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

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.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Xiao Jin Zhang 1, 2 and Xian Ming Hou 3, 4

Academic Editor:Guozhen Lu

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

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

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.

1. Introduction

The one-sided commutators considered in this paper are related to the commutators studied by Calderon in [1]. Cohen [2] defined the Cohen type commutators of Calderon-Zygmund singular integrals (for convenience, we only consider the 1 -dimensional case) by [figure omitted; refer to PDF] where Ω satisfies certain homogeneity, smoothness, and symmetry conditions. Chen and Lu [3] proved the boundedness of the commutators ^{T A} from Lebesgue spaces to Triebel-Lizorkin spaces for ^{A ( k )} ∈ Li _{p α} ( R ) ( k = 0,1 ) . A function A ∈ Li _{p α} ( R ) , 0 < α < 1 , if it satisfies [figure omitted; refer to PDF] Similar to an unbounded function log ... | x | ∈ BMO , the functions in Li _{p α} are not necessarily bounded either (e.g., | x ^{| α} ∈ Li _{p α} ). 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 _{T A} 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 K is called a one-sided Calderon-Zygmund kernel (OCZK) if K satisfies [figure omitted; refer to PDF] with support in ^{R -} = ( - ∞ , 0 ) or ^{R +} = ( 0 , + ∞ ) . An example of such a kernel is [figure omitted; refer to PDF] where _{χ E} denotes the characteristic function of a set E . In [15], Aimar et al. introduced the one-sided Calderon-Zygmund singular integrals which are defined by [figure omitted; refer to PDF] where the kernels K 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: [figure omitted; refer to PDF]

Recently, Sawyer [13] introduced the one-sided _{A p} classes ^{A p +} , ^{A p -} , which are defined by [figure omitted; refer to PDF] when 1 < p < ∞ ; also, for p = 1 , [figure omitted; refer to PDF] for some constant C .

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

Definition 1.

For 0 < α < 1 , 1 < p < ∞ , and an appropriate weight ω , the weighted one-sided Triebel-Lizorkin spaces ^{F p , + α , ∞} ( ω ) and ^{F p , - α , ∞} ( ω ) are defined by [figure omitted; refer to PDF]

In [16], the authors proved the boundedness for the one-sided commutators (with symbols A ∈ Li _{p α} ( R ) ) of Calderon-Zygmund singular integral, ^{T A +} f , and fractional integral, ^{I β , A +} f , respectively. ^{T A +} f and ^{I β , A +} f are defined as follows: [figure omitted; refer to PDF]

Let A ( x ) be locally integrable functions on R . Denote by _{R m} ( A ; x , y ) the m th order remainder of the Taylor series of A at x about y , precisely: [figure omitted; refer to PDF]

Cohen's commutators of one-sided singular integrals are defined by [figure omitted; refer to PDF]

Obviously, when m = 1 , ^{T A , m +} = ^{T A +} . Therefore, the results of this paper are the extension of [16].

Theorem 2.

Assume that 1 < p < q < ∞ and 1 / p - 1 / q = α . Let ^{A ( k )} ( x ) ∈ L i _{p α} ( R ) , k = 0,1 , ... , m - 1 . Then, one gets the following.

(i) If ω ∈ ^{A ( p , q ) +} , there exists the constant C > 0 such that [figure omitted; refer to PDF]

(ii) If ω ∈ ^{A ( p , q ) -} , there exists the constant C > 0 such that [figure omitted; refer to PDF]

Theorem 3.

Assume that 1 < p < ∞ and 0 < α < 1 . Let ^{A ( k )} ( x ) ∈ L i _{p α} ( R ) , k = 0,1 , ... , m - 1 . Then one gets the following.

(i) If ω ∈ ^{A p +} , there exists the constant C > 0 such that [figure omitted; refer to PDF]

(ii) If ω ∈ ^{A p -} , there exists the constant C > 0 such that [figure omitted; refer to PDF]

The other main objects in this paper are one-sided Cohen's commutators of fractional integral operators, which are defined by [figure omitted; refer to PDF] Obviously, when m = 1 , ^{I β , A + , m} = ^{I β , A +} .

Theorem 4.

Assume that 1 < p < q < ∞ and 1 / p - 1 / q = α + β . Let ^{A ( k )} ( x ) ∈ L i _{p α} ( R ) , k = 0,1 , ... , m - 1 . Then one gets the following.

(i) If ω ∈ ^{A ( p , q ) +} , there exists the constant C > 0 such that [figure omitted; refer to PDF]

(ii) If ω ∈ ^{A ( p , q ) -} , there exists the constant C > 0 such that [figure omitted; refer to PDF]

Theorem 5.

