Research Article Product of Extended Cesàro Operator and Composition Operator from Lipschitz Space to F p, q, s Space on the Unit Ball

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1 Abstract and Applied Analysis Volume 2011, Article ID , 9 pages doi: /2011/ Research Article Product of Extended Cesàro Operator and Composition Operator from Lipschitz Space to F p, q, s Space on the Unit all Yu-Xia Liang and Ze-Hua Zhou Department of Mathematics, Tianjin University, Tianjin , China Correspondence should be addressed to Ze-Hua Zhou, zehuazhou2003@yahoo.com.cn Received 16 January 2011; Accepted 16 March 2011 Academic Editor: Ljubisa Kocinac Copyright q 2011 Y.-X. Liang and Z.-H. Zhou. 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. This paper characterizes the boundedness and compactness of the product of extended Cesàro operator and composition operator from Lipschitz space to F p, q, s space on the unit ball of C n. 1. Introduction Let be the unit ball in the n-dimensional complex space C n,theclosureof will be written as.ydv we denote the Lebesgue measure on normalized so that v 1andbydσ the normalized rotation invariant measure on the boundary S of.leth be the class of all holomorphic functions on and S the collection of all the holomorphic selfmappings of.denotebya the unit ball algebra of all continuous functions on that are holomorphic on. For f H, let Rf z n j 1 z j f z j z 1.1 be the radial derivative of f. We recall that the α-loch space α for α 0 consists of all f H such that ( ) ( α f 1 z 2) α Rf z <. 1.2 z

2 2 Abstract and Applied Analysis The expression α f defines a seminorm while the natural norm is given by f α f 0 α f. Thisnormmakes α into a anach space. When α 1, 1 is the well known loch space. For α 0, 1, L α denotes the holomorphic Lipschitz space of order α which is the set of all f H such that, for some C>0, f z f w C z w α 1.3 for every z, w. It is clear that each space L α contains the polynomials and is contained in the ball algebra A. ItiswellknownthatL α is endowed with a complete norm Lα that is given by f Lα f 0 sup z / w;z,w { } f z f w z w α. 1.4 See 1, 2 for more information of the Lipschitz spaces on. For a,letg z, a log ϕ a z 1 be Green s function on with logarithmic singularity at a, whereϕ a is the Möbius transformation of with ϕ a 0 a, ϕ a a 0, and ϕ a ϕ 1 a. Let 0 <p,s<, n 1 <q<, afunctionf H is said to belong to F p, q, s if see, e.g., 3 5 f p F p,q,s f 0 p Rf z p( 1 z 2) q g s z, a dυ z <. 1.5 If X is a anach space of holomorphic functions on a domain Ω and if ϕ is a holomorphic self-map of Ω, the composition operator of symbol ϕ is defined by C ϕ f f ϕ. The study of composition operators consists in the comparison of the properties of the operator C ϕ with that of the function ϕ itself, which is called the symbol of C ϕ.onecan characterize boundedness and compactness of C ϕ and many other properties. We refer to the books in 6, 7 and to some recent papers in 4, 5, 8 tolearnmuchmoreonthissubject. Let h H, the following integral-type operator was first introduced in 9 T h f z 1 0 f tz Rh tz dt, f H, z. 1.6 t This operator is called generalized Cesàro operator. It has been well studied in many papers, see, for example, 3, 9 24 as well as the related references therein. It is natural to discuss the product of extended Cesàro operator and composition operator. For h H and ϕ S, the product can be expressed as T h C ϕ f z 1 0 f ( ϕ tz ) Rh tz dt, f H, z. 1.7 t

