#### 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 [3] 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 [3–9] 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 [3] 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 [9].

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 [10]). Also, similar type results can be found in [11–13]. Specifically, distance from Bloch function to -type space is given in [11]; to the little Bloch space is obtained in [12], and to the space of the ball is characterized in [13]. 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 [14]. The distance from a function to Campanato-Morrey space was given in [15] 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 [14].

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 [16].

Lemma 5. *Suppose and . Then,
**
for all .*

The following lemma, quoted from Lemma 1 in [9], is an extension of Lemma 5. See also [17].

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 [3]. 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 [18]. Indeed, Qiu and Wu proved the case . Specially, the case is just Lemma in [14].

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 [18] 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 Int_{1}, 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 [14], 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).