Composition operators on weighted Fock spaces induced by $A_{\infty }$-type weights

Abstract

In this paper, we study the composition operators $C_{\varphi }$ acting on the weighted Fock spaces $F^p_{\alpha ,w}$, where w is a weight satisfying some restricted $A_{\infty }$-conditions. We first characterize the boundedness and compactness of the composition operators $C_{\varphi }:F^p_{\alpha ,w}\rightarrow F^q_{\beta ,v}$ for all $0<p,q<\infty$ in terms of certain Berezin type integral transforms. A new condition for the bounded embedding $I_d:F^p_{\alpha ,w}\rightarrow L^q(\mathbb {C},\mu )$ in the case $p>q$ is also obtained. Then, in the case that $w(z)=(1+|z|)^{mp}$ for $m\in \mathbb {R}$, using some Taylor coefficient estimates, we establish an upper bound for the approximation numbers of composition operators acting on $F^p_{\alpha ,w}$.

1 Introduction

A nonnegative function w on the complex plane $\mathbb {C}$ is said to be a weight if it is locally integrable on $\mathbb {C}$. For $0<p,\alpha <\infty$ and a weight w, the weighted space $L^p_{\alpha ,w}$ consists of measurable functions f on $\mathbb {C}$, such that

$$\begin{aligned} \Vert f\Vert ^p_{L^p_{\alpha ,w}}:=\int _{\mathbb {C}}|f(z)|^pe^{-\frac{p\alpha }{2}|z|^2}w(z)\textrm{d}A(z)<\infty , \end{aligned}$$

where $\textrm{d}A$ is the Lebesgue measure on $\mathbb {C}$. Let $\mathcal {H}(\mathbb {C})$ be the space of entire functions on $\mathbb {C}$. Then, the weighted Fock space $F^p_{\alpha ,w}$ is defined by

$$\begin{aligned} F^p_{\alpha ,w}:=L^p_{\alpha ,w}\cap \mathcal {H}(\mathbb {C}) \end{aligned}$$

with inherited (quasi-)norm. If $w\equiv 1$, then we obtain the classical Fock spaces $F^p_{\alpha }$ [21].

We now introduce the class of weights to work on. We always use Q to denote a square in $\mathbb {C}$ with sides parallel to the coordinate axes, and write l(Q) for its side length. As usual, $p'$ denotes the conjugate exponent of p, i.e., $1/p+1/p'=1$, for $1\le p\le \infty$. Given $1<p<\infty$, we say a weight w belongs to the class $A^{\textrm{restricted}}_p$ if $w(z)>0$ a.e. on $\mathbb {C}$ and for some fixed $r>0$

$$\begin{aligned} \mathcal {C}_{p,r}(w):=\sup _{Q,l(Q)=r}\left( \frac{1}{A(Q)}\int _{Q}w\textrm{d}A\right) \left( \frac{1}{A(Q)}\int _{Q}w^{-p'/p}\textrm{d}A\right) ^{p/p'}<\infty . \end{aligned}$$

(1.1)

The class $A^{\textrm{restricted}}_p$ was introduced by Isralowitz [11], and it was shown that the condition (1.1) is independent of the choice of r: if $\mathcal {C}_{p,r_0}(w)<\infty$ for some $r_0>0$, then $\mathcal {C}_{p,r}(w)<\infty$ for any $r>0$. Isralowitz [11, Theorem 3.1] proved that for $p>1$, the classical Fock projection $P_{\alpha }$ defined by

$$\begin{aligned} P_{\alpha }(f)(z):=\frac{\alpha }{\pi }\int _{\mathbb {C}}f(u)e^{\alpha \bar{u}z}e^{-\alpha |u|^2}\textrm{d}A(u) \end{aligned}$$

is bounded on $L^p_{\alpha ,w}$ if and only if $w\in A^{\textrm{restricted}}_p$. Later on, Cascante, Fàbrega and Peláez [3] introduced the class $A^{\textrm{restricted}}_1$, consisting of weights w, such that $w(z)>0$ a.e. on $\mathbb {C}$ and for some fixed $r>0$

$$\begin{aligned} \mathcal {C}_{1,r}(w):=\sup _{Q,l(Q)=r}\frac{\int _Qw\textrm{d}A}{A(Q)\textrm{ess}\inf _{u\in Q}w(u)}<\infty , \end{aligned}$$

and generalized Isralowitz’s result to the case $p=1$. Similarly to the Muckenhoupt weights, we denote

$$\begin{aligned} A^{\textrm{restricted}}_{\infty }:=\bigcup _{1\le p<\infty }A^{\textrm{restricted}}_p. \end{aligned}$$

Cascante, Fàbrega, and Peláez [3] established some Littlewood–Paley-type formulas for the weighted Fock spaces $F^p_{\alpha ,w}$ with $w\in A^{\textrm{restricted}}_{\infty }$, and characterized the boundedness of embedding derivatives from $F^p_{\alpha ,w}$ to the Lebesgue spaces $L^q(\mathbb {C},\mu )$.

In this paper, we are interested in the composition operators acting on the weighted Fock spaces $F^p_{\alpha ,w}$ with $w\in A^{\textrm{restricted}}_{\infty }$. Given $\varphi \in \mathcal {H}(\mathbb {C})$, the induced composition operator $C_{\varphi }$ is defined by $C_{\varphi }f=f\circ \varphi$, $f\in \mathcal {H}(\mathbb {C})$. The properties of composition operators acting on weighted Fock spaces have been attracting many attentions (see, for instance, [2, 4, 6, 10, 17, 18, 20]).

We first concentrate on boundedness and compactness. The results are expressed in terms of the following Berezin type integral transform:

$$\begin{aligned} B_{\alpha }^{\beta ,q,v}(\varphi )(u):=\int _{\mathbb {C}}e^{q\alpha \Re (\bar{u}\varphi (z))- \frac{q\alpha }{2}|u|^2-\frac{q\beta }{2}|z|^2}v(z)\textrm{d}A(z),\quad u\in \mathbb {C}, \end{aligned}$$

where $0<q,\alpha ,\beta <\infty$ and v is a weight. We use D(u, r) to denote the Euclidean disk of center u and radius $r>0$, and write $v(E)=\int _Ev\textrm{d}A$ for Borel sets $E\subset \mathbb {C}$. The boundedness and compactness of composition operators are characterized as follows.

Theorem 1.1

Suppose that $0<p\le q<\infty$, $\alpha ,\beta >0$, $w\in A^{\textrm{restricted}}_{\infty }$, and v is a weight. Let $\varphi \in \mathcal {H}(\mathbb {C})$. Then, the following assertions hold.

• (1) $C_{\varphi }:F^p_{\alpha ,w}\rightarrow F^q_{\beta ,v}$ is bounded if and only if the function

  $$\begin{aligned} G(u):=\frac{B_{\alpha }^{\beta ,q,v}(\varphi )(u)}{w(D(u,1))^{q/p}} \end{aligned}$$

  is bounded on $\mathbb {C}$. Moreover

  $$\begin{aligned} \Vert C_{\varphi }\Vert ^q_{F^p_{\alpha ,w}\rightarrow F^q_{\beta ,v}}\asymp \sup _{u\in \mathbb {C}}G(u). \end{aligned}$$

• (2) $C_{\varphi }:F^p_{\alpha ,w}\rightarrow F^q_{\beta ,v}$ is compact if and only if $\lim _{|u|\rightarrow \infty }G(u)=0$.

Theorem 1.2

Suppose that $0<q<p<\infty$, $\alpha ,\beta >0$, $w\in A^{\textrm{restricted}}_{\infty }$, and v is a weight. Let $\varphi \in \mathcal {H}(\mathbb {C})$. Then, the following conditions are equivalent.

• (a) $C_{\varphi }:F^p_{\alpha ,w}\rightarrow F^q_{\beta ,v}$ is bounded.

• (b) $C_{\varphi }:F^p_{\alpha ,w}\rightarrow F^q_{\beta ,v}$ is compact.

• (c) The function

  $$\begin{aligned} H(u):=\frac{B_{\alpha }^{\beta ,q,v}(\varphi )(u)}{w(D(u,1))} \end{aligned}$$

  belongs to $L^{\frac{p}{p-q}}(\mathbb {C},w)$.

Moreover, if one of these conditions holds, then

$$\begin{aligned} \Vert C_{\varphi }\Vert ^q_{F^p_{\alpha ,w}\rightarrow F^q_{\beta ,v}}\asymp \Vert H\Vert _{L^{\frac{p}{p-q}}(\mathbb {C},w)}. \end{aligned}$$

These two theorems are proved in Sect. 2. The proof of Theorem 1.1 is mainly based on some pointwise estimates of functions in $F^p_{\alpha ,w}$ and some estimates on the weight w. To prove Theorem 1.2, a new characterization on the positive Borel measures $\mu$, such that the embedding operator $I_d:F^p_{\alpha ,w}\rightarrow L^q(\mathbb {C},\mu )$ is bounded in the case $p>q$ is established (see Theorem 2.4).

After the boundedness and compactness are settled, we intend to study the behavior of approximation numbers of composition operators. Let $T:X\rightarrow Y$ be a bounded linear operator between Banach spaces X and Y. For $n\ge 1$, the nth approximation number $a_n(T)$ of T is defined by

$$\begin{aligned} a_n(T):=\inf _{\textrm{rank}\ R<n}\Vert T-R\Vert _{X\rightarrow Y}. \end{aligned}$$

It is clear that $a_1(T)=\Vert T\Vert _{X\rightarrow Y}$ and $\{a_n(T)\}$ is a non-increasing sequence, and in the case that Y has the approximation property, $T:X\rightarrow Y$ is compact if and only if $a_n(T)\rightarrow 0$.

Recently, the investigation of approximation numbers of composition operators on various spaces of holomorphic functions has drawn many attentions (see [8, 12,13,14,15,16]). In particular, Doan et al. [8] established some estimates for the approximation numbers of composition operators acting on the Hilbertian Fock spaces $F^2_{\alpha }$. Given $m\in \mathbb {R}$, we now consider the approximation numbers of composition operators acting on $F^p_{\alpha ,m}:=F^p_{\alpha ,w_{mp}}$, where $w_{mp}(z):=(1+|z|)^{mp}$. Note that the weighted Fock spaces $F^p_{\alpha ,m}$ are actually the Fock–Sobolev spaces that were studied in [5, 7]. By [3, Lemma 2.1], $w_{mp}\in A^{\textrm{restricted}}_{\infty }$ for any $m\in \mathbb {R}$. As an application of Theorem 1.1, the boundedness and compactness of composition operators acting on $F^p_{\alpha ,m}$ can be characterized as follows (see also [6, 17]).

