\documentstyle{SibMatJ}
% \TestXML
\topmatter

  \Author                   Liu
    \Initial                Y.
    \Gender                 he
    \Sign                   Youqi Liu
    \Email                  yqliu2021\@163.com
    \AffilRef               1
  \endAuthor

  \Affil 1
    \Division               School of Mathematics and Statistics
    \Organization           Chongqing Technology and Business University
    \City                   Chongqing
    \Country                China
  \endAffil

  \datesubmitted            December 29, 2025\enddatesubmitted
  \daterevised              January 20, 2026\enddaterevised
  \dateaccepted             February 16, 2026\enddateaccepted

  \UDclass
    ??? %Primary 47B33; Secondary 30H20
  \endUDclass

  \thanks
    This work was partially supported by the Chongqing Technology and Business University (no. 950323046). %\?grant, номер убрать
  \endthanks

  \title
    Generalized Weighted Composition Operators on Weighted Fock Spaces Induced by $A_{\infty}$-Type Weights
  \endtitle

  \abstract
    In this paper, we characterize the boundedness and compactness of generalized weighted composition operators on weighted Fock spaces $\Cal{F}_{\alpha,\omega}^{p}$ with $\omega\in A_{\infty}^{\roman{restricted}}$. We also study the essential norm of these operators on the weighted Fock spaces.
  \endabstract

  \keywords
    weighted Fock spaces,
    generalized weighted composition operators,
    boundedness,
    essential norm
  \endkeywords

\endtopmatter

\let\epsilon\varepsilon %\? \varepsilon не было, только \epsilon всюду

\head
  1. Introduction
\endhead

Let $\Cal{H}(\Bbb{C})$ be the space of entire functions on the complex plane $\Bbb{C}$. A function $\omega:\Bbb{C}\rightarrow [0,\infty)$ is a~weight if $\omega$ belongs to $L_{\loc}^{1}(\Bbb{C},dA)$, where $dA$ denotes the Lebesgue measure on $\Bbb{C}$.
For $0<p,\alpha<\infty$ and a weight $\omega$, the weighted space $\Cal{L}_{\alpha,\omega}^{p}$ consists of measurable functions $f$ on $\Bbb{C}$ satisfying
$$
\|f\|_{\Cal{L}_{\alpha,\omega}^{p}}^{p}=\int\limits_{\Bbb{C}}|f(z)|^{p}e^{-\frac{p\alpha}{2}|z|^{2}}\omega(z)dA(z)<\infty,
$$
and the weighted Fock space $\Cal{F}_{\alpha,\omega}^{p}=\Cal{L}_{\alpha,\omega}^{p} \cap \Cal{H}(\Bbb{C})$. Clearly, if $\omega\equiv1$, then $\Cal{F}_{\alpha,\omega}^{p}$ is the classical Fock spaces $F_{\alpha}^{p}$ (see, for instance, [1] and the references therein).

Throughout the paper, we use $Q$ to denote a square in $\Bbb{C}$ with sides parallel to the coordinate axes, and write $l(Q)$ for its side length.
If $E\subset\Bbb{C}$ is measurable, we denote by $|E|$ its Lebesgue area measure and let $\omega(E)=\int\nolimits_{E}\omega(z)dA(z)$.
For $1\leqslant p\leqslant\infty$, let $p'$ denote the conjugate exponent of $p$, that is, $\frac{1}{p}+\frac{1}{p'}=1$ with $1\leqslant p\leqslant\infty$.

Let $1<p<\infty$. We say a weight $\omega$ belongs to the class $A_{p}^{\roman{restricted}}$ if $\omega(z)>0$ a.e. on $\Bbb{C}$ and, for some $r>0$,
$$
\Cal{C}_{p,r}(\omega):=\sup\limits_{Q,l(Q)=r}\biggl(\frac{1}{|Q|}\int\limits_{Q}\omega dA \biggr) \biggl( \frac{1}{|Q|}\int\limits_{Q}\omega^{-\frac{p'}{p}} dA \biggr)^{\frac{p}{p'}}<\infty.
$$
The pioneering work about the weight class $A_{p}^{\roman{restricted}}$ was done by Isralowitz [2]. In fact, it follows from [2, Theorem~3.1] that these classes of weights are independent of the choice of $r$,
and $\omega\in A_{p}^{\roman{restricted}}$ if and only if the orthogonal projection $P_{\alpha}$ is bounded on $\Cal{L}_{\alpha,\omega}^{p}$. The boundedness of projections on $L^{p}$ spaces is a classical topic, which has many applications on operator theory, such as [1,\,3,\,4].
After that, Cascante, F\`abrega, and Pel\'aez [5] investigated a new class of weight $A_{1}^{\roman{restricted}}$. That is, for fixed $r>0$, a weight $\omega\in A_{1}^{\roman{restricted}}$ if $\omega(z)>0$ a.e. on $\Bbb{C}$ and
$$
\Cal{C}_{1,r}(\omega):=\sup\limits_{Q,l(Q)=r}\frac{\omega(Q)}{|Q|\text{infess}_{u\in Q}\omega(u)}<\infty.
$$

Similarly to %\?
Muckenhoupt weights, we denote
$$
A_{\infty}^{\roman{restricted}}:=\bigcup\limits_{1\leqslant p<\infty}A_{p}^{\roman{restricted}}.
$$

Furthermore, the authors in [5] characterize Littlewood--Paley formulas for weighted Fock spaces $\Cal{F}_{\alpha,\omega}^{p}$ with $\omega\in A_{\infty}^{\roman{restricted}}$. Using Littlewood--Paley formulas, the boundedness of the differentiation and integration operators are also obtained (see [5, Theorem~1.2]).
We denote $D^{n}(f)=f^{(n)}$ with $n\in\Bbb{N}$ and $D^{0}(f)=f$.
For a positive Borel measure $\mu$ on $\Bbb{C}$, we supplement the compactness of differentiation operator $D^{n}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow L^{q}(\mu)$ with $0<p,q<\infty$;
see \Par{Theorem 10}{Theorems 10} and \Par{Theorem 11}{11} below.
For more interesting results on the weighted Fock spaces we refer to [6--8].

Let $\varphi$ be an analytic self-map of $\Bbb{C}$, let $u$ be an entire function on $\Bbb{C}$, and let $n\in \Bbb{N}\cup\{0\}$.
The generalized weighted composition operator is defined by
$$
D^{n}_{\varphi,u}f=u\cdot f^{(n)}\circ\varphi.
$$
Clearly, if $n=0$ and $u(z)\equiv1$, then the generalized weighted composition operator is the well-known composition operator $C_{\varphi}f=f\circ\varphi$. And if $n=0$, then
we get the weighted composition operator $uC_{\varphi}f=u\cdot f\circ\varphi$.
These operators have been studied in various spaces (see, for example, [6,\,9--12] and the references therein).
In [13], Zhu characterized the boundedness and compactness of the operator $D^{n}_{\varphi,u}$ between different standard weighted Bergman spaces.
Similar problems for the generalized weighted composition operator on Bergman spaces induced by doubling weights were considered in [14].
The bounded and compact properties of the generalized weighted composition operator on classical Fock spaces have been studied in [15].

In this article, we continue those lines of research for this class of operators. To state our main results, we need some more notations.
Let $\mu$ be a positive Borel measure on $\Bbb{C}$ and let $h$ be a~measurable function on~$\Bbb{C}$.
Let $\varphi$ be an analytic self-map of $\Bbb{C}$, we define the measure related to $h$ by
$$
\varphi_{*}(h)(E)=\int\limits_{\varphi^{-1}(E)}hd\mu
$$
for every Borel subset $E$ of $\Bbb{C}$.
We denote by $D(z,r)$ the Euclidean disk of center $z$ and radius $r>0$.
We write $\Re(z)$ for the real part of a complex number $z$,
and $\omega_{\gamma}(z):=\frac{\omega(z)}{(1+|z|)^{\gamma}}$ with $\gamma\in\Bbb{R}$ (see \Tag(3) for more details).

For a positive Borel measure $\nu$ and $0<q<p<\infty$, [14, Theorem~1.1] describes the bounded and compact operators $D^{n}_{\varphi,u}$ acting from weighted Bergman space $A_{\omega}^{p}$ to $L_{\nu}^{q}$.
As an application of \Par*{Theorem~10}, we deduce the following consequence, which generalizes this result on weighted Fock spaces $\Cal{F}_{\alpha,\omega}^{p}$ with $\omega\in A_{\infty}^{\roman{restricted}}$.

\proclaim{Theorem 1}
Suppose that $0<q<p<\infty$, $\alpha\in(0,\infty)$, $\omega\in A_{\infty}^{\roman{restricted}}$, $n\in \Bbb{N}\cup\{0\}$,
and $\nu$ is a positive Borel measure on $\Bbb{C}$.
Let $\varphi$ be an analytic self-map of $\Bbb{C}$ and $u\in L_{\nu}^{q}$.
Then the following conditions are equivalent:
\Item (i) $D^{n}_{\varphi,u}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow L_{\nu}^{q}$ is bounded;
\Item (ii) $D^{n}_{\varphi,u}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow L_{\nu}^{q}$ is compact;
\Item (iii) The function
$$
\frac{1}{\omega_{np}(D(u,1))}\int\limits_{D(u,1)}e^{\frac{q\alpha |z|^{2}}{2}}d\varphi_{*}(|u|^{q}\nu)(z)
$$
is in $L^{\frac{p}{p-q}}(\Bbb{C},\omega_{np})$.
\endproclaim

When $0<p\leqslant q<\infty$, Liu studied the boundedness and compactness of the operator $D^{n}_{\varphi,u}$ acting between different weighted Bergman spaces $A_{\omega}^{p}$ (see [14, Theorem~1.3]).
Recently, Chen characterized the boundedness and compactness of composition operator $C_{\varphi}$ acting between different weighted Fock spaces~$\Cal{F}_{\alpha,\omega}^{p}$ with $\omega\in A_{\infty}^{\roman{restricted}}$; see [6, Theorem~1.1].
We extend these results to our weighted Fock spaces as follows.

