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

Abstract

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

Keywords

• Variation operator
• Oscillation operator
• Multilinear operator
• Hörmander condition
• Singular integral
• BMO