Abstract and Applied Analysis

Volume 2014 (2014), Article ID 107560, 10 pages

http://dx.doi.org/10.1155/2014/107560

Research Article

Weighted Composition Operator from Mixed Norm Space to Bloch-Type Space on the Unit Ball

Yu-Xia Liang¹ and Ren-Yu Chen²

¹School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, China

²Department of Mathematics, Tianjin University, Tianjin 300072, China

Received 13 May 2014; Accepted 19 July 2014; Published 7 August 2014

Academic Editor: Henrik Kalisch

Copyright © 2014 Yu-Xia Liang and Ren-Yu Chen. 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.

We discuss the boundedness and compactness of the weighted composition operator from mixed norm space to Bloch-type space on the unit ball of .

1. Introduction

Let be the class of all holomorphic functions on and the collection of all the holomorphic self-mappings of , where is the unit ball in the -dimensional complex space . Let denote the Lebesegue measure on normalized so that and the normalized rotation invariant measure on the boundary of . For , let

be the radial derivative of .

A positive continuous function on is called normal (see, e.g., [1]) if there exist three constants , and , such that for

In the rest of this paper we always assume that is normal on , and from now on if we say that a function is normal we will also suppose that it is radial on , that is, for .

Let , , and be normal on . is said to belong to the mixed norm space if is a measurable function on and , where

If , then is just the space is measurable function on .

Let . If , then is just the weighted Bergman space . In particular, is Bergman space if and . Otherwise, if and   , then is the Dirichlet-type space.

For , , let ; it is easy to see that the mixed norm space , written by , consists of all such that

Now is said to belong to Bloch-type space if

where is the complex gradient of .

It is clear that is a Banach space with norm . For , we denote

where

It was proved that , , and are equivalent for in [2, 3].

Let , ; the composition operator induced by is defined by

and the weighted composition operator is defined by

for . We can regard this operator as a generalization of a multiplication operator and a composition operator . That is, when , we obtain and when we obtain .

It is interesting to provide a function theoretic characterization when and induce a bounded or compact weighted composition operator between some spaces of holomorphic functions on . Recently, this operator is well studied by many papers; see, for example, [3–17] and their references therein. In particular, Stević [18] gave some conditions of weighted composition operators between mixed-norm spaces and spaces on the unit ball. Zhou and Chen [19] discussed weighted composition operators from to Bloch-type spaces on the unit ball. More recently, the weighted composition operator from Bers-type space to Bloch-type space on the unit ball was studied in [6]. Now in this paper, we will continue this line of research and characterize the boundedness and compactness of the weighted composition operator acting from mixed-norm spaces to Bloch-type space on the unit ball of . The paper is organized as follows. In Section 2, we give some lemmas. The main results are given in Section 3.

Throughout the remainder of this paper, will denote a positive constant; the exact value of which will vary from one appearance to the next. The notation means that there is a positive constant such that .

2. Some Lemmas

Lemma 1. Assume that , , and . Then there is a positive constant which is independent of such that

Proof. We first prove (10). By the monotonicity of the integral means and [20, Theorem 1.12] we have that

from which the desired result (10) follows.

Next we prove (11). By the monotonicity of the integral means, using the well-known asymptotic formula (e.g., [21, Theorem 2]), we obtain that

By [20, Theorem 1.12], it follows that

Then the desired result (11) follows. This completes the proof.

From the above lemma, when , then

For ,  , denote the Bergman metric of by

Lemma 2. Let and . Then

for all , where denotes the Jacobian matrix of and

Proof. Let . If , the desired result is obvious. If , from the definition of ,

Thus

The desired result follows from (20). The proof is completed.

The proof of the next lemma is standard; see, for example, [4, Proposition 3.11]. Hence, it is omitted.

Lemma 3. Assume that , , is a normal function, and , . Then is compact if and only if for any bounded sequence in which converges to zero uniformly on compact subsets of as ; then , as .