Assume that 1 < p , q < ∞ , 0 < β < 1 , and 1 / p - 1 / q = β . Let ^{A ( k )} ( x ) ∈ L i _{p α} ( R ) , k = 0,1 , ... , m - 1 . Then one gets the following.

(i) If ω ∈ ^{A ( p , q ) +} , there exists the constant C > 0 such that [figure omitted; refer to PDF]

(ii) If ω ∈ ^{A ( p , q ) -} , there exists the constant C > 0 such that [figure omitted; refer to PDF]

Throughout this paper the letter C 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 A is a function with derivatives of order m - 1 in L i _{p α} ( R ) ( 0 < α ...4; 1 ) , then, for the m th remainder of A , there is a constant C > 0 such that [figure omitted; refer to PDF]

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

Lemma 7.

Let T be a sublinear operator defined in ^{C c ∞} ( R ) satisfying [figure omitted; refer to PDF] for all ^{ω - 1} ∈ ^{A 1 -} . Then, for 1 < p < ∞ , [figure omitted; refer to PDF] holds whenever w ∈ ^{A p +} .

Lemma 8 (see [9]).

Suppose that ω ∈ ^{A 1 -} ; then there exists _{[straight epsilon] 1} > 0 such that, for all 1 < r ...4; 1 + _{[straight epsilon] 1} , ^{w r} ∈ ^{A 1 -} .

Proof of Theorem 2.

For convenience, we only prove case ( i ) . By Lemma 6 and assumption (4), it is easy to prove that [figure omitted; refer to PDF] where 1 / p - 1 / q = α . In the last inequality, we use the boundedness of ^{I α +} that appeared in [19].

Proof of Theorem 3.

Without loss of generality, we only prove case (i). Let x ∈ R , h > 0 , and J = [ x , x + 8 h ] . Write f = _{f 1} + _{f 2} , where _{f 1} = f _{χ J} . Then [figure omitted; refer to PDF]

To estimate I I ( x ) , we have [figure omitted; refer to PDF]

Noting the fact that | x + 2 h - z | ~ | y - z | and using Lemma 6 and (4) of kernel K ( x ) , we consider I _{I 1} ( x ) , I _{I 2} ( x ) , and I _{I 3} ( x ) , respectively: [figure omitted; refer to PDF] where we use the differential mean value theorem for I _{I 3} .

Combining the above estimate, we have [figure omitted; refer to PDF]

Consider the following two sublinear operators defined on ^{C c ∞} ( R ) : [figure omitted; refer to PDF] The above inequalities imply that [figure omitted; refer to PDF] Thus, we will discuss the boundedness of these two operators.

For I ( x ) , by Hölder's inequality and Theorem 2, we get [figure omitted; refer to PDF] where 1 / s - 1 / r = α , ^{ω - s} ∈ ^{A 1 -} for ^{ω - 1} ∈ ^{A 1 -} .

Next, we have [figure omitted; refer to PDF] By Lemma 7, we obtain [figure omitted; refer to PDF] for all ω ∈ ^{A p +} .

For ^{M 2 +} , set _{I j} = [ x , x + ^{2 j} h ] , j ∈ ^{Z +} . Then [figure omitted; refer to PDF] where 0 < α < 1 , ^{ω - t} ∈ ^{A 1 -} for ^{ω - 1} ∈ ^{A 1 -} . Then [figure omitted; refer to PDF] By Lemma 7, we have [figure omitted; refer to PDF] for all ω ∈ ^{A p +} .

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 1 < _{p 0} < ∞ and let T be sublinear operator defined in ^{C c ∞} ( R ) satisfying [figure omitted; refer to PDF] for every x ∈ R and ω ∈ ^{A ( _{p 0} , ∞ ) +} ; then, for every 1 < p < _{p 0} , 1 / p - 1 / q = 1 / _{p 0} , and ω ∈ ^{A ( p , q ) +} , the inequality [figure omitted; refer to PDF] holds.

Lemma 10 (see [20]).

Suppose that ω ∈ ^{A ( p , q ) +} ; then ^{ω q} ∈ ^{A q +} and ^{ω p} ∈ ^{A p +} for all 1 < p < q ...4; ∞ .

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 ^{I α + β +} that appeared in [19], we get [figure omitted; refer to PDF] where 1 / p - 1 / q = α + β .

Proof of Theorem 5.