\proclaim{Theorem 2}
Suppose that $0<p\leqslant q<\infty$, $\alpha,\beta>0$, $\omega\in A_{\infty}^{\roman{restricted}}$, $n\in \Bbb{N}\cup\{0\}$, and $\nu$ is a weight.
Let $\varphi$ be an analytic self-map of $\Bbb{C}$ and $u\in \Cal{F}_{\beta,\nu}^{q}$.
Then:
\Item (i) $D^{n}_{\varphi,u}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow \Cal{F}_{\beta,\nu}^{q}$ is bounded if and only if
$$
\sup\limits_{a\in \Bbb{C}}\int\limits_{\Bbb{C}}\frac{|u(z)|^{q}}{\omega_{np}(D(a,1))^{q/p}} e^{q\alpha \Re(\overline{a}\varphi(z))-\frac{q\alpha}{2}|a|^{2}-\frac{q\beta}{2}|z|^{2}}\nu(z)dA(z)<\infty.
$$
\Item (ii)
$D^{n}_{\varphi,u}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow \Cal{F}_{\beta,\nu}^{q}$ is compact if and only if
$$
\lim\limits_{|a|\rightarrow\infty}\int\limits_{\Bbb{C}}\frac{|u(z)|^{q}}{\omega_{np}(D(a,1))^{q/p}} e^{q\alpha \Re(\overline{a}\varphi(z))-\frac{q\alpha}{2}|a|^{2}-\frac{q\beta}{2}|z|^{2}}\nu(z)dA(z)=0.
$$
\endproclaim

In addition to boundedness and compactness, we intend to consider the essential norm of the generalized weighted composition operator $D^{n}_{\varphi,u}$.
For classical Fock spaces $F_{\alpha}^{p}$, Tien and Khoi [12] provided lower and upper estimates for essential norm of weighted composition operator $uC_{\varphi}$.
Later on, Hu, Li, and Qu [15] extended these estimates for the operator $D^{n}_{\varphi,u}$ on the Fock spaces $F_{\alpha}^{p}$.
Inspired by these papers, we investigate lower and upper estimates for essential norm of the operator $D^{n}_{\varphi,u}$ on weighted Fock spaces $\Cal{F}_{\alpha,\omega}^{p}$ via Littlewood--Paley formulas (see [5, Theorem~1.1]).
Combining \Par*{Theorem 16} with \Par*{Corollary 17} below, we offer these estimates of $D^{n}_{\varphi,u}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow \Cal{F}_{\alpha,\omega}^{p}$ whenever $\varphi(z)=az+b$ with $0<|a|<1$.

\proclaim{Theorem 3}
Suppose that $1<p<\infty$, $\alpha>0$, $\omega\in A_{\infty}^{\roman{restricted}}$, $n\in \Bbb{N}\cup\{0\}$.
Let $u\in \Cal{F}_{\alpha,\omega}^{p}$ be a~nonzero function and $\varphi(z)=az+b$ with $0<|a|<1$.
If $D^{n}_{\varphi,u}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow \Cal{F}_{\alpha,\omega}^{p}$ is bounded,
then
$$
\limsup\limits_{|z|\rightarrow\infty}m_{z,n}(u,\varphi)\lesssim \|D^{n}_{\varphi,u}\|_{e}
\lesssim |a|^{-\frac{2}{p}}\limsup\limits_{|z|\rightarrow\infty}\frac{m_{z,n}(u,\varphi)\omega(z)}{\omega(\varphi(z))}.
$$
\endproclaim

The rest of paper contains the proofs of the conclusions as above.
In \Sec*{Section 2}, we provide some auxiliary results.
The proofs of \Par{Theorem 1}{Theorems 1} and \Par{Theorem 2}{2} are given in \Sec*{Section 3}.
Finally, we establish lower and upper estimates for essential norm of the operator $D^{n}_{\varphi,u}$ on weighted Fock spaces $\Cal{F}_{\alpha,\omega}^{p}$ in \Sec*{Section 4}.

In the following, the notation $U(z)\lesssim V(z)$ (or equivalently $V(z)\gtrsim U(z)$) means that there is a~constant $C$ such that $U(z)\leqslant CV(z)$ holds for all $z$ in the set of a question. Furthermore, we write $U(z)\asymp V(z)$ if both $U(z)\lesssim V(z)$ and $V(z)\lesssim U(z)$.

\head
2. Preliminaries
\endhead

By [5, Remark~2.3], if $\omega\in A_{\infty}^{\roman{restricted}}$, for any $t>0$ and $N\in \Bbb{N}$ there is $C=C(\omega,N,p)$ such that
$$
\omega(D(a,t))\leqslant \omega(D(a,Nt))\leqslant C\omega(D(a,t)), \quad a\in\Bbb{C},
$$
and for any $z,w\in \Bbb{C}$ such that $|z-w|<L~(L>0)$,
$$
\omega(D(z,r))\asymp \omega(D(w,R)), \quad C^{-1}\leqslant r,R\leqslant C.
\tag1
$$
Combining these results with [2, Lemma~3.4], [5, Lemma~B], and [6, Lemma~2.2], %\?[2, Lemma~3.4; 5, Lemma~B; and 6, Lemma~2.2],
we have the following lemma.
As mentioned earlier, the notation $Q_{1}(z)$ means the square of center $z\in\Bbb{C}$ with side length $l(Q)=1$. Besides, since $\Bbb{R}^{2}$ can be canonically identified with $\Bbb{C}$, we will treat $\Bbb{Z}^{2}$ as a subset of $\Bbb{C}$.

\proclaim{Lemma 4}
Suppose that $\omega\in A_{\infty}^{\roman{restricted}}$.
\Item (i) There exists $C>0$ such that for any $v,v'\in\Bbb{Z}^{2}$,
$$
\frac{\omega(Q_{1}(v))}{\omega(Q_{1}(v'))}\leqslant C^{|v-v'|}.
$$
\Item (ii) For any $z,u\in\Bbb{C}$ such that $|z-u|<1$,
$$
\omega(Q_{1}(z))\asymp\omega(Q_{1}(u))\asymp\omega(D(z,1))\asymp\omega(D(u,1)).
$$
\endproclaim

Some preliminary results, which will be used to prove our main consequences in the later.

\proclaim{Lemma 5 \rm [5, Lemma~2.8]}
Assume $\omega\in A_{\infty}^{\roman{restricted}}$. Then there exists $r_{0}$ such that for any $r\in(0,r_{0})$ and for any $\beta>0$,
$$
\int\limits_{\Bbb{C}}e^{-\beta|z|^{2}}\omega(z)dA(z)\leqslant C(\omega,r,\beta)\omega(Q_{r}(0))<\infty.
$$
\endproclaim

\proclaim{Lemma 6 \rm [5, Lemma~3.1]}
Let $0<p,t<\infty$, $\alpha>0$, and $\omega\in A_{\infty}^{\roman{restricted}}$. Then, there exists a~constant $C=C(\alpha,p,\omega,t)$ such that
$$
|f(z)|^{p}e^{-\frac{p\alpha}{2}|z|^{2}}\leqslant \frac{C}{\omega(D(z,t))}\int\limits_{D(z,t)}|f(u)|^{p}e^{-\frac{p\alpha}{2}|u|^{2}}\omega(u)dA(u)
$$
for any $z\in\Bbb{C}$ and $f\in \Cal{H}(\Bbb{C})$.
\endproclaim

The following two lemmas generalize
[5, Lemma~3.1 and Corollary~3.2] and [15, Lemmas~2.2 and~2.3] %\?[5, Lemma~3.1 and Corollary~3.2; 15, Lemmas~2.2 and~2.3]
to the weighted Fock spaces $\Cal{F}_{\alpha,\omega}^{p}$.

\proclaim{Lemma 7}
Let $0<p,t<\infty$, $\alpha>0$, $\omega\in A_{\infty}^{\roman{restricted}}$, and $n\in \Bbb{N}\cup\{0\}$. Then there exists a constant $C=C(\alpha,p,\omega,t)$ such that
$$
|f^{(n)}(z)|\leqslant Cn!e^{\frac{\alpha}{2}|z|^{2}}\frac{(1+|z|)^{n}}{\omega(D(z,t))^{\frac{1}{p}}}\|f\|_{\Cal{F}_{\alpha,\omega}^{p}}
$$
for any $z\in\Bbb{C}$ and $f\in \Cal{H}(\Bbb{C})$.
\endproclaim

