Product-Type Operators on the Space of Fractional Cauchy Transforms

Fan, Zeng; Guo, Xin

doi:https://doi.org/10.1155/2022/2644844

Journal of Function Spaces

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Abstract Introduction Preliminaries Proofs Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 2644844 | https://doi.org/10.1155/2022/2644844

Product-Type Operators on the Space of Fractional Cauchy Transforms

Zeng Fan¹and Xin Guo²

Academic Editor: Humberto Rafeiro

Received09 Dec 2021

Accepted31 Jan 2022

Published09 Mar 2022

Abstract

The space of fractional Cauchy transforms plays a central role in classical complex analysis, harmonic analysis, and geometric measure theory. In this paper, we study the boundedness and compactness of product-type operators from the space of fractional Cauchy transforms to the Zygmund-type space in terms of the function theoretic characterization of Julia–Carathéodory type.

1. Introduction

Let be the class of all holomorphic self-maps of the unit disk of the complex plane , be the boundary of , be the set of all nonnegative integers, and be the set of all positive integers. Denote by the space of all holomorphic functions on .

We first recall the spaces we work on. Let be a real number and be the space of all complex Borel measures on endowed with the total variation norm. The family of fractional Cauchy transforms is the collection of holomorphic functions in for whichfor some . The space is a Banach space, with respect to the norm given bywhere the infimum extends over all measures .

The fractional Cauchy transforms space plays a central role in classical complex analysis, harmonic analysis, and geometric measure theory which has phenomenal development in connection with the Calderon–Zygmund-type singular integral theory. The space may be identified with , the quotient of the Banach space by , and the subspace of consisting of functions with mean value 0 whose conjugate belongs to the Hardy space . Hence, is isometrically isomorphic to . Furthermore, admits a decomposition , where is the space of Borel measures, which are singular with respect to Lebesgue measure, and . According to the Lebesgue decomposition theorem, each can be written as , where is absolutely continuous with respect to the Lebesgue measure and is singular with respect to the Lebesgue measure . Consequently, is isometrically isomorphic to . Hence, can be written as , where is isomorphic to , the closed subspace of of absolutely continuous measures, and is isomorphic to , the subspace of of singular measures. For further results about the space of fractional Cauchy transforms, we refer to [1–12] and references therein.

Let be a weight, that is, is a positive continuous function on . A positive continuous function on the interval is said to be normal if there are and and , such that

In this paper, we assume the normal weighted function is also radial, i.e., . Now, the Zygmund-type space consists of all such that

With the norm , the Zygmund-type space is a Banach space.

For and , the weighted composition operator, which plays an important role in the isometry theory of Banach spaces, induced by and is given by

We can regard this operator as a generalization for a multiplication operator induced by and a composition operator induced by , where and . An extensive study concerning the theory of (weighted) composition operators has been established during the past four decades on various settings. We refer to standard references [13–15] for various aspects about the theory of composition operators acting on holomorphic function spaces, especially the problems of relating operator-theoretic properties of to function theoretic properties of . The differentiation operator, on , is defined by

Note that is typically unbounded on many familiar spaces of holomorphic functions. The differential operator plays an important role in various fields such as dynamical system theory and operator theory.

The products of any two of , , and can be obtained in six ways, i.e., , , , , , and . Similarly, the products of all of , , and can also be obtained in six ways, i.e., , , , , , and . In order to treat above product-type operators in a unified manner, Stević et al. [16], for the first time, introduced the so-called Stević–Sharma operator:for , . This operator is related to the various products of multiplication, composition, and differentiation operators. It is clear that all products of multiplication, composition, and differentiation operator in the following six ways can be obtained from the operator by choosing different and . More specially, we have