Lemma 4. For and , one has

Proof.

This completes the proof.

3. The Boundedness and Compactness of

Theorem 5. Assume that , , is a normal function, and , . Then is bounded if and only if

Proof  

Sufficiency. Assume that (23) and (24) hold. Then for any , if for , by Lemma 1 and Lemma 2, it follows that

When for . From (23) we can easily obtain

Combining (25) and (26), the boundedness of follows.

Necessity. Suppose that is bounded. Firstly, we assume that and , where and .

If , where , choose the function

By [20, Theorem 1.12] and Lemma 4 we have that

Then and . Moreover, and

Thus

By the definition of and (30) it follows that

This shows that when , (24) follows.

On the other hand, if . For , let and , when ; otherwise when . Take

By [20, Theorem 1.12] and Lemma 4 we obtain that

Hence and . Moreover and

Similar to the proof of (30), we obtain that

It follows from (35) that

That is, when , (24) follows. Combining the above two cases, the desired result (24) holds.

For the general situation, we can use some unitary transform to make and we can prove (11) by taking the function . By the linearity of the unitary transform , , and the normalized rotation invariant measure on the boundary , we get that

Next we prove (23). Set the function

for fixed and . Then,

By [20, Theorem 1.12], it follows that

Applying Lemma 4 we have that

Therefore , and . Besides,

Therefore,

It follows from (43) and (24) that

Combining (44) and (45), the desired result (23) holds. This completes the proof.

Theorem 6. Assume that , , is a normal function, and , . Then is compact if and only if the followings are all satisfied:

(a) and for ;(b)

(c)

Proof  

Sufficiency. Suppose that , , and hold. Then for any , there is , such that

when .

Let be any sequence which converges to uniformly on compact subsets of satisfying . Then and converge to uniformly on . Hence

If and , by Lemma 1 and Lemma 2, we have

When ,

Combining (50) and (51) we obtain that

If , by (a), we have that

Combining (49), (52), (53), and Lemma 4, it follows that the is compact.

Necessity. Assume that is compact. It is obvious that is bounded. Then taking and by the boundedness of , it follows that

This shows that .

On the other hand, for , take the function . By the boundedness of , we get that

That is, for . Hence we obtain (a).

Next we prove (b) and (c). Let be a sequence in such that as . We can still suppose , where and is the vector . That is, , .

If , where . Let

From Theorem 5 we know that , and we notice that converges to 0 uniformly on compact subsets of when . By Lemma 3 we have . Then by a similar proof of (30) in Theorem 5 we have

And similar to the proofs of (31) and (57) we get that

On the other hand, we consider the case of . For , let and , when ; otherwise when . Take

Then ,  , and converges to 0 uniformly on compact subsets of when . By Lemma 3 we have . Notice that and

By a similar proof of (30), it follows that

And similar to the proofs of (31) and (61), we obtain

Combining (58) and (62), (47) holds under the two cases.

For the general situation, if there exists such that , then there is a unitary transformation such that , . And we can prove (47) by taking the function sequence and the details are omitted.

Next we prove (46). Let be a sequence in such that as . Choose

Then ,  , and . It is obvious that uniformly on compact subsets of as . By Lemma 3 we have that . Then by the similar proof of (44) we obtain

From the similar proof of (45) it follows that

Combining (64) and (65) we obtain (46). This completes the proof.

Corollary 7. Assume that ,  , is a normal function, and . Then is bounded if and only if

Corollary 8. Assume that , , is a normal function, and . Then is compact if and only if

And   for .

Corollary 9. Assume that , , is a normal function, and . Then is bounded if and only if

Corollary 10. Assume that , , is a normal function, and . Then is compact if and only if the following are all satisfied:

(a) and for any ;(b)

(c)

Conflict of Interests

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

Acknowledgment

This work was supported in part by the National Natural Science Foundation of China (Grants nos. 11371276; 11301373; and 11201331).

