The difference of weighted composition operators on Fock spaces

Abstract

In this note, we solve the open problem posted by Tien and Khoi (Monatsh Math 188:183–193, 2019). We prove that when $0<q<p<\infty$, the difference of two weighted composition operators between Fock spaces $W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}:\mathcal {F}^p\rightarrow \mathcal {F}^q$ is bounded if and only if both $W_{\psi _1,\varphi _1}$ and $W_{\psi _2,\varphi _2}$ are bounded. Furthermore, we prove that the same conclusion holds for the differences of a weighted composition operator and a weighted composition-differential operator on $\mathcal {F}^p$.

1 Introduction

Let $\mathbb {C}$ be the complex plane and $H(\mathbb {C})$ be the space of all entire functions on $\mathbb {C}$. For $0<p<\infty$, the classical Fock space $\mathcal {F}^p$ is defined as

$$\begin{aligned} \mathcal {F}^p=\left\{ f\in H(\mathbb {C}):\Vert f\Vert _p^p=\frac{p}{2\pi }\int _{\mathbb {C}}|f(z)|^pe^{-\frac{p}{2}|z|^2}dA(z)<\infty \right\} , \end{aligned}$$

where dA is the Lebesgue measure on $\mathbb {C}$. Furthermore, the space $\mathcal {F}^{\infty }$ consists of all functions $f\in H(\mathbb {C})$ such that

$$\begin{aligned}\Vert f\Vert _{\infty }=\sup _{z\in \mathbb {C}}|f(z)|e^{-\frac{|z|^2}{2}}<\infty .\end{aligned}$$

It is known that $\mathcal {F}^p$ is a Banach space for $1\le p\le \infty$. When $0<p<1$, $\mathcal {F}^p$ is a complete metric space with distance $d(f,g)=\Vert f-g\Vert _p^p$. In particular, $\mathcal {F}^2$ is a Hilbert space with the following inner product

$$\begin{aligned}\langle f,g \rangle =\frac{1}{\pi }\int _{\mathbb {C}}f(z)\overline{g(z)}e^{-|z|^2}dA(z).\end{aligned}$$

For each $w\in \mathbb {C}$, the linear point evaluation of nth order $f\mapsto f^{(n)}(w)$ is continuous on $\mathcal {F}^2$. It follows from the Riesz representation theorem in Hilbert space theory that there exsits a unique function $K_w^{[n]}$ in $\mathcal {F}^2$ such that

$$\begin{aligned}f^{(n)}(w)=\langle f,K_w^{[n]}\rangle \end{aligned}$$

for all $f\in \mathcal {F}^2$. $K_w^{[n]}$ is called the reproducing kernel function in $\mathcal {F}^2$ at w of order n. It is known that $K_{w}^{[0]}(z)=e^{\overline{w}z}$ and

$$\begin{aligned}K_{w}^{[n]}(z)=\frac{\partial ^nK_{w}^{[0]}}{\partial \overline{w}^n}(z)=z^ne^{\overline{w}z}\end{aligned}$$

for $n\ge 1$. Moreover, $\Vert K_{w}^{[0]}\Vert _p=e^{\frac{|w|^2}{2}}$ and $\Vert K_{w}^{[n]}\Vert _p\asymp (1+|w|)^{n}e^{\frac{|w|^2}{2}}$ for all $w\in \mathbb {C}$ and $0<p\le \infty$. Let $k_w(z)=e^{\overline{w}z-\frac{|w|^2}{2}}$, then each $k_w$ is a unit vector in $\mathcal {F}^p$ and converges to 0 uniformly on compact subsets of $\mathbb {C}$ as $|w|\rightarrow \infty$. One can refer to the monograph by Zhu [15] for more information about Fock spaces.

If $\varphi , \psi \in H(\mathbb {C})$, the weighted composition operator $W_{\psi ,\varphi }$ on $H(\mathbb {C})$ is defined by $W_{\psi ,\varphi }f=\psi \cdot (f\circ \varphi )$. When $\psi =1$, it reduces to the composition operator $C_{\varphi }$. The relationship between the operator-theoretic properties of $C_{\varphi }$ and the function-theoretic properties of $\varphi$ has been studied extensively during the past several decades. We refer the readers to monographs by Cowen and MacCluer [3] and by Shapiro [13] for more details. The boundedness and compactness of $W_{\psi ,\varphi }$ between Fock spaces have been completely characterized in [7, 8, 10]. One could also see [12] for the case in several variables and see [1] for large Fock spaces. Let $Df=f'$ be the differentiation operator on $H(\mathbb {C})$ and $D^n$ be the nth iterate of D. Write $W_{\psi ,\varphi }^{(n)}$ for the product of $D^n$ and $W_{\psi ,\varphi }$, i.e.

$$\begin{aligned}W_{\psi ,\varphi }^{(n)}f=W_{\psi ,\varphi }D^nf=\psi \cdot f^{(n)}\circ \varphi .\end{aligned}$$

$W_{\psi ,\varphi }^{(n)}$ is called a weighted composition-differential operator of order n. It is clear that $W_{\psi ,\varphi }$ is the special case $n=0$. When $n\ge 1$, the boundedness and compactness of $W_{\psi ,\varphi }^{(n)}$ between Fock spaces have been studied completely in [4].

In [9], Moorhouse characterized compactness of the difference of two composition operators on classical weighted Bergman spaces over the unit disk. Moorhouse showed that the difference of two composition operators is compact when suitable cancelation occurs and also that there exist two non-compact composition operators whose their difference is compact. However, no cancelation phenomenon exists on Fock spaces. Precisely, Choe et al. [2] showed that a linear sum of two composition operators is bounded (compact, resp.) on the Hilbert Fock spaces if and only if both composition operators are bounded (compact, resp.) Tien and Khoi [11] studied the differences of weighted composition operators between different Fock spaces and also showed that no cancelation exists. They proved that $W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}:\mathcal {F}^p\rightarrow \mathcal {F}^q$ is bounded (compact, resp.) if and only if both $W_{\psi _1,\varphi _1}$ and $W_{\psi _2,\varphi _2}:\mathcal {F}^p\rightarrow \mathcal {F}^q$ are bounded (compact, resp.) for $0<p\le q<\infty$. But this problem for the case $0<q<p<\infty$ is left open. In this paper, we completely solve this problem by using Khinchine’s inequality. Our first main result reads as follows.