Recently, product-type operators on some spaces of holomorphic functions on the unit disk have become a subject of increasing interest (see [17–19] and references therein). Hibschweiler et al. [20] first characterized the boundedness and compactness of between Bergman spaces and Hardy spaces. Liu and Yu [21] investigated the boundedness and compactness of the operator from and Bloch spaces to Zygmund spaces. Ohno [22] considered the boundedness and compactness of on Hardy space . Zhu [23] studied the boundedness and compactness of linear operators which are obtained by taking products of multiplication, composition, and differentiation operators from Bergman-type spaces to Bers-type spaces. Quite recently, Zhang and Liu [24] presented the boundedness and compactness of the operator from Hardy spaces to Zygmund-type spaces. Liu and Yu [25] gave the complete characterizations for the boundedness and compactness of the operator from Hardy spaces to the logarithmic Bloch spaces. Liu et al. [26] investigated the compactness of the operator on logarithmic Bloch spaces. Yu and Liu [27] characterized the boundedness and compactness of the operator from space to the logarithmic Bloch spaces. Jiang [28] considered the boundedness and compactness of the operator from the Zygmund spaces to the Bloch–Orlicz spaces. Li and Guo [29] studied the boundedness and compactness of the operator from Zygmund-type spaces to Bloch–Orlicz spaces.

Inspired by the above results, the purpose of the paper is devoted to the boundedness and compactness of the operator from the fractional Cauchy transforms’ spaces to the Zygmund-type spaces over the unit disk in terms of the function theoretic characterization of Julia–Carathéodory type. As the applications of our main results, readers easily can obtain the boundedness and compactness characterizations of all six product-type operators:from the space of fractional Cauchy transforms to the Zygmund-type spaces.

2. Preliminaries

In this section, we recall some basic facts and preliminary results to be used in the sequel.

Suppose and are two Banach spaces with norms and , respectively. Recall that a linear operator from to is bounded if there is a positive constant such that , for all in . The bounded operator is said to be compact if the image of every bounded set of is relatively compact in . Equivalently, is compact if and only if the image of every bounded sequence in has a subsequence that converges in .

The following lemma gives a convenient compactness criterion for the Stević–Sharma operator acting from the space of fractional Cauchy transforms to the Zygmund-type spaces .

Lemma 1. Suppose . Then, the operator is compact if and only if is bounded and in , for any bounded sequence in , such that uniformly on compact subsets of .

A proof can be found in Proposition 3.11 of [13] for a single composition operator over the unit disk, and it can be easily modified for the operator on .

The following lemma is taken from [30] which is vital to construct the test functions on the space of fractional Cauchy transforms.

Lemma 2. Let (1)If , then there exists a such that (2)If , then and (3)If , then , and there exists a such that (4)If and , then and

Based on Lemma 2, we can obtain the following lemma, see Lemma 2 of [31], for the detailed proof.

Lemma 3. Let , , and . Put

Then, and .

Furthermore, we need the following lemma to prove our main results.

Lemma 4. Let . Suppose that and . Then, there is a positive constant independent of such that

Proof. For , there is a such that (1) holds. Then, we haveThus, we haveTaking infimum over all measures, , for which (1) holds; the proof is complete.

3. Main Results and Proofs

In this section, we devote to investigating the boundedness and compactness of the operator acting from the spaces of fractional Cauchy transforms to the Zygmund-type spaces in terms of the function theoretic characterization of Julia–Carathéodory type.

Theorem 1. Let . Suppose . Then, is bounded if and only if the following conditions are satisfied:

Proof. Suppose that (14)–(17) hold. Let with . Using Lemma 4, we haveApplying (14)–(17), it follows from the last above inequality that is bounded.
Conversely, assume that is bounded. Then, there exists a constant such thatfor all . It is elementary to deduce that , for . First, take the function , we obtainThen, put , and we apply (20) to haveNext, taking , (20) and (21) yield thatFurthermore, putting , we deduce thatApplying (20)–(22) gives thatFix and . Consider the following test function:Lemma 3 gives that and . A straightforward calculation shows thatPut in (25) such thatThus, we haveHence,namely, (14) holds.
Fix and . PutIt follows from Lemma 3 that and . In addition,Put in (30) such thatThus, we haveHence,which implies thatApplying (21) yields thatCombining (20) and (36), we can obtainwhich means that (15) holds.
Fix and . Consider the following test function:Lemma 3 yields that and . Furthermore, we obtain thatPut in (38) such thatThus, we haveHence,which implies thatApplying (22) yields thatTogether (43) with (44), we can obtainwhich means that (16) holds.
Fix and . Consider the following test function:Applying Lemma 3 yields that and . A straightforward calculation shows thatPut in (46) such thatThus, we haveHence,which implies thatApplying (24) yields thatTogether (51) with (52), we can obtainwhich means that (17) holds. The proof is complete.

