# Vanishing Carleson measures and power compact weighted composition operators

Aakriti Sharma  Department of Mathematics, Central University of Jammu, Bagla,
Rahya-Suchani, Samba 181143, INDIA<EMAIL_ADDRESS>,  Ajay K. Sharma
Department of Mathematics, Central University of Jammu, Bagla, Rahya-Suchani,
Samba 181143, INDIA<EMAIL_ADDRESS>and  M. Mursaleen  Department of
Mathematics, Aligarh Muslim University, Aligarh 202002, India
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we characterize Carleson measure and vanishing Carleson measure
on Bergman spaces with admissible weights in terms of t-Berezin transform and
averaging function as key tools. Moreover, power bounded and power compact
weighted composition operators are characterized as application of Carleson
measure and vanishing Carleson measure respectively on Bergman spaces with
admissible weights.

###### Key words and phrases:

weighted composition operator, Carleson measure, power bounded, power compact
and admissible weight.

###### 2020 MSC Mathematics Subject Classification:

47B38, 47B33, 46E10, 46B50

The first author is thankful to CSIR(India), Grant Number (File no.
09/1231(0001)/2019-EMR-I).
The second author is thankful to DST(SERB) for the project grant under MATRICS
scheme. Grant No. MTR/2018/000479.

##  1\. Introduction

Let $\mathcal{H}(\mathbb{D})$ denote the space of analytic functions on the
unit disk $\mathbb{D}=\\{z{\in}\mathbb{C}:|z|<1\\}$. Given a positive
integrable function $\sigma\in C^{2}[0,1]$, we extend it on $\mathbb{D}$ by
defining $\sigma(z)=\sigma(|z|),z\in\mathbb{D}$ and call such $\sigma$ a
weight function.
For $0<p<{\infty}$ and a positive Borel measure $\Omega$, the space
$L^{p}({\Omega})$ consists of all measurable functions $f$ on $\mathbb{D}$ for
which

$\|f\|^{p}_{L^{p}({\Omega})}=\int_{\mathbb{D}}|f(z)|^{p}d\Omega(z)<\infty.$

In the case $p=\infty$, the space of all complex-valued measurable functions
$f$ on $\mathbb{D}$ is defined as

$L^{\infty}({\Omega})=\\{f\in\mathcal{H}(\mathbb{D}):\|f\|_{\infty}={\rm
ess\>sup}_{z\in\mathbb{D}}|f(z)|<\infty\\},$

where the essential supremum is taken with respect to the measure $\Omega.$ A
sequence $\\{f_{n}\\}_{n\in{\mathbb{N}}}$ is norm bounded in
$L^{\infty}({\Omega})$ if ${\rm sup}_{n\in\mathbb{N}}\|f_{n}\|_{\infty}$ is
finite. Let $dA(z)=\frac{dxdy}{\pi}$ be the normalized Lebesgue area measure
on $\mathbb{D}$, we define the weighted Bergman space as

$A_{\sigma}^{p}=\\{f\in\mathcal{H}(\mathbb{D}):\|f\|_{A_{\sigma}^{p}}^{p}=\int_{\mathbb{D}}|f(z)|^{p}\sigma{(z)}dA(z)<\infty\\}.$

Note that $A_{\sigma}^{2}$ is a closed subspace of $L^{2}(\sigma dA)$ and
hence is a Hilbert space endowed with the inner product

$\langle
f,g\rangle=\int_{\mathbb{D}}f(z)\overline{g(z)}\sigma(z)dA(z)\;\;\;\;f,g\in
A_{\sigma}^{2}.$

Throughout this paper, we will consider $\sigma$ as admissible weight
function. Recall that if a weight function $\sigma$ is non-increasing on
$[0,1)$ and $\sigma(r)(1-r)^{-(1+\delta)}$ is non decreasing on $[0,1)$ for
some $\delta>0$, then $\sigma$ is called admissible weight.
We refer the readers [12] for useful fact over pseudohyperbolic metric. The
pseudohyperbolic metric is defined as $\rho(a,z)=|\phi_{a}(z)|$, where
$\phi_{a}(z)=\frac{a-z}{1-\bar{a}z}$ is möbius transformation. For $r$ in
$(0,1)$ and $a$ in $\mathbb{D}$,
$E(a,r)=\\{z\in\mathbb{D}:\rho(z,a)<r\\}=\phi_{a}(E(0,r))=\phi_{a}(\\{z:|z|<r\\})$
denote the pseudohyperbolic disk centre at $a$ and radius $r\in(0,1)$. It
turns out $E(a,r)$ is a Euclidean disk with center
$\dfrac{1-r^{2}}{1-|z|^{2}r^{2}}z$ and radius
$\dfrac{(1-|z|^{2})r}{1-|z|^{2}r^{2}}$.
For every $z\in E(a,r)$,
$(1-|a|^{2})^{2}\asymp(1-|z|^{2})^{2}\asymp|{1-\bar{a}z}|^{2}$ and area of
$E(a,r)$ is denoted by $|E(a,r)|$ and $|E(a,r)|\asymp(1-|a|^{2})^{2}$ are well
known facts. Here the symbol $"\asymp"$ denotes that the left hand side is
bounded above and below by constant multiples of the right hand side, where
the constants are positive and independent of variables. Given $r$ in $(0,1)$,
a sequence $\\{z_{k}\\}_{k=1}^{\infty}\subset\mathbb{D}$ is said to be an
r-lattice if the disk $\\{E(z_{k},r)\\}_{k=1}^{\infty}$ cover $\mathbb{D}$ and
there is some integer $M>0$ such that each $z$ in $\mathbb{D}$ belongs to at
most $M$ of the disks $\\{E(z_{k},\frac{1+r}{2})\\}_{k=1}^{\infty}$.
Equivalently,

$1\leq\sum_{k=1}^{\infty}\chi_{E(z_{k},\frac{1+r}{2})}(z)\leq M.$ (1.1)

Recall that,

$K^{\alpha}(z,w)=\dfrac{1}{(1-\bar{w}z)^{\alpha+2}},\;\;\;z\in\mathbb{D}$

is the reproducing kernel in standard weighted Bergman space
$A_{\sigma}^{p}=A_{\alpha}^{p}$, where standard weight
$\sigma(z)=(1-|z|^{2})^{\alpha+2}$, $\alpha>-1$. The normalized reproducing
kernel of $A_{\alpha}^{p}$ is defined as

$k^{\alpha}_{z}(.)=\dfrac{K^{\alpha}(z,.)}{\sqrt{K^{\alpha}(z,z)}}$

where $K^{\alpha}(z,z)=\dfrac{1}{(1-|z|^{2})^{\alpha+2}}$.
For a finite positive Borel measure $\Omega$ on $\mathbb{D}$, the t-Berezin
transform is defined to be

$\widetilde{\Omega}_{t}(z)=\int_{\mathbb{D}}\left(|k_{z}^{\alpha}(w)|\right)^{t}d\Omega(w),\;\;\;z\in\mathbb{D}$
(1.2)

Note that for $t=2$, the classical Berezin transform is denoted by
$\widetilde{\Omega}_{2}$. Given $r$ in $(0,1)$, the Averaging function of
$\Omega$ is defined to be

$\widehat{\Omega}_{r}(z)=\frac{\Omega(E(z,r))}{|E(z,r)|},\;\;\;z\in\mathbb{D}$
(1.3)

If we set $d\Omega=fdA$, for a Lebesgue measurable function $f$, then we can
write $\widetilde{f_{t}}=\widetilde{\Omega}_{t}$ and
$\widehat{f}_{r}=\widehat{\Omega}_{r}$ for simplicity.
Motivated by [21] and [9], in this article we study the power bounded and
power compact weighted composition operator on $A_{\sigma}^{p}$ by using
Carleson measure characterization in terms of t-Berezin transform and
averaging function as a key tool. An operator $T$ on a normed linear space
$(X,\|.\|_{X})$ is called power bounded if $\\{T^{n}\\}$ is a bounded sequence
in the space of all bounded operators from $X$ to itself. Also, recall that an
operator $T$ on Banach space $(X,\|.\|_{X})$ is said to be ${\textit{power
compact}},$ see [2] if there exist some integer $m>0$ such that $T^{m}$ is
compact from $X$ to itself. Denote by $\Lambda^{2}(\mathbb{C})$, the linear
space of all double sequences with complex entries. A double sequence
$\\{\gamma_{j,k}\\}_{j,k\in\mathbb{N}}$ of complex numbers is bounded if there
exists some $M>0$ such that $\sup_{j,k}|\gamma_{j,k}|\leq M$. The space
$\Lambda^{2}_{\infty}$ of all bounded double sequences is defined as

$\Lambda^{2}_{\infty}=\\{\gamma_{jk}=\\{\gamma_{j,k}\\}_{j,k\in\mathbb{N}}\in\Lambda^{2}(\mathbb{C}):\|\gamma_{jk}\|_{\Lambda^{2}_{\infty}}=\sup_{j,k}|\gamma_{j,k}|<\infty\\}.$

Let $C_{\psi,\phi}$ denoted the well known weighted composition operator on
the space $\mathcal{H}(\mathbb{D})$ is defined as

$C_{\psi,\phi}(f)=\psi{(f\circ{\phi})}$

where $\psi\in\mathcal{H}(\mathbb{D})$ and $\phi$ is an analytic self map of
$\mathbb{D}$. If $\phi(z)=z$ and $\psi=1$, then $C_{\psi,\phi}$ becomes the
multiplication operator $M_{\psi}$ and the composition operator $C_{\phi}$
respectively. Denote by $\phi_{n}$ the $n$th iteration of $\phi,$ that is

$\phi_{n}=\underbrace{\phi\circ\phi\cdots\phi}_{n-\text{times}}.$

Note that any power of $C_{\psi,\phi}$ on $\mathcal{H}(\mathbb{D})$ is a
weighted composition operators which is defined as

$C_{\psi,\phi}^{n}f=\prod_{j=0}^{n-1}(\psi\circ\phi_{j})f\circ\phi_{n}.$

For the sake of simplicity, we set

$\langle\psi,\phi,n\rangle=\prod_{j=0}^{n-1}\psi\circ\phi_{j}.$

Thus,

$\|C_{\psi,\phi}^{n}f\|_{L^{p}({\Omega})}^{p}=\int_{\mathbb{D}}|\langle\psi,\phi,n\rangle(z)f\circ\phi_{n}(z)|^{p}d\Omega(z).$

We define $d\Omega_{n}=|\langle\psi,\phi,n\rangle|d\Omega\circ\phi^{-1}_{n}$,
one can easily see that $\Omega_{n}$ is a measure and therefore

$\|C_{\psi,\phi}^{n}f\|_{L^{p}({\Omega})}^{p}=\int_{\mathbb{D}}|f(z)|^{p}d\Omega_{n}$

.For $t>0$, the $t$-Berezin transform and for $0<r<1$, the averaging function
of $\Omega_{n}$ are defined as

$\widetilde{\Omega}_{n,t}(z)=\int_{\mathbb{D}}\big{(}|k_{z}^{\alpha}(w)|\big{)}^{t}d\Omega_{n}(z),\;\;\;z\in\mathbb{D}$

and

$\widehat{\Omega}_{n,r}(z)=\dfrac{\Omega_{n}(E(z,r))}{|E(z,r)|},\;\;\;z\in\mathbb{D}$

respectively. For more about weighted composition operators, Carleson measures
and vanishing Carleson measures, we refer to [16]-[12]. Throughout the paper,
the expression $E\lesssim F$ means that there exists a constant $C$ such that
$E\leq CF$.