Theorem A

Let $0<q<p<\infty$ and $\varphi _1\ne \varphi _2$. Then the following conditions are equivalent:

• (a) $W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}:\mathcal {F}^p\rightarrow \mathcal {F}^q$ is bounded;

• (b) $W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}:\mathcal {F}^p\rightarrow \mathcal {F}^q$ is compact;

• (c) Both $W_{\psi _1,\varphi _1}$ and $W_{\psi _2,\varphi _2}:\mathcal {F}^p\rightarrow \mathcal {F}^q$ are bounded;

• (d) Both $W_{\psi _1,\varphi _1}$ and $W_{\psi _2,\varphi _2}:\mathcal {F}^p\rightarrow \mathcal {F}^q$ are compact.

Actually, we can characterize the differences of a weighted composition operator and a weighted composition-differential operator. To the best of our knowledge, no prior results on describing the compactness of two such operators on Fock spaces $\mathcal {F}^p$, and even analytic function spaces on any other domains. Our second main result reads as follows.

Theorem B

Let n be a positive integer and $\varphi _1\ne \varphi _2$. Then $W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}^{(n)}$ is bounded (compact, resp.) on $\mathcal {F}^p$ if and only if both $W_{\psi _1,\varphi _1}$ and $W_{\psi _2,\varphi _2}^{(n)}$ are bounded (compact, resp.) on $\mathcal {F}^p$.

The paper is organized as follows. In Sect. 2, we present some known facts and auxiliary lemmas that will be needed later. Section 3 and Sect. 4 are devoted to the proof of Theorem A and B, respectively.

Throughout the paper, we write $A\lesssim B$ if there exists an absolute constant $C>0$ such that $A\le CB$. As usual, $A\asymp B$ means $A\lesssim B$ and $B\lesssim A$. We will be more specific if the dependence of such constants on certain parameters becomes critical.

2 Preliminaries

In this section, we collet some preliminary facts and auxiliary lemmas which will be used later. Firstly, according to [5] and [14], we have the following characterizations for Fock spaces via higher order derivatives.

Lemma 2.1

Let $0<p\le \infty$ and $f\in H(\mathbb {C})$. Then $f\in \mathcal {F}^p$ if and only if

$$\begin{aligned} \frac{f^{(n)}(z)}{(1+|z|)^n}e^{-\frac{|z|^2}{2}}\in L^p(\mathbb {C},dA) \end{aligned}$$

for any non-negative integer n. Moreover,

$$\begin{aligned} \Vert f\Vert _p\asymp \sum _{j=0}^{n-1}|f^{(j)}(0)|+\left( \int _{\mathbb {C}}\frac{|f^{(n)}(z)|^p}{(1+|z|)^{np}}e^{-\frac{p}{2}|z|^2}dA(z)\right) ^{1/p} \end{aligned}$$

for $0<p<\infty$. And

$$\begin{aligned} \Vert f\Vert _{\infty }\asymp \sum _{j=0}^{n-1}|f^{(j)}(0)|+\sup _{z\in \mathbb {C}}\frac{|f^{(n)}(z)|}{(1+|z|)^{n}}e^{-\frac{|z|^2}{2}}. \end{aligned}$$

By Lemma 2.1, we can easily get the following estimate for derivatives of functions in Fock space. See also [4] for details.

Lemma 2.2

Let $0<p\le \infty$ and n be a non-negative integer. Then

$$\begin{aligned}|f^{(n)}(z)|\lesssim (1+|z|)^ne^{\frac{|z|^2}{2}}\Vert f\Vert _p\end{aligned}$$

for all $f\in \mathcal {F}^p$ and $z\in \mathbb {C}$.

For $z\in \mathbb {C}$ and $r>0$, write

$$\begin{aligned}D(z,r)=\{w\in \mathbb {C}:|w-z|<r\}\end{aligned}$$

for the Euclidean disk centered at z with radius r. A sequence $\{a_j\}$ in $\mathbb {C}$ is called an r-lattice if the following conditions are satisfied:

• (i) $\cup _{j=1}^{\infty }D(a_j,r)=\mathbb {C}$,

• (ii) $\{D(a_j,\frac{r}{2})\}_{j=1}^{\infty }$ are pairwise disjoint.

For example, the set of points on $r\mathbb {Z}^2$ is an r-lattice. With hypotheses $(\mathrm i)$ and $(\mathrm ii)$, it is easy to check that

• (iii) there exists a positive integer N (depending only on r) such that every point in $\mathbb {C}$ belongs to at most N of the sets $\{D(a_j,r)\}$.

Lemma 2.3

([6]) Let $1\le s\le \infty$, $\mu$ be a positive Borel measure on $\mathbb {C}$ and $\{a_j\}$ be an r-lattice in $\mathbb {C}$. Then the following conditions are equivalent:

• (i) $\{\mu (D(a_j,r))\}\in l^s$,

• (ii) $\mu (D(\cdot ,\delta ))\in L^s(\mathbb {C},dA)$ for some (or any) $\delta >0$.

The boundedness and compactness of $W_{\psi ,\varphi }:\mathcal {F}^p\rightarrow \mathcal {F}^q$ for $0<q<p<\infty$ has been charaterized in [10]. We state the results as follows.

