A result about the atomic decomposition of Bloch-type space in the polydisk

1.   Introduction

Let ${\mathbb C}$ be the complex plane. Denote by $D(a, r)$ the open disk in ${\mathbb C}$ centered at $a$ with radius $r$, by ${\mathbb C}^N$ the $N$-dimensional complex Euclidean space with the inner product $\langle z, w\rangle = \sum_{j = 1}^Nz_j\overline{w}_j$, by ${\mathbb D}$ the open unit disk in ${\mathbb C}$, by ${\mathbb D}^N$ the open unit polydisk in ${\mathbb C}^N$, and by ${\mathbb{B}}^N$ the open unit ball in ${\mathbb C}^N$. For given $z\in {\mathbb C}^N$, write $|z|_\infty = \max_{1\leq j\leq N}|z_j|$. Let $H({\mathbb D}^N)$ be the space of all holomorphic functions on ${\mathbb D}^N$ and $H^\infty({\mathbb D}^N)$ the space of all bounded holomorphic functions on ${\mathbb D}^N$ with the supremum norm $\|f\|_\infty = \sup_{z\in {\mathbb D}^N}|f(z)|$.

A positive continuous radial function $\mu$ on the interval $[0, 1)$ is called normal (see, for example, ^[9]), if there are $\lambda \in[0, 1)$, $a$ and $b$ ($0 < a < b$) such that

For such function, the following examples were given in ^[9]:

and

The following fact can be used to prove that there exist lots of non-normal functions. It follows from ^[14] that if $\mu$ is normal, then for each $s\in(0, 1)$ there exists a positive constant $C = C(s)$ such that

for $0\leq r\leq t\leq r+s(1-r)$. From (1.1), it is easy to check the following functions are non-normal

and

where

The functions $\{\mu, \nu\}$ will be called a normal pair if $\mu$ is normal and for $b$ in above definition of normal function, there exists $\beta > b$ such that

If $\mu$ is normal, then there exists $\nu$ such that $\{\mu, \nu\}$ is normal pair (see ^[17]). Note that if $\{\mu, \nu\}$ is a normal pair, then $\nu$ is also normal. One of the purposes of introducing normal pair is to characterize the duality of spaces defined by the normal function (see ^[4,25]).

Given a normal function $\mu$, the Bloch-type space $\mathcal{B}_{\mu}({\mathbb D}^N)$ consists of all $f\in H({\mathbb D}^N)$ such that

Endowed with the norm $\|f\|_{\mathcal{B}_{\mu}({\mathbb D}^N)} = |f(0)|+{\beta}_{\mu}(f)$, it is a Banach space. There were a handful of studies on the space ${\mathcal {B}}_\mu({\mathbb D}^N)$ (see, for example, ^[5,6]). But the spaces with some special normal functions defined on the unit ball or unit disk and some operators have been extensively studied (see, for example, ^{[3,12,19,20,28]}, where ^[28] contains the elementary knowledge of such space).

Given a normal function $\mu$, the weighted-type space $H^\infty_{\mu}({\mathbb D}^N)$ consists of all $f\in H({\mathbb D}^N)$ such that

$H^\infty_{\mu}({\mathbb D}^N)$ is a Banach space with the norm $\|\cdot\|_{H^\infty_{\mu}({\mathbb D}^N)}$. There are lots of studies about such spaces on the unit ball or unit disk and some operators (see, for example, ^{[7,8,10,11,21,22,24]}).

Let $dA(z) = \frac{1}{\pi}dxdy$ be the normalized Lebesgue measure on ${\mathbb D}$ and $dA_{\alpha}(z) = ({\alpha}+1)(1-|z|^2)^{\alpha} dA(z)$ the weighted Lebesgue measure on ${\mathbb D}$, where $-1 < {\alpha} < +\infty$. For given $\vec{{\alpha}} = ({\alpha}_1, \ldots, {\alpha}_N)$, $-1 < {\alpha}_j < +\infty$, $j = 1, \ldots, N$, and $0 < p < +\infty$, the weighted Bergman space $A_{\vec{{\alpha}}}^p({\mathbb D}^N)$ consists of all $f\in H({\mathbb D}^N)$ such that

where $dA_{\vec{{\alpha}}}(z) = dA_{{\alpha}_1}(z_1)\ldots dA_{{\alpha}_N}(z_N)$. For some information about this space, see, for example, ^[23]. When $p\geq1$, the weighted Bergman space with the norm $\|\cdot\|_{A^p_{\vec{{\alpha}}}({\mathbb D}^N)}$ becomes a Banach space. While $p\in(0, 1)$, it is a Fréchet space with the translation invariant metric