##  2\. Preliminaries

In this section, we prove and collect some useful facts and lemmas that are
required for the proof of our main results. The next lemma gives a growth
estimate for functions in $A_{\sigma}^{p}$ and an asymptotic estimate for norm
of $K^{\alpha}(z,\cdot)$ already proved in [1].

######  Lemma 1.

Let $p>0$, $\sigma$ be an admissible weight and $\alpha>-1$. Then

  1. (1)

for each $z\in\mathbb{D},$ we have that

$|f(z)|^{p}\lesssim\dfrac{\|f\|_{A_{\sigma}^{p}}^{p}}{\sigma(z)(1-|z|^{2})^{2}}\;\text{
for all }\;f\in A_{\sigma}^{p}.$ (2.1)

  2. (2)

for each $z\in\mathbb{D},$ we have that

$\|K^{\alpha}(z,\cdot)\|_{A_{\sigma}^{p}}\asymp\dfrac{1}{(\sigma(z))^{\frac{1}{p}}(1-|z|^{2})^{(\alpha+2)-\frac{2}{p}}}$
(2.2)

######  Lemma 2.

Let $1\leq p\leq{\infty}$ and $r\in(0,1)$. Then the operator
$f\longmapsto\widehat{f}_{r}$ is bounded from $L^{p}(\mathbb{D})$ to
$L^{p}(\mathbb{D})$.

###### Proof.

Since $\chi_{E(z,r)}(\xi)=\chi_{E(\xi,r)}(z),\;\;z,\xi\in\mathbb{D}$. By
Fubini’s theorem for all $f\in L^{1}(\mathbb{D})$, we have

$\displaystyle\|\widehat{f}_{r}\|_{1}$
$\displaystyle\leq\int_{\mathbb{D}}\bigg{|}\frac{1}{|E(z,r)|}\int_{E(z,r)}|f(\xi)|dA(\xi)\bigg{|}dA(z)$
$\displaystyle\leq\int_{\mathbb{D}}\frac{dA(z)}{|E(z,r)|}\int_{E(z,r)}|f(\xi)|dA(\xi)$
$\displaystyle=\int_{\mathbb{D}}|f(\xi)|dA(\xi)\int_{E(\xi,r)}\frac{dA(z)}{|E(z,r)|}$
$\displaystyle\leq C\|f\|_{1}$

The boundedness of the operator $f\longmapsto\widehat{f}_{r}$ is trivially
holds for $p=\infty$ , that is
$\|\widehat{f}_{r}\|_{\infty}\leq\|f\|_{\infty}$ and also holds for
$1<p<\infty$, by complex interpolation. ∎

######  Lemma 3.

Suppose $0<p<{\infty}$, $r\in(0,1)$ and $\Omega$ be a finite positive Borel
measure. Then there exists some constant $C$ such that

$\int_{\mathbb{D}}h(z)d\Omega(z)\leq
C\int_{\mathbb{D}}h(z)\widehat{\Omega}_{r}(z)dA(z)$

for all non-negative subharmonic function
$h:\mathbb{D}\longrightarrow[0,\infty)$.

###### Proof.

Since $h$ be non negative subharmonic function
$h:\mathbb{D}\longrightarrow[0,\infty)$. Then, we have

$h(z)\leq\frac{C}{|E(z,r)|}\int_{E(z,r)}h(\xi)dA(\xi)$

for all $z\in\mathbb{D}$, see page no. 125 of [12]. Using above inequality,
Fubini’s theorem and (1.3), we have that

$\displaystyle\int_{\mathbb{D}}h(z)d\Omega(z)$ $\displaystyle\leq
C\int_{\mathbb{D}}\frac{d\Omega(z)}{|E(z,r)|}\int_{E(z,r)}h(\xi)dA(\xi)$
$\displaystyle=C\int_{\mathbb{D}}h(\xi)dA(\xi)\int_{E(\xi,r)}\frac{d\Omega(z)}{|E(z,r)|}$
$\displaystyle\leq C\int_{\mathbb{D}}h(z)\widehat{\Omega}_{r}(z)dA(z)$

This accomplished the result.∎

Note that lemma 4 and lemma 5 already prove in [21] and [22] in more general
sense. For the sake of completeness, we include the proofs. Note that lemma 4
reveals that the function $\Omega(E(z,r))$ behaves like a subharmonic
function. For $r=R$, the following lemma, lemma 4 was first obtained in [ZK1].

######  Lemma 4.

Suppose $r,R>0$ and $\Omega$ be a positive borel measure on $\mathbb{D}$, then
there exists a constant $C$ such that

$\Omega(E(a,R))\leq\dfrac{C}{|E(a,R)|}\int_{E(a,R)}\Omega(E(z,r))dA(z)$ (2.3)

###### Proof.

For $r,R>0$, we have

$\displaystyle\int_{E(a,R)}\Omega(E(z,r))dA(z)$
$\displaystyle=\int_{\mathbb{D}}\chi_{E(a,R)}(z)dA(z)\int_{\mathbb{D}}\chi_{E(z,r)}(w)d\Omega(w)$
$\displaystyle=\int_{\mathbb{D}}d\Omega(w)\int_{E(a,R)}\chi_{(E(z,r)}(w)dA(w)$

Since $\chi_{E(z,r)}(w)=\chi_{E(w,r)}(z)$ for all $z,w\in\mathbb{D}$, then

$\displaystyle\int_{E(a,R)}\Omega(E(z,r))dA(z)$
$\displaystyle=\int_{\mathbb{D}}d\Omega(w)\int_{E(a,R)}\chi_{E(w,r)}(z)dA(z)$
$\displaystyle\geq\int_{E(a,R)}|(E(a,R)\cap E(w,r))|d\Omega(w)$
$\displaystyle\geq\Omega(E(a,R))\inf_{w\in E(a,R)}\\{|(E(a,R)\cap E(w,r))|\\}$

For $w\in(E(a,R)$, then there exists a Euclidean disk with diameter
$\frac{1}{2}\min\\{r,R\\}$ contained in $(E(a,R)\cap E(w,r))$. Therefore (2.3)
holds.

∎

######  Lemma 5.

Suppose $1\leq p\leq{\infty}$ and $s\in\mathbb{R}$. Then
$\dfrac{\widehat{\Omega}_{r}(z)}{\sigma^{s}(z)\left(1-|z|^{2}\right)^{2(s-1)}}\in
L^{p}(\mathbb{D})$ for some $r$, $0<r<1$ if and only if
$\dfrac{\widehat{\Omega}_{R}(z)}{\sigma^{s}(z)\left(1-|z|^{2}\right)^{2(s-1)}}\in
L^{p}(\mathbb{D})$ for all $0<R<1$. Moreover,
$\bigg{\|}\dfrac{\widehat{\Omega}_{r}(z)}{\sigma^{s}(z)\left(1-|z|^{2})\right)^{2(s-1)}}\bigg{\|}_{p}\asymp\bigg{\|}\dfrac{\widehat{\Omega}_{R}(z)}{\sigma^{s}(z)\left(1-|z|^{2}\right)^{2(s-1)}}\bigg{\|}_{p}$.

###### Proof.

For $r$ and $R$, by lemma 1 there exists a constant $C$ such that

$\widehat{\Omega}_{R}(z)\leq\dfrac{C}{|E(z,R)|}\int_{E(z,R)}\widehat{\Omega}_{r}(\xi)dA(\xi)$

for all $z\in\mathbb{D}$. Above inequality and (1.3) implies that,

$\displaystyle G_{R}(z)$
$\displaystyle\leq\dfrac{C}{|E(z,R)|}\int_{E(z,R)}G_{r}(\xi)dA(\xi)$
$\displaystyle\asymp\widehat{G_{r}}(z),$

where
$G_{l}(z)=\frac{\widehat{\Omega}_{l}(z)}{\sigma^{s}(z)\left(1-|z|^{2}\right)^{2(s-1)}}$.
Combining this with Lemma 3, we find that

$\bigg{\|}\frac{\widehat{\Omega}_{R}(z)}{\sigma^{s}(z)\left(1-|z|^{2}\right)^{2(s-1)}}\bigg{\|}_{p}\leq\bigg{\|}\widehat{G_{r}}(z)\bigg{\|}_{p}\leq\bigg{\|}G_{r}(z)\bigg{\|}_{p}\;\;\;\;\text{for}\;1\leq
p\leq\infty.$

Interchanging the role of $r$ and $R$, we get the desired result. ∎

######  Lemma 6.

Suppose $1<p<\infty$ and $t>0$. Then the integral operator

$T_{t,s}f(z)=\sigma^{s}(z)\big{(}1-|z|^{2}\big{)}^{(\alpha+2)t-2s}\int_{\mathbb{D}}\dfrac{|K^{\alpha}(z,\xi)|^{t}}{\sigma^{s}(\xi)\left(1-|\xi|^{2}\right)^{2(s-1)}}f(\xi)dA(\xi)$

is bounded on $L^{p}(\mathbb{D})$ whenever
$\frac{1-p}{p}<s<(\alpha+2)\frac{t}{2}+\frac{1}{p}$.

###### Proof.

Let $P=\frac{1-p}{p}(s+1)$, $Q=\frac{p-1}{p}\\{(\alpha+2)t-{2}-s\\}$,
$U=\frac{s+1-(\alpha+2)t}{p}$ and $V=\frac{s}{p}$. Then claim that the
intervals $(P,Q)$ and $(U,V)$ are non-empty. By using hypothesis one can
easily find that

$\displaystyle Q-P$
$\displaystyle=(\alpha+2)(1-\frac{1}{p})(t-\frac{1}{\alpha+2})>0$

and

$\displaystyle V-U$
$\displaystyle=\frac{(\alpha+2)}{p}(t-\frac{1}{\alpha+2})>0$

Implies that the intervals $(P,Q)$ and $(U,V)$ are non-empty. Also,

$\displaystyle V-P$ $\displaystyle=s-\frac{1-p}{p}>0$

and

$\displaystyle Q-U$ $\displaystyle=(\alpha+2)t-2+\frac{1}{p}-s>0$

implies that $P<V$ and $U<Q$. Thus, $(P,Q)\cap(U,V)$ is non-empty. For some
$m\in(P,Q)\cap(U,V)$ and take
$h(\xi)=\big{(}\sigma(\xi)(1-|\xi|^{2})\big{)}^{m}$ and
$\frac{1}{p}+\frac{1}{p^{{}^{\prime}}}=1$. From Lemma 1, there is a positive
constant $C$ such that

$\sigma^{s}(z)\big{(}1-|z|^{2}\big{)}^{(\alpha+2)t-2s}\int_{\mathbb{D}}\dfrac{|K^{\alpha}(z,\xi)|^{t}}{\sigma^{s}(\xi)\left(1-|\xi|^{2}\right)^{2(s-1)}}h^{p^{{}^{\prime}}}(\xi)dA(\xi)\leq
Ch^{p^{{}^{\prime}}}(\xi),\;\;\;z\in\mathbb{D}$

and

$\sigma^{-s}(z)\left(1-|z|^{2}\right)^{2(1-s)}\int_{\mathbb{D}}|K^{\alpha}(z,\xi)|^{t}\sigma^{s}(\xi)\big{(}1-|\xi|^{2}\big{)}^{(\alpha+2)t-2s}h^{p}(\xi)dA(\xi)\leq
Ch^{p}(\xi)$

$z\in\mathbb{D}$. By Schur’s test, boundedness of the operator $T_{t,s}$ on
$L^{p}(\mathbb{D})$ holds. ∎

######  Lemma 7.

Let $\\{z_{k}\\}_{k=1}^{\infty}$ be an $r$\- lattice. For $1<p<\infty$ and
$\\{\lambda_{k}\\}_{k=1}^{\infty}\in l^{p}$, let

$f(z)=\sum_{k=1}^{\infty}\lambda_{k}\dfrac{K^{\alpha}({z_{k}},z)}{\sigma^{\frac{1}{p}}(z_{k})(1-|z_{k}|^{2})^{\frac{2}{p}-(\alpha+2)p}}$
(2.4)

where $\alpha>-1$. Then $f\in A_{\sigma}^{p}(\mathbb{D})$ and
$\|f\|_{p,\sigma}\leq C\|\\{\lambda_{k}\\}_{k=1}^{\infty}\|_{l^{p}}$.