Corollary 1.3

Suppose that $0<p,\alpha <\infty$, $m\in \mathbb {R}$ and $\varphi \in \mathcal {H}(\mathbb {C})$.

• (1) $C_{\varphi }$ is bounded on $F^p_{\alpha ,m}$ if and only if $\varphi (z)=az+b$ for some $a,b\in \mathbb {C}$ with $|a|<1$, or $|a|=1$ and $b=0$.

• (2) $C_{\varphi }$ is compact on $F^p_{\alpha ,m}$ if and only if $\varphi (z)=az+b$ for some $a,b\in \mathbb {C}$ with $|a|<1$.

To study the behavior of the approximation numbers of composition operators on $F^p_{\alpha ,m}$, we establish some coefficient estimates of functions in $F^p_{\alpha ,m}$ in Sect. 3, including the Hardy–Littlewood type and the Hausdorff–Young type theorems. Based on these estimates, we obtain the following upper bound for the approximation numbers of compact composition operators acting on $F^p_{\alpha ,m}$.

Theorem 1.4

Let $1\le p<\infty$, $\alpha >0$, $m\in \mathbb {R}$ and $C_{\varphi }$ be a compact composition operator on $F^p_{\alpha ,m}$ induced by $\varphi (z)=az+b$, where $0<|a|<1$. Then, for sufficiently large positive integer n, we have

$$\begin{aligned} a_{n}(C_{\varphi })\lesssim |a|^ne^{c\sqrt{m+n}}, \end{aligned}$$

where $c=|a|^{-1}|b|\sqrt{\alpha }$.

This theorem is proved in Sect. 4. As an application, we obtain that for any $0<p<\infty$, any compact composition operator on $F^2_{\alpha ,m}$ belongs to the Schatten class $S_p(F^2_{\alpha ,m})$. Using a generalization of Weyl’s inequality, we also show that if $\varphi$ is not constant, then the approximation numbers of $C_{\varphi }$ acting on $F^p_{\alpha ,m}$ cannot decrease faster than geometrically.

Throughout the paper, the notation $A\lesssim B$ (or $B\gtrsim A$) means that there exists a nonessential constant $C>0$, such that $A\le CB$. If $A\lesssim B\lesssim A$, then we write $A\asymp B$.

2 Bounded and compact composition operators

In this section, we study the boundedness and compactness of composition operators. Let v be a weight and $\varphi$ be an entire function. We use $\mu ^{\beta ,q,v}_{\varphi }$ to denote the positive pull-back measure on $\mathbb {C}$ defined by

$$\begin{aligned} \mu ^{\beta ,q,v}_{\varphi }(E):=\int _{\varphi ^{-1}(E)}e^{-\frac{q\beta }{2}|z|^2}v(z)\textrm{d}A(z) \end{aligned}$$

for every Borel subset E of $\mathbb {C}$. Then, the non-univalent change of variables formula gives that for any entire function f

$$\begin{aligned} \Vert C_{\varphi }f\Vert ^q_{F^q_{\beta ,v}}=\int _{\mathbb {C}}|f(\varphi (z))|^qe^{-\frac{q\beta }{2}|z|^2}v(z)\textrm{d}A(z) =\int _{\mathbb {C}}|f(z)|^q\textrm{d}\mu _{\varphi }^{\beta ,q,v}(z). \end{aligned}$$

(2.1)

The following lemma follows from an elementary computation (see [20, Lemma 2] for a proof).

Lemma 2.1

Let $0<q,\alpha ,\beta <\infty$, v be a weight, and $\varphi \in \mathcal {H}(\mathbb {C})$. Then

$$\begin{aligned} \int _{D(u,1)}e^{\frac{q\alpha }{2}|z|^2}\textrm{d}\mu ^{\beta ,q,v}_{\varphi }(z)\lesssim B_{\alpha }^{\beta ,q,v}(\varphi )(u). \end{aligned}$$

We will need the following estimates for weights in the class $A^{\textrm{restricted}}_{\infty }$, which can be found in [3, Remark 2.3] and [11, Lemma 3.4]. Here, we write $Q_1(z)$ for the square of center $z\in \mathbb {C}$ with side length $l(Q)=1$. Moreover, since $\mathbb {R}^2$ can be canonically identified with $\mathbb {C}$, we will treat $\mathbb {Z}^2$ as a subset of $\mathbb {C}$.

Lemma 2.2

Let $w\in A^{\textrm{restricted}}_{\infty }$.

• (1) There exists $C>0$, such that for any $\nu ,\nu '\in \mathbb {Z}^2$,

  $$\begin{aligned} \frac{w(Q_1(\nu ))}{w(Q_1(\nu '))}\le C^{|\nu -\nu '|}. \end{aligned}$$

• (2) For any $z,u\in \mathbb {C}$, such that $|z-u|<1$

  $$\begin{aligned} w(Q_1(z))\asymp w(Q_1(u))\asymp w(D(z,1))\asymp w(D(u,1)). \end{aligned}$$

The following lemma gives a family of test functions in $F^p_{\alpha ,w}$.

Lemma 2.3

Let $0<p,\alpha <\infty$ and $w\in A^{\textrm{restricted}}_{\infty }$. For $u\in \mathbb {C}$, define

$$\begin{aligned} f_u(z):=w(D(u,1))^{-1/p}e^{\alpha \bar{u}z-\frac{\alpha }{2}|u|^2},\quad z\in \mathbb {C}. \end{aligned}$$

Then, $\Vert f_u\Vert _{F^p_{\alpha ,w}}\asymp 1$ and $f_u\rightarrow 0$ uniformly on compact subsets of $\mathbb {C}$ as $|u|\rightarrow \infty$.

Proof

The norm estimate follows from [3, Proposition 4.1]. We now prove the second assertion. By Lemma 2.2, we may find $C>0$, such that

$$\begin{aligned} \frac{w(D(u,1))}{w(D(0,1))}\ge C^{-|u|}. \end{aligned}$$

Therefore, if $|z|\le R$ for some $R>0$, then

$$\begin{aligned} |f_u(z)|&\le w(D(0,1))^{-1/p}C^{|u|/p}e^{\alpha \Re (\bar{u}z)-\frac{\alpha }{2}|u|^2}\\&=w(D(0,1))^{-1/p}C^{|u|/p}e^{\frac{\alpha }{2}|z|^2-\frac{\alpha }{2}|u-z|^2}\\&\le w(D(0,1))^{-1/p}e^{\alpha R^2}C^{|u|/p}e^{-\frac{\alpha }{4}|u|^2}\rightarrow 0 \end{aligned}$$

as $|u|\rightarrow \infty$, which completes the proof. $\square$

We are now ready to prove Theorem 1.1.

Proof of Theorem 1.1

For $u\in \mathbb {C}$, write

$$\begin{aligned} f_u(z)=w(D(u,1))^{-1/p}e^{\alpha \bar{u}z-\frac{\alpha }{2}|u|^2},\quad z\in \mathbb {C}. \end{aligned}$$

(1) Assume first that $C_{\varphi }:F^p_{\alpha ,w}\rightarrow F^q_{\beta ,v}$ is bounded. Then, by Lemma 2.3, we have

$$\begin{aligned} \Vert C_{\varphi }\Vert ^q_{F^p_{\alpha ,w}\rightarrow F^q_{\beta ,v}} \gtrsim \Vert C_{\varphi }f_u\Vert ^q_{F^q_{\beta ,v}}=G(u), \end{aligned}$$

(2.2)

which gives that $\Vert C_{\varphi }\Vert ^q_{F^p_{\alpha ,w}\rightarrow F^q_{\beta ,v}}\gtrsim \sup _{u\in \mathbb {C}}G(u)$.

Conversely, assume that the function G is bounded on $\mathbb {C}$. For any $f\in F^p_{\alpha ,w}$, by Eq. (2.1), [3, Lemma 3.1], Fubini’s theorem, Lemmas 2.1 and 2.2, we obtain that

$$\begin{aligned} \Vert C_{\varphi }f\Vert ^q_{F^q_{\beta ,v}}&=\int _{\mathbb {C}}|f(z)|^q\textrm{d}\mu _{\varphi }^{\beta ,q,v}(z)\nonumber \\&\lesssim \int _{\mathbb {C}}\frac{e^{\frac{q\alpha }{2}|z|^2}}{w(D(z,1))} \int _{D(z,1)}|f(u)|^qe^{-\frac{q\alpha }{2}|u|^2}w(u)\textrm{d}A(u)\textrm{d}\mu ^{\beta ,q,v}_{\varphi }(z)\nonumber \\&\asymp \int _{\mathbb {C}}|f(u)|^qe^{-\frac{q\alpha }{2}|u|^2}\frac{w(u)}{w(D(u,1))} \int _{D(u,1)}e^{\frac{q\alpha }{2}|z|^2}\textrm{d}\mu ^{\beta ,q,v}_{\varphi }(z)\textrm{d}A(u)\nonumber \\&\lesssim \int _{\mathbb {C}}|f(u)|^qe^{-\frac{q\alpha }{2}|u|^2}w(u)\frac{B_{\alpha }^{\beta ,q,v}(\varphi )(u)}{w(D(u,1))}\textrm{d}A(u)\nonumber \\&\le \left( \sup _{u\in \mathbb {C}}G(u)\right) \int _{\mathbb {C}}|f(u)|^qe^{-\frac{q\alpha }{2}|u|^2}\frac{w(u)}{w(D(u,1))^{(p-q)/p}}\textrm{d}A(u). \end{aligned}$$

(2.3)

Applying [3, Corollary 3.2] yields that

$$\begin{aligned}&\int _{\mathbb {C}}|f(u)|^qe^{-\frac{q\alpha }{2}|u|^2}\frac{w(u)}{w(D(u,1))^{(p-q)/p}}\textrm{d}A(u)\nonumber \\&\quad =\int _{\mathbb {C}}|f(u)|^pe^{-\frac{p\alpha }{2}|u|^2}w(u) \left( |f(u)|e^{-\frac{\alpha }{2}|u|^2}w(D(u,1))^{1/p}\right) ^{q-p}\textrm{d}A(u)\lesssim \Vert f\Vert ^q_{F^p_{\alpha ,w}}. \end{aligned}$$

(2.4)

Therefore, $C_{\varphi }:F^p_{\alpha ,w}\rightarrow F^q_{\beta ,v}$ is bounded, and $\Vert C_{\varphi }\Vert ^q_{F^p_{\alpha ,w}\rightarrow F^q_{\beta ,v}}\lesssim \sup _{u\in \mathbb {C}}G(u)$.

