theorem 3.1.

Suppose $f\in H(\mathbb{D})$, $0<r<\infty$, $0<p<\infty$, and
${\omega_{[p-2]}}\in\hat{\mathcal{D}}$. Then the following conditions are
equivalent:

  1. (i)

$f\in B_{p}(\omega)$;

  2. (ii)

the function $z\mapsto|f(D(z,r))|^{1/2}$ belongs to
$L^{p}(\mathbb{D},wd\lambda)$;

  3. (iii)

$O_{r}(f)\in L^{p}(\mathbb{D},wd\lambda)$;

  4. (iv)

$MO_{r}(f)\in L^{p}(\mathbb{D},wd\lambda)$.

###### Proof.

$(i)\Rightarrow(ii)$. Let $f\in B_{p}(\omega)$. By the proof of Theorem 2.4,
there exist real numbers $x$ and $\tau$ such that

$\displaystyle\int_{\mathbb{D}}|f^{\prime}(z)|^{p}\left(\int_{\mathbb{D}}\frac{(1-|\zeta|^{2})^{x}\omega(\zeta)dA(\zeta)}{|1-\overline{\zeta}z|^{\tau+x+4}}\right)dA_{\tau+p}(z)$
$\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{D}}|f^{\prime}(z)|^{p}{\omega_{[p-2]}}(z)dA(z)<\infty.$
(3.1)

Note that for $z\in\mathbb{D}$ the integral
$\int_{D(z,r)}|f^{\prime}(\zeta)|^{2}dA(\zeta)$, familiar from the theory of
the classical Dirichlet space, is equal to the area of $f(D(z,r))$ counting
multiplicities. The quantity $|f(D(z,r))|$ disregards multiplicities. Hence

$\displaystyle|f(D(z,r))|\leq\int_{D(z,r)}|f^{\prime}(\zeta)|^{2}dA(\zeta).$
(3.2)

It is well known that $|D(z,r)|\thickapprox(1-|z|^{2})^{2}$ for all
$z\in\mathbb{D}$. For $\zeta\in D(z,r)$, then $D(\zeta,r)\subseteq D(z,2r)$,
and $1-|z|\thickapprox 1-|\zeta|\thickapprox|1-\overline{z}\zeta|$. Hence,

$\displaystyle|f^{\prime}(\zeta)|^{p}$
$\displaystyle\lesssim\frac{1}{|D(\zeta,r)|}\int_{D(\zeta,r)}|f^{\prime}(u)|^{p}dA(u)$
$\displaystyle\lesssim\frac{1}{|D(z,r)|}\int_{D(z,2r)}|f^{\prime}(u)|^{p}dA(u)$
(3.3)

for all $\zeta\in D(z,r)$. Using (3.3), we deduce

$\displaystyle\left[\int_{D(z,r)}|f^{\prime}(\zeta)|^{2}dA(\zeta)\right]^{p/2}$
$\displaystyle\leq$
$\displaystyle|D(z,r)|^{p/2}\sup\\{|f^{\prime}(\zeta)|^{p}:\ \ \zeta\in
D(z,r)\\}$ $\displaystyle\lesssim$
$\displaystyle(1-|z|)^{p-2}\int_{D(z,2r)}|f^{\prime}(u)|^{p}dA(u)$
$\displaystyle\thickapprox$
$\displaystyle\int_{D(z,2r)}\frac{|f^{\prime}(u)|^{p}(1-|u|^{2})^{\tau+p}(1-|z|^{2})^{x+2}}{|1-\overline{z}u|^{4+\tau+x}}dA(u)$

for all $z\in\mathbb{D}$. By this estimate, the Fubini theorem and (3.1), we
get

$\displaystyle\int_{\mathbb{D}}\left[\int_{D(z,r)}|f^{\prime}(\zeta)|^{2}dA(\zeta)\right]^{p/2}\omega(z)d\lambda(z)$
$\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f^{\prime}(u)|^{p}(1-|u|^{2})^{\tau+p}(1-|z|^{2})^{x+2}}{|1-\overline{z}u|^{4+\tau+x}}dA(u)\omega(z)d\lambda(z)$
$\displaystyle\thickapprox$
$\displaystyle\int_{\mathbb{D}}|f^{\prime}(u)|^{p}(1-|u|^{2})^{\tau+p}\int_{\mathbb{D}}\frac{(1-|z|^{2})^{x}\omega(z)}{|1-\overline{z}u|^{4+\tau+x}}dA(z)dA(u)$
$\displaystyle<$ $\displaystyle\infty.$

Combining this with (3.2), we get that the function
$z\mapsto|f(D(z,r))|^{1/2}$ belongs to $L^{p}(\mathbb{D},wd\lambda)$.

$(ii)\Rightarrow(i)$. It follows from [6, p. 322] that
$(1-|z|^{2})|f^{\prime}(z)|\lesssim|f(D(z,r))|^{1/2}$ for all
$z\in\mathbb{D}$. Thus this implication is true.

$(i)\Rightarrow(iii)$. Let $f\in B_{p}(\omega)$. By Theorem 2.4, there exist
real numbers $\sigma$ and $\tau$ such that

$\displaystyle\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f(z)-f(\zeta)|^{p}}{|1-\overline{\zeta}z|^{4+\tau+\sigma}}\omega(\zeta)dA_{\tau}(z)A_{\sigma}(\zeta)\lesssim\int_{\mathbb{D}}|f^{\prime}(z)|^{p}\omega(z)dA_{p-2}(z)$
(3.4)

For $z$ and $\zeta$ in $\mathbb{D}$, the subharmonicity of the function
$u\mapsto|f(u)-f(z)|^{p}$ on $\mathbb{D}$ yields

$|f(\zeta)-f(z)|^{p}\leq\frac{C}{|D(\zeta,r)|}\int_{D(\zeta,r)}|f(u)-f(z)|^{p}dA(u),$

where $C$ is a positive constant depending only on $r$ (cf. [33, Proposition
4.13]). If $\zeta\in D(z,r)$, then $D(\zeta,r)\subseteq D(z,2r)$, and
$1-|z|\thickapprox 1-|\zeta|\thickapprox|1-\overline{z}\zeta|$. Consequently,

$\displaystyle[O_{r}(f)]^{p}(z)$
$\displaystyle\lesssim\frac{1}{|D(z,2r)|}\int_{D(z,2r)}|f(u)-f(z)|^{p}dA(u)$
$\displaystyle\thickapprox\int_{D(z,2r)}|f(u)-f(z)|^{p}\frac{(1-|z|^{2})^{2+\sigma}(1-|u|^{2})^{\tau}}{|1-z\overline{u}|^{4+\tau+\sigma}}dA(u)$
$\displaystyle\lesssim\int_{\mathbb{D}}|f(u)-f(z)|^{p}\frac{(1-|z|^{2})^{2+\sigma}(1-|u|^{2})^{\tau}}{|1-z\overline{u}|^{4+\tau+\sigma}}dA(u).$

Combining this with (3.4), we get $O_{r}(f)\in L^{p}(\mathbb{D},wd\lambda)$.

$(iii)\Rightarrow(iv)$. Let $O_{r}(f)\in L^{p}(\mathbb{D},wd\lambda)$. For
$z\in\mathbb{D}$, a direct calculation gives

$\displaystyle MO_{r}(f)(z)$
$\displaystyle\leq\frac{1}{|D(z,r)|^{2}}\int_{D(z,r)}\int_{D(z,r)}|f(\zeta)-f(u)|dA(\zeta)dA(u)$
$\displaystyle\leq\frac{2}{|D(z,r)|}\int_{D(z,r)}|f(\zeta)-f(z)|dA(\zeta)$
$\displaystyle\leq 2O_{r}(f)(z).$

Hence $MO_{r}(f)\in L^{p}(\mathbb{D},wd\lambda)$.

$(iv)\Rightarrow(i)$. By [34, p. 330], $(1-|z|^{2})|f^{\prime}(z)|\lesssim
MO_{r}(f)(z)$ for all $z\in\mathbb{D}$. Then the desired result holds. We
finish the proof. ∎

##  4\. Hankel type operators related to $B_{p}(\omega)$ spaces with radial
Békollé-Bonami weights

In this section, applying double integral estimates for Besov type spaces
$B_{p}(\omega)$, we characterize the boundedness of Hankel type operators
related to $B_{p}(\omega)$ with radial Békollé-Bonami weights. This result is
new for $B_{p}(\alpha)$ when $1<p<\infty$ and $1/2<\alpha<p/2$. Our result on
$B_{2}(\omega)$ also improves a previous result from the literature. The
compactness of these Hankel type operators is also considered.

For $p>1$ and $s>-1$, the Békollé-Bonami class $\mathcal{B}_{p,s}$ consists of
nonnegative and integrable functions $\eta$ on $\mathbb{D}$ with the property
that there exists a positive constant $C$ satisfying

$\left(\int_{S(a)}\eta(z)dA(z)\right)\left(\int_{S(a)}\left(\frac{\eta(z)}{(1-|z|^{2})^{s}}\right)^{-\frac{p^{\prime}}{p}}dA_{s}(z)\right)^{\frac{p}{p^{\prime}}}\leq
C(A_{s}(S(a)))^{p}$

for all Carleson boxes $S(a)$. Here and in what follows, $p^{\prime}$ is the
real number with $1/p+1/p^{\prime}=1$. D. Békollé and A. Bonami [11] proved
that $\eta\in\mathcal{B}_{p,s}$ if and only if the Bergman projection

$P_{s}f(z)=(s+1)\int_{\mathbb{D}}\frac{f(\zeta)}{(1-\overline{\zeta}z)^{s+2}}dA_{s}(\zeta)$

is bounded from $L^{p}(\mathbb{D},\eta dA)$ to $A^{p}_{\eta}$. Note that
weights in $\mathcal{B}_{p,s}$ are not necessarily radial. From Proposition 6
in [24], any radial weight in $\mathcal{B}_{p,s}$ belongs to $\mathcal{D}$.
See [10, 11] for these weights.

