Journal of Function Spaces and Applications

Volume 2013, Article ID 167514, 13 pages

http://dx.doi.org/10.1155/2013/167514

## Maximal and Area Integral Characterizations of Bergman Spaces in the Unit Ball of

^{1}Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, 30 West District, Xiao-Hong-Shan, Wuhan 430071, China^{2}Graduate University of Chinese Academy of Sciences, Beijing 100049, China

Received 15 May 2013; Accepted 17 July 2013

Academic Editor: Pei De Liu

Copyright © 2013 Zeqian Chen and Wei Ouyang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We present maximal and area integral characterizations of Bergman spaces in the unit ball of . The characterizations are in terms of maximal functions and area integral functions on Bergman balls involving the radial derivative, the complex gradient, and the invariant gradient. As an application, we obtain new maximal and area integral characterizations of Besov spaces. Moreover, we give an atomic decomposition of real-variable type with respect to Carleson tubes for Bergman spaces.

#### 1. Introduction and Main Results

Let denote the set of complex numbers. Throughout the paper we fix a positive integer , and let denote the Euclidean space of complex dimension . Addition, scalar multiplication, and conjugation are defined on component wise. For and in , we write where is the complex conjugate of . We also write The open unit ball in is the set The boundary of will be denoted by and is called the unit sphere in , that is, Also, we denote by the closed unit ball, that is, The automorphism group of , denoted by , consists of all biholomorphic mappings of . Traditionally, bi-holomorphic mappings are also called automorphisms.

For , the weighted Lebesgue measure on is defined by where for and if , which is a normalizing constant so that is a probability measure on . In the case of , we denote the resulting measure by and call it the invariant measure on , since for any automorphism of .

For and , the (weighted) Bergman space consists of holomorphic functions in with where the weighted Lebesgue measure on is defined by and is a normalizing constant so that is a probability measure on . Thus, where is the space of all holomorphic functions in . When , we simply write for . These are the usual Bergman spaces. Note that for is a Banach space under the norm . If , the space is a quasi-Banach space with -norm .

Recall that denotes the Bergman metric ball at with , where is the Bergman metric on . It is known that whereafter is the bijective holomorphic mapping in , which satisfies , , and .

As is well known, maximal functions play a crucial role in the real-variable theory of Hardy spaces (cf. [1]). In this paper, we first establish a maximal-function characterization for the Bergman spaces. To this end, we define for each and : Then we have the following result.

Theorem 1. *Suppose and . Let . Then for any if and only if . Moreover,
**
where “” depends only on , and .*

The norm appearing on the right-hand side of (15) can be viewed as an analogue of the so-called nontangential maximal function in Hardy spaces. The proof of Theorem 1 is fairly elementary (see Section 2), using some basic facts and estimates on the Bergman balls.

In order to state the area integral characterizations of the Bergman spaces, we require some more notation. For any and , we define and call it the radial derivative of at . The complex and invariant gradients of at are, respectively, defined as

Now, for fixed and , we define for each and :(1)the radial area integral function (2)the complex gradient area integral function (3)the invariant gradient area integral function

We state the second main result of this paper as follows.

Theorem 2. *Suppose , and . Let . Then, for any , the following conditions are equivalent:*(a)*,*(b)* is in ,*(c)* is in ,*(d)* is in .**
Moreover, the quantities
**
are all comparable to , where the comparable constants depend only on , and .*

For and , we fix a nonnegative integer with and define the generalized Bergman space as introduced in [2] to be the space of all such that . One then easily observes that is independent of the choice of and consistent with the traditional definition when . Let be the smallest nonnegative integer such that . Put Equipped with (22), becomes a Banach space when and a quasi-Banach space for .

clized Bergman spaces covers most of the spaces of holomorphic functions in the unit ball of , such as the classical diagonal Besov space and the Sobolev space , which has been extensively studied before in the literature under different names (see e.g., [2] for an overview).

There are various characterizations for or involving complex-variable quantities in terms of radical derivatives, complex and invariant gradients, and fractional differential operators (for a review and details see [2] and references therein). However, as an application of Theorems 1 and 2, we obtain new maximal and area integral characterizations of the Besov spaces as follows, which can be considered as a unified characterization for such spaces involving real-variable quantities.

Corollary 3. *Suppose and . Let and a positive integer such that . Then for any if and only if , where
**
Moreover,
**
where “” depends only on , and .*

Corollary 4. *Suppose , and . Let and a nonnegative integer such that . Then for any if and only if is in , where
**
Moreover,
**
where “” depends only on , and .*

To prove Corollaries 3 and 4, one merely notices that if and only if and applies Theorems 1 and 2, respectively, to with the help of Lemma 5 below.