\demo{Proof}
Firstly, let $n$ be a positive integer.
For $|z|\leqslant1$, \Par*{Lemma 6} and Cauchy integral formula %\?всюду артикль
indicate
$$
\align
|f^{(n)}(z)|&\leqslant \frac{n!}{2\pi}\int\limits_{|z-\zeta|=1}\frac{|f(\zeta)|}{|\zeta-z|^{n+1}}|d\zeta|
\leqslant n!\max\limits_{|z-\zeta|=1}|f(\zeta)|\\
&\leqslant C_{1}n!\|f\|_{\Cal{F}_{\alpha,\omega}^{p}}\max\limits_{|z-\zeta|=1}\frac{e^{\frac{\alpha}{2}|\zeta|^{2}}}{\omega(D(\zeta,t_{1}))^{\frac{1}{p}}}
\leqslant C_{1}C_{2}n!e^{2\alpha}\|f\|_{\Cal{F}_{\alpha,\omega}^{p}}\omega(D(z,t_{2}))^{-\frac{1}{p}}.
\endalign
$$
For $|z|>1$, we conclude from Cauchy integral formula again that
$$
\align
|f^{(n)}(z)|&\leqslant \frac{n!}{2\pi}\int\limits_{|z-\zeta|=|z|^{-1}}\frac{|f(\zeta)|}{|\zeta-z|^{n+1}}|d\zeta|
\leqslant n!|z|^{n}\max\limits_{|z-\zeta|=|z|^{-1}}|f(\zeta)|\\
&\leqslant C_{1}n!|z|^{n}\|f\|_{\Cal{F}_{\alpha,\omega}^{p}}\max\limits_{|\zeta|\leqslant |z|+|z|^{-1}}\frac{e^{\frac{\alpha}{2}|\zeta|^{2}}}{\omega(D(\zeta,t_{1}))^{\frac{1}{p}}}\\
&\leqslant C_{1}C_{2}n!e^{2\alpha}(1+|z|)^{n}\|f\|_{\Cal{F}_{\alpha,\omega}^{p}}
\frac{e^{\frac{\alpha}{2}|z|^{2}}}{\omega(D(z,t_{2}))^{\frac{1}{p}}}.
\endalign
$$
The case $n=0$ follows from [5, Corollary~3.2], which ends this proof.
\qed\enddemo

\proclaim{Lemma 8}
Let $0<p,t<\infty$, $\alpha>0$, $\omega\in A_{\infty}^{\roman{restricted}}$, and let $n$ be a positive integer. Then there exists a constant $C=C(\alpha,p,\omega,t)$ such that
$$
|f^{(n)}(z)e^{-\frac{\alpha}{2}|z|^{2}}|\leqslant Cn!\frac{(1+|z|)^{n}}{\omega(D(z,t))^{\frac{1}{p}}}
\biggl(\int\limits_{D(z,t)}|f(\xi)e^{-\frac{\alpha}{2}|\xi|^{2}}|^{p}\omega(\xi)dA(\xi)\biggr)^{\frac{1}{p}}
$$
for any $z\in\Bbb{C}$ and $f\in \Cal{H}(\Bbb{C})$.
\endproclaim

\demo{Proof}
Let first $|z|\leqslant1$. By the joining of \Par*{Lemma 6} and the proof of \Par*{Lemma 7}, we have
$$
\align
|f^{(n)}(z)e^{-\frac{\alpha}{2}|z|^{2}}|
&\leqslant |f^{(n)}(z)|
\leqslant n!(1+|z|)^{n}\max\limits_{|z-\zeta|=1}|f(\zeta)|\\
&\leqslant C_{1}n!(1+|z|)^{n}\max\limits_{|\zeta|\leqslant2}
\biggl(\omega(D(\zeta,r_{0}))^{-1}\int\limits_{D(\zeta,r_{0})}|f(u)e^{-\frac{\alpha}{2}|u|^{2}}|^{p}\omega(u)dA(u)\biggr)^{\frac{1}{p}}\\
&\leqslant C_{1}C_{2}n!(1+|z|)^{n}\omega(D(z,3))^{-\frac{1}{p}}
\biggl(\ \int\limits_{D(z,3)}|f(u)e^{-\frac{\alpha}{2}|u|^{2}}|^{p}\omega(u)dA(u)\biggr)^{\frac{1}{p}}.
\endalign
$$
If $|z|>1$, then we obtain
$$
\align
|f^{(n)}(z)e^{-\frac{\alpha}{2}|z|^{2}}|
&\leqslant n!|z|^{n}e^{-\frac{\alpha|z|^{2}}{2}}\max\limits_{|z-\zeta|\leqslant|z|^{-1}}|f(\zeta)|\\
&\leqslant C_{1}n!|z|^{n}e^{-\frac{\alpha|z|^{2}}{2}}\max\limits_{|z-\zeta|\leqslant|z|^{-1}}
\frac{e^{\frac{\alpha}{2}|\zeta|^{2}}}{\omega(D(\zeta,r_{1}))^{\frac{1}{p}}}
\biggl(\ \int\limits_{D(\zeta,r_{1})}|f(u)e^{-\frac{\alpha}{2}|u|^{2}}|^{p}\omega(u)dA(u)\biggr)^{\frac{1}{p}}\\
&\lesssim n!\frac{(1+|z|)^{n}}{\omega(D(z,r_{1}))^{\frac{1}{p}}}\max\limits_{|z-\zeta|\leqslant|z|^{-1}}
\biggl(\ \int\limits_{D(\zeta,r_{1})}|f(u)e^{-\frac{\alpha}{2}|u|^{2}}|^{p}\omega(u)dA(u)\biggr)^{\frac{1}{p}}\\
&\lesssim n!\frac{(1+|z|)^{n}}{\omega(D(z,r_{2}))^{\frac{1}{p}}}
\biggl(\ \int\limits_{D(z,r_{1}+1)}|f(u)e^{-\frac{\alpha}{2}|u|^{2}}|^{p}\omega(u)dA(u)\biggr)^{\frac{1}{p}}.
\endalign
$$
This completes the proof.
\qed\enddemo

The following  lemma, see [16, Lemma~4.10] or [17, Lemma~3.7], plays a key role to study compactness of linear operators.

\proclaim{Lemma 9}
Let $X$ and $Y$ be two Banach {\rm(}or quasi-Banach{\rm)} spaces of entire functions on $\Bbb{C}$, and let $T: X\rightarrow Y$ be a linear operator. Suppose that the following conditions are satisfied:
\Item (i) The point evaluation functionals on $Y$ are bounded.
\Item (ii) The closed unit ball of $X$ is a compact subset of $\Cal{H}(\Bbb{C})$, where $\Cal{H}(\Bbb{C})$ is endowed with the topology of uniform convergence on compacta.
\Item (iii) $T: X\rightarrow Y$ is continuous, where both $X$ and $Y$ are endowed with the topology of uniform convergence on compacta.\par
Then $T: X\rightarrow Y$ is a compact operator if and only if for any bounded sequence $\{f_{j}\}$ in $X$ such that $f_{j}\rightarrow0$ uniformly on compact subset %\?
of $\Bbb{C}$ as $j\rightarrow\infty$, the sequence $\{Tf_{j}\}$ converges to zero in the norm of $Y$ as $j\rightarrow\infty$.
\endproclaim

\head
3. Proofs of \Par{Theorem 1}{Theorems~1} and~\Par{Theorem 2}{2}
\endhead

In this section, we will provide the boundedness and compactness of generalized weighted composition operator %\?the
$D^{n}_{\varphi,u}$ between different spaces.
At first, we consider the compactness of differentiation operator %\?
$D^{n}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow L^{q}(\mu)$ with $0<q<p<\infty$. This result is a supplement of [5, Theorem~1.2].

\proclaim{Theorem 10}\Label{T10}
Let $\alpha\in(0,\infty)$, $\omega\in A_{\infty}^{\roman{restricted}}$, $n\in \Bbb{N}\cup\{0\}$, and let $\mu$ be
a positive Borel measure on~$\Bbb{C}$. For $0<q<p<\infty$, the following conditions are equivalent:
\Item (i) $D^{n}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow L^{q}(\mu)$ is bounded.
\Item (ii) $D^{n}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow L^{q}(\mu)$ is compact.
\Item (iii) The function
$$
\frac{1}{\omega_{np}(D(u,1))}\int\limits_{D(u,1)}e^{\frac{q\alpha |z|^{2}}{2}}d\mu(z)
$$
is in $L^{\frac{p}{p-q}}(\Bbb{C},\omega_{np})$.
\endproclaim