Let x ∈ R , h > 0 . Write f = _{f 1} + _{f 2} , where _{f 1} = f _{χ J} , J = [ x , x + 4 h ] . Then [figure omitted; refer to PDF] By Lemma 6, we have [figure omitted; refer to PDF]

Consider the following two sublinear operators defined on ^{C c ∞} ( R ) : [figure omitted; refer to PDF]

We conclude from (43) and (45) that [figure omitted; refer to PDF]

If ω ∈ ^{A ( 1 / β , ∞ ) +} , then ^{ω - 1 / 1 - β} ∈ ^{A 1 -} ; see [16]. By Lemma 8, there exists t > 1 , such that ^{ω - t / ( 1 - β )} ∈ ^{A 1 -} . Let r = t / ( 1 - β ) and 1 / s - 1 / r = β . Using the boundedness of ^{I β +} from ^{L s} to ^{L r} , together with Lemma 6, we have [figure omitted; refer to PDF] In the last inequality, we use the fact ^{ω - r} = ^{ω - t / ( 1 - β )} ∈ ^{A 1 -} for all ω ∈ ^{A ( 1 / β , ∞ ) +} .

By Lemma 9, there exists _{p 0} = 1 / β such that [figure omitted; refer to PDF] thus [figure omitted; refer to PDF] for all ω ∈ ^{A ( p , q ) +} .

For ^{M 4 +} , let _{I j} = [ x , x + ^{2 j} h ] , j ∈ ^{Z +} . Then [figure omitted; refer to PDF] where ^{ω - r} = ^{ω - t / ( 1 - β )} ∈ ^{A 1 -} ( t > 1 ) for all ω ∈ ^{A ( 1 / β , ∞ ) +} . Then [figure omitted; refer to PDF] By Lemma 9, the inequlity [figure omitted; refer to PDF] holds for ω ∈ ^{A ( p , q ) +} .

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 ω ∈ ^{A ( p , q ) -} instead of ω ∈ ^{A ( p , q ) +} .

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).

Conflict of Interests

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

References

[1] A. P. Calderon, "Commutators of singular integral operators," Proceedings of the National Academy of Sciences of the United States of America , vol. 53, pp. 1092-1099, 1965.

[2] J. Cohen, "A sharp estimate for a multilinear singular integral in ^{R n} ," Indiana University Mathematics Journal , vol. 30, no. 5, pp. 693-702, 1981.

[3] W. G. Chen, S. Z. Lu, "On multilinear singular integrals in ^{R n} ," Advances in Mathematics (China) , vol. 29, no. 4, pp. 325-330, 2000.

[4] Z. Fu, S. Lu, Y. Pan, S. Shi, "Some one-sided estimates for oscillatory singular integrals," Nonlinear Analysis: Theory, Methods & Applications , vol. 108, pp. 144-160, 2014.

[5] Z. Fu, S. Lu, S. Sato, S. Shi, "On weighted weak type norm inequalities for one-sided oscillatory singular integrals," Studia Mathematica , vol. 207, no. 2, pp. 137-151, 2011.

[6] Z. W. Fu, G. L. Wang, 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.

[7] Z. Fu, S. Lu, Y. Pan, S. Shi, "Boundedness of one-sided oscillatory integral operators on weighted Lebesgue spaces," Abstract and Applied Analysis , vol. 2014, 2014.

[8] M. Lorente, 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.

[9] R. A. Macias, 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.

[10] F. J. Martin-Reyes, A. de la Torre, "One-sided BMO spaces," Journal of the London Mathematical Society , vol. 49, no. 3, pp. 529-542, 1994.

[11] C. Perez, "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.

[12] S. Shi, Z. Fu, "Estimates of some operators on one-sided weighted Morrey spaces," Abstract and Applied Analysis , vol. 2013, 2013.

[13] 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.

[14] A. de la Torre, J. L. Torrea, "One-sided discrete square function," Studia Mathematica , vol. 156, no. 3, pp. 243-260, 2003.

[15] H. Aimar, L. Forzani, F. J. Martin-Reyes, "On weighted inequalities for singular integrals," Proceedings of the American Mathematical Society , vol. 125, no. 7, pp. 2057-2064, 1997.

[16] Z. W. Fu, S. Z. Lu, "One-sided Triebel-Lizorkin space and its applications," Scientia Sinica Mathematica , vol. 41, pp. 43-52, 2011.

[17] Y. Ding, S. Z. Lu, K. Yabuta, "Multilinear singular and fractional integrals," Acta Mathematica Sinica , vol. 22, no. 2, pp. 347-356, 2006.

[18] M. Lorente, M. S. Riveros, "Two extrapolation theorems for related weights and applications," Mathematical Inequalities & Applications , vol. 10, no. 3, pp. 643-660, 2007.

[19] K. F. Andersen, 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.

[20] F. J. Martin-Reyes, L. Pick, A. de la Torre, " ^{A ∞ +} condition," Canadian Journal of Mathematics , vol. 45, no. 6, pp. 1231-1244, 1993.