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Some new characterizations of the Bloch space
Journal of Inequalities and Applications volume 2014, Article number: 459 (2014)
Abstract
We obtain some new characterizations for the Bloch space on the open unit disk in the complex plane ℂ and the open unit ball of .
MSC:32A18.
1 Introduction
Let be the open unit disk in the complex plane ℂ, , the open unit ball of the complex vector space , the class of all holomorphic functions on and the class of all holomorphic functions on .
For , the invariant gradient is defined by
where
is the complex gradient of f.
A holomorphic function f in is said to belong to the Bloch space if
Under the above norm, is a Banach space (see, e.g. [1]). On the unit ball, the Bloch space , which was introduced by Hahn in [2], is the space of all such that
For some classical results on Bloch spaces see [3] and [4].
It is well known that if and only if (see, e.g. [5])
For , an is said to belong to the α-Bloch space, denoted by , if
When , the α-Bloch space is the classical Bloch space.
It is of some importance to give new characterizations for a function space, since, for example, it can be useful in the study of operators acting on the space. For example, by using the last expression, it is difficult to study composition operators on α-Bloch space. However, in [6] Zhang and Xu introduced the metric
and proved that if and only if
and by using (1), they completely characterized the boundedness and compactness of composition operators on α-Bloch spaces. For some other results on operators on Bloch-type spaces see, for example, [7–21] and the references therein.
For , in [22] was proved that if and only if
Somewhat later, in [23] it was proved that if and only if
while in [24] it was proved that if and only if
These characterizations can be seen as derivative-free characterizations of the Bloch space on the unit ball. For the case of the unit polydisk, see [25] and [26]. For more characterizations of Bloch-type spaces in the unit disk, unit polydisk and unit ball, see, for example, [5, 22–24, 27–49].
In this paper, we give some new characterizations for the Bloch space. In Section 2, we give some preliminary results which are used in the proofs of our main results. In Section 3, we give two new characterizations for the Bloch space on . In Section 4, we give four new characterizations for the Bloch space in the unit ball , which, among others, generalize the corollaries in Section 3.
Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to the next. We say that two quantities and are comparable, if there are positive constants and independent of variable x such that
2 Preliminaries and auxiliary results
Let and be points in the complex vector space and . Let be the group of all biholomorphic selfmaps of . It is well known that is generated by the unitary operators on and the involutions of the form
where , is the orthogonal projection into the space spanned by , i.e.,
and . See [5, 50] for more properties of .
Recall that the weighted Bergman space , where and , consists of those functions for which
where , dv is the normalized Lebesgue measure of (i.e. ). When , we denote by . When , we get the classical Bergman space, which will be denoted by .
Let
For any and ,
Thus is a Möbius invariant measure (see, e.g. [36]).
Next, we quote some well-known results that will be used in the proofs of our main results. We begin with the following characterization of the Bloch space in the unit ball (see [5, 32, 34]).
Lemma 2.1 Let . A holomorphic function f is in the Bloch space if and only if
In [51], are proved the following two characterizations for the weighted Bergman space in the unit ball.
Lemma 2.2 Assume that , and . If β and γ are real parameters such that
and
then the following statements are equivalent:
-
(a)
;
-
(b)
(7)
-
(c)
(8)
Moreover, the quantities , , and are comparable.
Lemma 2.3 [32]
Assume that , , , , , and . Then for ,
Lemma 2.4 [32]
Assume that and . Then if and only if
The following well-known result can be found in [50] or [5].
Lemma 2.5 Let and . Then there is a positive constant C such that
for all .
3 Characterizations of the Bloch space in the unit disk
In this section, we give two characterizations for the Bloch space in the unit disk as follows.
Theorem 3.1 Assume that and . If β and γ are real parameters satisfying the following conditions:
then the following statements are equivalent:
-
(a)
;
-
(b)
-
(c)
Proof By taking , in Lemma 2.2, we see that if and only if
and if and only if
when the conditions in (11) hold.
Replacing f by in (12) and (13), and using Lemma 2.1, we conclude that if and only if
which is equivalent to
Using the change of variables , and the following equalities (see, e.g. [1]):
and
we see that the double integrals on the left side of (14) and (15) are equivalent to
and
respectively. Therefore, if and only if (b) holds, and if and only if (c) holds, as desired. □
Taking in Theorem 3.1, we easily get the following corollary.
Corollary 3.1 Assume that and . Then the following statements are equivalent:
-
(a)
;
-
(b)
-
(c)
where .
Taking , in Theorem 3.1, we can easily get the following result.
Corollary 3.2 Assume that and . Then the following statements are equivalent:
-
(a)
;
-
(b)
-
(c)
where .
4 Characterizations of the Bloch space in the unit ball
In this section, we generalize Corollaries 3.1 and 3.2 in the setting of the unit ball.
Theorem 4.1 Assume that and . Then if and only if
where .
Proof Let and in Lemma 2.2. We see that if and only if
where .
From this and by Lemma 2.1 we see that if and only if
Using the change of variables , and (4), we see that the left side of the last inequality is equivalent to
The result follows by using the equalities (see, e.g. [5])
and
in (16). □
Theorem 4.2 Assume that and . Then if and only if
where .
Proof Suppose that (17) holds. Since
Then by Theorem 4.1 and (18) we see that .
Conversely, suppose that . By using the change of variables , Lemma 2.3 and the following equality (see [5]):
we have
where
Employing the change of variables and using the fact that we have
It follows from Lemma 2.5 and the fact that (see [5]) that
Combining (19) with (20), the result follows from Lemma 2.4. □
By choosing , , and in Lemma 2.2, similarly to the proof of Theorem 4.1 is obtained the following result.
Theorem 4.3 Assume that and . Then if and only if
where .
Using Theorem 4.3, similarly to the proof of Theorem 4.2 is proved the following result.
Theorem 4.4 Assume that and . Then if and only if
where .
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Acknowledgements
The first author of this paper is supported by the project of Department of Education of Guangdong Province (No. 2013KJCX0170) and NSF of Guangdong, China (S2013010011978).
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Li, S., Stević, S. Some new characterizations of the Bloch space. J Inequal Appl 2014, 459 (2014). https://doi.org/10.1186/1029-242X-2014-459
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DOI: https://doi.org/10.1186/1029-242X-2014-459