\demo{Proof}
It is clear that \Par{T10}{(ii)}$\Rightarrow$\Par{T10}{(i)}, and the proof of \Par{T10}{(i)}$\Rightarrow$\Par{T10}{(iii)} follows by
[5, Theorem~1.2].
Next, we show %\?that
\Par{T10}{(iii)} implies \Par{T10}{(ii)}.
It is enough to give that for any bounded sequences $\{f_{k}\}\in\Cal{F}_{\alpha,\omega}^{p}$ which tends to zero uniformly on compact subsets of $\Bbb{C}$ as $k\rightarrow\infty$, we deduce $\|D^{n}f_{k}\|_{L^{q}(\mu)}^{q}\rightarrow0$ as $k\rightarrow\infty$. Set $\|f_{k}\|_{\Cal{F}_{\alpha,\omega}^{p}}\leqslant M<\infty$. By \Par*{Lemma 8} with $t=1$, we have
$$
|f^{(n)}(z)|^{q}\leqslant C(n!)^{q}\frac{(1+|z|)^{n}e^{\frac{q\alpha}{2}|z|^{2}}}{\omega(D(z,1))}
\int\limits_{D(z,1)}|f(\xi)e^{-\frac{\alpha}{2}|\xi|^{2}}|^{q}\omega(\xi)dA(\xi).
$$
Note that $1+|\xi|\asymp 1+|z|$ for $\xi\in D(z,1)$, and then by Fubini's theorem and \Par*{Lemma 4}, we get
$$
\|D^{n}f_{k}\|^{q}_{L^{q}(\mu)}\lesssim
\int\limits_{\Bbb{C}}|f_{k}(\xi)e^{-\frac{\alpha}{2}|\xi|^{2}}|^{q}\frac{(1+|\xi|)^{nq}}{\omega(D(\xi,1))}
\int\limits_{D(\xi,1)}e^{\frac{q\alpha}{2}|z|^{2}}d\mu(z)\omega(\xi)dA(\xi).
$$
For any $\epsilon>0$. Due to the hypothesis,  %\?
there is $R>0$ such that
$$
\int\limits_{\Bbb{C}\setminus \overline{D(0,R)}}\biggl(\frac{(1+|\xi|)^{nq}}{\omega(D(\xi,1))}
\int\limits_{D(\xi,1)}e^{\frac{q\alpha}{2}|z|^{2}}d\mu(z)\biggr)^{p/(p-q)}\omega(\xi)dA(\xi)\leqslant \epsilon^{p/(p-q)}.
$$
Since $\{f_{k}\}$ uniformly converge %\?
on compact subsets of $\Bbb{C}$ as $k\rightarrow\infty$, we may choose $k_{0}\in \Bbb{N}$ such that
$|f_{k}(\xi)|^{q}<\epsilon$ whenever $k\geqslant k_{0}$ and $\xi\in \overline{D(0,R)}$.
Therefore,
$$
\align
\|D^{n}f_{k}\|^{q}_{L^{q}(d\mu)}
&\lesssim
\biggl(\ \int\limits_{\overline{D(0,R)}}+\int\limits_{\Bbb{C}\setminus \overline{D(0,R)}}\biggr)
|f_{k}(\xi)e^{-\frac{\alpha}{2}|\xi|^{2}}|^{q}\\
&\cdot %\?\times
\frac{(1+|\xi|)^{nq}}{\omega(D(\xi,1))}\int\limits_{D(\xi,1)}e^{\frac{q\alpha}{2}|z|^{2}}d\mu(z)\omega(\xi)dA(\xi)\\
&\lesssim \sup\limits_{\xi\in \overline{D(0,R)}}\frac{(1+|\xi|)^{nq}}{\omega(D(\xi,1))}\int\limits_{D(\xi,1)}e^{\frac{q\alpha}{2}|z|^{2}}d\mu(z)
\int\limits_{\overline{D(0,R)}}|f_{k}(\xi)|^{q}\omega(\xi)dA(\xi)\\
&\qquad +\biggl(\int\limits_{\Bbb{C}}|f_{k}(\xi)|^{p}\omega(\xi)dA(\xi)\biggr)^{q/p}\\
& \cdot %\?
\biggl(\ \int\limits_{\Bbb{C}\setminus \overline{D(0,R)}}\biggl(\frac{(1+|\xi|)^{nq}}{\omega(D(\xi,1))}
\int\limits_{D(\xi,1)}e^{\frac{q\alpha}{2}|z|^{2}}d\mu(z)\biggr)^{\frac{p}{p-q}}\omega(\xi)dA(\xi)\biggr)^{\frac{p-q}{p}}\\
&\lesssim \epsilon+\epsilon\|f_{k}\|_{\Cal{F}_{\alpha,\omega}^{p}}\leqslant (1+M)\epsilon.
\endalign
$$
And hence, $D^{n}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow L^{q}(\mu)$ is compact, which finishes the proof.
\qed\enddemo

With the help of the above conclusion, we can now provide the proof of \Par*{Theorem 1}.

\demo{Proof of \Par*{Theorem 1}}
Note that the non-univalent %\?
change of variables formula indicates $\|D_{\varphi,u}^{n}f\|_{L_{\nu}^{q}}=\|f^{(n)}\|_{L_{\varphi_{*}(|u|^{q}\nu)}^{q}}$.
This together with \Par*{Theorem 10} ends the proof.
\qed\enddemo

The following theorem concerns the compactness of differentiation operator $D^{n}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow L^{q}(\mu)$ with $0<p\leqslant q<\infty$.

\proclaim{Theorem 11}\Label{T11}
Let $\alpha\in(0,\infty)$, $\omega\in A_{\infty}^{\roman{restricted}}$, $n\in \Bbb{N}\cup\{0\}$,
and let $\mu$ be a positive Borel measure on~$\Bbb{C}$. For $0<p\leqslant q<\infty$, the following conditions are equivalent:
\Item (i) $D^{n}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow L^{q}(\mu)$ is compact;
\Item (ii)
$$
\lim\limits_{|a|\rightarrow\infty}\frac{1}{(\omega_{np}(D(a,1)))^{\frac{q}{p}}}\int\limits_{D(a,1)}e^{\frac{q\alpha |z|^{2}}{2}}d\mu(z)=0.
$$
\endproclaim

\demo{Proof}
We begin with the case $n=0$.
Assume that $I_{d}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow L^{q}(\mu)$ is compact.
Let
$$
f_{a}(z)=\omega(D(a,1))^{-1/p}e^{\alpha \overline{a}z-\frac{\alpha}{2}|a|^{2}}, \quad z\in \Bbb{C}.
\tag2
$$
It follows from [6, Lemma~2.3] that $\|f_{a}\|_{\Cal{F}_{\alpha,\omega}^{p}}\asymp1$ and $f_{a}$ tends to zero uniformly on compact subsets of $\Bbb{C}$ as $|a|$ goes to $\infty$.
Hence the closure of the set $\{f_{a}:a\in\Bbb{C}\}$ is compact in $L^{q}(\mu)$. For any $\epsilon>0$, the open balls $B(f_{a},\epsilon)=\{f\in L^{q}(\mu):\|f_{a}-f\|_{L^{q}(\mu)}<\epsilon\}$ cover the closure of the set $\{f_{a}:a\in\Bbb{C}\}$.
Then there is a~finite subcover $\{B(f_{a_{n}},\epsilon):n=1,\dots,N=N(\epsilon)\}$. For each $a\in \Bbb{C}$, let $j=j(a)\in \{1,\dots,N\}$ such that $f_{a}\in B(f_{a_{j}},\epsilon)$. Therefore, for each $R>0$,
$$
\align
\int\limits_{\Bbb{C}\setminus D(0,R)}|f_{a}(z)|^{q}d\mu(z)
&\lesssim
\int\limits_{\Bbb{C}\setminus D(0,R)}|f_{a}(z)-f_{a_{j}}(z)|^{q}d\mu(z)+\int\limits_{\Bbb{C}\setminus D(0,R)}|f_{a_{j}}(z)|^{q}d\mu(z)\\
&\leqslant \|f_{a}-f_{a_{j}}\|^{q}_{L^{q}(\mu)}+\max\limits_{n=1,\dots,N}\int\limits_{\Bbb{C}\setminus D(0,R)}|f_{a_{n}}(z)|^{q}d\mu(z),
\endalign
$$
and so
$$
\lim\limits_{R\rightarrow\infty}\int\limits_{\Bbb{C}\setminus D(0,R)}|f_{a}(z)|^{q}d\mu(z)=0
$$
uniformly in $a$. Combining with the uniform convergence, we deduce
$$
\align
0=\lim\limits_{|a|\rightarrow\infty}\|f_{a}\|^{q}_{L^{q}(\mu)}
&\geqslant\lim\limits_{|a|\rightarrow\infty}\int\limits_{D(a,1)}|f_{a}(z)|^{q}d\mu(z)\\
&=\lim\limits_{|a|\rightarrow\infty}\omega(D(a,1))^{-\frac{q}{p}}
\int\limits_{D(a,1)}e^{\frac{q\alpha}{2}|z|^{2}-\frac{q\alpha}{2}|z-a|^{2}}d\mu(z)\\
&\gtrsim\lim\limits_{|a|\rightarrow\infty}\omega(D(a,1))^{-\frac{q}{p}}
\int\limits_{D(a,1)}e^{\frac{q\alpha}{2}|z|^{2}}d\mu(z).
\endalign
$$

Conversely, assume \Par{T11}{(ii)} holds, let $\{f_{k}\}_{k}\in \Cal{F}_{\alpha,\omega}^{p}$ be a sequence such that $\sup\nolimits_{k\in \Bbb{N}}\|f_{k}\|_{\Cal{F}_{\alpha,\omega}^{p}}=M<\infty$. Then we conclude from [5, Corollary~3.2] that $\{f_{k}\}_{k}$ is uniformly bounded on compact subsets of $\Bbb{C}$. With the help of Montel's theorem and Weierstrass' theorem, there is a subsequence $\{f_{k_{j}}\}_{j}$ converges uniformly %\?which converges uniformly or that converges uniformly
on compact subsets of $\Bbb{C}$ to a function $f$ which is in $\Cal{H}(\Bbb{C})$. And thus $f\in \Cal{F}_{\alpha,\omega}^{p}$ by Fatou's lemma.
Set $|a_{n}|\rightarrow\infty$ as $n\rightarrow\infty$. Then the hypothesis gives that, for any $\epsilon>0$, there exists $N=N(\epsilon)\in\Bbb{N}$ such that
$$
\frac{1}{(\omega(D(a_{n},1)))^{\frac{q}{p}}}\int\limits_{D(a_{n},1)}e^{\frac{q\alpha |z|^{2}}{2}}d\mu(z)<\epsilon.
$$
This together with \Tag(1), \Par*{Lemma 6}, and Minkowski's inequality in continuous form (Fubini's theorem in the case $p=q$)
shows %\?that
$$
\align
\sum\limits_{n=N}^{\infty}
&\int\limits_{D(a_{n},1)}|f(z)-f_{k_{j}}(z)|^{q}d\mu(z)\\
&\lesssim \sum\limits_{n=N}^{\infty}\int\limits_{D(a_{n},1)}\frac{e^{\frac{q\alpha}{2}|z|^{2}}}{\omega(D(z,1))^{\frac{q}{p}}}
\biggl(\ \int\limits_{D(z,1)}|f(u)-f_{k_{j}}(u)|^{p}e^{-\frac{p\alpha}{2}|u|^{2}}\omega(u)dA(u)\biggr)^{\frac{q}{p}}d\mu(z)\\
&\leqslant \sum\limits_{n=N}^{\infty}
\biggl[\,\int\limits_{\{u:D(z,1)\cap D(a_{n},1)\neq\emptyset\}}|f(u)-f_{k_{j}}(u)|^{p}e^{-\frac{p\alpha}{2}|u|^{2}}
\biggl(\frac{\int_{D(a_{n},1)}e^{\frac{q\alpha}{2}|z|^{2}}d\mu(z)}{\omega(D(z,1))^{\frac{q}{p}}}
\biggr)^{\frac{p}{q}}\omega(u)dA(u)\biggr]^{\frac{q}{p}}\\
&\lesssim \epsilon\sum\limits_{n=N}^{\infty}
\biggl(\ \int\limits_{\{u:D(z,1)\cap D(a_{n},1)\neq\emptyset\}}|f(u)-f_{k_{j}}(u)|^{p}e^{-\frac{p\alpha}{2}|u|^{2}}
\omega(u)dA(u)\biggr)^{\frac{q}{p}}\\
&\lesssim \epsilon\|f-f_{k_{j}}\|_{\Cal{F}_{\alpha,\omega}^{p}}^{q}\leqslant \epsilon M^{q}.
\endalign
$$

