Variation and oscillation for the multilinear singular integrals satisfying Hörmander type conditions

Suppose that the kernel K satisfies a certain Hörmander type condition. Let b be a function satisfying \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\alpha}b\in BMO(\mathbb{R}^{n})$\end{document}Dαb∈BMO(Rn) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\vert \alpha \vert =m$\end{document}|α|=m, and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T^{b}=\{T^{b}_{\epsilon}\}_{\epsilon>0}$\end{document}Tb={Tϵb}ϵ>0 be a family of multilinear singular integral operators, i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} T^{b}_{\epsilon}f(x)= \int_{ \vert x-y \vert >\epsilon}\frac{ R_{m+1}(b;x,y)}{ \vert x-y \vert ^{m}}K(x,y)f(y)\,dy. \end{aligned}$$ \end{document}Tϵbf(x)=∫|x−y|>ϵRm+1(b;x,y)|x−y|mK(x,y)f(y)dy. The main purpose of this paper is to establish the weighted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{p}$\end{document}Lp-boundedness of the variation operator and the oscillation operator for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T^{b}$\end{document}Tb.


Introduction and results
Let K be a singular kernel in R n and satisfy K(x) ≤ C |x| n for |x| > 0, (1) where C is a fixed constant. Consider the family of operators T = {T } >0 , where The ρ-variation operator is defined by where the supremum is taken over all sequences of real numbers { i } decreasing to zero. The oscillation operator is defined by where {t i } is a fixed sequence which decreases to zero. Variation and oscillation can be used to measure the speed of convergence of certain convergent families of operators. The operators of variation and oscillation have attracted many researchers' attention in probability, ergodic theory, and harmonic analysis. Bourgain [1] obtained variation inequality for the ergodic averages of a dynamic system, his work has launched a new research direction in harmonic analysis. Campbell, Jones, Reinhold, and Wierdl in [2] established the L p -boundedness of variation operator and oscillation operator for the Hilbert transform. Recently, many other publications have enriched this research direction [3][4][5][6][7].
Let 1 ≤ r ≤ ∞, m ∈ N ∪ {0}. We say that the kernel K satisfies the H r,m -Hörmander condition if there exist the constants c ≥ 1 and C r,m > 0 such that, for all y ∈ R n and R > c|x|, We notice that H r,0 is L r -Hörmander condition, which was studied in depth in [8].
Let K satisfy (1) and H r,1 -Hörmander condition, and let T b = {T ,b } >0 , where T ,b is the commutator of T and b, Suppose r > 1, ρ > 2, and b ∈ BMO(R n ). Zhang and Wu proved in [9] that, if V ρ (T) and Given m is a positive integer, and b is a function on R n . Let R m+1 (b; x, y) be the m + 1th Taylor series remainder of b at x expander about y, i.e., We consider the family of operators T b = {T b } >0 , where T b are the multilinear singular integral operators of T as follows: Note that when m = 0, T b is just the commutator of T and b. However, when m > 0, T b is a non-trivial generation of the commutator. It is well known that multilinear operators have been widely studied by many authors (see [10][11][12][13][14]).
In [15], Hu and Wang 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. In this paper, if K satisfies (1) and H r,1 -Hörmander condition, we will study the bounded behaviors of variation and oscillation operators for the family of the multilinear singular integrals defined by (4) Our main results can be formulated as follows.

Corollary 1 ([9]) Let K satisfy
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.

A p (R n ) weight
A weight ω belongs to A p (R n ) for 1 < p < ∞ if there exists a constant C such that, for every ball B ⊂ R n , where p is the dual of p such that 1/p + 1/p = 1. A weight ω belongs to

Function of BMO(R n )
Following [16], a locally integrable function b is said to be in BMO(R n ) if The function of BMO(R n ) has the following property.

Maximal function
The Hardy-Littlewood maximal operator is defined by We also define the maximal function [17]).

Taylor series remainder
By definition, it is obvious that The following lemma gives an estimate on Taylor series remainder.