The proof is an easy modification of arguments in Theorem 4.1 in [1]. We omit
the details.

######  Lemma 8.

Suppose $\Omega\geq 0$, $1\leq p\leq\infty$, $t>0$ and $s\in\mathbb{R}$
satisfies $t<s+\frac{1}{p}<1$. Then the following are equivalent

  1. (a)

$\widetilde{M}_{t,s}(z)=\dfrac{\widetilde{\Omega}_{t}(z)}{\sigma^{s}(z)(1-|z|^{2})^{2s-(\frac{\alpha+2}{2})t}}\in
L^{p}(\mathbb{D})$.

  2. (b)

$\widehat{M}_{R,s}(z)=\dfrac{\widehat{\Omega}_{R}(z)}{\sigma^{s}(z)(1-|z|^{2})^{2(s-1)}}\in
L^{p}(\mathbb{D})$ for some (or any) $R$, $0<R<1$.

  3. (c)

The sequence
$\bigg{\\{}\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{s}(z_{k})(1-|z_{k}|^{2})^{2(s-1-\frac{1}{p})}}\bigg{\\}}_{k=1}^{\infty}$
belongs to $l^{p}$ for some (or any) $r$-lattice $\\{z_{k}\\}_{k=1}^{\infty}$
with $0<r<1$.

Moreover, we have

$\|\widetilde{M}_{t,s}\|_{p}\asymp\|\widehat{M}_{R,s}\|_{p}\asymp\bigg{\|}\bigg{\\{}\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{s}(z_{k})(1-|z_{k}|^{2})^{2(s-1-\frac{1}{p})}}\bigg{\\}}_{k=1}^{\infty}\bigg{\|}_{l^{p}}.$

###### Proof.

We will prove the result in the order: $(a)\Leftrightarrow(b)$ and
$(b)\Leftrightarrow(c).$
$(a)\Rightarrow(b)$. For any $R\in(0,1)$, there exist positive constant
$C_{R}$ such that for any $z\in\mathbb{D}$ kernel estimate holds. Thus for
$s\in\mathbb{R}$ , we have

$\displaystyle\widehat{M}_{R,s}(z)$
$\displaystyle=\dfrac{\Omega(E(z,R))}{\sigma^{s}(z)(1-|z|^{2})^{2s}}$
$\displaystyle\leq
C_{R}\dfrac{1}{\sigma^{s}(z)(1-|z|^{2})^{2s-(\frac{\alpha+2}{2})t}}\int_{E(z,R)}|k_{z}^{\alpha}(\xi)|^{t}d\Omega(\xi)$
$\displaystyle\leq
C_{R}\dfrac{1}{\sigma^{s}(z)(1-|z|^{2})^{2s-(\frac{\alpha+2}{2})t}}\int_{\mathbb{D}}|k_{z}^{\alpha}(\xi)|^{t}d\Omega(\xi)$
$\displaystyle\leq C_{R}\widetilde{M}_{t,s}(z)$

Above implies that, $\|\widehat{M}_{R,s}\|_{p}\leq
C\|\widetilde{M}_{t,s}\|_{p}$.

$(b)\Rightarrow(a)$. By Lemma 2 and Lemma 5, there is a positive constant $C$
such that for any $z\in\mathbb{D}$ and $s\in\mathbb{R}$, we have

$\displaystyle\widetilde{M}_{t,s}(z)$
$\displaystyle=\dfrac{\widetilde{\Omega}_{t}(z)}{\sigma^{s}(z)(1-|z|^{2})^{2s-(\frac{\alpha+2}{2})t}}$
$\displaystyle=\dfrac{1}{\sigma^{s}(z)(1-|z|^{2})^{2s-(\frac{\alpha+2}{2})t}}\int_{\mathbb{D}}|k_{z}^{\alpha}(\xi)|^{t}d\Omega(\xi)$
$\displaystyle\leq\dfrac{C}{\sigma^{s}(z)(1-|z|^{2})^{2s-(\frac{\alpha+2}{2})t}}\int_{\mathbb{D}}|k_{z}^{\alpha}(\xi)|^{t}\widehat{\Omega}_{R}(\xi)dA(\xi)$
$\displaystyle\leq\dfrac{C}{\sigma^{s}(z)(1-|z|^{2})^{2s-(\alpha+2)t}}\int_{\mathbb{D}}|K^{\alpha}(z,\xi)|^{t}\sigma^{s}(\xi)(1-|\xi|^{2})^{2(s-1)}\widehat{M}_{R,s}(\xi)dA(\xi)$
$\displaystyle\leq CT_{t,-s}(\widehat{M}_{R,s})(z)$

Since $t<s+\frac{1}{p}<1$ and Lemma 6 implies that

$\|\widetilde{M}_{t,s}\|_{p}\leq\|T_{t,-s}(\widehat{M}_{R,s})\|_{p}\leq
C\|\widehat{M}_{R,s}\|_{p}.$

$(b)\Rightarrow(c)$. Assume that $\widehat{M}_{R,s}\in L^{p}(\mathbb{D})$ for
some $R$, $0<R<1$. Let $\\{z_{k}\\}_{k=1}^{\infty}$ be any $r$-lattice. By
Lemma 5, we may assume $R<r$. By triangle inequality, we have
$E(z_{k},r)\subset E(z,2r)$ for $z\in E(z_{k},r)$ and for all $k$. Thus, we
have

$\displaystyle\dfrac{\widetilde{\Omega}_{r}(z_{k})}{\sigma^{s}(z_{k})(1-|z_{k}|^{2})^{2(s-1)}}$
$\displaystyle\leq C\dfrac{\Omega(E(z,2r))}{\sigma^{s}(z)(1-|z|^{2})^{2s}}$
$\displaystyle\asymp\dfrac{\widehat{\Omega}_{2r}(z)}{\sigma^{s}(z)(1-|z|^{2})^{2(s-1)}}$
(2.5)

whenever $z\in E(z_{k},r)$. Therefore, we have

$\displaystyle\sum_{k=1}^{\infty}\dfrac{\widehat{\Omega}_{r}^{p}(z_{k})}{\sigma^{sp}(z_{k})(1-|z_{k}|^{2})^{2(s-1)p-2}}$
$\displaystyle\leq
C\sum_{k=1}^{\infty}\int_{E(z_{k},r)}\dfrac{\widehat{\Omega}_{2r}^{p}(z)}{\sigma^{sp}(z)(1-|z|^{2})^{2(s-1)p}}dA(z)$
$\displaystyle\leq
CM\int_{\mathbb{D}}\dfrac{\widehat{\Omega}_{2r}^{p}(z)}{\sigma^{sp}(z)(1-|z|^{2})^{2(s-1)p}}dA(z)$

Thus by Lemma 5, we have

$\bigg{\|}\bigg{\\{}\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{s}(z_{k})(1-|z_{k}|^{2})^{2(s-1-\frac{1}{p})}}\bigg{\\}}_{k}\bigg{\|}_{l^{p}}\leq
C\|\widehat{M}_{2r,s}\|_{p}\asymp\|\widehat{M}_{R,s}\|_{p}$

.

$(c)\Rightarrow(b)$. Finally, Suppose
$\bigg{\\{}\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{s}(z_{k})(1-|z_{k}|^{2})^{2(s-1-\frac{1}{p})}}\bigg{\\}}_{k}\in{l^{p}}$
for some $r$-lattice $\\{z_{k}\\}_{k=1}^{\infty}$. Similar to (2)
$\widehat{\Omega}_{r}(z)\leq C\widehat{\Omega}_{2r}(z_{k})$ for $z\in
E(z_{k},r)$. Therefore, we have

$\displaystyle\int_{\mathbb{D}}\bigg{(}\dfrac{\widehat{\Omega}_{r}(\xi)}{\sigma^{s}(\xi)(1-|\xi|^{2})^{2(s-1)}}\bigg{)}^{p}dA(\xi)$
$\displaystyle\leq\sum_{k=1}^{\infty}\int_{E(z_{k},r)}\dfrac{\widehat{\Omega}^{p}_{r}(\xi)}{\sigma^{sp}(\xi)(1-|\xi|^{2})^{2(s-1)p}}dA(\xi)$
$\displaystyle\leq
C\sum_{k=1}^{\infty}\dfrac{\widehat{\Omega}^{p}_{2r}(z_{k})}{\sigma^{sp}(z_{k})(1-|z_{k}|^{2})^{2(s-1)p-2}}$
$\displaystyle\leq
C\sum_{k=1}^{\infty}\dfrac{\widehat{\Omega}^{p}_{r}(z_{k})}{\sigma^{sp}(z_{k})(1-|z_{k}|^{2})^{2(s-1)p-2}}$

Above inequality and Lemma 5 implies that

$\|\widehat{M}_{R,s}\|_{p}\asymp\|\widehat{M}_{r,s}\|_{p}\leq
C\bigg{\|}\bigg{\\{}\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{s}(z_{k})(1-|z_{k}|^{2})^{2(s-1-\frac{1}{p})}}\bigg{\\}}_{k}\bigg{\|}_{l^{p}}$

for any $R\in(0,1)$. ∎

##  3\. Carleson measure characterizations

In this section, we are using averaging function and t-Berezin transform as
our main tools to characterize the $(p,q,\sigma)-$Bergman Carleson measure for
$0<p,q<\infty$ and $t>0$. Let $\Omega$ be a finite positive Borel measure.
Recall that,

  * •

$\Omega$ is a $(p,q,\sigma)-$Bergman Carleson measure if the embedding
$i:A_{\sigma}^{p}\longrightarrow L^{q}(\Omega)$ is bounded. In other words, we
can say $\Omega$ is a $(p,q,\sigma)-$Bergman Carleson measure if there exists
a finite constant $C>0$ such that

$\int_{\mathbb{D}}|f|^{q}d\Omega\leq C\|f\|_{A_{\sigma}^{p}}^{q}$

for all $f\in A_{\sigma}^{p}$.

