Abstract and Applied Analysis

Volume 2014 (2014), Article ID 832713, 14 pages

http://dx.doi.org/10.1155/2014/832713

Research Article

Weighted Differentiation Composition Operator from Logarithmic Bloch Spaces to Zygmund-Type Spaces

Huiying Qu, Yongmin Liu, and Shulei Cheng

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China

Received 7 January 2014; Accepted 10 March 2014; Published 17 April 2014

Academic Editor: Giovanni P. Galdi

Copyright © 2014 Huiying Qu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let denote the space of all holomorphic functions on the unit disk of , and let  n be a positive integer, a holomorphic self-map of , and a weight. In this paper, we investigate the boundedness and compactness of a weighted differentiation composition operator , from the logarithmic Bloch spaces to the Zygmund-type spaces.

1. Introduction

Let denote the open unit disk of the complex plane and the space of all analytic functions in .

The logarithmic Bloch space is defined as follows:

The space is a Banach space under the norm . Let denote the subspace of consisting of those such that

It is obvious that there are unbounded functions. For example, consider the function . There are also bounded functions that do not belong to . In fact, the interpolating Blaschke products do not belong to . It is easily proved that, for , . first appeared in the study of boundedness of the Hankel operators on the Bergman space. Attele in [1] proved that, for , the Hankel operator is bounded if and only if , thus giving one reason, and not the only reason, why log-Bloch-type spaces are of interest. Ye in [2] proved that is a closed subspace of . Galanopoulos in [3] characterized the boundedness and compactness of the composition operator and the boundedness and compactness of the weighted composition operator . Ye in [4] characterized the boundedness and compactness of the weighted composition operator between the logarithmic Bloch space and the -Bloch spaces on the unit disk and the boundedness and compactness of the weighted composition operator between the little logarithmic Bloch space and the little -Bloch spaces on the unit disk. Li in [5] characterized the boundedness and compactness of the weighted composition operator from Bergman spaces into the logarithmic Bloch space on the unit disk. Ye in [6] characterized the boundedness and compactness of the weighted composition operator from the general function space into the logarithmic Bloch space on the unit disk. Colonna and Li in [7] studied the boundedness and compactness of the weighted composition operators from Hardy space into the logarithmic Bloch space and the little logarithmic Bloch space. Petrov in [8] obtains sharp reverse estimates for the logarithmic Bloch spaces on the unit disk. Castillo et al. in [9] characterized the boundedness and compactness of the composition operator from the logarithmic Bloch spaces into weighted Bloch spaces. García Ortiz and Ramos-Fernández in [10] characterized the boundedness and compactness of the composition operators from logarithmic Bloch spaces into Bloch-type spaces.

Let be a weight; that is, is a positive continuous function on . The Zygmund-type space consists of all such that

With the norm , it becomes a Banach space. The little Zygmund-type space is a subspace of consisting of those such that

When , the Zygmund-type space becomes the Zygmund space [11], while the little Zygmund-type space becomes the little Zygmund space .

Let be the differentiation operator; that is, . If , then the operator is defined by .

The weighted differentiation composition operator, denoted by , is defined as follows [12, 13]:

where and is a nonconstant holomorphic self-map of .

If , then becomes the weighted composition operator , defined by

which, for , is reduced to the composition operator for some recent articles on weighted composition operators on some -type spaces, for example, [14–16] and references therein. If , , then , which was studied in [17–21]. When , then , which was studied in [17, 19]. If , , then , that is, the product of differentiation operator and multiplication operator defined by . Zhu in [13] completely characterized the boundedness and compactness of linear operators which are obtained by taking products of differentiation, composition, and multiplication operators from Bergman type spaces to Bers spaces. Stević in [12] studied the boundedness and compactness of the weighted differentiation composition operator from mixed-norm spaces to weighted-type spaces or the little weighted-type space (see also [22–24]). Zhu in [25] studied the boundedness and compactness of the generalized weighted composition operator on weighted Bergman spaces. Yang in [21] studied the boundedness and compactness of the operator and from to and spaces. Liu and Yu in [18] studied the boundedness and compactness of the operator between and Zygmund spaces. Ye and Zhou in [26] studied the boundedness and compactness of the weighted composition operators from Hardy to Zygmund type spaces. Stević in [27] studied the boundedness and compactness of the generalized composition operator from mixed-norm space to the Bloch-type space, the little Bloch-type space, the Zygmund space, and the little Zygmund space. For other recently introduced products of operators on spaces of holomorphic functions see [13, 16]. Motivated by the results [12, 18, 23, 24, 27], we consider the boundedness and compactness of the operators from the logarithmic Bloch spaces to the Zygmund-type spaces and the little Zygmund-type spaces. For the proof, we need different test functions and some complex calculations kills.