3 Abstract and Applied Analysis 3 It is interesting to characterize the boundedness and compactness of the product operator on all kinds of function spaces. Even on the disk of C, some properties are not easily managed; see some recent papers in 18, uilding on those foundations, the present paper continues this line of research and discusses the operator in high dimension. The remainder is assembled as follows: in Section 2, we state a couple of lemmas. In Section 3, we characterize the boundedness and compactness of the product T h C ϕ of extended Cesàro operator and composition operator from Lipschitz spaces to F p, q, s spaces on the unit ball of C n. Throughout the remainder of this paper, C will denote a positive constant, the exact valueofwhichwillvaryfromoneappearancetothenext.thenotationa means that there is a positive constant C such that /C A C. 2. Some Lemmas To begin the discussion, let us state a couple of lemmas, which are used in the proofs of the main results. Lemma 2.1. Suppose that f, h H. Then, R [ T h C ϕ ( f )] z f ) Rh z. 2.1 Proof. The proof of this Lemma follows by standard arguments see, e.g., 9, 29, 30. Lemma 2.2 see 2, 31. If 0 <α<1,then 1 α L α ; furthermore, f 1 α f Lα 2.2 as f varies through L α. The following criterion for compactness follows from standard arguments similar to the corresponding lemma in 6. Hence, we omit the details. Lemma 2.3. Assume that h H and ϕ S. Suppose that X or Y is one of the following spaces L α, F p, q, s. Then,T h C ϕ : X Y is compact if and only if T h C ϕ : X Y is bounded, and for any bounded sequence {f k } k N in X which converges to zero uniformly on compact subsets of as k,onehas T h C ϕ f k Y 0 as k. Lemma 2.4 see 4, 5. If f α,then f z Cf a, 0 <α<1, 2.3 f z Cf e a ln 1 z, 2 α 1, 2.3 f z C f a ( 1 z 2) α 1, α > The next lemma was obtained in 32.

4 4 Abstract and Applied Analysis Lemma 2.5. If a>0, b>0, then the elementary inequality holds a p b p, 0 <p<1, a b p 2 p 1 a p b p, p It is obvious that Lemma 2.5 holds for the sum of finite number k, thatis, ( ) a 1 a k p C a p 1 ap k, 2.5 where a 1,...,a k > 0andC is a positive constant. Lemma 2.6 see 4, 5. For 0 <p, s<, n 1 <q<, q s> 1, thereexistsc>0 such that for every ω. sup (1 w 2) p 1 z, w n 1 q p ( 1 z 2) q C 2.6 Lemma 2.7 see 4. There is a constant C>0 so that, for all t> 1 and z,onehas ln 1 1 z, w 2 (1 w 2) t ( ) 2 1 dν z C ln. 2.7 n 1 t 1 z, w 1 z 2 Lemma 2.8 see 4, 5. Suppose that 0 <p, s<, n 1 <q<,andq s > 1.Iff F p, q, s, then f n 1 q /p,and f n 1 q /p C f F p,q,s. Lemma 2.9. Let {f k } k N be a bounded sequence in F p, q, s which converges to zero uniformly on compact subsets of the unit ball,where n 1 q /p < 1.Then,lim k sup z f k z 0. Proof. It follows from Lemma 2.8 that F p, q, s n 1 q /p and f n 1 q /p C f F p,q,s for any f F p, q, s. So,when n 1 q /p < 1, the proof of this lemma is similar to that of Lemma 3.6 of 33, hence the proof is omitted. 3. The oundedness and Compactness of the Operator T h C ϕ : L α F p, q, s Theorem 3.1. Assume that α 0, 1, 0 <p, s<, n 1 <q<, q s> 1, ϕ S, and h H. Then,T h C ϕ : L α F p, q, s is bounded if and only if h F p, q, s. Proof. Assume that h F p, q, s.since0< 1 α<1, by Lemmas 2.2 and 2.4, foranyf L α, we have f z Cf 1 α C flα. 3.1

