Distance from Bloch-Type Functions to the Analytic Space
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.
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 [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  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 [11–13]. 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
Notation. Throughout this paper, we only write (or ) for for a positive constant , and moreover for both and .
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 .
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
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 ,
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.
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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