Carleson measures and Berezin-type operators on Fock spaces

Abstract

We characterize (vanishing) Fock–Carleson measures by products of functions in Fock spaces. We also study the boundedness of Berezin-type operators from a weighted Fock space to a Lebesgue space. Due to the special properties of Fock–Carleson measures, the boundedness of Berezin-type operators on Fock spaces is different from the corresponding results on Bergman spaces.

1 Introduction

Carleson measures are important research objects and play powerful role in function spaces and the theory of operators. The concept of it was first introduced by Carleson in [5, 6], where it is used to study interpolating sequences and the corona problem of all bounded analytic functions on the unit disc. After that, there are a large amount of works on characterizations of Carleson measures in or between spaces, including Hardy spaces and Bergman spaces, on various domains, such as Hardy–Carleson measures on the unit disc [9, 22, 31] and on the unit ball [11, 24, 27], Bergman–Carleson measures on the unit ball [19,20,21, 23] and on the polydisc [10] and so on. Then it has been further investigated with the development of the theory of Toeplitz operators, such as harmonic Bergman–Carleson measures on the upper half space [7], holomorphic Bergman-Carleson measures on the strongly pseudoconvex domain [1,2,3,4, 15] and on the Siegel upper half space [17, 30], Besov–Carleson measures on the unit ball [16, 26, 29], and Fock–Carleson measures in $\mathbb {C}^n$ [12,13,14, 33].

In 2010, Zhao [34] gave a characterization of Carleson measures using products of functions in Hardy spaces on the unit disc. In 2015, Pau and Zhao [25] overcame the lack of Riesz factorization theorem on weighted Bergman spaces and gave a corresponding result on the unit ball. Recently, Abate, Mongodi, and Raissy [1, 4] extended Pau and Zhao’s results to strongly pseudoconvex domains and gave a characterization of skew Carleson measures through products of functions in weighted Bergman spaces. They also obtained the boundedness and compactness of a larger class of Toeplitz operators from one weighted Bergman space to another. Wang and Zhou [32] investigated tent Carleson measures in terms of products of functions in Hardy-type tent spaces and also obtained the boundedness and compactness of Toeplitz operators between distinct Hardy-type tent spaces. Motivated by these results, we aim to give some characterizations of Fock–Carleson measures involving products of functions in Fock spaces.

Let $\mathbb {C}^n$ be the n-dimensional complex Euclidean space. For $0<p<\infty$ and $0<\alpha <\infty$, the function space $L_{\alpha }^{p}$ consists of all measurable functions f on $\mathbb {C}^n$ for which

$$\begin{aligned} \Vert f\Vert _{p,\alpha }=\left( \int _{\mathbb {C}^n}\left| f(z) e^{-\frac{\alpha }{2}|z|^{2}}\right| ^{p} \textrm{d}v(z)\right) ^{\frac{1}{p}}<\infty , \end{aligned}$$

where $\textrm{d}v$ denotes the Lebesgue volume measure. When $\alpha =0$, we write $L_{0}^p=L^p$. Let $H\left( \mathbb {C}^n\right)$ be the family of all holomorphic functions on $\mathbb {C}^n$. Fock space $F_\alpha ^p$ is defined by

$$\begin{aligned} F_{\alpha }^{p}:=L_{\alpha }^{p} \cap H\left( \mathbb {C}^n\right) . \end{aligned}$$

It is clear that $F_{\alpha }^{p}$ is a Banach space under the norm $\Vert \cdot \Vert _{p,\alpha }$ for $1\le p<\infty$ and is an Fréchet space under $d(f,g)=\Vert f-g\Vert _{p,\alpha }^p$ for $0<p < 1$.

For $0< p, q<\infty$, a positive Borel measure $\mu$ on $\mathbb {C}^n$ is called a (p, q)-Fock–Carleson measure if there is a constant $C>0$ such that

$$\begin{aligned} \int _{\mathbb {C}^n}\left| f(z) e^{-\frac{\alpha }{2}|z|^2}\right| ^q {\text {d}}\mu (z) \le C \Vert f\Vert _{p,\alpha }^q \end{aligned}$$

for any $f \in F_{\alpha }^{p}$. We say $\mu$ is a vanishing (p, q)-Fock–Carleson measure if

$$\begin{aligned} \lim _{k \rightarrow \infty } \int _{\mathbb {C}^n}\left| f_k(z) e^{-\frac{\alpha }{2}|z|^2}\right| ^q {\text {d}} \mu (z)=0 \end{aligned}$$

whenever $\left\{ f_k\right\}$ is a bounded sequence in $F_{\alpha }^p$ that converges to 0 uniformly on any compact subset of $\mathbb {C}^n$ as $k \rightarrow \infty$. Since (p, q)-Fock–Carleson measures only depend on the ratio $\lambda :=q/p$, we simply say (vanishing) $(\lambda ,\alpha )$-Fock–Carleson measures for (vanishing) (p, q)-Fock–Carleson measures. We refer the reader to [12] for details. Our main results are the following three theorems.

Theorem 1.1

Let $\mu$ be a positive Borel measure on $\mathbb {C}^n$, $0<p_i,q_i<\infty$ and $0<\alpha _i<\infty$, where $i=1, 2, \ldots , k$ and $k\ge 1$. Set

$$\begin{aligned} \lambda = \sum \limits _{i=1}^{k} \frac{q_i}{p_i}, \quad \gamma = \sum \limits _{i=1}^{k} \alpha _i. \end{aligned}$$

(1.1)

Then $\mu$ is a ($\lambda ,\gamma$)-Fock–Carleson measure if and only if there is a constant $C>0$ such that for any $f_i \in F_{\alpha _i}^{p_i}$,

$$\begin{aligned} \int _{\mathbb {C}^n} \prod _{i=1}^{k}\left| f_{i}(z)\right| ^{q_i} e^{- \frac{\alpha _i}{2} |z|^2 q_i} {\text {d}} \mu (z) \le C \prod _{i=1}^{k}\left\| f_{i}\right\| _{p_i,\alpha _i}^{q_i}. \end{aligned}$$

(1.2)

Theorem 1.2

Let $\mu$ be a positive Borel measure on $\mathbb {C}^n$, $0<p_i,q_i<\infty$ and $0<\alpha _i<\infty$, where $i=1, 2, \ldots , k$ and $k\ge 1$. Let

$$\begin{aligned} \lambda = \sum \limits _{i=1}^{k} \frac{q_i}{p_i}, \quad \gamma = \sum \limits _{i=1}^{k} \alpha _i. \end{aligned}$$

Then the following statements are equivalent:

• (i) $\mu$ is a vanishing $(\lambda , \gamma )$-Fock–Carleson measure;

• (ii) For any sequence $\left\{ f_{1, l}\right\}$ in $F_{\alpha _1}^{p_1}$ that is convergent to 0 uniformly on any compact subset of $\mathbb {C}^n$,

  $$\begin{aligned} \lim _{l \rightarrow \infty } F(l)=0, \end{aligned}$$

  where

  $$\begin{aligned} \begin{aligned} F(l)= \sup _{\begin{array}{c} \left\| f_i\right\| _{p_i, \alpha _i} \le 1\\ i=2, \ldots , k \end{array}} \left\{ \int _{\mathbb {C}^n}\left| f_{1, l}(z)e^{-\frac{\alpha _1}{2}|z|^2}\right| ^{q_1} \prod _{i=2}^k\left| f_i(z)e^{-\frac{\alpha _i}{2}|z|^2}\right| ^{q_i} \text {d} \mu (z)\right\} ; \end{aligned} \end{aligned}$$

