Journal of Function Spaces
Volume 2014 (2014), Article ID 720875, 7 pages
http://dx.doi.org/10.1155/2014/720875

Research Article

Lipschitz Estimates for One-Sided Cohen’s Commutators on Weighted One-Sided Triebel-Lizorkin Spaces

Xiao Jin Zhang1,2 and Xian Ming Hou3,4

¹School of Sciences, China University of Mining & Technology, Beijing 100083, China
²Department of Basic Curriculum, North China Institute of Science and Technology, Hebei 065201, China
³Department of Mathematics, Linyi University, Linyi 276005, China
⁴School of Sciences, Shandong Normal University, Jinan 250014, China

Received 19 September 2014; Accepted 20 November 2014; Published 8 December 2014

Academic Editor: Guozhen Lu

Copyright © 2014 Xiao Jin Zhang and Xian Ming Hou. 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 introduce one-sided Cohen’s commutators of singular integral operators and fractional integral operators, respectively. Using the extrapolation of one-sided weights, we establish the boundedness of these operators from weighted Lebesgue spaces to weighted one-sided Triebel-Lizorkin spaces.

1. Introduction

The one-sided commutators considered in this paper are related to the commutators studied by Calderón in [1]. Cohen [2] defined the Cohen type commutators of Calderón-Zygmund singular integrals (for convenience, we only consider the -dimensional case) by

where satisfies certain homogeneity, smoothness, and symmetry conditions. Chen and Lu [3] proved the boundedness of the commutators from Lebesgue spaces to Triebel-Lizorkin spaces for . A function , , if it satisfies

Similar to an unbounded function , the functions in are not necessarily bounded either (e.g., ). Therefore, it is also nontrivial to investigate the commutators generated by operators and Lipschitz functions.

In the one-sided case, we will study the weighted boundedness of the commutators from weighted Lebesgue spaces to weighted Triebel-Lizorkin spaces. The one-sided operators were motivated as not only the generalization of the theory of both-sided ones but also the requirement in ergodic theory. Lots of results show that, for a class of smaller operators (one-sided operators) and a class of wider weights (one-sided weights), many results in harmonic analysis still hold; see [4–14]. However, for one-sided weights, classical reverse Hölder’s inequality does not hold.

A function is called a one-sided Calderón-Zygmund kernel (OCZK) if satisfies

with support in or . An example of such a kernel is

where denotes the characteristic function of a set . In [15], Aimar et al. introduced the one-sided Calderón-Zygmund singular integrals which are defined by

where the kernels are OCZKs.

The study of weights for one-sided operators is motivated by their natural appearance in harmonic analysis, such as the one-sided Hardy-Littlewood maximal operator:

Recently, Sawyer [13] introduced the one-sided classes , , which are defined by

when ; also, for ,

for some constant .

Very recently, Fu and Lu [16] introduced a class of one-sided Triebel-Lizorkin spaces and their weighted version.

Definition 1. For , , and an appropriate weight , the weighted one-sided Triebel-Lizorkin spaces and are defined by

In [16], the authors proved the boundedness for the one-sided commutators (with symbols ) of Calderón-Zygmund singular integral, , and fractional integral, , respectively. and are defined as follows:

Let be locally integrable functions on . Denote by the th order remainder of the Taylor series of at about , precisely:

Cohen’s commutators of one-sided singular integrals are defined by

Obviously, when , . Therefore, the results of this paper are the extension of [16].

Theorem 2. Assume that and . Let , . Then, one gets the following.

(i)If , there exists the constant such that

(ii)If , there exists the constant such that

Theorem 3. Assume that and . Let , . Then one gets the following.

(i)If , there exists the constant such that

(ii)If , there exists the constant such that

The other main objects in this paper are one-sided Cohen’s commutators of fractional integral operators, which are defined by

Obviously, when , .

Theorem 4. Assume that and . Let , . Then one gets the following.

(i)If , there exists the constant such that

(ii)If , there exists the constant such that

Theorem 5. Assume that , and . Let , . Then one gets the following.

(i)If , there exists the constant such that

(ii)If , there exists the constant such that

Throughout this paper the letter will denote a positive constant that may vary from line to line.

2. Estimates for the One-Sided Cohen Type Commutators of Singular Integrals

This section begins with some necessary lemmas.

Lemma 6 (see [17]). If is a function with derivatives of order in , then, for the th remainder of , there is a constant such that

The primary tool in the proof of Theorem 3 is an extrapolation theorem that appeared in [18].

Lemma 7. Let be a sublinear operator defined in satisfying

for all . Then, for ,

holds whenever .

Lemma 8 (see [9]). Suppose that ; then there exists such that, for all , .

Proof of Theorem 2. For convenience, we only prove case . By Lemma 6 and assumption (4), it is easy to prove that

where . In the last inequality, we use the boundedness of that appeared in [19].

Proof of Theorem 3. Without loss of generality, we only prove case (i). Let , , and . Write , where . Then

To estimate , we have

Noting the fact that and using Lemma 6 and (4) of kernel , we consider , , and , respectively:

where we use the differential mean value theorem for .
Combining the above estimate, we have

Consider the following two sublinear operators defined on :

The above inequalities imply that

Thus, we will discuss the boundedness of these two operators.
For , by Hölder’s inequality and Theorem 2, we get

where , for .
Next, we have

By Lemma 7, we obtain

for all .
For , set , . Then

where , for . Then

By Lemma 7, we have

for all .
Combining estimates (36) and (39), the proof is completed.