Therefore,
$$
\align
&\limsup\limits_{j\rightarrow\infty}\int\limits_{\Bbb{C}}|f(z)-f_{k_{j}}(z)|^{q}d\mu(z)\\
&\qquad\leqslant \limsup\limits_{j\rightarrow\infty}\Biggl(\sum\limits_{n=1}^{N-1}\int\limits_{D(a_{n},1)}|f(z)-f_{k_{j}}(z)|^{q}d\mu(z)
+\sum\limits_{n=N}^{\infty}\int\limits_{D(a_{n},1)}|f(z)-f_{k_{j}}(z)|^{q}d\mu(z)\Biggr)\lesssim\epsilon,
\endalign
$$
because of
$$
\lim\limits_{j\rightarrow\infty}\sum\limits_{n=1}^{N-1}\int\limits_{D(a_{n},1)}|f(z)-f_{k_{j}}(z)|^{q}d\mu(z)=0
$$
by the uniform convergence in compact subsets.
And hence, for any $\epsilon>0$, we get
$$
\limsup\limits_{j\rightarrow\infty}\int\limits_{\Bbb{C}}|f(z)-f_{k_{j}}(z)|^{q}d\mu(z)=0,
$$
which concludes $I_{d}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow L^{q}(\mu)$ is compact.
Next, by [5, Lemma~2.1], we have
$$
\omega\in A_{\infty}^{\roman{restricted}} \Leftrightarrow \omega_{\gamma}(z)=\frac{\omega(z)}{(1+|z|)^{\gamma}}\in A_{\infty}^{\roman{restricted}}.
\tag3
$$
Now, combining \Par*{Lemma 9}, the proof for case $n\in\Bbb{N}$ is an analog of above, %\?the
we omit the details here, and the desired consequences are complete.
\qed\enddemo

We use $\mu_{\varphi}^{\beta,\nu}(|u|^{q})$ to denote the positive measure on $\Bbb{C}$ defined by
$$
\mu_{\varphi}^{\beta,\nu}(|u|^{q})(E)=\int\limits_{\varphi^{-1}(E)}|u(z)|^{q}e^{-\frac{q\beta}{2}|z|^{2}}\nu(z)dA(z)
$$
for every Borel subset $E$ of $\Bbb{C}$.
Next, we study the boundedness and compactness of $D^{n}_{\varphi,u}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow \Cal{F}_{\beta,\nu}^{q}$ with $0<p\leqslant q<\infty$. For this goal, we will divide \Par*{Theorem 2} into two theorems as follows.

\proclaim{Theorem 12}
Suppose that $0<p\leqslant q<\infty$, $\alpha,\beta>0$, $\omega\in A_{\infty}^{\roman{restricted}}$, $n\in \Bbb{N}\cup\{0\}$, and $\nu$ is a weight.
Let $\varphi$ be an analytic self-map of $\Bbb{C}$ and $u\in \Cal{F}_{\beta,\nu}^{q}$.
Then $D^{n}_{\varphi,u}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow \Cal{F}_{\beta,\nu}^{q}$ is bounded if and only if
$$
\sup\limits_{a\in \Bbb{C}}\int\limits_{\Bbb{C}}\frac{|u(z)|^{q}}{\omega_{np}(D(a,1))^{q/p}} e^{q\alpha \Re(\overline{a}\varphi(z))-\frac{q\alpha}{2}|a|^{2}-\frac{q\beta}{2}|z|^{2}}\nu(z)dA(z)<\infty.
\tag4
$$
\endproclaim

\demo{Proof}
Assume that $D^{n}_{\varphi,u}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow \Cal{F}_{\beta,\nu}^{q}$ is bounded. It follows that
$$
\align
\|D^{n}_{\varphi,u}f_{a}\|_{\Cal{F}_{\beta,\nu}^{q}}^{q}
&=(\alpha|a|)^{qn}\int\limits_{\Bbb{C}}
\frac{|u(z)|^{q}}{\omega(D(a,1))^{q/p}} e^{q\alpha \Re(\overline{a}\varphi(z))-\frac{q\alpha}{2}|a|^{2}-\frac{q\beta}{2}|z|^{2}}\nu(z)dA(z)\\
&\asymp \int\limits_{\Bbb{C}}
\frac{|u(z)|^{q}}{\omega_{np}(D(a,1))^{q/p}} e^{q\alpha \Re(\overline{a}\varphi(z))-\frac{q\alpha}{2}|a|^{2}-\frac{q\beta}{2}|z|^{2}}\nu(z)dA(z)
\leqslant C\|f_{a}\|_{\Cal{F}_{\alpha,\omega}^{p}}^{q}<\infty,
\endalign
$$
which concludes \Tag(4).

Conversely, suppose that \Tag(4) holds. Thus,
$$
\align
\infty&>\int\limits_{\Bbb{C}}\frac{|u(z)|^{q}}{\omega_{np}(D(a,1))^{q/p}} e^{q\alpha \Re(\overline{a}\varphi(z))-\frac{q\alpha}{2}|a|^{2}-\frac{q\beta}{2}|z|^{2}}\nu(z)dA(z)\\
&\geqslant \int\limits_{\varphi^{-1}(D(a,1))}\frac{|u(z)|^{q}}{\omega_{np}(D(a,1))^{q/p}} e^{q\alpha \Re(\overline{a}\varphi(z))-\frac{q\alpha}{2}|a|^{2}-\frac{q\beta}{2}|z|^{2}}\nu(z)dA(z)\\
&= \int\limits_{D(a,1)}\omega_{np}(D(a,1))^{-q/p} e^{\frac{q\alpha}{2}|z|^{2} -\frac{q\alpha}{2}|z-a|^{2}}d\mu_{\varphi}^{\beta,\nu}(|u|^{q})(z)\\
&\gtrsim \frac{1}{\omega_{np}(D(a,1))^{q/p}}\int\limits_{D(a,1)}e^{\frac{q\alpha |z|^{2}}{2}}d\mu_{\varphi}^{\beta,\nu}(|u|^{q})(z).
\endalign
$$
Using Theorem [5, Theorem~1.2], %\?
we deduce $\|f^{(n)}\|_{L_{\mu_{\varphi}^{\beta,\nu}(|u|^{q})}^{q}}\lesssim \|f\|_{\Cal{F}_{\alpha,\omega}^{p}}$ with $f\in\Cal{F}_{\alpha,\omega}^{p}$, and hence
$D^{n}_{\varphi,u}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow \Cal{F}_{\beta,\nu}^{q}$ is bounded since
$\|D_{\varphi,u}^{n}f\|_{\Cal{F}_{\beta,\nu}^{q}}= \|f^{(n)}\|_{L_{\mu_{\varphi}^{\beta,\nu}(|u|^{q})}^{q}}$.
This finishes the proof.
\qed\enddemo

\proclaim{Theorem 13}
Suppose that $0<p\leqslant q<\infty$, $\alpha,\beta>0$, $\omega\in A_{\infty}^{\roman{restricted}}$, $n\in \Bbb{N}\cup\{0\}$, and $\nu$ is a weight.
Let $\varphi$ be an analytic self-map of $\Bbb{C}$ and $u\in \Cal{F}_{\beta,\nu}^{q}$.
Then $D^{n}_{\varphi,u}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow \Cal{F}_{\beta,\nu}^{q}$ is compact if and only if
$$
\lim\limits_{|a|\rightarrow\infty}\int\limits_{\Bbb{C}}\frac{|u(z)|^{q}}{\omega_{np}(D(a,1))^{q/p}} e^{q\alpha \Re(\overline{a}\varphi(z))-\frac{q\alpha}{2}|a|^{2}-\frac{q\beta}{2}|z|^{2}}\nu(z)dA(z)=0.
\tag5
$$
\endproclaim

\demo{Proof}
Assume that $D^{n}_{\varphi,u}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow \Cal{F}_{\beta,\nu}^{q}$ is compact. Set the function $f_{a}$ as in \Tag(2).
Then we obtain $\|f_{a}\|_{\Cal{F}_{\alpha,\omega}^{p}}\asymp1$ and $f_{a}$ tends to zero uniformly on compact subsets of $\Bbb{C}$ as $|a|$ goes to $\infty$.
And thus, $\|D^{n}_{\varphi,u}f_{a}\|_{\Cal{F}_{\beta,\nu}^{q}}\rightarrow0$ when $|a|\rightarrow\infty$.
It follows that
$$
\align
\lim\limits_{|a|\rightarrow\infty}\|D^{n}_{\varphi,u}f_{a}\|_{\Cal{F}_{\beta,\nu}^{q}}^{q}
&=\lim\limits_{|a|\rightarrow\infty}(\alpha|a|)^{qn}\int\limits_{\Bbb{C}}
\frac{|u(z)|^{q}}{\omega(D(a,1))^{q/p}} e^{q\alpha \Re(\overline{a}\varphi(z))-\frac{q\alpha}{2}|a|^{2}-\frac{q\beta}{2}|z|^{2}}\nu(z)dA(z)\\
&\asymp \lim\limits_{|a|\rightarrow\infty}\int\limits_{\Bbb{C}}
\frac{|u(z)|^{q}}{\omega_{np}(D(a,1))^{q/p}} e^{q\alpha \Re(\overline{a}\varphi(z))-\frac{q\alpha}{2}|a|^{2}-\frac{q\beta}{2}|z|^{2}}\nu(z)dA(z)=0,
\endalign
$$
which indicates \Tag(5).

