Schatten class operators on exponential weighted Bergman spaces

Abstract

In this paper, we study Toeplitz and Hankel operators on exponential weighted Bergman spaces. For $0< p<\infty$, we obtain sufficient and necessary conditions for Toeplitz and Hankel operators to belong to Schatten-p class by the averaging functions of symbols. For a continuous increasing convex function h, the Schatten-h class Toeplitz and Hankel operators are also characterized.

1 Introduction

Let $\mathbb{D}=\{z\in \mathbb{C}~|~|z|<1\}$ be the unit disk in the complex plane $\mathbb{C}$ and $dA(z)=\frac{dxdy}{\pi}$ be the normalized Lebesgue area measure on $\mathbb{D}$. Let $\mathcal{L}$ denote a class (see [2, 13] for more details about the class). A function $\rho (z)$ is said to be in $\mathcal{L}$ if $\rho (z)$ is positive on $\mathbb{D}$ satisfying the following conditions:

(a) For any $z\in \mathbb{D}$, there is a constant $c_{1}>0$ such that $\rho (z)\leq c_{1} (1-|z|)$.

(b) There is a constant $c_{2}>0$ such that $|\rho (z)-\rho (w)|\leq c_{2} |z-w|$, where $z,w\in \mathbb{D}$.

Write $A\lesssim B$ for two quantities A and B if there is a constant $C>0$ such that $A\leq CB$. Furthermore, $A\asymp B$ means that both $A\lesssim B$ and $B\lesssim A$ are satisfied. A subharmonic function $\varphi (z)\in C^{2}(\mathbb{D})$ satisfying $(\Delta \varphi (z))^{-1/2}\asymp \rho (z)$ is called $\varphi \in \mathcal{L}^{*}$, where $\rho (z)\in \mathcal{L}$ and Δ is the standard Laplace operator.

The Lebesgue space $L^{p}_{\varphi}~(0< p<\infty )$ consists of all measurable functions f on $\mathbb{D}$ such that

$$ \Vert f \Vert _{\varphi ,p}= \biggl( \int _{\mathbb{D}} \bigl\vert f(z)e^{-\varphi (z)} \bigr\vert ^{p}\,dA(z) \biggr)^{1/p}< \infty . $$

In particular, $L^{\infty}_{\varphi}$ consists of all measurable functions f on $\mathbb{D}$ such that

$$ \Vert f \Vert _{\varphi ,\infty}=\mathrm{esssup}_{z\in \mathbb{D}} \bigl\vert f(z)e^{-\varphi (z)} \bigr\vert < \infty . $$

Now let $\mathcal {H}(\mathbb{D})$ be the space of analytic functions in the unit disk $\mathbb{D}$. The exponential weighted Bergman spaces $A^{p}_{\varphi}=L^{p}_{\varphi}\cap \mathcal{H}(\mathbb{D})$. When $1\leq p\leq \infty$, $A^{p}_{\varphi}$ is a Banach space, and $A^{p}_{\varphi}$ is a Fréchet space if $0< p<1$.

Notice that $A_{\varphi}^{2}$ is a reproducing kernel Hilbert space, and hence there is a function $K_{\varphi ,z}\in A_{\varphi}^{2}$ such that the orthogonal projection P from $L_{\varphi}^{2}$ to $A_{\varphi}^{2}$ can be represented as

$$\begin{aligned} P(f) (z)= \int _{\mathbb{D}} f(w)\overline{K_{\varphi ,z}(w)}e^{-2 \varphi (w)}\,dA(w), \quad z\in \mathbb{D}. \end{aligned}$$

See [3, 13]. The function $K_{\varphi ,z}(\cdot )$ is called the reproducing kernel of Bergman space $A_{\varphi}^{2}$ and has the property that $K_{\varphi ,z}(w)=\overline{K_{\varphi ,w}(z)}$ for every $z,w\in \mathbb{D}$. It follows from [3, Theorems 4.1 and 4.2] that, for $\varphi \in \mathcal{E}$ and $1\leq p\leq \infty$, the Bergman projection $P: L^{p}_{\varphi}\rightarrow A^{p}_{\varphi}$ is bounded.

For a positive Borel measure μ on $\mathbb{D}$ and a measurable function f, the Toeplitz operator and Hankel operator are defined respectively by

$$ T_{\mu}(g) (z)= \int _{\mathbb{D}} g(w)K_{\varphi}(z,w)e^{-2\varphi (w)}\,d \mu (w), \quad g\in A^{p}_{\varphi} $$

and

$$ H_{f}(g) (z)= \int _{\mathbb{D}} \bigl(f(z)g(w)-f(w)g(w) \bigr)K_{ \varphi}(z,w)e^{-2\varphi (w)}\,dA(w), \quad g\in A^{p}_{\varphi}. $$

The pioneering work on this class of exponential weighted Bergman spaces was done by Oleinik and Perelman [14]. Throughout this paper, we call these spaces $\mathcal {O}\mathcal {P}\mathcal {S}$. Later, has attracted much attention. In [12], Lin and Rochberg characterized the boundedness and compactness of Hankel operators on exponential weighted Bergman spaces. To further study these spaces, Lin and Rochberg [13] gave the necessary and sufficient conditions for Schatten-p class Toeplitz (or Hankel) operators when $1\leq p<\infty$. Furthermore, for $0< p<1$, the sufficient condition for Schatten class membership of the Toeplitz operator was obtained as well. In [3, 4], Arroussi and Pau studied the dual space and estimates of the reproducing kernel.

Borichev, Dhuez, and Kellay [5] introduced another exponential weighted Bergman spaces. The authors, in [2], showed the Schatten class membership of the Toeplitz operator on spaces introduced by [5]. Hu, Lv, and Schuster [8] characterized a new kind of space, which contains these exponential weighted Bergman spaces considered in [5], write $\mathcal{HLS}$ for simplicity. Indeed, the spaces $\mathcal{HLS}$ differ from the spaces in this paper, see [8]. In [9], Hu and Pau gave bounded and compact Hankel operators associated with general symbols. Zhang, Wang and Hu [17] showed the boundedness and compactness of Toeplitz operators with positive symbols acting between different spaces $\mathcal{HLS}$, and Schatten-p class membership. Recently, in [16], the authors studied the sufficient and necessary conditions for Schatten-p class membership of Hankel operators associated with general symbols on $\mathcal{HLS}$.

For $0< p<\infty$, by using averaging functions, we obtain the sufficient and necessary conditions for Schatten-p class membership of Toeplitz operators with positive symbols and Hankel operators with general symbols on $\mathcal {O}\mathcal {P}\mathcal {S}$. These results fill the research gap of [13]. Generally speaking, the difficulty in such problems lies in the characterization of $0< p<1$. For this goal, we need more tools than [13]. Schatten-h class membership of operators is an important generalization of Schatten-p class operators, and it is interesting to study Schatten-h class membership. We refer to [1] and the relevant references therein for a brief account on Schatten-h class. In this paper, we explore Schatten-h class Toeplitz and Hankel operators on the spaces. Such properties of Hankel operators are not yet known in the existing literature.

By [8, Theorem 3.2], the following estimate holds for the reproducing kernel in this space: there exist constants $C,\sigma >0$ such that

$$ \bigl\vert K(z,w) \bigr\vert \leq C \frac{e^{\varphi (z)+\varphi (w)}}{\rho (z)\rho (w)}e^{- \sigma\,d_{\rho}(z,w)}, \quad z,w\in \mathbb{D}, $$

where $d_{\rho}(z,w)$ is the Bergman metric induced by reproducing kernel. However, the reproducing kernel in $\mathcal {O}\mathcal {P}\mathcal {S}$ does not have the similar estimate, which brings more obstacles to the research in this paper.

The paper is organized as follows. In Sect. 2, we give some basic notation and lemmas. In Sect. 3, we show the sufficient and necessary conditions for Schatten-p class membership of Toeplitz operators with positive symbols, and give the characterization for Schatten-h class membership of Toeplitz operators induced by continuous increasing convex functions. Finally, in Sect. 4, we investigate membership in Schatten-p class Hankel operators with general symbols, and also obtain Schatten-h class properties of Hankel operators.

2 Preliminaries

We begin with giving some basic notation and lemmas. For $z \in \mathbb{D}$ and $r>0$, let $D(z,r)=\{w:|w-z|< r\}$ be the Euclidean disk with radius r and center z. Also, we use $D^{r}(z)=D(z,r\rho (z))$ to denote the disk with radius $r\rho (z)$ and center z.

The following lemma is from [3, (2.1)].

Lemma 2.1

Suppose $\rho \in \mathcal{L}$, $z \in \mathbb{D}$ and $w \in D^{\alpha }(z)$, where $0<\alpha <m_{\rho}=\frac{ \min \{1, c^{-1}_{1} ,c^{-1}_{2}\}}{4}$. Then

$$\begin{aligned} \frac{1}{2}\rho (w)< \rho (z) < 2\rho (w). \end{aligned}$$

(2.1)

It is from [3, Lemma A] that we have the following pointwise estimate.

Lemma 2.2

Suppose $\varphi \in \mathcal{L}^{*}$, $0< p<\infty$, $\beta \in \mathbb{R}$ and $z\in \mathbb{D}$. Then there exists a constant $M\geq 1$, for $f\in \mathcal{H}(\mathbb{D})$ and small enough $\delta >0$, such that

$$\begin{aligned} \bigl\vert f(z) \bigr\vert ^{p} e^{-\beta \varphi (z)}\leq \frac{M}{\delta ^{2}\rho (z)^{2}} \int _{D^{\delta}(z)} \bigl\vert f(\zeta ) \bigr\vert ^{p} e^{- \beta \varphi (\zeta )}\,dA(\zeta ). \end{aligned}$$

(2.2)

As we known, the covering lemma is useful for studying Bergman spaces, so does exponential weighted Bergman spaces. The following lemma comes from [2, Lemma B].

Lemma 2.3

Suppose $\rho \in \mathcal{L}$ and $0< r< m_{\rho}$. Then there exists a sequence $\{a_{j}\}_{j=1}^{\infty} \subseteq \mathbb{D}$ satisfying

(a) $a_{j}\notin D^{r}(a_{k})$, $k\neq j$.

(b) $\mathbb{D}=\bigcup _{j=1}^{\infty }D^{r}(a_{j})$.

(c) $\tilde{D}^{r}(a_{j}) \subseteq D^{3r}(a_{j})$, where $\tilde{D}^{r}(a_{j})=\bigcup _{z\in D^{r}(a_{j}) }D ^{r}(z)$.

(d) $\{D^{3r}(a_{j})\}_{j=1}^{\infty}$ is a covering of $\mathbb{D}$ of finite multiplicity, that is, for any $z \in \mathbb{D}$,

$$\begin{aligned} 1\leq \sum_{j=1}^{\infty} \chi _{D^{3r}(a_{j})}(z)\leq N, \end{aligned}$$

(2.3)

where N is a positive constant integer.

A sequence $\{a_{j}\}_{j=1}^{\infty}$ satisfying the above lemma is called the $(\rho ,r)$-lattice. Furthermore, the conditions ${\mathrm{(a)}}$ and ${\mathrm{(c)}}$ indicate there is a $s>0$ such that

$$\begin{aligned} D^{sr}(a_{j}) \cap D^{sr}(a_{k})= \emptyset , \quad j \neq k. \end{aligned}$$

It is important to investigate pointwise and norm estimates of the reproducing kernels $K_{\varphi ,z}$ on $A_{\varphi}^{2}$. The following results are from [3, Lemma B, Theorem 3.1 and (3.1)].

If $\varphi \in \mathcal{L}^{*}$, $0< r< m_{\rho}$ and $w\in D^{r}(z)$, then we have

$$\begin{aligned} \bigl\vert K_{\varphi , z}(w) \bigr\vert \asymp \Vert K_{\varphi , z} \Vert _{\varphi ,2} \Vert K_{ \varphi , w} \Vert _{\varphi ,2}. \end{aligned}$$

(2.4)

Lemma 2.4

Suppose $\varphi \in \mathcal{L}^{*}$ and function ρ satisfies that, if there exist $b>0$ and $0< t<1$, for $z,w\in \mathbb{D}$ and $|z-w|>b\rho (w)$, such that

$$ \rho (z)\leq \rho (w)+t \vert z-w \vert , $$

then

$$\begin{aligned} \Vert K_{\varphi ,z} \Vert ^{2}_{\varphi ,2} \asymp e^{2\varphi (z)}\rho ^{-2}(z). \end{aligned}$$

(2.5)

Definition 2.5

The weight $\varphi \in \mathcal{L}^{*}$ is called $\varphi \in \mathcal{E}$ if the function ρ satisfies, for any $m\geq 1$, there exist constants $b_{m}>0$ and $0< t_{m}<1/m$, when $|z-w|>b_{m} \rho (w)$, such that

$$ \rho (z)\leq \rho (w)+t_{m} \vert z-w \vert . $$

Theorem 2.6

If $\varphi \in \mathcal{E}$, then for any $M \geq 1$ there is a constant $C>0$ such that

$$\begin{aligned} \bigl\vert K_{\varphi ,w}(z) \bigr\vert \leq C e^{\varphi (z)}e^{\varphi (w)} \frac{1}{\rho (z) }\frac{1}{\rho (w)} \biggl( \frac{\min \{\rho (z),\rho (w)\}}{ \vert z-w \vert } \biggr)^{M}, \quad z,w\in \mathbb{D}. \end{aligned}$$

(2.6)

Proof

See [3, Theorem 3.1]. □

With the help of estimates for the reproducing kernels, we get the following atomic decomposition.