Theorem 2. Let . Suppose . Then, is compact if and only if is bounded, and the following conditions are satisfied:

Proof. Assume that is bounded and (54)–(57) hold. Due to Lemma 1, in order to prove that is compact, it suffices to show that, for any bounded sequence in with uniformly on compact subsets of , . We may assume that , for all . By (54)–(57), we have that, for any , there exists such thatfor . From the boundedness of the operator and the proof of Theorem 1, we obtain that (20)–(24) hold. Since uniformly on compact subsets of , Cauchy estimates show that converges to 0 uniformly on compact subsets of . Then, there exists , for , such thatFrom (58)–(61) and Lemma 4, we havewhich implies that is compact.
Conversely, it is clear that the compactness of implies that the boundedness of . If , it is clear that (54)–(57) is vacuous and obviously hold. Hence, assume that . Let be a sequence in such that as . Take the test functionswhere is defined in (25). From the proof of Theorem 1, we have that andIt is obvious that converges to 0 uniformly on compact subsets of . From Lemma 1 and the compactness of , we haveas , which implies that (54) holds.
Take the test functions:where is defined in (30). From the proof of Theorem 1, we have that andIt is obvious that converges to 0 uniformly on compact subsets of . From Lemma 1 and the compactness of , we haveas . Thus, for , we havewhich implies (55) holds.
Analogously, choosing the test functions and which is defined in (38) and (46), same to the above approach, we can deduce (56) and (57). Consequently, the proof is complete.

Remark 1. Due to Theorems 1 and 2, we may easily obtain the characterizations for the boundedness and compactness of all six product-type operators:from the space of fractional Cauchy transforms to the Zygmund-type spaces. We leave the details to the readers interested in this research area.

Data Availability

No data were used to support the findings of the study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The research was supported by China Postdoctoral Science Foundation (no. 2020M672399).