Conversely, suppose that \Tag(5) holds. By the joining of \Par*{Theorem 11} and the proof of \Par*{Theorem 12}, it is enough to show
$$
\lim\limits_{|a|\rightarrow\infty}\frac{1}{\omega_{np}(D(a,1))^{q/p}}\int\limits_{D(a,1)}e^{\frac{q\alpha |z|^{2}}{2}}d\mu_{\varphi}^{\beta,\nu}(|u|^{q})(z)=0.
$$
Since
$$
\align
0&=\lim\limits_{|a|\rightarrow\infty}\int\limits_{\Bbb{C}}\frac{|u(z)|^{q}}{\omega_{np}(D(a,1))^{q/p}} e^{q\alpha \Re(\overline{a}\varphi(z))-\frac{q\alpha}{2}|a|^{2}-\frac{q\beta}{2}|z|^{2}}\nu(z)dA(z)\\
&\geqslant \lim\limits_{|a|\rightarrow\infty}\int\limits_{\varphi^{-1}(D(a,1))}\frac{|u(z)|^{q}}{\omega_{np}(D(a,1))^{q/p}} e^{q\alpha \Re(\overline{a}\varphi(z))-\frac{q\alpha}{2}|a|^{2}-\frac{q\beta}{2}|z|^{2}}\nu(z)dA(z)\\
&= \lim\limits_{|a|\rightarrow\infty}\int\limits_{D(a,1)}\omega_{np}(D(a,1))^{-q/p} e^{\frac{q\alpha}{2}|z|^{2} -\frac{q\alpha}{2}|z-a|^{2}}d\mu_{\varphi}^{\beta,\nu}(|u|^{q})(z)\\
&\gtrsim \lim\limits_{|a|\rightarrow\infty}\frac{1}{\omega_{np}(D(a,1))^{q/p}}\int\limits_{D(a,1)}e^{\frac{q\alpha |z|^{2}}{2}}d\mu_{\varphi}^{\beta,\nu}(|u|^{q})(z),
\endalign
$$
we deduce that
$D^{n}_{\varphi,u}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow \Cal{F}_{\beta,\nu}^{q}$ is compact, and the proof is complete.
\qed\enddemo

Suppose $0<p,q<\infty$. We call $\mu$ a $(p,q)$-Carleson measure %\?\it нет
if the embedding operator $i:\Cal{F}_{\alpha,\omega}^{p}\rightarrow L^{q}(\mu)$ is bounded, that is, there exists some constant $C$ such that for $f\in \Cal{F}_{\alpha,\omega}^{p}$,
$$
\biggl(\ \int\limits_{\Bbb{C}}|f(z)|^{q}d\mu(z) \biggr)^{\frac{1}{q}}\leqslant C\|f\|_{\Cal{F}_{\alpha,\omega}^{p}}.
$$

For a positive Borel measure $\mu$ on $\Bbb{C}$ and $0<p,\alpha<\infty$, let $\widetilde{\mu}_{\alpha,q}$ be the function
$$
\widetilde{\mu}_{\alpha,q}(z)=\int\limits_{\Bbb{C}}e^{q\alpha \Re(\overline{z}u)-\frac{q\alpha}{2}|z|^{2}}d\mu(u),\quad z\in \Bbb{C}.
$$
The following theorem establishes a new characterization about boundedness of the embedding operator with $0<p\leqslant q<\infty$.

\proclaim{Theorem 14}
Let $\alpha\in(0,\infty)$, $\omega\in A_{\infty}^{\roman{restricted}}$, and let $\mu$ be a positive Borel measure on $\Bbb{C}$. For $0<p\leqslant q<\infty$, the following conditions are equivalent:
\Item (i) $\mu$ is a $(p,q)$-Carleson measure;
\Item (ii) $\Psi_{\mu}(a)=\frac{\tilde{\mu}_{\alpha,q}(a)}{\omega(D(a,1))^{\frac{q}{p}}}\in L^{\infty}$, $a\in \Bbb{C}$.
\endproclaim

\demo{Proof}
Let $0<p\leqslant q<\infty$. Then
$$
\int\limits_{D(a,1)}e^{\frac{q\alpha}{2}|z|^{2}}d\mu(z)\lesssim(\omega(D(a,1)))^{\frac{q}{p}}, \quad a\in \Bbb{C},
\tag6
$$
if and only if $\mu$ is a $(p,q)$-Carleson measure by [5, Theorem~4.3].

For any $a\in \Bbb{C}$, it is easy to see that
$$
\widetilde{\mu}_{\alpha,q}(a)=\int\limits_{\Bbb{C}}e^{\frac{q\alpha}{2}|z|^{2}-\frac{q\alpha}{2}|a-z|}d\mu(z)
\gtrsim \int\limits_{D(a,1)}e^{\frac{q\alpha}{2}|z|^{2}}d\mu(z).
$$
This gives \Tag(6) if $\Psi_{\mu}(a)\in L^{\infty}$ for $a\in \Bbb{C}$, and thus $\mu$ is a $(p,q)$-Carleson measure.

Let $K_{a}(z)=e^{\alpha \overline{a}z}$. Assume that $\mu$ is a $(p,q)$-Carleson measure. By [5, Theorem~4.3] again, we have
$\|K_{a}\|_{L^{q}(\mu)}\lesssim \|K_{a}\|_{\Cal{F}_{\alpha,\omega}^{p}}$.
For any $a\in \Bbb{C}$, it follows from [5, Proposition~4.1] that
$$
e^{-\frac{q\alpha}{2}|a|^{2}}\int\limits_{\Bbb{C}}e^{q\alpha \Re(\overline{a}z)}d\mu(z)
=\int\limits_{\Bbb{C}}e^{\frac{q\alpha}{2}|z|^{2}-\frac{q\alpha}{2}|z-a|^{2}}d\mu(z)
\lesssim (\omega(D(a,1)))^{\frac{q}{p}},
$$
which indicates $\Psi_{\mu}(a)\in L^{\infty}$ with $a\in \Bbb{C}$. This finishes the proof.
\qed\enddemo

\head
4. Proof of \Par*{Theorem~3}
\endhead

Following the ideas in [12,\,15]  with some modifications, we will offer the proof of \Par*{Theorem 3} in this section.

Recall first that the essential norms for bounded linear operators. Let $X$ and $Y$ be Banach spaces, and let $\Cal{K}(X,Y)$ be the set of all compact operators from $X$ into $Y$.
The essential norm of a bounded linear operator $L: X\rightarrow Y$ is denoted by $\|L\|_{e}$, where
$$
\|L\|_{e}=\inf\{\|L-T\|: T\in\Cal{K}(X,Y)\}.
$$
It is clear that $L$ is compact if and only if $\|L\|_{e}=0$. The following auxiliary conclusion will be used later.

\proclaim{Lemma 15 \rm [11, Lemma~2.1]}
Let $f$ and $\varphi$ be two entire functions on $\Bbb{C}$ such that $f\not\equiv0$. Suppose that there is a positive constant $C$ such that
$$
|f(z)|^{2}e^{|\varphi(z)|^{2}-|z|^{2}}\leqslant C
$$
for all $z\in\Bbb{C}$. Then $\varphi(z)=\varphi(0)+\lambda z$ for some $|\lambda|\leq1$. If $|\lambda|=1$, then $f(z)=f(0)e^{-\overline{\beta}z}$, where $\beta=\overline{\lambda}\varphi(0)$. Furthermore, if
$$
\lim\limits_{|z|\rightarrow\infty}|f(z)|^{2}e^{|\varphi(z)|^{2}-|z|^{2}}=0,
$$
then $\varphi(z)=\lambda z+b$ with $|\lambda|<1$.
\endproclaim

Before embarking on the proof of \Par*{Theorem 3}, we deduce some necessary conditions as follows.