Throughout this paper, we will use the letter to denote a positive constant that can change its value at each occurrence.

2. Auxiliary Results

Here we prove and quote some auxiliary results which will be used in the proofs of the main results in this paper.

Lemma 1. Let be a positive integer. Suppose ; there exists a constant such that

Proof. We use induction on . Using the definition of the logarithmic Bloch spaces we have

the case holds for . Assume the case holds; since

let ; then we have , so

By the Cauchy integral formula we obtain

Note that

we have

for every . Hence the case holds. The desired result follows. The proof of this lemma is complete.

Lemma 2 (see [4, 28]). Let

then .

The following criterion for the compactness is a useful tool and it follows from standard arguments (e.g., [29, Proposition 3.11] or [30, Lemma 2.10]).

Lemma 3. Let , and let n be a nonnegative integer, a holomorphic self-map of , and a weight. Then is compact if and only if is bounded and, for any bounded sequence in which converges to zero uniformly on compact subsets of as , we have as .

Lemma 4. A closed set in is compact if and only if is bounded and satisfies

The proof is similar to that of Lemma 1 in [31]; hence we omit it.

3. Boundedness and Compactness of from to Spaces

In this section, we study the boundedness and compactness of .

Theorem 5. Let , and let n be a nonnegative integer, a holomorphic self-map of , and a weight. Then the following statements are equivalent:

(1) is bounded;(2) is bounded;(3)

Proof. . Suppose that (17), (18), and (19) hold. Then, for every and , by Lemma 1, we have

On the other hand, we have

Applying conditions (20) and (21), we deduce that the operator is bounded.

. This implication is clear.

. Assume that is bounded; that is, there exists a constant , such that

for all . For , we have that

Taking ; we have that

By (23), (24), and the boundedness of the function , we get

In the same way, taking , we have that

By (23), (25), (26), and the boundedness of the function , we have that

For a fixed , set

We get that

By Lemma 2 we have

Hence, and .

On the other hand for each fix , by (30), we obtain that

it follows that for each fix . From (29), we have and

Hence

By (33), we obtain that

And from (23), we have

Thus combining (35) with (34) we get the condition (17).

For a fixed , set

It is easy to see that

Using Lemma 2, we easily get that and with a direct calculation. From (37), we have ,

Hence

From (39), we obtain that

By (25), we have

Thus combining (40) with (41) we get the condition (18).

Next, we prove (19). To see this, for a fixed , put

It is easy to see that

From Lemma 2 we obtain that and with a direct calculation. From (43), we have ,

Hence

By (45), we obtain that

By (27), we have

Thus combining (47) with (46) we get the condition (19), finishing the proof of the theorem.

Theorem 6. Let , and let n be a nonnegative integer, a holomorphic self-map of , and a weight. Then the following statements are equivalent:

(1) is compact;(2) is compact;(3) is bounded and

Proof. . Assume that is bounded and that conditions (48), (49), and (50) hold. For any bounded sequence in which converges to zero uniformly on compact subsets of . To establish the assertion, it suffices, in view of Lemma 3, to show that

We assume that . From (48), (49), and (50) we have that, for any , there exists ; when , we have

From the boundedness of by Theorem 5, we see that (23), (25), and (27) hold. Since uniformly on compact subsets of , Cauchy's estimate gives that , , and converge to uniformly on compact subsets of ; there exists a such that implies that

From (52) and (53) and Lemma 1 we have

when . It follows that the operator is compact.

 ⇒ . It is obvious.

 ⇒ . Assume that is compact. Then it is clear that is bounded. By Theorem 5 we get that is bounded. Let be a sequence in such that as . We can use the test functions

Note that

for . We see that converges to uniformly on ; hence, converges to uniformly on compact subsets of and from (30) and (33) we have , . Then is a bounded sequence in which converges to 0 uniformly on compact subsets of . By Lemma 3, we have

Note that

From (33) and using the compactness of we obtain

From (59) and , it follows that

and consequently (48) holds.

Next, let

By a direct calculation, we obtain that converges to uniformly on compact subsets of , , and . By Lemma 3, we have

Note that

From (39) and using the compactness of we obtain

From (64) and , we have

it implies that (49) holds.