  * •

$\Omega$ is a vanishing $(p,q,\sigma)-$Bergman Carleson measure if
$\int_{\mathbb{D}}|f_{n}|^{q}d\Omega\longrightarrow 0$ as
$n\longrightarrow\infty$ whenever $\\{f_{n}\\}$ is a bounded sequence in
$A_{\sigma}^{p}$ which converges to $0$ uniformly on any compact subset of
$\mathbb{D}$.

Note that, by taking $p=q$ and $\sigma(z)=1$, $\Omega$ becomes a Bergman
Carleson measure and vanishing Carleson measure. We divide our result into two
cases: $0<p\leq q<{\infty}$ and $0<q<p<{\infty}$.

######  Theorem 1.

Let $\Omega$ be a finite positive Borel measure and $0<p\leq q<{\infty}$. Then
the following statements are equivalent:

  1. (a)

$\Omega$ is a $(p,q,\sigma)-$Bergman Carleson measure.

  2. (b)

The function
$\dfrac{\widetilde{\Omega}_{t}(z)}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2\frac{q}{p}-(\frac{\alpha+2}{2})t}}$
is bounded on $\mathbb{D}$ for $t>\frac{2q}{p(\alpha+2)}$.

  3. (c)

The function
$\dfrac{\widehat{\Omega}_{R}(z)}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2(\frac{q-p}{p})}}$
is bounded on $\mathbb{D}$ for some (or any) $R\in(0,1)$.

  4. (d)

The sequence
$\bigg{\\{}\dfrac{\widehat{\Omega}_{r}(a_{k})}{\sigma^{\frac{q}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{q-p}{p})}}\bigg{\\}}_{k=1}^{\infty}$
is bounded for some (or any) $r$-lattice $\\{z_{k}\\}_{k=1}^{\infty}$ with
$0<r<1$.

Furthermore,

$\displaystyle\|i\|_{A_{\sigma}^{p}\rightarrow L^{p}(\Omega)}$
$\displaystyle\asymp\sup_{z\in\mathbb{D}}\dfrac{\widetilde{\Omega}_{t}(z)}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2\frac{q}{p}-(\frac{\alpha+2}{2})t}}$
$\displaystyle\asymp\sup_{z\in\mathbb{D}}\dfrac{\widehat{\Omega}_{R}(z)}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2(\frac{q-p}{p})}}$
$\displaystyle\asymp\sup_{k}\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{\frac{q}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{q-p}{p})}}$
(3.1)

###### Proof.

$(d)\Rightarrow(a)$ We assume that $\\{z_{k}\\}_{k=1}^{\infty}$ be an
$r$-lattice. We use the elementary inequality
$\sum_{k=1}^{\infty}u_{k}^{l}\leq(\sum_{k=1}^{\infty}u_{k})^{l}$, $u_{k}\geq
0$, $k=1,2,\cdots$, by taking $l=\frac{p}{q}\geq 1$, using Lemma 1 and (1.1),
we obtain

$\displaystyle\int_{\mathbb{D}}|f(z)|^{q}d{\Omega}$
$\displaystyle=\sum_{k=1}^{\infty}\int_{E(z_{k},r)}|f(z)|^{q}d{\Omega}$
$\displaystyle\leq\sum_{k=1}^{\infty}\widehat{\Omega}_{r}(z_{k})|E(z_{k},r)|\bigg{(}\sup_{z\in
E(z_{k},r)}|f(z)|^{p}\bigg{)}^{\frac{q}{p}}$ $\displaystyle\leq
C\sum_{k=1}^{\infty}\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{\frac{q}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{q-p}{p})}}\bigg{(}\int_{E(z_{k},\frac{1+r}{2})}|f(\xi)|^{p}\sigma(\xi)dA(\xi)\bigg{)}^{\frac{q}{p}}$
$\displaystyle\leq
C\sup_{k}\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{\frac{q}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{q-p}{p})}}\bigg{(}\sum_{k=1}^{\infty}\int_{E(z_{k},\frac{1+r}{2})}|f(\xi)|^{p}\sigma(\xi)dA(\xi)\bigg{)}^{\frac{q}{p}}$
$\displaystyle\leq
CM^{\frac{q}{p}}\sup_{k}\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{\frac{q}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{q-p}{p})}}\|f\|_{p,\sigma}^{q}$
(3.2)

Above inequality reveals that

$\|i\|_{A_{\sigma}^{p}\rightarrow L^{q}(\Omega)}\leq
C\sup_{k}\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{\frac{q}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{q-p}{p})}}$

.

$(a)\Rightarrow(c)$. Set $f_{z}(w)=K^{\alpha}(z,w),\;w\in\mathbb{D}$. By Lemma
1 and statement$(a)$, we have

$\displaystyle\dfrac{\widehat{\Omega}_{R}(z)}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2(\frac{q-p}{p})}}$
$\displaystyle=\dfrac{\Omega(E(z,R))}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2\frac{q}{p}}}$
$\displaystyle\leq
C\dfrac{(1-|z|^{2})^{q(\alpha+2-\frac{2}{p})}}{\sigma^{\frac{q}{p}}(z)}\int_{E(z,R)}|f_{z}(\xi)|^{q}d\Omega(\xi)$
$\displaystyle\leq
C\dfrac{(1-|z|^{2})^{q(\alpha+2-\frac{2}{p})}}{\sigma^{\frac{q}{p}}(z)}\int_{\mathbb{D}}|f_{z}(\xi)|^{q}d\Omega(\xi)$
(3.3) $\displaystyle\leq
C\dfrac{(1-|z|^{2})^{q(\alpha+2-\frac{2}{p})}}{\sigma^{\frac{q}{p}}(z)}\|i\|_{A_{\sigma}^{p}\rightarrow
L^{q}(\Omega)}^{q}\|f_{z}\|_{A_{\sigma}^{p}}^{q}$ $\displaystyle\leq
C\|i\|_{A_{\sigma}^{p}\rightarrow L^{q}(\Omega)}^{q}$

Above inequality reveals that

$\sup_{z\in\mathbb{D}}\dfrac{\widehat{\Omega}_{R}(z)}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2(\frac{q-p}{p})}}\leq\|i\|_{A_{\sigma}^{p}\rightarrow
L^{q}(\Omega)}.$ (3.4)

The equivalence of $(a)$,$(c)$ and $(d)$ follows from above proof of
implications. Moreover,

$\displaystyle\|i\|_{A_{\sigma}^{p}\rightarrow L^{p}(\Omega)}$
$\displaystyle\asymp\sup_{z\in\mathbb{D}}\dfrac{\widehat{\Omega}_{R}(z)}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2(\frac{q-p}{p})}}$
$\displaystyle\asymp\sup_{k}\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{\frac{q}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{q-p}{p})}}$
(3.5)

$(b)\Rightarrow(c)$. For any $R$, $0<R<1$, Lemma 8 yields

$\dfrac{\widehat{\Omega}_{R}(z)}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2(\frac{q-p}{p})}}\leq
C\dfrac{\widetilde{\Omega}_{t}(z)}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2\frac{q}{p}-(\frac{\alpha+2}{2})t}}$
(3.6)

.

$(a)\Rightarrow(b)$. The estimate (3) reveals that the embedding operator
$i:A_{\sigma}^{p}\longrightarrow L^{q}(\Omega)$ is bounded for some $0<p\leq
q<{\infty}$ if and only if $i:A_{\sigma}^{p_{1}}\longrightarrow
L^{q_{1}}(\Omega)$ is bounded for some $0<p_{1}\leq q_{1}<{\infty}$ with
$\frac{q_{1}}{p_{1}}=\frac{q}{p}$. Since $\Omega$ is a $(p,q,\sigma)$-carleson
measure $i:A_{\sigma}^{Np}\longrightarrow L^{Nq}(\Omega)$ where $N$ is some
integer with $Np>\frac{4}{(\alpha+2)}$. Let $f_{z}(.)=K_{z}^{\alpha}(.)$,
$z\in\mathbb{D}$ and (3) tells us that

$\displaystyle\dfrac{\widetilde{\Omega}_{Nq}(z)}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2\frac{q}{p}-(\frac{\alpha+2}{2})Nq}}$
$\displaystyle\asymp\dfrac{1}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2\frac{q}{p}-(\frac{\alpha+2}{2})Nq}}\int_{\mathbb{D}}|f_{z}(\xi)|^{Nq}d{\Omega}(\xi)$
(3.7) $\displaystyle\lesssim\dfrac{\|i\|^{Nq}_{A_{\sigma}^{Np}\rightarrow
L^{Nq}(\Omega)}\|f_{z}\|_{Np,\sigma}^{Nq}}{{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2\frac{q}{p}-(\frac{\alpha+2}{2})Nq}}}$
$\displaystyle\leq
C\sup_{k}\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{\frac{q}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{q-p}{p})}}$

Hence, for any $z\in\mathbb{D}$

$\displaystyle\dfrac{\widetilde{\Omega}_{Nq}(z)}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2\frac{q}{p}-(\frac{\alpha+2}{2})Nq}}$
$\displaystyle\leq C\|i\|^{Nq}_{A_{\sigma}^{Np}\rightarrow L^{Nq}(\Omega)}$
$\displaystyle\asymp
C\sup_{k}\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{\frac{q}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{q-p}{p})}}$

Statement $(a)$ shows that the operator
$i:A_{\sigma}^{\frac{tp}{Nq}}\rightarrow L^{\frac{t}{N}}(\Omega)$ is bounded.
Since $t>\frac{2q}{p(\alpha+2)}$, we have $\frac{tp(\alpha+2)}{q}>2$. The
above calculations show that

$\displaystyle\sup_{z\in\mathbb{D}}\dfrac{\widetilde{\Omega}_{t}(z)}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2\frac{q}{p}-(\frac{\alpha+2}{2})t}}$
$\displaystyle\leq C\|i\|^{Nq}_{A_{\sigma}^{Np}\rightarrow L^{Nq}(\Omega)}$
$\displaystyle\asymp
C\sup_{k}\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{\frac{q}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{q-p}{p})}}$

This completes the proof. ∎

######  Corollary 1.

Let $t>0$, $0<p<{\infty}$, $d\Omega=\sigma dA$ is a positive measure and
${C}_{\psi,\varphi}:A_{\sigma}^{p}\to L^{p}(\Omega)$ be a bounded operator.
Then the following statements are equivalent.