The paper is organized as follows. In Section 2 we will prove Theorems 1 and 2. An atomic decomposition of real-variable type with respect to Carleson tubes for Bergman spaces will be presented in Section 3 via duality method. Finally, in Section 4, we will prove Theorem 2 through using the real-variable atomic decomposition of Bergman spaces established in the preceding section.

In what follows, always denotes a constant depending (possibly) on , or but not on , which may be different in different places. For two nonnegative (possibly infinite) quantities and , by we mean that there exists a constant such that and by that and . Any notation and terminology not otherwise explained are as used in [3] for spaces of holomorphic functions in the unit ball of .

#### 2. Proofs of Theorems 1 and 2

For the sake of convenience, we collect some elementary facts on the Bergman metric and holomorphic functions in the unit ball of as follows.

Lemma 5 (cf. [3, Lemma 2.20]). * For each ,
**
for all and in with .*

Lemma 6 (cf. [3, Lemma 2.24]). *Suppose , and . Then there exists a constant such that for any ,
*

Lemma 7 (cf. [3, Lemma 2.27]). *For each ,
**
for all in and in with .*

##### 2.1. Proof of Theorem 1

One needs the following result (cf. [4, Lemma 5]).

Lemma 8. *For fixed , there exist a positive integer and a sequence in such that*(1)*, and*(2)*each belongs to at most of the sets .*

*Proof of Theorem 1. *Let . By Lemmas 8, 6, and 5, one has
where is the constant in Lemma 8 depending only on and .

##### 2.2. Proof of Theorem 2

Recall that is defined as the space of all so that with the norm is a Banach space and called the Bloch space. Then, the following interpolation result holds.

Lemma 9 (cf. [3, Theorem 3.25]). *Let . Suppose and
**
for and . Then
**
with equivalent norms. *

Moreover, to prove Theorem 2 for the case , one will use atom decomposition for Bergman spaces due to Coifman and Rochberg [5] (see also [3, Theorem 2.30]) as follows.

Proposition 10. *Suppose , and . Then there exists a sequence in such that consists exactly of functions of the form
**
where belongs to the sequence space and the series converges in the norm topology of . Moreover,
**
where the infimum runs over all the above decompositions. *

Also, we need a characterization of Carleson type measures for Bergman spaces as follows, which can be found in [2, Theorem 45].

Proposition 11. *Suppose and is a positive Borel measure on . Then, there exists a constant such that
**
if and only if for each , there exists a constant such that
**
for all .*

We are now ready to prove Theorem 2. Note that for any , (cf. [3, Lemma 2.14]). We have that (d) implies (c) and (c) implies (b) in Theorem 2. Then, it remains to prove that (b) implies (a) and (a) implies (d).

*Proof of . *Since is holomorphic, by Lemma 6 we have
Then,
Hence, for any , if then is in , which implies that (cf. [3, Theorem 2.16]).

The proof of is divided into two steps. We first prove the case using the atomic decomposition and then the remaining case via complex interpolation.

*Proof of . *To this end, we write
An immediate computation yields that
Then we have
By Lemmas 5 and 7, one has
where we have used the fact . Note that (cf. [3, Corollary 5.24]); by Proposition 11 we have
Hence, for , we have for with ,
This concludes that
The proof is complete.

*Proof of for . *Set . Consider the operator
Note that and the measure is invariant under any automorphism of (cf. [3, Proposition 1.13]); we have
On the other hand,
This follows that is bounded from into . Notice that we have proved that is bounded from to . Thus, by Lemma 9 and the well known fact that
we conclude that is bounded from into for any ; that is,
where depends only on , , , , and . The proof is complete.

*Remark 12. *From the proofs of that and that for , we find that Theorem 2 still holds true for the Bloch space. That is, for any if and only if one (or equivalently, all) of , and is (or, are) in . Moreover,
where “” depends only on , and .

#### 3. Atomic Decomposition for Bergman Spaces

We let It is known that satisfies the triangle inequality and the restriction of to is a metric. As usual, is called the nonisotropic metric.

For any and , the set is called a Carleson tube with respect to the nonisotropic metric . We usually write in short.

As usual, we define the atoms with respect to the Carleson tube as follows: for is said to be a -atom if there is a Carleson tube such that(1) is supported in ;(2); (3).

The constant function (1) is also considered to be a -atom. Note that for any -atom ,

Recall that is the orthogonal projection from onto , which can be expressed as where extends to a bounded projection from onto ().

We have the following useful estimates.