5 Abstract and Applied Analysis 5 Since T h C ϕ f 0 0, by using Lemma 2.1 and relations 2.3 and 3.1,wehave Th C ϕ f p F p,q,s C sup f ) Rh z p ( 1 z 2) q f p 1 α C h p f p F p,q,s L <. α 3.2 Thus T h C ϕ : L α F p, q, s is bounded. Conversely, suppose that T h C ϕ : L α F p, q, s is bounded. Taking the function f z 1 L α,then Th C ϕ f p F p,q,s Th C ϕ f 0 p R ( T h C ϕ f ) z p( 1 z 2) q f ) Rh z p ( 1 z 2) q h p F p,q,s. 3.3 From which, the boundedness of T h C ϕ implies that h F p, q, s. This completes the proof of this theorem. Next, we characterize the compactness of T h C ϕ : L α F p, q, s. Theorem 3.2. Assume that α 0, 1, 0 <p, s<, n 1 <q<, q s> 1, ϕ S, and h H. Then,T h C ϕ : L α F p, q, s is compact if and only if T h C ϕ : L α F p, q, s is bounded, and lim sup 0. r 1 { ϕ z >r} 3.4 Proof. Assume that T h C ϕ : L α F p, q, s is bounded and 3.4 holds. It follows from Theorem 3.1 that h F p, q, s. Now, let {f j } j N be a bounded sequence of functions in L α such that f j 0uniformly on the compact subsets of as j. Suppose that sup j N f j Lα L. It follows from 3.4 that, for any ε>0, there exists r 0 0, 1 such that, for every r 0 <r<1, sup <ε. { ϕ z >r} 3.5

6 6 Abstract and Applied Analysis Set r 0 <r<1, then T h C ϕ f j p F p,q,s sup f j ) p { ϕ z r} { ϕ z >r} f j ) p fj ) p 3.6 I 1 I 2, where I 1 : I 2 : { ϕ z r} { ϕ z >r} fj ) p, f j ) p. 3.7 Let K {w : w r}, thenk is a compact subset of.sincef j compact subsets of as j and h F p, q, s,weget 0uniformlyon I 1 supf j w p sup w K { ϕ z r} h p F p,q,s fj w p C sup fj w p 0, j. w K w K 3.8 On the other hand, by 3.5 and Lemmas 2.2 and 2.4, it follows that I 2 C fj p sup 1 α { ϕ z >r} C f j p L α ε CL p ε. 3.9 Since ε is arbitrary, from the above inequalities, we get lim Th C ϕ f F p,q,s j 0. j 3.10 Hence, by 3.10 and Lemma 2.3,weconcludethatT h C ϕ : L α F p, q, s is compact. For the converse direction, we suppose that T h C ϕ : L α F p, q, s is compact. It is obvious that T h C ϕ : L α F p, q, s is bounded. Now, we prove 3.4. Setting the test functions f m z z m for fixed l {1,...,n}, l l where z z 1,...,z n and m 1, 2,... It is easy to check that f m l Lα C, andf m l 0

7 Abstract and Applied Analysis 7 uniformly on the compact subsets of as m.writeϕ ϕ 1,...,ϕ n,sincet h C ϕ : L α F p, q, s is compact, by Lemma 2.3, itfollowsthat,asm, T h C ϕ f m l p F p,q,s ϕl z mp Note that ϕ z 2 ϕ 1 z 2 ϕ n z 2 ϕ 1 z ϕ n z 2 ;bytherelation 3.11 and Lemma 2.5,wehave sup ϕ z mp sup ( n ϕl z mp ) l 1 ( n C sup ϕ l z ) Rh z mp p( 1 z 2) q 0, m. l This means that, for every ε>0, there is m 0 N such that, for every r 0, 1, r m0p sup sup sup { ϕ z >r} { ϕ z >r} { ϕ z >r} r m 0p ϕ z m 0 p ϕ z m 0 p 3.13 <ε. Thus, when r>2 1/m 0p, by the above inequality, we obtain sup < 2ε. { ϕ z >r} 3.14 From which, the desired result 3.4 holds. This completes the proof of this theorem. Remark 3.3. When ϕ z z, the product of extended Cesàro operator T h C ϕ is the generalized extended Cesàro operator T h ;thus,bytheorems3.1 and 3.2, wehavethefollowingtwo corollaries. Corollary 3.4. Assume that α 0, 1, 0 <p, s<, n 1 <q<, q s> 1, andh H. Then, T h : L α F p, q, s is bounded if and only if h F p, q, s.