The reason why people study the atomic decompositions for holomorphic function spaces is that it is very useful in operator theory. In particular, it can be used to describe dual spaces or to study basic questions such as the boundedness, the compactness or the Schatten class membership of concrete operators (see, for example, ^{[1,13,14,15,18]}). The atomic decompositions for holomorphic function spaces have been studied. For example, Coifman and Rochberg in ^[2] studied this problem on the weighted Bergman space. Zhu in ^[28] modified the proof of ^[2] and gave the atomic decomposition for the weighted Bloch space on the unit ball. Motivated by the previous studies, Zhang et al. in ^[26] characterized the atomic decomposition for the $\mu$-Bergman space on the unit ball. Later, Zhang et al. in ^[27] also considered the atomic decomposition for $\mu$-Bloch space on the unit ball. The works of ^[26,27] extended the corresponding results in ^[28].

We find that in above-mentioned works, the following result have been used many times in the atomic decompositions (see ^[28] for some details):

For any $p > 0$ and ${\alpha} > -1$, there exists a positive constant $C$ independent of the separation constants $r$ and $\eta$ such that

for all $r\in(0, 1)$, $z\in{\mathbb{B}}^N$ and $f\in H({\mathbb{B}}^N)$, where $\sigma$ is some constant related to $r$ and $\eta$, and the operator $S$ on $H({\mathbb{B}}^N)$ is defined by

A very natural problem is to extend this useful result to some other domains in ${\mathbb C}^N$. For example, here we will consider the result on ${\mathbb D}^N$. Another reason of our extension is that ${\mathbb D}^N$ and ${\mathbb{B}}^N$ are completely different domains in ${\mathbb C}^N$ (see, for example, ^[16]), which may produce some differences in methods and techniques.

In this paper, positive numbers are denoted by $C$, and they may vary in different situations. The notation $a\lesssim b$ (resp. $a\gtrsim b$) means that there is a positive number $C$ such that $a\leq Cb$ (resp. $a\geq Cb$). When $a\lesssim b$ and $b\gtrsim a$, we write $a\asymp b$.

2.   Preliminary results

In this section, we will obtain several elementary results which are used to prove the main results. We first have the following basic one (see, for example, ^[28]).

Lemma 2.1. There exists a positive integer $\widehat{N}$ such that for any $0 < r < 1$, we can find a sequence $\{a_k\}$ in ${\mathbb D}$ and a set $\{D_k\}$ consisting of Lebesgue measurable sets satisfying the following conditions:

(1) ${\mathbb D} = \cup_{k = 1}^\infty D(a_k, r)$;

(2) $D_k\cap D_j = \emptyset$ for $k\neq j$;

(3) $D(a_k, \frac{r}{4})\subset D_k\subset D(a_k, r)$ for every $k\in {\mathbb N}$;

(4) Each point $z\in {\mathbb D}$ belongs to at most $\widehat{N}$ of the sets $D(a_k, 4r)$.

Let $\vec{r} = (r_1, \ldots, r_N)$, $0 < r_j < 1$, $j = 1, \ldots, N$. Then for each fixed $r_j$ there exist a sequence $\{a_{jk}\}$ and a set $\{D_{jk}\}$ satisfy Lemma 2.1. For convenience, we denote by $a_k = (a_{1k}, \ldots, a_{Nk})$, by $\widehat{D}(a_k, \vec{r}) = D(a_{1k}, r_{1})\times\cdots\times D(a_{Nk}, r_N)$, and by $\widehat{D}_k = D_{1k}\times\cdots \times D_{Nk}$.

By using Lemma 2.1, we obtain the following similar disjoint decomposition of ${\mathbb D}^N$.

