Some inequalities for the multilinear singular integrals with Lipschitz functions on weighted Morrey spaces

The aim of this paper is to prove the boundedness of the oscillation and variation operators for the multilinear singular integrals with Lipschitz functions on weighted Morrey spaces.


Introduction
We first say that there exists a continuous function K(x, y) defined on Ω = {(x, y) ∈ R × R : x = y} and C > 0 if K admits the following representation: and for all x, x 0 , y ∈ R with |x -y| > 2|xx 0 | K(x, y) -K(x 0 , y) + K(y, x) -K(y, x 0 ) where 1 > β > 0. Then K is said to be a Calderón-Zygmund standard kernel.
Suppose that K satisfies (1) and (2). Then Zhang and Wu [12] considered the family of and In this sense, following [12], the definition of the oscillation operator of T is given by where {t i } is a decreasing fixed sequence of positive numbers converging to 0 and a related ρ-variation operator is defined by where the supremum is taken over all sequences of real number { i } decreasing to 0. We also take into account the operator On the other hand, it is obvious that That is, Recently, Campbell et al. in [1] proved the oscillation and variation inequalities for the Hilbert transform in L p (1 < p < ∞) and then following [1], we denote by E the mixed norm Banach space of the two-variable function h defined on R × N such that Given T := {T } >0 is a family operators such that lim →0 T f (x) = Tf (x) exists almost everywhere for certain class of functions f , where T defined as (3). For a fixed decreasing sequence {t i } with t i 0, let J i = (t i+1 , t i ] and define the E-valued operator U(T) : f → U(T)f given by We denote by F ρ the mixed norm space of twovariable functions g(i, β) such that We also take into account the Thus, Given m is a positive integer, and b is a function on R. Let R m+1 (b; x, y) be the m + 1th order Taylor series remainder of b at x about y, that is, In this paper, we consider the family of operators T b := {T b } >0 given by [6], where T b are the multilinear singular integral operators of T as follows: Thus, if m = 0, then T b is just the commutator of T and b, which is given by (4). But, if m > 0, then T b are non-trivial generation of the commutators. The theory of multilinear analysis was received extensive studies in the last 3 decades (see [2,5] for example). Hu and Wang [6] proved that the weighted (L p , L q )-boundedness of the oscillation and variation operators for T b when the mth derivative of b belongs to the homogeneous Lipschitz spaceΛ β . In this sense, we recall the definition of homogeneous Lipschitz spaceΛ β as follows.
Besides the Lebesgue space L q (R), the Morrey space M q p (R) is another important function space with definition as follows.
On the other hand, for a given weight function w and any interval I, we also denote the Lebesgue measure of I by |I| and set weighted measure A weight w is said to belong to the Muckenhoupt class A p for 1 < p < ∞ such that where p = p p-1 . The condition (6) is called the A p -condition, and the weights which satisfy it are called A p -weights. The expression [w] A p is also called characteristic constant of w.
Here and after, A p denotes the Muckenhoupt classes (see [5,7]). The A p class of weights characterizes the L p (w) boundedness of the maximal function as Muckenhoupt [9] established in the 1970s. Subsequent work of Muckenhoupt [9] himself Muckenhoupt and Wheeden [10,11], Coifman and Fefferman [3] was devoted to exploring the connection of the A p class with weighted estimates for singular integrals. However, it was not until the 2000s that the quantitative dependence on the so called A p constant, namely [w] A p , became a trending topic.
When p = 1, w ∈ A 1 if there exists C > 1 such that, for almost every x, and the infimum of C satisfying the inequality (7) is denoted by [w] A 1 , where M is the classical Hardy-Littlewood maximal operator. When p = ∞, we define A ∞ (R) = 1≤p<∞ A p (R). That is, the A ∞ constant is given by where we utilize the notation M(χ I w) to denote the Hardy-Littlewood maximal function of a function χ I w by A weight function w belongs to A p,q (Muckenhoupt-Wheeden class) for 1 < From the definition of A p,q , we know that w(x) ∈ A p,q (R) implies w(x) q ∈ A q (R) and w(x) p ∈ A p (R). Now, we begin with some lemmas. These lemmas are very necessary for the proof of the main result. for any interval I.
More precisely, for all λ > 1 we have where C is a constant independent of I or λ and w(I) = I w(x) dx.
Next, in 2009, the weighted Morrey space L p,κ (w) was defined by Komori and Shirai [7] as follows.
Definition 3 (Weighted Morrey space) Let 1 ≤ p < ∞, 0 < κ < 1 and w be a weight function. Then the weighted Morrey space L p,κ (w) ≡ L p,κ (R, w) is defined by When w ≡ 1 and κ = 1 -p q with 1 < p ≤ q < ∞, then Finally, we recall the definition of the weighted Morrey space with two weights as follows.
Definition 4 (Weighted Morrey space with two weights) Let 1 ≤ p < ∞ and 0 < κ < 1. Then for two weights u and v, the weighted Morrey space L p,κ (u, v) ≡ L p,κ (R, u, v) is defined by It is obvious that In 2016, Zhang and Wu [12] gave the boundedness of the oscillation and variation operators for Calderón-Zygmund singular integrals and the corresponding commutators on the weighted Morrey spaces. In 2017, Hu and Wang [6] established the weighted (L p , L q )inequalities of the variation and oscillation operators for the multilinear Calderón-Zygmund singular integral with a Lipschitz function in R. Inspired of these results [6,12], we investigate the boundedness of the oscillation and variation operators for the family of the multilinear singular integral defined by (5) on weighted Morrey spaces when the mth derivative of b belongs to the homogeneous Lipschitz spaceΛ β in this work.
Throughout this paper, C always means a positive constant independent of the main parameters involved, and may change from one occurrence to another. We also use the notation F G to mean F ≤ CG for an appropriate constant C > 0, and F ≈ G to mean F G and G F.

The proof of Theorem 1
Proof We consider the proof related to O(T b ) firstly. Fix an interval I = (x 0l, x 0 + l), and we write as f = f 1 + f 2 , where f 1 = f χ 2I , χ 2I denotes the characteristic function of 2I. Thus, it is sufficient to show that the conclusion holds for every interval I ⊂ R. Then First, we use (9) to estimate F 1 , and we obtain Thus, Second, for x ∈ I, k = 1, 2, . . . , m ∈ N, let A k = {y : 2 k l ≤ |y -x| < 2 k+1 l}, B k = {y : |y -x| < 2 k+1 l}, and By [2], for any y ∈ A k , it is obvious that Hence, by Lemma 2 and (11) Also, following [12], we have Thus, the estimate of F 2 can be obtained as follows: For G 1 , since pβ and using Hölder's inequality, we have Since w ∈ A p,q , we have w q ∈ A ∞ . Thus, Lemma 1 implies w q (B k ) ≤ (C) k w q (I), C > 1, i.e., with κ p -1 q < 0. This implies Let y ∈ A k . For G 2 , by (11), (12) and (13) we get Thus, by (14) and (15), we obtain b Λ β f L p,κ (w p ,w q ) .
As a result, by (10) and (16), we get Similarly, V ρ (T b ) has the same estimate as above (here we omit the details), thus the inequality is valid. Therefore, Theorem 1 is completely proved.