Lemma 13. *For and , there exists a constant such that
**
for any -atom .*

To prove Lemma 13, we need first to show an inequality for reproducing kernel associated with , which is essentially borrowed from [6, Proposition 2.13].

Lemma 14. *For , there exists a constant such that for all satisfying , we have
*

*Proof. *Note that
We have

Write and , where and are parallel to , while and are perpendicular to . Then
and so
Since ,
and similarly
we have
This concludes that there is such that
whenever . Then, we have
Therefore,
and the lemma is proved.

*Proof of Lemma 13. *When is the constant function , the result is clear. Thus we may suppose that is a -atom. Let be supported in a Carleson tuber and , where is the constant in Lemma 14. Since is a bounded operator on , we have
Next, if then
Then
where we have used the fact that in the third inequality (cf. [3, Corollary 5.24]). Thus, we get
where depends only on , and .

Now we turn to the atomic decomposition of with respect to the Carleson tubes. Recall that for any -atom . Then, we define as the space of all which admits a decomposition where for each , is an -atom and so that . We equip this space with the norm where the infimum is taken over all decompositions of described above.

It is easy to see that is a Banach space. By Lemma 13, we have the contractive inclusion . We will prove in what follows that these two spaces coincide. That establishes the “real-variable” atomic decomposition of the Bergman space . In fact, we will show the remaining inclusion by duality.

Theorem 15. *Let and . For every there exist a sequence of -atoms and a sequence of complex numbers such that
**
Moreover,
**
where the infimum is taken over all decompositions of described above and “” depends only on , and .*

Recall that the dual space of is the Bloch space (we refer to [3] for details). The Banach dual of can be identified with (with equivalent norms) under the integral pairing (cf. [3, Theorem 3.17].)

In order to prove Theorem 15, we need the following result, which can be found in [4] (see also [3, Theorem 5.25]).

Lemma 16. *Suppose and . Then, for any , is in if and only if there exists a constant depending only on and such that
**
for all and all , where
**
Moreover,
**
where “” depends only on , and .*

As noted above, we will prove Theorem 15 via duality. To this end, we first prove the following duality theorem.

Proposition 17. * For any and , one has isometrically. More precisely,*(i)

*every*

*defines a continuous linear functional*

*on*

*by*(ii)

*conversely*,

*each*

*is given as*(83)

*by some*

*.*

*Moreover, we have*

*Proof. *Let be the conjugate index of , that is, . We first show . Let . For any -atom , by Lemma 16 one has
On the other hand, for the constant function , we have and so
Thus, we deduce that
for any finite linear combination of -atoms. Hence, defines a continuous linear functional on a dense subspace of , and extends to a continuous linear functional on such that
for all .

Next let be a bounded linear functional on . Note that
Then, is a bounded linear functional on . By duality there exists such that
Let be a Carleson tube. For any supported in , it is easy to check that
is a -atom. Then, and so
Hence, for any , we have
This concludes that
By Lemma 16, we have that and . Therefore, is given as (83) by with .

Now we are ready to prove Theorem 15.

*Proof of Theorem 15. *By Lemma 13 we know that . On the other hand, by Proposition 17 we have . Hence, by duality we have .

*Remark 18. *(1) One would like to expect that when also admits an atomic decomposition in terms of atoms with respect to Carleson tubes. However, the proof of Theorem 15 via duality cannot be extended to the case . At the time of this writing, this problem is entirely open.

(2) The real-variable atomic decomposition of Bergman spaces should be known to specialists in the case . Indeed, based on their theory of harmonic analysis on homogeneous spaces, Coifman and Weiss [7] claimed that the Bergman space admits an atomic decomposition in terms of atoms with respect to , where
This also applies to because is a homogeneous space for (see e.g. [6]). However, the approach of Coifman and Weiss is again based on duality and therefore not constructive and cannot be applied to the case . Recently, the present authors [8] extend this result to the case through using a constructive method.

#### 4. Area Integral Inequalities: Real-Variable Methods

In this section, we will prove the area integral inequality for the Bergman space via atomic decomposition established in Section 3.

Theorem 19. *Suppose , and . Then,
*

This is the assertion of Theorem 2 in the case . The novelty of the proof here is to involve a real-variable method.

The following lemma is elementary.

Lemma 20. *Suppose and . If , then
**
where “” depends only on , and .*

*Proof. *Note that . Then
In the last step, we have used [3, Theorem 2.16(b)].

*Proof of (96). *By Theorem 15, it suffices to show that for , and , there exists such that
for all -atoms . Given an -atom supported in , by Lemma 20, we have
where . On the other hand,where .

An immediate computation yields that
Moreover,Then, we have