/ / Article

Research Article | Open Access

Volume 2014 |Article ID 610237 | 7 pages | https://doi.org/10.1155/2014/610237

# Distance from Bloch-Type Functions to the Analytic Space

Revised09 Jul 2014
Accepted04 Aug 2014
Published19 Aug 2014

#### Abstract

The analytic space can be embedded into a Bloch-type space. We establish a distance formula from Bloch-type functions to , which generalizes the distance formula from Bloch functions to BMOA by Peter Jones, and to by Zhao.

#### 1. Introduction

Let denote the unit disc of the complex plane and let be its boundary. As usual, denotes the space of all analytic functions on .

Recall that, for , the Bloch-type space is the space of analytic functions on satisfying The little Bloch-type space is the subspace of all with It is well known that is a Banach space under the norm In particular, when , becomes the classic Bloch space , which is the maximal Möbius invariant Banach space that has a decent linear functional; see [1, 2] for more details on the Bloch spaces.

For , the involution of the unit disk is denoted by . It is well known and easy to check that

Let , , , and . The space , introduced by Zhao in  and known as the general family of function spaces, is defined as the set of for which where is the normalized area measure on . The space consists of all such that For appropriate parameter values , , and , coincides with several classical function spaces. For instance, if . The space is the classical Bergman space , and is the classical Besov space . The spaces are the spaces, in particular, , and the function space of bounded mean oscillation. See  for these basic facts.

For , we say that a nonnegative Borel measure defined on is an -Carleson measure if where the supremum ranges over all subarcs of , denotes the arc length of , and is the Carleson square based on a subarc . We write for the class of all -Carleson measures. Moreover, is said to be a vanishing -Carleson measure if

For an analytic function on , we define It was proved in  that if and only if is an -Carleson measure and if and only if is a vanishing -Carleson measure.

Let be an analytic function space. The distance from a Bloch-type function to is defined by

The following result is obtained by Zhao in .

Theorem 1. Suppose , , and . The following two quantities are equivalent: (1);(2),where and denotes the characteristic function of a set.

When and , the above characterization is Peter Jone’s distance formula from a Bloch function to BMOA (Peter Jone never published his result but a proof was provided in ). Also, similar type results can be found in . Specifically, distance from Bloch function to -type space is given in ; to the little Bloch space is obtained in , and to the space of the ball is characterized in . All these spaces are Möbius invariant.

This paper is dedicated to characterize the distance from to , which extends Zhao’s result. The main result is following.

Theorem 2. Suppose , , , and . Then where

The strategy in this paper follows from Theorem   in . The distance from a function to Campanato-Morrey space was given in  with similar idea.

Notation. Throughout this paper, we only write (or ) for for a positive constant , and moreover for both and .

#### 2. Preliminaries

We begin with a lemma quoted from Lemma   in .

Lemma 3. Let , , and be nonnegative Radon measures on . Then, if and only if

According to Lemma 3 and the fact that if and only if is an -Carleson measure, we can easily get the following corollary.

Corollary 4. Let be an analytic function on . if and only if there exists an such that

We will also need the following standard result from .

Lemma 5. Suppose and . Then, for all .

The following lemma, quoted from Lemma  1 in , is an extension of Lemma 5. See also .

Lemma 6. Suppose and , . If , then

Next, we see that is contained in . We thank Zhao for pointing out that the following result is firstly proved in . Here, we give another proof with a different approach.

Lemma 7. For , , and , . In particular, if , then .

Proof. We can use the reproducing formula for to get that for some constant , where is a real number greater than ; see, for example, [14, page 55].
Let . If , denote ; it follows from the Hölder’s inequality and (15) that Apparently, we have used Lemma 5 in the last inequality. This gives that when .
If , then Recall that and . We can easily use (4) to check that Thus, when .
Now, suppose and let , then for all . It follows that Again, the above inequality follows from Lemma 5. This completes the proof.

Our strategy relies on an integral operator preserving the -Carleson measures. For , we define the integral operator as

The following lemma is similar to Theorem  2.5 in . Indeed, Qiu and Wu proved the case . Specially, the case is just Lemma   in .

Lemma 8. Assume , , and . Let , let , and let be Lebesgue measurable on . If belongs to , then also belongs to .

Proof. We firstly prove the case and then sketch the outline argument of the case modified from  for the completeness.
When , according to Lemma 3, it is sufficient to show that for some . That is to show is finite. By Fubini’s theorem, it is enough to verify that is finite.
Choosing such that , we can use Lemma 6 to control the last integral by Since is an -Carleson measure, we can complete the proof by using Lemma 3 again.
When , we need to verify that
holds for any arc . In order to make this estimate, let , be the biggest integer satisfying , and let , , denotes the arcs on with the same center as and length , and is just . We can control and decompose the integral as
In order to estimate Int1, we define the linear operator as where If we choose a test function , then Schur’s lemma combines with Lemma 5 implying that Hence, is a bounded operator. Letting , then with Thus,
To handle , first note that, for , if and , then . Further, it is easy to check that, for any fixed , Now, splitting as we have Recall that . It follows from Hölder’s inequality that
Now, an easy computation gives that since and . This completes the proof.

#### 3. Proof of the Main Result

Proof of Theorem 2. For , it is easy to establish the following formula (see, e.g., [19, (1.1)] or [14, page 55]. Notice that it is a special case of the -order derivative of , as in , which holds for all holomorphic on ). Consider Define, for each , Then, Write Then,
So, if is in , Lemma 8 implies that By Corollary 4, . Meanwhile, recall that, for and , we can use Lemma 5 to obtain This means that To summarize the above argument, we have , (by (47)), and (by (49)), and is an -Carleson measure for each . Thus,
In order to prove the other direction of the inequality, we assume that equals the right-hand quantity of the last inequality and We only consider the case . Then, there exists an such that Hence, by definition, we can find a function such that Now, for any , we have that is not in . But, according to (53), we get and so This implies that does not belong to . But, it follows from (13) that . Therefore, Since , Corollary 4 implies that is in . This means that is in , and so is . This contradicts (57). Thus, we must have as required.

Remark 9. Theorem 2 characterizes the closure of in the norm. That is, for , is in the closure of in the norm if and only if, for every , for any Carleson square .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors would like to thank the referee for her/his helpful comments and suggestions which improved this paper. Cheng Yuan is supported by NSFC 11226086 of China and Tianjin Advanced Education Development Fund 20111005; Cezhong Tong is supported by the National Natural Science Foundation of China (Grant nos. 11301132 and 11171087) and Natural Science Foundation of Hebei Province (Grant no. A2013202265).

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