(2) If $C_{\varphi }:F^p_{\alpha ,w}\rightarrow F^q_{\beta ,v}$ is compact, then by Lemma 2.3 and (2.2), we obtain that

$$\begin{aligned} G(u)=\Vert C_{\varphi }f_u\Vert ^q_{F^q_{\beta ,v}}\rightarrow 0 \end{aligned}$$

as $|u|\rightarrow \infty$. Assume now that $\lim _{|u|\rightarrow \infty }G(u)=0$. Then, for any $\epsilon >0$, there exists $R>0$, such that $G(u)<\epsilon ^q$ whenever $|u|>R$. Let $\{f_n\}\subset F^p_{\alpha ,w}$ be a bounded sequence that converges to 0 uniformly on compact subsets of $\mathbb {C}$. Then, by (2.3) and (2.4)

$$\begin{aligned}&\Vert C_{\varphi }f_n\Vert ^q_{F^q_{\beta ,v}}\\&\quad \lesssim \left( \int _{|u|\le R}+\int _{|u|>R}\right) |f_n(u)|^qe^{-\frac{q\alpha }{2}|u|^2}w(u) w(D(u,1))^{\frac{q-p}{p}}G(u)\textrm{d}A(u)\\&\quad \lesssim \left( \sup _{u\in \mathbb {C}}G(u)\right) \int _{|u|\le R}|f_n(u)|^qe^{-\frac{q\alpha }{2}|u|^2}w(u) w(D(u,1))^{\frac{q-p}{p}}\textrm{d}A(u)+\epsilon ^q. \end{aligned}$$

Since $f_n\rightarrow 0$ uniformly on the set $\{|u|\le R\}$, we obtain that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\Vert C_{\varphi }f_n\Vert _{F^q_{\beta ,v}}\lesssim \epsilon . \end{aligned}$$

The arbitrariness of $\epsilon >0$ then implies the desired compactness. $\square$

To study the boundedness and compactness of $C_{\varphi }:F^p_{\alpha ,w}\rightarrow F^q_{\beta ,v}$ in the case $p>q$, we need a new characterization of positive Borel measures $\mu$, such that the embedding operator $I_d:F^p_{\alpha ,w}\rightarrow L^q(\mathbb {C},\mu )$ is bounded. For a positive Borel measure $\mu$ on $\mathbb {C}$ and $0<p,\alpha <\infty$, let $\widetilde{\mu }_{\alpha ,q}$ be the function defined by

$$\begin{aligned} \widetilde{\mu }_{\alpha ,q}(z):=\int _{\mathbb {C}}e^{q\alpha \Re (\bar{z}u)-\frac{q\alpha }{2}|z|^2}\textrm{d}\mu (u),\quad z\in \mathbb {C}. \end{aligned}$$

The following theorem characterizes the boundedness of $I_d:F^p_{\alpha ,w}\rightarrow L^q(\mathbb {C},\mu )$ in the case $p>q$ via the functions $\widetilde{\mu }_{\alpha ,q}$.

Theorem 2.4

Suppose that $0<q<p<\infty$, $\alpha >0$, $w\in A^{\textrm{restricted}}_{\infty }$ and $\mu$ is a positive Borel measure on $\mathbb {C}$. Then, the embedding operator $I_d:F^p_{\alpha ,w}\rightarrow L^q(\mathbb {C},\mu )$ is bounded if and only if the function

$$\begin{aligned} \Psi _{\mu }(z):=\frac{\widetilde{\mu }_{\alpha ,q}(z)}{w(D(z,1))} \end{aligned}$$

belongs to $L^{\frac{p}{p-q}}(\mathbb {C},w)$. Moreover

$$\begin{aligned} \Vert I_d\Vert ^q_{F^p_{\alpha ,w}\rightarrow L^q(\mathbb {C},\mu )}\asymp \Vert \Psi _{\mu }\Vert _{L^{\frac{p}{p-q}}(\mathbb {C},w)}. \end{aligned}$$

Proof

Assume first that $\Psi _{\mu }\in L^{\frac{p}{p-q}}(\mathbb {C},w)$. It is easy to see that for $z\in \mathbb {C}$

$$\begin{aligned} \widetilde{\mu }_{\alpha ,q}(z)=\int _{\mathbb {C}}e^{\frac{q\alpha }{2}|u|^2-\frac{q\alpha }{2}|z-u|^2}\textrm{d}\mu (u) \gtrsim \int _{D(z,1)}e^{\frac{q\alpha }{2}|u|^2}\textrm{d}\mu (u). \end{aligned}$$

Therefore, it follows from [3, Theorem 1.2] that $I_d:F^p_{\alpha ,w}\rightarrow L^q(\mathbb {C},\mu )$ is bounded, and $\Vert I_d\Vert ^q_{F^p_{\alpha ,w}\rightarrow L^q(\mathbb {C},\mu )}\lesssim \Vert \Psi _{\mu }\Vert _{L^{\frac{p}{p-q}}(\mathbb {C},w)}$.

Conversely, assume that $I_d:F^p_{\alpha ,w}\rightarrow L^q(\mathbb {C},\mu )$ is bounded. Then, the proof of [3, Theorem 4.4] yields that

$$\begin{aligned} \sum _{\nu \in \mathbb {Z}^2}\left( \frac{\int _{D(\nu ,2)}e^{\frac{q\alpha }{2}|z|^2}\textrm{d}\mu (z)}{w(D(\nu ,1))^{q/p}}\right) ^{\frac{p}{p-q}}\lesssim \Vert I_d\Vert ^{\frac{pq}{p-q}}_{F^p_{\alpha ,w}\rightarrow L^q(\mathbb {C},\mu )}. \end{aligned}$$

(2.5)

Using some elementary computations and Lemma 2.2 gives that

$$\begin{aligned}&\Vert \Psi _{\mu }\Vert ^{\frac{p}{p-q}}_{L^{\frac{p}{p-q}}(\mathbb {C},w)}\nonumber \\&\quad =\int _{\mathbb {C}}\left( \frac{1}{w(D(z,1))}\int _{\mathbb {C}}e^{q\alpha \Re (\bar{z}u)-\frac{q\alpha }{2}|z|^2} \textrm{d}\mu (u)\right) ^{\frac{p}{p-q}}w(z)\textrm{d}A(z)\nonumber \\&\quad =\sum _{\nu \in \mathbb {Z}^2}\int _{Q_1(\nu )}\left( \frac{1}{w(D(z,1))}\sum _{\nu '\in \mathbb {Z}^2} \int _{Q_1(\nu ')}e^{\frac{q\alpha }{2}|u|^2-\frac{q\alpha }{2}|z-u|^2}\textrm{d}\mu (u)\right) ^{\frac{p}{p-q}}w(z)\textrm{d}A(z)\nonumber \\&\quad \lesssim \sum _{\nu \in \mathbb {Z}^2}\int _{Q_1(\nu )}\left( \frac{1}{w(D(z,1))}\sum _{\nu '\in \mathbb {Z}^2} e^{-\frac{q\alpha }{4}|\nu -\nu '|^2}\int _{Q_1(\nu ')}e^{\frac{q\alpha }{2}|u|^2}\textrm{d}\mu (u)\right) ^{\frac{p}{p-q}} w(z)\textrm{d}A(z)\nonumber \\&\quad \asymp \sum _{\nu \in \mathbb {Z}^2}\frac{1}{w(D(\nu ,1))^{\frac{q}{p-q}}} \left( \sum _{\nu '\in \mathbb {Z}^2}e^{-\frac{q\alpha }{4}|\nu -\nu '|^2} \int _{Q_1(\nu ')}e^{\frac{q\alpha }{2}|u|^2}\textrm{d}\mu (u)\right) ^{\frac{p}{p-q}}. \end{aligned}$$

(2.6)

By Hölder’s inequality, we have that

$$\begin{aligned}&\left( \sum _{\nu '\in \mathbb {Z}^2}e^{-\frac{q\alpha }{4}|\nu -\nu '|^2} \int _{Q_1(\nu ')}e^{\frac{q\alpha }{2}|u|^2}\textrm{d}\mu (u)\right) ^{\frac{p}{p-q}}\\&\quad \le \left( \sum _{\nu '\in \mathbb {Z}^2}e^{-\frac{p\alpha }{8}|\nu -\nu '|^2}\right) ^{\frac{q}{p-q}} \sum _{\nu '\in \mathbb {Z}^2}e^{-\frac{pq\alpha }{8(p-q)}|\nu -\nu '|^2} \left( \int _{Q_1(\nu ')}e^{\frac{q\alpha }{2}|u|^2}\textrm{d}\mu (u)\right) ^{\frac{p}{p-q}}\\&\quad \lesssim \sum _{\nu '\in \mathbb {Z}^2}e^{-\frac{pq\alpha }{8(p-q)}|\nu -\nu '|^2} \left( \int _{Q_1(\nu ')}e^{\frac{q\alpha }{2}|u|^2}\textrm{d}\mu (u)\right) ^{\frac{p}{p-q}}. \end{aligned}$$

Inserting this into (2.6) and using Fubini’s theorem, we obtain that

$$\begin{aligned}&\Vert \Psi _{\mu }\Vert ^{\frac{p}{p-q}}_{L^{\frac{p}{p-q}}(\mathbb {C},w)}\\&\quad \lesssim \sum _{\nu \in \mathbb {Z}^2}\frac{1}{w(D(\nu ,1))^{\frac{q}{p-q}}} \sum _{\nu '\in \mathbb {Z}^2}e^{-\frac{pq\alpha }{8(p-q)}|\nu -\nu '|^2} \left( \int _{Q_1(\nu ')}e^{\frac{q\alpha }{2}|u|^2}\textrm{d}\mu (u)\right) ^{\frac{p}{p-q}}\\&\quad \le \sum _{\nu '\in \mathbb {Z}^2}\sum _{\nu \in \mathbb {Z}^2} \left( \frac{w(D(\nu ',1))}{w(D(\nu ,1))}\right) ^{\frac{q}{p-q}}e^{-\frac{pq\alpha }{8(p-q)}|\nu -\nu '|^2} \left( \frac{\int _{D(\nu ',2)}e^{\frac{q\alpha }{2}|u|^2}\textrm{d}\mu (u)}{w(D(\nu ',1))^{q/p}}\right) ^{\frac{p}{p-q}}. \end{aligned}$$

