Variation and oscillation for the multilinear singular integrals satisfying H\xf6rmander type conditions

Yinhong Xia

doi:10.1186/s13660-018-1689-8

Abstract

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

Highlights

Introduction and results LetK be a singular kernel in Rn and satisfy K (x) ≤C |x|n for |x| > 0, (1)where C is a fixed constant
We consider the family of operators Tb = {Tb} >0, where Tb are the multilinear singular integral operators of T as follows:
If K satisfies (1) and Hr,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) in Lp(Rn, ω(x) dx) when Dαb(y) – (Dαb) ∈ BMO(Rn) for |α| = m

Summary

Introduction and results Let K be a singular kernel in Rn and satisfy

Let K satisfy (1) and Hr,1-Hörmander condition, and let Tb = {T ,b} >0, where T ,b is the commutator of T and b,. We consider the family of operators Tb = {Tb} >0, where Tb are the multilinear singular integral operators of T as follows: y). In [15], Hu and Wang established the weighted (Lp, Lq) inequalities of the variation and oscillation operators for the multilinear Calderón–Zygmund singular integral with a Lipschitz function in R. If K satisfies (1) and Hr,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) in Lp(Rn, ω(x) dx) when Dαb ∈ BMO(Rn) for |α| = m.

Maximal function The Hardy–Littlewood maximal operator is defined by

Conclusion