\proclaim{Theorem 16}
Suppose that $0<p,q<\infty$, $\alpha>0$, $\omega\in A_{\infty}^{\roman{restricted}}$, $n\in \Bbb{N}\cup\{0\}$, and $\varphi$ is an analytic self-map of $\Bbb{C}$.
Let $u$ be a nonzero entire function on $\Bbb{C}$.
If $D^{n}_{\varphi,u}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow \Cal{F}_{\alpha,\omega}^{q}$ is bounded,
then $u\in \Cal{F}_{\alpha,\omega}^{q}$. In this case,
$\varphi(z)=az+b$ with $|a|<1$ and
$$
|u(z)||\varphi(z)|^{n}e^{\frac{3\alpha|\varphi(z)|^{2}}{4}-\frac{3\alpha|z|^{2}}{4}}
\lesssim \|D^{n}_{\varphi,u}f_{\varphi(z)}\|_{\Cal{F}_{\alpha,\omega}^{q}}\leqslant \|D^{n}_{\varphi,u}\|
$$
for any $z\in \Bbb{C}$.
\endproclaim

\demo{Proof}
If $f(z)=z^{n}$, then we have $u\in \Cal{F}_{\alpha,\omega}^{q}$ by the boundedness of $D^{n}_{\varphi,u}$.
Setting $f_{a}$ as in \Tag(2).
For any $a,z\in \Bbb{C}$, we conclude from \Par*{Lemma 4} that there are constants $C_{1},C_{2}>0$ such that
$$
\omega(D(z,1))\geqslant C_{1}^{-|z|}\omega(D(0,1))\quad\text{and}\quad \omega(D(a,1))\leqslant C_{2}^{|a|}\omega(D(0,1)).
\tag7
$$
This together with \Par*{Lemma 7} yields
$$
\align
\|D^{n}_{\varphi,u}f_{a}\|_{\Cal{F}_{\alpha,\omega}^{q}}&\gtrsim |D^{n}_{\varphi,u}f_{a}(z)|\omega(D(z,1))^{\frac{1}{q}}e^{-\frac{\alpha|z|^{2}}{2}}\\
&=|u(z)|\frac{\omega(D(z,1))^{\frac{1}{q}}}{\omega(D(a,1))^{\frac{1}{p}}}(\alpha|a|)^{n}e^{\alpha \Re(\overline{a}\varphi(z))-\frac{\alpha|a|^{2}}{2}-\frac{\alpha|z|^{2}}{2}}\\
&\geqslant C_{1}^{-\frac{|z|}{q}}C_{2}^{-\frac{|a|}{p}}|u(z)|\omega(D(0,1))^{\frac{1}{q}-\frac{1}{p}}(\alpha|a|)^{n}e^{\alpha \Re(\overline{a}\varphi(z))-\frac{\alpha|a|^{2}}{2}-\frac{\alpha|z|^{2}}{2}}.
\endalign
$$
Setting $a=\varphi(z)$, we deduce
$$
C_{1}^{-\frac{|z|}{q}}C_{2}^{-\frac{|\varphi(z)|}{p}}|u(z)||\varphi(z)|^{n}e^{\frac{\alpha|\varphi(z)|^{2}}{2}-\frac{\alpha|z|^{2}}{2}}
\lesssim \|D^{n}_{\varphi,u}f_{\varphi(z)}\|_{\Cal{F}_{\alpha,\omega}^{q}}\leqslant \|D^{n}_{\varphi,u}\|.
$$
Note that
$$
C_{1}^{-\frac{|z|}{q}}C_{2}^{-\frac{|\varphi(z)|}{p}}e^{\frac{\alpha|\varphi(z)|^{2}}{2}-\frac{\alpha|z|^{2}}{2}}
\geqslant C_{1}^{-\frac{|z|}{q}}e^{-\frac{\alpha|z|^{2}}{4}}C_{2}^{-\frac{|\varphi(z)|}{p}}e^{\frac{-\alpha|\varphi(z)|^{2}}{4}}
e^{\frac{3\alpha|\varphi(z)|^{2}}{4}-\frac{3\alpha|z|^{2}}{4}},
$$
it follows that
$$
|u(z)||\varphi(z)|^{n}e^{\frac{3\alpha|\varphi(z)|^{2}}{4}-\frac{3\alpha|z|^{2}}{4}}
\lesssim \|D^{n}_{\varphi,u}f_{\varphi(z)}\|_{\Cal{F}_{\alpha,\omega}^{q}}\leqslant \|D^{n}_{\varphi,u}\|.
$$
Hence, by \Par*{Lemma 15} we get $\varphi(z)=az+b$ with $|a|\leqslant1$. However, if $|a|=1$, then Liouville's theorem and \Par*{Lemma 15} indicate $u(z)\varphi(z)^{n}\equiv0$, and hence $u(z)=0$ since $\varphi(z)\neq0$.
The proof of this theorem is complete.
\qed\enddemo

\proclaim{Corollary 17}
Suppose that $0<p,q<\infty$, $\alpha>0$, $\omega\in A_{\infty}^{\roman{restricted}}$, $n\in \Bbb{N}\cup\{0\}$,
and $u$ is a nonzero analytic function in $\Cal{F}_{\alpha,\omega}^{q}$.
If $\varphi(z)=b$, then $D^{n}_{\varphi,u}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow \Cal{F}_{\alpha,\omega}^{q}$ is compact, and
there are constants $C_{1},C_{2}>0$ such that
$$
C_{1}^{-\frac{|b|}{p}}|b|^{n}e^{\frac{\alpha|b|^{2}}{2}}\|u\|_{\Cal{F}_{\alpha,\omega}^{q}}\lesssim \|D^{n}_{\varphi,u}\|
\lesssim C_{2}^{\frac{|b|}{p}}(1+|b|)^{n}e^{\frac{\alpha|b|^{2}}{2}}\|u\|_{\Cal{F}_{\alpha,\omega}^{q}}.
$$
\endproclaim

\demo{Proof}
For $g\in \Cal{F}_{\alpha,\omega}^{p}$, it follows from \Par*{Lemma 7} and \Tag(7) that
$$
\|D^{n}_{\varphi,u}g\|_{\Cal{F}_{\alpha,\omega}^{q}}
\lesssim C_{2}^{\frac{|b|}{p}}(1+|b|)^{n}e^{\frac{\alpha|b|^{2}}{2}}\|g\|_{\Cal{F}_{\alpha,\omega}^{p}}\|u\|_{\Cal{F}_{\alpha,\omega}^{q}}.
$$
Letting $f_{b}$ as in \Tag(2). %\?
Using \Tag(7) again, we obtain
$$
\|D^{n}_{\varphi,u}f_{b}\|_{\Cal{F}_{\alpha,\omega}^{q}}
\gtrsim C_{1}^{-\frac{|b|}{p}}|b|^{n}e^{\frac{\alpha|b|^{2}}{2}}\|u\|_{\Cal{F}_{\alpha,\omega}^{q}}.
$$
Furthermore, $D^{n}_{\varphi,u}$ is a finite rank operator with rank 1, and so it is compact.
This finishes the proof.
\qed\enddemo

For entire functions $u$ and $\varphi$ on $\Bbb{C}$, we denote
$$
m_{z,n}(u,\varphi)=|u(z)||\varphi(z)|^{n}e^{\frac{|\varphi(z)|^{2}-|z|^{2}}{2}}, \quad z\in\Bbb{C}.
$$
In view of \Par*{Theorem 16} and \Par*{Corollary 17}, we study the essential norm of $D^{n}_{\varphi,u}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow \Cal{F}_{\alpha,\omega}^{p}$ when
$\varphi(z)=az+b$ with $0<|a|<1$.

\proclaim{Theorem 18}
Suppose that $1<p\leqslant q<\infty$, $\alpha>0$, $\omega\in A_{\infty}^{\roman{restricted}}$, $n\in \Bbb{N}\cup\{0\}$.
Let $u\in \Cal{F}_{\alpha,\omega}^{q}$ be a nonzero function and $\varphi(z)=az+b$ with $0<|a|<1$. If $D^{n}_{\varphi,u}: \Cal{F}_{\alpha,\omega}^{p}\rightarrow \Cal{F}_{\alpha,\omega}^{q}$ is bounded,
then
$$
\limsup\limits_{|z|\rightarrow\infty}m_{z,n}(u,\varphi)\lesssim \|D^{n}_{\varphi,u}\|_{e}.
$$
\endproclaim

\demo{Proof}
It follows from \Par*{Theorem 16} that $\limsup\nolimits_{|z|\rightarrow\infty}m_{z,n}(u,\varphi)<\infty$, and there is a constant $C_{0}>0$ so that $m_{z,n}(u,\varphi)\leqslant C_{0}\|D^{n}_{\varphi,u}f_{\varphi(z)}\|_{\Cal{F}_{\alpha,\omega}^{q}}$. Assume that
$$
C_{0}\|D^{n}_{\varphi,u}\|_{e}<\limsup\limits_{|z|\rightarrow\infty}m_{z,n}(u,\varphi).
$$
Then there exist positive constants $C_{1},C_{2}$ and a compact operators $T:\Cal{F}_{\alpha,\omega}^{p}\rightarrow \Cal{F}_{\alpha,\omega}^{q}$ such that
$$
C_{0}\|D^{n}_{\varphi,u}-T\|<C_{1}<C_{2}<\limsup\limits_{|z|\rightarrow\infty}m_{z,n}(u,\varphi).
$$
In fact, there is a sequence $\{z_{j}\}$, satisfying $|z_{j}|\rightarrow\infty$ whenever $j\rightarrow\infty$, such that
$$
\limsup\limits_{j\rightarrow\infty}m_{z_{j},n}(u,\varphi)=\limsup\limits_{|z|\rightarrow\infty}m_{z,n}(u,\varphi)>C_{2}.
\tag8
$$
It follows that
$$
\aligned
\|D^{n}_{\varphi,u}-T\|&\geqslant \|D^{n}_{\varphi,u}f_{\varphi(z_{j})}-Tf_{\varphi(z_{j})}\|_{\Cal{F}_{\alpha,\omega}^{q}}
\geqslant \|D^{n}_{\varphi,u}f_{\varphi(z_{j})}\|_{\Cal{F}_{\alpha,\omega}^{q}}-\|Tf_{\varphi(z_{j})}\|_{\Cal{F}_{\alpha,\omega}^{q}}\\
&\geqslant C_{0}^{-1} m_{z_{j},n}(u,\varphi)-\|Tf_{\varphi(z_{j})}\|_{\Cal{F}_{\alpha,\omega}^{q}}.
\endaligned
\tag9
$$
Note that $|\varphi(z_{j})|\rightarrow\infty$ as $j\rightarrow\infty$.
Then by the similar reason as in [12, Lemmas~2.3 and~2.4],
we deduce that $\|Tf_{\varphi(z_{j})}\|_{\Cal{F}_{\alpha,\omega}^{q}}\rightarrow0$ when $j\rightarrow\infty$.
Combining \Tag(8) with \Tag(9), for sufficiently large $j$ we get
$$
C_{1}>C_{0}\|D^{n}_{\varphi,u}-T\|\geqslant m_{z_{j},n}(u,\varphi)>C_{2},
$$
which is a contradiction, and hence
$$
\limsup\limits_{|z|\rightarrow\infty}m_{z,n}(u,\varphi)\lesssim \|D^{n}_{\varphi,u}\|_{e}.
$$
This completes the proof.
\qed\enddemo

Combining \Par*{Theorem 18} with Littlewood--Paley formulas for weighted Fock spaces $\Cal{F}_{\alpha,\omega}^{p}$ (see [5, Theorem~1.1]),
we can now provide the proof of \Par*{Theorem 3}. Clearly, only the upper estimate for essential norm of the operator $D^{n}_{\varphi,u}$ on weighted Fock spaces $\Cal{F}_{\alpha,\omega}^{p}$ needs to be proven.