By Lemma 2.2, we may find $C>0$, such that

$$\begin{aligned} \sum _{\nu \in \mathbb {Z}^2} \left( \frac{w(D(\nu ',1))}{w(D(\nu ,1))}\right) ^{\frac{q}{p-q}}e^{-\frac{pq\alpha }{8(p-q)}|\nu -\nu '|^2} \lesssim \sum _{\nu \in \mathbb {Z}^2}C^{|\nu -\nu '|}e^{-\frac{pq\alpha }{8(p-q)}|\nu -\nu '|^2}<\infty . \end{aligned}$$

Therefore

$$\begin{aligned} \Vert \Psi _{\mu }\Vert ^{\frac{p}{p-q}}_{L^{\frac{p}{p-q}}(\mathbb {C},w)}\lesssim \sum _{\nu '\in \mathbb {Z}^2} \left( \frac{\int _{D(\nu ',2)}e^{\frac{q\alpha }{2}|u|^2}\textrm{d}\mu (u)}{w(D(\nu ',1))^{q/p}}\right) ^{\frac{p}{p-q}}, \end{aligned}$$

which, in conjunction with (2.5), finishes the proof. $\square$

We are now ready to prove Theorem 1.2.

Proof of Theorem 1.2

The implication (b)$\Rightarrow$(a) is trivial. We now prove the implications (a)$\Rightarrow$(c)$\Rightarrow$(b).

Assume first that (a) holds. Then, by Eq. (2.1), the embedding operator $I_d:F^p_{\alpha ,w}\rightarrow L^q(\mathbb {C},\mu ^{\beta ,q,v}_{\varphi })$ is bounded, and

$$\begin{aligned} \Vert I_d\Vert _{F^p_{\alpha ,w}\rightarrow L^q(\mathbb {C},\mu ^{\beta ,q,v}_{\varphi })}=\Vert C_{\varphi }\Vert _{F^p_{\alpha ,w}\rightarrow F^q_{\beta ,v}}. \end{aligned}$$

Consequently, Theorem 2.4 implies that $\frac{\widetilde{(\mu ^{\beta ,q,v}_{\varphi })}_{\alpha ,q}(\cdot )}{w(D(\cdot ,1))}\in L^{\frac{p}{p-q}}(\mathbb {C},w)$, and

$$\begin{aligned} \left\| \frac{\widetilde{(\mu ^{\beta ,q,v}_{\varphi })}_{\alpha ,q}(\cdot )}{w(D(\cdot ,1))}\right\| _{L^{\frac{p}{p-q}}(\mathbb {C},w)}\asymp \Vert I_d\Vert ^q_{F^p_{\alpha ,w}\rightarrow L^q(\mathbb {C},\mu ^{\beta ,q,v}_{\varphi })} =\Vert C_{\varphi }\Vert ^q_{F^p_{\alpha ,w}\rightarrow F^q_{\beta ,v}}. \end{aligned}$$

The definition of $\mu ^{\beta ,q,v}_{\varphi }$ gives that

$$\begin{aligned} \widetilde{(\mu ^{\beta ,q,v}_{\varphi })}_{\alpha ,q}(z)= \int _{\mathbb {C}}e^{q\alpha \Re (\bar{z}u)-\frac{q\alpha }{2}|z|^2}\textrm{d}\mu ^{\beta ,q,v}_{\varphi }(u) =B_{\alpha }^{\beta ,q,v}(\varphi )(z),\quad z\in \mathbb {C}. \end{aligned}$$

Therefore, $H\in L^{\frac{p}{p-q}}(\mathbb {C},w)$ with $\Vert H\Vert _{L^{\frac{p}{p-q}}(\mathbb {C},w)}\asymp \Vert C_{\varphi }\Vert ^q_{F^p_{\alpha ,w}\rightarrow F^q_{\beta ,v}}$, that is, (c) holds.

Assume now that (c) holds. Then, for any $\epsilon >0$, we may choose $R>0$, such that

$$\begin{aligned} \int _{|u|>R}H(u)^{\frac{p}{p-q}}w(u)\textrm{d}A(u)<\epsilon ^{\frac{pq}{p-q}}. \end{aligned}$$

Let $\{f_n\}\subset F^p_{\alpha ,w}$ be a bounded sequence that converges to 0 uniformly on compact subsets of $\mathbb {C}$. Then, by (2.3) and Hölder’s inequality

$$\begin{aligned} \Vert C_{\varphi }f_n\Vert ^q_{F^q_{\beta ,v}}&\lesssim \left( \int _{|u|\le R}+\int _{|u|>R}\right) |f_n(u)|^qe^{-\frac{q\alpha }{2}|u|^2}w(u)H(u)\textrm{d}A(u)\\&\le \left( \int _{|u|\le R}|f_n(u)|^pe^{-\frac{p\alpha }{2}|u|^2}w(u)\textrm{d}A(u)\right) ^{q/p} \Vert H\Vert _{L^{\frac{p}{p-q}}(\mathbb {C},w)}\\&\qquad +\Vert f_n\Vert ^q_{F^p_{\alpha ,w}}\left( \int _{|u|>R}H(u)^{\frac{p}{p-q}}w(u)\textrm{d}A(u)\right) ^{\frac{p-q}{p}}\\&\lesssim \left( \int _{|u|\le R}|f_n(u)|^pe^{-\frac{p\alpha }{2}|u|^2}w(u)\textrm{d}A(u)\right) ^{q/p}+\epsilon ^q. \end{aligned}$$

Since $f_n\rightarrow 0$ uniformly on the set $\{|u|\le R\}$, we obtain that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\Vert C_{\varphi }f_n\Vert _{F^q_{\beta ,v}}\lesssim \epsilon . \end{aligned}$$

The arbitrariness of $\epsilon$ then implies (b). $\square$

To prove Corollary 1.3, note that for any $\gamma \in \mathbb {R}$

$$\begin{aligned} w_{\gamma }(D(u,1))=\int _{D(u,1)}(1+|z|)^{\gamma }\textrm{d}A(z)\asymp (1+|u|)^{\gamma },\quad u\in \mathbb {C}. \end{aligned}$$

(2.7)

Proof of Corollary 1.3

By Theorem 1.1 and (2.7), $C_{\varphi }$ is bounded on $F^p_{\alpha ,m}$ if and only if the function

$$\begin{aligned} G_{\varphi }(u):=\frac{1}{(1+|u|)^{mp}}\int _{\mathbb {C}}\left| e^{\alpha \bar{u}\varphi (z)}\right| ^p e^{-\frac{p\alpha }{2}|u|^2-\frac{p\alpha }{2}|z|^2}(1+|z|)^{mp}\textrm{d}A(z) \end{aligned}$$

is bounded on $\mathbb {C}$, and $C_{\varphi }$ is compact on $F^p_{\alpha ,m}$ if and only if $\lim _{|u|\rightarrow \infty }G_{\varphi }(u)=0$.

Assume first that $C_{\varphi }$ is bounded on $F^p_{\alpha ,m}$. Then, using the subharmonic property, we have for any $u\in \mathbb {C}$

$$\begin{aligned} \sup _{\mathbb {C}} G_{\varphi }&\ge \frac{1}{(1+|u|)^{mp}}\int _{D(u,1)}\left| e^{\alpha \bar{u}\varphi (z)}\right| ^p e^{-p\alpha \Re (\bar{u}z)-\frac{p\alpha }{2}|z-u|^2}(1+|z|)^{mp}\textrm{d}A(z)\nonumber \\&\gtrsim \int _{D(u,1)}\left| e^{\alpha \bar{u}(\varphi (z)-z)}\right| ^p\textrm{d}A(z) \gtrsim e^{p\alpha (\Re (\bar{u}\varphi (u))-|u|^2)}, \end{aligned}$$

(2.8)

which implies that

$$\begin{aligned} \limsup _{|u|\rightarrow \infty }\Re \left( \frac{\varphi (u)}{u}\right) \le 1. \end{aligned}$$

Consequently, $\varphi (u)=au+b$ for some $a,b\in \mathbb {C}$ with $\Re a\le 1$. Since the boundedness of $C_{\varphi }$ on $F^p_{\alpha ,m}$ implies that for any $\theta \in [0,2\pi )$, $C_{\varphi _{e^{i\theta }}}$ is bounded on $F^p_{\alpha ,m}$, where $\varphi _{e^{i\theta }}(z):=\varphi (e^{i\theta }z)$, we actually have $|a|\le 1$. If $|a|=1$, then applying (2.8) to the symbol $\varphi _{\bar{a}}$ gives that

$$\begin{aligned} \sup _{\mathbb {C}}G_{\varphi }=\sup _{\mathbb {C}}G_{\varphi _{\bar{a}}}\gtrsim e^{p\alpha (\Re (\bar{u}\varphi (\bar{a}u))-|u|^2)} =e^{p\alpha \Re (\bar{u}b)},\quad \forall u\in \mathbb {C}, \end{aligned}$$

which gives $b=0$. Therefore, we obtain that if $C_{\varphi }$ is bounded on $F^p_{\alpha ,m}$, then $\varphi (z)=az+b$ for some $a,b\in \mathbb {C}$ with $|a|<1$, or $|a|=1$ and $b=0$.

If $\varphi (z)=az$ for some unimodular constant a, then the rotation invariance of the Lebesgue measure $\textrm{d}A$ implies that $C_{\varphi }$ is an isometric isomorphism on $F^p_{\alpha ,m}$, which is unquestionably non-compact.