For $s>-1$, by the Bergman projection $P_{s}$, it is possible to define a
(small) Hankel type operator $h_{s,f}$ on $\mathcal{P}$ by

$h_{s,f}g=\overline{P_{s}(f\overline{g})},\ \ \ g\in\mathcal{P}.$

We refer to [27, 33] for more results on Hankel type operators.

Let $p>0$ and let $\omega_{[p-2]}$ be a weight. Denote by $S_{p}(\omega)$ the
Sobolev type space of smooth functions $u:\mathbb{D}\rightarrow\mathbb{C}$
such that

$\|u\|_{S_{p}(\omega)}=|u(0)|+\left(\int_{\mathbb{D}}|\nabla
u(z)|^{p}(1-|z|^{2})^{p-2}\omega(z)dA(z)\right)^{1/p}<\infty.$

Also, if $\omega(z)=(1-|z|^{2})^{1-2\alpha}$, then we write $S_{p}(\omega)$ as
$S_{p}(\alpha)$. Clearly, $B_{p}(\omega)$ is a subset of all analytic
functions in $S_{p}(\omega)$. Denote by $\mathcal{P}$ the class of polynomials
on $\mathbb{D}$. If $w$ is radial, it is well known that $\mathcal{P}$ is
dense in $B_{p}(\omega)$ (cf. [18]). For $s>-1$, consider the operator
$\widetilde{P}_{s}$ given by

$\widetilde{P}_{s}u(z)=u(0)+\int_{\mathbb{D}}\frac{\partial u}{\partial
w}(w)\frac{1-(1-\overline{w}z)^{1+s}}{\overline{w}(1-\overline{w}z)^{1+s}}(1-|w|^{2})^{s}dA(w).$

Then one can define a (small) Hankel type operator on $\mathcal{P}$ with
certain symbol $f$ by

$h_{f}^{s}g=\overline{\widetilde{P}_{s}(f\overline{g})},\ \ \
g\in\mathcal{P}.$

As explained by D. Blasi and J. Pau [13, p. 402], $\widetilde{P}_{1-2\alpha}$
defines a projection from $S_{p}(\alpha)$ to $B_{p}(\alpha)$. For $f$ analytic
in $\mathbb{D}$, the boundedness of $h_{f}^{1-2\alpha}$ from $B_{p}(\alpha)$
to $S_{p}(\alpha)$ was also studied in [13]. Considering the same topic with
weights $\omega$, one should define a Hankel type operator by the projection
from $S_{p}(\omega)$ to $B_{p}(\omega)$. From this way, the results in Section
2 will not be used in the study. Our purpose of this section is to present how
to apply double integral estimates for $B_{p}(\omega)$ obtained in Section 2.
Thus we still focus on the operator $h_{f}^{s}$ in this section. In
particular, some special cases of our main result can complete or improve some
results before.

###  4.1. The boundedness of Hankel type operators from $B_{p}(\omega)$ to
$S_{p}(\omega)$

In this subsection, we investigate the boundedness of
$h^{s}_{f}:B_{p}(\omega)\to S_{p}(\omega)$ with analytic symbol $f$.

We first recall Proposition 5 in [26] as follows.

######  Proposition G.

Let $0<p<\infty$, $\omega\in\hat{\mathcal{D}}$ and write
$\tilde{w}(r)=\hat{\omega}(r)/(1-r)$ for all $0\leq r<1$. Then
$\omega\in\check{\mathcal{D}}$ if and only if
$\|f\|_{A^{p}_{\omega}}\thickapprox\|f\|_{A^{p}_{\tilde{\omega}}}$ for all
$f\in H(\mathbb{D})$.

Remark . In fact, by the proof of the proposition above in [26], if
$\omega\in\mathcal{D}$, then $\tilde{\omega}\in\mathcal{R}$. Here
$\mathcal{R}$ is the regular class of weights consisting of all
$\omega\in\mathcal{D}$ such that

$\hat{\omega}(r)\thickapprox(1-r)\omega(r),\,\,\,r\in(0,1).$

Also, if $\omega\in\mathcal{D}$, then
$\hat{\tilde{\omega}}(r)\thickapprox\hat{\omega}(r)$ for all $0\leq r<1$. Then
it is also clear that $U(\omega)=U(\tilde{\omega})$ and
$L(\omega)=L(\tilde{\omega})$.

We also need the following result (cf. [21, Lemma 3.1]).

######  Lemma H.

Let $0<p<\infty$ and $\omega\in\hat{\mathcal{D}}$. Then there is real number
$\lambda_{0}=\lambda_{0}(\omega)$ such that for any $\lambda\geq\lambda_{0}$
and each $a\in\mathbb{D}$ the function
$F_{a,p}(z)=\left(\frac{1-|a|^{2}}{1-\overline{a}z}\right)^{(\lambda+1)/p}$ is
analytic in $\mathbb{D}$ and satisfies

$|F_{a,p}(z)|\thickapprox 1,\ \ z\in S(a),\ \ a\in\mathbb{D},$

and

$\|F_{a,p}\|^{p}_{A^{p}_{\omega}}\thickapprox\int_{S(a)}\omega dA,\ \
a\in\mathbb{D}.$

Now we give the following lemma.

######  Lemma 4.1.

Suppose $1<p<\infty$, $-1<s<\infty$, and $\eta$ is a radial weight in
$\mathcal{B}_{p,s}$. Let $f\in A_{\eta}^{p}$ and let $h_{s,f}$ be a bounded
operator from $B_{p}(\eta_{[2-p]})$ to $L_{\eta}^{p}$. Then

$\sup_{a\in\mathbb{D}}(1-|a|^{2})|f(a)|<\infty.$

###### Proof.

Bear in mind that any radial weight in $\mathcal{B}_{p,s}$ belongs to
$\mathcal{D}$. By Proposition G, we have

$\displaystyle\|h_{s,f}g\|_{L^{p}(\mathbb{D},\eta dA)}$
$\displaystyle=\|P_{s}(f\overline{g})\|_{L^{p}(\mathbb{D},\eta dA)}$
$\displaystyle=\|P_{s}(f\overline{g})\|_{A_{\eta}^{p}}$
$\displaystyle\thickapprox\|P_{s}(f\overline{g})\|_{A_{\tilde{\eta}}^{p}}=\|h_{s,f}g\|_{L^{p}(\mathbb{D},\tilde{\eta}dA)}$
(4.1)

for all $g\in B_{p}(\eta_{[2-p]})$. For any $a\in\mathbb{D}$, set

$g_{a}(z)=\frac{z^{n+1}}{(1-\overline{a}z)^{n+1}},\ \ \ z\in\mathbb{D},$

where $n$ is a positive integer. By Lemma H, if $n$ is large enough,

$\displaystyle\|g_{a}\|_{B_{p}(\eta_{[2-p]})}\lesssim\frac{(1-|a|)^{\frac{1}{p}}\hat{\eta}(a)^{\frac{1}{p}}}{(1-|a|)^{n+2}}$
(4.2)

for all $a\in\mathbb{D}$. Note that $h_{s,f}$ is bounded from
$B_{p}(\eta_{[2-p]})$ to $L_{\eta}^{p}$, and $P_{s}f=\overline{h_{s,f}1}$ .
Hence $P_{s}f$ belongs to $A^{p}_{\eta}$. Then $P_{s}f=f$. For any fixed
$\lambda>-1$, we obtain

$\displaystyle f^{(n+1)}(a)$ $\displaystyle=$
$\displaystyle(s+1)\left(\int_{\mathbb{D}}\frac{f(z)(1-|z|^{2})^{s}dA(z)}{(1-a\overline{z})^{s+2}}\right)^{(n+1)}$
$\displaystyle=$ $\displaystyle
C(n,s)(s+1)\int_{\mathbb{D}}\frac{\overline{z}^{n+1}f(z)}{(1-a\overline{z})^{n+1}}\frac{1}{(1-a\overline{z})^{2+s}}(1-|z|^{2})^{s}dA(z)$
$\displaystyle=$ $\displaystyle
C(n,s,\lambda)(s+1)\int_{\mathbb{D}}\frac{\overline{z}^{n+1}f(z)}{(1-a\overline{z})^{n+1}}\left(\int_{\mathbb{D}}\frac{dA_{\lambda}(w)}{(1-w\overline{z})^{2+s}(1-a\overline{w})^{\lambda+2}}\right)dA_{s}(z)$
$\displaystyle=$ $\displaystyle
C(n,s,\lambda)\int_{\mathbb{D}}\left((s+1)\int_{\mathbb{D}}\frac{\overline{z}^{n+1}f(z)}{(1-a\overline{z})^{n+1}}\frac{dA_{s}(z)}{(1-w\overline{z})^{2+s}}\right)\frac{dA_{\lambda}(w)}{(1-a\overline{w})^{\lambda+2}}$
$\displaystyle=$ $\displaystyle
C(n,s,\lambda)\int_{\mathbb{D}}\overline{h_{s,f}(g_{a})(w)}\frac{dA_{\lambda}(w)}{(1-a\overline{w})^{\lambda+2}},$

where $C(n,s)$ is a constant depending on $n$ and $s$, and $C(n,s,\lambda)$ is
a constant depending on $n$, $s$ and $\lambda$. It follows from Hölder’s
inequality, the boundedness of $h_{s,f}$, (4.1) and (4.2) that