Lemma 2.7

Suppose $\varphi \in \mathcal{E}$ and $\{a_{j}\}_{j=1}^{\infty}$ is a (ρ, r)-lattice, where $0< r\leq m_{\rho}$. Then, if $\{\lambda _{j}\}_{j=1}^{\infty }\in l^{2}$, we have $F(z)=\sum_{j=1}^{\infty}\lambda _{j}k_{\varphi ,a_{j}}(z) \in A^{2}_{ \varphi}$ and

$$ \Biggl\Vert \sum_{j=1}^{\infty}\lambda _{j}k_{\varphi ,a_{j}} \Biggr\Vert _{ \varphi ,2} \leq C \bigl\Vert \{\lambda _{j}\}_{j=1}^{\infty} \bigr\Vert _{l^{2}}, $$

where $k_{\varphi ,w}(z)= \frac{K_{\varphi}(z,w)}{\|K_{\varphi ,w}\|_{\varphi ,2}}$ is called normalized reproducing kernel.

Proof

By (2.5) and Hölder’s inequality, we have

$$\begin{aligned} \bigl\Vert F(z) \bigr\Vert ^{2}_{\varphi ,2} \lesssim & \int _{\mathbb{D}} \Biggl(\sum_{j=1}^{ \infty} \vert \lambda _{j} \vert e^{-\varphi (a_{j})} \rho (a_{j}) \bigl\vert K_{\varphi ,a_{j}}(z) \bigr\vert \Biggr)^{2}e^{-2\varphi (z)}\,dA(z) \\ \lesssim & \int _{\mathbb{D}} \Biggl(\sum_{j=1}^{\infty} \vert \lambda _{j} \vert ^{2} e^{-\varphi (a_{j})} \bigl\vert K_{\varphi ,a_{j}}(z) \bigr\vert \Biggr) M(z)e^{-2 \varphi (z)}\,dA(z), \end{aligned}$$

(2.7)

where

$$ M(z)=\sum_{j=1}^{\infty }\rho (a_{j})^{2} \bigl\vert K_{\varphi ,a_{j}}(z) \bigr\vert e^{- \varphi (a_{j})}. $$

It follows from (2.2), (2.5), and [3, Lemma 3.3] that

$$\begin{aligned} M(z)\lesssim \sum_{j=1}^{\infty } \int _{D^{r}(a_{j})} \bigl\vert K_{\varphi ,z}(w) \bigr\vert e^{- \varphi (w)}\,dA(w)\lesssim \int _{\mathbb{D}} \bigl\vert K_{\varphi ,z}(w) \bigr\vert e^{- \varphi (w)}\,dA(w)\lesssim e^{\varphi (z)}. \end{aligned}$$

(2.8)

This together with (2.7), (2.8), and (2.5) implies that

$$\begin{aligned} \bigl\Vert F(z) \bigr\Vert ^{2}_{\varphi ,2} \lesssim & \int _{\mathbb{D}} \Biggl(\sum_{j=1}^{ \infty} \vert \lambda _{j} \vert ^{2} e^{-\varphi (a_{j})} \bigl\vert K_{\varphi ,a_{j}}(z) \bigr\vert \Biggr) e^{- \varphi (z)}\,dA(z) \\ \lesssim & \sum_{j=1}^{\infty} \vert \lambda _{j} \vert ^{2} e^{-\varphi (a_{j})} \int _{\mathbb{D}} \bigl\vert K_{\varphi ,a_{j}}(z) \bigr\vert e^{- \varphi (z)}\,dA(z) \\ \lesssim & \sum_{j=1}^{\infty} \vert \lambda _{j} \vert ^{2} = \bigl\Vert \{\lambda _{j} \}_{j=1}^{\infty } \bigr\Vert ^{2}_{l^{2}}, \end{aligned}$$

which ends the proof. □

To describe the Schatten-p membership of Hankel operators, we need some auxiliary conclusions. For $z,w\in \mathbb{D}$, we write

$$ d_{\rho}(z,w)=\frac{ \vert z-w \vert }{\min (\rho (z), \rho (w))}. $$

Lemma 2.8

([2, Lemma 4.4])

Let $\rho \in \mathcal{L}$ and $\{a_{j}\}_{j}$ be a (ρ, r)-lattice on $\mathbb{D}$. Then for any $w\in \mathbb{D}$, the set

$$ D_{m}(w)=\bigl\{ z\in \mathbb{D}~|~d_{\rho}(z,w)< 2^{m} r\bigr\} $$

contains at most K points of the lattice, where K depends on the positive integer m, but not on the point w.

Lemma 2.9

([2, Lemma 4.5])

Let $\rho \in \mathcal{L}$, $r \in (0,m_{\rho}]$ and $k\in \mathbb{N}^{+}$. Any $(\rho ,r)$-lattice $\{a_{j}\}_{j=1}^{\infty}$ on $\mathbb{D}$, can be partitioned into M subsequences such that, if $a_{i}$ and $a_{j}$ are different points in the same subsequence, then $|a_{i}-a_{j}|\geq 2^{m}r\min \{\rho (a_{i}),\rho (a_{j}) \}$.

Given a positive Borel measure μ on $\mathbb{D}$ and $r>0$, the averaging function $\hat{\mu}_{r}$ with respect to measure μ is defined by

$$ \widehat{\mu}_{r}(z)=\frac{\int _{D^{r}(z)}\,d\mu}{ \vert D^{r}(z) \vert }. $$

Lemma 2.10

If μ is a positive Borel measure, $0< p<\infty$ and $r \in (0,m_{\rho}]$, then

$$\begin{aligned} \int _{\mathbb{D}} \bigl\vert g(z)e^{-\varphi (z)} \bigr\vert ^{p}\,d\mu (z) \lesssim \int _{ \mathbb{D}} \bigl\vert g(z)e^{-\varphi (z)} \bigr\vert ^{p}\widehat{\mu}_{r}(z)\,dA(z), \end{aligned}$$

(2.9)

where $g \in \mathcal{H}(\mathbb{D})$.

Proof

See [7, Lemma 2.4]. □

3 Schatten class Toeplitz operators

In this section, for $0< p<\infty$, we investigate the sufficient and necessary conditions for Schatten-p class membership of Toeplitz operators with positive measure symbols on $\mathcal {O}\mathcal {P}\mathcal {S}$. Also, we give the characterization for Schatten-h class membership of Toeplitz operators where h is a continuous increasing convex function.

Let $T:H_{1} \to H_{2}$ be a bounded linear operator, and write $s_{j}(T)$ for the singular values of T, where

$$ s_{j}(T)=\inf \bigl\{ \Vert T-K \Vert :K:H_{1} \to H_{2} , \text{ rank } (K) \leq j \bigr\} . $$

Here $\operatorname{rank} (K)$ means the rank of operator K. Recall that the operator T is compact if and only if $s_{j}(T) \to 0$ whenever $j \to \infty$. For $0< p<\infty$, it is called T is in $S_{p}$ if

$$ \Vert T \Vert ^{p}_{S_{p}}=\sum _{j=1}^{\infty }s_{j}(T)^{p} < \infty , $$

and we write $T \in S_{p}(H_{1},H_{2})$. Futhermore, $\|\cdot \|_{S_{p}}$ is actually a norm when $1 \leq p <\infty$ and $\|\cdot \|_{S_{p}}$ is not, if $0< p <1$.

Using

$$\begin{aligned} \Vert S+T \Vert _{S_{p}}\leq \Vert S \Vert _{S_{p}}+ \Vert T \Vert _{S_{p}},\quad 1\leq p< \infty , \end{aligned}$$

(3.1)

and

$$\begin{aligned} \Vert S+T \Vert _{S_{p}}^{p}\leq \Vert S \Vert _{S_{p}}^{p}+ \Vert T \Vert _{S_{p}}^{p}, \quad 0< p< 1, \end{aligned}$$

(3.2)

it is easy to see $T\in S_{p}$ if and only if $T^{*}T\in S_{\frac{p}{2}}$.

As we known, the Schatten class of Toeplitz operators with positive measure symbols is an important problem in operator theory, which has been described in many papers (see, for example, [2, 13, 17]). The following theorem is closely related to the main result [2, Theorem 1.2]. To Study the Schatten class of Toeplitz operators, we define the measure $d\lambda _{\rho}$ by

$$ d\lambda _{\rho}(z)=\frac{dA(z)}{\rho (z)^{2}},\quad z\in \mathbb{D}. $$

Theorem 3.1

Suppose $\varphi \in \mathcal{E}$, $0< p<\infty$, and μ is a finite positive Borel measure on $\mathbb{D}$. Then following statements are equivalent:

(a) $T_{\mu}\in \mathcal{S}_{p}(A_{\varphi}^{2})$.

(b) $\widehat{\mu}_{\delta}\in L^{p}(\mathbb{D},d\lambda _{\rho})$, where $\delta \in (0, \alpha _{m}]$.

(c) $\{\widehat{\mu}_{r}(w_{n})\}_{n}\in l^{p}$, where $\{\widehat{\mu}_{r}(w_{n})\}_{n}$ is a $(\rho , r)$-lattice with $r\in (0, \alpha _{m}]$.

(d) $\widetilde{\mu}\in L^{p}(\mathbb{D}, d\lambda _{\rho})$, where $\widetilde{\mu}(w)=\int _{\mathbb{D}}|k_{\varphi ,w}(z)|^{2}\,d\mu (z)$ is the Berezin transform of μ.

Proof

The proof of ${\mathrm{(b)}} \Leftrightarrow {\mathrm{(c)}}\Leftrightarrow {\mathrm{(d)}}$ is similar to [17, Proposition 2.5], and we omit the details here. Indeed, this proof indicates the $L^{p}$ behavior of averaging function $\hat{\mu}_{r}$ is independent of r. (That is, for small enough r, $\|\hat{\mu}_{\delta}\|_{L^{p}}\asymp \|\hat{\mu}_{r}\|_{L^{p}}$ with small enough δ.) The rest part is an analogue of [17, Theorem 5.1], and for the convenience of readers, we give the proof for implication ${\mathrm{(a)}}\Rightarrow{\mathrm{(c)}}$ when $0< p<1$.

Assume the Toeplitz operator $T_{\mu}$ is in $\mathcal{S}_{p}(A_{\varphi}^{2})$. Let $\{w_{n} \}$ be a $(\rho ,r)$-lattice with $r\in (0, m_{\rho}]$ sufficiently small. Set a large enough integer $m\geq 2$, by Lemma 2.9, the lattice $\{w_{n} \}$ can be devided into Γ subsequences such that

$$ \vert w_{i}-w_{j} \vert \geq 2^{m}r\min \bigl(\rho (w_{i}),\rho (w_{j}) \bigr), $$

where $w_{i}$ and $w_{j}$ are in the same subsequence. Let $\{a_{n} \}$ be such a subsequence, and measure ν be defined by

$$ d\nu = \biggl(\sum_{n}\chi _{n} \biggr)\,d\mu , $$

where $\chi _{n}$ is the characteristic function of $D^{r}(a_{n})$. Disks $D^{r}(a_{n})$ are pairwise disjoints since $m\geq 2$. Note that $T_{\mu}\in \mathcal{S}_{p}(A_{\varphi}^{2})$ and $0 \leq \nu \leq \mu$, thus $0 \leq T_{\nu} \leq T_{\mu}$, and then $T_{\nu}\in \mathcal{S}_{p}(A_{\varphi}^{2})$ and $\Vert T_{\nu} \Vert _{\mathcal{S}_{p}(A_{\varphi}^{2})} \leq \Vert T_{\mu} \Vert _{\mathcal{S}_{p}(A_{\varphi}^{2})}$.