Assume now that $\varphi (z)=az+b$ for some $a,b\in \mathbb {C}$ with $|a|<1$. A straightforward computation gives that

$$\begin{aligned} G_{\varphi }(u)&=\frac{1}{(1+|u|)^{mp}}\int _{\mathbb {C}}\left| e^{\alpha \bar{u}(az+b)}\right| ^p e^{-\frac{p\alpha }{2}|u|^2-\frac{p\alpha }{2}|z|^2}(1+|z|)^{mp}\textrm{d}A(z)\\&=\frac{e^{p\alpha \Re (\bar{u}b)}}{(1+|u|)^{mp}}\int _{\mathbb {C}} e^{-\frac{p\alpha }{2}|\bar{a}u-z|^2+\frac{p\alpha }{2}|au|^2-\frac{p\alpha }{2}|u|^2}(1+|z|)^{mp}\textrm{d}A(z)\\&=\frac{e^{-\frac{p\alpha }{2}(1-|a|^2)|u|^2+p\alpha \Re (\bar{u}b)}}{(1+|u|)^{mp}} \int _{\mathbb {C}}e^{-\frac{p\alpha }{2}|z|^2}(1+|z-\bar{a}u|)^{mp}\textrm{d}A(z). \end{aligned}$$

In the case $m\ge 0$, we have

$$\begin{aligned} G_{\varphi }(u)&\le e^{-\frac{p\alpha }{2}(1-|a|^2)|u|^2+p\alpha \Re (\bar{u}b)} \int _{\mathbb {C}}e^{-\frac{p\alpha }{2}|z|^2}(1+|z|)^{mp}\textrm{d}A(z)\\&\lesssim e^{-\frac{p\alpha }{4}(1-|a|^2)|u|^2}\rightarrow 0 \end{aligned}$$

as $|u|\rightarrow \infty$. In the case $m<0$, we have

$$\begin{aligned}&\sup _{z,u\in \mathbb {C}}\left( \frac{1+|z-\bar{a}u|}{1+|u|}\right) ^{mp}e^{-\frac{p\alpha }{4}(1-|a|^2)|u|^2}\\&\quad \le \sup _{u\in \mathbb {C}}(1+|u|)^{-mp}e^{-\frac{p\alpha }{4}(1-|a|^2)|u|^2}<\infty . \end{aligned}$$

Consequently

$$\begin{aligned} G_{\varphi }(u)\lesssim e^{-\frac{p\alpha }{4}(1-|a|^2)|u|^2+p\alpha \Re (\bar{u}b)} \int _{\mathbb {C}}e^{-\frac{p\alpha }{2}|z|^2}\textrm{d}A(z)\lesssim e^{-\frac{p\alpha }{8}(1-|a|^2)|u|^2}\rightarrow 0 \end{aligned}$$

as $|u|\rightarrow \infty$. Therefore, in both cases, $C_{\varphi }$ is compact on $F^p_{\alpha ,m}$. The proof is complete. $\square$

3 Taylor coefficients of functions in $F^p_{\alpha ,m}$

In this section, we establish some estimates for the Taylor coefficients of functions in weighted Fock spaces $F^p_{\alpha ,m}$, which generalize some results on $F^p_{\alpha }$ in [19].

We will need the following integral estimate.

Lemma 3.1

Let $0<p,\alpha <\infty$ and $m\in \mathbb {R}$. For any positive integer n with $n+m>0$, we have

$$\begin{aligned} \int _0^{\infty }r^{np+1}(r+1)^{mp}e^{-\frac{p\alpha }{2}r^2}\textrm{d}r\asymp \bigg (\frac{n!}{\alpha ^n}\bigg )^{\frac{p}{2}}n^{\frac{(2m-1)p+2}{4}}, \end{aligned}$$

where the implicit constant is independent of n.

Proof

By a change of variables, we have that

$$\begin{aligned} \int _0^{\infty }&r^{np+1}(r+1)^{mp}e^{-\frac{p\alpha }{2}r^2}\textrm{d}r\\&\lesssim \int _0^1e^{-\frac{p\alpha }{2}r^2}\textrm{d}r+ \int _1^{\infty }r^{(m+n)p+1}e^{-\frac{p\alpha }{2}r^2}\textrm{d}r\\&\lesssim 1+\frac{1}{2}\left( \frac{2}{p\alpha }\right) ^{\frac{(m+n)p+2}{2}} \int _{\frac{p\alpha }{2}}^{\infty }t^{\frac{(m+n)p}{2}}e^{-t}\textrm{d}t\\&\lesssim 1+\left( \frac{2}{p\alpha }\right) ^{\frac{(m+n)p}{2}} \Gamma \left( \frac{(m+n)p}{2}+1\right) . \end{aligned}$$

Similarly

$$\begin{aligned} \int _0^{\infty }&r^{np+1}(r+1)^{mp}e^{-\frac{p\alpha }{2}r^2}\textrm{d}r\\&\gtrsim \int _1^{\infty }r^{(m+n)p+1}e^{-\frac{p\alpha }{2}r^2}\textrm{d}r\\&=\frac{1}{2}\left( \frac{2}{p\alpha }\right) ^{\frac{(m+n)p+2}{2}} \int _{\frac{p\alpha }{2}}^{\infty }t^{\frac{(m+n)p}{2}}e^{-t}\textrm{d}t\\&\asymp \left( \frac{2}{p\alpha }\right) ^{\frac{(m+n)p}{2}} \int _{0}^{\infty }\left( t+\frac{p\alpha }{2}\right) ^{\frac{(m+n)p}{2}} e^{-t-\frac{p\alpha }{2}}\textrm{d}t\\&\gtrsim \left( \frac{2}{p\alpha }\right) ^{\frac{(m+n)p}{2}} \Gamma \left( \frac{(m+n)p}{2}+1\right) . \end{aligned}$$

Using Stirling’s formula twice yields that

$$\begin{aligned} \left( \frac{2}{p\alpha }\right) ^{\frac{(m+n)p}{2}} \Gamma \left( \frac{(m+n)p}{2}+1\right)&\asymp \left( \frac{m+n}{\alpha e}\right) ^{\frac{(m+n)p}{2}}\sqrt{2\pi (m+n)}\nonumber \\&\asymp \bigg (\frac{n!}{\alpha ^n}\bigg )^{\frac{p}{2}}n^{\frac{(2m-1)p+2}{4}}, \end{aligned}$$

(3.1)

which completes the proof. $\square$

The following result gives the Hardy–Littlewood type theorem on Taylor coefficients of functions in $F^p_{\alpha ,m}$.

Theorem 3.2

Let $f(z)=\sum _{n\ge 0}a_nz^n$ be an entire function. Fix $\alpha >0$ and $m\in \mathbb {R}$, and consider the following conditions:

$$\begin{aligned} \sum _{n=1}^{\infty }|a_n|^p\bigg (\frac{n!}{\alpha ^n}\bigg )^{\frac{p}{2}} n^{\frac{(2m-1)p+2}{4}}<\infty \end{aligned}$$

(3.2)

and

$$\begin{aligned} \sum _{n=1}^{\infty }|a_n|^p\bigg (\frac{n!}{\alpha ^n}\bigg )^{\frac{p}{2}} n^{\frac{(2m+3)p-6}{4}}<\infty . \end{aligned}$$

(3.3)

• (1) If $0<p\le 2$, then (3.2) implies $f\in F^p_{\alpha ,m}$, and $f\in F^p_{\alpha ,m}$ implies (3.3).

• (2) If $2\le p<\infty$, then (3.3) implies $f\in F^p_{\alpha ,m}$, and $f\in F^p_{\alpha ,m}$ implies (3.2).

Proof

Fix $r>0$. By Hölder’s inequality and the Hardy–Littlewood theorem for Hardy spaces (see, for instance, [9, Theorems 6.2 and 6.3]), we have

$$\begin{aligned} \sum _{n=0}^{\infty }(n+1)^{p-2}|a_n|^pr^{np}\lesssim \int _0^{2\pi }|f(re^{i\theta })|^p\textrm{d}\theta \lesssim \sum _{n=0}^{\infty }|a_n|^pr^{np}, \end{aligned}$$

in the case $0<p\le 2$, and

$$\begin{aligned} \sum _{n=0}^{\infty }|a_n|^pr^{np}\lesssim \int _0^{2\pi }|f(re^{i\theta })|^p\textrm{d}\theta \lesssim \sum _{n=0}^{\infty }(n+1)^{p-2}|a_n|^pr^{np} \end{aligned}$$

in the case $2\le p<\infty$, where the implicit constants are all independent of r. Computing the norm of f in $F^p_{\alpha ,m}$ by polar coordinates, and combining the above inequalities with Lemma 3.1, we can obtain the desired results. $\square$

As an immediate consequence of Theorem 3.2, we have for any entire function $f(z)=\sum _{n\ge 0}a_nz^n$

$$\begin{aligned} \Vert f\Vert ^2_{F^2_{\alpha ,m}}\asymp |a_0|^2+\sum _{n=1}^{\infty }|a_n|^2\frac{n!n^m}{\alpha ^n}. \end{aligned}$$

(3.4)

Our next aim is to establish the Hausdorff–Young theorem for the spaces $F^p_{\alpha ,m}$. To this end, we recall some facts on integral operators induced by reproducing kernels of $F^2_{\alpha ,m}$ (see [5, 7] for details). Given an entire function $f(z)=\sum _{n\ge 0}a_nz^n$ and a real number s, write

$$\begin{aligned} f^+_s(z):=\sum _{n>|s|}a_nz^n\quad \text {and} \quad f^-_s(z):=\sum _{0\le n\le |s|}a_nz^n. \end{aligned}$$

For $m\in \mathbb {R}$ and $\alpha >0$, let $\langle \cdot ,\cdot \rangle _{\alpha ,m}$ be the pairing defined for $f,g\in F^2_{\alpha ,m}$ by

$$\begin{aligned} \langle f,g\rangle _{\alpha ,m}:=\int _{\mathbb {C}}f(z)\overline{g(z)}|z|^{2m}e^{-\alpha |z|^2}\textrm{d}A(z) \end{aligned}$$

if $m>-1$, and

$$\begin{aligned} \langle f,g\rangle _{\alpha ,m}:=\int _{\mathbb {C}}\left( f^-_m(z)\overline{g^-_m(z)}+ f^+_m(z)\overline{g^+_m(z)}|z|^{2m}\right) e^{-\alpha |z|^2}\textrm{d}A(z) \end{aligned}$$

if $m\le -1$. Then, $\langle \cdot ,\cdot \rangle _{\alpha ,m}$ is an inner product on $F^2_{\alpha ,m}$ that induces an equivalent norm. For any $u\in \mathbb {C}$, let $K^{\alpha ,m}_u$ be the reproducing kernel of the Hilbert space $F^2_{\alpha ,m}$ endowed with the inner product $\langle \cdot ,\cdot \rangle _{\alpha ,m}$, and write $K^{\alpha ,m+}_u:=(K^{\alpha ,m}_u)^+_{2m}$. For $f\in L^p_{\alpha ,m}:=L^p_{\alpha ,w_{mp}}$, define

$$\begin{aligned} P_{\alpha ,m}f(z):=\int _{\mathbb {C}}f(u)\overline{K^{\alpha ,m}_z(u)}|u|^{2m}e^{-\alpha |u|^2}\textrm{d}A(u),\quad z\in \mathbb {C}, \end{aligned}$$

in the case $m>-1$, and define

$$\begin{aligned} P^+_{\alpha ,m}f(z):=\int _{\mathbb {C}}f(u)\overline{K^{\alpha ,m+}_z(u)}|u|^{2m} e^{-\alpha |u|^2}\textrm{d}A(u),\quad z\in \mathbb {C} \end{aligned}$$

in the case $m\le -1$. The following result can be found in [7, Theorem 5.3].