• (iii) For any sequences $\left\{ f_{1, l}\right\} ,\left\{ f_{2, l}\right\} , \ldots ,\left\{ f_{k, l}\right\}$ in $F_{\alpha _1}^{p_1}, F_{\alpha _2}^{p_2}$, $\ldots , F_{\alpha _k}^{p_k}$, respectively, that are all convergent to 0 uniformly on any compact subset of $\mathbb {C}^n$,

  $$\begin{aligned} \lim _{l \rightarrow \infty } \int _{\mathbb {C}^n}\prod _{i=1}^k\left| f_{i, l}(z)e^{-\frac{\alpha _i}{2}|z|^2}\right| ^{q_i} {\text {d}} \mu (z)=0. \end{aligned}$$

The proof of Theorem 1.1, especially the necessity for the case of $0<\lambda <1$, is related to the boundedness of a class of integral operators defined by

$$\begin{aligned} S_{\mu }^{r, t} f(z) = \int _{\mathbb {C}^n} e^{-\frac{\alpha _2 t}{2} |z-w|^2 } |f(w)|^r e^{- \frac{\alpha _1}{2} |w|^2 r}{\text {d}}\mu (w) \end{aligned}$$

for $0<r, t<\infty$. When $r=0$, it is that

$$\begin{aligned} S_{\mu }^{0, t}(z):= \widetilde{\mu }_{t}(z)=\int _{\mathbb {C}^{n}} e^{-\frac{\alpha t}{2}|z-w|^{2}} {\text {d}} \mu (w). \end{aligned}$$

(1.3)

It is just the t-Berezin transform of $\mu$ defined in [12]. And we see (2.1) in detail. Therefore, we call $S_{\mu }^{r, t}$ a Berezin-type operator of $\mu$ on Fock spaces.

Here and follows, we say that the Berezin-type operator $S_\mu ^{r, t}$ is bounded from a weighted Fock space $F_{\alpha _1}^{p_1}$ to a Lebesgue space $L^{p_2}$ if there is a constant $C>0$ such that $\Vert S_{\mu }^{r, t} f\Vert _{{p_2}} \le C\Vert f\Vert _{p_1,\alpha _1}^r$ for any $f \in F_{\alpha _1}^{p_1}$.

Theorem 1.3

Let $\mu$ be a positive Borel measure on $\mathbb {C}^{n}$, $0< p_1, p_2<\infty$ and $0<\alpha _1, \alpha _2<\infty$. Let

$$\begin{aligned} \lambda =1+\frac{r}{p_1}-\frac{1}{p_2}, \quad \gamma =\alpha _1 + \alpha _2. \end{aligned}$$

Suppose $\lambda >0$. If $S_\mu ^{r, t}$ is bounded from $F_{\alpha _1}^{p_1}$ to $L^{p_2}$, then $\mu$ is a $(\lambda , \gamma )$-Fock–Carleson measure.

A similar class of integral operators on the unit ball of $\mathbb {C}^n$ was studied in [25, Lemma 3.3], and then its generalizations were studied in [18, 28]. Since the characterization of the Fock–Carleson measure is different from that of the Bergman–Carleson measure on the unit ball, we only give the necessity of the boundedness of $S_\mu ^{r, t}$ from $F_{\alpha _1}^{p_1}$ to $L^{p_2}$. This is enough for proving Theorem 1.1, but we do not know whether the sufficiency of it holds or not. We list it as an open problem.

OpenProblem

Let $\mu$ be a positive Borel measure on $\mathbb {C}^{n}$, $0< p_1, p_2<\infty$ and $0<\lambda , \gamma <\infty$. If $\mu$ is a $(\lambda , \gamma )$-Fock–Carleson measure, then how is the boundedness of $S_\mu ^{r, t}$ from $F_{\alpha _1}^{p_1}$ to $L^{p_2}$?

The paper is organized as follows. We collect some notations and preliminary results in Sect. 2. We are devoted to proving Theorem 1.3 in Sect. 3 and proving Theorems 1.1 and 1.2 in Sect. 4.

In what follows, the positive constant C may change from line to line but does not depend on the functions. The notation $A \lesssim B$ means that there is a constant C such that $A \le C B$, and $A \simeq B$ means that $A \lesssim B$ and $B \lesssim A$.

2 Preliminaries

We list some lemmas in this section. Given $r>0$ and $z \in \mathbb {C}^{n}$, the Euclidean ball centered at z with radius r is denoted by

$$\begin{aligned} B(z, r)=\left\{ w \in \mathbb {C}^{n}:|w-z|<r\right\} . \end{aligned}$$

Lemma 2.1

There exists a positive integer N such that for any $r>0$, we can find a sequence $\left\{ a_{k}\right\}$ in $\mathbb {C}^{n}$ with the following properties:

• (i) $\bigcup _{k=1}^{\infty } B\left( a_{k}, r\right) =\mathbb {C}^{n}$;

• (ii) $\left\{ B\left( a_{k}, \frac{r}{2}\right) \right\} _{k=1}^{\infty }$ are pairwise disjoint;

• (iii) Each point $z\in \mathbb {C}^{n}$ belongs to at most N of the sets $B\left( a_{k}, 2r\right)$.

The sequence $\left\{ a_{k}\right\}$ satisfying the conditions of Lemma 2.1 is called an r-lattice. We write $B_k = B(a_{k}, r)$, ${\widetilde{B}}_k = B(a_{k}, 2r)$ for convenience throughout the paper.

Let $\mu$ be a positive Borel measure on $\mathbb {C}^{n}$. Notice that v(B(z, r)) is a constant independent of z, the average function of $\mu$ is defined by

$$\begin{aligned} \widehat{\mu }_{r}(z):=\mu (B(z, r)) \end{aligned}$$

for $z\in \mathbb {C}^n$. Let $K_{\alpha }(z, w)=e^{\alpha \langle z, w\rangle }$ denote the reproducing kernel of the Fock space $F_{\alpha }^{2}$. For $t>0$, the t-Berezin transform of $\mu$ is defined by

$$\begin{aligned} \widetilde{\mu }_{t}(z)=~&\int _{\mathbb {C}^{n}}\left| \frac{K_{\alpha }(z, w)}{\sqrt{K_{\alpha }(z, z) K_{\alpha }(w, w)}}\right| ^{t} \textrm{d} \mu (w)\nonumber \\ =~&\int _{\mathbb {C}^{n}} e^{-\frac{\alpha t}{2}|z-w|^{2}} {\text {d}} \mu (w). \end{aligned}$$

(2.1)

Lemma 2.2

[12, Theorem 3.1] Let $\mu$ be a positive Borel measure on $\mathbb {C}^n$. Set $\lambda =q / p$ and $1 \le \lambda <\infty$. Then the following statements are equivalent:

• (i) $\mu$ is a $(\lambda , \alpha )$-Fock–Carleson measure;

• (ii) $\widetilde{\mu }_{t}$ is bounded on $\mathbb {C}^{n}$ for some (or any) $t>0$;

• (iii) $\mu (B(\cdot , \delta ))$ is bounded on $\mathbb {C}^{n}$ for some (or any) $\delta >0$;

• (iv) For some (or any) $r>0$, the sequence $\left\{ \mu \left( B\left( a_{k}, r\right) \right) \right\} _{k=1}^{\infty }$ is bounded.