Let $\{e_{n}\}$ be an orthonormal basis for $A_{\varphi}^{2}$. Consider an operator G on $A_{\varphi}^{2}$ as

$$\begin{aligned} Gf=\sum_{n}\langle f,e_{n}\rangle _{A_{\varphi}^{2}} k_{\varphi ,a_{n}}, \quad f\in A_{\varphi}^{2}. \end{aligned}$$

(3.3)

It follows from Lemma 2.7 that G is bounded on $A_{\varphi}^{2}$, then $T=G^{\ast}T_{\nu}G$ is in $\mathcal{S}_{p}(A_{\varphi}^{2})$ and

$$\begin{aligned} \Vert T \Vert _{\mathcal{S}_{p}(A_{\varphi}^{2})} \leq \Vert G \Vert ^{2} \cdot \Vert T_{\nu} \Vert _{\mathcal{S}_{p}(A_{\varphi}^{2})} \lesssim \Vert T_{\mu} \Vert _{\mathcal{S}_{p}(A_{\varphi}^{2})}. \end{aligned}$$

(3.4)

By (3.3) and

$$ \langle Tf, g\rangle _{A_{\varphi}^{2}}=\langle T_{\nu}Gf, Gg\rangle _{A_{ \varphi}^{2}},\quad f,g \in A_{\varphi}^{2}, $$

we have

$$ T f=\sum_{n, j} \langle T_{\nu} k_{\varphi ,a_{n}}, k_{\varphi ,a_{j}} \rangle _{A_{\varphi}^{2}} \langle f, e_{n} \rangle _{A_{\varphi}^{2}} e_{j}, \quad f \in A_{\varphi}^{2}. $$

We now take a decomposition of the operator T as $T=T_{1}+T_{2}$, where $T_{1}$ is the diagonal operator defined by

$$ T_{1} f=\sum_{n} \langle T_{\nu} k_{\varphi ,a_{n}}, k_{ \varphi ,a_{n}} \rangle _{A_{\varphi}^{2}} \langle f, e_{n} \rangle _{A_{\varphi}^{2}} e_{n}, \quad f \in A_{\varphi}^{2}, $$

and $T_{2}=T-T_{1}$ is the non-diagonal part. Using Rotfel’d inequality (see [15]), we see

$$\begin{aligned} \Vert T \Vert _{\mathcal{S}_{p}(A_{\varphi}^{2})}^{p} \geq \Vert T_{1} \Vert _{ \mathcal{S}_{p}(A_{\varphi}^{2})}^{p} - \Vert T_{2} \Vert _{\mathcal{S}_{p}(A_{ \varphi}^{2})}^{p}. \end{aligned}$$

(3.5)

Notice that $T_{1}$ is a positive diagonal operator, this together with the definition of ν, (2.1), (2.4), and (2.5) gives

$$\begin{aligned} \Vert T_{1} \Vert _{\mathcal{S}_{p}(A_{\varphi}^{2})}^{p} &=\sum_{n} \langle T_{\nu} k_{\varphi ,a_{n}}, k_{\varphi ,a_{n}} \rangle _{A_{\varphi}^{2}}^{p} =\sum _{n} \biggl( \int _{ \mathbb{D}} \bigl\vert k_{\varphi ,a_{n}}(z) \bigr\vert ^{2} e^{-2\varphi (z)}\,d \nu (z) \biggr)^{p} \\ &\gtrsim \sum_{n} \biggl( \int _{D^{r}(a_{n})}\frac{1}{\rho (z)^{2}}\,d \mu (z) \biggr)^{p} \gtrsim \sum_{n}\widehat{\mu}_{r}(a_{n})^{p}. \end{aligned}$$

(3.6)

For $0 < p < 1$, [18, Proposition 1.29] and Lemma 2.3 show

$$\begin{aligned} \Vert T_{2} \Vert _{\mathcal{S}_{p}(A_{\varphi}^{2})}^{p} &\leq \sum_{n} \sum _{k} \langle T_{2} e_{n}, e_{k} \rangle _{A_{\varphi}^{2}}^{p}=\sum _{ k \neq n} \langle T_{\nu} k_{ \varphi ,a_{n}}, k_{\varphi ,a_{k}} \rangle _{A_{\varphi}^{2}}^{p} \\ &\leq \sum_{ k \neq n} \biggl( \int _{\mathbb{D}} \bigl\vert k_{\varphi ,a_{n}}( \xi ) \bigr\vert \bigl\vert k_{\varphi ,a_{k}}(\xi ) \bigr\vert e^{-2\varphi ( \xi )}\,d \nu (\xi ) \biggr)^{p} \\ &\leq \sum_{ k \neq n} \biggl(\sum _{j} \int _{D^{r}(a_{j})} \bigl\vert k_{ \varphi ,a_{n}}(\xi ) \bigr\vert \bigl\vert k_{\varphi ,a_{k}}(\xi ) \bigr\vert e^{-2 \varphi (\xi )}\,d \mu (\xi ) \biggr)^{p}. \end{aligned}$$

(3.7)

If $n \neq k$, then $|a_{n}-a_{k}|\geq 2^{m}r\min (\rho (a_{n}),\rho (a_{k}) )$. Hence, for $\xi \in D^{r}(a_{j})$, we get either

$$ \vert a_{n}-\xi \vert \geq 2^{m-2}r\min \bigl(\rho (a_{n}),\rho (\xi ) \bigr) \quad \text{or}\quad \vert \xi -a_{k} \vert \geq 2^{m-2}r\min \bigl(\rho (\xi ),\rho (a_{k}) \bigr). $$

Therefore, for any $\xi \in D^{r}(a_{j})$, we may assume $|a_{n}-\xi |\geq 2^{m-2}r\min (\rho (a_{n}),\rho (\xi ) )$.

For any $n,k\in \mathbb{N}^{+}$, set

$$ J_{n k}(\mu )=\sum_{j} \int _{D^{r} (a_{j} )} \bigl\vert k_{ \varphi ,a_{n}}(\xi ) \bigr\vert \bigl\vert k_{\varphi ,a_{k}}(\xi ) \bigr\vert e^{-2 \varphi (\xi )}\,d\mu (\xi ). $$

This, combined with (3.7), yields

$$\begin{aligned} \Vert T_{2} \Vert _{\mathcal{S}_{p}(A_{\varphi}^{2})}^{p} \leq \sum_{n, k : k \neq n} J_{n k}(\mu )^{p}. \end{aligned}$$

(3.8)

Let M be large enough. Here M is from Theorem 2.6. Applying $|a_{n}-\xi |\geq 2^{m-2}r \min (\rho (a_{n}),\rho (\xi ) )$, we have

$$\begin{aligned} \bigl\vert k_{a_{n}}(\xi ) \bigr\vert e^{-\varphi (\xi )} \lesssim \frac{1}{\rho (\xi )} \biggl( \frac{\min (\rho (a_{n}),\rho (\xi ) )}{ \vert a_{n}-\xi \vert } \biggr)^{M}\lesssim \frac{1}{\rho (\xi )}2^{-M m}. \end{aligned}$$

And hence,

$$\begin{aligned} \bigl\vert k_{\varphi ,a_{n}}(\xi ) \bigr\vert = \bigl\vert k_{\varphi ,a_{n}}( \xi ) \bigr\vert ^{1/2} \bigl\vert k_{\varphi ,a_{n}}(\xi ) \bigr\vert ^{1/2} \lesssim 2^{-Mm/2} \frac{e^{\varphi (\xi )/2}}{ \rho (\xi )^{1/2}} \bigl\vert k_{\varphi ,a_{n}}(\xi ) \bigr\vert ^{1/2}. \end{aligned}$$

(3.9)

It follows from (2.4), (2.5), and (2.6) that

$$\begin{aligned} \bigl\vert k_{\varphi ,a_{k}}(\xi ) \bigr\vert = \frac{ \vert K_{\varphi}(\xi ,a_{k}) \vert ^{1/2}}{ \Vert K_{\varphi ,a_{k}} \Vert _{A_{\varphi}^{2}}^{1/2}} \bigl\vert k_{\varphi ,a_{k}}( \xi ) \bigr\vert ^{1/2} \lesssim \frac{e^{\varphi (\xi )/2}}{ \rho (\xi )^{1/2}} \bigl\vert k_{\varphi ,a_{k}}(\xi ) \bigr\vert ^{1/2}. \end{aligned}$$

(3.10)

By joining (3.9), (3.10), and Lemma 2.1, we obtain

$$ J_{n k}(\mu ) \lesssim 2^{\frac{-M m}{2}} \sum _{j} \frac{1}{\rho (a_{j})} \int _{D^{r} ( a_{j} )} \bigl\vert k_{ \varphi ,a_{n}}(\xi ) \bigr\vert ^{1 / 2} \bigl\vert k_{\varphi ,a_{k}}(\xi ) \bigr\vert ^{1 / 2} e^{-\varphi (\xi )}\,d \mu (\xi ). $$

Applying Lemmas 2.1, 2.2, and 2.3 (c), for $\xi \in D^{r}(a_{j})$, we conclude

$$\begin{aligned} \bigl\vert k_{\varphi ,a_{n}}(\xi ) \bigr\vert ^{1 / 2} e^{-\varphi (\xi )/ 2} \lesssim \biggl(\frac{1}{\rho (\xi )^{2}} \int _{D^{r} (\xi )} \bigl\vert k_{\varphi ,a_{n}}(z) \bigr\vert ^{p / 2} e^{-p\varphi (z) / 2}\,dA(z) \biggr)^{1/p} \lesssim \rho (a_{j})^{-2/p}S_{n}(a_{j})^{1/p}, \end{aligned}$$

where

$$ S_{n}(\cdot )= \int _{D^{3\alpha}(\cdot )} \bigl\vert k_{\varphi ,a_{n}}(z) \bigr\vert ^{p / 2} e^{-p\varphi (z) / 2}\,dA(z). $$

The analogous reasons indicate

$$\begin{aligned} \bigl\vert k_{\varphi ,a_{k}}(\xi ) \bigr\vert ^{1 / 2} e^{-\varphi (\xi )/ 2} \lesssim \rho (a_{j})^{-2/p}S_{k}(a_{j})^{1/p}. \end{aligned}$$

So, for M large enough, we have

$$\begin{aligned} J_{n k}(\mu )&\lesssim 2^{-M m/2} \sum _{j} \frac{\rho (a_{j})^{-4/p}}{\rho (a_{j})} S_{n}(a_{j})^{1/p} S_{k}(a_{j})^{1/p} \mu \bigl(D^{r}(a_{j}) \bigr) \\ &\leq 2^{-m} \sum_{j} \rho (a_{j})^{1-4/p}S_{n}(a_{j})^{1/p}S_{k}(a_{j})^{1/p} \widehat{\mu}_{r}(a_{j}). \end{aligned}$$

And hence, for $0 < p < 1$,

$$\begin{aligned} J_{n k}(\mu )^{p}\lesssim 2^{-m p} \sum _{j} \rho (a_{j})^{p-4} S_{n}(a_{j}) S_{k}(a_{j}) \widehat{ \mu}_{r}(a_{j})^{p}. \end{aligned}$$

Now (3.8) can be estimated further as

$$\begin{aligned} \Vert T_{2} \Vert _{\mathcal{S}_{p}(A_{\varphi}^{2})}^{p} \lesssim 2^{-m p} \sum_{j} \rho (a_{j})^{p-4} \widehat{\mu}_{r}(a_{j})^{p} \biggl(\sum_{n} S_{n}(a_{j}) \biggr) \biggl(\sum_{k} S_{k}(a_{j}) \biggr). \end{aligned}$$

(3.11)

On the other hand, by the definition of $S_{k}(a_{j})$, we see

$$\begin{aligned} \sum_{k}S_{k}(a_{j}) = \int _{D^{3\alpha}(a_{j})} \biggl(\sum_{k} \bigl\vert k_{\varphi ,a_{k}}(z) \bigr\vert ^{p/2} \biggr)e^{-p\varphi (z)/2}\,dA(z). \end{aligned}$$

(3.12)

We claim that

$$\begin{aligned} \sum_{k} \bigl\vert k_{\varphi ,a_{k}}(z) \bigr\vert ^{p/2}\lesssim e^{p \varphi (z)/2}\rho (z)^{-p/2}. \end{aligned}$$

(3.13)

For this goal, by (2.4), (2.1), and Lemma 2.3 (d), for some $r_{0}>0$, we get

$$\begin{aligned} \sum_{a_{k}\in D^{r_{0}}(z)} \bigl\vert k_{\varphi ,a_{k}}(z) \bigr\vert ^{p/2} \lesssim e^{p\varphi (z)/2}\sum _{a_{k}\in D^{r_{0}}(z)}\rho (a_{k})^{-p/2} \lesssim e^{p\varphi (z)/2}\rho (z)^{-p/2}. \end{aligned}$$

(3.14)

Taking M in Theorem 2.6 such that $Mp/2-2>0$, then

$$\begin{aligned} \sum_{a_{k}\notin D^{r_{0}}(z)} \bigl\vert k_{\varphi ,a_{k}}(z) \bigr\vert ^{p/2} &\lesssim e^{p\varphi (z)/2}\rho (z)^{Mp/2-p/2-2}\sum _{a_{k}\notin D^{r_{0}}(z)} \frac{\rho (a_{k})^{2}}{ \vert z-a_{k} \vert ^{Mp/2}} \\ &=e^{p\varphi (z)/2}\rho (z)^{Mp/2-p/2-2}\sum_{j=0}^{\infty} \sum_{a_{k} \in R_{j}(z)}\frac{\rho (a_{k})^{2}}{ \vert z-a_{k} \vert ^{Mp/2}}, \end{aligned}$$

where

$$ R_{j}(z)= \bigl\{ \zeta \in \mathbb{D} : 2^{j} r_{0} \rho (z)\leq \vert \zeta -z \vert < 2^{j+1} r_{0} \rho (z) \bigr\} ,\quad j=0,1,2 \ldots . $$