Lemma 3.3

Let $1\le p<\infty$, $\alpha >0$ and $m\in \mathbb {R}$.

• (1) If $m>-1$, then $P_{\alpha ,m}:L^p_{\alpha ,m}\rightarrow F^p_{\alpha ,m}$ is a bounded projection.

• (2) If $m\le -1$, then $P^+_{\alpha ,m}:L^p_{\alpha ,m}\rightarrow F^{p,+}_{\alpha ,m}$ is a bounded projection, where

  $$\begin{aligned} F^{p,+}_{\alpha ,m}:=\{f\in F^p_{\alpha ,m}:f^-_{2m}=0\}. \end{aligned}$$

We are now ready to establish the Hausdorff–Young theorem for $F^p_{\alpha ,m}$.

Theorem 3.4

Let $1\le p\le \infty$, $\alpha >0$, and $m\in \mathbb {R}$, and let $f(z)=\sum _{n\ge 0}a_nz^n$ be an entire function. Consider the following condition for $1<p\le \infty$:

$$\begin{aligned} \sum _{n=1}^{\infty }|a_n|^{p'}\bigg (\frac{n!}{\alpha ^n}\bigg )^{\frac{p'}{2}} n^{\frac{(2m+1)p'-2}{4}}<\infty . \end{aligned}$$

(3.5)

• (1) If $2\le p\le \infty$ and (3.5) holds, then $f\in F^p_{\alpha ,m}$.

• (2) If $1<p\le 2$ and $f\in F^p_{\alpha ,m}$, then (3.5) holds.

• (3) If $f\in F^1_{\alpha ,m}$, then

  $$\begin{aligned} \sup _{n\ge 1}|a_n|\sqrt{\frac{n!}{\alpha ^n}}n^{\frac{2m+1}{4}}<\infty . \end{aligned}$$

Proof

We consider the Lebesgue spaces $L^p(\mathbb {C}):=L^p(\mathbb {C},\textrm{d}A)$ and $l^p(\mu ):=L^p(\mathbb {N}_0,\mu )$, where $\mu$ is the discrete measure defined by

$$\begin{aligned} \mu (\{0\}):=1 \quad \text {and} \quad \mu (\{n\}):=\frac{1}{\sqrt{n}},\ n=1,2,\ldots . \end{aligned}$$

(1) For any sequence $\sigma =\{\sigma _n\}_{n\ge 0}$, define

$$\begin{aligned} T(\sigma )(z):=\left( \sigma _0+\sum _{n=1}^{\infty }\sqrt{\frac{\alpha ^n}{n!n^{m+\frac{1}{2}}}}\sigma _nz^n\right) (1+|z|)^me^{-\frac{\alpha }{2}|z|^2},\quad z\in \mathbb {C}. \end{aligned}$$

It is easy to see that for any $\sigma \in l^{p'}(\mu )$, $T(\sigma )$ is well defined and the sum inside the parentheses is an entire function. We now prove that $T:l^{p'}(\mu )\rightarrow L^p(\mathbb {C})$ is bounded for any $2\le p\le \infty$. By the Riesz–Thorin theorem, it is sufficient to show that $T:l^2(\mu )\rightarrow L^2(\mathbb {C})$ and $T:l^1(\mu )\rightarrow L^{\infty }(\mathbb {C})$ are both bounded.

For the boundedness of $T:l^2(\mu )\rightarrow L^2(\mathbb {C})$, by (3.4), it is easy to see that

$$\begin{aligned} \Vert T(\sigma )\Vert ^2_{L^2(\mathbb {C})}&=\bigg \Vert \sigma _0+\sum _{n=1}^{\infty }\sqrt{\frac{\alpha ^n}{n!n^{m+\frac{1}{2}}}}\sigma _nz^n\bigg \Vert ^2_{F^2_{\alpha ,m}}\\&\asymp |\sigma _0|^2+\sum _{n=1}^{\infty }\frac{|\sigma _n|^2}{\sqrt{n}}=\Vert \sigma \Vert ^2_{l^2(\mu )}. \end{aligned}$$

We next consider the boundedness of $T:l^1(\mu )\rightarrow L^{\infty }(\mathbb {C})$. It is clear that

$$\begin{aligned} \Vert T(\sigma )\Vert _{L^{\infty }(\mathbb {C})}\lesssim |\sigma _0|+ \sum _{n=1}^{\infty }\sqrt{\frac{\alpha ^n}{n!n^{m+\frac{1}{2}}}}|\sigma _n|\sup _{z\in \mathbb {C}} \Big (|z|^n(1+|z|)^me^{-\frac{\alpha }{2}|z|^2}\Big ). \end{aligned}$$

In the case $m\le 0$, for any $n>-m$, by (3.1), we have that

$$\begin{aligned} \sup _{z\in \mathbb {C}}\Big (|z|^n(1+|z|)^me^{-\frac{\alpha }{2}|z|^2}\Big )&\le \sup _{z\in \mathbb {C}}\Big (|z|^{m+n}e^{-\frac{\alpha }{2}|z|^2}\Big )\\&=\bigg (\frac{m+n}{\alpha e}\bigg )^{\frac{m+n}{2}}\asymp \sqrt{\frac{n!n^{m-\frac{1}{2}}}{\alpha ^n}}. \end{aligned}$$

Similarly, in the case $m>0$, for any $n\ge 1$

$$\begin{aligned} \sup _{z\in \mathbb {C}}\Big (|z|^n(1+|z|)^me^{-\frac{\alpha }{2}|z|^2}\Big )&\lesssim \sup _{z\in \mathbb {C}}\Big (|z|^n(1+|z|^m)e^{-\frac{\alpha }{2}|z|^2}\Big )\\&\le \bigg (\frac{n}{\alpha e}\bigg )^{\frac{n}{2}}+ \bigg (\frac{m+n}{\alpha e}\bigg )^{\frac{m+n}{2}}\\&\lesssim \sqrt{\frac{n!n^{m-\frac{1}{2}}}{\alpha ^n}}. \end{aligned}$$

Therefore

$$\begin{aligned} \Vert T(\sigma )\Vert _{L^{\infty }(\mathbb {C})}\lesssim |\sigma _0|+ \sum _{n=1}^{\infty }\frac{|\sigma _n|}{\sqrt{n}}=\Vert \sigma \Vert _{l^1(\mu )}. \end{aligned}$$

Consequently, for any $2\le p\le \infty$, the operator $T:l^{p'}(\mu )\rightarrow L^p(\mathbb {C})$ is bounded.

Assume now that (3.5) holds. Then, the sequence $\sigma =\{\sigma _n\}_{n\ge 0}$ defined by

$$\begin{aligned} \sigma _0:=a_0 \quad \text {and} \quad \sigma _n:=\sqrt{\frac{n!n^{m+\frac{1}{2}}}{\alpha ^n}}a_n,\ n=1,2,\ldots \end{aligned}$$

is in $l^{p'}(\mu )$. Hence, the boundedness of $T:l^{p'}(\mu )\rightarrow L^p(\mathbb {C})$ gives that

$$\begin{aligned} \Vert f\Vert ^p_{F^p_{\alpha ,m}}&=\int _{\mathbb {C}}\bigg |\bigg (\sum _{n=0}^{\infty } a_nz^n\bigg )(1+|z|)^me^{-\frac{\alpha }{2}|z|^2}\bigg |^p\textrm{d}A(z)\\&=\int _{\mathbb {C}}\bigg |\bigg (\sigma _0+\sum _{n=1}^{\infty }\sqrt{\frac{\alpha ^n}{n!n^{m+\frac{1}{2}}}}\sigma _nz^n\bigg )(1+|z|)^me^{-\frac{\alpha }{2}|z|^2}\bigg |^p\textrm{d}A(z)\\&=\Vert T(\sigma )\Vert ^p_{L^p(\mathbb {C})}\lesssim \Vert \sigma \Vert ^{p}_{l^{p'}(\mu )}<\infty , \end{aligned}$$

which is $f\in F^p_{\alpha ,m}$.