Furthermore,

$$\begin{aligned} \Vert \mu \Vert \simeq \left\| \widetilde{\mu }_{t}\right\| _{\infty } \simeq \left\| \mu (B(\cdot , \delta ))\right\| _{\infty } \simeq \left\| \left\{ \mu \left( B\left( a_{k}, r\right) \right) \right\} \right\| _{\ell ^{\infty }}, \end{aligned}$$

where

$$\begin{aligned} \Vert \mu \Vert = \sup _{f \in F_{\alpha }^p,\Vert f\Vert _{p, \alpha } \le 1}\int _{\mathbb {C}^n}\left| f(z)e^{-\frac{\alpha }{2}|z|^2}\right| ^q {\text {d}}\mu (z). \end{aligned}$$

Lemma 2.3

[12, Theorem 3.2] Let $\mu$ be a positive Borel measure on $\mathbb {C}^n$. Set $\lambda =q / p$ and $1 \le \lambda <\infty$. Then the following statements are equivalent:

• (i) $\mu$ is a vanishing $(\lambda , \alpha )$-Fock–Carleson measure;

• (ii) $\widetilde{\mu }_t(z) \rightarrow 0$ as $z \rightarrow \infty$ for some (or any) $t>0$;

• (iii) $\mu (B(z, \delta )) \rightarrow 0$ as $z \rightarrow \infty$ for some (or any) $\delta >0$;

• (iv) $\mu \left( B\left( a_k, r\right) \right) \rightarrow 0$ as $k \rightarrow \infty$ for some (or any) $r>0$.

Lemma 2.4

[12, Theorem 3.3] Let $\mu$ be a positive Borel measure on $\mathbb {C}^n$. Set $\lambda =q / p$ and $0<\lambda <1$. Then the following statements are equivalent:

• (i) $\mu$ is a $(\lambda , \alpha )$-Fock–Carleson measure;

• (ii) $\mu$ is a vanishing $(\lambda , \alpha )$-Fock–Carleson measure;

• (iii) $\widetilde{\mu }_{t} \in L^{1/(1-\lambda )}$ for some (or any) $t>0$;

• (iv) $\mu (B(\cdot , \delta )) \in L^{1/(1-\lambda )}$ for some (or any) $\delta >0$;

• (v) $\sum _{k=1}^{\infty } \mu \left( B\left( a_{k}, r\right) \right) ^{1/(1-\lambda )}<\infty$ for some (or any) $r>0$.

Furthermore,

$$\begin{aligned} \Vert \mu \Vert \simeq \left\| \widetilde{\mu }_{t}\right\| _{1/(1-\lambda )} \simeq \Vert \mu (B(\cdot , \delta ))\Vert _{1/(1-\lambda )} \simeq \left\| \left\{ \mu \left( B\left( a_{k}, r\right) \right) \right\} \right\| _{\ell ^{1/(1-\lambda )}}. \end{aligned}$$

Lemma 2.5

[8, Lemma 3] Suppose $0<p<\infty$. Then

$$\begin{aligned} \Vert K_\alpha (\cdot , z)\Vert _{p, \alpha }\simeq e^{\frac{\alpha }{2} |z|^2}. \end{aligned}$$

(2.2)

Lemma 2.6

Let $\nu$ be a positive Borel measure on $\mathbb {C}^{n}$ and $0<p, t<\infty$. If $\{f_k\}$ is a sequence of Lebesgue measurable functions such that

$$\begin{aligned} \begin{aligned} \int _{\mathbb {C}^n}\bigg (\sum _{k=1}^\infty |f_k(z)|^t\bigg )^{p/t}{\text{ d }}\nu (z)<\infty , \end{aligned} \end{aligned}$$

then

$$\begin{aligned} \begin{aligned} \sum _{k=1}^\infty \int _{\widetilde{B}_k} |f_k(z)|^{p} {\text{ d }}\nu (z)\le \max \{1, N^{(t-p)/t}\} \int _{\mathbb {C}^n}\bigg (\sum _{k=1}^\infty |f_k(z)|^t\bigg )^{p/t}{\text{ d }}\nu (z). \end{aligned} \end{aligned}$$

Proof

The proof is divided into two cases. If $p/t\ge 1$, then $\ell ^t$ injects continuously into $\ell ^p$. Thus, we obtain

$$\begin{aligned} \begin{aligned} \int _{\mathbb {C}^n}\sum _{k=1}^\infty |f_k(z)|^p \chi _{\widetilde{B}_k}(z) {\text {d}}\nu (z) \le ~&\int _{\mathbb {C}^n}\left( \sum _{k=1}^\infty |f_k(z)|^t\chi _{\widetilde{B}_k}(z)\right) ^{p/t} {\text {d}}\nu (z)\\ \le ~&\int _{\mathbb {C}^n}\left( \sum _{k=1}^\infty |f_k(z)|^t\right) ^{p/t} {\text {d}}\nu (z). \end{aligned} \end{aligned}$$

If $0<p/t<1$, then $t/p> 1$. Using Hölder’s inequality, we have

$$\begin{aligned} \begin{aligned} \int _{\mathbb {C}^n}\sum _{k=1}^\infty |f_k(z)|^p \chi _{\widetilde{B}_k}(z) {\text {d}}\nu (z) \le ~&\int _{\mathbb {C}^n}\left( \sum _{k=1}^\infty |f_k(z)|^t\right) ^{p/t} \bigg (\sum _{k=1}^\infty \chi _{\widetilde{B}_k}(z)\bigg )^{(t-p)/t} {\text {d}}\nu (z)\\ \le ~&N^{(t-p)/t}\int _{\mathbb {C}^n}\left( \sum _{k=1}^\infty |f_k(z)|^t\right) ^{p/t}{\text {d}}\nu (z). \end{aligned} \end{aligned}$$

The proof is complete. $\square$

Lemma 2.7

Let $\left\{ a_k\right\}$ be an r-lattice, $0 < p \le \infty$ and $\left\{ \lambda _k\right\} \in \ell ^p$. If

$$\begin{aligned} f(z)=\sum _{k=1}^{\infty }\lambda _k e^{\alpha \langle z,a_k \rangle -\frac{\alpha }{2}|a_k|^2}, \end{aligned}$$

then $f \in F_{\alpha }^{p}$ and $\Vert f\Vert _{p,\alpha } \lesssim \Vert \left\{ \lambda _k\right\} \Vert _{\ell ^p}$.

Proof

Since the result has been proven in [12, Lemma 2.4] for $1\le p\le \infty$, it suffices to prove the case $0<p<1$. If $0<p<1$, then we get

$$\begin{aligned} \Vert f\Vert _{p,\alpha }^p&\le \sum _{k=1}^{\infty }|\lambda _k|^p \int _{\mathbb {C}^n}\left| e^{\alpha \langle z,a_k \rangle -\frac{\alpha }{2}|a_k|^2} e^{-\frac{\alpha }{2}|z|^2} \right| ^p \textrm{d}v(z) \\ \le&C\sum _{k=1}^{\infty }|\lambda _k|^p\int _{\mathbb {C}^n} e^{-\frac{\alpha p}{2}|z-a_k|^2}\textrm{d}v(z)\\ \lesssim&\sum _{k=1}^{\infty }|\lambda _k|^p. \end{aligned}$$

The proof is complete. $\square$

3 Proof of Theorem 1.3

We are devoted to proving Theorem 1.3 in this section. Two inequalities related to Rademacher functions will be used. We use the notation $r_k$ to denote Rademacher functions as follows

$$\begin{aligned} r_0(s)&~=~{\left\{ \begin{array}{ll} 1, &{} 0\le s-[s]<1/2, \\ -1, &{} 1/2\le s-[s]<1, \end{array}\right. }\\ r_k(s)&~=~ r_0(2^ks) \ \textrm{for}\ k=1, 2, \ldots . \end{aligned}$$

Here [s] denotes the largest integer not greater than s. One is Khinchine’s inequality. For $0<p<\infty$, there exist constants $0<A_p \le B_p<\infty$ such that

$$\begin{aligned} \begin{aligned} A_p\left( \sum _{k=1}^m\left| c_k\right| ^2\right) ^{p / 2} \le \int _0^1\left| \sum _{k=1}^m c_k r_k(s)\right| ^p {\text{ d }}s \le B_p\left( \sum _{k=1}^m\left| c_k\right| ^2\right) ^{p / 2} \end{aligned} \end{aligned}$$

for all nonnegative integers m and all complex numbers $c_1, c_2, \ldots , c_m$. The other one is Khinchine-Kahane-Kalton inequality. If $0<p, q<\infty$, then

$$\begin{aligned} \begin{aligned} \left( \int _0^1\left\| \sum _{k=1}^\infty x_k r_k(s)\right\| _X^q {\text{ d }}s\right) ^{\frac{1}{q}} \simeq \left( \int _0^1\left\| \sum _{k=1}^\infty x_k r_k(s)\right\| _X^p {\text{ d }} s\right) ^{\frac{1}{p}} \end{aligned} \end{aligned}$$

for any sequence $\left\{ x_k\right\} \subset X$, where X is a quasi-Banach space with quasi-norm $\Vert \cdot \Vert _X$. See [23] in detail.

Proof of Theorem 1.3

We divide the proof into two cases: $\lambda \ge 1$ and $0<\lambda <1$.

Case I: $\lambda \ge 1$. For fixed $a \in \mathbb {C}^n$, set

$$\begin{aligned} f_a(z)=\frac{e^{\alpha _1 \langle z,a \rangle }}{e^{\frac{\alpha _1}{2}|a|^2}}, \quad z\in \mathbb {C}^n. \end{aligned}$$

It follows from Lemma 2.5 that $f_a\in F_{\alpha _1}^{p_1}$ and $\Vert f_a\Vert _{p_1, \alpha _1}\lesssim 1$. Since for any $z, w\in B(a, \delta )$ with some given $\delta >0$, we have

$$\begin{aligned} |z-w|\le |z-a|+|w-a|<2\delta . \end{aligned}$$

Therefore, for any $z \in B(a,\delta )$, we get

$$\begin{aligned} \mu (B(a,\delta ))&= \int _{B(a, \delta )}{\text {d}}\mu (w) \nonumber \\ {}&\le e^{\frac{\alpha _1 r}{2} \delta ^2} \cdot e^{\frac{\alpha _2 t}{2}(2\delta )^2} \cdot \int _{B(a,\delta )}e^{-\frac{\alpha _2 t}{2}|z-w|^2}\cdot e^{-\frac{\alpha _1 r}{2}|w-a|^2}{\text {d}}\mu (w)\nonumber \\&= e^{\frac{\alpha _1 r}{2} \delta ^2+\frac{\alpha _2 t}{2}(2\delta )^2 }\int _{B(a, \delta )}e^{-\frac{\alpha _2 t}{2}|z-w|^2}|f_a(w)|^r e^{-\frac{\alpha _1}{2}|w|^2 r}{\text {d}}\mu (w) \nonumber \\ {}&\le e^{\frac{\alpha _1 r}{2} \delta ^2+\frac{\alpha _2 t}{2}(2\delta )^2 }S_{\mu }^{r, t} f_a(z). \end{aligned}$$

(3.1)

Thus,

$$\begin{aligned} \mu (B(a,\delta ))^{p_2}&\lesssim \int _{B(a,\delta )} \mu (B(a,\delta ))^{p_2}\textrm{d}v(z) \lesssim \int _{B(a,\delta )}S_{\mu }^{r, t} f_a(z) ^{p_2}\textrm{d}v(z) \\&\le \int _{\mathbb {C}^n}\left| S_{\mu }^{r, t} f_a(z) \right| ^{p_2}\textrm{d}v(z) \le C \Vert f_a\Vert _{p_1,\alpha _1}^{rp_2} \lesssim C. \end{aligned}$$

This shows that $\mu$ is a $(\lambda , \gamma )$-Fock–Carleson measure by Lemma 2.2.

Case II: $0<\lambda <1$. Let $\left\{ a_k\right\}$ be an r-lattice on $\mathbb {C}^n$. For any sequence of real numbers $\left\{ \lambda _k\right\} \in \ell ^{p_1}$, set

$$\begin{aligned} f_s(z)=\sum _{k=1}^{\infty } \lambda _k r_k(s)g_k(z), \end{aligned}$$

where

$$\begin{aligned} g_k(z)=e^{\alpha _1 \langle z, a_k \rangle - \frac{\alpha _1}{2}|a_k|^2}. \end{aligned}$$

Then Lemma 2.7 implies $f_s(z)\in F_{\alpha _1}^{p_1}$ and $\left\| f_s\right\| _{{p_1},\alpha _1} \lesssim \left\| \left\{ \lambda _k\right\} \right\| _{\ell ^{p_1}}$ for almost every $s\in (0,1)$. Hence, the condition shows that

$$\begin{aligned} \left\| S_{\mu }^{r, t} f_s\right\| _{{p_2}}^{p_2} = \int _{\mathbb {C}^n}\left| S_{\mu }^{r, t} f_s(z) \right| ^{p_2}\textrm{d}v(z) \le C^{p_2}\Vert f_s\Vert _{{p_1},\alpha _1}^{rp_2} \lesssim C^{p_2} \left\| \left\{ \lambda _k\right\} \right\| _{\ell ^{p_1}}^{rp_2} \end{aligned}$$

for almost every $s \in (0,1)$. Integrating both sides with respect to s from 0 to 1, we have

$$\begin{aligned} \begin{aligned} \int _{0}^1 \left\| S_{\mu }^{r,t} f_s\right\| _{{p_2}}^{p_2} {\text {d}}s \lesssim C^{p_2}\left\| \left\{ \lambda _k\right\} \right\| _{\ell ^{p_1}}^{rp_2}. \end{aligned} \end{aligned}$$

(3.2)

