- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

ISRN Mathematical Analysis

Volume 2013 (2013), Article ID 492052, 6 pages

http://dx.doi.org/10.1155/2013/492052

## Norm of a Volterra Integral Operator on Some Analytic Function Spaces

^{1}College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China^{2}Department of Mathematics, Jiaying University, Meizhou, Guangdong 514015, China

Received 27 June 2013; Accepted 30 July 2013

Academic Editors: O. Miyagaki and M. Winter

Copyright © 2013 Hao Li and Songxiao Li. 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 be an analytic function in the unit disc . The Volterra integral operator is defined as follows: In this paper, we compute the norm of on some analytic function spaces.

#### 1. Introduction

Let be the unit disk of complex plane and the class of functions analytic in . Denote by the normalized Lebesgue area measure in and the Green function with logarithmic singularity at ; that is, , where is the Möbius transformation of .

Let . The is the space of all functions such that From [1, 2], we see that = BMOA, the space of all analytic functions of bounded mean oscillation. When , the space is the same and equal to the Bloch space , which consists of all for which See [3, 4] for the theory of Bloch functions.

For , the -Bloch space, denoted by , is the space of all such that It is clear that for .

Let and let . The mean Lipschitz space consists of those functions for which It is obvious that is just the Bloch space , which is contained in for every . Note that increases with . We refer to [5] for more information of mean Lipschitz spaces.

For , we say that an belongs to the growth space if It is easy to see that .

For , an is said to belong to the space if

For , the Besov space is defined to be the space of all analytic functions in such that

Let . The Volterra integral operators and are defined as follows: It is easy to see that where denotes the multiplication operator; that is, . If is a constant, then all results about , , or are trivial. In this paper, we assume that is a nonconstant. Both operators have been studied by many authors. See [6–23] and the references therein.

Norms of some special operators, such as composition operator, weighted composition operator, and some integral operators, have been studied by many authors. The interested readers can refer [13, 24–32], for example. Recently, Liu and Xiong studied the norm of integral operators and on the Bloch space, Dirichlet space, BMOA space, and so on in [13]. In this paper, we study the norm of integral operator on some function spaces in the unit disk.

#### 2. Main Results

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.

Lemma 1 (see [5, page 144]). *If , then , , and the inequality is sharp for each fixed .*

Lemma 2. *Let and . For any , the following one has:
**
where is any point in .*

*Proof. *For any , taking and the subharmonicity of , we get
and so
For any , let . Replacing by and applying the change of variable formula give the following:
The proof is complete.

Theorem 3. *Let . The integral operator is bounded on if and only if . Moreover, one has
*

*Proof. *If , by (4), we have
Thus .

On the other hand, denote . Given any , there exists such that . Let
Then we have . In fact, taking and and using Poisson integral, we get
Taking , we obtain
So and . Thus Theorem 2.6 in [5] yields
By Theorem 2.12 in [5], we have
so the arbitrariness of gives and the proof is complete.

Lemma 3 in [13] gives the norm of on Dirichlet space. Here, we consider the norm of on -Dirichlet space .

Theorem 4. *Let and . Then is bounded on if and only if . Moreover, one has
*

*Proof. *First, we assume that . Let . Then (6) gives
and so we have .

Now we need only to show the reverse inequality. Denote . Given any , there exists such that . Let
where is any path in from to . By Theorem 13.11 in [33, page 274], we know is an analytic function in and ′. Also it is easy to check that . Indeed, by using the method of the proof of Lemma 4.2.2 in [4], we have
Let , and so . Thus by Lemma 2 we have
Since is arbitrary, we get
which implies the desired result.

Theorem 5. *Let and let . The integral operator is bounded on if and only if . Moreover, one has
*

*Proof. *If , then by (7), we have
and so .

Now we need only to show the reverse inequality. Denote . Given any , there exists such that . Let
We see that . Indeed,
Let . Then . Thus by Lemma 2, we have
Since is arbitrary, we get . The proof is complete.