By Lemma 2.3, for $j=0,1,2, \ldots$ , when $a_{k} \in D^{r_{0}2^{j+1}}(z)$, we obtain

$$ D^{r_{0}}(a_{k}) \subset D^{20 r_{0}2^{j}}(z). $$

So

$$ \sum_{a_{k} \in R_{j}(z)} \rho (a_{k} )^{2} \lesssim \bigl\vert D^{20 r_{0}2^{j}}(z) \bigr\vert \lesssim 2^{2 j} \rho (z)^{2}, $$

and hence (3.13) holds by (3.14) and the following estimate

$$\begin{aligned} \sum_{a_{k}\notin D^{r_{0}}(z)} \bigl\vert k_{\varphi ,a_{k}}(z) \bigr\vert ^{p/2} &\lesssim e^{p\varphi (z)/2}\rho (z)^{-p/2-2}\sum _{j=0}^{\infty}2^{-M pj/2} \sum _{a_{k} \in R_{j}(z)}\rho (a_{k})^{2} \\ &\lesssim e^{p\varphi (z)/2}\rho (z)^{-p/2}\sum _{j=0}^{\infty}2^{ \frac{(4-Mp)}{2}j}\lesssim e^{p\varphi (z)/2} \rho (z)^{-p / 2}. \end{aligned}$$

Bearing in mind (3.13), (3.12) can be estimated as

$$\begin{aligned} \sum_{k} S_{k}(a_{j}) \lesssim \rho (a_{j})^{2-p / 2}. \end{aligned}$$

(3.15)

Similarly,

$$\begin{aligned} \sum_{n}S_{n}(a_{j}) \lesssim \rho (a_{j})^{2-p/ 2}. \end{aligned}$$

(3.16)

By joining (3.15), (3.16), and (3.11), for integer $m>0$ large enough, we get

$$ \Vert T_{2} \Vert _{\mathcal{S}_{p}(A_{\varphi}^{2})}^{p} \lesssim 2^{-m p} \sum_{j} \widehat{ \mu}_{r}(a_{j})^{p} \leq \frac{1}{2} \sum _{j} \widehat{\mu}_{r}(a_{j})^{p}. $$

This together with (3.6) and (3.5) yields

$$ \sum_{j} \widehat{\mu}_{r}(a_{j})^{p} \lesssim \Vert T \Vert _{\mathcal{S}_{p}(A_{ \varphi}^{2})}^{p}. $$

Since the above estimate holds for each of the Γ subsequences $\{w_{n}\}$, we finally obtain

$$\begin{aligned} \sum_{n} \widehat{\mu}_{r}(w_{n})^{p} \lesssim M \Vert T \Vert _{\mathcal{S}_{p}(A_{ \varphi}^{2})}^{p}\lesssim M \Vert T_{\mu} \Vert _{\mathcal{S}_{p}(A_{ \varphi}^{2})}^{p}< \infty \end{aligned}$$

by (3.4), which finishes this proof. □

We are going to describe the Schatten-h class Toeplitz operators. See [1] and the references therein for details about the Schatten-h class. We give first the following analogous definition.

Definition 3.2

Let T be a compact operator and $h:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ be a continuous increasing convex function. We say that $T\in S_{h}$ if there is a positive constant c such that

$$ \sum^{\infty}_{j=1} h\bigl(c\cdot s_{j}(T)\bigr)< \infty . $$

Similar to [1], we get the following consequence.

Theorem 3.3

Suppose $h:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ is a continuous increasing convex function, and μ is a positive Borel measure such that Toeplitz operator $T_{\mu}:A^{2}_{\varphi}\rightarrow A^{2}_{\varphi}$ is compact. Then $T_{\mu}\in S_{h}$ if and only if there exists a constant $c>0$ such that

$$ \int _{\mathbb{D}}h\bigl(c\tilde{\mu}(z)\bigr)\rho ^{-2}(z)\,dA(z)< \infty . $$

Proof

Assume that $T_{\mu}\in S_{h}$. Then there exists $c>0$ such that

$$ \sum^{\infty}_{j=1} h\bigl(cs_{j}(T) \bigr)< \infty . $$

Let $\{e_{k}\}_{k=1}^{\infty}$ be an orthonormal basis for $A^{2}_{\varphi}$, and

$$ T_{\mu}f=\sum^{\infty}_{k=1} s_{k}\langle f,e_{k} \rangle _{A^{2}_{\varphi}} e_{k}, $$

where $s_{k}$ is the singular value sequence of $T_{\mu}$. With the help of

$$ \sum^{\infty}_{k=1} \bigl\vert \langle K_{\varphi ,z},e_{k} \rangle _{A^{2}_{\varphi}} \bigr\vert ^{2}=1, $$

the convexity of h, Jensen’s inequality, (2.4), and (2.5), we have

$$ \begin{aligned} \int _{\mathbb{D}}h\bigl(c\widetilde{\mu}(z)\bigr)\rho ^{-2}(z)\,dA(z) &= \int _{ \mathbb{D}}h\bigl(c\langle T_{\mu}k_{\varphi ,z},k_{\varphi ,z} \rangle _{A^{2}_{ \varphi}}\bigr)\rho ^{-2}(z)\,dA(z) \\ &= \int _{\mathbb{D}}h\Biggl(\sum^{\infty}_{k=1}cs_{k} \bigl\vert \langle k_{\varphi ,z},e_{k}\rangle _{A^{2}_{\varphi}} \bigr\vert ^{2}\Biggr)\rho ^{-2}(z)\,dA(z) \\ &\leq \int _{\mathbb{D}}\sum^{\infty}_{k=1} h(cs_{k}) \bigl\vert \langle k_{\varphi ,z},e_{k} \rangle _{A^{2}_{\varphi}} \bigr\vert ^{2}\rho ^{-2}(z)\,dA(z) \\ &= \int _{\mathbb{D}}\sum^{\infty}_{k=1} h(cs_{k}) \Vert K_{\varphi ,z} \Vert ^{-2}_{\varphi ,2} \bigl\vert e_{k}(z) \bigr\vert ^{2}\rho ^{-2}(z)\,dA(z) \\ &\lesssim \sum^{\infty}_{k=1} h(cs_{k}) \int _{ \mathbb{D}} \bigl\vert e_{k}(z) \bigr\vert ^{2}e^{-2\varphi (z)}\,dA(z) \\ &=\sum^{\infty}_{k=1} h(cs_{k})< \infty . \end{aligned} $$

Conversely, if there exists $c>0$ such that $\int _{\mathbb{D}} h(c\tilde{\mu}(z))\rho ^{-2}(z)\,dA(z)<\infty$, then it follows from (2.4) and (2.5) that

$$ \begin{aligned} \widehat{\mu}_{r}(z)&= \int _{D^{r}(z)}\rho ^{-2}(z)\,d\mu (w) \\ &\asymp \int _{D^{r}(z)} \bigl\vert k_{\varphi ,z}(w) \bigr\vert ^{2}e^{-2\varphi (w)}\,d\mu (w) \leq \widetilde{\mu}(z). \end{aligned} $$

Notice that

$$ \begin{aligned} \langle T_{\mu}e_{k},e_{k} \rangle _{A^{2}_{\varphi}}&= \int _{ \mathbb{D}} \bigl\vert e_{k}(z) \bigr\vert ^{2}e^{-2\varphi (z)}\,d\mu (z) \\ &\lesssim \int _{\mathbb{D}} \widehat{\mu}_{r}(z) \bigl\vert e_{k}(z) \bigr\vert ^{2}e^{-2 \varphi (z)}\,dA(z) \\ &\lesssim \int _{\mathbb{D}}\widetilde{\mu}(z) \bigl\vert e_{k}(z) \bigr\vert ^{2}e^{-2 \varphi (z)}\,dA(z), \end{aligned} $$

then by Jensen’s inequality again we get

$$ \begin{aligned} &\sum^{\infty}_{k=1} h\bigl(c\langle T_{\mu}e_{k},e_{k} \rangle _{A^{2}_{\varphi}}\bigr) \\ &\quad\leq \int _{\mathbb{D}} h\bigl(c\widetilde{\mu}(z)\bigr) \Biggl( \sum ^{\infty}_{k=1} \bigl\vert e_{k}(z) \bigr\vert ^{2}\Biggr)e^{-2\varphi (z)}\,dA(z) \\ &\quad= \int _{\mathbb{D}} h\bigl(c\widetilde{\mu}(z)\bigr) \Vert K_{\varphi ,z} \Vert ^{2}_{ \varphi ,2}e^{-2\varphi (z)}\,dA(z) \\ &\quad= \int _{\mathbb{D}} h\bigl(c\widetilde{\mu}(z)\bigr)\rho ^{-2}(z)\,dA(z)< \infty , \end{aligned} $$

which gives $T_{\mu}\in S_{h}$. This completes the proof. □

4 Schatten class Hankel operators

This section devotes to studying membership in Schatten ideals of Hankel operators with general symbols. First, when $0< p<\infty$, we get the sufficient and necessary conditions for Hankel operators are in Schatten-p class. Here we mainly discuss case $0< p<1$, see case $1\leq p<\infty$ in [13]. Next, for a continuous increasing convex function h, we obtain the sufficient and necessary conditions for Hankel operators to be in Schatten-h class. This kind of problem is new for Hankel operators.

Lemma 4.1

If A and B are bounded linear operators, $p \in (0,1)$, then

$$\begin{aligned} \Vert AB \Vert ^{p}_{S_{p}} \leq \vert \vert B \vert \vert ^{p} \vert \vert A \vert \vert ^{p}_{S_{p}} \quad {\mathrm{and}} \quad \vert \vert AB \vert \vert ^{p}_{S_{p}} \leq \vert \vert A \vert \vert ^{p} \vert \vert B \vert \vert ^{p}_{S_{p}}. \end{aligned}$$

(4.1)

Proof

See [6]. □

Let $L^{2}_{\mathrm{loc}}(\mathbb{D})$ denote the space consists of locally square integrable Lebesgue measurable functions on $\mathbb{D}$. If $f \in L^{2}_{\mathrm{loc}}(\mathbb{D})$ and $z \in \mathbb{D}$, $G_{r}(f)(z)$ is defined by

$$ G_{r}(f) (z)=\inf \biggl\{ \biggl(\frac{1}{ \vert D^{r}(z) \vert } \int _{D^{r}(z)} \vert f-h \vert ^{2}\,dA \biggr)^{1/2}: h \in \mathcal{H}\bigl(D^{r}(z)\bigr) \biggr\} , $$

where $\mathcal{H}(D^{r}(z))$ is the analytic functions space on $D^{r}(z)$. For $z \in \mathbb{D}$, $f \in L^{2}(D^{r}(z),dA)$ and $r>0$, the averaging function of $|f|$ on $D^{r}(z)$ is defined by

$$ M_{r}(f) (z)= \biggl(\frac{1}{ \vert D^{r}(z) \vert } \int _{D^{r}(z)} \vert f \vert ^{2}\,dA \biggr)^{1/2}. $$

Indeed, $M_{r}(f)(z)=(\widehat{|f|^{2}}_{r})^{1/2}$.

Lemma 4.2

For $z \in \mathbb{D}$, $f \in L^{2}(D^{r}(z),dA)$, and $r>0$, there exists an $h \in \mathcal{H}(D^{r}(z))$ such that

$$\begin{aligned} M_{r}(f-h) (z)=G_{r}(f) (z). \end{aligned}$$

(4.2)

Proof

The proof is similar to [11, Lemma 3.3]. □

For $z \in \mathbb{D}$ and $r>0$, let

$$ A^{2}\bigl(D^{r}(z),dA\bigr)=L^{2} \bigl(D^{r}(z),dA\bigr)\cap \mathcal{H}\bigl(D^{r}(z)\bigr) $$

denote the Bergman space on $D^{r}(z)$. Let $B_{z,r}$ denote Bergman projection induced by the reproducing kernel of $A^{2}(D^{r}(z),dA)$. As we known, $B_{z,r}$ is bounded and $B_{z,r}h=h$, where $h \in A^{2}(D^{r}(z),dA)$. The following consequence is similar to [11, Lemma 3.4] with $q=2$.