$\displaystyle(1-|a|^{2})^{n+2}|f^{(n+1)}(a)|$ $\displaystyle\lesssim$
$\displaystyle(1-|a|^{2})^{n+2}\|h_{s,f}g_{a}\|_{L^{p}(\mathbb{D},\tilde{\eta}dA)}\left(\int_{\mathbb{D}}\frac{(1-|w|^{2})^{{p^{\prime}}\lambda}dA(w)}{\tilde{\eta}(w)^{\frac{{p^{\prime}}}{p}}|1-a\overline{w}|^{(\lambda+2){p^{\prime}}}}\right)^{\frac{1}{{p^{\prime}}}}$
$\displaystyle\lesssim$ $\displaystyle\|h_{s,f}\|_{B_{p}(\eta_{[2-p]})\to
L^{p}(\mathbb{D},\eta
dA)}(1-|a|)^{\frac{1}{p}}{\hat{\eta}}(a)^{\frac{1}{p}}\left(\int_{0}^{1}\frac{(1-r)^{{p^{\prime}}\lambda}dr}{\tilde{\eta}(r)^{\frac{{p^{\prime}}}{p}}(1-|a|r)^{(\lambda+2){p^{\prime}}-1}}\right)^{\frac{1}{{p^{\prime}}}}$
(4.3)

for all $a\in\mathbb{D}$. Since $\tilde{\eta}\in\mathcal{R}$, there exist
$-1<\alpha,\beta<\infty$ such that $\frac{\tilde{\eta}(t)}{(1-t)^{\alpha}}$
and $\frac{\tilde{\eta}(t)}{(1-t)^{\beta}}$ are essentially increasing and
essentially decreasing respectively (cf. [23, p. 11]). Note that
$-\frac{p^{\prime}\beta}{p}-2p^{\prime}+2=(\beta+2)(1-p^{\prime})<0$. If
$\lambda$ is large enough, then

$\displaystyle\int_{0}^{|a|}\frac{(1-r)^{{p^{\prime}}\lambda}dr}{{\tilde{\eta}}(r)^{p^{\prime}/p}(1-|a|r)^{(\lambda+2){p^{\prime}}-1}}$
$\displaystyle\lesssim$
$\displaystyle\left(\frac{(1-|a|)^{\beta}}{{\tilde{\eta}}(|a|)}\right)^{p^{\prime}/p}\int_{0}^{|a|}\frac{(1-r)^{{p^{\prime}}\lambda-\frac{{p^{\prime}}\beta}{p}}dr}{(1-|a|r)^{(\lambda+2){p^{\prime}}-1}}$
$\displaystyle\lesssim$
$\displaystyle\left(\frac{(1-|a|)^{\beta}}{{\tilde{\eta}}(|a|)}\right)^{p^{\prime}/p}\int_{0}^{|a|}(1-r)^{-\frac{{p^{\prime}}\beta}{p}-2{p^{\prime}}+1}dr$
$\displaystyle\lesssim$
$\displaystyle\frac{(1-|a|)^{-2{p^{\prime}}+2}}{{\tilde{\eta}}(a)^{p^{\prime}/p}},$

and

$\displaystyle\int_{|a|}^{1}\frac{(1-r)^{{p^{\prime}}\lambda}dr}{{\tilde{\eta}}(r)^{p^{\prime}/p}(1-|a|r)^{(\lambda+2){p^{\prime}}-1}}\leq$
$\displaystyle\left(\frac{(1-|a|)^{\alpha}}{{\tilde{\eta}}(|a|)}\right)^{p^{\prime}/p}\int_{|a|}^{1}\frac{(1-r)^{{p^{\prime}}\lambda-\frac{{p^{\prime}}\alpha}{p}}dr}{(1-|a|r)^{(\lambda+2){p^{\prime}}-1}}$
$\displaystyle\lesssim$
$\displaystyle\frac{(1-|a|)^{-2{p^{\prime}}+2}}{{\tilde{\eta}}(a)^{p^{\prime}/p}}$

for all $a\in\mathbb{D}$. Combining this with (4.3), we get

$\sup_{a\in\mathbb{D}}(1-|a|^{2})^{n+2}|f^{(n+1)}(a)|<\infty,$

which is equivalent to

$\sup_{a\in\mathbb{D}}(1-|a|^{2})|f(a)|<\infty.$

The proof is complete. ∎

The following theorem is the main result in this section and the double
integral estimates for Besov type spaces will be used in its proof.

######  Theorem 4.2.

Suppose $1<p<\infty$, $-1<s<\infty$, $f\in H(\mathbb{D})$ and
$\eta\in\mathcal{B}_{p,s}$ is a radial weight. Let $\eta,p,s$ satisfy further
one of the following conditions:

  1. (a)

$U(\eta)<p-1$, $p\leq 2$, and
$U(\eta)-\frac{L(\eta)}{p-1}<p-1+\frac{sp}{p-1}$;

  2. (b)

$U(\eta)<p-1$, $p\geq 2$, and $U(\eta)-L(\eta)<ps+1$;

  3. (c)

$p-1\leq U(\eta)<ps+p$, $s>0$, and $L(\eta)>p-1-ps$.

Then the following conditions are equivalent:

  1. (i)

$h_{s,f}:B_{p}(\eta_{[2-p]})\to L^{p}(\mathbb{D},\eta dA)$ is a bounded
operator;

  2. (ii)

$h_{F}^{s}:B_{p}(\eta_{[2-p]})\to S_{p}(\eta_{[2-p]})$ is a bounded operator,
where $F$ is in $H(\mathbb{D})$ satisfying $F^{\prime}=f$ on $\mathbb{D}$;

  3. (iii)

$d\mu(z)=|f(z)|^{p}\eta(z)dA(z)$ is a $p$-Carleson measure for
$B_{p}(\eta_{[2-p]})$.

###### Proof.

$(i)\Leftrightarrow(ii)$. For $g\in\mathcal{P}$, it is easy to check that for
any $z\in\mathbb{D}$,

$\frac{\partial h_{F}^{s}g}{\partial z}(z)=0$

and

$\displaystyle\frac{\partial h_{F}^{s}g}{\partial\overline{z}}(z)$
$\displaystyle=(s+1)\overline{\int_{\mathbb{D}}\overline{g(w)}f(w)\frac{(1-|w|^{2})^{s}}{(1-\overline{w}z)^{s+2}}dA(w)}$
$\displaystyle=h_{s,f}g(z).$

Hence

$|\nabla h_{F}^{s}g(z)|\thickapprox|h_{s,f}g(z)|$

for all $z\in\mathbb{D}$. Then the equivalence between (i) and (ii) follows.

$(iii)\Rightarrow(i)$. Suppose $\mu$ is a $p$-Carleson measure for
$B_{p}(\eta_{[2-p]})$. Since $\eta\in\mathcal{B}_{p,s}$, the Bergman
projection $P_{s}:L^{p}(\mathbb{D},\eta dA)\to A_{\eta}^{p}$ is bounded. Then,
for any $g\in B_{p}(\eta_{[2-p]})$, we get

$\displaystyle\|h_{s,f}g\|_{L^{p}(\mathbb{D},\eta dA)}^{p}$
$\displaystyle=\|P_{s}(f\overline{g})\|_{L^{p}(\mathbb{D},\eta
dA)}^{p}\lesssim\|fg\|_{L^{p}(\mathbb{D},\eta
dA)}^{p}\lesssim\|g\|_{B_{p}(\eta_{[2-p]})}^{p}.$

Thus $h_{s,f}$ is bounded.

$(i)\Rightarrow(iii)$. Suppose $h_{s,f}:B_{p}(\eta_{[2-p]})\to
L^{p}(\mathbb{D},\eta dA)$ is bounded. Note that any radial weight in
$\mathcal{B}_{p,s}$ belongs to $\mathcal{D}$. By (4.1), Proposition G and the
remark after it, we have

$\tilde{\eta}\in\mathcal{R},\,\,\,L(\tilde{\eta})=L(\eta),\,\,\,U(\tilde{\eta})=U(\eta),$

and

$\|h_{s,f}\|_{B_{p}(\tilde{\eta}_{[2-p]})\to
L^{p}(\mathbb{D},\tilde{\eta}dA)}\thickapprox\|h_{s,f}\|_{B_{p}(\eta_{[2-p]})\to
L^{p}(\mathbb{D},\eta dA)}.$

Thus $h_{s,f}:\ B_{p}(\tilde{\eta}_{[2-p]})\to
L^{p}(\mathbb{D},\tilde{\eta}dA)$ is also a bounded operator. Here,

$\tilde{\eta}_{[2-p]}(z)=(1-|z|^{2})^{2-p}\tilde{\eta}(z)=(1-|z|^{2})^{2-p}\frac{\hat{\eta}(z)}{1-|z|},\,\,\,z\in\mathbb{D}.$

We claim that

$\|f\overline{g}-\overline{h_{s,f}g}\|_{L^{p}(\mathbb{D},\tilde{\eta}dA)}\lesssim\|g\|_{B_{p}(\tilde{\eta}_{[2-p]})}$

for all $g\in B_{p}(\tilde{\eta}_{[2-p]})$. By this claim, the boundedness of
$h_{s,f}:B_{p}(\tilde{\eta}_{[2-p]})\to L^{p}(\mathbb{D},\tilde{\eta}dA)$, and
Proposition G, we see that $\mu$ is a $p$-Carleson measure for
$B_{p}(\eta_{[2-p]})$. Thus, it suffices to prove the claim for case (a), case
(b) and case (c) respectively. As explained in Lemma 4.1, $P_{s}f=f$. This,
together with (i), yields

$\displaystyle
f(z)\overline{g(z)}-\overline{h_{s,f}g(z)}=(s+1)\int_{\mathbb{D}}\frac{f(w)\overline{(g(z)-g(w))}}{(1-\overline{w}z)^{2+s}}dA_{s}(w).$
(4.4)

for any $z\in\mathbb{D}$.

Case (a). Suppose

$\displaystyle U(\eta)<p-1,\,\,\,\,1<p\leq
2,\,\,\,U(\eta)-\frac{L(\eta)}{p-1}<p-1+\frac{sp}{p-1}.$ (4.5)

For $1<p<2$, by $L(\eta)\leq U(\eta)$ and
$U(\eta)-\frac{L(\eta)}{p-1}<p-1+\frac{sp}{p-1}$, we get
$U(\eta)>\frac{(p-1)^{2}+sp}{p-2}$. Note that $U(\eta)<p-1$. Then
$\frac{(p-1)^{2}+sp}{p-2}<p-1$; that is, $\frac{ps}{p-1}>-1$. Hence
$U(\eta)<p-1<p+\frac{ps}{p-1}$. For $p=2$, condition (4.5) also yields

$U(\eta)<1+2s+L(\eta)\leq 1+2s+U(\eta)<2+2s=p+\frac{ps}{p-1}.$

Thus for $1<p\leq 2$,

$U(\eta)<p-1<p+\frac{ps}{p-1}.$

Due to $s>-1$, it is clear that $U(\eta)<p+s$. By a simple calculation,
$U(\eta)<p-1$ is equivalent to

$(2+s)p<4+(p-2+s)p-(U(\eta)-1)(p-1)+p-2;$ (4.6)

$U(\eta)<p+s$ is equivalent to

$(p-2+s)p-(U(\eta)-1)(p-1)>U(\eta)-p-1;$ (4.7)

$U(\eta)<p+\frac{ps}{p-1}$ is equivalent to

$(p-2+s)p-(U(\eta)-1)(p-1)>-1;$ (4.8)

$U(\eta)-\frac{L(\eta)}{p-1}<p-1+\frac{sp}{p-1}$ is equivalent to

$(p-2+s)p-(U(\eta)-1)(p-1)>p-2-L(\eta).$ (4.9)

Note that $p>1$. By (4.6), (4.7), (4.8), (4.9) and $L(\eta)>0$, there exits
$\varepsilon>0$ such that

$\displaystyle(2+s)p\leq 4+(p-2+s)p-(U(\eta)-1+\varepsilon)(p-1)+p-2,$ (4.10)
$\displaystyle(p-2+s)p-(U(\eta)-1+\varepsilon)(p-1)>\max\\{U(\eta)-p-1,-1\\},$
(4.11)

and

$\displaystyle\min\left\\{(p-2+s)p-(U(\eta)-1+\varepsilon)(p-1),p-2\right\\}>p-2-L(\eta).$
(4.12)

$U(\eta)>0$ and (4.8) yield

$\displaystyle p-2+s>\frac{(U(\eta)-1)(p-1)-1}{p}>-1$ (4.13)

Because of $\tilde{\eta}\in\mathcal{R}$, for any positive number
$\varepsilon_{1}$, $\frac{\tilde{\eta}(t)}{(1-t)^{U(\eta)-1+\varepsilon_{1}}}$
is essentially increasing on $[0,1)$. Choose $\varepsilon_{1}\leq\varepsilon$.
Combining this with (4.11), we obtain

$\displaystyle\int_{\mathbb{D}}\frac{(1-|w|^{2})^{(p-2+s)p}}{\tilde{\eta}(w)^{p-1}}dA(w)$
$\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{D}}(1-|w|^{2})^{(p-2+s)p-(U(\eta)-1+\varepsilon_{1})(p-1)}dA(w)<\infty.$
(4.14)

It is known that $f=\overline{h_{s,f}1}\in A_{\eta}^{p}$. In fact, $f$ is also
in $A_{\tilde{\eta}}^{p}$. By Lemma 4.1,

$\displaystyle M_{f}:=\sup_{a\in\mathbb{D}}(1-|a|^{2})|f(a)|<\infty.$ (4.15)

By $1<p\leq 2$, (4.13), (4.1), (4.15), the Hölder inequality and the
essentially increasing property of
$\frac{\tilde{\eta}(t)}{(1-t)^{U(\eta)-1+\varepsilon}}$ on $[0,1)$, we deduce

$\displaystyle\left|\int_{\mathbb{D}}\frac{f(w)\overline{(g(z)-g(w))}}{(1-\overline{w}z)^{2+s}}dA_{s}(w)\right|^{p}$
$\displaystyle\leq$ $\displaystyle
M_{f}^{(2-p)p}\left(\int_{\mathbb{D}}\frac{|f(w)|^{p-1}|g(z)-g(w)|}{|1-\overline{w}z|^{2+s}}dA_{p-2+s}(w)\right)^{p}$
$\displaystyle\leq$ $\displaystyle
M_{f}^{(2-p)p}\|f\|_{A_{\tilde{\eta}}^{p}}^{p(p-1)}\int_{\mathbb{D}}\frac{|g(z)-g(w)|^{p}(1-|w|^{2})^{(p-2+s)p}}{|1-\overline{w}z|^{(2+s)p}{\tilde{\eta}}(w)^{p-1}}dA(w)$
$\displaystyle\lesssim$ $\displaystyle
M_{f}^{(2-p)p}\|f\|_{A_{\tilde{\eta}}^{p}}^{p(p-1)}\int_{\mathbb{D}}\frac{|g(z)-g(w)|^{p}(1-|w|^{2})^{(p-2+s)p-(U(\eta)-1+\varepsilon)(p-1)}}{|1-\overline{w}z|^{(2+s)p}}dA(w).$
(4.16)

Then, it follows from (4.4), (4.10) and (4.16) that

$\displaystyle\|f\overline{g}-\overline{h_{s,f}g}\|_{L^{p}(\mathbb{D},\tilde{\eta}dA)}^{p}$
$\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|g(z)-g(w)|^{p}(1-|w|^{2})^{(p-2+s)p-(U(\eta)-1+\varepsilon)(p-1)}}{|1-\overline{w}z|^{(2+s)p}}{\tilde{\eta}}(z)dA(w)dA(z)$
$\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|g(z)-g(w)|^{p}(1-|w|^{2})^{(p-2+s)p-(U(\eta)-1+\varepsilon)(p-1)}}{|1-\overline{w}z|^{4+(p-2+s)p-(U(\eta)-1+\varepsilon)(p-1)+p-2}}{\tilde{\eta}}(z)dA(w)dA(z).$
(4.17)

Bear in mind (4.11) and (4.12). Applying Proposition 2.5 by setting
$\omega(z)=\tilde{\eta}(z)(1-|z|^{2})^{2-p}$, $\sigma=p-2$ and
$\tau=(p-2+s)p-(U(\eta)-1+\varepsilon)(p-1)$, we see that

$\displaystyle\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|g(z)-g(w)|^{p}(1-|w|^{2})^{(p-2+s)p-(U(\eta)-1+\varepsilon)(p-1)}}{|1-\overline{w}z|^{4+(p-2+s)p-(U(\eta)-1+\varepsilon)(p-1)+p-2}}{\tilde{\eta}}(z)dA(w)dA(z)$
$\displaystyle\lesssim$
$\displaystyle\|g\|_{B_{p}({\tilde{\eta}}_{[2-p]})}^{p}.$

This, together with (4.17), yields that our claim holds.

Case (b). The result for $p=2$ has been proved in case (a). Now suppose

$\displaystyle U(\eta)<p-1,\,\,\,p>2,\,\,\,s>-1,\ \ U(\eta)-L(\eta)<ps+1.$
(4.18)

Note that $0<L(\eta)\leq U(\eta)<\infty$. Then $ps>U(\eta)-L(\eta)-1\geq-1$,
and hence $U(\eta)<p-1<p+ps$. Also, $2U(\eta)<2p-2<2p+ps$. By these
inequalities and (4.18), there exist real numbers $x$, $y$, and positive
number $\varepsilon$ such that

$\displaystyle U(\eta)\leq p-1-\varepsilon,$ (4.19)
$\displaystyle\max\\{U(\eta)-p+1+\varepsilon,\ \ 1-p\\}$ $\displaystyle<$
$\displaystyle(p^{\prime}+x)(p-1)$ $\displaystyle<$
$\displaystyle\min\left\\{ps+p+1-U(\eta),ps+1,ps-p+2+L(\eta)\right\\},$ (4.20)

and

$\displaystyle
2p-3-U(\eta)-\varepsilon<(y-(p^{\prime}+x))(p-1)<2p-3-U(\eta)+L(\eta)-\varepsilon.$
(4.21)

From (4.1), $p^{\prime}+x>-1$ and $ps-(p-1)(p^{\prime}+x)>-1$. Using Hölder’s
inequality, we get

