Abstract and Applied Analysis

Volume 2009, Article ID 246521, 9 pages

http://dx.doi.org/10.1155/2009/246521

## Extended Cesàro Operators from Logarithmic-Type Spaces to Bloch-Type Spaces

Department of Mathematics, JiaYing University, Meizhou, Guangdong 514015, China

Received 27 March 2009; Accepted 12 April 2009

Academic Editor: Stevo Stević

Copyright © 2009 Dinggui Gu. 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

The boundedness and compactness of the extended Cesàro operator from logarithmic-type spaces to Bloch-type spaces on the unit ball are completely characterized in this paper.

#### 1. Introduction

Let be the unit ball of the unit sphere of , the space of all holomorphic functions in , and the space of all bounded holomorphic functions on . For , let denote the radial derivative of .

A positive continuous function on the interval is called normal if there is and and , such that If we say that is normal we will also assume that

Let be normal. The Bloch-type space is the space of all functions such that is Banach space with the norm The little Bloch-type space consists of all such that It is easy to see that is a closed subspace of . When , , we obtain so-called -Bloch spaces and little -Bloch spaces, respectively, which for are reduced to classical Bloch spaces (see, e.g., [1–4] and the references therein). When , we obtain the logarithmic Bloch space and the little logarithmic Bloch space , respectively, (see [5]). It was shown that is a multiplier of if and only if and in [6].

An is said to belong to the logarithmic-type space , if It is easy to see that becomes a Banach space under the norm , and that the inclusions hold. For some information of the space see [7, 8].

Let . The extended Cesàro operator on is defined by This operator is a natural extension of a one-dimensional operator defined in [9]. Some other results on the one-dimensional operator can be found, for example, in [10, 11] (see also the references therein). For some extensions of operator (1.6) on the unit disk see [12–16]. On related operators on the unit polydisk see, for example, [17–21] and references therein. The boundedness and compactness of operator (1.6) between various spaces of holomorphic functions has been extensively studied recently, see, [17, 22–36]. For some integral operators on spaces of harmonic functions see, for example, [37] as well as the references therein. A new extension of operator (1.6) in the unit ball case have been recently introduced by Stević in [38] (see also [5, 39]).

In this paper, we study the extended Cesàro operator from to Bloch-type spaces and . Sufficient and necessary conditions for the extended Cesàro operator to be bounded and compact are given.

Throughout the paper, constants are denoted by , they are positive and may not be the same in every occurrence.

#### 2. Main Results and Proofs

In this section, we give our main results and their proofs. Before stating these results, we need some auxiliary results, which are incorporated in the lemmas which follows.

Lemma 2.1. *Assume that and are normal. 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 , one has as *

The proof of Lemma 2.1 follows by standard arguments (see, e.g., Lemmas in [20, 21, 29]). Hence, we omit the details.

Lemma 2.2. *Assume that is normal. A closed set in is compact if and only if it is bounded and satisfies
*

This lemma can be found in [5], and its proof is similar to the proof of Lemma in [40]. Hence, it will be omitted.

The following result was proved in [8].

Lemma 2.3. *There exist two functions such that
*

Now we are in a position to state and prove our main results.

Theorem 2.4. *Assume that and is normal. Then is bounded if and only if
**
Moreover, if is bounded then the following asymptotic relation holds
*

*Proof. *Assume that (2.3) holds. Then, for any , we have
In addition, it is easy to see that . Therefore we have
as desired.

Conversely, assume that is bounded. For , set
It is easy to see that and .

For any , we have
from which (2.3) follows, moreover
From (2.6) and (2.9), we see that (2.4) holds. The proof is completed.

Theorem 2.5. *Assume that and is normal. Then is compact if and only if
*

*Proof. *Suppose that is compact. Let be a sequence in such that . Set
It is easy to see that . Moreover uniformly on compact subsets of as . By Lemma 2.1,
In addition,
which together with (2.12) implies that
From the above inequality we see that (2.10) holds.

Conversely, assume that (2.10) holds. From Theorem 2.4 we see that is bounded. In order to prove that is compact, according to Lemma 2.1, it suffices to show that if is a bounded sequence in converging to uniformly on compact subsets of , then

Let be a bounded sequence in such that uniformly on compact subsets of as By (2.10) we have that for any , there is a constant such that
whenever . Let . From (2.10) we see that . Equality (2.16) along with the fact that implies
Observe that is a compact subset of , so that
Therefore
Since is an arbitrary positive number it follows that the last limit is equal to zero. Therefore, is compact. The proof is completed.