Applying Fubini’s theorem and Khinchine-Kahane-Kalton inequality, we obtain

$$\begin{aligned} \int _{0}^1 \left\| S_{\mu }^{r,t} f_s\right\| _{{p_2}}^{p_2} \textrm{d}s&=\int _{\mathbb {C}^n}\int _{0}^1 \left| S_{\mu }^{r,t} f_s(z) \right| ^{p_2}\textrm{d}s\textrm{d}v(z)\\&=\int _{\mathbb {C}^n}\int _{0}^1\left( \int _{\mathbb {C}^n} e^{-\frac{\alpha _2 t}{2}|z-w|^2}\big |\sum _{k=1}^{\infty } \lambda _k r_k(s)g_k(w)\big |^r e^{-\frac{\alpha _1}{2}|w|^2 r} {\text {d}}\mu (w)\right) ^{p_2}\textrm{d}s \textrm{d}v(z)\\& > rsim \int _{\mathbb {C}^n}\left( \int _{0}^1\int _{\mathbb {C}^n} e^{-\frac{\alpha _2 t}{2}|z-w|^2}\big |\sum _{k=1}^{\infty } \lambda _k r_k(s)g_k(w)\big |^r e^{-\frac{\alpha _1}{2}|w|^2 r} {\text {d}}\mu (w) \textrm{d}s\right) ^{p_2} \textrm{d}v(z). \end{aligned}$$

Therefore, using Fubini’s theorem and Khinchine’s inequality, we have

$$\begin{aligned} \int _{0}^1 \left\| S_{\mu }^{r,t} f_s\right\| _{{p_2}}^{p_2} \textrm{d}s& > rsim \int _{\mathbb {C}^n}\left( \int _{\mathbb {C}^n}\left( \sum \limits _{k=1}^{\infty }|\lambda _k|^2 |g_k(w)|^2\right) ^{r/2} e^{-\frac{\alpha _2 t}{2}|z-w|^2} e^{-\frac{\alpha _1}{2}|w|^2 r} {\text {d}}\mu (w)\right) ^{p_2} \textrm{d}v(z). \end{aligned}$$

(3.3)

Remember that $B_k:=B(a_k, \delta )$ and $\widetilde{B}_k:=B(a_k, 2\delta )$ with $\delta >0$. A similar discussion to (3.1) implies that

$$\begin{aligned} \mu (B_k) \lesssim ~&\int _{B_k}e^{-\frac{\alpha _2 t}{2}|z-w|^2}|g_k(w)|^re^{-\frac{\alpha _1}{2}|w|^2r}{\text {d}}\mu (w) \end{aligned}$$

for $z\in B_k$. Thus,

$$\begin{aligned} \mu (B_k)^{p_2}\lesssim ~&\bigg ( \int _{\widetilde{B}_k}e^{-\frac{\alpha _2 t}{2}|z-w|^2}|g_k(w)|^re^{-\frac{\alpha _1}{2}|w|^2r}{\text {d}}\mu (w)\bigg )^{p_2}\\ \lesssim ~&\int _{B_k}\bigg ( \int _{\widetilde{B}_k}e^{-\frac{\alpha _2 t}{2}|z-w|^2}|g_k(w)|^re^{-\frac{\alpha _1}{2}|w|^2r}{\text {d}}\mu (w)\bigg )^{p_2}\textrm{d}v(z)\\ \lesssim ~&\int _{\widetilde{B}_k}\bigg ( \int _{\widetilde{B}_k}e^{-\frac{\alpha _2 t}{2}|z-w|^2}|g_k(w)|^re^{-\frac{\alpha _1}{2}|w|^2r}{\text {d}}\mu (w)\bigg )^{p_2}\textrm{d}v(z). \end{aligned}$$

It follows that

$$\begin{aligned} \sum _{k=1}^{\infty }|\lambda _k|^{rp_2}{\mu (B_k)^{p_2}}&\lesssim \sum _{k=1}^\infty \int _{\widetilde{B}_k}|\lambda _k|^{rp_2}\bigg (\int _{{\widetilde{B}}_k}e^{-\frac{\alpha _2 t}{2}|z-w|^2}|g_k(w)|^re^{-\frac{\alpha _1}{2}|w|^2r}{\text {d}}\mu (w)\bigg )^{p_2}\textrm{d}v(z). \end{aligned}$$

Using Lemma 2.6 twice, we obtain

$$\begin{aligned} \sum _{k=1}^{\infty }|\lambda _k|^{rp_2}{\mu (B_k)^{p_2}}&\lesssim \int _{\mathbb {C}^n}\bigg (\sum _{k=1}^\infty |\lambda _k|^{r}\int _{{\widetilde{B}}_k} e^{-\frac{\alpha _2 t}{2}|z-w|^2}|g_k(w)|^re^{-\frac{\alpha _1}{2}|w|^2r}{\text {d}}\mu (w)\bigg )^{p_2}\textrm{d}v(z)\\&\lesssim \int _{\mathbb {C}^n}\left( \int _{\mathbb {C}^n}\left( \sum \limits _{k=1}^{\infty }|\lambda _k|^2 |g_k(w)|^2\right) ^{r/2} e^{-\frac{\alpha _2 t}{2}|z-w|^2} e^{-\frac{\alpha _1}{2}|w|^2 r} {\text {d}}\mu (w)\right) ^{p_2}\textrm{d}v(z). \end{aligned}$$

This, together with (3.2) and (3.3), yields

$$\begin{aligned} \sum _{k=1}^{\infty }|\lambda _k|^{rp_2}{\mu (B_k)^{p_2}} \lesssim \left\| \left\{ \lambda _k\right\} \right\| _{\ell ^{p_1}}^{rp_2}. \end{aligned}$$

The duality of $l^{p_1/(r p_2)}$ and $l^{1/((1-\lambda )p_2)}$ gives

$$\begin{aligned} \left\{ \mu (B_k)\right\} \in \ell ^{1 /(1-\lambda )}. \end{aligned}$$

This shows $\mu$ is a $(\lambda , \gamma )$-Fock–Carleson measure in view of Lemma 2.4. The proof is complete. $\square$

4 Proofs of Theorems 1.1 and 1.2

Lemma 4.1

Let $0<p_{i}, q_{i}<\infty$, $0<\alpha _i<\infty$, $f_{i} \in F_{\alpha _{i}}^{p_{i} / q_{i}}$, where $i=1,2, \ldots , k$. If

$$\begin{aligned} \lambda = \sum \limits _{i=1}^{k} \frac{q_i}{p_i}, \quad \gamma = \sum \limits _{i=1}^{k} \alpha _i, \end{aligned}$$

then $\prod \limits _{i=1}^{k} f_{i} \in F_{\gamma }^{1 / \lambda }$ and

$$\begin{aligned} \left\| \prod _{i=1}^{k} f_{i}\right\| _{{1 / \lambda },{\gamma }} \lesssim \prod _{i=1}^{k}\left\| f_{i}\right\| _{{p_{i} / q_{i}, \alpha _{i}}}. \end{aligned}$$

(4.1)