In order to prove (50), choose

By a direct calculation, we may easily prove that converges to uniformly on compact subsets of , , and . By Lemma 3, we have

Note that

From (45) and using the compactness of we obtain

From (69) and , it follows that

and consequently (50) holds, finishing the proof of the theorem.

Theorem 7. Let , and let n be a nonnegative integer, a holomorphic self-map of , and a weight. Then is a bounded operator provided that the following conditions are satisfied:

Proof. Suppose that (71), (72), and (73) hold. It is clear that (17), (18), and (19) hold. By Theorem 5 we have that is bounded. In order to prove is bounded, it is enough to show that, for any , . Using (71), (72), and (73) we have that, for any , there is a constant , such that implies

Then, for any , from Lemma 1 we obtain that

when . Hence for all , completing the proof of the theorem.

Theorem 8. Let , and let n be a nonnegative integer, a holomorphic self-map of , and a weight. If is a bounded operator, then (17), (18), and (19) hold and the following conditions are satisfied:

Proof. Assume that is bounded; it is clear that is bounded. By Theorem 5 we have that (17), (18), and (19) hold. On the other hand, for all , . Take ; we have that

then (76) holds. Let ; we have that

By (80), (76), and the boundedness of the function , we get

Hence, (77) holds. In the same way, let ; we have that

By (76), (77), (82), and the boundedness of the function , we have that

That is, (78) holds. The proof is completed.

Theorem 9. Let , and let n be a nonnegative integer, a holomorphic self-map of , and a weight. Then is a bounded operator if and only if is a bounded operator and (76), (77), and (78) hold.

Proof. Assume that is a bounded operator; it is clear that is a bounded operator. On the other hand, for all , . Taking , we have that

then (76) holds. Let ; we have that

By (85), (76), and the boundedness of the function , we get

Hence, (77) holds. In the same way, take ; we have that

By (76), (77), (87), and the boundedness of the function , we have that

That is, (78) holds.

Conversely, suppose that is a bounded operator and (76), (77), and (78) hold. For each polynomial we get that

Hence, . On the other hand, since polynomials are dense in , thus, for each , there is a sequence of polynomials such that

Since is a bounded operator, by Theorem 5 we have is a bounded operator. Since

and is the closed subset of , we see that ; thus is a bounded operator. The proof is completed.

Theorem 10. Let , and let n be a nonnegative integer, a holomorphic self-map of , and a weight. Then the following statements are equivalent:

(1) is compact;(2) is compact;(3)(71), (72), and (73) hold.

Proof.  ⇒ . Suppose that (71) (72), and (73) hold. By Theorem 7 we know that is bounded. Taking the supremum in inequality (20) over all such that and letting yields

Hence, by Lemma 4, we see that the operator is compact.

 ⇒ . This implication is clear.

 ⇒ . Assume that is compact. Firstly, it is obvious is bounded. By Theorem 9 we have that (76), (77), and (78) hold. On the other hand, we have that is compact. By Theorem 6 we have that (48), (49), and (50) hold. We prove that (76) and (48) imply (71). The proof of (72) and (73) is similar; hence, it will be omitted. From (48), it follows that, for every , there exists such that

when . Using (76) we see that there exists such that

when . Therefore when and , by (93), we have

On the other hand, when and , by (94), we obtain

From (95) and (96) we have

we obtain that (71) holds, as desired. The proof is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referees for their thoughtful comments and helpful suggestions. This research was supported by the Natural Science Foundation of China (11171285) and the Priority Academic Program Development of Jiangsu Higher Education Institutions.

References

K. R. M. Attele, “Toeplitz and Hankel operators on Bergman one space,” Hokkaido Mathematical Journal, vol. 21, no. 2, pp. 279–293, 1992.

S. Ye, “Multipliers and cyclic vectors on the weighted Bloch space,” Mathematical Journal of Okayama University, vol. 48, pp. 135–143, 2006.

P. Galanopoulos, “On to pullbacks,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 712–725, 2008.

S. L. Ye, “A weighted composition operator between different weighted Bloch-type spaces,” Acta Mathematica Sinica, vol. 50, no. 4, pp. 927–942, 2007 (Chinese).

S. Li, “Weighted composition operators from Bergman spaces into weighted Bloch spaces,” Communications of the Korean Mathematical Society, vol. 20, no. 1, pp. 63–70, 2005.

S. Ye, “Weighted composition operators from into logarithmic Bloch space,” Journal of the Korean Mathematical Society, vol. 45, no. 4, pp. 977–991, 2008.