(2) We first consider the case $m>-1$. By Lemma 3.3, $P_{\alpha ,m}:L^p_{\alpha ,m}\rightarrow F^p_{\alpha ,m}$ is a bounded projection. For any function $g\in L^p(\mathbb {C})$, define

$$\begin{aligned} \sigma _0:=\int _{\mathbb {C}}P_{\alpha ,m}Vg(z)(1+|z|)^{2m}e^{-\alpha |z|^2}\textrm{d}A(z) \end{aligned}$$

and

$$\begin{aligned} \sigma _n:=\sqrt{\frac{\alpha ^n\sqrt{n}}{n!n^m}}\int _{\mathbb {C}}\bar{z}^nP_{\alpha ,m}Vg(z) (1+|z|)^{2m}e^{-\alpha |z|^2}\textrm{d}A(z),\quad n\ge 1, \end{aligned}$$

and let $S(g)=\sigma :=\{\sigma _n\}_{n\ge 0}$, where V is the isometry from $L^p(\mathbb {C})$ to $L^p_{\alpha ,m}$ defined by

$$\begin{aligned} Vg(z):=g(z)(1+|z|)^{-m}e^{\frac{\alpha }{2}|z|^2},\quad z\in \mathbb {C}. \end{aligned}$$

We claim that $S:L^1(\mathbb {C})\rightarrow l^{\infty }(\mu )$ is bounded. In fact, it is obvious that

$$\begin{aligned} |\sigma _0|\lesssim \Vert P_{\alpha ,m}Vg\Vert _{F^1_{\alpha ,m}}\lesssim \Vert Vg\Vert _{L^1_{\alpha ,m}} =\Vert g\Vert _{L^1(\mathbb {C})}. \end{aligned}$$

For any positive integer n, arguing as (1), we have that

$$\begin{aligned} |\sigma _n|&\le \sqrt{\frac{\alpha ^n\sqrt{n}}{n!n^m}}\sup _{z\in \mathbb {C}} \left( |z|^n(1+|z|)^me^{-\frac{\alpha }{2}|z|^2}\right) \Vert P_{\alpha ,m}Vg\Vert _{F^1_{\alpha ,m}}\\&\lesssim \sqrt{\frac{\alpha ^n\sqrt{n}}{n!n^m}}\bigg (\frac{m+n}{\alpha e}\bigg )^{\frac{m+n}{2}}\Vert g\Vert _{L^1(\mathbb {C})}\asymp \Vert g\Vert _{L^1(\mathbb {C})}. \end{aligned}$$

Hence, $S:L^1(\mathbb {C})\rightarrow l^{\infty }(\mu )$ is bounded. We next prove that $S:L^2(\mathbb {C})\rightarrow l^2(\mu )$ is bounded. Suppose that $P_{\alpha ,m}Vg(z)=\sum _{n\ge 0}c_nz^n$. Then, it is clear that $|\sigma _0|\asymp |c_0|$ and by Lemma 3.1

$$\begin{aligned} |\sigma _n|=|c_n|\sqrt{\frac{\alpha ^n\sqrt{n}}{n!n^m}}\int _{\mathbb {C}}|z|^{2n} (1+|z|)^{2m}e^{-\alpha |z|^2}\textrm{d}A(z)\asymp |c_n|\sqrt{\frac{n!}{\alpha ^n}}n^{\frac{2m+1}{4}} \end{aligned}$$

for any positive integer n. Consequently, by (3.4) and the boundedness of $P_{\alpha ,m}:L^2_{\alpha ,m}\rightarrow F^2_{\alpha ,m}$

$$\begin{aligned} \Vert \sigma \Vert ^2_{l^2(\mu )}&=|\sigma _0|^2+\sum _{n=1}^{\infty }\frac{|\sigma _n|^2}{\sqrt{n}} \asymp |c_0|^2+\sum _{n=1}^{\infty }|c_n|^2\frac{n!n^m}{\alpha ^n}\\&\asymp \Vert P_{\alpha ,m}Vg\Vert ^2_{F^2_{\alpha ,m}}\lesssim \Vert Vg\Vert ^2_{L^2_{\alpha ,m}}=\Vert g\Vert ^2_{L^2(\mathbb {C})}, \end{aligned}$$

which gives the boundedness of $S:L^2(\mathbb {C})\rightarrow l^2(\mu )$. Therefore, for any $1\le p\le 2$, $S:L^p(\mathbb {C})\rightarrow l^{p'}(\mu )$ is bounded by the Riesz–Thorin theorem.

We now assume that $1<p\le 2$ and $f(z)=\sum _{n\ge 0}a_nz^n$ belongs to $F^p_{\alpha ,m}$. Then, $f=Vg$ for some $g\in L^p(\mathbb {C})$ with $\Vert f\Vert _{F^p_{\alpha ,m}}=\Vert g\Vert _{L^p(\mathbb {C})}$, and $P_{\alpha ,m}Vg=P_{\alpha ,m}f=f$. Thus, we have $S(g)=\{\sigma _n\}$, where

$$\begin{aligned} |\sigma _0|\asymp |a_0|,\quad |\sigma _n|\asymp |a_n|\sqrt{\frac{n!}{\alpha ^n}}n^{\frac{2m+1}{4}},\quad n\ge 1. \end{aligned}$$

Therefore

$$\begin{aligned} \sum _{n=1}^{\infty }|a_n|^{p'}\bigg (\frac{n!}{\alpha ^n}\bigg )^{\frac{p'}{2}} n^{\frac{(2m+1)p'-2}{4}}&\asymp \sum _{n=1}^{\infty }\frac{|\sigma _n|^{p'}}{\sqrt{n}} \le \Vert S(g)\Vert ^{p'}_{l^{p'}(\mu )}\\&\lesssim \Vert g\Vert ^{p'}_{L^p(\mathbb {C})}=\Vert f\Vert ^{p'}_{F^p_{\alpha ,m}}<\infty , \end{aligned}$$

that is, (3.5) holds.

In the case $m\le -1$, we use the bounded projection $P^+_{\alpha ,m}:L^p_{\alpha ,m}\rightarrow F^{p,+}_{\alpha ,m}$ instead of $P_{\alpha ,m}$. The rest part is the same and so is omitted.

(3) As in (2), this follows from the boundedness of $S:L^1(\mathbb {C})\rightarrow l^{\infty }(\mu )$. $\square$

4 Approximation numbers of composition operators

In this section, we study the behavior of approximation numbers of composition operators $C_{\varphi }$ acting on $F^p_{\alpha ,m}$. To this end, we need to estimate the norms of $\varphi ^n$, which relies on the following lemma.

Lemma 4.1

Let $0<p,c<\infty$. Then, for sufficiently large $s>0$

$$\begin{aligned} I_s:=\int _c^{\infty }x^{sp+1}e^{-\frac{p}{2}(x-c)^2}\textrm{d}x\lesssim \big (\Gamma (s)\big )^{\frac{p}{2}}s^{\frac{1}{2}+\frac{p}{4}}e^{pc\sqrt{s}}, \end{aligned}$$

where the implicit constant is independent of s.

Proof

Write $h_s(x):=(sp+1)\log x-\frac{p}{2}(x-c)^2$. Then, we have

$$\begin{aligned} h'_s(x)=\frac{sp+1}{x}-p(x-c) \end{aligned}$$

and

$$\begin{aligned} h''_s(x)=-\frac{sp+1}{x^2}-p\le -p. \end{aligned}$$

(4.1)

Letting $h'_s(x)=0$, we obtain

$$\begin{aligned} x=x_s:=\frac{pc+\sqrt{p^2c^2+4p(sp+1)}}{2p}, \end{aligned}$$

and

$$\begin{aligned} h_s(x_s)&=(sp+1)\log \frac{pc+\sqrt{p^2c^2+4p(sp+1)}}{2p}\\&\quad -\frac{p}{2}s+ \frac{c\sqrt{p^2c^2+4p(sp+1)}}{4}-\frac{pc^2}{4}-\frac{1}{2}. \end{aligned}$$

By an elementary computation, we have

$$\begin{aligned} \lim _{s\rightarrow +\infty }\biggl ((sp+1)\biggl (\log \frac{pc+\sqrt{p^2c^2+4p(sp+1)}}{2p} -\log \sqrt{s}\biggr )-\frac{pc}{2}\sqrt{s}\biggr )=\frac{1}{2}, \end{aligned}$$

so we can find $C>0$, independent of s, such that for sufficiently large s

$$\begin{aligned} h_s(x_s)\le \frac{sp+1}{2}\log s-\frac{sp}{2}+pc\sqrt{s}+C, \end{aligned}$$

which implies that

$$\begin{aligned} e^{h_s(x_s)}\lesssim s^{\frac{sp+1}{2}}e^{-\frac{sp}{2}}e^{pc\sqrt{s}}. \end{aligned}$$

Stirling’s formula then yields

$$\begin{aligned} e^{h_s(x_s)}\lesssim \big (\Gamma (s)\big )^{\frac{p}{2}}s^{\frac{1}{2}+\frac{p}{4}}e^{pc\sqrt{s}}. \end{aligned}$$

Since for any $x\in [c,\infty )$, by (4.1), there exists $\xi$, such that

$$\begin{aligned} h_s(x)=h_s(x_s)+\frac{(x-x_s)^2}{2}h''_s(\xi )\le h_s(x_s)-\frac{p}{2}(x-x_s)^2, \end{aligned}$$

we obtain that

$$\begin{aligned} I_s&=\int _c^{\infty }e^{h_s(x)}\textrm{d}x\le e^{h_s(x_s)}\int _c^{\infty }e^{-\frac{p}{2}(x-x_s)^2}\textrm{d}x\\&\lesssim e^{h_s(x_s)}\lesssim \big (\Gamma (s)\big )^{\frac{p}{2}}s^{\frac{1}{2}+\frac{p}{4}}e^{pc\sqrt{s}}. \end{aligned}$$

The proof is complete. $\square$

Lemma 4.2

Let $0<p,\alpha <\infty$, $m\in \mathbb {R}$, and $\varphi (z)=az+b$, $a\ne 0$. Then, for sufficiently large positive integer n

$$\begin{aligned} \Vert \varphi ^n\Vert ^p_{F^p_{\alpha ,m}}\lesssim |a|^{np}\bigg (\frac{n!}{\alpha ^n}\bigg ) ^{\frac{p}{2}}n^{\frac{(2m-1)p+2}{4}}e^{pc\sqrt{m+n}}, \end{aligned}$$

where $c=|a|^{-1}|b|\sqrt{\alpha }$, and the implicit constant is independent of n.