Proof

Let $f_{i} \in F_{\alpha _{i}}^{p_{i} / q_{i}}$, where $i=1, 2, \ldots , k$. Since $\sum _{i=1}^k q_i/(p_i\lambda )=1$, it follows from Hölder’s inequality that

$$\begin{aligned} \begin{aligned} \Big \Vert \prod _{i=1}^{k} f_{i} \Big \Vert _{{1 / \lambda },{\gamma }}&=\left( \int _{\mathbb {C}^n} \left| \prod _{i=1}^{k} |f_{i}(z)| e^{- \frac{\gamma }{2} |z|^2}\right| ^{1 / \lambda } {\text {d}} v(z)\right) ^{\lambda }\\ {}&= \left( \int _{\mathbb {C}^n} \prod _{i=1}^{k}\left| f_{i}(z)\right| ^{1 / \lambda } e^{- \frac{\alpha _i}{2} |z|^2/\lambda } {\text {d}} v(z)\right) ^{\lambda } \\ {}&\lesssim \prod _{i=1}^{k}\left( \int _{\mathbb {C}^n}\left| f_{i}(z)\right| ^{(1 / \lambda )(p_{i} \lambda / q_{i})} e^{(- \frac{\alpha _i}{2} |z|^2/\lambda )(p_{i} \lambda / q_{i})} {\text {d}} v(z)\right) ^{{q_{i} / {p_{i}}}} \\ {}&= \prod _{i=1}^{k}\left( \int _{\mathbb {C}^n}\left| f_{i}(z)\right| ^{p_{i} / q_{i}} e^{-{\frac{\alpha _i}{2} |z|^2} ({p_i} / {q_i}) } {\text {d}} v(z)\right) ^{q_{i} / p_{i}} \\ {}&\lesssim \prod _{i=1}^{k}\left\| f_{i}\right\| _{{p_{i} / q_{i},{\alpha _{i}}}}. \end{aligned} \end{aligned}$$

The proof is complete. $\square$

Proof of Theorem 1.1

First, we prove the necessity. Assume that $\mu$ is a ($\lambda ,\gamma$)-Fock–Carleson measure. It suffices to prove $k\ge 2$, since the result is just the definition when $k=1$. It follows from Lemma 4.1 that if $h_i \in F_{\alpha _{i}}^{p_{i} / q_{i}}$ for any $i=1,2,\ldots ,k$, then $\prod \nolimits _{i=1}^{k} h_i \in F_{\gamma }^{1 / \lambda }$. Because $\mu$ is a ($\lambda ,\gamma$)-Fock–Carleson measure, we have

$$\begin{aligned} \int _{\mathbb {C}^n} \left| \prod _{i=1}^{k} h_{i}(z)\right| e^{{- \frac{\gamma }{2}} |z|^2} {\text {d}} \mu (z)&\le C\Vert \prod _{i=1}^{k} h_{i} \Vert _{{1 / \lambda }, {\gamma }}. \nonumber \end{aligned}$$

This, together with (4.1), gives

$$\begin{aligned} \int _{\mathbb {C}^n} \prod _{i=1}^{k}\bigg (|h_i(z)| e^{- \frac{\alpha _i}{2} |z|^2}\bigg ){\text {d}}\mu (z) \le C\prod _{i=1}^{k}\left\| h_{i}\right\| _{{p_{i} / q_{i}}, {\alpha _i}}. \end{aligned}$$

(4.2)

Let

$$\begin{aligned} {\text {d}}\mu _1 = \left( \prod _{i=2}^{k}|h_i| e^{- \frac{\alpha _i}{2} |z|^2}{\text {d}}\mu \right) \Big / \left( \prod _{i=2}^{k} \Vert h_i\Vert _{{p_{i} / q_{i}},{\alpha _i}} \right) . \end{aligned}$$

Then (4.2) is equivalent to

$$\begin{aligned} \int _{\mathbb {C}^n} \left| h_1(z)\right| e^{- \frac{\alpha _1}{2} |z|^2}{\text {d}}\mu _1(z) \le C\Vert h_1\Vert _{{p_1/q_1}, {\alpha _1}}. \end{aligned}$$

Thus, $\mu _1$ is a ($q_1 / p_1,\alpha _1$)-Fock–Carleson measure. Therefore, for any $f_1 \in F_{\alpha _1}^{p_1}$,

$$\begin{aligned} \int _{\mathbb {C}^n}\left| f_1(z)\right| ^{q_1} e^{{-\frac{\alpha _1}{2} |z|^2}q_1}{\text {d}}\mu _1(z) \le C {\Vert f_1\Vert _{{p_1},{\alpha _1}}^{q_1}}, \end{aligned}$$

which is the same as

$$\begin{aligned} \int _{\mathbb {C}^n}\left| f_1(z)\right| ^{q_1} e^{-{\frac{\alpha _1}{2}}|z|^2{q_1}} \prod _{i=2}^{k} |h_{i}(z)| e^{- \frac{\alpha _i}{2} |z|^2}{\text {d}}\mu (z) \le C \Vert f_1\Vert _{{p_1},{\alpha _1}}^{q_1} \prod _{i=2}^{k}\left\| h_{i}\right\| _{{p_{i} / q_{i}},{\alpha _i}}. \end{aligned}$$

(4.3)

Let

$$\begin{aligned} {\text {d}}\mu _2 =\left( |f_1|^{q_1} e^{{-\frac{\alpha _1}{2} |z|^2}q_1} \prod _{i=3}^{k}|h_i| e^{- \frac{\alpha _i}{2} |z|^2}{\text {d}}\mu \right) \Big / \left( \Vert f_1\Vert _{{p_1},{\alpha _1}}^{q_1} \prod _{i=3}^{k}\left\| h_{i}\right\| _{{p_{i} / q_{i}},{\alpha _i}} \right) . \end{aligned}$$

Then (4.3) is the same as

$$\begin{aligned} \int _{\mathbb {C}^n} \left| h_2(z)\right| e^{- \frac{\alpha _2}{2} |z|^2}{\text {d}}\mu _2(z) \le C\Vert h_2\Vert _{{p_2/q_2},{\alpha _2}}. \end{aligned}$$

This means that $\mu _2$ is a ($q_2 / p_2,\alpha _2$)-Fock–Carleson measure. Therefore, for any $f_2 \in F_{\alpha _2}^{p_2}$,

$$\begin{aligned} \int _{\mathbb {C}^n}\left| f_2(z)\right| ^{q_2} e^{-{\frac{\alpha _2}{2}}|z|^2{q_2}}{\text {d}}\mu _2(z) \le C\Vert f_2\Vert _{{p_2},{\alpha _2}}^{q_2}, \end{aligned}$$

that is

$$\begin{aligned}&\int _{\mathbb {C}^n}\left| f_1(z)\right| ^{q_1} e^{-{\frac{\alpha _1}{2}}|z|^2{q_1}} \left| f_2(z)\right| ^{q_2} e^{-{\frac{\alpha _2}{2}}|z|^2{q_2}}\prod _{i=3}^{k} |h_{i}(z)| e^{- \frac{\alpha _i}{2} |z|^2}{\text {d}}\mu (z)\\ {}&~\le C \Vert f_1\Vert _{{p_1},{\alpha _1}}^{q_1} \Vert f_2\Vert _{{p_2},{\alpha _2}}^{q_2} \prod _{i=3}^{k}\left\| h_{i}\right\| _{{p_{i} / q_{i}},{\alpha _i}}. \end{aligned}$$

Continuing this process, we eventually have

$$\begin{aligned} \int _{\mathbb {C}^n} \prod _{i=1}^{k}\left| f_{i}(z)\right| ^{q_i} e^{- \frac{\alpha _i}{2} |z|^2 q_i} {\text {d}} \mu (z) \le C \prod _{i=1}^{k}\left\| f_{i}\right\| _{{p_{i}},{\alpha _i}}^{q_i}. \end{aligned}$$

Hence, we obtain (1.2). The proof of the necessity for Theorem 1.1 is complete.

