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

Similar to an unbounded function log |x| ∈ BMO, the functions in Lip α are not necessarily bounded either (e.g., |x| α ∈ Lip 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 Calderón-Zygmund kernel (OCZK) if K satisfies


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 1-dimensional case) by where Ω 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  () ∈ Lip  (R) ( = 0, 1).A function  ∈ Lip  (R), 0 <  < 1, if it satisfies Similar to an unbounded function log || ∈ BMO, the functions in Lip  are not necessarily bounded either (e.g., ||  ∈ Lip  ).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][5][6][7][8][9][10][11][12][13][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 Journal of Function Spaces with support in R − = (−∞, 0) or R + = (0, +∞).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 1 <  < ∞; also, for  = 1, for some constant .
Very recently, Fu and Lu [16] introduced a class of onesided Triebel-Lizorkin spaces and their weighted version.
In [16], the authors proved the boundedness for the onesided commutators (with symbols  ∈ Lip  (R)) of Calderón-Zygmund singular integral,  +  , and fractional integral,  + , , respectively. +   and  + ,  are defined as follows: Let () be locally integrable functions on R. 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  = 1,  + , =  +  .Therefore, the results of this paper are the extension of [16].
Then one gets the following.
The other main objects in this paper are one-sided Cohen's commutators of fractional integral operators, which are defined by Obviously, when  = 1,  +, , =  + , .
Then one gets the following.
Then one gets the following.
(i) If  ∈  + (,) , there exists the constant  > 0 such that (ii) If  ∈  − (,) , there exists the constant  > 0 such that Throughout this paper the letter  will denote a positive constant that may vary from line to line.

Estimates for the One-Sided Cohen Type Commutators of Singular Integrals
This section begins with some necessary lemmas.
The primary tool in the proof of Theorem 3 is an extrapolation theorem that appeared in [18] holds whenever  ∈  +  .
By Lemma 6 and assumption (4), it is easy to prove that where 1/ − 1/ = .In the last inequality, we use the boundedness of  +  that appeared in [19].
Combining the above estimate, we have

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.