Lemma 2.2. There exists a positive integer $\widehat{N}$ such that for any $\vec{r} = (r_1, \ldots, r_N)$, $0 < r_j < 1$, $j = 1, \ldots, N$, we can find a sequence $\{a_k\}$ in ${\mathbb D}^N$ and a set $\{\widehat{D}_k\}$ consisting of Lebesgue measurable sets satisfying the following conditions:

(1) ${\mathbb D}^N = \cup_{k = 1}^\infty \widehat{D}(a_k, \vec{r})$;

(2) $\widehat{D}_k\cap\widehat{D}_j = \emptyset$ for $k\neq j$;

(3) $\widehat{D}(a_k, \frac{1}{4}\vec{r})\subset\widehat{D}_k\subset \widehat{D}(a_k, \vec{r})$ for every $k\in {\mathbb N}$;

(4) Each point $z\in {\mathbb D}^N$ belongs to at most $\widehat{N}^N$ sets $\widehat{D}(a_k, 4\vec{r})$.

The following integral expression for the functions in $A^1_{{\alpha}}({\mathbb{B}}^N)$ is well known (see ^[28]).

Lemma 2.3. If ${\alpha} > -1$ and $f\in A^1_{{\alpha}}({\mathbb{B}}^N)$, then

Let $x^* = (x_1, \ldots, x_{j-1})$ and $x_* = (x_{j+1}, \ldots, x_N)$, $j = 1, \ldots, N$. As an application of Lemma 2.3 for $N = 1$, if we regard the function $f(z_1, \ldots, z_N)\in A^1_{\vec{{\alpha}}}({\mathbb D}^N)$ as a one-variable function $f(z^*, z_j, z_*)\in A^1_{{\alpha}_j}({\mathbb D})$, $j = 1, \ldots, N$, then we have the following integral expression on $A^1_{\vec{{\alpha}}}({\mathbb D}^N)$.

Lemma 2.4. If ${\alpha}_j > -1$, $j = 1, \ldots, N$, and $f\in A^1_{\vec{{\alpha}}}({\mathbb D}^N)$, then

Remark 2.1. In order to obtain Theorem 3.1, we need give a key decomposition for ${\mathbb D}^N$. We first further partition the sets $\{\widehat{D}_k\}$ in Lemma 2.2. Actually, we partition the set $\widehat{D}_1$ and use automorphisms to carry the partition to other $\widehat{D}_k$'s. To this end, we let $\vec{\eta} = (\eta_1, \ldots, \eta_N)$, where each $\eta_j$ denotes a positive number that is much smaller than the separation constant $r_j$ in $\vec{r} = (r_1, \ldots, r_N)$, in the sense that the quotient $\eta_{j}/r_{j}$ is small. We fix a finite sequence $\{z_1, \ldots, z_J\}$ in $\widehat{D}(0, \vec{r})$, depending on $\vec{\eta}$, such that $\{\widehat{D}(z_j, \vec{\eta})\}$ covers $\widehat{D}(0, \vec{r})$ and that $\{\widehat{D}(z_j, \frac14\vec{\eta})\}$ are disjoint. We then can enlarge each set $\widehat{D}(z_j, \frac14\vec{\eta})\cap\widehat{D}(0, \vec{r})$ to a Borel set $E_j$ in the way that $E_j\subset\widehat{D}(z_j, \vec{\eta})$ and that

is a disjoint union. Here we just give a instruction. If you would like, you can see the proof of Lemma 2.28 in ^[28] for how to achieve this.

Let $w = (w_1, \ldots, w_N)\in {\mathbb D}^N$ and ${\varphi}_w(z)$ be the involutive automorphism of ${\mathbb D}^N$. Then

where

is the involutive automorphism of ${\mathbb D}$.

