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Weighted composition followed and proceeded by differentiation operators fromZygmund spaces to Bloch-type spaces
Journal of Inequalities and Applications volume 2014, Article number: 152 (2014)
Abstract
The boundedness and compactness of the weighted composition followed andproceeded by differentiation operators from Zygmund spaces to Bloch-type spacesand little Bloch-type spaces are characterized.
MSC: 47B38, 30H30, 32C15.
1 Introduction
Let be the open unit disc in the complex plane ℂ,and let be the class of all analytic functions on Δ. Theα-Bloch space () is, by definition, the set of all functionsf in such that
Under the above norm, is a Banach space. When , is the well-known Bloch space. Let denote the subspace of , i.e.,
This space is called the little α-Bloch space (see [1]).
Assume that μ is a positive continuous function on, having the property that there exist positivenumbers s and t, , and , such that
Then μ is called a normal function (see [2]).
It is well known that is a Banach space with the norm (see [4]).
Let denote the subspace of , i.e.,
This space is called the little Bloch-type space. When , the induced space becomes the α-Bloch space.
An f in is said to belong to the Zygmund space, denoted by, if
where the supremum is taken over all and . By Theorem 5.3 in [6], we see that if and only if
It is easy to check that is a Banach space under the above norm. For every, by using a result in [7], we have
Let denote the subspace of consisting of those for which
The space is called the little Zygmund space. For thecorresponding n-dimensional Zygmund space see, e.g., [8] and [9].
Let φ be a nonconstant analytic self-map of Δ, and letϕ be an analytic function in Δ. We define the linearoperators
They are called weighted composition followed and proceeded by differentiationoperators, respectively, where and D are composition and differentiationoperators respectively. Associated with φ is the composition operator and weighted composition operator for and . It is interesting to provide a function theoreticcharacterization for φ inducing a bounded or compact compositionoperator, weighted composition operator and related ones on various spaces (see,e.g., [10–19]). For example, it is well known that is bounded on the classical Hardy, Bloch and Bergmanspaces. Operators and as well as some other products of linear operatorswere studied, for example, in [20–29] (see also the references therein). There has been some considerablerecent interest in investigation various type of operators from or to Zygmund typespaces (see, [7, 11, 23, 30–37]).
In this paper, we investigate the operators and from Zygmund spaces to Bloch-type spaces and littleBloch-type spaces. Some sufficient and necessary conditions for the boundedness andcompactness of these operators are given.
Throughout this paper, constants are denoted by C, they are positive and maydiffer from one occurrence to the other. The notation means that there is a positive constant Csuch that .
2 Main results and proofs
In this section, we state and prove our main results. In order to formulate our mainresults, we quote several lemmas which will be used in the proofs of the mainresults in this paper. The following lemma can be proved in a standard way (see,e.g., Proposition 3.11 in [10]). Hence we omit the details.
Lemma 2.1 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal. Thenis compact if and only ifis bounded and for any bounded sequencein(or) which converges to zero uniformly on compactsubsets of Δ as, and (or) as.
Lemma 2.2[25]
A closed set of is compact if and only if it is bounded and satisfied
Theorem 2.3 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal. Then the following statements are equivalent.
-
(i)
is bounded;
-
(ii)
is bounded;
-
(iii)
(2.1)
and
Proof of Theorem 2.3 (i) ⇒ (ii). This implication is obvious.
-
(ii)
⇒ (iii). Assume that is bounded, i.e., there exists a constant C such that
for all . Taking the functions and respectively, we get
Using these facts and the boundedness of function φ, we have
Set
and
for . It is known that (see [7]). Since
and
for , we have
Hence
For , set
Then . It is easy to see that
and
Therefore
From (2.8) and (2.10), we have
On the other hand, from the first inequality in (2.3), we have
Hence, from (2.3), (2.11), and (2.12), we obtain (2.2). Further, from (2.10), wehave
On the other hand, by (2.4), we have
Combining (2.13) and (2.14), (2.1) follows.
-
(iii)
⇒ (i). Assume that (2.1) and (2.2) hold. Then, for every , from (1.4), we have
(2.15)
Taking the supremum in (2.15) for , and employing (2.1) and (2.2), we deduce that is bounded. The proof of Theorem 2.3 iscompleted. □
Theorem 2.4 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal. Then the following statements are equivalent.
-
(i)
is compact;
-
(ii)
is compact;
-
(iii)
is bounded,
(2.16)
and
Proof of Theorem 2.4 (i) ⇒ (ii). This implication is clear.