8 8 Abstract and Applied Analysis Corollary 3.5. Assume that α 0, 1, 0 <p, s<, n 1 <q<, q s> 1, andh H. Then, T h : L α F p, q, s is compact if and only if T h : L α F p, q, s is bounded, and lim sup g s z, a 0. r 1 z >r 3.15 Acknowledgments The authors would like to thank the editor and referees for carefully reading the paper and providing corrections and suggestions for improvements. Z.-H. Zhou was supported in part by the National Natural Science Foundation of China Grant nos and References 1 W. Rudin, Function Theory in the Unit all of C n, vol. 241 of Fundamental Principles of Mathematical Science, Springer, erlin, Germany, K. Zhu, Spaces of Holomorphic Functions in the Unit all, vol. 226 of Graduate Texts in Mathematics, Springer, New York, NY, USA, S. Li and S. Stević, Compactness of Riemann-Stieltjes operators between F(p, q, s) spaces and α-loch spaces, Publicationes Mathematicae Debrecen, vol. 72, no. 1-2, pp , Z.-H. Zhou and R.-Y. Chen, Weighted composition operators from F(p, q, s) to loch type spaces on the unit ball, International Mathematics, vol. 19, no. 8, pp , H.-G. Zeng and Z.-H. Zhou, An estimate of the essential norm of a composition operator from F(p, q, s) to β α in the unit ball, journal of Inequalities and Applications, vol. 2010, Article ID , 22 pages, C. C. Cowen and. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Studiesin Advanced Mathematics, CRC Press, oca Raton, Fla, USA, J. H. Shapiro, Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics, Springer, New York, NY, USA, Z. Zhou and J. Shi, Compactness of composition operators on the loch space in classical bounded symmetric domains, The Michigan Mathematical Journal, vol. 50, no. 2, pp , Z. Hu, Extended Cesàro operators on mixed norm spaces, Proceedings of the American Mathematical Society, vol. 131, no. 7, pp , K. Avetisyan and S. Stević, Extended Cesàro operators between different Hardy spaces, Applied Mathematics and Computation, vol. 207, no. 2, pp , Z.-S. Fang and Z.-H. Zhou, Extended Cesáro operators on MOA spaces in the unit ball, Mathematical Inequalities, vol. 4, no. 1, pp , Z.-S. Fang and Z.-H. Zhou, Extended Cesàro operators from generally weighted loch spaces to Zygmund space, Mathematical Analysis and Applications, vol. 359, no. 2, pp , Z.-S. Fang and Z.-H. Zhou, Extended Cesáro operators on Zygmund spaces in the unit ball, Journal of Computational Analysis and Applications, vol. 11, no. 3, pp , D. Gu, Extended Cesàro operators from logarithmic-type spaces to loch-type spaces, Abstract and Applied Analysis, vol. 2009, Article ID , 9 pages, Z. Hu, Extended Cesàro operators on ergman spaces, Mathematical Analysis and Applications, vol. 296, no. 2, pp , S. Li and S. Stević, Riemann-Stieltjes operators on Hardy spaces in the unit ball of C n, ulletin of the elgian Mathematical Society, vol. 14, no. 4, pp , S. Li and S. Stević, Riemann-Stieltjes-type integral operators on the unit ball in C n, Complex Variables and Elliptic Equations, vol. 52, no. 6, pp , S. Li and S. Stević, Riemann-Stieltjes operators between different weighted ergman spaces, ulletin of the elgian Mathematical Society, vol. 15, no. 4, pp , 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 , 2009.

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