For fixed $j\in\{1, \ldots, J\}$ and $a_k = (a_{1k}, \ldots, a_{Nk})$, we define $b_{jk} = {\varphi}_{a_{k}}(z_{j})$. For $l\in\{1, \ldots, N\}$, we define $E_{lj} = \{z_l\in {\mathbb D}:(z^*, z_l, z_*)\in E_j\}$. Let $B_{ljk} = D_{lk}\cap{\varphi}_{a_{lk}}(E_{lj})$, where sets $\{D_{lk}\}$ are in Lemma 2.1 for $\{a_{lk}\}$ and $l\in\{1, \ldots, N\}$. Since

is a disjoint union for every $l$ and $k$, we obtain a disjoint decomposition

of ${\mathbb D}^N$, where $\widehat{B}_{jk}$ is the Cartesian product of $B_{ljk}$, $l = 1, \ldots, N$.

By using the decomposition of ${\mathbb D}^N$, we give the following definition, which is similar to those of (1.2).

Definition 2.1. Let $b_l > 1$ and ${\beta}_l = b_l-2$, $l = 1, \ldots, N$. Then the operator $S$ on $H({\mathbb D}^N)$ is defined by

Remark 2.2. From the definition of $dA_{\vec{{\beta}}}$, it follows that

3.   Main results and proofs

After discussions in Part 2, we now return to the following result.

Theorem 3.1. For $p > 0$, ${\alpha}_l > -1$, $l = 1, \ldots, N$, there exists a positive constant $C$ independent of the separation constants $\vec{r}$, $\vec{\eta}$, $f\in H({\mathbb D}^N)$ and $z\in {\mathbb D}^N$, such that

for all $\vec{r} = (r_1, \ldots, r_N)$, $0 < r_l < 1$, $l = 1, \ldots, N$, where

and $\tanh (\cdot)$ is the hyperbolic tangent function.

Proof. Without loss of generality, we may assume that $f\in A_{\vec{{\beta}}}^1({\mathbb D}^N)$. By the integral expression of the functions in $A_{\vec{{\beta}}}^1({\mathbb D}^N)$, we have

Since $\{\widehat{B}_{jk}\}$ is a partition of ${\mathbb D}^N$, we can write

Since $\widehat{B}_{jk}$ is the Cartesian product of $B_{ljk}$, $l = 1, \ldots, N$, it follows that

By using the triangle inequality in (3.1), we have

where by (3.2) we get

and

We first estimate $I(z)$. Let

By a change of variables in (3.3), we have

where

For $w_l\in E_{ljk}$, the quantities $(1-|w_l|^2)^{{\beta}_j+2}$ and $|1-\overline{b}_{ljk}w_l|$ are both bounded from below and from above. Also, since each $b_{ljk}\in D(a_{lk}, r_l)$, the quantities $1-|b_{ljk}|^2$ and $1-|a_{lk}|^2$ are comparable. Therefore, there exists a positive constant $C$ independent of $\vec{r}$ and $\vec{\eta}$, such that

Write $r_l' = \tanh(r_l)$, $\eta_l' = \tanh(\eta_l)$. Since each $\eta_l$ is much smaller than $r_l$ for $l\in\{1, \ldots, N\}$, we may as well assume that $R = \max_{1\leq l\leq N}\eta_l'/r_l'\leq\frac{1}{2}$. By Lemma 2.4, there exists a positive constant $C$ such that

for $g\in H({\mathbb D}^N)$. Let $\vec{r'} = (r_1', \ldots, r_N')$ and $\vec{\eta'} = (\eta_1', \ldots, \eta_N')$. Consider $g(z) = h(\vec{r'}z)$, where $h(z) = f\circ{\varphi}_{b_{jk}}(z)$, $\vec{r'}z = (r'_1z_1, \ldots, r'_Nz_N)$, $z\in {\mathbb D}^N$. After a change of variables, we obtain

for all $|z|\leq R$. That is,

for all $z\in\widehat{D}(0, \vec{\eta})$. For any $w\in \widehat{E}_{jk}\subset\widehat{D}(0, \vec{\eta})$, the identity

directly leads to

From (3.5) and (3.7), we therefore have

where $\widehat{E}_{jk}$ is the Cartesian product of $E_{ljk}$, $l = 1, \ldots, N$. Combining (3.6) with (3.8), we obtain