-
(ii)
⇒ (iii). Assume that is compact. Then it is clear that is bounded. By Theorem 2.3 we know that is bounded. Let be a sequence in Δ such that as and , (if such a sequence does not exist then (2.16) and (2.17) are vacuously satisfied). Set
(2.18)
Then from the proof of Theorem 2.3, we see that for each . Moreover uniformly on compact subsets of Δ as and
Since is compact, by Lemma 2.1, we have
On the other hand, similar to the proof of Theorem 2.3, we have
which implies that
if one of these two limits exists.
Next, set
Then and converges to 0 uniformly on compact subsets of Δas (see [7]). Since
we have , for every and
By using these facts, since is compact, and from Lemma 2.1, we find that
Therefore
which implies (2.16). From this and (2.19), we have
From (2.21), it follows that , which altogether imply (2.17).
-
(iii)
⇒ (i). Suppose that is bounded and conditions (2.16) and (2.17) hold. From Theorem 2.3, it follows that
(2.22)
By the assumption, for every , there is a , such that
whenever .
Assume that is a sequence in such that and converges to 0 uniformly on compact subsets of Δas . Let . Then by (1.4), (2.22), and (2.23), we have
i.e. we obtain
Since converges to 0 uniformly on compact subsets of Δas , from Cauchy’s estimate, it follows that and as on compact subsets of Δ. Hence, letting in (2.24), and using the fact that ε isan arbitrary positive number, we obtain
By combining this with Lemma 2.1 the result easily follows. The proof ofTheorem 2.4 is completed. □
Theorem 2.5 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal. Thenis bounded if and only ifis bounded and
Proof of Theorem 2.5 Assume that is bounded. Then, it is clear that is bounded. Taking the test functions and respectively, we obtain (2.25).
Conversely, assume that is bounded and (2.25) holds. Then for each polynomialp, we have
In view of the facts
from (2.25) and (2.26), it follows that . Since the set of all polynomials is dense in (see [23]), it follows that for every , there is a sequence of polynomials such that as . Hence
as . Since the operator is bounded, we have , which implies the boundedness of. □
Theorem 2.6 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal. Then the following statements are equivalent.
-
(i)
is compact;
-
(ii)
is compact;
-
(iii)
(2.27)
and
Proof of Theorem 2.6 (i) ⇒ (ii). This implication is trivial.
-
(ii)
⇒ (iii). Assume that is compact. Then is bounded. From the proof of Theorem 2.5, we have
(2.29)
and
Hence, if , from (2.29) and (2.30), we obtain
and
from which the result follows in this case.
Now assume that . Let be a sequence such that as . Since is compact, by Theorem 2.4, we have
and
From (2.30) and (2.31), it follows that for every , there exists an such that , when , and there exists a such that , when . Therefore, when and , we have
On the other hand, if and , we obtain
Inequality (2.33) together with (2.34) gives the (2.27). Similarly, (2.29) and(2.32) imply (2.28).
-
(iii)
⇒ (i). Let . Then we have
Taking the supremum in this inequality over all such that , then letting , and using (2.27) and (2.28), we obtain
From Lemma 2.2 it follows that the operator is compact. □
Similarly to the proofs of Theorems 2.3-2.6, we can get the following results; weomit the proof.
Theorem 2.7 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal. Then the following statements are equivalent.
-
(i)
is bounded;
-
(ii)
is bounded;
-
(iii)
and
Theorem 2.8 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal. Then the following statements are equivalent.
-
(i)
is compact;
-
(ii)
is compact;
-
(iii)
is bounded,
and
Theorem 2.9 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal. Thenis bounded if and only ifis bounded and
Theorem 2.10 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal. Then the following statements are equivalent.
-
(i)
is compact;
-
(ii)
is compact;
-
(iii)
and
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Acknowledgements
The authors would like to thank the referees for their valuable suggestions whichgreatly improve the present article. The work was supported by the UnitedTechnology Foundation of Science and Technology Department of Guizhou Provinceand Guizhou Normal University (Grant No. LKS[2012]12), and National NaturalScience Foundation of China (Grant No. 11171080, Grant No. 11171277).
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Long, J., Qiu, C. & Wu, P. Weighted composition followed and proceeded by differentiation operators fromZygmund spaces to Bloch-type spaces. J Inequal Appl 2014, 152 (2014). https://doi.org/10.1186/1029-242X-2014-152
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DOI: https://doi.org/10.1186/1029-242X-2014-152