  1. (a)

${C}_{\psi,\varphi}$ is power bounded, that is,

$Q_{1}=\sup_{n\in\mathbb{N}}\|C^{n}_{\psi,\varphi}\|^{p}<\infty.$

  2. (b)

The sequence of functions $\\{f_{n}\\}_{n=1}^{\infty}$ defined on $\mathbb{D}$
as

$f_{n}(z)=\frac{\widetilde{{\Omega}}_{n,t}(z)}{\sigma(z)\delta^{2-({\alpha+2})t/{2}}(z)},\;z\in\mathbb{D}$

is a norm bounded family in $L^{\infty}(\mathbb{D})$ for $t>{2}/{(\alpha+2)}$,
that is,

$Q_{2}=\sup_{n\in\mathbb{N}}\|f_{n}\|_{\infty}={\rm sup}_{n\in\mathbb{N}}{\rm
ess\>sup}_{z\in\mathbb{D}}\frac{\widetilde{{\Omega}}_{n,t}(z)}{\sigma(z)\delta^{2-({\alpha+2})t/{2}}(z)}<\infty.$

  3. (c)

The sequence of functions $\\{g_{n}\\}_{n=1}^{\infty}$ defined on $\mathbb{D}$
as

$g_{n}(z)=\frac{\widehat{\Omega}_{n,R}(z)}{\sigma(z)},\;z\in\mathbb{D}$

is a norm bounded family in $L^{\infty}(\mathbb{D})$ for some (or any)
$R\in(0,1)$, that is,

$Q_{3}=\sup_{n\in\mathbb{N}}\|g_{n}\|_{\infty}={\rm sup}_{n\in\mathbb{N}}{\rm
ess\>sup}_{z\in\mathbb{D}}\frac{\widehat{\Omega}_{n,R}(z)}{\sigma(z)}<\infty.$

  4. (d)

The double sequence $\gamma_{nk}=\\{\gamma_{n,k}\\}_{n,k}$, where

$\gamma_{n,k}=\frac{\widehat{\Omega}_{n,r}(z_{k})}{\sigma(z_{k})}$

is bounded for some (any) $r$-lattice $\\{z_{k}\\}_{k=1}^{\infty}$ with fixed
$r$, $0<r<1$, that is,

$Q_{4}=\|\gamma_{nk}\|_{\Lambda^{2}_{\infty}}=\sup_{n,k\in\mathbb{N}}\frac{\widehat{\Omega}_{n,r}(z_{k})}{\sigma(z_{k})}<\infty.$

Moreover, $Q_{1}\asymp Q_{2}\asymp Q_{3}\asymp Q_{4}.$

######  Theorem 2.

Let $\Omega\geq 0$ and $0<p\leq q<{\infty}$. Then the following statement are
equivalent:

  1. (a)

$\Omega$ is a vanishing $(p,q,\sigma)$-Bergman Carleson measure.

  2. (b)

For $t>\frac{2q}{p(\alpha+2)}$, we have

$\dfrac{\widetilde{\Omega}_{t}(z)}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2\frac{q}{p}-(\frac{\alpha+2}{2})t}}\longrightarrow
0$

as $z\longrightarrow\partial\mathbb{D}$.

  3. (c)

For some $($or any$)$ $R\in(0,1)$, we have

$\dfrac{\widehat{\Omega}_{R}(z)}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2(\frac{q-p}{p})}}\longrightarrow
0$

as $z\longrightarrow\partial\mathbb{D}$.

  4. (d)

For some (or any) $r$-lattice $\\{z_{k}\\}_{k=1}^{\infty}$ with $r\in(0,1)$,
we have

$\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{\frac{q}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{q-p}{p})}}\longrightarrow
0$

as $k\longrightarrow 0$.

###### Proof.

The implication $(c)\Rightarrow(d)$ is trivial because
$z_{k}\rightarrow\partial\mathbb{D}$ as $k\rightarrow{\infty}$ whenever
$\\{z_{k}\\}_{k=1}^{\infty}$ is an $r$-lattice. It follows from (3.6) that
$(b)\Rightarrow(c)$.
$(a)\Rightarrow(c)$ Given $0<R<1$. For $z\in\mathbb{D}$, we set
$f_{z}(\xi)=\dfrac{K_{z}^{\alpha}(\xi)}{\sigma^{\frac{1}{p}}(z)(1-|z|^{2})^{\frac{2}{p}-(\alpha+2)}}$,
$\xi\in\mathbb{D}$. One can easily find that $f_{z}\in A_{\sigma}^{p}$,
$\|f_{z}\|_{p,\sigma}\leq C$ and $f_{z}\rightarrow 0$ uniformly on any compact
subset of $\mathbb{D}$ as $z\rightarrow\partial\mathbb{D}$. Since $\Omega$ is
a vanishing $(p,q,\sigma)-$Bergman Carleson measure, it follows from (3.3)
that

$\dfrac{\widehat{\Omega}_{R}(z)}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2(\frac{q-p}{p})}}\leq
C\dfrac{(1-|z|^{2})^{q(\alpha+2-\frac{2}{p})}}{\sigma^{\frac{q}{p}}(z)}\int_{\mathbb{D}}|f_{z}(\xi)|^{q}d\Omega(\xi)\longrightarrow
0$

as $z\rightarrow\partial\mathbb{D}.$

$(d)\Rightarrow(a)$. Suppose $(d)$ holds. For any $\epsilon>0$, there exists a
positive integer $k_{0}$ such that
$\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{\frac{q}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{q-p}{p})}}<\epsilon$,
whenever $k>k_{0}$. Notice that $\cup_{k=1}^{k_{0}}E(z_{k},r)$ is relatively
compact in $\mathbb{D}$. Let us consider a bounded sequence
$\\{f_{j}\\}_{j=1}^{\infty}$ in $A_{\sigma}^{p}$ such that
$f_{j}\longrightarrow 0$ uniformly on any compact subset of $\mathbb{D}$ as
$j\rightarrow{\infty}$. Similar to the proof of (3.2) and if $j$ is large
enough, we have that

$\displaystyle\int_{\mathbb{D}}|f_{j}(z)|^{q}d\Omega(z)$
$\displaystyle\leq\int_{\cup_{k=1}^{k_{0}}E(z_{k},r)}|f_{j}(z)|^{q}d\Omega(z)+\sum_{k=k_{0}+1}^{\infty}\int_{E(z_{k},r)}|f_{j}(z)|^{q}d\Omega(z)$
$\displaystyle\leq C\epsilon\|f_{j}\|_{p,\sigma}^{q}$ $\displaystyle\leq
C\epsilon.$ (3.8)

where $C$ is independent of $\epsilon$.

$(a)\Rightarrow(b)$ The equivalence of $(a)$, $(c)$ and $(d)$ shows that the
measure $\Omega$ is a vanishing $(Np,Nq,\sigma)$-Bergman Carleson measure if
$\Omega$ is a vanishing $(p,q,\sigma)$-Bergman Carleson measure. For
$z\in\mathbb{D}$, set
$f_{z}(\xi)=\dfrac{K_{z}^{\alpha}(\xi)}{\sigma^{\frac{1}{p}}(z)(1-|z|^{2})^{\frac{2}{p}-(\alpha+2)}}$,
$\xi\in\mathbb{D}$. One can easily find that $f_{z}\in A_{\sigma}^{p}$,
$\|f_{z}\|_{p,\sigma}\leq C$ and $f_{z}\rightarrow 0$ uniformly on any compact
subset of $\mathbb{D}$ as $z\rightarrow\partial\mathbb{D}$. Since
$Np>\frac{4}{(\alpha+2)}$ and it follows from (3.7), we have that

$\dfrac{\widetilde{\Omega}_{Nq}(z)}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2\frac{q}{p}-(\frac{\alpha+2}{2})Nq}}\asymp\dfrac{1}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2\frac{q}{p}-(\frac{\alpha+2}{2})Nq}}\int_{\mathbb{D}}|f_{z}(\xi)|^{Nq}d{\Omega}(\xi).$

Statement $(a)$ yields that $\Omega$ is a vanishing
$(\frac{tp}{Nq},\frac{t}{N},\sigma)-$Bergman Carleson measure. Therefore

$\lim_{z\rightarrow\partial\mathbb{D}}\dfrac{\widetilde{\Omega}_{Nq}(z)}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2\frac{q}{p}-(\frac{\alpha+2}{2})Nq}}=0.$

The proof is completed. ∎

######  Corollary 2.

Let $t>0$, $0<p<{\infty}$, $d\Omega=\sigma dA$ is a positive measure and
${C}_{\psi,\varphi}:A_{\sigma}^{p}\to L^{p}(\Omega)$ be a bounded operator.
Then the following statements are equivalent.

  1. (a)

${C}_{\psi,\varphi}$ is power compact, that is, ${C}^{n}_{\psi,\varphi}$ is
compact, for some $n\in{\mathbb{N}}$.

  2. (b)

For $t>\frac{2q}{p(\alpha+2)}$, we have

$\dfrac{\widetilde{\Omega}_{n,t}(z)}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2\frac{q}{p}-(\frac{\alpha+2}{2})t}}\longrightarrow
0$

as $z\longrightarrow\partial\mathbb{D}$.

  3. (c)

For some $($or any$)$ $R\in(0,1)$, we have

$\dfrac{\widehat{\Omega}_{n,R}(z)}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2(\frac{q-p}{p})}}\longrightarrow
0$

as $z\longrightarrow\partial\mathbb{D}$.

  4. (d)

For some (or any) $r$-lattice $\\{z_{k}\\}_{k=1}^{\infty}$ with $r\in(0,1)$,
we have

$\dfrac{\widehat{\Omega}_{n,r}(z_{k})}{\sigma^{\frac{q}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{q-p}{p})}}\longrightarrow
0$

as $k\longrightarrow 0$.

######  Theorem 3.

Let $\Omega$ be a finite positive Borel measure and $0<q<p<{\infty}$. Then the
following statements are equivalent:

  1. (a)

$\Omega$ is a $(p,q,\sigma)-$Bergman Carleson measure.

  2. (b)

$\Omega$ is a vanishing $(p,q,\sigma)-$Bergman Carleson measure.