*Remark 2.6. *From [24] and Theorems 2.4 and 2.5, we see that is bounded if and only if is bounded; is compact if and only if is compact.

Theorem 2.7. *Assume that and is normal. Then the following statements are equivalent:*(a)* is bounded;*(b)* is compact;*(c)

*Proof. *(b)(a). This implication is obvious.

(a)(c). Assume that is bounded. Now we prove that (2.20) holds. Note that (2.20) is equivalent with
Hence we only need to show that (2.21) holds. This can be done by contradiction. Now assume that the condition (2.21) does not hold. If it was, then it would exist and a sequence such that , and
for sufficiently large . We may assume that and also
According to Lemma 2.3 we know that there exist two functions such that
Let
Then clearly . By the boundedness of we have
On the other hand,
for sufficiently large . Since , from the above inequality we obtain that , which is a contradiction.

(c)(b). From (1.5) we have that
Taking the supremum in the above inequality over all such that then letting , by (2.20) we arrive at
From this and by employing Lemma 2.2, we see that is compact. The proof is completed.

From Theorems 2.4, 2.5, and 2.7, we have the following corollary.

Corollary 2.8. *Let . Then*(1)* is bounded if and only if
*(2)* is compact if and only if is bounded if and only if is compact if and only if
*

#### Acknowledgment

The author is supported partly by the Natural Science Foundation of Guangdong Province (no. 07006700).