Lemma 2.4

([10]) Let $0<q<p<\infty$, $\psi \in \mathcal {F}^q$ and $\varphi (z)=az+b$ with $0<|a|\le 1$. Then the following conditions are equivalent:

• (i) $W_{\psi ,\varphi }:\mathcal {F}^p\rightarrow \mathcal {F}^q$ is bounded;

• (ii) $W_{\psi ,\varphi }:\mathcal {F}^p\rightarrow \mathcal {F}^q$ is compact;

• (iii) $|\psi (z)|e^{\frac{|\varphi (z)|^2-|z|^2}{2}}\in L^{\frac{pq}{p-q}}(\mathbb {C},dA)$.

In order to study the difference $W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}^{(n)}$, we need to estimate the norm of the reproducing kernels from below. The following lemmas play important roles in our proof.

Lemma 2.5

Let n be a positive integer, $\alpha ,\beta \in \mathbb {C}\backslash \{0\}$ and $\varepsilon >0$. Then there exists a contant $C=C(\varepsilon ,n)>0$, independent of $\alpha$ and $\beta$, such that

$$\begin{aligned} \Vert \alpha K_{w_1}+\beta K_{w_2}^{[n]}\Vert _{\infty }\ge C\left( |\alpha |\Vert K_{w_1}\Vert _{\infty }+|\beta |\Vert K_{w_2}^{[n]}\Vert _{\infty }\right) \end{aligned}$$

for all $w_1, w_2\in \mathbb {C}$ with $|w_1-w_2|\ge \varepsilon$.

Proof

Put $L=\Vert \alpha K_{w_1}+\beta K_{w_2}^{[n]}\Vert _{\infty }$ for brevity. It is enough to prove that $L\ge C|\alpha |\Vert K_{w_1}\Vert _{\infty }$ for some $C=C(\varepsilon ,n)>0$. After achieving that, it follows immediately from the triangle inequality that

$$\begin{aligned} \begin{aligned} |\beta |\Vert K_{w_2}^{[n]}\Vert _{\infty }&\le \Vert \alpha K_{w_1}+\beta K_{w_2}^{[n]}\Vert _{\infty }+|\alpha | \Vert K_{w_1}\Vert _{\infty }\\&\le (1+\frac{1}{C})L. \end{aligned} \end{aligned}$$

Firstly, we know that

$$\begin{aligned} L\ge |\alpha K_{w_1}(z)+\beta K_{w_2}^{[n]}(z)|e^{-\frac{|z|^2}{2}}=|\alpha e^{\overline{w_1}z}+\beta z^ne^{\overline{w_2}z}|e^{-\frac{|z|^2}{2}} \end{aligned}$$

(1)

for all $z\in \mathbb {C}$. By Lemma 2.1, there exists a constant $C_1=C_1(n)>0$ such that

$$\begin{aligned} \begin{aligned} C_1L&\ge \frac{|\alpha K_{w_1}^{(n)}(z)+\beta (K_{w_2}^{[n]})^{(n)}(z)|}{(n+|z|)^n}e^{-\frac{|z|^2}{2}}\\&=\frac{\left| \alpha \overline{w_1}^ne^{\overline{w_1}z}+\beta h(\overline{w_2}z)e^{\overline{w_2}z}\right| }{(n+|z|)^n}e^{-\frac{|z|^2}{2}} \end{aligned} \end{aligned}$$

(2)

for all $z\in \mathbb {C}$, where $h(x)=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) \frac{n!}{(n-k)!}x^{n-k}$. Let $R>4$ be a sufficiently large number satisfying

$$\begin{aligned}\frac{R^n}{(n+R)^n}>\frac{1}{2}(1+e^{-\frac{\varepsilon ^2}{2}}).\end{aligned}$$

If $|w_1|\le R$, then

$$\begin{aligned}L\ge |\alpha |\ge e^{-\frac{R^2}{2}}|\alpha |e^{\frac{|w_1|^2}{2}}.\end{aligned}$$

If $|w_1|>R$ and $|w_2|\le 1$, then taking $z=w_2+\zeta$ in (1), where $\zeta =w_2/|w_2|$ if $w_2\ne 0$ and $\zeta =1$ if $w_2=0$, we get

$$\begin{aligned} \begin{aligned} L&\ge e^{-\frac{1}{2}}|\beta |(1+|w_2|)^ne^{\frac{|w_2|^2}{2}}-|\alpha |e^{\frac{|w_1|^2}{2}}e^{-\frac{|w_1-w_2-\zeta |^2}{2}}\\&\ge e^{-\frac{1}{2}}|\beta |(1+|w_2|)^ne^{\frac{|w_2|^2}{2}}-e^{-2}|\alpha |e^{\frac{|w_1|^2}{2}}. \end{aligned} \end{aligned}$$

(3)

Taking $z=w_1$ in (2), we have

$$\begin{aligned} \begin{aligned} C_1L&\ge |\alpha |\frac{|w_1|^n}{(n+|w_1|)^n}e^{\frac{|w_1|^2}{2}}-|\beta |\frac{h(|w_1w_2|)}{(n+|w_1|)^n} e^{\frac{|w_2|^2}{2}}e^{-\frac{|w_1-w_2|^2}{2}}\\&\ge \frac{1}{2}|\alpha |e^{\frac{|w_1|^2}{2}}-e^{-\frac{9}{2}}|\beta |(1+|w_2|)^ne^{\frac{|w_2|^2}{2}}. \end{aligned} \end{aligned}$$

(4)

Adding (3) and (4), we obtain that

$$\begin{aligned}(1+C_1)L\ge (\frac{1}{2}-\frac{1}{e^2})|\alpha |e^{\frac{|w_1|^2}{2}}.\end{aligned}$$

If $|w_1|>R$ and $|w_2|>1$, then taking $z=w_2$ in (1) and taing $z=w_1$ in (2), we get

$$\begin{aligned} \begin{aligned} L&\ge |\beta ||w_2|^ne^{\frac{|w_2|^2}{2}}-|\alpha |e^{\frac{|w_1|^2}{2}}e^{-\frac{|w_1-w_2|^2}{2}}\\&\ge |\beta ||w_2|^ne^{\frac{|w_2|^2}{2}}-e^{-\frac{\varepsilon ^2}{2}}|\alpha |e^{\frac{|w_1|^2}{2}} \end{aligned} \end{aligned}$$