Theorem 6. *Let and let . The integral operator is bounded from to if and only if . Moreover, one has
*

*Proof. *If , then by (3), we have
Hence .

For the converse, denote . Given any , there exists such that . Set
where is any path in from to . By Theorem 13.11 in [33, page 274], we know that is an analytic function in and , and it is easy to check that . Thus
Since is arbitrary, we obtain the desired result. The proof is complete.

Theorem 7. *Let . The integral operator is bounded from to if and only if . Moreover, one has
*

*Proof. *If , then by Lemma 1, we have
and hence .

For the converse, denote . Given any , there exists such that . Let
Then by the proof of Theorem 3, we see that . In the meantime, we know that ′, which gives
Since is arbitrary, we get the desired result. The proof is complete.

Finally, we consider the norm of from to some Banach spaces.

Theorem 8. *If , then the following assertions hold.*(1)*Let . The integral operator is bounded from space to space if and only if satisfies
Moreover, one has
*(2)*Let and let . The integral operator is bounded from space to space if and only if satisfies
Moreover, one has
*(3)*Let . The integral operator is bounded from space to space if and only if satisfies . Moreover, one has
*(4)*Let . The integral operator is bounded from space to space if and only if satisfies . Moreover, one has
*

*Proof. *The assertion (1) will be proved only here, and the conclusions of (2), (3), and (4) follow by using the similar arguments to that used in proving (1), and so the proofs are omitted.

If , then by (1), we have
and so

For the converse, let . It is easy to see that . Thus
The desired result follows by (47) and (48). The proof is complete.

#### Acknowledgments

Hao Li is supported by the National Natural Science Foundation of China (no. 11126284). Songxiao Li is supported by the project of Department of Education of Guangdong Province (no. 2012KJCX0096).