$\displaystyle\left|\int_{\mathbb{D}}\frac{f(w)\overline{(g(z)-g(w))}}{(1-\overline{w}z)^{2+s}}dA_{s}(w)\right|^{p}$
$\displaystyle\leq$
$\displaystyle\left(\int_{\mathbb{D}}\frac{|f(w)|^{p^{\prime}}}{|1-\overline{w}z|^{y}}dA_{p^{\prime}+x}(w)\right)^{p-1}\int_{\mathbb{D}}\frac{|g(z)-g(w)|^{p}dA_{ps-(p-1)(p^{\prime}+x)}(w)}{|1-\overline{w}z|^{(2+s)p-(p-1)y}}.$
(4.22)

Note that $\frac{1}{\frac{p-1}{p-2}}+\frac{1}{p-1}=1$, $\frac{p-1}{p-2}>1$ and
$p-1>1$. It follows from the Hölder inequality again that

$\displaystyle\left(\int_{\mathbb{D}}\frac{|f(w)|^{p^{\prime}}}{|1-\overline{w}z|^{y}}dA_{p^{\prime}+x}(w)\right)^{p-1}$
$\displaystyle\leq$
$\displaystyle\|f\|_{A_{\tilde{\eta}}^{p}}^{p}\left(\int_{\mathbb{D}}\left(\frac{(1-|w|^{2})^{p^{\prime}+x}}{|1-\overline{w}z|^{y}}\right)^{\frac{p-1}{p-2}}{\tilde{\eta}}(w)^{-\frac{1}{p-2}}dA(w)\right)^{p-2}.$
(4.23)

Because of $\tilde{\eta}\in\mathcal{R}$,
$\frac{\tilde{\eta}(t)}{(1-t)^{U(\eta)-1+\varepsilon}}$ is essentially
increasing on $[0,1)$. Hence

$\displaystyle\sup\limits_{w\in\mathbb{D}}\frac{(1-|w|)^{U(\eta)-1+\varepsilon}}{\tilde{\eta}(w)}<\infty.$
(4.24)

Due to (4.1) and (4.21), we get

$\displaystyle\frac{(p^{\prime}+x)(p-1)-(U(\eta)-1+\varepsilon)}{p-2}>-1,$
(4.25)

and

$\displaystyle\frac{y(p-1)}{p-2}-\frac{(p^{\prime}+x)(p-1)-(U(\eta)-1+\varepsilon)}{p-2}>2,$
(4.26)

By (4.1), (4.24), (4.25), (4.26) and Lemma C, one gets

$\displaystyle\left(\int_{\mathbb{D}}\frac{|f(w)|^{p^{\prime}}}{|1-\overline{w}z|^{y}}dA_{p^{\prime}+x}(w)\right)^{p-1}$
$\displaystyle\lesssim$
$\displaystyle\|f\|_{A_{\tilde{\eta}}^{p}}^{p}\left(\int_{\mathbb{D}}\frac{(1-|w|^{2})^{\frac{(p^{\prime}+x)(p-1)-(U(\eta)-1+\varepsilon)}{p-2}}}{|1-\overline{w}z|^{\frac{y(p-1)}{p-2}}}dA(w)\right)^{p-2}$
$\displaystyle\lesssim$
$\displaystyle\frac{\|f\|_{A_{\tilde{\eta}}^{p}}^{p}}{(1-|z|)^{(y-(p^{\prime}+x))(p-1)+(U(\eta)-1+\varepsilon)-2(p-2)}}.$

Combining this with (4.1), we obtain

$\displaystyle\int_{\mathbb{D}}\left|\int_{\mathbb{D}}\frac{f(w)\overline{(g(z)-g(w))}}{(1-\overline{w}z)^{2+s}}dA_{s}(w)\right|^{p}\tilde{\eta}(z)dA(z)$
$\displaystyle\lesssim$
$\displaystyle\|f\|_{A_{\tilde{\eta}}^{p}}^{p}\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|g(z)-g(w)|^{p}}{|1-\overline{w}z|^{(2+s)p-(p-1)y}}dA_{ps-(p-1)(p^{\prime}+x)}(w)$
$\displaystyle\times\tilde{\eta}(z)dA_{2(p-2)-(y-(p^{\prime}+x))(p-1)-(U(\eta)-1+\varepsilon)}(z).$
(4.27)

Set $\omega(z)=\tilde{\eta}(z)(1-|z|^{2})^{2-p}$,
$\sigma=3(p-2)-(y-(p^{\prime}+x))(p-1)-(U(\eta)-1+\varepsilon)$, and
$\tau=ps-(p-1)(p^{\prime}+x)$. Note that (4.19) holds if and only if

$(2+s)p-(p-1)y\leq 4+\sigma+\tau.$

Then (4.1) gives

$\displaystyle\int_{\mathbb{D}}\left|\int_{\mathbb{D}}\frac{f(w)\overline{(g(z)-g(w))}}{(1-\overline{w}z)^{2+s}}dA_{s}(w)\right|^{p}\tilde{\eta}(z)dA(z)$
$\displaystyle\lesssim$
$\displaystyle\|f\|_{A_{\tilde{\eta}}^{p}}^{p}\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|g(z)-g(w)|^{p}}{|1-\overline{w}z|^{4+\sigma+\tau}}\omega(z)dA_{\tau}(w)dA_{\sigma}(z).$
(4.28)

By (4.1) and (4.21), condition (2.30) in Proposition 2.5 holds. By (4.21),
$L(\eta)=L(\tilde{\eta})$, and $\tilde{\eta}\in\mathcal{R}$, we get

$\displaystyle\int_{\mathbb{D}}\omega_{[\sigma]}(z)dA(z)$
$\displaystyle=\int_{\mathbb{D}}\tilde{\eta}(z)(1-|z|^{2})^{2(p-2)-(y-(p^{\prime}+x))(p-1)-(U(\eta)-1+\varepsilon)}dA(z)$
$\displaystyle\lesssim\int_{\mathbb{D}}\tilde{\eta}(z)(1-|z|^{2})^{-L(\eta)}dA(z)<\infty,$

which means that $\omega_{[\sigma]}$ is a weight. Thus, Proposition 2.5 yields

$\displaystyle\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|g(z)-g(w)|^{p}}{|1-\overline{w}z|^{4+\sigma+\tau}}\omega(z)dA_{\tau}(w)dA_{\sigma}(z)\lesssim\|g\|_{B_{p}({\tilde{\eta}}_{[2-p]})}^{p}.$
(4.29)

From (4.4), (4.1) and (4.29), our claim holds.

Case (c). Note that

$\displaystyle U(\eta)<ps+p,\,\,\,p>1,\,\,\,s>0,\,\,\,L(\eta)>p-1-ps.$

Then there exits a real number $x$ satisfying

$-1<x<\min\left\\{\frac{ps-U(\eta)+1}{p-1},\frac{ps-p+1}{p-1},\frac{ps-2p+2+L(\eta)}{p-1}\right\\}.$
(4.30)

Clearly, there is another real number $y$ such that

$0<y-x-2<\min\left\\{\frac{p}{p-1},\frac{L(\eta)}{p-1}\right\\}.$ (4.31)

Note that $p^{\prime}+x>x>-1$ and $ps-p-(p-1)x>-1$. From Hölder’s inequality,
(4.15), (4.31), and Lemma C, one gets

$\displaystyle\left|\int_{\mathbb{D}}\frac{f(w)\overline{(g(z)-g(w))}}{(1-\overline{w}z)^{2+s}}dA_{s}(w)\right|^{p}$
$\displaystyle\leq$
$\displaystyle\left(\int_{\mathbb{D}}\frac{|f(w)|^{p^{\prime}}}{|1-\overline{w}z|^{y}}dA_{{p^{\prime}}+x}(w)\right)^{p-1}\int_{\mathbb{D}}\frac{|g(z)-g(w)|^{p}}{|1-\overline{w}z|^{(2+s)p-(p-1)y}}dA_{ps-p-(p-1)x}(w)$
$\displaystyle\lesssim$ $\displaystyle
M_{f}^{p}\left(\int_{\mathbb{D}}\frac{dA_{x}(w)}{|1-\overline{w}z|^{y}}\right)^{p-1}\int_{\mathbb{D}}\frac{|g(z)-g(w)|^{p}}{|1-\overline{w}z|^{(2+s)p-(p-1)y}}dA_{ps-p-(p-1)x}(w)$
$\displaystyle\lesssim$ $\displaystyle
M_{f}^{p}(1-|z|)^{-(y-x-2)(p-1)}\int_{\mathbb{D}}\frac{|g(z)-g(w)|^{p}}{|1-\overline{w}z|^{(2+s)p-(p-1)y}}dA_{ps-p-(p-1)x}(w).$
(4.32)

Set $\tau=ps-p-(p-1)x$, $\sigma=p-2-(y-x-2)(p-1)$ and
$\omega(z)=\tilde{\eta}(z)(1-|z|^{2})^{2-p}$. Then
$(2+s)p-(p-1)y=4+\tau+\sigma$. Also, (4.30) and (4.31) yield condition (2.30)
in Proposition 2.5 holds. Note that $0<y-x-2<\frac{L(\eta)}{p-1}$ and
$\tilde{\eta}\in\mathcal{R}$. Then

$\displaystyle\int_{\mathbb{D}}\omega_{[\sigma]}(z)dA(z)$
$\displaystyle=\int_{\mathbb{D}}\tilde{\eta}(z)(1-|z|^{2})^{-(y-x-2)(p-1)}dA(z)$
$\displaystyle\lesssim\int_{\mathbb{D}}\tilde{\eta}(z)(1-|z|^{2})^{-L(\eta)}dA(z)<\infty;$

that is, $\omega_{[\sigma]}$ is a weight. By (4.1) and applying Proposition
2.5, we have