(5)

and

$$\begin{aligned} \begin{aligned} C_1L&\ge |\alpha |\frac{|w_1|^n}{(n+|w_1|)^n}e^{\frac{|w_1|^2}{2}}-|\beta |\frac{h(|w_1w_2|)}{(n+|w_1|)^n} e^{\frac{|w_2|^2}{2}}e^{-\frac{|w_1-w_2|^2}{2}}\\&\ge \frac{1}{2}(1+e^{-\frac{\varepsilon ^2}{2}})|\alpha |e^{\frac{|w_1|^2}{2}}-|\beta ||w_2|^ne^{\frac{|w_2|^2}{2}}. \end{aligned} \end{aligned}$$

(6)

Adding (5) and (6), we obtain that

$$\begin{aligned}(1+C_1)L\ge \frac{1}{2}\left( 1-e^{-\frac{\varepsilon ^2}{2}}\right) |\alpha |e^{\frac{|w_1|^2}{2}}.\end{aligned}$$

Therefore, letting $C=\min \{e^{-\frac{R^2}{2}},\frac{1}{1+C_1}(\frac{1}{2}-\frac{1}{e^2}),\frac{1}{2(1+C_1)}(1-e^{-\frac{\varepsilon ^2}{2}})\}$, we have

$$\begin{aligned}L\ge C|\alpha |e^{\frac{|w_1|^2}{2}}=C|\alpha |\Vert K_{w_1}\Vert _{\infty }.\end{aligned}$$

The proof is complete. $\square$

Lemma 2.6

Let n be a positive integer and $\alpha ,\beta \in \mathbb {C}\backslash \{0\}$. Then there exists a contant $C>0$, independent of $\alpha$ and $\beta$, such that

$$\begin{aligned} \Vert \alpha K_{w_1}+\beta K_{w_2}^{[n]}\Vert _{\infty }^2\ge C|\alpha ||\beta |e^{\frac{|w_1|^2+|w_2|^2}{2}} \frac{|w_1-w_2|^4}{(1+|w_1-w_2|^2)^2}. \end{aligned}$$

for all $w_1,w_2\in \mathbb {C}$.

Proof

Put $L=\Vert \alpha K_{w_1}+\beta K_{w_2}^{[n]}\Vert _{\infty }$ for brevity. Let $R>4$ be a sufficiently large number satisfying

$$\begin{aligned}\frac{R^n}{(n+R)^n}>\frac{1}{2}(1+e^{-\frac{1}{2}}).\end{aligned}$$

If $|w_1|\le R$ or $|w_1|>R$ with $|w_1-w_2|\ge 1$, then by Lemma 2.5, we have

$$\begin{aligned}L\ge C \left( |\alpha | \Vert K_{w_1}\Vert _{\infty }+|\beta |\Vert K_{w_2}^{[n]}\Vert _{\infty }\right) \ge C(|\alpha |e^{\frac{|w_1|^2}{2}}+|\beta |e^{\frac{|w_2|^2}{2}}),\end{aligned}$$

where C is independent of $\alpha$ and $\beta$. It follow that $L^2\ge C|\alpha ||\beta |e^{\frac{|w_1|^2+|w_2|^2}{2}}$.

It remains to prove the results for the case where $|w_1|>R$ and $|w_1-w_2|<1$. If $|w_2|\ge |w_1|$, then taking $z=w_1$ in (1), we have

$$\begin{aligned} L\ge |\alpha e^{|w_1|^2}+\beta w_1^ne^{\overline{w_2}w_1}|e^{-\frac{|w_1|^2}{2}} \end{aligned}$$

(7)

And taking $z=w_2$ in (1), we have

$$\begin{aligned} L\ge |\alpha e^{\overline{w_1}w_2}+\beta w_2^n e^{|w_2|^2}|e^{-\frac{|w_2|^2}{2}}. \end{aligned}$$

(8)

Adding (7) and (8), we obtain that

$$\begin{aligned} \begin{aligned} L&\ge C|\beta ||w_2|^ne^{\frac{|w_2|^2}{2}}\left( 1-\left| \frac{w_1}{w_2}\right| ^ne^{-\frac{|w_1-w_2|^2}{2}}\right) \\&\ge C|\beta ||w_2|^ne^{\frac{|w_2|^2}{2}}\left( e^{\frac{|w_1-w_2|^2}{2}}-1\right) \\&\ge C|\beta ||w_2|^ne^{\frac{|w_2|^2}{2}}|w_1-w_2|^2 \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} L&\ge C|\alpha |\frac{1}{|w_1|^n}e^{\frac{|w_1|^2}{2}}\left( 1-\left| \frac{w_1}{w_2}\right| ^ne^{-\frac{|w_1-w_2|^2}{2}}\right) \\&\ge C|\alpha |\frac{1}{|w_1|^n}e^{\frac{|w_1|^2}{2}}\left( e^{\frac{|w_1-w_2|^2}{2}}-1\right) \\&\ge C|\alpha |\frac{1}{|w_1|^n}e^{\frac{|w_1|^2}{2}}|w_1-w_2|^2. \end{aligned} \end{aligned}$$

Thus, $L^2\ge C|\alpha ||\beta |\left| \frac{w_2}{w_1}\right| ^ne^{\frac{|w_1|^2+|w_2|^2}{2}}\ge C|\alpha ||\beta |e^{\frac{|w_1|^2+|w_2|^2}{2}}$.