F. Colonna and S. Li, “Weighted composition operators from Hardy spaces into logarithmic Bloch spaces,” Journal of Function Spaces and Applications, vol. 2012, Article ID 454820, 20 pages, 2012.

A. N. Petrov, “Reverse estimates in logarithmic Bloch spaces,” Archiv der Mathematik, vol. 100, no. 6, pp. 551–560, 2013.

R. E. Castillo, D. D. Clahane, J. F. Farías López, and J. C. Ramos Fernández, “Composition operators from logarithmic Bloch spaces to weighted Bloch spaces,” Applied Mathematics and Computation, vol. 219, no. 12, pp. 6692–6706, 2013.

A. J. García Ortiz and J. C. Ramos-Fernández, “Composition operators from logarithmic Bloch spaces to Bloch-type spaces,” Georgian Mathematical Journal, vol. 20, no. 4, pp. 671–686, 2013.

S. Li and S. Stević, “Volterra-type operators on Zygmund spaces,” Journal of Inequalities and Applications, vol. 2007, Article ID 32124, 10 pages, 2007.

S. Stević, “Weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces,” Applied Mathematics and Computation, vol. 211, no. 1, pp. 222–233, 2009.

X. Zhu, “Products of differentiation, composition and multiplication from Bergman type spaces to Bers type spaces,” Integral Transforms and Special Functions, vol. 18, no. 3-4, pp. 223–231, 2007.

S. Li and S. Stević, “Weighted composition operators between and -Bloch spaces in the unit ball,” Taiwanese Journal of Mathematics, vol. 12, no. 7, pp. 1625–1639, 2008.

S. Stević, “Norm of weighted composition operators from Bloch space to on the unit ball,” Ars Combinatoria, vol. 88, pp. 125–127, 2008.

S. Stević, “Norm and essential norm of composition followed by differentiation from -Bloch spaces to ,” Applied Mathematics and Computation, vol. 207, no. 1, pp. 225–229, 2009.

R. A. Hibschweiler and N. Portnoy, “Composition followed by differentiation between Bergman and Hardy spaces,” The Rocky Mountain Journal of Mathematics, vol. 35, no. 3, pp. 843–855, 2005.

Y. Liu and Y. Yu, “Composition followed by differentiation between and Zygmund spaces,” Complex Analysis and Operator Theory, vol. 6, no. 1, pp. 121–137, 2012.

S. Ohno, “Products of composition and differentiation between Hardy spaces,” Bulletin of the Australian Mathematical Society, vol. 73, no. 2, pp. 235–243, 2006.

S. Stević, “Composition followed by differentiation from and the Bloch space to th weighted-type spaces on the unit disk,” Applied Mathematics and Computation, vol. 216, no. 12, pp. 3450–3458, 2010.

W. Yang, “Products of composition and differentiation operators from spaces to Bloch-type spaces,” Abstract and Applied Analysis, vol. 2009, Article ID 741920, 14 pages, 2009.

H. Li and X. Fu, “A new characterization of generalized weighted composition operators from the Bloch space into the Zygmund space,” Journal of Function Spaces and Applications, vol. 2013, Article ID 925901, 6 pages, 2013.

S. Stević, “Weighted differentiation composition operators from the mixed-norm space to the th weighted-type space on the unit disk,” Abstract and Applied Analysis, vol. 2010, Article ID 246287, 15 pages, 2010.

S. Stević, “Weighted differentiation composition operators from and Bloch spaces to th weighted-type spaces on the unit disk,” Applied Mathematics and Computation, vol. 216, no. 12, pp. 3634–3641, 2010.

X. Zhu, “Generalized weighted composition operators on weighted Bergman spaces,” Numerical Functional Analysis and Optimization, vol. 30, no. 7-8, pp. 881–893, 2009.

S. Ye and Z. Zhuo, “Weighted composition operators from Hardy to Zygmund type spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 365286, 10 pages, 2013.

S. Stević, “Generalized composition operators between mixed-norm and some weighted spaces,” Numerical Functional Analysis and Optimization, vol. 29, no. 7-8, pp. 959–978, 2008.

S. Ye and J. Gao, “Extended Cesáro operators between different weighted Bloch-type spaces,” Acta Mathematica Scientia A, vol. 28, no. 2, pp. 349–358, 2008 (Chinese).

C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1995.

M. Tjani, Compact composition operators on some Moebius invariant Banach spaces [Ph.D. thesis], Michigan State University, 1996.

K. Madigan and A. Matheson, “Compact composition operators on the Bloch space,” Transactions of the American Mathematical Society, vol. 347, no. 7, pp. 2679–2687, 1995.