Lemma 4.3

For $z \in \mathbb{D}$ and $r>0$, if $f \in L^{2}(D^{r}(z),dA)$, then we have

$$\begin{aligned} M_{r}\bigl(f-B_{z,r}(f)\bigr) (z) \asymp G_{r}(f) (z). \end{aligned}$$

(4.3)

Proof

Taking h from Lemma 4.2, we have $h \in A^{2}(D^{r}(z),dA)$ since $f \in L^{2}_{\mathrm{loc}}(\mathbb{D})$. Then $B_{z,r}h=h$. By trigonometric inequality and Lemma 4.2,

$$\begin{aligned} M_{r}\bigl(f-B_{z,r}(f)\bigr) (z) &\leq M_{r}(f-h) (z)+M_{r}\bigl(h-B_{z,r}(f)\bigr) (z) \\ &=M_{r}(f-h) (z)+M_{r}\bigl(B_{z,r}(h-f)\bigr) (z) \\ &\lesssim M_{r}(f-h) (z)=G_{r}(f) (z). \end{aligned}$$

It is obvious that $G_{r}(f)(z) \leq M_{r}(f-B_{z,r}(f))(z)$, and hence this proof is complete. □

Given $r>0$, let $\{a_{j}\}_{j=1}^{\infty}$ be a $(\rho ,r/3)$-lattice, $J_{z}=\{j:z \in D^{r}(a_{j})\}$, and $|J_{z}|$ be the number of elements of $J_{z}$. By (2.3), $1 \leq |J_{z}| \leq N$. Let $\{\psi _{j}\}_{j=1}^{\infty}$ denote the unit decomposition induced by $\{D^{r/3}(a_{j})\}_{j=1}^{\infty}$, that is,

$$ \psi _{j} \in C^{\infty}(\mathbb{D}),\quad \operatorname{supp} \psi _{j} \subseteq D^{r/3}(a_{j}), \qquad \vert \bar{ \partial}\psi _{j} \vert \leq C\rho (a_{j})^{-1}, \qquad \sum_{j=1}^{\infty} \psi _{j}=1, \quad \psi _{j}\geq 0. $$

By (2.1), it is easy to see

$$ \rho (z) \bigl\vert \bar{\partial}\psi _{j}(z) \bigr\vert \leq C, \quad \text{for any } j=1,2, \dots \text{ and } z\in \mathbb{D}. $$

Given $f \in L^{2}_{\mathrm{loc}}(\mathbb{D})$, for $j=1,2,\dots$, taking $h_{j} \in \mathcal{H}(D^{r}(a_{j}))$ in Lemma 4.2 such that

$$\begin{aligned} M_{r}(f-h_{j})=G_{r}(f) (a_{j}). \end{aligned}$$

Definition 4.4

By the decomposition above, we define

$$\begin{aligned} f_{1}=\sum_{j=1}^{\infty}h_{j} \psi _{j} \quad \text{and }\quad f_{2}=f-f_{1}. \end{aligned}$$

(4.4)

Note that $f_{1}(z)$ is actually a finite summation for any $z \in \mathbb{D}$, and by $\operatorname{supp} \psi _{j} \subseteq D^{r/3}(a_{j}) \subseteq D^{r}(a_{j})$, then $f_{1}$ is well-defined.

Lemma 4.5

Let $f \in L^{2}_{\mathrm{loc}}(\mathbb{D})$ and $r>0$. By (4.4), f admits a decomposition $f=f_{1}+f_{2}$. Then $f_{1} \in C^{1}(\mathbb{D})$ and

$$\begin{aligned} \bigl\vert \rho (z)\bar{\partial}f_{1}(z) \bigr\vert +M_{r/9}(\rho \bar{\partial}f_{1}) (z)+M_{r/9}(f_{2}) (z) \leq CG_{9r}(f) (z), \end{aligned}$$

(4.5)

where $z \in \mathbb{D}$ and $C>0$ is independent of f.

Proof

Since $h_{j} \in \mathcal{H}(D^{r}(a_{j}))$ and $\psi _{j} \in C^{\infty}(\mathbb{D})$, $f_{1} \in C^{1}(\mathbb{D})$. For $z \in \mathbb{D}$, without loss of generality, we may assume $z \in D^{r/3}(a_{1})$. It is easy to check that $D^{r/9}(z) \subseteq D^{r}(a_{j})$ whenever $z \in D^{r/3}(a_{j})$. By $\sum_{j=1}^{\infty}\bar{\partial} \psi _{j}(z)=0$ and the subharmonic property of $|h_{j}-h_{1}|$ on $D^{r/9}(z) \subseteq D^{r}(a_{j})$,

$$\begin{aligned} \bigl\vert \rho (z)\bar{\partial}f_{1}(z) \bigr\vert &= \Biggl\vert \sum_{j=1}^{\infty}\bigl(h_{j}(z)-h_{1}(z) \bigr) \rho (z)\bar{\partial}\psi _{j}(z) \Biggr\vert \\ &\leq \sum_{j=1}^{\infty} \bigl\vert h_{j}(z)-h_{1}(z) \bigr\vert \bigl\vert \rho (z)\bar{ \partial} \psi _{j}(z) \bigr\vert \\ &\leq C\sum_{\{j:z \in D^{r/3}(a_{j})\}}M_{r/9}(h_{j}-h_{1}) (z) \\ &\leq C\sum_{\{j:z \in D^{r/3}(a_{j})\}}\bigl[M_{r/9}(f-h_{j}) (z)+M_{r/9}(f-h_{1}) (z)\bigr] \\ &\lesssim \sum_{\{j:z \in D^{r/3}(a_{j})\}}G_{r}(f) (a_{j}). \end{aligned}$$

If $z \in D^{r/3}(a_{j})$, then we have $D^{r}(a_{j}) \subseteq D^{9r}(z)$, and

$$ G_{r}(f) (a_{j}) \leq C G_{9r}(f) (z). $$

Hence,

$$\begin{aligned} \bigl\vert \rho (z)\bar{\partial}f_{1}(z) \bigr\vert \leq CG_{9r}(f) (z). \end{aligned}$$

(4.6)

If $w \in D^{r/9}(z)$, then $D^{3r}(w) \subseteq D^{9r}(z)$. Thus, similar to (4.6),

$$\begin{aligned} M_{r/9}(\rho \bar{\partial}f_{1}) (z)^{2} &\leq C\rho (z)^{-2} \int _{D^{r/9}(z)}G_{3r}(f) (w)^{2}\,dA(w) \\ &\leq CG_{9r}(f) (z)^{2}, \end{aligned}$$

(4.7)

since $G_{3r}(f)(w) \leq CG_{9r}(f)(z)$ for $w\in D^{r/9}(z)$.

Using Cauchy–Schwarz inequality,

$$ \bigl\vert f_{2}(z) \bigr\vert ^{2} \leq \sum _{j=1}^{\infty} \bigl\vert f(z)-h_{j}(z) \bigr\vert ^{2}\psi _{j}(z). $$

Therefore,

$$\begin{aligned} M_{r/9}(f_{2}) (z)^{2} &\leq \sum _{j=1}^{\infty} \frac{1}{ \vert D^{r/9}(z) \vert } \int _{D^{r/9}(z)} \vert f-h_{j} \vert ^{2} \psi _{j}\,dA \\ &\leq C \sum_{\{j:z \in D^{r/3}(a_{j})\}}\frac{1}{ \vert D^{r}(a_{j}) \vert } \int _{D^{r}(a_{j})} \vert f-h_{j} \vert ^{2}\,dA \\ &=C \sum_{\{j:z \in D^{r}(a_{j})\}}G_{r}(f) (a_{j})^{2} \\ &\leq CG_{9r}(f) (z)^{2}. \end{aligned}$$

This finishes the proof. □

Lemma 4.6

Let $0< p<\infty$ and $f \in L^{2}_{\mathrm{loc}}(\mathbb{D})$. Then following statements are equivalent:

(a) For some (or any) $r \leq m_{\rho}$, $M_{r}(f)(z) \in L^{p}(\mathbb{D},\rho ^{-2}\,dA)$.

(b) For some (or any) $r \leq m_{\rho}$, $\{a_{j}\}_{j=1}^{\infty}$ is a $(\rho ,\delta )$-lattice with $\delta \leq r$, then the sequence $\{M_{\delta}(f)(a_{j})\}_{j=1}^{\infty}\in l^{p}$, and

$$\begin{aligned} \bigl\Vert M_{r}(f) \bigr\Vert _{L^{p}(\mathbb{D},\rho ^{-2}\,dA)} \asymp \bigl\Vert \bigl\{ M_{\delta}(f) (a_{j}) \bigr\} _{j=1}^{\infty} \bigr\Vert _{l^{p}}. \end{aligned}$$

(4.8)

Proof

The proof is an analogue of [17, Proposition 2.4]. □

For $z \in \mathbb{D}$ and $r>0$, we denote $L^{2}(D^{r}(z),e^{-2\varphi}\,dA)=L^{2}_{\varphi}(D^{r}(z))$ and $A^{2}_{\varphi}(D^{r}(z))=L^{2}_{\varphi}(D^{r}(z))\cap \mathcal{H}(D^{r}(z))$. Let $P_{z,r}:L^{2}_{\varphi}(D^{r}(z))\rightarrow A^{2}_{\varphi}(D^{r}(z))$ be the projection. Given $f \in L^{2}_{\varphi}(D^{r}(z))$, we may assume $P_{z,r}(f)(w)=0$, when $w \in \mathbb{D}\setminus D^{r}(z)$, it follows that $P_{z,r}(f)$ is a natural extension on $\mathbb{D}$. If $f,g \in L^{2}_{\varphi}$, then it is easy to see $f,g \in L^{2}_{\varphi}(D^{r}(z))$. Then, for $f,g \in L^{2}_{\varphi}$, we have $P^{2}_{z,r}(f)=P_{z,r}(f)$ and $\langle f,P_{z,r}(g)\rangle =\langle P_{z,r}(f),g\rangle$. Also, if $h \in A^{2}_{\varphi}$, then $P_{z,r}(h)=\chi _{D^{r}(z)}h$, and hence

$$\begin{aligned} \langle h,\chi _{D^{r}(z)}g\rangle &=\langle \chi _{D^{r}(z)}h,g \rangle =\bigl\langle P_{z,r}(h),g\bigr\rangle =\bigl\langle h,P_{z,r}(g)\bigr\rangle , \quad g \in L^{2}_{\varphi}. \end{aligned}$$

Equivalently,

$$\begin{aligned} \bigl\langle h,\chi _{D^{r}(z)}g-P_{z,r}(g)\bigr\rangle =0. \end{aligned}$$

(4.9)

Lemma 4.7

If $f,g \in L^{2}_{\varphi}$, then

$$\begin{aligned} \bigl\langle f-P(f),\chi _{D^{r}(z)}g-P_{z,r}(g)\bigr\rangle = \bigl\langle \chi _{D^{r}(z)}f-P_{z,r}(f), \chi _{D^{r}(z)}g-P_{z,r}(g)\bigr\rangle . \end{aligned}$$

Proof

See [10, Lemma 5.1]. □

By [13], $H_{f}:A^{2}_{\varphi }\to L^{2}_{\varphi}$ is bounded if and only if $G_{r}(f) \in L^{\infty}$. In fact, $G_{r}(f) \in L^{\infty}$ is independent of r. Further, $\|G_{r}(f)\|_{L^{\infty}} \asymp \|G_{\delta}(f)\|_{L^{\infty}}$. Suppose $G_{r}(f) \in L^{\infty}$, it is from Lemma 4.5 that

$$\begin{aligned} \vert \vert M_{r}(f_{2}) \vert \vert _{L^{\infty}} \lesssim \vert \vert G_{r}(f) \vert \vert _{L^{\infty}}. \end{aligned}$$

(4.10)

Hence, the condition $G_{r}(f) \in L^{\infty}$ is natural in the study of Schatten class membership of Hankel operators.

Lemma 4.8

Suppose $\varphi \in \mathcal{E}$, $r\in (0, m_{\rho}]$, $H_{f}$ is densely defined satisfying $G_{r}(f) \in L^{\infty}$ and the decomposition $f=f_{1}+f_{2}$ by Lemma 4.5. Then both $H_{f_{1}}$ and $H_{f_{2}}$ are bounded, and

$$ \vert \vert H_{f_{1}}(g) \vert \vert _{L^{2}_{\varphi}} \lesssim \vert \vert g\rho \bar{\partial}f_{1} \vert \vert _{L^{2}_{ \varphi}} \quad {\mathrm{and}} \quad \vert \vert H_{f_{2}}(g) \vert \vert _{L^{2}_{\varphi}} \lesssim \vert \vert f_{2}g \vert \vert _{L^{2}_{\varphi}}. $$

Proof

See [13, Theorem 3.1]. □

Now we are ready for the characterization of Schatten class Hankel operators.

Theorem 4.9

Suppose $\varphi \in \mathcal{E}$, $0< p<\infty$, $0< r\leq m_{\rho}$ and $H_{f}$ is densely defined satisfying $G_{r}(f) \in L^{\infty}$. Then following statements are equivalent:

(a) The Hankel operator $H_{f}$ is in $S_{p}$.

(b) For some (or any) (ρ, r)-lattice $\{a_{j}\}_{j=1}^{\infty}$, $\{G_{r}(f)(a_{j})\}_{j=1}^{\infty }\in l^{p}$.

(c) For some (or any) r, $G_{r}(f) \in L^{p}(\mathbb{D},\rho ^{-2}\,dA)$.

(d) For some (or any) r, f admits a decomposition $f=f_{1}+f_{2}$ such that $f_{1} \in C^{1}(\mathbb{D})$, $M_{r}(\rho \bar{\partial}f_{1}) \in L^{p}(\mathbb{D},\rho ^{-2}\,dA)$ and $M_{r}(f_{2}) \in L^{p}(\mathbb{D},\rho ^{-2}\,dA)$.