  3. (c)

For $t>\frac{2(q+p)}{p(\alpha+2)}$, we have

$\widetilde{M}_{t}(z)=\dfrac{\widetilde{\Omega}_{t}(z)}{\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{2\frac{q}{p}+1-(\frac{\alpha+2}{2})t}}\in
L^{\frac{p}{p-q}}(\mathbb{D}).$

  4. (d)

For some $($or any$)$ $R\in(0,1)$, we have

$\widehat{M}_{R}(z)=\dfrac{\widehat{\Omega}_{R}(z)}{\sigma^{\frac{q}{p}}(z)}\in
L^{\frac{p}{p-q}}(\mathbb{D}).$

  5. (e)

For some (or any) $r$-lattice $\\{z_{k}\\}_{k=1}^{p}$ with $r\in(0,1)$, we
have

$\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{\frac{q}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{q-p}{p})}}\in
l^{\frac{p}{p-q}}(\mathbb{D}).$

Moreover, we have

$\displaystyle\|i\|_{A_{\sigma}^{p}\rightarrow L^{p}(\Omega)}^{q}$
$\displaystyle\asymp\|\widetilde{M}_{t}\|_{\frac{p}{p-q}}$
$\displaystyle\asymp\|\widehat{M}_{R}\|_{\frac{p}{p-q}}$
$\displaystyle\asymp\bigg{\|}\bigg{\\{}\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{\frac{q}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{q-p}{p})}}\bigg{\\}}_{k=1}^{\infty}\bigg{\|}_{l^{\frac{p}{p-q}}}.$
(3.9)

###### Proof.

Since Lemma 8 implies that the statements $(c)$, $(d)$ and $(e)$ are
equivalent with the corresponding norm estimate (3.9) and the implication
$(b)\Rightarrow(a)$ is trivially true, it is sufficient to prove that
$(d)\Rightarrow(a)$, $(a)\Rightarrow(e)$ and $(a)\Rightarrow(b)$.
$(d)\Rightarrow(a)$. Since $0<q<p<{\infty}$ implies $\frac{p}{q}>1$ ,therefore
the conjugate exponent of $\frac{p}{q}$ is $\frac{p}{p-q}$. For $f\in
A_{\sigma}^{p}$, we have

$\displaystyle\int_{\mathbb{D}}|f(\xi)|^{q}d\Omega(\xi)$ $\displaystyle\leq
C\int_{\mathbb{D}}|f(\xi)|^{q}\widehat{\Omega}_{R}(\xi)dA(\xi)$
$\displaystyle\leq
C\bigg{\|}\widehat{M}_{R}\bigg{\|}_{\frac{p}{p-q}}\|f\|_{p,\sigma}^{q}.$

Above inequality follows from Lemma 3 and Holder’s inequality shows that
$\Omega$ is a $(p,q,\sigma)-$Bergman Carleson measure and
$\|i\|_{A_{\sigma}^{p}\rightarrow L^{p}(\Omega)}^{q}\leq
C\|\widehat{M}_{R}\|{\frac{p}{p-q}}$.
$(a)\Rightarrow(e)$. Let $\\{\lambda_{k}\\}_{k=1}^{\infty}\in l^{p}$ and set
$f$ as in Lemma 7. Statement $(a)$ and Lemma 7 implies that

$\int_{\mathbb{D}}\bigg{|}\sum_{k=1}^{\infty}\lambda_{k}\dfrac{K_{z_{k}}^{\alpha}(z)}{\sigma^{\frac{1}{p}}(z_{k})(1-|z_{k}|^{2})^{\frac{2}{p}-(\alpha+2)p}}\bigg{|}^{q}d\Omega(z)\leq
C\|i\|_{A_{\sigma}^{p}\rightarrow
L^{q}}^{q}\|\\{\lambda_{k}\\}_{k}\|_{l^{p}}^{q}.$ (3.10)

Recall that Rademacher functions $\psi_{k}$ are defined by

$\psi_{0}(t)=\left\\{\begin{array}[]{ll}\;1,&\text{ if }0\leq t-[t]<{1}/{2}\\\
-1,&\text{ if }{1}/{2}\leq t-[t]<1\end{array}\right.$

and $\psi_{k}(t)=\psi_{0}(2^{k}t)$ for $k=1,2,\cdots$, where $[t]$ denotes the
greatest integer less than or equal to $t$. For $0<q<\infty$, Khinchine’s
inequality is given as

$C_{1}\bigg{(}\displaystyle\sum_{k=1}^{m}|b_{k}|^{2}\bigg{)}^{{q}/{2}}\leq\int_{0}^{1}\bigg{|}\displaystyle\sum_{k=1}^{m}b_{k}\psi_{k}(t)\bigg{|}^{q}dt\leq
C_{2}\bigg{(}\sum_{k=1}^{m}|b_{k}|^{2}\bigg{)}^{{q}/{2}},$

which holds for all $m\geq 1$ and all complex numbers $b_{1},b_{2},\cdots
b_{m}$. Let $\psi_{k}(t)$ be the $k$th Rademacher function on $[0,1]$.
Replacing $\lambda_{k}$ with $\psi_{k}(t)\lambda_{k}$, integrating w.r.t $t$
from $0$ to $1$ and applying Khinchine’s inequality in [3.10], we see that

$\int_{\mathbb{D}}\bigg{(}\sum_{k=1}^{\infty}|\lambda_{k}|^{2}\dfrac{|K_{z_{k}}^{\alpha}(z)|^{2}}{\sigma^{\frac{2}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{2}{p}-(\alpha+2))}}\bigg{)}^{\frac{q}{2}}d\Omega(z)\leq
C\|i\|_{A_{\sigma}^{p}\rightarrow
L^{q}}^{q}\|\\{\lambda_{k}\\}_{k}\|_{l^{p}}^{q}.$

Thus, we have

$\displaystyle\sum_{k>k_{0}}|\lambda_{k}|^{q}\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{\frac{q}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{q-p}{p})}}$
$\displaystyle\asymp\sum_{k>k_{0}}\int_{E(z_{k},r)}|\lambda_{k}|^{q}|K_{z_{k}}^{\alpha}(z)|^{q}\sigma^{\frac{q}{p}}(z)(1-|z|^{2})^{(\alpha+2)q-\frac{2q}{p}}d\Omega(z)$
$\displaystyle\leq
C\sum_{k>k_{0}}\int_{E(z_{k},r)}\bigg{(}\sum_{k=1}^{\infty}|\lambda_{k}|^{2}\dfrac{|K_{z_{k}}^{\alpha}(z)|^{2}}{\sigma^{\frac{2}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{2}{p}-(\alpha+2))}}\bigg{)}^{\frac{q}{2}}d\Omega(z)$
$\displaystyle\leq
C\int_{\mathbb{D}}\bigg{(}\sum_{k=1}^{\infty}|\lambda_{k}|^{2}\dfrac{|K_{z_{k}}^{\alpha}(z)|^{2}}{\sigma^{\frac{2}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{2}{p}-(\alpha+2))}}\bigg{)}^{\frac{q}{2}}d\Omega(z)$
$\displaystyle\leq C\|i\|_{A_{\sigma}^{p}\rightarrow
L^{q}}^{q}\|\\{\lambda_{k}\\}_{k}\|_{l^{p}}^{q}.$ (3.11)

Since $\cup_{k=1}^{k_{0}}E(z_{k},R)$ is relatively compact in $\mathbb{D}$, so

$\sum_{k=0}^{\infty}|\lambda_{k}|^{q}\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{\frac{q}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{q-p}{p})}}\leq
C\|i\|_{A_{\sigma}^{p}\rightarrow
L^{q}}^{q}\|\\{\lambda_{k}\\}_{k}\|_{l^{p}}^{q}.$

Setting $c_{k}=|\lambda_{k}|^{q}$, for each $k$, then
$\\{c_{k}\\}_{k=1}^{\infty}\in l^{\frac{p}{q}}$ because
$\\{\lambda_{k}\\}_{k=1}^{\infty}\in l^{p}$ implies that

$\sum_{k=0}^{\infty}c_{k}\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{\frac{q}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{q-p}{p})}}\leq
C\|i\|_{A_{\sigma}^{p}\rightarrow
L^{q}}^{q}\|\\{c_{k}\\}\|_{l^{\frac{p}{q}}}.$

Hence by duality argument,we have

$\bigg{\|}\bigg{\\{}\dfrac{\widehat{\Omega}_{r}(z_{k})}{\sigma^{\frac{q}{p}}(z_{k})(1-|z_{k}|^{2})^{2(\frac{q-p}{p})}}\bigg{\\}}_{k=1}^{\infty}\bigg{\|}_{l^{\frac{p}{p-q}}}\leq
C\|i\|_{A_{\sigma}^{p}\rightarrow L^{p}(\Omega)}^{q}.$

Finally, we will prove the implication $(a)\Rightarrow(b)$. Let us consider a
bounded sequence $\\{f_{n}\\}_{n=1}^{\infty}$ in $A_{\sigma}^{p}$ such that
$f_{n}\longrightarrow 0$ uniformly on each compact subset of $\mathbb{D}$. Let
$F$ be any compact subset of $\mathbb{D}$ and $\Omega_{F}$ be the restriction
of $\Omega$ to $F$. Then we have

$\displaystyle\int_{\mathbb{D}}|f_{n}(z)|^{q}d\Omega(z)$
$\displaystyle=\int_{F}+\int_{\mathbb{D}\setminus F}|f_{n}(z)|^{q}d\Omega(z)$
$\displaystyle=I_{1}+I_{2}.$ (3.12)

Since $f_{n}\rightarrow 0$ uniformly on $F$ as $n\rightarrow\infty$, we have

$\displaystyle I_{1}$ $\displaystyle=\int_{F}|f_{n}(z)|^{q}d\Omega(z)$
$\displaystyle\leq C\sup_{z\in F}|f_{n}(z)|^{q}\longrightarrow 0.$

Fix $R\in(0,1)$ and $z\in\mathbb{D}$, and we obtain that

$\dfrac{\widehat{(\Omega_{F})}_{R}(z)}{\sigma^{\frac{q}{p}}(z)}\longrightarrow
0$

as $F$ extended to $\mathbb{D}$. By the equivalence of $(a)$ and $(d)$, we
have

$\dfrac{\widehat{(\Omega_{F})}_{R}^{\frac{p}{p-q}}(z)}{\sigma^{\frac{q}{p-q}}(z)}\leq\dfrac{\widehat{(\Omega_{F})}_{R}^{\frac{p}{p-q}}(z)}{\sigma^{\frac{q}{p-q}}(z)}\in
L^{1}(\mathbb{D}).$

Therefore,

$\displaystyle I_{2}$
$\displaystyle=\int_{\mathbb{D}}|f_{n}(z)|^{q}d\Omega_{F}(z)$
$\displaystyle\leq
C\sup_{n}\|f_{n}\|_{p,\sigma}^{q}\bigg{\|}\dfrac{\widehat{(\Omega_{F})}_{R}(z)}{\sigma^{\frac{q}{p}}(z)}\bigg{\|}_{\frac{p}{p-q}}$
$\displaystyle\longrightarrow 0\;\;\;\text{as}\;F\;\text{extended
to}\;\mathbb{D},$ (3.13)

above follows from (1.1) and the dominated convergence theorem. Hence

$\lim_{n\rightarrow\infty}\int_{\mathbb{D}}|f_{n}(z)|^{q}d\Omega(z)=0$

implies that $\Omega$ is a vanishing $(p,q,\sigma)-$Bergman Carleson measure.
This completes the proof. ∎

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# Towards Large-Scale and Spatio-temporally Resolved Diagnosis of
Electronic Density of States

Qiyu Zeng  Bo Chen  Xiaoxiang Yu  Shen Zhang  Dongdong Kang  Department of
Physics, National University of Defense Technology,Changsha, Hunan 410073, P.
R. China  Han Wang<EMAIL_ADDRESS>Laboratory of Computational Physics,
Institute of Applied Physics and Computational Mathematics, Beijing 100088, P.
R. China  Jiayu Dai<EMAIL_ADDRESS>Department of Physics, National
University of Defense Technology,Changsha, Hunan 410073, P. R. China

(August 30, 2024)

###### Abstract

Modern laboratory techniques like ultrafast laser excitation and shock
compression can bring matter into highly nonequilibrium states with complex
structural transformation, metallization and dissociation dynamics. To
understand and model the dramatic change of both electronic structures and ion
dynamics during such dynamic processes, the traditional method faces
difficulties. Here, we demonstrate the ability of deep neural network (DNN) to
capture the atomic local-environment dependence of electronic density of
states (DOS) for both multicomponent system under exoplanet thermodynamic
condition and nonequilibrium system during super-heated melting process. Large
scale and time-resolved diagnosis of DOS can be efficiently achieved within
the accuracy of ab initio method. Moreover, the atomic contribution to DOS
given by DNN model accurately reveals the information of local neighborhood
for selected atom, thus can serve as robust order parameters to identify
different phases and intermediate local structures, strongly highlights the
efficacy of this DNN model in studying dynamic processes.