Now suppose $|w_1|>R$, $|w_1-w_2|<1$ and $|w_2|<|w_1|$. Let $\zeta =\sqrt{3}n\frac{w_1-w_2}{|w_1-w_2|}i$ or $\zeta =-\sqrt{3}n\frac{w_1-w_2}{|w_1-w_2|}i$ such that the included angle between $\zeta$ and $w_2$ is not more than $\frac{\pi }{2}$. Taking $z=w_2+\zeta$ in (1), we get

$$\begin{aligned} \begin{aligned} L&\ge \left| \alpha e^{\overline{w_1}(w_2+\zeta )}+\beta (w_2+\zeta )^ne^{\overline{w_2}(w_2+\zeta )}\right| e^{\frac{|w_2+\zeta |^2}{2}}\\&\ge (|\beta ||w_2+\zeta |^ne^{\frac{|w_2|^2}{2}}-|\alpha ||e^{\overline{w_1}w_2}|e^{-\frac{|w_2|^2}{2}}|e^{\overline{w_1-w_2}\zeta }|)e^{-\frac{|\zeta |^2}{2}}\\&\ge C\left( |\beta |\left( |w_2|+\frac{n}{|w_2|}\right) ^ne^{\frac{|w_2|^2}{2}}-|\alpha |e^{\frac{|w_1|^2}{2}}e^{-\frac{|w_1-w_2|^2}{2}}\right) . \end{aligned} \end{aligned}$$

(9)

By Lemma 2.1, we have

$$\begin{aligned} \begin{aligned} L&\ge C\sup _{|z|>1}\frac{\left| \alpha \overline{w_1}^ne^{\overline{w_1}z}+\beta h(\overline{w_2}z)e^{\overline{w_z}z}\right| }{|z|^n}e^{-\frac{|z|^2}{2}}\\&\ge C\frac{|\alpha \overline{w_1}^ne^{|w_1|^2}+\beta h(\overline{w_2}w_1)e^{\overline{w_2}w_1}|}{|w_1|^n}e^{-\frac{|w_1|^2}{2}}\\&\ge C|\alpha |e^{\frac{|w_1|^2}{2}}-|\beta |\left( |w_2|+\frac{n}{|w_2|}\right) ^ne^{\frac{|w|^2}{2}}e^{-\frac{|w_1-w_2|^2}{2}} \end{aligned} \end{aligned}$$

(10)

Thus, adding (9) and (10), we obtain that

$$\begin{aligned} \begin{aligned} L&\ge C(|\alpha |e^{\frac{|w_1|^2}{2}}+|\beta ||w_2|^ne^{\frac{|w_2|^2}{2}})(1-e^{-\frac{|w_1-w_2|^2}{2}})\\&\ge C (|\alpha |e^{\frac{|w_1|^2}{2}}+|\beta |e^{\frac{|w_2|^2}{2}})\frac{|w_1-w_2|^2}{1+|w_1-w_2|^2}. \end{aligned} \end{aligned}$$

It follows that $L^2\ge C|\alpha ||\beta |e^{\frac{|w_1|^2+|w_2|^2}{2}}\frac{|w_1-w_2|^4}{(1+|w_1-w_2|^2)^2}$. The proof is complete. $\square$

3 The proof of Theorem A

In order to prove Theorem A for the case $0<q<p<\infty$, we also need to use the classical Khinchine’s inequality, which is an important tool in complex and functional analysis. Here we recall the basic facts about this inequality.

Let $\{r_k(t)\}$ denotes the sequence of Rademacher functions defined by

$$\begin{aligned}r_0(t)={\left\{ \begin{array}{ll}1, &{}\text{ if }\ 0\le t-[t]<\frac{1}{2}\\ -1, &{}\text{ if }\ \frac{1}{2}\le t-[t]<1,\end{array}\right. }\end{aligned}$$

where [t] denotes the largest integer not greater than t and $r_k(t)=r_0(2^kt)$ for $k=1,2,\cdots$. If $0<p<\infty$, then Khinchine’s inequality states that

$$\begin{aligned}\left( \sum _{k}\left| c_{k}\right| ^{2}\right) ^{p / 2} \asymp \int _{0}^{1}\left| \sum _{k} c_{k} r_{k}(t) \right| ^{p} dt\end{aligned}$$

for complex sequences $\left\{ c_{k}\right\}$.

Now we are ready to prove Theorem A.

Proof of Theorem A

It is obvious that (d) implies (b) and (b) implies (a). The equivalence of (c) and (d) follows from Proposition 3.1, Corollary 3.2 in [10] and Lemma 2.4. Thus we only need to prove (a) implies (d).

Suppose $W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}:\mathcal {F}^p\rightarrow \mathcal {F}^q$ is bounded. Let

$$\begin{aligned}M_i=\sup _{z\in \mathbb {C}}|\psi _i(z)|e^{\frac{|\varphi _i(z)|^2-|z|^2}{2}},\quad i=1,2.\end{aligned}$$

By [11, Proposition 2.2], we have $M_i<\infty$ and $\varphi _i(z)=a_iz+b_i$ with $|a_i|\le 1$ for $i=1,2$. We consider the following three cases.

Case 1. $a_1=a_2=0$, i.e. $\varphi _1(z)\equiv b_1$ and $\varphi _2(z)\equiv b_2$ with $b_1\ne b_2$ since $\varphi _1\ne \varphi _2$. Then

$$\begin{aligned} \psi _1-\psi _2=(W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2})1\in \mathcal {F}^q \end{aligned}$$

(11)

and $b_1\psi _1-b_2\psi _2=(W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2})z\in \mathcal {F}^q$. Thus $\psi _1,\psi _2\in \mathcal {F}^q$. It follows that both $W_{\psi _1,\varphi _1}$ and $W_{\psi _2,\varphi _2}:\mathcal {F}^p\rightarrow \mathcal {F}^q$ are compact by [10, Corollary 3.2].