(e) For some (or any) (ρ, r)-lattice $\{a_{j}\}_{j=1}^{\infty}$, f admits a decomposition $f=f_{1}+f_{2}$ such that $f_{1} \in C^{1}(\mathbb{D})$, $\{M_{r}(\rho \bar{\partial}f_{1})(a_{j}) \}_{j=1}^{\infty}\in l^{p}$ and $\{M_{r}(f_{2})(a_{j})\}_{j=1}^{\infty }\in l^{p}$.

Proof

(a) ⇒ (b). We give only the case $0< p<1$. Let $\{a_{j}\}_{j=1}^{\infty}$ be a (ρ, r)-lattice. By Lemma 2.9, $\{a_{j}\}_{j=1}^{\infty}$ can be devided into N subsequences, if $a_{i}$ and $a_{j}$ are in the same subsequence, then

$$\begin{aligned} \vert a_{i}-a_{j} \vert \geq 2^{k}r\min \bigl(\rho (a_{i}),\rho (a_{j}) \bigr). \end{aligned}$$

(4.11)

In fact, just consider one of subsequences here. Without loss of generality, it is assumed that $\{a_{j}\}_{j=1}^{\infty}$. For any finite subset $J\subseteq \mathbb{N}^{+}$, let $\{e_{j}\}_{j=1}^{\infty}$ be an orthonormal basis for $A^{2}_{\varphi}$, and

$$ A(g)=\sum_{j \in J} \langle g,e_{j} \rangle k_{\varphi ,a_{j}}, \quad g\in A^{2}_{\varphi}. $$

Then, by Parseval’s equality,

$$ \sum_{j\in J} \bigl\vert \langle g,e_{j} \rangle \bigr\vert ^{2} \leq \sum_{j=1}^{\infty} \bigl\vert \langle g,e_{j} \rangle \bigr\vert ^{2}= \Vert g \Vert ^{2}_{\varphi}. $$

It follows from Lemma 2.7 that A is bounded on $A^{2}_{\varphi}$.

If $\|\chi _{D^{r}(a_{j})}gk_{\varphi ,a_{j}}-P_{a_{j},r}(gk_{\varphi ,a_{j}})\|_{L^{2}_{ \varphi}} \neq 0$, we let

$$\begin{aligned} h_{j}= \frac{\chi _{D^{r}(a_{j})}fk_{\varphi ,a_{j}}-P_{a_{j},r}(fk_{\varphi ,a_{j}})}{ \Vert \chi _{D^{r}(a_{j})}fk_{\varphi ,a_{j}} -P_{a_{j},r}(fk_{\varphi ,a_{j}}) \Vert _{L^{2}_{\varphi}}}, \end{aligned}$$

and $h_{j}=0$ otherwise. It is easy to see $\|h_{j}\|^{2}_{\varphi }\leq 1$. Assume $D^{r}(a_{i}) \cap D^{r}(a_{j}) \neq \emptyset$, then $|a_{i}-a_{j}| \leq 3r\min \{\rho (a_{i}),\rho (a_{j})\}$. For k large enough, we have $D^{r}(a_{i}) \cap D^{r}(a_{j}) = \emptyset$ whenever $i \neq j$. Hence, $\langle h_{i},h_{j} \rangle =0$ if $i \neq j$.

Let $\{c_{j}\}_{j \in J}$ denote nonnegative sequence, we define the operator B by

$$ B(g)=\sum_{j\in J}c_{j} \langle g,h_{j} \rangle e_{j}. $$

It is easy to check that B is bounded on $A^{2}_{\varphi}$, and $\|B\| \leq \sup_{j \in J}\{c_{j}\}$. It follows that

$$\begin{aligned} BH_{f}A(g)&=\sum_{j\in J}c_{j} \bigl\langle H_{f}A(g), h_{j} \bigr\rangle e_{j} \\ &=\sum_{j\in J}\sum_{i\in J}c_{j} \langle H_{f}k_{\varphi ,a_{i}}, h_{j} \rangle \langle g,e_{i} \rangle e_{j}. \end{aligned}$$

The application of Lemma 4.1 gives

$$\begin{aligned} \Vert BH_{f}A \Vert ^{p}_{S_{p}} \leq \vert \vert B \vert \vert ^{p} \vert \vert H_{f} \vert \vert ^{p}_{S_{p}} \vert \vert A \vert \vert ^{p} \leq C\sup_{j \in J}c^{p}_{j}. \end{aligned}$$

Taking a decomposition of the operator $BH_{f}A$ as the diagonal part

$$ Y(g)=\sum_{j\in J}c_{j} \langle H_{f}k_{\varphi ,a_{j}}, h_{j} \rangle \langle g,e_{j} \rangle e_{j} $$

and the non-diagonal part

$$ Z(g)=\sum_{j,i\in J:i\neq j}c_{j} \langle H_{f}k_{\varphi ,a_{i}}, h_{j} \rangle \langle g,e_{i} \rangle e_{j}, $$

we have, by (3.2),

$$ \vert \vert Y \vert \vert ^{p}_{S_{p}} \lesssim \vert \vert BH_{f}A \vert \vert ^{p}_{S_{p}}+ \vert \vert Z \vert \vert ^{p}_{S_{p}}. $$

By Lemma 2.4, there exists a constant $C>0$ such that for $z \in D^{r}(a_{j})$

$$ \bigl\vert k_{\varphi ,a_{j}}(z) \bigr\vert \geq Ce^{\varphi (z)}\rho (a_{j})^{-1}>0, $$

and hence $k_{\varphi ,a_{j}}^{-1} \in \mathcal {H}(D^{r}(a_{j}))$. According to Lemma 4.7 and (2.5),

$$\begin{aligned} \Vert Y \Vert ^{p}_{S_{p}}&=\sum _{j\in J}c_{j}^{p} \bigl\vert \langle H_{f}k_{\varphi ,a_{j}},h_{j} \rangle \bigr\vert ^{p}=\sum_{j\in J}c_{j}^{p} \bigl\vert \bigl\langle fk_{\varphi ,a_{j}}-P(fk_{ \varphi ,a_{j}}),h_{j} \bigr\rangle \bigr\vert ^{p} \\ &=\sum_{j\in J}c_{j}^{p} \bigl\vert \bigl\langle \chi _{D^{r}(a_{j})}fk_{\varphi ,a_{j}}-P_{a_{j},r}(fk_{ \varphi ,a_{j}}),h_{j} \bigr\rangle \bigr\vert ^{p} \\ &=\sum_{j\in J}c_{j}^{p} \bigl\Vert \chi _{D^{r}(a_{j})}fk_{\varphi ,a_{j}}-P_{a_{j},r}(fk_{ \varphi ,a_{j}}) \bigr\Vert _{L^{2}_{\varphi}}^{p} \\ &=\sum_{j\in J}c_{j}^{p} \biggl\{ \int _{D^{r}(a_{j})} \bigl\vert fk_{\varphi ,a_{j}}-P_{a_{j},r}(fk_{ \varphi ,a_{j}}) \bigr\vert ^{2}e^{-2\varphi}\,dA \biggr\} ^{p/2} \\ &=\sum_{j\in J}c_{j}^{p} \biggl\{ \int _{D^{r}(a_{j})} \vert k_{\varphi ,a_{j}} \vert ^{2}e^{-2 \varphi} \bigl\vert f-k_{\varphi ,a_{j}}^{-1}P_{a_{j},r}(fk_{\varphi ,a_{j}}) \bigr\vert ^{2}\,dA \biggr\} ^{p/2} \\ &\asymp \sum_{j\in J}c_{j}^{p} \biggl\{ \frac{1}{ \vert D^{r}(a_{j}) \vert } \int _{D^{r}(a_{j})} \bigl\vert f-k_{ \varphi ,a_{j}}^{-1}P_{a_{j},r}(fk_{\varphi ,a_{j}}) \bigr\vert ^{2}\,dA \biggr\} ^{p/2} \\ &\geq \sum_{j\in J}c_{j}^{p}G_{r}(f) (a_{j})^{p}. \end{aligned}$$

This together with [18, Proposition 1.29], Lemma 4.7, and Cauchy–Schwarz inequality yields

$$\begin{aligned} \Vert Z \Vert ^{p}_{S_{p}} &\leq \sum _{n=1}^{\infty}\sum_{m=1}^{\infty } \bigl\vert \langle Ze_{n}, e_{m} \rangle \bigr\vert ^{p}=\sum_{j,i\in J:i\neq j}c_{j}^{p} \bigl\vert \langle H_{f}k_{\varphi ,a_{i}}, h_{j} \rangle \bigr\vert ^{p} \\ &=\sum_{j,i\in J:i\neq j}c_{j}^{p} \bigl\vert \langle \chi _{D^{r}(a_{j})}fk_{ \varphi ,a_{i}}-P_{a_{j},r}fk_{\varphi ,a_{i}}, h_{j} \rangle \bigr\vert ^{p} \\ & \leq \sum_{j,i\in J:i\neq j}c_{j}^{p} \Vert \chi _{D^{r}(a_{j})}fk_{ \varphi ,a_{i}}-P_{a_{j},r}fk_{\varphi ,a_{i}} \Vert _{\varphi ,2}^{p} \\ &=\sum_{j,i\in J:i\neq j}c_{j}^{p} \biggl\{ \int _{D^{r}(a_{j})} \bigl\vert fk_{ \varphi ,a_{i}}-P_{a_{j},r}(fk_{\varphi ,a_{i}}) \bigr\vert ^{2}e^{-2\varphi}\,dA \biggr\} ^{p/2} \\ &\leq \sum_{j,i\in J:i\neq j}c_{j}^{p} \biggl\{ \int _{D^{r}(a_{j})} \bigl\vert fk_{ \varphi ,a_{i}}-k_{\varphi ,a_{i}}B_{a_{j},r}(f) \bigr\vert ^{2}e^{-2\varphi}\,dA \biggr\} ^{p/2}, \end{aligned}$$

where $B_{z,r}$ is the projection from $L^{2}(D^{r}(z))$ to $A^{2}(D^{r}(z))$. Hence, by Lemma 4.3,

$$\begin{aligned} \Vert Z \Vert ^{p}_{S_{p}}\leq{}& \sum _{j \in J}c_{j}^{p} \biggl\{ \int _{D^{r}(a_{j})} \sum_{i\in J:i\neq j} \bigl( \vert k_{\varphi ,a_{i}} \vert ^{2}e^{-2\varphi} \bigr) \bigl\vert f-B_{a_{j},r}(f) \bigr\vert ^{2}\,dA \biggr\} ^{p/2} \\ \lesssim{}& \sum_{j\in J}c_{j}^{p} \sup_{z\in D^{r}(a_{j})} \biggl( \sum_{i\in J:i\neq j} \bigl\vert k_{\varphi ,a_{i}}(z) \bigr\vert ^{2}e^{-2\varphi (z)} \biggr)^{p/2}\rho (a_{j})^{p} \\ &{} \cdot \biggl\{ \frac{1}{ \vert D^{r}(a_{j}) \vert } \int _{D^{r}(a_{j})} \bigl\vert f-B_{a_{j},r}(f) \bigr\vert ^{2}\,dA \biggr\} ^{p/2} \\ \asymp {}&\sum_{j\in J}c_{j}^{p}G_{r}(f) (a_{j})^{p}\rho (a_{j})^{p} \sup _{z\in D^{r}(a_{j})} \biggl(\sum_{i\in J:i\neq j} \bigl\vert k_{\varphi ,a_{i}}(z) \bigr\vert ^{2}e^{-2 \varphi (z)} \biggr)^{p/2}. \end{aligned}$$

(4.12)

Let $i,j \in J$ and $i \neq j$. Then there exists $w_{j,i} \in \overline{D^{r}(a_{j})}$ such that

$$ \vert w_{j,i}-a_{i} \vert =\inf_{z \in D^{r}(a_{j})} \vert z-a_{i} \vert . $$

This combined with (2.6) and (2.1), for $z \in D^{r}(a_{j})$, implies

$$\begin{aligned} \bigl\vert k_{\varphi ,a_{i}}(z) \bigr\vert ^{2}e^{-2\varphi (z)} & \leq C \frac{1}{\rho (z)} \biggl( \frac{\min (\rho (z),\rho (a_{i}))}{ \vert z-a_{i} \vert } \biggr)^{N} \bigl\vert k_{ \varphi ,a_{i}}(z) \bigr\vert e^{-\varphi (z)} \\ &\lesssim \frac{1}{\rho (a_{j})} \biggl( \frac{\min (2\rho (w_{j,i}),\rho (a_{i}))}{ \vert w_{j,i}-a_{i} \vert } \biggr)^{N} \bigl\vert k_{ \varphi ,a_{i}}(z) \bigr\vert e^{-\varphi (z)} \\ &\lesssim \frac{1}{\rho (a_{j})} \biggl( \frac{\min (\rho (w_{j,i}),\rho (a_{i}))}{ \vert w_{j,i}-a_{i} \vert } \biggr)^{N} \bigl\vert k_{ \varphi ,a_{i}}(z) \bigr\vert e^{-\varphi (z)}. \end{aligned}$$