Proof

In the case $b\ne 0$, we have $|z|+1\asymp |z|+|a|^{-1}|b|$ for any $z\in \mathbb {C}$. Consequently

$$\begin{aligned} \Vert \varphi ^n\Vert ^p_{F^p_{\alpha ,m}}&=\int _{\mathbb {C}}|az+b|^{np}(|z|+1)^{mp} e^{-\frac{p\alpha }{2}|z|^2}\textrm{d}A(z)\\&\lesssim |a|^{np}\int _{\mathbb {C}}(|z|+|a|^{-1}|b|)^{(m+n)p} e^{-\frac{p\alpha }{2}|z|^2}\textrm{d}A(z)\\&\lesssim |a|^{np}\int _0^{\infty }(r+|a|^{-1}|b|)^{(m+n)p+1}e^{-\frac{p\alpha }{2}r^2}dr\\&=|a|^{np}\alpha ^{-\frac{(m+n)p+2}{2}}\int _c^{\infty }x^{(m+n)p+1}e^{-\frac{p}{2}(x-c)^2}\textrm{d}x. \end{aligned}$$

Therefore, by Lemma 4.1 and Stirling’s formula, we obtain that

$$\begin{aligned} \Vert \varphi ^n\Vert ^p_{F^p_{\alpha ,m}}&\lesssim \alpha ^{-\frac{np}{2}}|a|^{np} \big (\Gamma (m+n)\big )^{\frac{p}{2}} (m+n)^{\frac{1}{2}+\frac{p}{4}}e^{pc\sqrt{m+n}}\\&\asymp |a|^{np}\bigg (\frac{n!}{\alpha ^n}\bigg )^{\frac{p}{2}} n^{\frac{(2m-1)p+2}{4}}e^{pc\sqrt{m+n}}. \end{aligned}$$

In the case $b=0$, by Lemma 3.1, we have that

$$\begin{aligned} \Vert \varphi ^n\Vert ^p_{F^p_{\alpha ,m}}&=|a|^{np}\int _{\mathbb {C}}|z|^{np}(|z|+1)^{mp} e^{-\frac{p\alpha }{2}|z|^2}\textrm{d}A(z)\\&=2\pi |a|^{np}\int _0^{\infty }r^{np+1}(r+1)^{mp} e^{-\frac{p\alpha }{2}r^2}dr\\&\asymp |a|^{np}\bigg (\frac{n!}{\alpha ^n}\bigg )^{\frac{p}{2}}n^{\frac{(2m-1)p+2}{4}}, \end{aligned}$$

which finishes the proof. $\square$

We are now in a position to prove Theorem 1.4.

Proof of Theorem 1.4

Assume that $n>-m$ is large enough. Define $K_n:F^p_{\alpha ,m}\rightarrow F^p_{\alpha ,m}$ by

$$\begin{aligned} K_n(f)(z)=\sum _{k=0}^{n-1}a_kz^k \end{aligned}$$

for $f(z)=\sum _{k\ge 0}a_kz^k$. Then, $\textrm{rank}(C_{\varphi }\circ K_n)\le n$, and we have

$$\begin{aligned} \Vert (C_{\varphi }-C_{\varphi }\circ K_n)f\Vert _{F^p_{\alpha ,m}}= \biggl \Vert \sum _{k=n}^{\infty }a_k\varphi ^k\biggr \Vert _{F^p_{\alpha ,m}} \le \sum _{k=n}^{\infty }|a_k|\Vert \varphi ^k\Vert _{F^p_{\alpha ,m}}. \end{aligned}$$

We first consider the case $1\le p\le 2$. In the case $p=1$, using Theorem 3.4 and Lemma 4.2 yields that

$$\begin{aligned} \Vert (C_{\varphi }-C_{\varphi }\circ K_n)f\Vert _{F^1_{\alpha ,m}}&\lesssim \Vert f\Vert _{F^1_{\alpha ,m}}\sum _{k=n}^{\infty }\alpha ^{\frac{k}{2}}(k!)^{-\frac{1}{2}} k^{-\frac{2m+1}{4}}\Vert \varphi ^k\Vert _{F^1_{\alpha ,m}}\\&\lesssim \Vert f\Vert _{F^1_{\alpha ,m}}\sum _{k=n}^{\infty }|a|^ke^{c\sqrt{m+k}}. \end{aligned}$$

In the case $1<p\le 2$, by Hölder’s inequality, Theorem 3.4, and Lemma 4.2, we obtain that

$$\begin{aligned}&\Vert (C_{\varphi }-C_{\varphi }\circ K_n)f\Vert _{F^p_{\alpha ,m}}\\&\quad \le \left( \sum _{k=n}^{\infty }|a_k|^{p'}\left( \frac{k!}{\alpha ^k}\right) ^{\frac{p'}{2}} k^{\frac{(2m+1)p'-2}{4}}\right) ^{\frac{1}{p'}} \\&\quad \times \left( \sum _{k=n}^{\infty }\Vert \varphi ^k\Vert ^p_{F^p_{\alpha ,m}}\left( \frac{\alpha ^k}{k!} \right) ^{\frac{p}{2}}k^{\frac{p}{2p'}-\frac{(2m+1)p}{4}}\right) ^{\frac{1}{p}}\\&\quad \lesssim \Vert f\Vert _{F^p_{\alpha ,m}}\left( \sum _{k=n}^{\infty } |a|^{kp}e^{pc\sqrt{m+k}}\right) ^{\frac{1}{p}}. \end{aligned}$$

Therefore, for any $1\le p\le 2$, we have

$$\begin{aligned} \Vert C_{\varphi }-C_{\varphi }\circ K_n\Vert _{F^p_{\alpha ,m}\rightarrow F^p_{\alpha ,m}}\lesssim \left( \sum _{k=n}^{\infty }|a|^{kp}e^{pc\sqrt{m+k}}\right) ^{\frac{1}{p}}. \end{aligned}$$

Noting that $\lim _{k\rightarrow \infty }(k+1)^2|a|^ke^{c\sqrt{k}}=0$, we establish that

$$\begin{aligned} a_{n+1}(C_{\varphi })&\le \Vert C_{\varphi }-C_{\varphi }\circ K_n\Vert _{F^p_{\alpha ,m}\rightarrow F^p_{\alpha ,m}}\\&\lesssim \left( \sum _{k=n}^{\infty }|a|^{kp}e^{pc\sqrt{m+k}}\right) ^{\frac{1}{p}}\\&=|a|^ne^{c\sqrt{m+n}}\left( \sum _{k=n}^{\infty }|a|^{p(k-n)} e^{pc(\sqrt{m+k}-\sqrt{m+n})}\right) ^{\frac{1}{p}}\\&\le |a|^ne^{c\sqrt{m+n}}\left( \sum _{k=n}^{\infty }|a|^{p(k-n)} e^{pc\sqrt{k-n}}\right) ^{\frac{1}{p}}\\&\lesssim |a|^ne^{c\sqrt{m+n}}\left( \sum _{k=n}^{\infty }(k-n+1)^{-2p}\right) ^{\frac{1}{p}}\\&\lesssim |a|^{n+1}e^{c\sqrt{m+n+1}}. \end{aligned}$$

In the case $2<p<\infty$, using Theorem 3.2 instead of Theorem 3.4, we can similarly establish that

$$\begin{aligned} a_{n+1}(C_{\varphi })&\le \Vert C_{\varphi }-C_{\varphi }\circ K_n\Vert _{F^p_{\alpha ,m}\rightarrow F^p_{\alpha ,m}}\\&\lesssim \biggl (\sum _{k=n}^{\infty }|a|^{kp'}e^{p'c\sqrt{m+k}}\biggr )^{\frac{1}{p'}} \lesssim |a|^{n+1}e^{c\sqrt{m+n+1}}, \end{aligned}$$

which finishes the proof. $\square$

Recall that for $0<p<\infty$, a compact operator T on a separated Hilbert space H is said to be in the Schatten class $S_p(H)$ if $\sum _{n=1}^{\infty }a_n(T)^p<\infty$. As an application of Theorem 1.4, we have the following corollary.

Corollary 4.3

Let $\alpha >0$ and $m\in \mathbb {R}$. Then, for any $0<p<\infty$, any compact composition operator $C_{\varphi }$ on $F^2_{\alpha ,m}$ belongs to the Schatten class $S_p(F^2_{\alpha ,m})$.

Proof

Since $C_{\varphi }$ is compact, by Corollary 1.3, we can write $\varphi (z)=az+b$, where $|a|<1$ and $b\in \mathbb {C}$. If $a=0$, then $C_{\varphi }$ is of rank 1, and consequently in any $S_p(F^2_{\alpha ,m})$. If $0<|a|<1$, by Theorem 1.4, we have for some $N>0$

$$\begin{aligned} \sum _{n>N}a_n(C_{\varphi })^p\lesssim \sum _{n>N}|a|^{np}e^{pc\sqrt{m+n}}<\infty . \end{aligned}$$

The proof is complete. $\square$

We now turn to the lower estimates for $a_n(C_{\varphi })$. Before proceeding, we recall the following generalization of Weyl’s inequality, which can be found in [1, Proposition 2].

Lemma 4.4

Let T be a compact operator on a complex Banach space X and $\{\lambda _n(T)\}_{n\ge 1}$ be the sequence of its eigenvalues, indexed, such that $|\lambda _1(T)|\ge |\lambda _2(T)|\ge \cdots$. Then, for $n=1,2,\ldots$ and $m=0,1,\ldots ,n-1$, one has

$$\begin{aligned} \prod _{k=1}^n|\lambda _k(T)|\le 16^n\Vert T\Vert _{X\rightarrow X}^ma_{m+1}(T)^{n-m}. \end{aligned}$$

We also need the following lemma (see [8, Lemma 3.3] for instance).

Lemma 4.5

If $\varphi (z)=az+b$ with $0<|a|<1$, then $\{1,a,a^2,\ldots \}$ are all the eigenvalues of the operator $C_{\varphi }:F^p_{\alpha ,m}\rightarrow F^p_{\alpha ,m}$.

The following proposition gives a lower estimate for $a_n(C_{\varphi })$, which indicates that in the non-trivial case, the approximation numbers of compact composition operators on $F^p_{\alpha ,m}$ cannot decrease faster than geometrically.

Proposition 4.6

Let $1\le p<\infty$, $\alpha >0$, $m\in \mathbb {R}$ and $C_{\varphi }$ be a compact operator on $F^p_{\alpha ,m}$ induced by $\varphi (z)=az+b$, where $0<|a|<1$. Then

$$\begin{aligned} a_n(C_{\varphi })\ge \frac{1}{256\Vert C_{\varphi }\Vert _{F^p_{\alpha ,m} \rightarrow F^p_{\alpha ,m}}}|a|^{2n-1}. \end{aligned}$$

Proof

By Lemmas 4.4 and 4.5, we have that

$$\begin{aligned} |a|^{n(2n-1)}&=\prod _{k=1}^{2n}|\lambda _k(C_{\varphi })| \le 16^{2n}\Vert C_{\varphi }\Vert _{F^p_{\alpha ,m}\rightarrow F^p_{\alpha ,m}}^{n-1}a_n(C_{\varphi })^{n+1}\\&\le 16^{2n}\Vert C_{\varphi }\Vert _{F^p_{\alpha ,m} \rightarrow F^p_{\alpha ,m}}^{n}a_n(C_{\varphi })^n, \end{aligned}$$

which gives the desired result. $\square$