#### References

- R. Aulaskari, J. Xiao, and R. H. Zhao, “On subspaces and subsets of BMOA and UBC,”
*Analysis*, vol. 15, no. 2, pp. 101–121, 1995. View at Zentralblatt MATH · View at MathSciNet - J. Xiao,
*Holomorphic Q Classes*, vol. 1767 of*Lecture Notes in Mathematics*, Springer, Berlin, Germany, 2001. View at Publisher · View at Google Scholar · View at MathSciNet - K. H. Zhu, “Bloch type spaces of analytic functions,”
*The Rocky Mountain Journal of Mathematics*, vol. 23, no. 3, pp. 1143–1177, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. H. Zhu,
*Operator Theory in Function Spaces*, vol. 139 of*Monographs and Textbooks in Pure and Applied Mathematics*, Marcel Dekker, New York, NY, USA, 1990. View at MathSciNet - P. L. Duren,
*Theory of H*, vol. 38 of^{p}Spaces*Pure and Applied Mathematics*, Academic Press, New York, NY, USA, 1970. View at MathSciNet - A. Aleman and A. G. Siskakis, “An integral operator on ${H}^{p}$,”
*Complex Variables*, vol. 28, no. 2, pp. 149–158, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Aleman and A. G. Siskakis, “Integration operators on Bergman spaces,”
*Indiana University Mathematics Journal*, vol. 46, no. 2, pp. 337–356, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Austin,
*Multiplication and integral operators on Banach spaces of analytic functions [Ph.D. thesis]*, University of Hawai, 2010. - S. Li and S. Stević, “Volterra-type operators on Zygmund spaces,”
*Journal of Inequalities and Applications*, vol. 2007, Article ID 32124, 10 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Li and S. Stević, “Generalized composition operators on Zygmund spaces and Bloch type spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 338, no. 2, pp. 1282–1295, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Li and S. Stević, “Products of integral-type operators and composition operators between Bloch-type spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 349, no. 2, pp. 596–610, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Liu, Z. Lou, and C. Xiong, “Essential norms of integral operators on spaces of analytic functions,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 75, no. 13, pp. 5145–5156, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - J. Liu and C. Xiong, “Norm-attaining integral operators on analytic function spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 399, no. 1, pp. 108–115, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Pan, “On an integral-type operator from ${Q}_{K}(p,q)$ spaces to
*α*-Bloch spaces,”*Filomat*, vol. 25, no. 3, pp. 163–173, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - Ch. Pommerenke, “Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation,”
*Commentarii Mathematici Helvetici*, vol. 52, no. 4, pp. 591–602, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. G. Siskakis and R. Zhao, “A Volterra type operator on spaces of analytic functions,” in
*Function Spaces (Edwardsville, IL, 1998)*, vol. 232 of*Contemp. Math.*, pp. 299–311, American Mathematical Society, Providence, RI, USA, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On an integral operator between Bloch-type spaces on the unit ball,”
*Bulletin des Sciences Mathématiques*, vol. 134, no. 4, pp. 329–339, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Yang, “On an integral-type operator between Bloch-type spaces,”
*Applied Mathematics and Computation*, vol. 215, no. 3, pp. 954–960, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Zhu, “Generalized composition operators from generalized weighted Bergman spaces to Bloch type spaces,”
*Journal of the Korean Mathematical Society*, vol. 46, no. 6, pp. 1219–1232, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Zhu, “Generalized composition operators and Volterra composition operators on Bloch spaces in the unit ball,”
*Complex Variables and Elliptic Equations*, vol. 54, no. 2, pp. 95–102, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Zhu, “An integral-type operator from ${H}^{\infty}$ to Zygmund-type spaces,”
*Bulletin of the Malaysian Mathematical Sciences Society*, vol. 35, no. 3, pp. 679–686, 2012. View at MathSciNet - S. Li, “On an integral-type operator from the Bloch space into the ${Q}_{K}(p,q)$ space,”
*Filomat*, vol. 26, pp. 125–133, 2012. View at Publisher · View at Google Scholar - P. S. Bourdon, E. E. Fry, C. Hammond, and C. H. Spofford, “Norms of linear-fractional composition operators,”
*Transactions of the American Mathematical Society*, vol. 356, no. 6, pp. 2459–2480, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Colonna, G. R. Easley, and D. Singman, “Norm of the multiplication operators from ${H}^{\infty}$ to the Bloch space of a bounded symmetric domain,”
*Journal of Mathematical Analysis and Applications*, vol. 382, no. 2, pp. 621–630, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Hammond, “The norm of a composition operator with linear symbol acting on the Dirichlet space,”
*Journal of Mathematical Analysis and Applications*, vol. 303, no. 2, pp. 499–508, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. J. Martín, “Norm-attaining composition operators on the Bloch spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 369, no. 1, pp. 15–21, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Norm of weighted composition operators from Bloch space to ${H}_{\mu}^{\infty}$ on the unit ball,”
*Ars Combinatoria*, vol. 88, pp. 125–127, 2008. View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Norms of some operators from Bergman spaces to weighted and Bloch-type spaces,”
*Utilitas Mathematica*, vol. 76, pp. 59–64, 2008. View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Norm of weighted composition operators from
*α*-Bloch spaces to weighted-type spaces,”*Applied Mathematics and Computation*, vol. 215, no. 2, pp. 818–820, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - S. Stević, “Norms of some operators on bounded symmetric domains,”
*Applied Mathematics and Computation*, vol. 216, no. 1, pp. 187–191, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Norm of an integral-type operator from Dirichlet to Bloch space on the unit disk,”
*Utilitas Mathematica*, vol. 83, pp. 301–303, 2010. View at Zentralblatt MATH · View at MathSciNet - W. Rudin,
*Real and Complex Analysis*, McGraw-Hill, New York, NY, USA, 3rd edition, 1987. View at MathSciNet