We claim that $|w_{j,i}-a_{i}|\geq 2^{k-2}r\min (\rho (w_{j,i}),\rho (a_{i}))$. If not, we assume $|w_{j,i}-a_{i}| \leq 2^{k-2}r \min (\rho (w_{j,i}),\rho (a_{i}))$. By (2.1) and the trigonometric inequality,

$$\begin{aligned} \vert a_{j}-a_{i} \vert &\leq \vert a_{j}-w_{j,i} \vert + \vert w_{j,i}-a_{i} \vert \leq r\rho (a_{j})+2^{k-2}r\rho (w_{j,i})< 2^{k}r\rho (a_{j}), \end{aligned}$$

and

$$\begin{aligned} \vert a_{j}-a_{i} \vert &\leq \vert a_{j}-w_{j,i} \vert + \vert w_{j,i}-a_{i} \vert \\ &\leq r\rho (a_{j})+2^{k-2}r\rho (w_{j,i}) \\ &\leq 2r\rho (w_{j,i})+2^{k-2}r\rho (w_{j,i}) \\ &\leq 4r\rho (a_{i})+2^{k-1}r\rho (a_{i})< 2^{k}r\rho (a_{i}). \end{aligned}$$

So $|a_{j}-a_{i}| < 2^{k}r\min (\rho (a_{j}),\rho (a_{i}))$, which causes a contradiction with (4.11). Thus, for $z \in D^{r}(a_{j})$,

$$\begin{aligned} \bigl\vert k_{\varphi ,a_{i}}(z) \bigr\vert ^{2}e^{-2\varphi (z)} &\lesssim \frac{1}{\rho (a_{j})} \biggl( \frac{1}{2^{k-2}r} \biggr)^{N} \bigl\vert k_{ \varphi ,a_{i}}(z) \bigr\vert e^{-\varphi (z)} \\ &\lesssim \frac{1}{\rho (a_{j})} \biggl(\frac{1}{2} \biggr)^{Nk} \bigl\vert k_{ \varphi ,a_{i}}(z) \bigr\vert e^{-\varphi (z)}. \end{aligned}$$

(4.13)

By joining (4.12) and (4.13), we obtain

$$\begin{aligned} \Vert Z \Vert ^{p}_{S_{p}} \lesssim \biggl( \frac{1}{2} \biggr)^{\frac{Npk}{2}} \sum_{j\in J}c_{j}^{p}G_{r}(f) (a_{j})^{p}\rho (a_{j})^{\frac{p}{2}} \sup _{z\in D^{r}(a_{j})} \biggl(\sum_{i\in J:i\neq j} \bigl\vert k_{\varphi ,a_{i}}(z) \bigr\vert e^{- \varphi (z)} \biggr)^{p/2}. \end{aligned}$$

(4.14)

Set $r_{0}=3r$. Fix $j \in J$, then, for any $z \in D^{r}(a_{j})$, we have

$$\begin{aligned} \sum_{i\in J:i\neq j} \bigl\vert k_{\varphi ,a_{i}}(z) \bigr\vert e^{-\varphi (z)} \leq{}& \sum_{i=1:i\neq j}^{\infty} \bigl\vert k_{\varphi ,a_{i}}(z) \bigr\vert e^{-\varphi (z)} \\ ={}&\sum_{\{a_{i}: \vert a_{j}-a_{i} \vert \leq r_{0}\rho (a_{j})\}} \bigl\vert k_{\varphi ,a_{i}}(z) \bigr\vert e^{- \varphi (z)}\\ &{}+\sum_{n=1}^{\infty } \sum_{\{a_{i}:2^{n}r_{0}\rho (a_{j})< \vert a_{j}-a_{i} \vert \leq 2^{n+1}r_{0}\rho (a_{j})\}} \bigl\vert k_{\varphi ,a_{i}}(z) \bigr\vert e^{-\varphi (z)}\\ ={}&I+II. \end{aligned}$$

It is from (2.1) that, for any $z \in D^{r}(a_{j})$,

$$\begin{aligned} I=\sum_{\{a_{i}: \vert a_{j}-a_{i} \vert \leq r_{0}\rho (a_{j})\}} \bigl\vert k_{\varphi ,a_{i}}(z) \bigr\vert e^{- \varphi (z)} \lesssim \sum _{\{a_{i}: \vert a_{j}-a_{i} \vert \leq r_{0}\rho (a_{j}) \}}\frac{1}{\rho (a_{i})} \lesssim \frac{1}{\rho (a_{j})}. \end{aligned}$$

(4.15)

When $2^{n}r_{0}\rho (a_{j})<|a_{j}-a_{i}| \leq 2^{n+1}r_{0}\rho (a_{j})$, we get

$$\begin{aligned} \vert w_{j,i}-a_{i} \vert &\geq \vert a_{j}-a_{i} \vert - \vert a_{i}-w_{j,i} \vert \\ &>2^{n}r_{0}\rho (a_{j})-r\rho (a_{j}) \\ &= \biggl(2^{n}-\frac{1}{3} \biggr)r_{0}\rho (a_{j})\geq \frac{1}{2} \cdot 2^{n}r_{0} \rho (a_{j}). \end{aligned}$$

For any $z \in D^{r}(a_{j})$,

$$\begin{aligned} II&=\sum_{n=1}^{\infty }\sum _{\{a_{i}:2^{n}r_{0}\rho (a_{j})< \vert a_{j}-a_{i} \vert \leq 2^{n+1}r_{0}\rho (a_{j})\}} \bigl\vert k_{\varphi ,a_{i}}(z) \bigr\vert e^{-\varphi (z)} \\ &\lesssim \sum_{n=1}^{\infty }\sum _{\{a_{i}:2^{n}r_{0}\rho (a_{j})< \vert a_{j}-a_{i} \vert \leq 2^{n+1}r_{0}\rho (a_{j})\}}\frac{1}{\rho (z)} \biggl( \frac{\min (\rho (a_{i}),\rho (z))}{ \vert w_{j,i}-a_{i} \vert } \biggr)^{N} \\ &\lesssim \sum_{n=1}^{\infty } \biggl( \frac{1}{2} \biggr)^{nN}\rho (a_{j})^{-1-N+(N-2)} \sum_{\{a_{i}:2^{n}r_{0}\rho (a_{j})< \vert a_{j}-a_{i} \vert \leq 2^{n+1}r_{0} \rho (a_{j})\}}\rho (a_{i})^{2}. \end{aligned}$$

It is clear that for $n=1,2,\dots$, if $a_{i} \in D^{r_{0}2^{n+1}}(a_{j})$, we have

$$ D^{r_{0}}(a_{i}) \subseteq D^{Cr_{0}2^{n}}(a_{j}). $$

Hence

$$\begin{aligned} \sum_{\{a_{i}:2^{n}r_{0}\rho (a_{j})< \vert a_{j}-a_{i} \vert \leq 2^{n+1}r_{0} \rho (a_{j})\}}\rho (a_{i})^{2} \lesssim \bigl\vert D^{Cr_{0}2^{n}}(a_{j}) \bigr\vert \asymp 2^{2n}\rho (a_{j})^{2}. \end{aligned}$$

Choose $N-2>1$ such that

$$ \sum_{n=1}^{\infty } \biggl(\frac{1}{2} \biggr)^{n(N-2)} \leq C. $$

So, for $z \in D^{r}(a_{j})$,

$$\begin{aligned} \sum_{n=1}^{\infty }\sum _{\{a_{i}:2^{n}r_{0}\rho (a_{j})< \vert a_{j}-a_{i} \vert \leq 2^{n+1}r_{0}\rho (a_{j})\}} \bigl\vert k_{\varphi ,a_{i}}(z) \bigr\vert e^{-\varphi (z)} \lesssim \frac{1}{\rho (a_{j})}. \end{aligned}$$

(4.16)

For any $z \in D^{r}(a_{j})$, by (4.15) and (4.16), we see

$$\begin{aligned} \sup_{z\in D^{r}(a_{j})} \biggl(\sum_{i\in J:i\neq j} \bigl\vert k_{\varphi ,a_{i}}(z) \bigr\vert e^{- \varphi (z)} \biggr)^{p/2}\lesssim \biggl(\frac{1}{\rho (a_{j})} \biggr)^{\frac{p}{2}}. \end{aligned}$$

Joining (4.14) and the above estimates, we get

$$\begin{aligned} \Vert Z \Vert ^{p}_{S_{p}} \lesssim \biggl( \frac{1}{2} \biggr)^{\frac{Npk}{2}} \sum_{j\in J}c_{j}^{p}G_{r}(f) (a_{j})^{p}. \end{aligned}$$

(4.17)

Choose k large enough such that

$$ \biggl(\frac{1}{2} \biggr)^{\frac{Npk}{2}} \to 0 \quad \text{as } k \to \infty . $$

Hence, for any J,

$$ \sum_{j \in J}c_{j}^{p}G_{r}(f) (a_{j})^{p} \lesssim \sup_{j \in J}c_{j}^{p}. $$

Therefore (a) ⇒ (b) since $l^{\infty}$ is the dual space of $l^{1}$.

(b) ⇒ (c). Notice that $\{a_{j}\}_{j=1}^{\infty}$ is a (ρ, 3r)-lattice if $\{a_{j}\}_{j=1}^{\infty}$ is a (ρ, r)-lattice. Assume $\sum_{j=1}^{\infty}G_{3r}(f)(a_{j})^{p} < \infty$. Since $z \in D^{r}(a_{j})$, $D^{r}(z) \subseteq D^{3r}(a_{j})$, and hence

$$\begin{aligned} \int _{\mathbb{D}}G_{r}(f) (z)^{p}\rho (z)^{-2}\,dA(z) &\leq \sum_{j=1}^{ \infty} \int _{D^{r}(a_{j})}G_{r}(f) (z)^{p}\rho (z)^{-2}\,dA(z) \\ &\lesssim \sum_{j=1}^{\infty}\sup _{z \in D^{r}(a_{j})}G_{r}(f) (z)^{p} \\ &\lesssim \sum_{j=1}^{\infty}G_{3r}(f) (a_{j})^{p} < \infty . \end{aligned}$$

(c) ⇒ (d). Suppose $G_{r}(f)(z) \in L^{p}(\mathbb{D},\rho ^{-2}\,dA)$ and the decomposition $f=f_{1}+f_{2}$ is from Lemma 4.5, then $f_{1} \in C^{1}(\mathbb{D})$ and

$$ \bigl\vert \rho (z)\bar{\partial}f_{1}(z) \bigr\vert +M_{r/81}(\rho \bar{\partial}f_{1}) (z)+M_{r/81}(f_{2}) (z) \leq CG_{r}(f) (z). $$

By Lemma 4.6,

$$ \bigl\Vert M_{r}(\rho \bar{\partial}f_{1}) \bigr\Vert _{L^{p}(\mathbb{D},\rho ^{-2}\,dA)} \asymp \bigl\Vert M_{r/28}(\rho \bar{ \partial}f_{1}) \bigr\Vert _{L^{p}(\mathbb{D},\rho ^{-2}\,dA)} \leq C \bigl\Vert G_{r}(f) \bigr\Vert _{L^{p}(\mathbb{D},\rho ^{-2}\,dA)} < \infty $$

and

$$ \bigl\Vert M_{r}(f_{2}) \bigr\Vert _{L^{p}(\mathbb{D},\rho ^{-2}\,dA)} \asymp \bigl\Vert M_{r/28}(f_{2}) \bigr\Vert _{L^{p}( \mathbb{D},\rho ^{-2}\,dA)}\leq C \bigl\Vert G_{r}(f) \bigr\Vert _{L^{p}(\mathbb{D},\rho ^{-2}\,dA)} < \infty . $$

(d) ⇔ (e). See Lemma 4.6.