$\displaystyle\int_{\mathbb{D}}\left|\int_{\mathbb{D}}\frac{f(w)\overline{(g(z)-g(w))}}{(1-\overline{w}z)^{2+s}}dA_{s}(w)\right|^{p}{\tilde{\eta}}(z)dA(z)$
$\displaystyle\lesssim$ $\displaystyle
M_{f}^{p}\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|g(z)-g(w)|^{p}}{|1-\overline{w}z|^{(2+s)p-(p-1)y}}dA_{ps-p-(p-1)x}(w){\tilde{\eta}}(z)dA_{-(y-x-2)(p-1)}(z)$
$\displaystyle\thickapprox$ $\displaystyle
M_{f}^{p}\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|g(z)-g(w)|^{p}}{|1-\overline{w}z|^{4+\tau+\sigma}}\omega(z)dA_{\tau}(w)dA_{\sigma}(z)$
$\displaystyle\lesssim$ $\displaystyle
M_{f}^{p}\int_{\mathbb{D}}|g^{\prime}(z)|^{p}{\tilde{\eta}}(z)dA(z).$

Combining this with (4.4), we see that our claim also holds for this case. The
proof is complete. ∎

Remark.  First, the proof of case (c) is valid for
$U(\eta)<ps+p,\,\,\,p>1,\,\,\,s>0,\,\,\,L(\eta)>p-1-ps$. It is easy to see
that the case of $U(\eta)<p-1,\,\,\,p>1,\,\,\,s>0,\,\,\,L(\eta)>p-1-ps$ is
contained in case (a) and case (b). Thus we use the now version of case (c) in
the expressions of Theorem 4.2. In addition, by the proof of Theorem 4.2,
$h_{s,f}:B_{p}(\eta_{[2-p]})\to L^{p}(\mathbb{D},\eta dA)$ is a bounded
operator if and only if $|f(z)|^{p}\tilde{\eta}(z)dA(z)$ is a $p$-Carleson
measure for $B_{p}(\tilde{\eta}_{[2-p]})$. When $p>1$ and $\omega$ is a
Békollé-Bonami weight satisfying one more condition including the case of
$\omega$ in $\mathcal{R}$, the description of $p$-Carleson measures for
$B_{p}(\omega)$ can be found in [5].

Checking the proof of case (a) and case (b) in (i)$\Rightarrow$(iii) of
Theorem 4.2, if $\eta$ satisfies that $\frac{\hat{\eta}(t)}{(1-t)^{U(\eta)}}$
is essentially increasing on [0, 1), we can choose $\varepsilon=0$ in the
proof. For such case, the condition $U(\eta)<p-1$ in both case (a) and case
(b) in Theorem 4.2 can be replaced by $U(\eta)\leq p-1$. We state this result
as follows.

######  Proposition 4.3.

Let $1<p<\infty$, $-1<s<\infty$, $f\in H(\mathbb{D})$ and let
$\eta\in\mathcal{B}_{p,s}$ be a radial weight such that
$\frac{\hat{\eta}(t)}{(1-t)^{U(\eta)}}$ is essentially increasing on [0, 1).
Suppose $\eta,p,s$ satisfies further one of the following conditions:

  1. (a)

$U(\eta)\leq p-1$, $p\leq 2$, and
$U(\eta)-\frac{L(\eta)}{p-1}<p-1+\frac{sp}{p-1}$;

  2. (b)

$U(\eta)\leq p-1$, $p\geq 2$, and $U(\eta)-L(\eta)<ps+1$.

Then the following conditions are equivalent:

  1. (i)

$h_{s,f}:B_{p}(\eta_{[2-p]})\to L^{p}(\mathbb{D},\eta dA)$ is a bounded
operator;

  2. (ii)

$h_{F}^{s}:B_{p}(\eta_{[2-p]})\to S_{p}(\eta_{[2-p]})$ is a bounded operator,
where $F$ is in $H(\mathbb{D})$ satisfying $F^{\prime}=f$ on $\mathbb{D}$;

  3. (iii)

$d\mu(z)=|f(z)|^{p}\eta(z)dA(z)$ is a $p$-Carleson measure for
$B_{p}(\eta_{[2-p]})$.

The following result is Theorem 4.2 in [13].

######  Theorem I.

Let $1<p<\infty$ and $\alpha\leq 1/2$, and let $s$ with
$s>\max\\{\frac{-1}{p},\frac{1-p}{p}\\}$ if $\alpha=1/2$ and
$s>\max\\{0,-2\alpha\\}$ if $\alpha<1/2$. Let $g$ be analytic on $\mathbb{D}$.
Then the operator $h_{s,g}$ is bounded from $B_{p}(\alpha)$ to
$L^{p}(\mathbb{D},(1-|z|^{2})^{p-1-2\alpha}dA(z))$ if and only if the measure
$|g(z)|^{p}(1-|z|^{2})^{p-1-2\alpha}dA(z)$ is a $p$-Carleson measure for
$B_{p}(\alpha)$.

Applying Theorem 4.2 and Proposition 4.3, we obtain Theorem 4.4 below which
optimizes the range of the parameter $\alpha$ in Theorem I.

######  Theorem 4.4.

Suppose $g\in H(\mathbb{D})$, $1<p<\infty$, $\alpha<\frac{p}{2}$, and
$s>-\frac{2\alpha}{p}$. Furthermore, if $p$, $\alpha$ and $s$ satisfy further
one of the following conditions:

  1. (a)

$\alpha\geq\frac{1}{2}$, $p\leq 2$ and $s>\frac{(4-2p)\alpha-1}{p}$;

  2. (b)

$\alpha\geq\frac{1}{2}$, $p>2$ and $s>-\frac{1}{p}$;

  3. (c)

$\alpha\leq\frac{1}{2}$ and $s>0$.

Then $h_{s,g}$ is bounded from $B_{p}(\alpha)$ to
$L^{p}(\mathbb{D},(1-|z|^{2})^{p-1-2\alpha}dA(z))$ if and only if the measure
$|g(z)|^{p}(1-|z|^{2})^{p-1-2\alpha}dA(z)$ is a $p$-Carleson measure for
$B_{p}(\alpha)$.

###### Proof.

Set $\eta(z)=(1-|z|^{2})^{p-1-2\alpha}$ with $p>2\alpha$. Then
$B_{p}(\eta_{[2-p]})=B_{p}(\alpha)$. Clearly, $U(\eta)=L(\eta)=p-2\alpha$. For
$-1<s<\infty$ and $a\in\mathbb{D}\setminus\\{0\\}$, we deduce

$\displaystyle\left(\int_{S(a)}\eta(z)dA(z)\right)\left(\int_{S(a)}\left(\frac{\eta(z)}{(1-|z|^{2})^{s}}\right)^{-\frac{p^{\prime}}{p}}dA_{s}(z)\right)^{\frac{p}{p^{\prime}}}$
$\displaystyle\thickapprox$
$\displaystyle(1-|a|)^{p+1-2\alpha}\left((1-|a|)\int_{|a|}^{1}(1-r)^{-1+s+\frac{2\alpha+s}{p-1}}dr\right)^{\frac{p}{p^{\prime}}}$
$\displaystyle\thickapprox$
$\displaystyle(1-|a|)^{(2+s)p}\thickapprox(A_{s}(S(a)))^{p},$

where we use $s+\frac{2\alpha+s}{p-1}=\frac{2\alpha+ps}{p-1}>0$. Hence
$\eta\in\mathcal{B}_{p,s}$ when $s>\frac{-2\alpha}{p}$. Note that
$\frac{-2\alpha}{p}>-1$. Then the desired result follows from Theorem 4.2 and
Proposition 4.3. ∎

Remark.  Theorem 4.4 contains the case of $1<p<\infty$ and $1/2<\alpha<p/2$
missing in Theorem I. For $1<p<\infty$ and $\alpha<1/2$, by (c) of Theorem
4.4, we get the same result in Theorem I. For $1<p<\infty$ and $\alpha=1/2$,
from (b) and (c) in Theorem 4.4, we also get the same result as Theorem I.

Z. Lou and R. Qian [17] investigated the boundedness of Hankel type operator
$h_{s,f}$ related to a class of Dirichlet type spaces $\mathfrak{D}_{\rho}$.
Let $\rho:[0,\infty)\rightarrow[0,\infty)$ be a right-continuous and
nondecreasing function. $\mathfrak{D}_{\rho}$ is equal to
$B_{2}(\omega_{\rho})$ with $\omega_{\rho}(|z|)=\rho(1-|z|)$. $\rho$ is said
to be upper (resp. lower) type $\gamma\in(0,\infty)$ (cf. [16]) if

$\rho(xy)\leq Cx^{\gamma}\rho(y),\ \ x\geq 1\ \ (\text{resp.}\ \ x\leq 1)\ \
\text{and}\ \ 0<y<\infty.$

Clearly, if $\rho$ is upper type $\gamma$ for some $\gamma>0$, then
$\rho(2y)\lesssim\rho(y)$ for all $y>0$.

The following result is Theorem 1 in [17].

######  Theorem J.

Let $0<\gamma<1$, $s>\frac{1+\gamma}{2}$ and $f\in H(\mathbb{D})$. Suppose
$\rho:[0,\infty)\rightarrow[0,\infty)$ is a right-continuous and nondecreasing
function of upper type $\gamma$. Then $h_{s,f}:B_{2}(\omega_{\rho})\to
L^{2}(\mathbb{D},\omega_{\rho}dA)$ is bounded if and only if
$|f(z)|^{2}\rho(1-|z|^{2})dA(z)$ is a $2$-Carleson measure for
$B_{2}(\omega_{\rho})$.