By a change of variables again,

It is easy to see that the quantities $1-|b_{ljk}|^2$ and $|1-\overline{b}_{ljk}w_l|$ are both comparable to $1-|a_{lk}|^2$ for $w_l\in D(b_{ljk}, r_l)$. This along with the fact that for each $l\in\{1, \ldots, N\}$ it follows that $D(b_{ljk}, r_l)\subset D(a_{lk}, 2r_l)$ shows that

Since $1-|a_{lk}|^2$ is comparable to $1-|w_{l}|^2$ for $w_{l}\in D(a_{lk}, 2r_l)$, we have

From (3.9)–(3.11), we have

Since

and

where the last inequality follows from Lemma 1.23 in ^[28], we have

From (3.12) and (3.13), it follows that

For each $k\in {\mathbb N}$ and $1\leq j\leq J$, it follows from Lemma 2.27 in ^[28] for $N = 1$ that $|1-\overline{b}_{ljk}z_l|^{b_l}$ is comparable to $|1-\overline{a}_{lk}z_l|^{b_l}$. Therefore,

Now we estimate $H(z)$. Let

By Lemma 2.27 for $N = 1$ in ^[28], we have

From ^[23], there is a positive constant $C$ independent of $f\in H({\mathbb D}^N)$ and $z\in {\mathbb D}^N$ such that

By the definition of $B_{ljk}$ and since $\widehat{B}_{jk}$ is the Cartesian product of $B_{ljk}$, $l = 1, \ldots, N$, we deduce that $\widehat{B}_{jk}\subseteq \widehat{D}_k$, and then by Lemma 2.2 we further obtain that $\widehat{B}_{jk}\subseteq\widehat{D}_k\subseteq \widehat{D}(a_k, 2\vec{r})$. From this and replacing $z$ by $a_k$ in (3.14), for every $w\in\widehat{B}_{jk}$ we have

Then

Since

we deduce that

From above same reasons, we have seen that $1-|b_{ljk}|^2$ and $1-|a_{lk}|^2$ are comparable, and $|1-\overline{b}_{ljk}z_l|$ and $|1-\overline{a}_{lk}z_l|$ are also comparable. It follows that

This completes the proof.

As an important application of Theorem 3.1, we obtain the next result, which shows that the operator $I-S$ is bounded on $H^\infty_\mu({\mathbb D}^N)$.

Theorem 3.2. Let $\mu$ be normal on $[0, 1)$. Then there exist a positive constant $C$ independent of $f\in H^\infty_{\mu}({\mathbb D}^N)$, $\vec{r}$ and $\vec{\eta}$, such that

Proof. We first choose $\vec{{\alpha}} = 0$ and $p = 1$ in Theorem 3.1. Then we obtain

for $\vec{r} = (r_1, \ldots, r_N)$, $0 < r_l < 1$, $l = 1, \ldots, N$, $z\in {\mathbb D}^N$. From Lemma 2.3 for $N = 1$ in ^[26], there exists a positive constant $A_l$ for each $0\leq l\leq N$ such that

Then, from (3.15) and (3.16), it follows that there exists a positive constant $C$ independent of $\vec{\eta}$ and $\vec{r}$ such that

Which shows that

By Lemma 2.2 in ^[26], for each $l\in\{1, \ldots, N\}$ we have

We therefore obtain

For convenience, write

then from (3.17) and (3.18), we have that

It is easy to see that there are $2^N$ terms in $\prod_{l = 1}^N(x_l^a+x_l^b)$. For arbitrary term $h(x_1, \ldots, x_N)$, without lose of generality, we may assume that

Substituting $x_l$ in (3.20), we obtain

Then by Theorem 2.12 in ^[28] we obtain

From this and (3.19), the desired result follows. This completes the proof.

4.   Conclusions

In this paper, a interesting result in the polydisk about the atomic decomposition of Bloch-type space has been obtained. It is well known that the existing similar results in the unit ball have been applied many times to the atomic decompositions of Bloch-type and weighted Bergman spaces. Hope that this study can attract people's more attention for the atomic decomposition of Bloch-type space.

Acknowledgments

The author would like to thank the anonymous referee for providing valuable comments that improved the presentation of the paper. This work was supported by the Sichuan Science and Technology Program (2022ZYD0010).

Conflict of interest

The authors declare that they have no competing interests.