Case 2. $a_1=0$, $a_2\ne 0$ (it is similar for the case $a_1\ne 0$, $a_2=0$). Then

$$\begin{aligned} \psi _2(z)\le M_2e^{\frac{|z|^2-|a_2z-b_2|^2}{2}}\lesssim M_2e^{\frac{(1-|a_2|^2)|z|^2}{2}}, \end{aligned}$$

(12)

which means that $\psi _2\in \mathcal {F}^q$. Combining this with (11), we also have $\psi _1\in \mathcal {F}^q$. By [10, Corollary 3.2] again, we have $W_{\psi _1,\varphi _1}:\mathcal {F}^p\rightarrow \mathcal {F}^q$ is compact, then so is $W_{\psi _2,\varphi _2}$.

Case 3. $a_1\ne 0$ and $a_2\ne 0$. In this case, a similar argument as (12) gives $\psi _1,\psi _2\in \mathcal {F}^q$. Let $\{a_j\}$ be an $r-$lattice in $\mathbb {C}$. For any $\{\lambda _j\}\in l^p$, we set

$$\begin{aligned}f(z)=\sum _{j}\lambda _j k_{a_j}(z).\end{aligned}$$

Then $f\in \mathcal {F}^p$ with $\Vert f\Vert _p\lesssim \Vert \{\lambda _j\}\Vert _{l^p}$. If $W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}:\mathcal {F}^p\rightarrow \mathcal {F}^q$ is bounded, then

$$\begin{aligned} \int _{\mathbb {C}}\left| \sum _{j}\lambda _j(W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2})k_{a_j}(z)\right| ^qe^{-\frac{q}{2}|z|^2}dA(z)\lesssim \Vert \{\lambda _j\}\Vert _{l^p}^q. \end{aligned}$$

(13)

In (13), we replace $\lambda _j$ by $r_j(t)\lambda _j$, so that the right-hand side does not change. Then we integrate both sides with respact to t from 0 to 1 to obtain

$$\begin{aligned} \int _{0}^{1}\int _{\mathbb {C}}\left| \sum _{j}r_j(t)\lambda _j(W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2})k_{a_j}(z)\right| ^qe^{-\frac{q}{2}|z|^2}dA(z)\lesssim \Vert \{\lambda _j\}\Vert _{l^p}^q. \end{aligned}$$

By Fubini’s Theorem and Khinchine’s inequality,

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {C}}\left( \sum _{j}|\lambda _j|^2|(W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2})k_{a_j}(z)|^2e^{-|z|^2}\right) ^{\frac{q}{2}}dA(z)\\&\lesssim \int _{\mathbb {C}}\int _{0}^{1}\left| \sum _{j}r_j(t)\lambda _j(W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2})k_{a_j}(z)e^{-\frac{1}{2}|z|^2}\right| ^qdtdA(z)\\&\lesssim \int _{0}^{1}\int _{\mathbb {C}}\left| \sum _{j}r_j(t)\lambda _j(W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2})k_{a_j}(z)e^{-\frac{1}{2}|z|^2}\right| ^qdA(z)dt\\&\lesssim \Vert \{\lambda _j\}\Vert _{l^p}^q. \end{aligned} \end{aligned}$$

Recall that there is a positive N such that each point $z\in \mathbb {C}$ belongs to at most N of the disks $\{D(a_j,r)\}$. Let

$$\begin{aligned}E=\{z\in \mathbb {C}:|\varphi _1(z)-\varphi _2(z)|> 3r\}.\end{aligned}$$

Applying Minkowski’s inequality if $\frac{2}{q}\le 1$ and Hölder’s inequality if $\frac{2}{q}>1$, we obtain

$$\begin{aligned} \begin{aligned}&\int _{\varphi _1^{-1}(D(a_j,r))}\sum _{j}|\lambda _j|^q|(W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2})k_{a_j}(z)|^qe^{-\frac{q}{2}|z|^2}\chi _{E}(z)dA(z)\\&\quad \le \max \{1,N^{1-\frac{q}{2}}\}\int _{\varphi _1^{-1}(D(a_j,r))\cap E}\left( \sum _{j}|\lambda _j|^2|(W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2})k_{a_j}(z)|^2e^{-|z|^2}\right) ^{\frac{q}{2}}dA(z)\\&\quad \lesssim \int _{\mathbb {C}}\left( \sum _{j}|\lambda _j|^2|(W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2})k_{a_j}(z)|^2e^{-|z|^2}\right) ^{\frac{q}{2}}dA(z)\\&\quad \lesssim \Vert \{\lambda _j\}\Vert _{l^p}^q. \end{aligned} \end{aligned}$$

It follows from duality argument that

$$\begin{aligned} \int _{\varphi _1^{-1}(D(a_j,r))}|(W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2})k_{a_j}(z)|^qe^{-\frac{q}{2}|z|^2}\chi _{E}(z)dA(z)\in l^{\frac{p}{p-q}}. \end{aligned}$$

If $z\in \varphi _1^{-1}(D(a_j,r))\cap E$, then

$$\begin{aligned} \begin{aligned}&|(W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2})k_{a_j}(z)|e^{-\frac{1}{2}|z|^2}\\&\quad \ge |\psi _1(z)|e^{\frac{|\varphi _1(z)|^2-|z|^2}{2}}e^{-\frac{|a_j-\varphi _1(z)|^2}{2}}-|\psi _2(z)| e^{\frac{|\varphi _1(z)|^2-|z|^2}{2}}e^{-\frac{|a_j-\varphi _2(z)|^2}{2}}\\&\quad \ge e^{-\frac{1}{2}r^2}\left( |\psi _1(z)|e^{\frac{|\varphi _1(z)|^2-|z|^2}{2}}-e^{-\frac{3}{2}r^2}|\psi _2(z)|e^{\frac{|\varphi _1(z)|^2-|z|^2}{2}}\right) \end{aligned} \end{aligned}$$

Thus

$$\begin{aligned} \int _{\varphi _1^{-1}(D(a_j,r))\cap E}\left( |\psi _1(z)|e^{\frac{|\varphi _1(z)|^2-|z|^2}{2}}-e^{-\frac{3}{2}r^2}|\psi _2(z)|e^{\frac{|\varphi _2(z)|^2-|z|^2}{2}}\right) ^qdA(z)\in l^{\frac{p}{p-q}}. \end{aligned}$$

