Oscillation and variation inequalities for the multilinear singular integrals related to Lipschitz functions

The main purpose of this paper is to establish the weighted (Lp,Lq)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(L^{p},L^{q})$\end{document} inequalities of the oscillation and variation operators for the multilinear Calderón-Zygmund singular integral with a Lipschitz function.

to zero. Following [], the oscillation operator is defined as  / and the ρ-variation operator is defined as where the sup is taken over all sequences of real number { i } decreasing to zero.
The oscillation and variation for some families of operators have been studied by many authors on probability, ergodic theory, and harmonic analysis; see [-]. Recently, some authors [-] researched the weighted estimates of the oscillation and variation operators for the commutators of singular integrals.
Let m be a positive integer, let b be a function on R, and let R m+ (b; x, y) be the m + th Taylor series remainder of b at x expander about y, i.e.
We Note that when m = , T b is just the commutator of T and b, which is denoted by T ,b , that is to say However, when m > , T b is a non-trivial generation of the commutator. It is well known that multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see In this paper, we will study the boundedness of oscillation and variation operators for the family of the multilinear singular integral related to a Lipschitz function defined by (.) in weighted Lebesgue space. Our main results are as follows.
In this paper, we shall use the symbol A B to indicate that there exists a universal positive constant C, independent of all important parameters, such that A ≤ CB. A ≈ B means that A B and B A.

Weight
A weight ω is a nonnegative, locally integrable function on R. The classical weight theories were introduced by Muckenhoupt

and Wheeden in [] and [].
A weight ω is said to belong to the Muckenhoup class A p (R) for  < p < ∞, if there exists a constant C such that for every interval I. The class A  (R) is defined by replacing the above inequality with  |I| I ω(x) dx ess inf x∈I w(x) for every ball I ⊂ R.
It is well known that if ω ∈ A p.q (R), then ω q ∈ A ∞ (R).

Function of Lip β (R)
The function of Lip β (R) has the following important properties.

Maximal function
We recall the definition of Hardy-Littlewood maximal operator and fractional maximal operator. The Hardy-Littlewood maximal operator is defined by The fractional maximal function is defined as In order to simplify the notation, we set for all f such that the left hand side is finite.

Taylor series remainder
The following lemma gives an estimate on Taylor series remainder.

Lemma . []
Let b be a function on R and b (m) ∈ L s (R) for any s > . Then

Oscillation and variation operators
We consider the operator It is easy to check that Following [], we denote by E the mixed norm Banach space of two variable function h defined on R × N such that We also consider the F ρ -valued operator V(T) : f → V(T)f given by Next, let B be a Banach space and ϕ be a B-valued function, we define the sharp maximal operator as follows: and Finally, let us recall some results about oscillation and variation operators.

The proof of main results
Note that if ω ∈ A p,q (R), then ω q ∈ A ∞ (R). By Lemma . and Lemma ., we only need to prove hold for any  < r < ∞.
We will prove only inequality (.), since (.) can be obtained by a similar argument. Fix f and x  with an interval I = (x l, x  + l). Write f = f  + f  = f χ I + f χ R\I , and let Then For x ∈ I, k = , -, -, . . . , let E k = {y :  k- · l ≤ |y -x| <  k · l}, let I k = {y : |y -x| <  k · l}, and let b k (z) = b(z) - m! (b (m) ) I k z m . By [] we have R m+ (b; x, y) = R m+ (b k ; x, y) for any y ∈ E k . By Lemma ., we know O (T) is bounded on L u (R) for u > . Then, using Hölder's inequality, we deduce By Lemma . and Lemma ., Since b (m) k (y) = b (m) (y) -(b (m) ) I k , then, applying Hölder's inequality and Lemma ., we get We now estimate M  . For x ∈ I, we have For k = , , , . . . , let F k = {y :  k · l ≤ |yx  | <  k+ · l}, let I k = {y : |yx  | <  k · l}, By Minkowski's inequalities and {χ {t i+ <|x-y|<s} } s∈J i ,i∈N E ≤ , we obtain From the mean value theorem, there exists η ∈ I such that For η, x ∈ I, y ∈ F k , we have |yx  | ≈ |y -x| ≈ |y -η| and |y -η| ≈ |yx  | ≤  k+ · l. By Lemma . and Lemma . we get Then Similar to the estimates for N  , we have Then Finally, let us estimate N  . Notice that the integral will be non-zero in the following cases: In case (ii) we have s < |x  -y| < l + s and in case (iv) we have s < |x -y| < l + s. By (.) and taking  < t < r, we have