Weighted Norm Inequalities for a Class of Multilinear Singular Integral Operators

DOI：10.3770/j.issn:2095-2651.2014.02.006

 作者 单位 王荣乾 河南工业职业技术学院基础教学部, 河南 南阳 473000

本文考虑多线性奇异积分算子$$T_{A}f(x)={\rm p.\,v.}\int_{\mathbb{R}^n}\frac{\Omega(x-y)}{|x-y|^{n+1}}\big(A(x)-A(y)-\nabla A(y)(x-y)\big)f(y)\,dy$$的带一般权函数的加权$L^p$估计,其中$\Omega$是零阶齐次函数, 具有一阶消失矩且属于${\rm Lip}_{\gamma}(S^{n-1})$ ($\gamma\in (0,\,1]$),$A$的所有的一阶偏微商属于${\rm BMO}(\mathbb{R}^n)$.

In this paper, weighted estimates with general weights are established for the multilinear singular integral operator defined by $$T_{A}f(x)={\rm p.\,v.}\int_{\mathbb{R}^n}\frac{\Omega(x-y)}{|x-y|^{n+1}}\big(A(x)-A(y)-\nabla A(y)(x-y)\big)f(y)\d y,$$ where $\Omega$ is homogeneous of degree zero, has vanishing moment of order one, and belongs to ${\rm Lip}_{\gamma}(S^{n-1})$ with $\gamma\in (0,\,1]$, $A$ has derivatives of order one in ${\rm BMO}(\mathbb{R}^n)$.