It follows from Lemma 2.3 that

$$\begin{aligned} \int _{\varphi _1^{-1}(D(w,r))\cap E}\left( |\psi _1(z)|e^{\frac{|\varphi _1(z)|^2-|z|^2}{2}}-e^{-\frac{3}{2}r^2}|\psi _2(z)|e^{\frac{|\varphi _2(z)|^2-|z|^2}{2}}\right) ^qdA(z)\in L^{\frac{p}{p-q}}(\mathbb {C},dA). \end{aligned}$$

This implies that

$$\begin{aligned} \int _{D(w,\frac{r}{|a_1|})}\left( |\psi _1(z)|e^{\frac{|\varphi _1(z)|^2-|z|^2}{2}}-e^{-\frac{3}{2}r^2}|\psi _2(z)|e^{\frac{|\varphi _2(z)|^2-|z|^2}{2}}\right) ^q\chi _E(z)dA(z)\in L^{\frac{p}{p-q}}(\mathbb {C},dA). \end{aligned}$$

(14)

Similarly, we have

$$\begin{aligned} \int _{D(w,\frac{r}{|a_2|})}\left( |\psi _2(z)|e^{\frac{|\varphi _2(z)|^2-|z|^2}{2}}-e^{-\frac{3}{2}r^2}|\psi _1(z)|e^{\frac{|\varphi _1(z)|^2-|z|^2}{2}}\right) ^q\chi _E(z)dA(z)\in L^{\frac{p}{p-q}}(\mathbb {C},dA). \end{aligned}$$

(15)

Letting $\delta =\min \{\frac{r}{|a_1|},\frac{r}{|a_2|}\}$ and adding (14) and (15), we obtain

$$\begin{aligned} \int _{D(w,\delta )}|\psi _i(z)|^qe^{\frac{q}{2}(|\varphi _i(z)|^2-|z|^2)}\chi _E(z)dA(z)\in L^{\frac{p}{p-q}}(\mathbb {C},dA),\quad i=1,2. \end{aligned}$$

It follows that

$$\begin{aligned} \int _{D(w,\delta )}|\psi _i(z)|^qe^{\frac{q}{2}(|\varphi _i(z)|^2-|z|^2)}dA(z)\in L^{\frac{p}{p-q}}(\mathbb {C},dA),\quad i=1,2 \end{aligned}$$

since $M_i<\infty$ and $E^{c}$ is bounded in $\mathbb {C}$. Notice that

$$\begin{aligned}|\psi _i(z)|e^{\frac{|\varphi _i(z)|^2-|z|^2}{2}}=|\psi _i(z)e^{a_i\overline{b_i}z}|e^{\frac{(|a_i|^2-1)|z|^2+|b_i|^2}{2}}.\end{aligned}$$

By [15, Lemma 2.32], there exists a constant $C=C(q,r)$ such that

$$\begin{aligned}\int _{D(w,\delta )}|\psi _i(z)|^qe^{\frac{q}{2}(|\varphi _i(z)|^2-|z|^2)}dA(z)\ge C|\psi _i(w)|^qe^{\frac{q}{2}(|\varphi _i(w)|^2-|w|^2)}\end{aligned}$$

for $i=1,2$. Therefore, $|\psi _i(w)|e^{\frac{|\varphi _i(w)|^2-|w|^2}{2}}\in L^{\frac{pq}{p-q}}(\mathbb {C},dA)$. Then the compactness of $W_{\psi _i,\varphi _i}$ for $i=1,2$ is established by Lemma 2.4. The proof is complete. $\square$

4 The proof of Theorem B

In this section, we give the proof of Theorem B.

Lemma 4.1

Let $0<p\le \infty$, n be a positive integer and $\varphi _1\ne \varphi _2$. If $W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}^{(n)}$ is bounded on $\mathcal {F}^p$, then $\varphi _1(z)-\varphi _2(z)=az+b$ for some $a,b\in \mathbb {C}$.

Proof

Suppose $W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}^{(n)}$ is bounded on $\mathcal {F}^p$. By Lemma 2.2, we have

$$\begin{aligned} \begin{aligned} \Vert W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}^{(n)}\Vert&\ge \Vert (W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}^{(n)})k_w\Vert _p\\&\ge |(W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}^{(n)})k_w(z)|e^{-\frac{|z|^2}{2}}\\&=|\psi _1(z)K_w(\varphi _1(z))-\psi _2(z)K_w^{(n)}(\varphi _2(z))|e^{-\frac{|w|^2}{2}}e^{-\frac{|z|^2}{2}}\\&=|\overline{\psi _1(z)}K_{\varphi _1(z)}(w)-\overline{\psi _2(z)}K_{\varphi _2(z)}^{[n]}(w)|e^{-\frac{|w|^2}{2}}e^{-\frac{|z|^2}{2}} \end{aligned} \end{aligned}$$

(16)

for all $z,w\in \mathbb {C}$. This means that

$$\begin{aligned} \Vert \overline{\psi _1(z)}K_{\varphi _1(z)}-\overline{\psi _2(z)}K_{\varphi _2(z)}^{[n]}\Vert _{\infty }e^{-\frac{|z|^2}{2}}\le \Vert W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}^{(n)}\Vert \end{aligned}$$

(17)

for all $z\in \mathbb {C}$. Combining this with Lemma 2.6, we get

$$\begin{aligned} |\psi _1(z)\psi _2(z)|e^{\frac{|\varphi _1(z)|^2+|\varphi _2(z)|^2}{2}-|z|^2}\frac{|\varphi _1(z)-\varphi _2(z)|^4}{1+|\varphi _1(z)-\varphi _2(z)|^4}\lesssim \Vert W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}^{(n)}\Vert \end{aligned}$$

for all $z\in \mathbb {C}$. Therefore

$$\begin{aligned} \sup _{z\in \mathbb {C}}|\psi _1(z)\psi _2(z)|\frac{|\varphi _1(z)-\varphi _2(z)|^4}{1+|\varphi _1(z)-\varphi _2(z)|^4}e^{\left| \frac{\varphi _1(z)-\varphi _2(z)}{2}\right| ^2-|z|^2}<\infty . \end{aligned}$$