In [17, p. 219, Remark 2], the authors mentioned that they failed to prove
Theorem J and other results in their paper without the condition
$s>\frac{1+\gamma}{2}$. Motivated by this remark, we apply Theorem 4.2 to give
the following result which means that the condition in Theorem J can be
improved.

######  Theorem 4.5.

Let $0<\gamma<\infty$, $s>\max\\{0,\frac{\gamma-1}{2}\\}$ and $f\in
H(\mathbb{D})$. Suppose $\rho:[0,\infty)\rightarrow[0,\infty)$ is a right-
continuous and nondecreasing function of upper type $\gamma$. Then
$h_{s,f}:B_{2}(\omega_{\rho})\to L^{2}(\mathbb{D},\omega_{\rho}dA)$ is bounded
if and only if $|f(z)|^{2}\rho(1-|z|^{2})dA(z)$ is a $2$-Carleson measure for
$B_{2}(\omega_{\rho})$.

###### Proof.

Since $\rho$ is a nondecreasing function, we obtain

$\widehat{\omega_{\rho}}(r)=\int_{r}^{1}\rho(1-s)ds\leq(1-r)\rho(1-r),$

and

$\widehat{\omega_{\rho}}(r)\geq\int_{r}^{\frac{1+r}{2}}\rho(1-s)ds\geq\frac{1-r}{2}\rho\left(\frac{1-r}{2}\right)\gtrsim(1-r)\rho(1-r)$

for all $r\in[0,1)$. Hence
$\widehat{\omega_{\rho}}(r)\thickapprox(1-r)\rho(1-r)$ for all $r\in[0,1)$.
Since $\rho$ is upper type $\gamma$,
$\rho(1-x)\lesssim(\frac{1-x}{1-y})^{\gamma}\rho(1-y)$ for all $0<x\leq y<1$.
Consequently, $\frac{\widehat{\omega_{\rho}}(r)}{(1-r)^{\gamma+1}}$ is
essentially increasing on $[0,1)$ and $\frac{\widehat{\omega_{\rho}}(r)}{1-r}$
is essentially decreasing on $[0,1)$. Thus $1\leq L(\omega_{\rho})\leq
U(\omega_{\rho})\leq\gamma+1$.

Since $s>\frac{\gamma-1}{2}$ and $\rho(1-x)/(1-x)^{\gamma}$ is essentially
increasing on $[0,1)$, we deduce

$\displaystyle\int_{S(a)}\rho(1-|z|)dA(z)\int_{S(a)}\left(\frac{\rho(1-|z|)}{(1-|z|^{2})^{s}}\right)^{-1}dA_{s}(z)$
$\displaystyle\thickapprox$
$\displaystyle(1-|a|)^{2}\widehat{\omega_{\rho}}(|a|)\int_{|a|}^{1}\frac{(1-r)^{2s+\gamma}}{\rho(1-r)(1-r)^{\gamma}}dr$
$\displaystyle\lesssim$
$\displaystyle(1-|a|)^{3+\gamma}\int_{|a|}^{1}(1-r)^{2s-\gamma}dr\thickapprox(1-|a|)^{4+2s}\thickapprox(A_{s}(S(a)))^{2}$

for all $a\in\mathbb{D}\setminus\\{0\\}$. This gives
$\omega_{\rho}\in\mathcal{B}_{2,s}$ when $s>\frac{\gamma-1}{2}$.

Set $p=2$ and $\eta=\omega_{\rho}$ in (c) of Theorem 4.2. Since
$s>\max\\{0,\frac{\gamma-1}{2}\\}$ and $1\leq L(\eta)\leq
U(\eta)\leq\gamma+1$, we see that (c) in Theorem 4.2 holds. Then we get the
desired result. ∎

###  4.2. The compactness of Hankel type operators from $B_{p}(\omega)$ to
$S_{p}(\omega)$

This subsection is devoted to consider the compactness of Hankel type
operators corresponding to Theorem 4.2. This result is also new for
$\omega(z)=(1-|z|^{2})^{1-2\alpha}$.

A linear operator from one normed linear space to another normed linear space
is compact if the image of the unit ball under the operator has compact
closure.

We begin with the following lemma.

######  Lemma 4.6.

Let $1\leq p<\infty$, $-1<s<\infty$ and $f\in H(\mathbb{D})$. Suppose $\eta$
is a weight and there exist two constants $r\in(0,1)$ and $C>0$ such that

$C^{-1}\eta(\zeta)\leq\eta(z)\leq C\eta(\zeta)$

for all $z$ and $\zeta$ satisfying $\rho(z,\zeta)<r$. Then the following
statements hold:

  1. (i)

$h_{s,f}:B_{p}(\eta_{[2-p]})\to L^{p}(\mathbb{D},\eta dA)$ is a compact
operator if and only if
$\|h_{s,f}g_{n}\|_{L^{p}(\mathbb{D},\eta dA)}\rightarrow 0$ as $n\to\infty$
whenever $\\{g_{n}\\}$ is a bounded sequence in $B_{p}(\eta_{[2-p]})$ that
converges to 0 uniformly on every compact subset of $\mathbb{D}$;

  2. (ii)

$I_{d}:B_{p}(\eta_{[2-p]})\to L^{p}(\mathbb{D},|f(z)|^{p}\eta(z)dA(z))$ is a
compact operator if and only if
$\|g_{n}\|_{L^{p}(\mathbb{D},|f(z)|^{p}\eta(z)dA(z))}\rightarrow 0$ as
$n\to\infty$ whenever $\\{g_{n}\\}$ is a bounded sequence in
$B_{p}(\eta_{[2-p]})$ that converges to 0 uniformly on every compact subset of
$\mathbb{D}$.

###### Proof.

For any $g\in H(\mathbb{D})$, the assumes on $\eta$ and the subharmonicity of
$|g|^{p}$ yield

$\displaystyle\int_{\mathbb{D}}|g^{\prime}(\zeta)|^{p}\eta(\zeta)dA(\zeta)$
$\displaystyle\geq\int_{\Delta(z,r)}|g^{\prime}(\zeta)|^{p}\eta(\zeta)dA(\zeta)$
$\displaystyle\geq
C^{-1}\eta(z)\int_{\Delta(z,r)}|g^{\prime}(\zeta)|^{p}dA(\zeta)$
$\displaystyle\geq C_{1}|g^{\prime}(z)|^{p}\eta(z)(1-|z|)^{2}$ (4.33)

for all $z\in\mathbb{D}$, where $C_{1}$ is a positive constant independent of
$z$ and $g$. Clearly, $g(z)=\int_{0}^{z}g^{\prime}(\zeta)d\zeta+g(0)$. Hence
the point evaluations on $B_{p}(\eta_{[2-p]})$ are bounded functionals.
Similarly, the point evaluations on $A^{p}_{\eta}$ are also bounded. For
$z\in\mathbb{D}$ satisfying $f(z)\not=0$, the point evaluation induced by $z$
on $A^{p}_{|f|^{p}\eta}$ is also bounded. Next, the proof of this lemma is
quite standard (cf. [14, Proposition 3.11]). We omit it. ∎

We also need the following lemma.

######  Lemma 4.7.

Let $1<p<\infty$. Suppose $\eta\in\check{\mathcal{D}}$ and there exist two
constants $r\in(0,1)$ and $C>0$ such that

$C^{-1}\eta(\zeta)\leq\eta(z)\leq C\eta(\zeta)$

for all $z$ and $\zeta$ satisfying $\rho(z,\zeta)<r$. Then
$I_{d}:B_{p}(\eta_{[2-p]})\to L^{p}(\mathbb{D},\eta(z)dA(z))$ is a compact
operator.

###### Proof.

Let $\\{g_{n}\\}$ be a bounded sequence in $B_{p}(\eta_{[2-p]})$ that
converges to 0 uniformly on every compact subset of $\mathbb{D}$. Similar to
(4.2), we get

$\displaystyle\infty>\sup_{n}\int_{\mathbb{D}}|g^{\prime}_{n}(\zeta)|^{p}\eta(\zeta)dA(\zeta)\gtrsim\sup_{n}\sup_{z\in\mathbb{D}}|g_{n}^{\prime}(z)|^{p}\eta(z)(1-|z|^{2})^{2};$

that is,

$\displaystyle|g_{n}^{\prime}(z)|^{p}\eta(z)\lesssim G(z)$

for all $n$ and all $z\in\mathbb{D}$, where $G(z)=(1-|z|^{2})^{-2}$. For
$p>1$, it is clear that the integral
$\int_{\mathbb{D}}G(\zeta)(1-|\zeta|^{2})^{p}dA(\zeta)$ is convergent. Then
Lebesgue’s Dominated Convergence Theorem gives

$\displaystyle\lim_{n\to\infty}\int_{\mathbb{D}}|g^{\prime}_{n}(\zeta)|^{p}\eta(\zeta)(1-|\zeta|^{2})^{p}dA(\zeta)=0.$

Note that $\eta\in\check{\mathcal{D}}$. By [22, p. 8],

$\displaystyle\int_{\mathbb{D}}|g_{n}(\zeta)|^{p}\eta(\zeta)dA(\zeta)\lesssim|g_{n}(0)|^{p}+\int_{\mathbb{D}}|g^{\prime}_{n}(\zeta)|^{p}\eta(\zeta)(1-|\zeta|^{2})^{p}dA(\zeta)$

for all $n$. Consequently,

$\displaystyle\lim_{n\to\infty}\int_{\mathbb{D}}|g_{n}(\zeta)|^{p}\eta(\zeta)dA(\zeta)=0.$

By Lemma 4.6, the proof is complete. ∎

Now we give the main result in this subsection.