Introduction. Modern laboratory techniques like ultrafast laser excitation and
dynamic compression can bring matter into highly nonequilibrium states with
complex structural transformation, metallization and dissociation dynamics[1,
2, 3, 4]. By increasing temperatures and pressures in a really short time
scale ($\sim 10^{-12}\ {\rm s}$), the behavior of both electrons and ions
presents time-dependent characteristics. Especially for the time-resolved
diagnosis of electronic structures, it strongly suffers from the complication
of phase transition associated with dramatic changes of thermodynamic state.
The demanding requirements for accurate description of electronic structure
during dynamic processes raise challenges for all existing methods, and serve
great importance for different communities like condensed matter physics,
laboratory astrophysics, planetary physics, material science and industrial
applications [5, 3, 6].

The classical molecular dynamics simulation (CMD), by integrating Newton’s
equations of motion for the many-body system, inherently describes the mass
transport, energy transport and phase transition dynamics of ions with a large
enough simulation size. However, the lack of explicit treatment of electrons
prevents such a method from describing the evolution of electronic structure
at the same time. While for ab initio method based on quantum mechanics like
density functional theory (DFT), it is too computational expensive to give the
electronic response associated with long-time phase transition dynamics
($10^{0}\sim 10^{3}\ {\rm ps}$) and a large enough system ($>10^{3}$ atoms)
that involve coexistence of different phases, gradients of temperatures and
densities.

In past decades, there has been a burst of attempts to exploit machine-
learning (ML) method to address challenges in the domain of molecular
modeling. By directly approximating the solutions of the Kohn-Sham
differential equations, ML models can efficiently yield interatomic potential
energy surface [7, 8] or electronic properties [9, 10, 11] with high fidelity.
Most recently, ongoing efforts have demonstrated the usefulness of ML method
on the electronic density [9], electronic density of states (DOS) [10] and the
local density of states (LDOS) [11]. Especially for the DOS, it serves as an
important quantity for identification of metallization transition and
determination of the electronic heat capacity, electron-phonon coupling and
optical/X-ray absorption spectra [12, 13, 14, 15]. Since the existing attempts
have been restricted to equilibrium system under ambient conditions, the
ability of ML model to study temporal and spatial evolution of electronic
properties during dynamic processes, has not been validated yet.

Figure 1: Workflow of DeepDOS scheme. (a) Generating the local environmental
matrices $\tilde{\mathcal{R}}^{i}$ from atomic configurations
$\\{\mathbf{r}_{i}\\}$ to preserve the translation invariance. (b) Mapping
smoothed relative distances $s(r_{ji})$ into $M$-dimension feature space to
obtain local embedding matrices $\mathcal{G}^{i}$ via embedding net, where
$s(...)$ the smooth function [16]. (c) Generating descriptor $\mathcal{D}^{i}$
by multiplication of local embedding matrices and local environmental matrices
to preserve the permutation and rotation invariance. (d) Mapping the
descriptor of atom $i$ into the atomic DOS $g^{i}(\epsilon)$ via fitting net.

Here, we utilize the deep neural network (DNN) to capture the atomic local
environment dependence of DOS, named DeepDOS model. The accuracy and
robustness of DNN model is verified with magnesium oxide (MgO) system along
the isotherm of 5000 kelvin, pressures ranging from 236 GPa to 598 GPa.
Trained with small-system datasets, the DNN model is able to reproduce the DOS
curves with high-fidelity for the unseen configurations generated by thermal
fluctuation and density variation. Further, this size-extensive model is
applied to efficiently study the temporal evolution of DOS curves during
dynamics processes within the accuracy of ab initio method. The isobaric
melting process of silicon crystalline at pressure of 5 GPa is chosen. With a
large enough system, the coexistence of solid and liquid phase can be
incorporated and the corresponding DOS is given. Moreover, the atomic
contribution to DOS accurately reveals the local structure for any selected
atoms in this non-equilibrium system, and the DNN-predicted site-projected DOS
can be used as fingerprint to characterize different local structures. This
DeepDOS approach provides an efficient tool to study both the electronic
structure and ionic local structure during dynamic processes created by shock
impact, ultrafast laser heating etc.

Method.  The machine-learning scheme for DOS prediction based on the two key
approximation [10]: (1) the DOS for a configuration can be decomposed into
“atomic contributions” so as to preserve the extensive property. (2) each
“atomic contribution” depends on the local environment of the atom,
specifically, the relative coordinates of its neighbors within a pre-defined
cut-off radius $r_{cut}$. The DNN-DOS model can be defined as

$\displaystyle
g(\epsilon)=\sum_{i}\mathcal{N}_{\alpha_{i}}(\mathcal{D}_{\alpha_{i}}(r_{i},\\{r_{j}\\}_{j\in
n(i)}))$ (1)

where $\mathcal{N}_{\alpha_{i}}$ denotes the neural network for specified
chemical species $\alpha_{i}$ of atom $i$, and the descriptor
$\mathcal{D}_{\alpha_{i}}$ describes the local environment of atom $i$ with
its neighbor list $n(i)=\\{j|r_{ji}<r_{cut}\\}$. Here we adopted the symmetry-
preserving scheme introduced by DeepPot-SE model [16, 17, 8] to generate the
descriptor.

As illustrated in the Fig.1, the sub-network consists of an embedding and a
fitting neural network. The embedding network is specially designed to map the
local environment $\tilde{\mathcal{R}^{i}}$ to an embedded feature space,
which preserves the translational, rotational, and permutational symmetries of
the system. The fitting network is a fairly standard fully-connected
feedforward neural network, which maps the descriptor to the atomic
contribution to total DOS, denoted as “atomic DOS” (ADOS).

The full parameters $\mathbf{w}$ of the DNN are determined by minimizing a
loss function, and the loss function is defined as the root mean square error
(RMSE) of ADOS between model prediction and dataset.

$\displaystyle\mathcal{L}(\mathbf{w})=\sqrt{\frac{1}{N}\sum_{i}^{N}\int_{\epsilon}(g^{i}_{pred,\mathbf{w}}(\epsilon)-g^{i}(\epsilon))^{2}d\epsilon}$
(2)

Dataset. To generate the dataset, we perform the Deep Potential MD (DPMD)
simulations to generate trajectories for two different systems with the LAMMPS
package [18], and label the configurations along the trajectories.

  * •

Magnesium oxide (MgO, B1 phase) under exoplanet conditions at temperature of
5000 kelvin, pressures ranging from 236 GPa to 598 GPa.

  * •

Heat-until-melt simulations for silicon (cubic diamond phase) under isobaric
conditions at pressure of 5 GPa.

The DP models used in the DPMD simulations for MgO and Si are generated with
DeePMD-kit packages [17]. To cover a wide range of thermodynamics conditions
and minimize the computational consumption, a concurrent learning scheme, Deep
Potential Generator (DP-GEN) [19], has been adopted to sample the most compact
and adequate data set that guarantees the uniform accuracy of DP in the
explored configuration space. More details can be found in the supplemental
material (SM) [20]. In DPMD simulations, 40-ps-long molecular dynamics
trajectories are generated with a timestep of 1.0 fs, and the snapshots are
collected with an interval of 0.2 ps. To obtain the corresponding site-
projected DOS along the trajectories, the QUANTUM ESPRESSO package [21] is
used. The Perdew-Burke-Erzerhof (PBE) exchange correlation functional is used
[22], and the pseudopotential takes the projector augmented-wave (PAW)
formalism [23, 24]. The sampling of Brillouin zone is chosen as 0.125
${\rm\mathring{A}^{-1}}$. In the calculation of DOS, the Gaussian smearing is
used and the width is chosen as 0.02 Ry for MgO and 0.01 Ry for silicon. To
train a DeepDOS model, the embedding network is composed of three layers (25,
50, and 100 nodes) while the fitting network has three hidden layers with 240
nodes in each layer. The radius cutoff $r_{c}$ is chosen to be $6.0\
{\rm\mathring{A}}$.

Figure 2: (a)(b)(c) Comparison of DOS/ADOS between KS-DFT calculation and DNN
prediction at $p=298\ {\rm GPa}$, $T=5000\ {\rm K}$. (d)(e) Parity plot of
ADOS/width of band gap between DFT and DNN prediction in the validation set,
where five thermodynamic conditions are included. In (d), the marker color
indicates the frequency. In (e), the blue and gray dots represent the
predictions of band gap by well-trained DNN model (training steps of $2\times
10^{6}$) and under-trained DNN model ($5\times 10^{4}$) respectively.

Capturing thermal fluctuation and density variation. For a finite-temperature
system, the ionic thermal motion plays an important role that drives thermal
fluctuations both in the atomic local environment and electronic structure.
Along the isotherm of MgO system at 5000 kelvin, ionic thermal motion can
introduce significant fluctuations in the $3p$ band of Mg atoms, $2s$ and $2p$
band of oxygen atoms, indicating that both Mg-ADOS and O-ADOS are sensitive to
atomic local environment even at exoplanet pressures (see Fig.S1 and Fig.S2 in
SM). The standard deviation of ADOS is $\sigma_{1}=0.136\ {\rm
eV^{-1}},\sigma_{2}=0.262\ {\rm eV^{-1}}$ for Mg atom and O atom respectively.
Such finite temperature effect can also lead to a large deviation in
determining the width of band gap, varying from $\sim 6\ {\rm eV}$ to $\sim 9\
{\rm eV}$.

Here, we first test the performance of DNN model on DOS prediction for this
multicomponent system with thermal fluctuated atomic configurations as input.
To train our DNN model, we used a dataset covering five different exoplanet
pressures from 237 GPa to 599 GPa, with each case containing 400 magnesium
site-projected DOS and 400 oxygen site-projected DOS. This dataset is randomly
divided into training set and validation set by ratio of 4:1. As presented in
the Fig.2(a)(b)(c), a well-trained DNN model can accurately reproduce the KS-
DFT results of both the total DOS and site-projected DOS for an unseen atomic
configuration. Fig.2(d) highlights the consistency between DNN model and ab
initio method in predicting the atomic contribution to the number of
electronic states $g^{i}(\epsilon)$ at specific energy interval
$\epsilon+d\epsilon$. The RMSE of Mg-ADOS and O-ADOS prediction are
$0.115\sigma_{1}$ and $0.059\sigma_{2}$ respectively, significantly lower than
the standard deviation of dataset.

Figure 3: (a) Radar plot of the test statistics chosen to measure the band gap
prediction accuracy for validation set and test set. (b) DNN-Predicted band
gaps versus KS-DFT band gaps on test set. Black dashed line is for visual
reference, showing what the exact band gap prediction corresponds to. (c)
Configuration-averaged band gaps at different pressures along the isotherm of
5000 kelvin, KS-DFT calculations (grey), DNN predictions on validation set
(blue) and on test set (red) are presented.