#### References

- D. D. Clahane and S. Stević, “Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball,”
*Journal of Inequalities and Applications*, vol. 2006, Article ID 61018, 11 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Li and S. Stević, “Some characterizations of the Besov space and the $\alpha $-Bloch space,”
*Journal of Mathematical Analysis and Applications*, vol. 346, no. 1, pp. 262–273, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - S. Li and H. Wulan, “Characterizations of $\alpha $-Bloch spaces on the unit ball,”
*Journal of Mathematical Analysis and Applications*, vol. 343, no. 1, pp. 58–63, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - K. Zhu,
*Spaces of Holomorphic Functions in the Unit Ball*, vol. 226 of*Graduate Texts in Mathematics*, Springer, New York, NY, USA, 2005. View at MathSciNet - S. Stević, “On a new operator from the logarithmic Bloch space to the Bloch-type space on the unit ball,”
*Applied Mathematics and Computation*, vol. 206, no. 1, pp. 313–320, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. H. Zhu, “Multipliers of BMO in the Bergman metric with applications to Toeplitz operators,”
*Journal of Functional Analysis*, vol. 87, no. 1, pp. 31–50, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Fu and X. Zhu, “Weighted composition operators on some weighted spaces in the unit ball,”
*Abstract and Applied Analysis*, vol. 2008, Article ID 605807, 8 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Girela, J. Á. Peláez, F. Pérez-González, and J. Rättyä, “Carleson measures for the Bloch space,”
*Integral Equations and Operator Theory*, vol. 61, no. 4, pp. 511–547, 2008. 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 - A. Aleman and J. A. Cima, “An integral operator on ${H}^{p}$ and Hardy's inequality,”
*Journal d'Analyse Mathématique*, vol. 85, pp. 157–176, 2001. 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*, vol. 232 of*Contemporary Mathematics*, pp. 299–311, American Mathematical Society, Providence, RI, USA, 1999. 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 composition and integral type operators from ${H}^{\infty}$ to the Bloch space,”
*Complex Variables and Elliptic Equations*, vol. 53, no. 5, pp. 463–474, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Li and S. Stević, “Products of Volterra type operator and composition operator from ${H}^{\infty}$ and Bloch spaces to Zygmund spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 345, no. 1, pp. 40–52, 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 - 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 Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D.-C. Chang, S. Li, and S. Stević, “On some integral operators on the unit polydisk and the unit ball,”
*Taiwanese Journal of Mathematics*, vol. 11, no. 5, pp. 1251–1285, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D.-C. Chang and S. Stević, “Estimates of an integral operator on function spaces,”
*Taiwanese Journal of Mathematics*, vol. 7, no. 3, pp. 423–432, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Cesàro averaging operators,”
*Mathematische Nachrichten*, vol. 248-249, no. 1, pp. 185–189, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Boundedness and compactness of an integral operator on a weighted space on the polydisc,”
*Indian Journal of Pure and Applied Mathematics*, vol. 37, no. 6, pp. 343–355, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Boundedness and compactness of an integral operator in a mixed norm space on the polydisk,”
*Sibirskiĭ Matematicheskiĭ Zhurnal*, vol. 48, no. 3, pp. 694–706, 2007. View at Google Scholar · View at MathSciNet - K. Avetisyan and S. Stević, “Extended Cesàro operators between different Hardy spaces,”
*Applied Mathematics and Computation*, vol. 207, no. 2, pp. 346–350, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Hu, “Extended Cesàro operators on mixed norm spaces,”
*Proceedings of the American Mathematical Society*, vol. 131, no. 7, pp. 2171–2179, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Hu, “Extended Cesáro operators on the Bloch space in the unit ball of ${\u2102}^{n}$,”
*Acta Mathematica Scientia. Series B*, vol. 23, no. 4, pp. 561–566, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Hu, “Extended Cesàro operators on Bergman spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 296, no. 2, pp. 435–454, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Li, “Riemann-Stieltjes operators from $F(p,q,s)$ spaces to $\alpha $-Bloch spaces on the unit ball,”
*Journal of Inequalities and Applications*, vol. 2006, Article ID 27874, 14 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Li and S. Stević, “Integral type operators from mixed-norm spaces to $\alpha $-Bloch spaces,”
*Integral Transforms and Special Functions*, vol. 18, no. 7-8, pp. 485–493, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Li and S. Stević, “Riemann-Stieltjes operators on Hardy spaces in the unit ball of ${\u2102}^{n}$,”
*Bulletin of the Belgian Mathematical Society. Simon Stevin*, vol. 14, no. 4, pp. 621–628, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Li and S. Stević, “Riemann-Stieltjes-type integral operators on the unit ball in ${\u2102}^{n}$,”
*Complex Variables and Elliptic Equations*, vol. 52, no. 6, pp. 495–517, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Li and S. Stević, “Compactness of Riemann-Stieltjes operators between $F(p,q,s)$ spaces and $\alpha $-Bloch spaces,”
*Publicationes Mathematicae Debrecen*, vol. 72, no. 1-2, pp. 111–128, 2008. View at Google Scholar · View at MathSciNet - S. Li and S. Stević, “Riemann-Stieltjes operators between different weighted Bergman spaces,”
*Bulletin of the Belgian Mathematical Society. Simon Stevin*, vol. 15, no. 4, pp. 677–686, 2008. View at Google Scholar · View at MathSciNet - S. Li and S. Stević, “Riemann-Stieltjes operators between mixed norm spaces,”
*Indian Journal of Mathematics*, vol. 50, no. 1, pp. 177–188, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Li and S. Stević, “Cesàro-type operators on some spaces of analytic functions on the unit ball,”
*Applied Mathematics and Computation*, vol. 208, no. 2, pp. 378–388, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On an integral operator on the unit ball in ${\u2102}^{n}$,”
*Journal of Inequalities and Applications*, vol. 2005, no. 1, pp. 81–88, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Tang, “Extended Cesàro operators between Bloch-type spaces in the unit ball of ${\u2102}^{n}$,”
*Journal of Mathematical Analysis and Applications*, vol. 326, no. 2, pp. 1199–1211, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Xiao, “Riemann-Stieltjes operators on weighted Bloch and Bergman spaces of the unit ball,”
*Journal of the London Mathematical Society. Second Series*, vol. 70, no. 1, pp. 199–214, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Compactness of the Hardy-Littlewood operator on some harmonic function spaces,”
*Siberian Mathematical Journal*, vol. 50, no. 1, pp. 167–180, 2009. View at Publisher · View at Google Scholar - S. Stević, “On a new operator from ${H}^{\infty}$ to the Bloch-type space on the unit ball,”
*Utilitas Mathematica*, vol. 77, pp. 257–263, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On a new integral-type operator from the weighted Bergman space to the Bloch-type space on the unit ball,”
*Discrete Dynamics in Nature and Society*, vol. 2008, Article ID 154263, 14 pages, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet