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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 252735, 6 pages
Predual of Spaces
School of Sciences, Anhui University of Science and Technology, Huainan, Anhui 232001, China
Received 19 February 2013; Accepted 22 April 2013
Academic Editor: Kehe Zhu
Copyright © 2013 Jizhen Zhou. 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.
A necessary and sufficient condition is given for a positive measure on to be a -Carleson measure. We give the predual of spaces in terms of this condition.
In study of a function space, it is of interest to study the dual and predual of function space. It is well known that Fefferman’s and Sarason’s theorems are and . Anderson et al. gave the similar results on the Bloch space in . The reader can refer to [2, 3] about the predual of spaces. We note that spaces are a kind of spaces. Now our question is what is the predual of spaces. But the technique that is used to prove the predual of spaces does not work for spaces. Enlightened by , we started from the characterizations of -Carleson measure by an integral operator which contains the normalized nonnegative Borel measure on the unit disk. In this paper, we obtain a principal result that the predual of spaces is the analytic space , which is introduced in Section 3. We now recall a few fundamental definitions and establish some notation.
Let be the Green function on the unit disk with logarithmic singularity at , where is the Möbius transformation of . Denote by the set of all analytic functions on .
Let be a right-continuous and nondecreasing function. The space consists of all functions satisfying where is an area measure on normalized so that .
Equipped with the norm , the space is Banach. It is easy to check that the space is Möbius invariant in the sense that for any and . See [5, 6] for a general theory of spaces. Note that the space gives if we choose for . See [7, 8] for a summary of recent research for spaces.
Recall that a function is said to belong to the Bloch space, denoted by , if
By , we know that
The following two conditions have played a crucial role in the study of spaces during the last years: where
Throughout the paper, satisfies the following condition:
Otherwise, the space only contains constant functions (cf. ). By Theorem 2.1 in , we may assume that is defined on and extend its domain to by setting for . As the discussion in , we may assume that .
For a subarc , the boundary of , let
If , then we set . A positive measure on is said to be a -Carleson measure if
By results in , we know that a function belongs to the space if and only if is a -Carleson measure.
In the paper, we say that (for two functions and ) if there is a constant (independent of and ) such that . We say (i.e., is comparable with ) whenever .
2. -Carleson Measure
For any and the nondecreasing , denote where is the poisson kernel and for , for .
Lemma 1. Let be a nonnegative measure on . Let satisfy condition (4). Then for any arc, holds if and only if is a -Carleson measure.
Proof. First we assume that (12) holds. Now we show that is a -Carleson measure. For any given , we have
The above inequality shows that is a -Carleson measure.
Conversely, suppose that is a -Carleson measure. For a nonnegative integer , we use for the arc in which has the same center as and length . For and , we have the following estimate: If , we have If , we have Then We have the desired result by condition (4).
Let be the set of all nonnegative measure on with the normalized condition . For , let For any , denote The following estimate can be found in : Then we have This shows that . Hence, the definition of is logical.
Theorem 2. Let satisfy condition (4). For all , if and only if is a -Carleson measure.
Proof. Suppose that is a -Carleson measure. For any , denote . Clearly, is an arc on with the midpoint . We obtain the following estimate by (23):
Note that for any . Then Therefore, by Lemma 1 and Fubini’s theorem,
Conversely, we assume that (25) holds. For any arc , let be the point in such that . Let be the point mass at . Then This and Lemma 1 give that The proof is complete.
3. Predual of Spaces
In this section, we will apply Theorem 2 to get the predual of spaces.
Definition 3. Let be a right-continuous and nondecreasing function. Denote by the set of all measurable functions on such that the measure is a -Carleson measure.
By Theorem 2, we can define It is easy to check that is a norm.
Denote by the set of all measurable functions on such that
Lemma 4. For the space , define where Then is a norm.
Proof. It is obvious that ., then . Conversely, if we set
This implies . We have
For any given , it is easy to see that
Given , we have The proof is complete.
Remark 5. Note that . In fact, for any , and , we have This shows that . Hence, .
Theorem 6. Let satisfy condition (4). If is equipped with the norm then under the pairing
Proof. For any given , it is easy to see that is a -Carleson measure. By Theorem 2, for any ,
By Lemma 4, we have
This shows that is a bounded functional on for . We have by the elementary knowledge of functional analysis, where is norm of . This gives .
Conversely, let be a bounded linear functional on . For any given , we have where is norm of . So for any fixed , we have where the space consists of all Lebesgue measure functions on such that Hence, can be extended to as a bounded linear functional on such that for any and . By the Hölder inequality, we obtain that the dual of is under the pair where the space consists of all Lebesgue measurable functions on such that Then there exists a such that
Note that the function is independent of . In fact, for any given which is different from , we have Given any , consider the Bergman disk . Define to be the test function. It is easy to see that . Then we have The above equalities show that Hence, . on for any given . This implies that . on . We now have a so that, for any , Theorem 2 shows that is a -Carleson measure. Hence, .
Definition 7. Let be a right-continuous and nondecreasing function. Let be the set of all nonnegative measure on with the normalized condition . Denote by the set of all analytic functions such that where is defined as in (22).
Remark 8. In fact, if and only if for any . Obviously, we have . See (47) about the definition of .
We need the following result to proof the main theorem (cf. ). Let . Define an operator on as
Proof. Choose . Then is a -Carleson measure. Theorem 2 gives that, for any ,
By the Hölder inequality, we have
This and Theorem 2 give that
Conversely, let be a bounded linear function on . Since , can be viewed as a bounded linear functional on ; that is, . By Theorem 6, there exists a such that for any .
Consider the Bergman projection from to the Bergman space : It is easy to see that
Hence, is a -Carleson measure by Lemma 9. This shows that is analytic and in . Let be the function satisfying . Then .
For , we have . The Bergman projection is self-adjoint. Hence, We obtain We complete the proof of the predual Theorem 10.
The author is supported by NSF (no. 11071153) and the Department of Education of Anhui Province of China (no. KJ2013A101).
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