Lemma 2 ([11])
Let b be a function on R n with mth order derivatives in L q (R) for some q > n. Then where Q(x, y) is the cube centered at x and having diameter 5 √ n|x -y|.

Variation and oscillation operators
This implies that On the other hand, we consider the operator It is easy to check that We denote by E the mixed norm Banach space of two-variable function h defined on R × N such that Let B be a Banach space and ϕ be a B-valued function, we define the sharp maximal operator as follows: Then Finally, let us recall some results about the boundedness of V ρ (T) and O(T). (1), ρ > 2, T = {T } >0 be given by (2). Suppose that K satisfies (1) and K ∈ H r,0 -Hörmander condition, V ρ (T) and O(T) are bounded on L p 0 (R n ) for some 1 < p 0 < ∞. Then V ρ (T) and O(T) are bounded on L p (R n ) for any max{r , p 0 } < p < ∞.

Proof of Theorem 1
By the fact that M s is bounded on L p (R n , ω) for 1 ≤ s < p < ∞ and ω ∈ A p (R n ), we need to prove and for any s > max{r , p 0 }. We will only need to prove (5), since (6) can be obtained by a similar argument. To prove (5), it suffices to prove that for any fixed x 0 ∈ R n and some F ρ -valued constant c 1 , such that for every ball B = B(x 0 , l) with radius l, centered at x 0 , the following inequality holds. We Then Let x 1 be a point at the boundary of 2B, and let For any x ∈ B, k ∈ Z, let E k = {y : 2 k · 3l ≤ |y -x| < 2 k+1 · 3l}, let F k = {y : |y -x| < 2 k+1 · 3l}, and let By [11] we have R m+1 (b; x, y) = R m+1 (b k ; x, y) for any y ∈ E k . From Lemma 3, we know V ρ (T) is bounded on L u (R n ) for max{r , p 0 } < u < s. Then, using Hölder's inequality, we deduce For any y ∈ E k , by Lemma 2 and Lemma 1, Then we have applying Hölder's inequality and Lemma 2, we get We now estimate M 2 . We write By Minkowski's inequality and Applying the formula ( [11] p. 448), we have Then, by Lemmas 1 and 2, we have By (1) we have Then For y ∈ E k , x ∈ B and x 1 being a point at the boundary of 2B, we get |x 1 -y| ≈ |x -y|. Thus Then As for N 13 , due to Let us estimate N 14 now. For y ∈ (10B) c , we have |y - Note that Q(x, y) ⊂ 2 √ nQ(x 1 , y), then for x ∈ B, y ∈ E k we have Taking R = 3l, then |xx 1 | < R. By Hölder's inequality and K ∈ H r,1 ⊂ H r,0 , we have Finally, let us estimate N 2 . Note that the integral will only be non-zero if either χ { i+1 <|x-y|< i } (y) = 1 and χ { i+1 <|x 1 -y|< i } (y) = 0 or vice versa. That means the integral will only be non-zero in the following cases: In case (i) we observe that i+1 < |x -y| ≤ |x 1 -x| + |x 1 -y| < 3l + i+1 as |xx 0 | < l. Similarly, in case (iii) we have i+1 < |x 1 -y| < 3l + i+1 as |xx 0 | < l. In case (ii) we have i < |x 1 -y| < 3l + i , and in case (iv) we have i < |x -y| < 3l + i . By (1) we have It is easy to check that |x -y| ≥ 9l, |x 1 -y| ≥ 8l, and 8 11 this means P 1 = P 3 = 0. Similarly, P 2 = P 4 = 0 for 3l ≥ i , i ∈ N. By Hölder's inequality with t satisfying 1 < t < min(r , ρ), we get Note that for 3l < i+1 , we have (3l + i+1 ) n -( i+1 ) n ( i+1 ) n-1 l. Then 3 4 i }<|x-y|< i } (y)

Conclusion
In this paper, we have established the weighted L p -boundedness of variation and oscillation operators for a family of multilinear singular integrals with kernels satisfying certain Hörmander type conditions. These results extend the corresponding work in [9] and [15].