\demo{Proof of \Par*{Theorem 3}}
Let $k$ be a positive integer, and $V_{r_{k}}f(z)=f(r_{k}z)$ with $r_{k}=\frac{k}{k+1}$.
Then $V_{r_{k}}$ is a compact operator on $\Cal{F}_{\alpha,\omega}^{p}$.

Let $I$ denote the identity operator on $\Cal{F}_{\alpha,\omega}^{p}$. Taking a number $R>0$. For each $k\in \Bbb{N}$, we get
$$
\align
\|D^{n}_{\varphi,u}\|_{e}
&\leqslant \|D^{n}_{\varphi,u}-D^{n}_{\varphi,u}\circ V_{r_{k}}\|
=\sup\limits_{\|f\|_{\Cal{F}_{\alpha,\omega}^{p}}\leqslant1}\|D^{n}_{\varphi,u}\circ (I-V_{r_{k}})f\|\\
&\leqslant \sup\limits_{\|f\|_{\Cal{F}_{\alpha,\omega}^{p}}\leqslant1}\biggl(\ \int\limits_{|z|\leqslant R}|D^{n}_{\varphi,u}\circ (I-V_{r_{k}})f(z)|^{p}e^{-\frac{p\alpha}{2}|z|^{2}}\omega(z)dA(z) \biggr)^{\frac{1}{p}}\\
&\qquad+\sup\limits_{\|f\|_{\Cal{F}_{\alpha,\omega}^{p}}\leqslant1}\biggl(\ \int\limits_{|z|>R}|D^{n}_{\varphi,u}\circ (I-V_{r_{k}})f(z)|^{p}e^{-\frac{p\alpha}{2}|z|^{2}}\omega(z)dA(z) \biggr)^{\frac{1}{p}}\\
&:=J_{1,k}+J_{2,k}.
\endalign
$$

On the one hand, it follows from [5, Theorem~1.1] that
$$
\align
J_{2,k}&\lesssim \Bigl( \sup\limits_{|z|>R}\frac{m_{z,n}(u,\varphi)\omega(z)}{\omega(\varphi(z))} \Bigr)\\
&\quad\cdot\sup\limits_{\|f\|_{\Cal{F}_{\alpha,\omega}^{p}}\leqslant1}\biggl(\ \int\limits_{|z|>R} \frac{\bigl|(((I-V_{r_{k}})f)^{(n)}\circ\varphi)(z)\bigr|^{p}e^{-\frac{p\alpha}{2}|\varphi(z)|^{2}}}{(1+|\varphi(z)|)^{pn}}  \omega(\varphi(z))dA(z) \biggr)^{\frac{1}{p}}\\
&\lesssim |a|^{-\frac{2}{p}}\Bigl(\sup\limits_{|z|>R}\frac{m_{z,n}(u,\varphi)\omega(z)}{\omega(\varphi(z))} \Bigr)\sup\limits_{\|f\|_{\Cal{F}_{\alpha,\omega}^{p}}\leqslant1}\|(I-V_{r_{k}})f\|\\
&\lesssim |a|^{-\frac{2}{p}}(1+\|V_{r_{k}}\|)\biggl( \sup\limits_{|z|>R}\frac{m_{z,n}(u,\varphi)\omega(z)}{\omega(\varphi(z))} \biggr)
\lesssim |a|^{-\frac{2}{p}} \sup\limits_{|z|>R}\frac{m_{z,n}(u,\varphi)\omega(z)}{\omega(\varphi(z))}.
\endalign
$$

On the other hand, for any $\delta\in(0,\alpha)$, we conclude from [5, Proposition~3.3] that
$$
\Cal{F}_{\alpha,\omega}^{p}\subset \Cal{F}_{\alpha+\delta}^{\infty}.
$$
This together with \Par*{Lemma 5} yields
$$
\align
J_{1,k}&\leqslant \bigl( \sup\limits_{|z|\leqslant R} |u(z)|\bigr)
\sup\limits_{\|f\|_{\Cal{F}_{\alpha,\omega}^{p}}\leqslant1} \biggl(\ \int\limits_{|z|\leqslant R}|(I-V_{r_{k}})f^{(n)}(\varphi(z))|^{p}e^{-\frac{p\alpha}{2}|z|^{2}}\omega(z)dA(z) \biggr)^{\frac{1}{p}}\\
&\leqslant \bigl(\sup\limits_{|z|\leqslant R} |u(z)|\bigr) \sup\limits_{\|f\|_{\Cal{F}_{\alpha,\omega}^{p}}\leqslant1} \sup\limits_{|z|\leqslant R}|(I-V_{r_{k}})f^{(n)}(\varphi(z))|
\biggl(\ \int\limits_{|z|\leqslant R}e^{-\frac{p\alpha}{2}|z|^{2}}\omega(z)dA(z) \biggr)^{\frac{1}{p}}\\
&\leqslant C\bigl( \sup\limits_{|z|\leqslant R} |u(z)|\bigr) \sup\limits_{\|f\|_{\Cal{F}_{\alpha+\delta}^{\infty}}\leqslant1} \sup\limits_{|z|\leqslant R}|(I-V_{r_{k}})f^{(n)}(\varphi(z))|.
\endalign
$$

For each $f(z)=\sum\nolimits_{j=0}^{\infty}a_{j}z^{j}$ with $\|f\|_{\Cal{F}_{\alpha+\delta}^{\infty}}\leqslant1$. By an elementary computation, we obtain
$$
|a_{j}|\leqslant \Bigl(\frac{e(\alpha+\delta)}{j}\Bigr)^{\frac{j}{2}}, \quad j\geqslant1.
$$
It follows that
$$
\align
J_{1,k}
&\lesssim \bigl( \sup\limits_{|z|\leqslant R} |u(z)|\bigr) \sup\limits_{\|f\|_{\Cal{F}_{\alpha+\delta}^{\infty}}\leqslant1} \sup\limits_{|z|\leqslant R_{\varphi}}|(I-V_{r_{k}})f^{(n)}(z)|
\leqslant \bigl( \sup\limits_{|z|\leqslant R} |u(z)|\bigr)\\
&\cdot %\?
\sup\limits_{\|f\|_{\Cal{F}_{\alpha+\delta}^{\infty}}\leqslant1} \sup\limits_{|z|\leqslant R_{\varphi}}
\sum\limits_{j=n+1}^{\infty}|a_{j}|j(j-1)\cdots(j-n+1)\Bigl(1-\Bigl(\frac{k}{k+1}\Bigr)^{j-n}\Bigr)|z|^{j-n}\\
&\leqslant \bigl( \sup\limits_{|z|\leqslant R} |u(z)|\bigr) \sup\limits_{\|f\|_{\Cal{F}_{\alpha+\delta}^{\infty}}\leqslant1} \sup\limits_{|z|\leqslant R_{\varphi}}
\sum\limits_{i=1}^{\infty}|a_{i+n}|\frac{(i+n)!}{i!}\Bigl(1-\Bigl(\frac{k}{k+1}\Bigr)^{i}\Bigr)|z|^{i}\\
&\leqslant \frac{1}{k+1} \bigl( \sup\limits_{|z|\leqslant R} |u(z)|\bigr)
\sum\limits_{i=1}^{\infty}\frac{(i+n)!}{(i-1)!}R_{\varphi}^{i}\Bigl(\frac{e(\alpha+\delta)}{i+n}\Bigr)^{\frac{i+n}{2}},
\endalign
$$
where $R_{\varphi}=\max\nolimits_{|z|\leqslant R}|\varphi(z)|$, and the series in the last term is convergent. Hence,
$$
\align
\|D^{n}_{\varphi,u}\|_{e}&\leqslant \limsup\limits_{k\rightarrow\infty}\|D^{n}_{\varphi,u}-D^{n}_{\varphi,u}\circ V_{r_{k}}\|
\leqslant \limsup\limits_{k\rightarrow\infty}J_{1,k}+\limsup\limits_{k\rightarrow\infty}J_{2,k}\\
&\lesssim |a|^{-\frac{2}{p}}\sup\limits_{|z|>R}\frac{m_{z,n}(u,\varphi)\omega(z)}{\omega(\varphi(z))}.
\endalign
$$
Therefore, the desired consequence holds by letting $R\rightarrow\infty$. This finishes the proof.
\qed\enddemo

\acknowledgment
The author would like to thank the referee for a detailed reading of the paper and suggestions.

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\enddocument