Next, we prove the sufficiency. Assume that (1.2) holds for any $f_i \in F_{\alpha _{i}}^{p_{i} / q_{i}}, i=1,2,\ldots ,k$. We aim to prove that $\mu$ is a ($\lambda ,\gamma$)-Fock–Carleson measure. The proof is divided into two cases: $\lambda \ge 1$ and $0<\lambda <1$.

Case I: $\lambda \ge 1$. For fixed $z\in \mathbb {C}^n$, set

$$\begin{aligned} f_{i,z}(w)= \frac{e^{{\alpha _i}\langle w, z\rangle }}{e^{\frac{\alpha _i}{2}{|z|^2}}},\; w \in \mathbb {C}^n, \end{aligned}$$

where $i=1,2,\ldots ,k$. By (2.2), we have $f_{i, z} \in F_{\alpha _i}^{p_i}$ with $\Vert f_{i,z}\Vert _{{p_i},{\alpha _i}}\simeq 1$. Thus, (1.2) implies

$$\begin{aligned} \int _{\mathbb {C}^n} \prod _{i=1}^{k}\left| f_{i,z}(w)\right| ^{q_i} e^{- \frac{\alpha _i}{2} |w|^2 q_i} {\text {d}} \mu (w) \le C \prod _{i=1}^{k}\left\| f_{i,z}\right\| _{{p_i},{\alpha _i}}^{q_i}. \end{aligned}$$

(4.4)

Let $t = \max \left\{ q_1,q_2,\ldots ,q_k\right\}$. Then

$$\begin{aligned} \widetilde{\mu _t}(z)&\simeq \int _{\mathbb {C}^n} \left| {\frac{e^{\gamma \langle w, z\rangle }}{e^{{\frac{\gamma }{2}}|z|^2}}} e^{- {\frac{\gamma }{2}}|w|^2}\right| ^t {\text {d}}\mu (w)\\&\le \int _{\mathbb {C}^n} \prod _{i=1}^{k} \left| \frac{e^{{\alpha _i}\langle w, z\rangle }}{e^{\frac{\alpha _i}{2} |z|^2}} e^{- \frac{\alpha _i}{2} |w|^2}\right| ^{q_i} {\text {d}}\mu (w)\\ =&\int _{\mathbb {C}^n} \prod _{i=1}^{k}\left| f_{i,z}(w)\right| ^{q_i} e^{- \frac{\alpha _i}{2} |w|^2 q_i} {\text {d}} \mu (w). \end{aligned}$$

This, together with (4.4), shows

$$\begin{aligned} \widetilde{\mu _t}(z)&\le C\prod _{i=1}^{k} \Vert f_{i,z}\Vert _{{p_i},{\alpha _i}}^{q_i}\lesssim C. \end{aligned}$$

Thus, $\widetilde{\mu _t}(z)$ is bounded. It follows from Lemma 2.2 that $\mu$ is a ($\lambda ,\gamma$)-Fock–Carleson measure.

Case II: $0<\lambda <1$. The proof is by induction on k. If $k = 1$, then (1.2) is just the definition of $(\lambda , \gamma )$-Fock–Carleson measure. Assume that the result holds for $k-1$ functions for $k \ge 2$. Set $\lambda _k = \lambda$, $\gamma _k = \gamma$ and

$$\begin{aligned} \lambda _{k-1} = \sum \limits _{i=1}^{k-1} \frac{q_i}{p_i}, \quad \gamma _{k-1}=\sum \limits _{i=1}^{k-1}{\alpha _i }. \end{aligned}$$

Denote

$$\begin{aligned} {\text {d}}\mu _k(z)=|f_k(z)|^{q_k} e^{- \frac{\alpha _k}{2} |z|^2 {q_k}}{\text {d}}\mu (z). \end{aligned}$$

Then we rewrite the condition (1.2) as

$$\begin{aligned} \int _{\mathbb {C}^n} \prod _{i=1}^{k-1}\left| f_{i}(z)\right| ^{q_i} e^{- \frac{\alpha _i}{2} |z|^2 q_i} {\text {d}} \mu _k(z) \le C (f_k) \prod _{i=1}^{k-1}\left\| f_{i}\right\| _{p_{i},{\alpha _i}}^{q_i} \end{aligned}$$

with $C(f_k) = C\Vert f_k\Vert _{{p_k},{\alpha _k}}^{q_k}$. It follows from the induction assumption that $\mu _k$ is a ($\lambda _{k-1},\gamma _{k-1}$)-Fock–Carleson measure with $\Vert \mu _k\Vert \lesssim C(f_k)$. Since $0<\lambda _{k-1}<\lambda <1$, it follows from Lemma 2.4 that $\widetilde{\mu }_{k,t} \in L^{1 / (1-\lambda _{k-1})}$ for any $t>0$ with

$$\begin{aligned} \Vert \widetilde{\mu }_{k,t}\Vert _{{1 / (1-\lambda _{k-1})}} \lesssim C\Vert f_k\Vert _{{p_k},{\alpha _k}}^{q_k}, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} \widetilde{\mu }_{k, t}=\int _{\mathbb {C}^n}e^{-\frac{t\gamma _{k-1}}{2}|z-w|^2}{\text{ d }}\mu _k(w). \end{aligned} \end{aligned}$$

That is,

$$\begin{aligned}&\bigg ( \int _{\mathbb {C}^n} \left| \int _{\mathbb {C}^n} e^{-\frac{\gamma _{k-1}t}{2}|z-w|^2} |f_k(w)|^{q_k} e^{- \frac{\alpha _k}{2} |w|^2 {q_k}} {\text {d}}\mu (w)\right| ^{1 / (1-\lambda _{k-1})} \textrm{d}v(z) \bigg )^{1-\lambda _{k-1}}\\ {}&\quad \lesssim C\Vert f_k\Vert _{{p_k},{\alpha _k}}^{q_k}. \end{aligned}$$

This, together with the definition of $S_{\mu }^{q_k,t}$ in Theorem 1.3, implies

$$\begin{aligned} \left\| S_{\mu }^{q_k, t} f_k \right\| _{{1/(1-\lambda _{k-1})}} \le C\Vert f_k\Vert _{{p_k},{\alpha _k}}^{q_k}, \end{aligned}$$

where $f_k \in F_{\alpha _k}^{p_k}$. Thus, by Theorem 1.3, $\mu$ is a $(\lambda ^{*},\gamma ^{*})$-Fock–Carleson measure, where

$$\begin{aligned} \lambda ^{*}=1+\frac{q_k}{p_k}-(1-\lambda _{k-1}),\quad \gamma ^{*}=\gamma _{k-1}+\alpha _k. \end{aligned}$$

It follows from the definitions of $\lambda _{k-1}, \gamma _{k-1}$ and (1.1) that $\lambda ^{*}=\lambda$ and $\gamma ^*=\gamma$. The proof is complete. $\square$

Proof of Theorem 1.2

If $0<\lambda <1$, then by Lemma 2.4 we know that $\mu$ is a $(\lambda , \alpha )$-Fock–Carleson measure if and only if $\mu$ is a vanishing $(\lambda , \alpha )$-Fock–Carleson measure. It is just a consequence of Theorem 1.1. And so it suffices to prove the theorem in the case of $\lambda >1$. Since (ii)$\Longrightarrow$(iii) is obvious, the theorem will be proved by showing (i) $\Longrightarrow$(ii) and (iii)$\Longrightarrow$(i).