(d) ⇒ (a). To finish this, we let $M_{f_{2}}$ and $M_{\rho \bar{\partial}f_{1}}$ denote multiplication operators. Let ϕ be $f_{2}$ or $\rho \bar{\partial}f_{1}$. By $G_{r}(f)(z) \in L^{\infty}$ and Lemma 4.6, $M_{r}(\phi )(z)\in L^{\infty }$. We next show the operator $M_{\phi}$ is bounded from $A^{2}_{\varphi}$ to $L^{2}_{\varphi}$. Indeed, by Lemma 2.10 with $p=2$, then for $g \in A^{2}_{\varphi}$ we have

$$\begin{aligned} \Vert M_{\phi }g \Vert ^{2}_{\varphi ,2}&= \int _{\mathbb{D}} \vert g \vert ^{2}e^{-2 \varphi} \vert \phi \vert ^{2}\,dA \\ &\lesssim \int _{\mathbb{D}} \bigl\vert g(z) \bigr\vert ^{2}e^{-2\varphi (z)} \widehat{ \vert \phi \vert ^{2}}_{r}(z)\,dA(z) \\ &= \int _{\mathbb{D}} \bigl\vert g(z) \bigr\vert ^{2}e^{-2\varphi (z)}M_{r}( \phi ) (z)^{2}\,dA(z) \\ &\leq \vert \vert M_{r}(\phi ) \vert \vert ^{2}_{L^{\infty}} \vert \vert g \vert \vert ^{2}_{L^{2}_{\varphi}}. \end{aligned}$$

For any $g,h \in A^{2}_{\varphi}$,

$$\begin{aligned} \bigl\langle M^{*}_{\phi }M_{\phi }g, h\bigr\rangle = \langle M_{\phi }g, M_{ \phi }h\rangle =\langle T_{|\phi |^{2}}g, h\rangle . \end{aligned}$$

This gives $M^{*}_{\phi }M_{\phi}=T_{|\phi |^{2}}$ on $A^{2}_{\varphi}$. Using [18, Theorem 1.26], we get $M_{\phi }\in S_{p}$ if and only if $M^{*}_{\phi }M_{\phi}=T_{|\phi |^{2}} \in S_{p/2}$. By Theorem 3.1, $T_{|\phi |^{2}} \in S_{p/2}$ if and only if $\widehat{|\phi |^{2}}_{r}(z) \in L^{p/2}(\mathbb{D}, \rho ^{-2}\,dA)$ if and only if $M_{r}(\phi )(z) \in L^{p}(\mathbb{D}, \rho ^{-2}\,dA)$, and so $M_{\phi }\in S_{p}$. Since $\|H_{f_{1}}(g)\|_{\varphi ,2} \lesssim \|g\rho \bar{\partial}f_{1}\|_{ \varphi ,2}$ and $\|H_{f_{2}}(g)\|_{\varphi ,2} \lesssim \|f_{2}g\|_{\varphi ,2}$, both $H_{f_{1}}$ and $H_{f_{2}}$ are in $S_{p}$, therefore $H_{f} \in S_{p}$. This finishes the proof. □

Theorem 4.10

Let $\varphi \in \mathcal{E}$ and $h(\sqrt{(\cdot )}):\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ be a continuous increasing convex function. Suppose $H_{f}$ is densely defined satisfying $G_{r}(f)\in L^{\infty}$. Then following statements are equivalent:

(a) The Hankel operator $H_{f}$ is in $S_{h}$.

(b) For some (or any) $0< r< m_{\rho}$, there exists a constant $c>0$ such that

$$ \int _{\mathbb{D}} h\bigl(cG_{r}(f) (z)\bigr)\rho ^{-2}(z)\,dA(z)< \infty . $$

Proof

(a) ⇒ (b). Let $\{e_{j}\}^{\infty}_{j=1}$ be an orthonormal basis for $A^{2}_{\varphi}$, define

$$ T_{e_{j}}= \frac{\chi _{D^{r}(a_{j})}H_{f}(k_{\varphi ,a_{j}})}{(\int _{D^{r}(a_{j})} \vert H_{f}(k_{\varphi ,a_{j}}) \vert ^{2}e^{-2\varphi}\,dA)^{\frac{1}{2}}}=t_{j} \chi _{D^{r}(a_{j})}H_{f}(k_{\varphi ,a_{j}}) $$

where $\{a_{j}\}$ is a $(\rho , \frac{r}{3})$-lattice. It is clear that $\|T_{g}\|^{2}_{2,\varphi}\lesssim \|g\|^{2}_{2,\varphi}$, and hence T is bounded. By convexity of $h(\sqrt{(\cdot )})$, $h(\cdot )$ is a convex function. Let

$$ A(g)=\sum^{\infty}_{j=1}\langle g,e_{j}\rangle k_{ \varphi ,a_{j}}, $$

we have

$$ \begin{aligned} &\int _{\mathbb{D}}h\bigl(cG_{\frac{r}{3}}(f) (z)\bigr)\rho ^{-2}(z)\,dA(z) \\ &\quad\leq \sum^{\infty}_{j=1} \int _{D^{r}(a_{j})}h\bigl(cG_{ \frac{r}{3}}(f) (z)\bigr)\rho ^{-2}(z)\,dA(z) \\ &\quad\leq \sum^{\infty}_{j=1} \sup _{z\in D^{\frac{r}{3}}(a_{j})}h\bigl(cG_{r}(f) (z)\bigr) \lesssim \sum ^{\infty}_{j=1} h\bigl(c_{1}G_{r}(f) (a_{j})\bigr) \\ &\quad\lesssim \sum^{\infty}_{j=1} h\biggl(\biggl( \frac{c_{2}}{ \vert D^{r}(a_{j}) \vert } \int _{D^{r}(a_{j})} \biggl\vert f- \frac{1}{k_{\varphi ,a_{j}}}P(fk_{\varphi ,a,j}) \biggr\vert ^{2}\,dA(z)\biggr)^{ \frac{1}{2}}\biggr) \\ &\quad \asymp \sum^{\infty}_{j=1} h\biggl( \biggl(c_{3} \int _{D^{r}(a_{j})} \biggl\vert f-\frac{1}{k_{\varphi ,a_{j}}}P(fk_{\varphi ,a_{j}}) \biggr\vert ^{2} \vert k_{\varphi ,a_{j}} \vert ^{2}e^{-2 \varphi (z)}\,dA(z)\biggr)^{\frac{1}{2}}\biggr) \\ &\quad= \sum^{\infty}_{j=1} h\biggl( \biggl(c_{3} \int _{{D}^{r}(a_{j})} \bigl\vert H_{f}(k_{ \varphi ,a_{j}}) \bigr\vert ^{2}e^{-2\varphi (z)}\,dA(z)\biggr)^{\frac{1}{2}}\biggr) \\ &\quad=\sum^{\infty}_{j=1} h\bigl(c_{3} \bigl\vert t_{j}\langle H_{f}k_{ \varphi ,a_{j}},\chi _{D^{r}(a_{j})}H_{f}k_{\varphi ,a_{j}}\rangle \bigr\vert \bigr) \leq \sum^{\infty}_{j=1} h\bigl(c_{3} \bigl\vert \bigl\langle T^{*}H_{f}Ae_{j},e_{j} \bigr\rangle \bigr\vert \bigr) \\ &\quad\leq \sum^{\infty}_{j=1} h \bigl(c_{4} s_{j}\bigl(T^{*}H_{f}A \bigr)\bigr) \lesssim \sum^{\infty}_{j=1} h \bigl(c_{5} s_{j}(H_{f})\bigr)< \infty . \end{aligned} $$

(b) ⇒ (a). Suppose $\int _{\mathbb{D}} h(cG_{r}(f)(z))\rho ^{-2}(z)\,dA(z)<\infty$, f admits a decomposition $f=f_{1}+f_{2}$, where

$$ f_{1}=\sum^{\infty}_{j=1}h_{j} \psi _{j} \quad \text{and}\quad f_{2}=f-f_{1}. $$

Here $\{\psi _{j}\}_{j=1}^{\infty}$ is the unit decomposition induced by $\{D^{\frac{r}{3}}(a_{j})\}^{\infty}_{j=1}$. Choose $h_{j}\in \mathcal{H}(D^{r}(a_{j}))$ and $f\in L^{p}_{\mathrm{loc}}(\mathbb{D})$, $j=1,2,\ldots$ , such that

$$ M_{r}(f-h_{j})=G_{r}(f) (a_{j}), $$

then $f_{1}\in C^{1}(\mathbb{D})$ and

$$ \bigl\vert \rho (z)\bar{\partial}f_{1}(z) \bigr\vert +M_{\frac{r}{27}}(\rho \bar{\partial}f_{1}) (z)+M_{ \frac{r}{27}}(f_{2}) (z)\leq cG_{r}(f) (z). $$

Hence

$$ \begin{aligned} \int _{\mathbb{D}}h\bigl(M_{r}(\rho \bar{ \partial}f_{1})\bigr)\rho (z)^{-2}\,dA(z)&< \int _{\mathbb{D}}h\bigl(cM_{\frac{r}{27}}(\rho \bar{ \partial}f_{1})\bigr)\rho (z)^{-2}\,dA(z) \\ &\leq \int _{\mathbb{D}}h\bigl(cG_{r}(f) (z)\bigr)\rho (z)^{-2}\,dA(z)< \infty \end{aligned} $$

and

$$ \begin{aligned} \int _{\mathbb{D}}h(M_{r}(f_{2})\rho (z)^{-2}\,dA(z)&< \int _{\mathbb{D}}h(cM_{ \frac{r}{27}}(f_{2})\rho (z)^{-2}\,dA(z) \\ &\leq \int _{\mathbb{D}}h\bigl(cG_{r}(f) (z)\bigr)\rho (z)^{-2}\,dA(z)< \infty . \end{aligned} $$

Let θ be $f_{2}$ or $\rho \bar{\partial}f_{1}$, and $M_{\theta}$ be multiplication operator. By $G_{r}(f)(z)\in L^{\infty}$, $M_{r}(\theta )(z)\in L^{\infty}$, and so the operator $M_{\theta}$ is bounded from $A^{2}_{\varphi}$ to $L^{2}_{\varphi}$. Note that, for $g,h\in A^{2}_{\varphi}$,

$$ \bigl\langle M^{*}_{\theta}M_{\theta}g,h\bigr\rangle = \langle M_{\theta}g,M_{ \theta}h\rangle =\langle T_{|\theta |^{2}}g,h \rangle . $$

Hence $M^{*}_{\theta}M_{\theta}=T_{|\theta |^{2}}$. Since $M_{\theta}\in S_{h}$ if and only if $M^{*}_{\theta}M_{\theta}=T_{|\theta |^{2}}\in S_{h(\sqrt{(\cdot )})}$. According to Theorem 3.3 and the convexity of $h(\sqrt{(\cdot )})$, $T_{|\theta |^{2}}\in S_{h(\sqrt{(\cdot )})}$ if and only if

$$ \int _{\mathbb{D}}h\bigl(c\bigl(\widetilde{ \vert \theta \vert ^{2}}(z)\bigr)^{\frac{1}{2}}\bigr) \rho (z)^{-2}\,dA(z)< \infty . $$

It is easy to check that $\widehat{\mu}_{r}(z)\leq \widetilde{\mu}(z)$, and we claim that

$$ \int _{\mathbb{D}}h\bigl(\widetilde{\mu}(z)\bigr)\,dA(z)\leq c \int _{\mathbb{D}}h\bigl(c \widehat{\mu}_{r}(w)\bigr)\,dA(w). $$

By Jensen’s inequality, the convexity of h and $\widetilde{\mu}(z)\leq \widetilde{\widehat{\mu}_{r}}(z)$,

$$ \begin{aligned} \int _{\mathbb{D}} h\bigl(\widetilde{\mu}(z)\bigr)\,dA(z)&\leq \int _{\mathbb{D}} h\bigl(c \widetilde{\widehat{\mu}_{r}}(z) \bigr)\,dA(z) \\ &\leq \int _{\mathbb{D}} h\biggl(c \int _{\mathbb{D}} \bigl\vert k_{\varphi ,z}e^{- \varphi (w)} \bigr\vert ^{2}\widehat{\mu}_{r}(w)\,dA(w)\biggr) \\ &\leq \int _{\mathbb{D}}\biggl( \int _{\mathbb{D}}h\bigl(c\widehat{\mu}_{r}(w)\bigr) \bigl\vert k_{ \varphi ,z}e^{-\varphi (w)} \bigr\vert ^{2}\,dA(w) \biggr)\,dA(z) \\ &\leq \int _{\mathbb{D}}h\bigl(c\widehat{\mu}_{r}(w)\bigr)\,dA(w) \int _{ \mathbb{D}} \bigl\vert k_{\varphi ,z}e^{-\varphi (w)} \bigr\vert ^{2}\,dA(z) \\ &\leq c \int _{\mathbb{D}}h\bigl(c\widehat{\mu}_{r}(w)\bigr)\,dA(w). \end{aligned} $$

Recall that

$$ \int _{\mathbb{D}}h\bigl(cM_{r}(\theta ) (z)\bigr)\rho (z)^{-2}\,dA(z)< \infty , $$

thus

$$ \int _{\mathbb{D}}h\bigl(c\bigl(\widehat{ \vert \theta \vert ^{2}}_{r}(z)\bigr)^{\frac{1}{2}}\bigr) \rho (z)^{-2}\,dA(z)< \infty , $$

and hence

$$ \int _{\mathbb{D}}h\bigl(c\bigl(\widetilde{ \vert \theta \vert ^{2}}(z)\bigr)^{\frac{1}{2}}\bigr) \rho (z)^{-2}\,dA(z)< \infty . $$

So $M_{\theta}\in S_{h}$. Since $\|H_{f_{1}}(g)\|_{L^{2}_{\varphi}}\lesssim \|g\rho \bar{\partial}f_{1} \|_{L^{2}_{\varphi}}$ and $\|H_{f_{2}}(g)\|_{L^{2}_{\varphi}}\lesssim \|f_{2}g\|_{L^{2}_{ \varphi}}$, we have that both $H_{f_{1}}$ and $H_{f_{2}}$ are in $S_{h}$, and therefore $H_{f}\in S_{h}$. This completes the proof. □