######  Theorem 4.8.

Suppose $1<p<\infty$, $-1<s<\infty$, $f\in H(\mathbb{D})$ and
$\eta\in\mathcal{B}_{p,s}$ is a radial weight. Let $\eta,p,s$ satisfy further
one of the following conditions:

  1. (a)

$U(\eta)<p-1$, $p\leq 2$, and
$U(\eta)-\frac{L(\eta)}{p-1}<p-1+\frac{sp}{p-1}$;

  2. (b)

$U(\eta)<p-1$, $p\geq 2$, and $U(\eta)-L(\eta)<ps+1$;

  3. (c)

$p-1\leq U(\eta)<ps+p$, $s>0$, and $L(\eta)>p-1-ps$.

Then the following conditions are equivalent:

  1. (i)

$h_{s,f}:B_{p}(\eta_{[2-p]})\to L^{p}(\mathbb{D},\eta dA)$ is a compact
operator;

  2. (ii)

$h_{F}^{s}:B_{p}(\eta_{[2-p]})\to S_{p}(\eta_{[2-p]})$ is a compact operator,
where $F$ is in $H(\mathbb{D})$ satisfying $F^{\prime}=f$ on $\mathbb{D}$;

  3. (iii)

$d\mu(z)=|f(z)|^{p}\eta(z)dA(z)$ is a vanishing $p$-Carleson measure for
$B_{p}(\eta_{[2-p]})$.

###### Proof.

Similar to the proof of Theorem 4.8, (i) and (ii) are equivalent.

Recall that a radial weight $\eta\in\mathcal{B}_{p,s}$ yields
$\eta\in\mathcal{D}$. Hence $\tilde{\eta}\in\mathcal{R}$. From [22, p. 8],
$\tilde{\eta}(z)\thickapprox\tilde{\eta}(\zeta)$ whenever $\rho(z,\zeta)<r$
with any fixed $r\in(0,1)$.

$(iii)\Rightarrow(i)$. It follows from (iii) and Proposition G that
$d\nu(z)=|f(z)|^{p}\tilde{\eta}(z)dA(z)$ is also a vanishing $p$-Carleson
measure for $B_{p}(\tilde{\eta}_{[2-p]})$. Let $\\{g_{n}\\}$ be a bounded
sequence in $B_{p}(\tilde{\eta}_{[2-p]})$ that converges to 0 uniformly on
every compact subset of $\mathbb{D}$. By Lemma 4.6,
$\|g_{n}\|_{L^{p}(\mathbb{D},d\nu(z))}\rightarrow 0$ as $n\to\infty$. From the
proof of “$(iii)\Rightarrow(i)$” in Theorem 4.8 and Proposition G again, we
know

$\|h_{s,f}g_{n}\|^{p}_{L^{p}(\mathbb{D},\tilde{\eta}dA)}\lesssim\|fg_{n}\|^{p}_{L^{p}(\mathbb{D},\tilde{\eta}dA)}=\|g_{n}\|^{p}_{L^{p}(\mathbb{D},d\nu)}.$

Then $\|h_{s,f}g_{n}\|^{p}_{L^{p}(\mathbb{D},\tilde{\eta}dA)}\to 0$ as
$n\to\infty$. From Lemma 4.6, $h_{s,f}:B_{p}(\tilde{\eta}_{[2-p]})\to
L^{p}(\mathbb{D},\tilde{\eta}dA)$ is a compact operator. Combining this with
Proposition G, we get the desired result.

$(i)\Rightarrow(iii)$. Case (a). Because of Proposition G, it suffices to show
that $d\nu(z)=|f(z)|^{p}\tilde{\eta}(z)dA(z)$ is a vanishing $p$-Carleson
measure for $B_{p}(\tilde{\eta}_{[2-p]})$. Suppose $\\{g_{n}\\}$ is a bounded
sequence in $B_{p}(\tilde{\eta}_{[2-p]})$ that converges to 0 uniformly on
every compact subset of $\mathbb{D}$. Note that
$h_{s,f}:B_{p}(\tilde{\eta}_{[2-p]})\to L^{p}(\mathbb{D},\tilde{\eta}dA)$ is
also a compact operator. Due to Lemma 4.6,

$\|h_{s,f}g_{n}\|^{p}_{L^{p}(\mathbb{D},\tilde{\eta}dA)}\to 0\ \ \text{as}\ \
n\to\infty.$ (4.34)

Note that $f\in A^{p}_{\tilde{\eta}}$. For any $\epsilon>0$, there exists $r$
in (0, 1) such that

$\left(\int_{\mathbb{D}\setminus
r\mathbb{D}}|f(z)|^{p}\tilde{\eta}(z)dA(z)\right)^{p-1}<\epsilon.$ (4.35)

Checking Case (a) in the proof of “$(i)\Rightarrow(iii)$” of Theorem 4.8, we
get

$\displaystyle\left|\int_{\mathbb{D}}\frac{f(w)\overline{(g_{n}(z)-g_{n}(w))}}{(1-\overline{w}z)^{2+s}}dA_{s}(w)\right|^{p}$
$\displaystyle\leq$ $\displaystyle
M_{f}^{(2-p)p}\left(\left(\int_{r\mathbb{D}}+\int_{\mathbb{D}\setminus
r\mathbb{D}}\right)\frac{|f(w)|^{p-1}|g_{n}(z)-g_{n}(w)|}{|1-\overline{w}z|^{2+s}}dA_{p-2+s}(w)\right)^{p}$
$\displaystyle\lesssim$ $\displaystyle
M_{f}^{(2-p)p}\|f\|_{A_{\tilde{\eta}}^{p}}^{p(p-1)}\int_{r\mathbb{D}}\frac{|g_{n}(z)-g_{n}(w)|^{p}(1-|w|^{2})^{(p-2+s)p}}{|1-\overline{w}z|^{(2+s)p}{\tilde{\eta}}(w)^{p-1}}dA(w)$
$\displaystyle+M_{f}^{(2-p)p}\left(\int_{\mathbb{D}\setminus
r\mathbb{D}}|f(z)|^{p}\tilde{\eta}(z)dA(z)\right)^{p-1}\int_{\mathbb{D}}\frac{|g_{n}(z)-g_{n}(w)|^{p}dA_{(p-2+s)p}(w)}{|1-\overline{w}z|^{(2+s)p}{\tilde{\eta}}(w)^{p-1}}$
(4.36)

for all $z\in\mathbb{D}$ and all $n$. By Lemma 4.7 and the fact that
$\\{g_{n}\\}$ converges to 0 uniformly on every compact subset of
$\mathbb{D}$, there exits $N>0$ such that

$\displaystyle\int_{\mathbb{D}}\int_{r\mathbb{D}}\frac{|g_{n}(z)-g_{n}(w)|^{p}(1-|w|^{2})^{(p-2+s)p}}{|1-\overline{w}z|^{(2+s)p}{\tilde{\eta}}(w)^{p-1}}dA(w)\tilde{\eta}(z)dA(z)$
$\displaystyle\thickapprox$
$\displaystyle\int_{\mathbb{D}}\int_{r\mathbb{D}}|g_{n}(z)-g_{n}(w)|^{p}dA(w)\tilde{\eta}(z)dA(z)$
$\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{D}}|g_{n}(z)|^{p}\tilde{\eta}(z)dA(z)+\int_{r\mathbb{D}}|g_{n}(w)|^{p}dA(w)<\epsilon$
(4.37)

for all $n>N$. From (4.35) and Case (a) in the proof of
“$(i)\Rightarrow(iii)$” of Theorem 4.8, we get

$\displaystyle\left(\int_{\mathbb{D}\setminus
r\mathbb{D}}|f(z)|^{p}\tilde{\eta}(z)dA(z)\right)^{p-1}\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|g_{n}(z)-g_{n}(w)|^{p}dA_{(p-2+s)p}(w)}{|1-\overline{w}z|^{(2+s)p}{\tilde{\eta}}(w)^{p-1}}\tilde{\eta}(z)dA(z)$
$\displaystyle\lesssim$
$\displaystyle\epsilon\|g_{n}\|^{p}_{B_{p}(\tilde{\eta}_{[2-p]})}\lesssim\epsilon$
(4.38)

for all $n$. From (4.4), (4.2), (4.2) and (4.2), we obtain

$\|f\overline{g_{n}}-\overline{h_{s,f}g_{n}}\|_{L^{p}(\mathbb{D},\tilde{\eta}dA)}\lesssim\epsilon.$

for all $n>N$. In other words,

$\lim_{n\to\infty}\|f\overline{g_{n}}-\overline{h_{s,f}g_{n}}\|_{L^{p}(\mathbb{D},\tilde{\eta}dA)}=0.$

Combining this with (4.34), we see that
$\|f\overline{g_{n}}\|^{p}_{L^{p}(\mathbb{D},\tilde{\eta}dA)}\to 0$ as
$n\to\infty$. Thus $d\nu(z)$ is a vanishing $p$-Carleson measure for
$B_{p}(\tilde{\eta}_{[2-p]})$. Case (b) and Case (c) can be proved in the
similar way with minor modifications. We finish the proof. ∎

By Theorem 4.8 and the explanations before Proposition 4.3, the corresponding
results about the compactness of operators in Proposition 4.3, Theorem 4.4 and
Theorem 4.5 also hold.

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