(i) $\Longrightarrow$(ii). Assume that $\mu$ is a vanishing $(\lambda , \gamma )$-Fock–Carleson measure. Let $\left\{ f_{1, l}\right\}$ be any bounded sequence in $F_{\alpha _1}^{p_1}$ and $f_{1, l} \rightarrow 0$ uniformly on each compact subset of $\mathbb {C}^n$ as $l \rightarrow \infty$. Suppose that $\left\{ f_i\right\}$ is an arbitrary sequence in $F_{\alpha _i}^{p_i}$ with $\Vert f_i\Vert _{p_i, \alpha _i}\le 1$ for $i=2,3, \ldots , k$. For $r>0$, denote $B_r:=B(0, r)$, and $\mu _r$ the restriction of $\mu$ to $\mathbb {C}^n \backslash B_r$. Then $\mu _r$ is also a $(\lambda , \gamma )$-Fock–Carleson measure, and

$$\begin{aligned} \lim _{r \rightarrow \infty }\Vert \mu _r\Vert =0. \end{aligned}$$

On one hand, by Theorem 1.1, we obtain that for any $\varepsilon >0$ small enough, there exists an r large enough such that

$$\begin{aligned}&\int _{\mathbb {C}^n\backslash B_r}\left| f_{1, l}(z) e^{-\frac{\alpha _1}{2}|z|^2}\right| ^{q_1}\prod _{i=2}^k\left| f_i(z)e^{-\frac{\alpha _i}{2}|z|^2} \right| ^{q_i} {\text {d}} \mu (z) \\&\quad = \int _{\mathbb {C}^n}\left| f_{1, l}(z) e^{-\frac{\alpha _1}{2}|z|^2}\right| ^{q_1}\prod _{i=2}^k\left| f_i(z)e^{-\frac{\alpha _i}{2}|z|^2} \right| ^{q_i} {\text {d}} \mu _r(z) \\ {}&\quad \le \Vert \mu _r\Vert \cdot \Vert f_{1, l}\Vert _{p_1, \alpha _1}^{q_1} \cdot \prod _{i=2}^k \left\| f_i\right\| _{p_i,\alpha _i}^{q_i} \lesssim \varepsilon . \end{aligned}$$

Fix this r. Since $\left\{ f_{1, l}\right\}$ converges to 0 uniformly on each compact subset of $\mathbb {C}^n$, there is a constant $K>0$ such that for any $l>K$, $\left| f_{1, l}(z)\right| <\varepsilon$ for any $z \in B_r$. Therefore, using Theorem 1.1 again, we have

$$\begin{aligned}&\int _{B_r}\left| f_{1, l}(z) e^{-\frac{\alpha _1}{2}|z|^2}\right| ^{q_1} \prod _{i=2}^k\left| f_i(z)e^{-\frac{\alpha _i}{2}|z|^2} \right| ^{q_i} {\text {d}} \mu (z) \\&\quad \le \varepsilon \int _{\mathbb {C}^n} 1 \cdot e^{-\frac{\alpha _1}{2}|z|^2q_1} \cdot \prod _{i=2}^k\left| f_i(z)e^{-\frac{\alpha _i}{2}|z|^2} \right| ^{q_i} {\text {d}} \mu (z) \\&\quad \lesssim \varepsilon \left\| 1\right\| _{p_1, \alpha _1}^{q_1}\prod _{i=2}^k \left\| f_i\right\| _{p_i,\alpha _i}^{q_i} \lesssim \varepsilon \end{aligned}$$

for $l>K$. Thus, we get

$$\begin{aligned}&\limsup _{l \rightarrow \infty }\int _{\mathbb {C}^n}\left| f_{1, l}(z) e^{-\frac{\alpha _1}{2}|z|^2}\right| ^{q_1} \prod _{i=2}^k\left| f_i(z)e^{-\frac{\alpha _i}{2}|z|^2} \right| ^{q_i} {\text {d}} \mu (z)\\&\quad = \limsup _{l \rightarrow \infty }\left( \int _{B_r}+\int _{\mathbb {C}^n \backslash B_r}\right) \left| f_{1, l}(z) e^{-\frac{\alpha _1}{2}|z|^2}\right| ^{q_1} \prod _{i=2}^k\left| f_i(z)e^{-\frac{\alpha _i}{2}|z|^2} \right| ^{q_i} {\text {d}} \mu (z) \lesssim \varepsilon . \end{aligned}$$

It follows from the arbitrariness of $\varepsilon$ that $\lim \limits _{l\rightarrow \infty } F(l)=0$.

(iii) $\Longrightarrow$ (i). For fixed $a\in \mathbb {C}^n$, take

$$\begin{aligned} f_{i, a}(z)=\frac{e^{{\alpha _i}\langle z, a\rangle }}{e^{\frac{\alpha _i}{2}{|a|^2}}},\quad z\in \mathbb {C}^n, \end{aligned}$$

where $i=1,2,\ldots ,k$. By (2.2), we know that $f_{i,a} \in F_{\alpha _i}^{p_i}$ with $\Vert f_{i,a}\Vert _{{p_i},{\alpha _i}}\simeq 1$. Furthermore, it is easy to check that $f_{i,a}\rightarrow 0$ uniformly on any compact subset of $\mathbb {C}^n$ as $|a|\rightarrow \infty$. Thus, (iii) implies

$$\begin{aligned} \begin{aligned} \lim _{|a| \rightarrow \infty } \int _{\mathbb {C}^n} \prod _{i=1}^k \left| \frac{e^{{\alpha _i}\langle z, a\rangle }}{e^{\frac{\alpha _i}{2}{|a|^2}}} e^{-\frac{\alpha _i}{2}|z|^2} \right| ^{q_i} {\text {d}} \mu (z)=0. \end{aligned} \end{aligned}$$

Let $t = \max \left\{ q_1,q_2,\ldots ,q_k\right\}$. Then

$$\begin{aligned} \begin{aligned} \lim _{|a| \rightarrow \infty }\int _{\mathbb {C}^n}e^{-\frac{\gamma t}{2}|z-a|^2}{\text{ d }} \mu (z) \le \lim _{|a| \rightarrow \infty } \int _{\mathbb {C}^n} \prod _{i=1}^k \left| \frac{e^{{\alpha _i}\langle z, a\rangle }}{e^{\frac{\alpha _i}{2}{|a|^2}}} e^{-\frac{\alpha _i}{2}|z|^2} \right| ^{q_i} {\text {d}} \mu (z)=0. \end{aligned} \end{aligned}$$

This shows $\mu$ is a vanishing ($\lambda ,\gamma$)-Fock–Carleson measure by Lemma 2.3. The proof is complete. $\square$