Moreover, the width of band gap is extracted from the predicted DOS curve,
defined as the interval between valence band maximum (VBM) and conduction band
minimum (CBM). As presented in the Fig.2(e), the alignment of the band gap
predictions along the diagonal exhibits small standard deviations for thermal
fluctuated atomic configurations. The RMSE on the validation set is $0.17\
{\rm eV}$, the mean absolute error (MAE) is $0.11\ {\rm eV}$ and the mean
absolute percentage error (MAPE) is $1.3\%$. The model performance on
validation set indicates that the DeepDOS model has the ability to capture the
changes of atomic contribution to electronic structure induced by thermal
fluctuation of local environment, not only the shape of DOS curve, but also
the detailed structure like the position of VBM and CBM.

For comparison, prediction given by an under-trained model (trained with
$5\times 10^{4}$ steps) is presented in Fig.2(e). Interestingly, for a given
thermodynamic condition, the under-trained model failed to describe the subtle
difference of electronic states between thermal-fluctuated atomic
configurations, leading to several discrete band gap prediction like a
classification model for different thermodynamic conditions.

Further, the variation of density is considered. Since the inhomogeneous
density distribution and temperature distribution can be created during
dynamic processes, the ability for DNN to predict the unseen configurations
for different densities not included in the training set, should be checked.

Figure 4: (a)(b)(c) Temporal evolution of temperature, pressure and density
during isobaric heating of cubic diamond silicon at 5 GPa (containing 64 atoms
atoms), the orange line indicates the onset of melting. (d) Electronic density
of states prediction for the unseen snapshots during isobaric melting process,
the solid (dashed) line indicates result from KS-DFT calculation (DNN-
prediction). The inset shows the snapshots of the atomic configurations
colored by ext-CNA method [25].

Four interpolated density points within the pressure range of training data
are chosen to generate the dataset, denoted as the test set. As presented in
the Fig.3(a), the performance of DeepDOS model in both validation set and test
set show similar accuracy. In test set, the RMSE of band gap prediction is
$0.23\ {\rm eV}$, the MAE is $0.18\ {\rm eV}$ and the MAPE is $2.0\%$. The
DNN-predicted band gaps in Fig.3(b) are centered around the correct values,
thus providing a low bias estimation. The predicted band gaps vs the
calculated values from KS-DFT can be fitted to a linear function $y=ax+b$ with
the slope $a=0.99$ and bias $b=0.06\ {\rm eV}$. The configuration-averaged
band gap at different pressures are presented in Fig.3(c), the DNN exhibits a
good generalization performance along the isotherm with pressure-sampling
interval of 50 GPa, thus can provide a robust prediction in the study of
dynamic processes.

Large-Scale and Time-Resolved DOS Prediction. By combining non-equilibrium
molecular dynamics with DeepDOS model, we investigate the temporal evolution
of DOS during the superheated melting process of silicon, which contains
solid-liquid coexistence, density gradients and thermal-fluctuated local
structure.

The dataset containing 1600 silicon site-projected DOS is prepared. The atomic
configurations are collected from the heat-until-melt simulation under
isobaric condition, where the cold lattice, thermal-distorted crystalline and
liquid phase are included. Different system size is considered, we chose 40
atomic configurations for 8-atom-system, 20 configurations for 16-atom-system
and 10 configurations for 32-atom-system. Moreover, 80 snapshots for 8-atom-
system equilibrated at T= 2000 kelvin, p = 5 GPa are used to improve the
description of disordered structure.

Figure 5: (a)(b)(c) Temporal evolution of temperature, pressure and density
during isobaric heating of cubic diamond silicon at 5 GPa (containing 4096
atoms atoms). (d) DOS prediction for the unseen snapshots of solid-liquid
coexistence phase during the process of nucleation and growth of liquid phase.

We validate our DeepDOS model by studying the melting process performed with
64 atom-system, where KS-DFT calculation is affordable. We use DPMD to
generate the trajectories for 64-atom system, and label the configurations
along the trajectories by KS-DFT to provide the test set. As shown in
Fig.4(a)(b)(c), by heating silicon system from 300 kelvin to 2000 kelvin at 5
GPa, the cubic diamond structure maintains until 38.80 ps. As the temperature
excesses the limit of thermal stability, the crystal collapses into disordered
structure immediately accompanied with dramatic change of volume.
Correspondingly, the abrupt changes of DOS curve induced by solid-liquid
transition can be observed from Fig.4(d), where DNN predictions show good
agreement with KS-DFT calculations for these unseen snapshots of 64-atom-
system. Trained with small-system datasets, the DNN model is size extensive to
reproduce the DOS curves with high-fidelity for large system.

Although the superheated melting is known to be proceeded by nucleation and
growth of liquid regions inside the crystal. Due to limited simulation size,
tens of or several hundred atoms are not enough to form the liquid cluster
according to classic nucleation theory[26, 27]. For removal of such size
effect, simulations is performed with 4096-atom-system, and the DOS evolution
show totally different change compared with 64-atom results.

As presented in the insets of Fig.5(d), when the temperature quickly rises
above the normal melting point, melting occurs when the superheated crystal
generates a sufficiently large number of destabilized particles [28].
Appearance and growth of small liquid regions inside the crystal proceeds the
melting process. From 35.00 ps to 37.80 ps, the system is coexisted with solid
and liquid phase, and well-defined crystal-liquid interfaces can be observed
in a snapshot at 36.40 ps, identified by an extension of the common neighbor
analysis method (ext-CNA) [25]. The behavior of electronic structure thus
exhibits a hybrid characteristics of both phases as shown in Fig.5(d). During
the coalescence of the liquid regions, we can observe a gradual change of DOS
from solid-like semi-metal phase to liquid-like metal phase within $\sim 2.8$
ps, different from the abrupt change in 64-atom system results. We note that
these results can hardly be achieved by traditional ab initio method, and the
DeepDOS model provides an opportunity to study the electronic structure for
non-equilibrium processes, where both long-time ion dynamics and large system
size are needed.

Figure 6: (a)(b) PCA scatterplot of 81 920 ADOS from 20 snapshots of
4096-atom-system during melting process, where the marker color indicates the
frequency and ADOS-derived order parameter $Q$ respectively. The cyan line in
(a) indicates the trajectories of averaged ADOS in PCA space during melting
process. In (b), a trajectory of Si-3182 atom in ADOS-PCA space is highlighted
with black line.

Characterizing the atomic contribution to total DOS. Based on the
parameterized high-dimensional mapping between local environment and atomic
contribution to total DOS, here we further investigate the temporal evolution
of ADOS during melting process. By applying principal component analysis
(PCA), 81 920 ADOS extracted from 20 snapshots of 4096-atom-system during
melting processes (from 34.0 ps to 40.0 ps) are projected onto two-dimensional
principal component space.

As shown in Fig.6(a), due to different characteristics in the ADOS curves, the
distribution of ADOS are separated into two clusters corresponding to the
diamond phase and liquid phase, while the transitional region between two
phases represents the intermediate local structure like strongly-distorted
crystalline generated during melting processes. The electronic feature of
these intermediate structures exhibits a combination of liquid and solid
phase, as reported in previous section.

Choosing three electronic states from local minimum in DOS at Fermi level
$E_{F}$, $\epsilon_{1}\sim E_{F}-8\ {\rm eV}$ and $\epsilon_{2}\sim E_{F}-5\
{\rm eV}$, a characteristic parameter
$Q=g(E_{F})+g(\epsilon_{1})+g(\epsilon_{2})$ for ADOS curves is given to
describe the degree of disordered local environment for a selected atom. As
presented in Fig. 6(b), the characteristic parameter gradually increases from
$Q_{min}\sim 0.19$ to $Q_{max}\sim 1.46$ as the representative point in PCA
subspace flows from diamond-region into liquid-region, indicating that such
ADOS-derived quantity can serve as a robust order parameter to identify the
liquid phase, diamond phase and intermediate structure.

Figure 7: (a) Snapshots of atomic configurations, colored by $Q$. (b)(c)
Temporal evolution of ADOS in PCA subspace colored by ADOS-derived order
parameter $Q$ and ext-CNA method respectively.

The temporal evolution of ADOS during melting process is presented in the
Fig.7(b). Before the onset of melting at 32.0 ps, the representative points in
PCA subspace concentrates to form the diamond-phase island. As melting occurs,
with the coalescence of liquid region and the movement of liquid-solid
interface, these representative points gradually flow away from the diamond
region, and finally settle down in the liquid-phase region, show good
consistency with the snapshots of atomic configurations in the Fig.7(a). We
note that a significant amount of representative points locate in the
transitional zone from 34.0 ps to 36.8 ps, with characteristic parameter $Q$
ranging from $\sim 0.6$ to $\sim 0.8$, indicating that the DNN-predicted ADOS
and ADOS-derived order parameter $Q$ accurately resolves the complex local
environment for this nonequilibrium system that contains diamond crystalline,
liquid and thermal-distorted intermediate structure.

For comparison, the traditional structure-identification method like ext-CNA
method is used to classify the local structure. Based on the geometric feature
of atomic local neighborhoods for selected atoms, the ext-CNA method is
sensitive to the structural noise driven by ionic thermal motion and phase
transition dynamics, thus failed to classify all these local structures
correctly. As presented in the Fig.7(c), the distributions of pre-defined
reference structures in the PCA subspace all overlap one another. These
results indicate that the local electronic characteristics can achieve more
accurate and robust performance in identifying different phases compared with
geometrical characteristics like coordination number and bond-orientational
order parameter.

Summary. For nonequilibrium system created by dynamic processes, both the
electronic structure and the atomic local neighborhoods can be strongly
complicated by ionic thermal motion, density fluctuations, inhomogeneity of
density distribution, coexistence and interface of different phases. Here we
demonstrate that our developed DeepDOS approach can not only serve as an
efficient tool to study the temporal-spatial evolution of DOS during dynamic
processes within the accuracy of ab initio method, but also provide a
possibility to distinguish different phases and identify hidden patterns
through local electronic features.

Specifically, the accuracy and robustness of the DeepDOS model are validated
by reproducing DOS curves and band gaps for unseen atomic configurations
generated by thermal fluctuation and density variations. Trained with small-
system dataset, this size extensive model is able to give DOS curves for a
large system. Therefore, time-resolved diagnosis of DOS under isobaric heating
condition can be efficiently achieved by this machine-learning method, and we
present the temporal evolution of DOS curves during melting processes that
proceed by nucleation and growth of liquid region out of the solid matrix.
Moreover, the electronic feature for local structures can also be accurately
captured. As compared with geometric characteristics like ext-CNA method, the
ADOS can give relatively robust identification for local structures that
significantly deviates from ideal crystal symmetry due to thermal fluctuation
and phase transition dynamics.

In conclusion, this DeepDOS model can strongly enhance the further study of
both electronic structure and ion dynamics during non-equilibrium processes
created by ultrafast laser heating, shock loading or a quasi-isentropic ramped
pressure drive.

##  I Acknowledgments

This work was supported by the National Key R&D Program of China under Grant
No. 2017YFA0403200, the National Natural Science Foundation of China under
Grant No. 11774429, 11874424, 11904401, 12047561, 12104507, the NSAF under
Grant No. U1830206, the Science and Technology Innovation Program of Hunan
Province under Grant No. 2020RC2038, 2021RC4026.

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