3. Estimates for the One-Sided Commutators of Cohen Type of Fractional Integrals

In order to prove Theorems 4 and 5, we will introduce the one-sided extrapolation lemma.

Lemma 9 (see [18]). Let and let be sublinear operator defined in satisfying

for every and ; then, for every , , and , the inequality

holds.

Lemma 10 (see [20]). Suppose that ; then and   for all .

Proof of Theorem 4. For convenience, we only give the proof of case (i). Using Lemma 6 and the boundedness of one-sided fractional integral operators that appeared in [19], we get

where .

Proof of Theorem 5. Let , . Write , where , . Then

By Lemma 6, we have

Consider the following two sublinear operators defined on :

We conclude from (43) and (45) that

If , then ; see [16]. By Lemma 8, there exists , such that . Let and . Using the boundedness of from to , together with Lemma 6, we have

In the last inequality, we use the fact for all .
By Lemma 9, there exists such that

thus

for all .
For , let , . Then

where () for all . Then

By Lemma 9, the inequlity

holds for .
This completes the proof of case (i). For case (ii), we omit the details since they are similar to those of the proof of (i) with instead of .

Conflict of Interests

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

Acknowledgments

The authors cordially thank the referees for their careful reading and helpful comments. This work was partially supported by FRFCU (Grant nos. 3142013027 and 3142014127), North China Institute of Science and Technology (Grant no. HKXJZD201402), and NSF of China (Grant nos. 11271175 and 11171345).

References

A. P. Calderón, “Commutators of singular integral operators,” Proceedings of the National Academy of Sciences of the United States of America, vol. 53, pp. 1092–1099, 1965.

J. Cohen, “A sharp estimate for a multilinear singular integral in ,” Indiana University Mathematics Journal, vol. 30, no. 5, pp. 693–702, 1981.

W. G. Chen and S. Z. Lu, “On multilinear singular integrals in ,” Advances in Mathematics (China), vol. 29, no. 4, pp. 325–330, 2000.

Z. Fu, S. Lu, Y. Pan, and S. Shi, “Some one-sided estimates for oscillatory singular integrals,” Nonlinear Analysis: Theory, Methods & Applications, vol. 108, pp. 144–160, 2014.

Z. Fu, S. Lu, S. Sato, and S. Shi, “On weighted weak type norm inequalities for one-sided oscillatory singular integrals,” Studia Mathematica, vol. 207, no. 2, pp. 137–151, 2011.

Z. W. Fu, G. L. Wang, and Q. Y. Wu, “Weighted Lipschitz estimates for commutators of one-sided operators on one-sided Triebel-LIZorkin spaces,” Mathematical Inequalities & Applications, vol. 17, no. 2, pp. 611–625, 2014.

Z. Fu, S. Lu, Y. Pan, and S. Shi, “Boundedness of one-sided oscillatory integral operators on weighted Lebesgue spaces,” Abstract and Applied Analysis, vol. 2014, Article ID 291397, 7 pages, 2014.

M. Lorente and M. S. Riveros, “Weights for commutators of the one-sided discrete square function, the Weyl fractional integral and other one-sided operators,” Proceedings of the Royal Society of Edinburgh, Section A: Mathematics, vol. 135, no. 4, pp. 845–862, 2005.

R. A. Macías and M. S. Riveros, “One-sided extrapolation at infinity and singular integrals,” Proceedings of the Royal Society of Edinburgh: Section A Mathematics, vol. 130, no. 5, pp. 1081–1102, 2000.

F. J. Martín-Reyes and A. de la Torre, “One-sided BMO spaces,” Journal of the London Mathematical Society, vol. 49, no. 3, pp. 529–542, 1994.

C. Pérez, “Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function,” The Journal of Fourier Analysis and Applications, vol. 3, no. 6, pp. 743–756, 1997.

S. Shi and Z. Fu, “Estimates of some operators on one-sided weighted Morrey spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 829218, 9 pages, 2013.

E. Sawyer, “Weighted inequalities for the one-sided Hardy-Littlewood maximal functions,” Transactions of the American Mathematical Society, vol. 297, no. 1, pp. 53–61, 1986.

A. de la Torre and J. L. Torrea, “One-sided discrete square function,” Studia Mathematica, vol. 156, no. 3, pp. 243–260, 2003.

H. Aimar, L. Forzani, and F. J. Martín-Reyes, “On weighted inequalities for singular integrals,” Proceedings of the American Mathematical Society, vol. 125, no. 7, pp. 2057–2064, 1997.

Z. W. Fu and S. Z. Lu, “One-sided Triebel-Lizorkin space and its applications,” Scientia Sinica Mathematica, vol. 41, pp. 43–52, 2011 (Chinese).

Y. Ding, S. Z. Lu, and K. Yabuta, “Multilinear singular and fractional integrals,” Acta Mathematica Sinica, vol. 22, no. 2, pp. 347–356, 2006.

M. Lorente and M. S. Riveros, “Two extrapolation theorems for related weights and applications,” Mathematical Inequalities & Applications, vol. 10, no. 3, pp. 643–660, 2007.

K. F. Andersen and E. T. Sawyer, “Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators,” Transactions of the American Mathematical Society, vol. 308, no. 2, pp. 547–558, 1988.

F. J. Martín-Reyes, L. Pick, and A. de la Torre, “ condition,” Canadian Journal of Mathematics, vol. 45, no. 6, pp. 1231–1244, 1993.