Modifying the proof of [8, Proposition 2.1], we obtain $\varphi _1(z)-\varphi _2(z)=az+b$ for some $a,b\in \mathbb {C}$. The proof is complete. $\square$

We are now ready to prove Theorem B. For any $t>0$, we denote

$$\begin{aligned} \Omega _t=\{z\in \mathbb {C}:|\varphi _1(z)-\varphi _2(z)|<t\}. \end{aligned}$$

Proof of Theorem B

The “if part” is trivial, we only need to prove the “only if part”.

First assume $W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}^{(n)}$ is bounded on $\mathcal {F}^p$. Lemma 4.1 tells us that $\varphi _1(z)-\varphi _2(z)=az+b$. Let $t=\frac{b}{2}$ if $a=0$ and $t=1$ if $a\ne 0$. Then there exists $R>0$ such that $\{|z|>R\}\subset \Omega _t^{c}$. By (17) and Lemma 2.5, we can find a constant $C>0$, independent of $\psi _1$ and $\psi _2$, such that

$$\begin{aligned}\Vert W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}^{(n)}\Vert \ge C|\psi _1(z)|e^{\frac{|\varphi _1(z)|^2-|z|^2}{2}}\end{aligned}$$

for all $z\in \Omega _t^{c}$. It follows that $\sup _{z\in \mathbb {C}}|\psi _1(z)|e^{\frac{|\varphi _1(z)|^2-|z|^2}{2}}<\infty$ since $\Omega _t$ is bounded in $\mathbb {C}$. Obviously, $\psi _1=(W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}^{(n)})1\in \mathcal {F}^q$. Thus according to [10, Theorem 3.4], $W_{\psi _1,\varphi _1}$ is bounded, then so is $W_{\psi _2,\varphi _2}^{(n)}$.

Now suppose $W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}^{(n)}$ is compact on $\mathcal {F}^p$, then both $W_{\psi _1,\varphi _1}$ and $W_{\psi _2,\varphi _2}^{(n)}$ are bounded. It follows from [10, Propsition 3.1] and [4, Lemma 4.3] that

$$\begin{aligned}M_1=\sup _{z\in \mathbb {C}}|\psi _1(z)|e^{\frac{|\varphi _1(z)|^2-|z|^2}{2}}<\infty \end{aligned}$$

and

$$\begin{aligned}M'_2=\sup _{z\in \mathbb {C}}|\psi _2(z)||\varphi _2(z)|^ne^{\frac{|\varphi _2(z)|^2-|z|^2}{2}}<\infty .\end{aligned}$$

Moreover, $\varphi _i(z)=a_iz+b_i$ for $i=1,2$. If $a_1=0$, then [10, Corollary 3.2] tells us that $W_{\psi _1,\varphi _1}$ is bounded on $\mathcal {F}^p$. So is $W_{\psi _2,\varphi _2}^{(n)}$. Now we consider the case $a_1\ne 0$.

Taking $w=\varphi _1(z)$ in (16), we obtain

$$\begin{aligned} \begin{aligned}&|\psi _1(z)|e^{\frac{|\varphi _1(z)|-|z|^2}{2}}-M'_2\left| \frac{\varphi _1(z)}{\varphi _2(z)}\right| ^ne^{-\frac{|\varphi _{1}(z)-\varphi _2(z)|^2}{2}}\\ \le&\Vert (W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}^{(n)})k_{\varphi _1(z)}\Vert _p. \end{aligned} \end{aligned}$$

(18)

The compactness of $W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}^{(n)}$ yields that $\Vert (W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}^{(n)})k_{\varphi _1(z)}\Vert _p\rightarrow 0$ as $|z|\rightarrow \infty$. If $a_1\ne a_2$, then $\lim _{|z|\rightarrow \infty }|\varphi _1(z)-\varphi _2(z)|=\infty$. Then by (18), we have

$$\begin{aligned}\lim _{|z|\rightarrow \infty }|\psi _1(z)|e^{\frac{|\varphi _1(z)|-|z|^2}{2}}=0.\end{aligned}$$

If $0\ne a_1=a_2$, then $b_1\ne b_2$ and $|\varphi _1(z)|,|\varphi _2(z)|\rightarrow \infty$ as $|z|\rightarrow \infty$. By the proof of Lemma 2.5, we obtain

$$\begin{aligned} \begin{aligned}&|\psi _1(z)|e^{\frac{|\varphi _1(z)|-|z|^2}{2}}\\ \lesssim&\Vert (W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}^{(n)}k_{\varphi _1(z)}\Vert _p+\Vert (W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}^{(n)}k_{\varphi _2(z)}\Vert _p \end{aligned} \end{aligned}$$

for |z| large enough. Then the compactness of $W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}^{(n)}$ yields that

$$\begin{aligned}\lim _{|z|\rightarrow \infty }|\psi _1(z)|e^{\frac{|\varphi _1(z)|-|z|^2}{2}}=0.\end{aligned}$$

Therefore, by [10, Theorem 3.4], $W_{\psi _1,\varphi _1}$ is compact. Then so is $W_{\psi _2,\varphi _2}^{(n)}$. The proof is complete. $\square$

Remark

The methods in this paper can be applied to study the difference $W_{\psi _1,\varphi _1}^{(m)}-W_{\psi _2,\varphi _2}^{(n)}:\mathcal {F}^p\rightarrow \mathcal {F}^q$ with $p\ne q$, $\varphi _1\ne \varphi _2$ and $m\ne n$ through more elaborate computations.