References

J. Cima and T. MacGregor, Cauchy Transforms of Measures and Univalent Functions, Springer, Berlin, Germany, pp. 78–88, 1987, Lecture Notes in Mathematics.
J. Cima and A. Matheson, “Cauchy transforms and composition operators,” Illinois Journal of Mathematics, vol. 4, pp. 58–69, 1998.
View at: Publisher Site | Google Scholar
J. Cima, A. Matheson, and W. Ross, “The Cauchy transform,” Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 2006.
View at: Google Scholar
X. Guo and M. Wang, “Difference of weighted composition operators on the space of Cauchy integral transforms,” Taiwanese Journal of Mathematics, vol. 22, pp. 1435–1450, 2018.
View at: Publisher Site | Google Scholar
R. Hibschweiler, “Weighted composition operators between the fractional Cauchy spaces and the bloch-type spaces,” Journal of Complex Analysis, vol. 2017, Article ID 9486907, 5 pages, 2017.
View at: Publisher Site | Google Scholar
R. Hibschweiler and T. MacGregor, “Closure properties of families of Cauchy-Stieltjes transforms,” Proceedings of the American Mathematical Society, vol. 105, pp. 615–621, 1989.
View at: Publisher Site | Google Scholar
R. Hibschweiler and T. MacGregor, “Multipliers of families of Cauchy-Stieltjes transforms,” Transactions of the American Mathematical Society, vol. 331, pp. 377–394, 1992.
View at: Publisher Site | Google Scholar
R. Hibschweiler and T. MacGregor, Fractional Cauchy Transforms, Chapman and Hall/CRC, Boca Raton, FL, USA, 2006.
A. Sharma and R. Krishan, “Difference of composition operators from the space of Cauchy integral transforms to the Dirichlet space,” Complex Analysis and Operator Theory, vol. 10, pp. 141–152, 2016.
View at: Publisher Site | Google Scholar
A. Sharma, R. Krishan, and E. Subhadarsini, “Difference of composition operators from the space of Cauchy integral transforms to Bloch-type spaces,” Integral Transforms and Special Functions, vol. 28, pp. 145–155, 2017.
View at: Publisher Site | Google Scholar
A. K. Sharma and A. Sharma, “Integration operators from Cauchy integral transforms to weighted Dirichlet spaces,” New Zealand Journal of Mathematics, vol. 44, pp. 93–101, 2014.
View at: Google Scholar
M. Wang and X. Guo, “Difference of differentiation composition operators on the fractional Cauchy transforms spaces,” Numerical Functional Analysis and Optimization, vol. 39, pp. 1291–1315, 2018.
View at: Publisher Site | Google Scholar
C. Cowen and B. MacCluer, Composition Operators on Spaces of Analytic Functions (Studies in Advanced Mathematics), CRC Press, Boca Raton, FL, USA, 1995.
J. Shapiro, Composition Operators and Classical Function Theory, Springer, Berlin, Germany, 1993.
K. Zhu, Operator Theory in Function Spaces, American Mathematical Society, Providence, RI, USA, 2nd edition, 2007.
S. Stević, A. K. Sharma, and A. Bhat, “Products of multiplication composition and differentiation operators on weighted Bergman spaces,” Applied Mathematics and Computation, vol. 217, pp. 8115–8125, 2011.
View at: Google Scholar
S. Li and S. Stevic, “Generalized composition operators on Zygmund spaces and Bloch type spaces,” Journal of Mathematical Analysis and Applications, vol. 338, pp. 1282–1295, 2008.
View at: Publisher Site | Google Scholar
S. Li and S. Stevic, “Products of integral-type operators and composition operators between bloch-type spaces,” Journal of Mathematical Analysis and Applications, vol. 349, pp. 596–610, 2009.
View at: Publisher Site | Google Scholar
S. Wang, M. Wang, and X. Guo, “Products of composition, multiplication and iterated differentiation operators between Banach spaces of holomorphic functions,” Taiwanese Journal of Mathematics, vol. 24, pp. 355–376, 2020.
View at: Publisher Site | Google Scholar
R. Hibschweiler and N. Portnoy, “Composition followed by differentiation between Bergman and Hardy spaces,” Rocky Mountain Journal of Mathematics, vol. 35, pp. 843–855, 2005.
View at: Publisher Site | Google Scholar
Y. Liu and Y. Yu, “Composition followed by differentiation between H^∝ and Zygmund spaces,” Complex Analysis and Operator Theory, vol. 6, pp. 121–137, 2012.
View at: Publisher Site | Google Scholar
S. Ohno, “Products of composition and differentiation between Hardy spaces,” Bulletin of the Australian Mathematical Society, vol. 73, pp. 235–243, 2006.
View at: Publisher Site | Google Scholar
X. Zhu, “Products of differentiation, composition and multiplication from Bergman type spaces to Bers type spaces,” Integral Transforms and Special Functions, vol. 18, pp. 223–231, 2007.
View at: Publisher Site | Google Scholar
F. Zhang and Y. Liu, “On a Stević-Sharma operator from Hardy spaces to Zygmund-type spaces on the unit disk,” Complex Analysis and Operator Theory, vol. 12, pp. 81–100, 2018.
View at: Publisher Site | Google Scholar
Y. Liu and Y. Yu, “On a Stević-Sharma operator from Hardy spaces to the logarithmic Bloch spaces,” Journal of Inequalities and Applications, vol. 22, p. 19, 2015.
View at: Publisher Site | Google Scholar
F. Zhang and Y. Liu, “On the compactness of the Stević-Sharma operator on the logarithmic Bloch spaces,” Mathematical Inequalities & Applications, vol. 19, pp. 625–642, 2016.
View at: Publisher Site | Google Scholar
Y. Yu and Y. liu, “On Stević type operator from H^∝ space to the logarithmic Bloch spaces,” Complex Analysis and Operator Theory, vol. 9, pp. 1759–1780, 2015.
View at: Publisher Site | Google Scholar
Z. Jiang, “On Stević-Sharma operator from the Zygmund space to the Bloch-Orlicz space,” Advances in Difference Equations, vol. 228, p. 12, 2015.
View at: Publisher Site | Google Scholar
H. Li and Z. Guo, “On a product-type operator from Zygmund-type spaces to Bloch-Orlicz spaces,” Journal of Inequalities and Applications, vol. 132, p. 18, 2015.
View at: Publisher Site | Google Scholar
R. Hibschweiler, “Composition operators on spaces of fractional Cauchy transforms,” Complex Analysis and Operator Theory, vol. 6, pp. 897–911, 2012.
View at: Publisher Site | Google Scholar
X. Guo and M. Wang, “Linear combination of composition operators on Cauchy transform type spaces,” Science China Mathematics, vol. 50, pp. 1733–1744, 2020.
View at: Google Scholar

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Copyright © 2022 Zeng Fan and Xin Guo. 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.

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