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Composition operators from Zygmund spaces into spaces
Journal of Inequalities and Applications volume 2013, Article number: 175 (2013)
Abstract
The boundedness and compactness of composition operators from Zygmund and little Zygmund spaces into and spaces are characterized in this paper.
MSC:47B33, 30H99.
1 Introduction
Let be the open unit disk of complex plane ℂ. Denote by the class of functions analytic in . Let denote the Green’s function with pole at , i.e., , where is a Möbius transformation of . An is said to belong to the Zygmund space, denoted by , if
where the supremum is taken over all and . By Theorem 5.3 in [1], we see that if and only if
It is easy to check that is a Banach space under the above norm. Let denote the subspace of consisting of those for which
The space is called the little Zygmund space. Throughout this paper, the closed unit ball in and will be denoted by and , respectively.
Let be a nondecreasing continuous function. We say that an belongs to the space if (see, e.g., [2–4])
Here, dA is the normalized Lebesgue area measure in . Modulo constants, is a Banach space under the norm and is Möbius invariant. When (see [2])
. Here, ℬ is the Bloch space defined as follows:
If , then (see [5, 6]). If such that
we say that f belongs to the space . If consists of just constant functions, we say that it is trivial. is nontrivial if and only if (see [2])
To avoid that is trivial, we assume from now that (4) is satisfied. See [2–4, 7–15] for the study of the space .
Let φ be an analytic self-map of . The composition operator is defined by
It is interesting to provide a function theoretic characterization of when φ induces a bounded or compact composition operator on various spaces. For a study of the composition operators, see [16] and [17].
The composition operator from Bloch spaces to and was studied in [9, 10, 18]. Some characterizations of the boundedness and compactness of the composition operator, as well as Volterra type operator, on the Zygmund space can be found in [19–23].
The purpose of this paper is to study the boundedness and compactness of the operator from the Zygmund space and little Zygmund space into and .
Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to the other.
2 Main results and proofs
In this section, we state and prove our main results. In order to formulate our main results, we need some auxiliary results which are incorporated in the following lemmas. The following lemma, can be proved in a standard way (see, e.g., Theorem 3.11 in [16]).
Lemma 1 Let K be a nonnegative nondecreasing function on . Assume that φ is an analytic self-map of . Then is compact if and only if is bounded and for every bounded sequence in which converges to 0 uniformly on compact subsets of as , .
By using the methods of [10] (see also [24]), we can obtain the following lemma. Since the proof is similar, we omit the details.
Lemma 2 Let K be a nonnegative nondecreasing function on . Assume that φ is an analytic self-map of . If is compact, then for any there exists a δ, , such that for all f in ,
holds whenever .
By modifying the proof of Theorem 3.1 of [7] (or see [25]), we can prove the following lemma. We omit the details.
Lemma 3 Let K be a nonnegative nondecreasing function on . Assume that φ is an analytic self-map of . Then is compact if and only if is bounded and
Lemma 4 [20]
Suppose that , then
Lemma 5 [26]
Suppose that is an increasing sequence of positive integers satisfying for all . Let . Then there are two positive constants and , depending only on p and λ such that
Now we are in a position to state and prove our main results in this paper.
Theorem 1 Let K be a nonnegative nondecreasing function on . Assume that φ is an analytic self-map of . Then the following statements hold:
-
(i)
If
(6)
then is bounded.
-
(ii)
If is bounded, then
(7)
Proof (i) Let . Then by the following result (see [20]):
we have
In addition, by the well-known fact that , we obtain
Therefore, is bounded, and hence is bounded.
-
(ii)
First, we suppose that is bounded. Let . By the boundedness of we have that . Hence, we have
(9)
For , such that . Let
Then by the fact that belongs to Bloch space (see [[27], Theorem 1]) and the relationship of Bloch function and Zygmund function, we see that . Let
Then and . We have
Since
by (10), Lemma 5 and Fubini’s theorem we have
For any , a calculation shows that
since the number of terms in the sum from to is . Therefore,
which together with (9) implies that (7) holds.
Now suppose that is bounded. Take the function given by the above. Set
Then . Then, as argued the same with the case of and let , we get the desired result. The proof of the theorem is finished. □
Theorem 2 Let K be a nonnegative nondecreasing function on . Assume that φ is an analytic self-map of . Then the following statements holds:
-
(i)
If and
(13)then is compact.
-
(ii)
If is compact, then and
(14)
Proof (i) Assume that and (13) holds. Let be a bounded sequence in which converges to 0 uniformly on compact subsets of . We need to show that converges to 0 in norm. By (13), for given , there is an , such that
Therefore, by (8), we have
From the assumption, we see that also converges to 0 uniformly on compact subsets of by Cauchy’s estimates. It follows that since and as . By Lemma 1, is compact, and hence is also compact.
(ii) We only need to prove the case of . Assume that is compact. By taking we get . Now we choose the function given in the proof of Theorem 1. Then . Choose a sequence in which converges to 1 as , and let for . Then, for all and . Let . Then . Replace f by in (5) and then integrate both sides with respect to θ. By Fubini’s theorem, we obtain
From the proof of Theorem 1, for and for sufficiently large j, (16) gives
By Fatou’s lemma, we get (14). □
Theorem 3 Let K be a nonnegative nondecreasing function on . Assume that φ is an analytic self-map of . Then the following statements hold:
-
(i)
If is bounded, then and
(17) -
(ii)
If and
(18)
then is bounded.
Proof (i) Assume that is bounded. Then it is obvious that is bounded. By Theorem 1, (17) holds. Taking and using the boundedness of , we get .
(ii) Suppose that and (18) holds. From Theorem 1, we see that is bounded. To prove that is bounded, it suffices to prove that for any . Let . By Lemma 4, for every , we can choose such that for all . Then by (8), we have
which together with the assumed conditions imply the desired result. □
Theorem 4 Let K be a nonnegative nondecreasing function on . Assume that φ is an analytic self-map of . Then the following statements holds:
-
(i)
If
(19)
then is compact.
-
(ii)
If is compact, then
(20)
Proof (i) Assume that (19) holds. Set
From the assumption, we have that for every , there is a such that for , . Similarly to the proof of Lemma 2.3 of [25], we see that is continuous on , hence is bounded on . Therefore, is bounded on . From Theorem 1, we see that is bounded.
For any , by (8), we have
which together with (19) imply that is bounded. Fix . The right-hand side of (21) tends to 0, as by (19). From Lemma 3, we see that is compact, and hence is compact.
(ii) From the assumption, we see that is bounded and is compact. From Theorems 2 and 3, we have and
Hence, for any given , there exists a such that
Therefore, by (23) and the fact that , we have
By the arbitrary of ε, we get the desired result. The proof of the theorem is completed. □
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Acknowledgements
The authors are supported by the project of Department of Education of Guangdong Province (No. 2012KJCX0096), NNSF of China (No. 11001107).
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Fu, X., Li, S. Composition operators from Zygmund spaces into spaces. J Inequal Appl 2013, 175 (2013). https://doi.org/10.1186/1029-242X-2013-175
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DOI: https://doi.org/10.1186/1029-242X-2013-175