References

A. L. Shields and D. L. Williams, “Bonded projections, duality, and multipliers in spaces of analytic functions,” Transactions of the American Mathematical Society, vol. 162, pp. 287–302, 1971.

H. Chen and P. Gauthier, “Boundedness from below of composition operators on -Bloch spaces,” Canadian Mathematical Bulletin, vol. 51, no. 2, pp. 195–204, 2008.

X. J. Zhang and J. C. Liu, “Composition operators from weighted Bergman spaces to μ-Bloch spaces,” Chinese Annals of Mathematics A, vol. 28, no. 2, pp. 255–266, 2007.

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.

L. Luo and S. Ueki, “Weighted composition operators between weighted Bergman spaces and Hardy spaces on the unit ball of ,” Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 88–100, 2007.

Y. Liang, Z. Zhou, and X. Dong, “Weighted composition operator from Bers-type space to Bloch-type space on the unit ball,” Bulletin of the Malaysian Mathematical Sciences Society II, vol. 36, no. 3, pp. 833–844, 2013.

B. D. MacCluer and R. H. Zhao, “Essential norms of weighted composition operators between Bloch-type spaces,” The Rocky Mountain Journal of Mathematics, vol. 33, no. 4, pp. 1437–1458, 2003.

S. Ohno, “Weighted composition operators between and the Bloch space,” Taiwanese Journal of Mathematics, vol. 5, no. 3, pp. 555–563, 2001.

S. Ohno, K. Stroethoff, and R. Zhao, “Weighted composition operators between Bloch-type spaces,” The Rocky Mountain Journal of Mathematics, vol. 33, no. 1, pp. 191–215, 2003.

A. Montes-Rodríguez, “Weighted composition operators on weighted Banach spaces of analytic functions,” Journal of the London Mathematical Society, vol. 61, no. 3, pp. 872–884, 2000.

S. Stevic and S. Ueki, “Weighted composition operators from the weighted Bergman space to the weighted Hardy space on the unit ball,” Applied Mathematics and Computation, vol. 215, no. 10, pp. 3526–3533, 2010.

S. Ueki and L. Luo, “Essential norms of weighted composition operators between weighted Bergman spaces of the ball,” Acta Scientiarum Mathematicarum, vol. 74, no. 3-4, pp. 827–841, 2008.

S. Ueki and L. Luo, “Compact weighted composition operators and multiplication operators between Hardy spaces,” Abstract and Applied Analysis, vol. 2008, Article ID 196498, 12 pages, 2008.

J. Xiao, “Composition operators associated with Bloch-type spaces,” Complex Variables. Theory and Application, vol. 46, no. 2, pp. 109–121, 2001.

X. J. Zhang, “Composition type operator from Bergman space to μ-BLOch type space in ,” Journal of Mathematical Analysis and Applications, vol. 298, no. 2, pp. 710–721, 2004.

Z. Zhou and J. Shi, “Compactness of composition operators on the Bloch space in classical bounded symmetric domains,” The Michigan Mathematical Journal, vol. 50, no. 2, pp. 381–405, 2002.

H. Zeng and Z. Zhou, “An estimate of the essential norm of a composition operator from to in the unit ball,” Journal of Inequalities and Applications, vol. 2010, Article ID 132970, 18 pages, 2010.

S. Stević, “Weighted composition operators between mixed norm spaces and spaces in the unit ball,” Journal of Inequalities and Applications, vol. 2007, Article ID 28629, 9 pages, 2007.

Z.-H. Zhou and R.-Y. Chen, “Weighted composition operators from to Bloch type spaces on the unit ball,” International Journal of Mathematics, vol. 19, no. 8, pp. 899–926, 2008.

K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, vol. 226 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2005.

Z. J. Hu, “Extended Cesàro operators on mixed norm spaces,” Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2171–2179, 2003.