Weighted vector-valued estimates for a non-standard Calderón-Zygmund operator

Abstract

In this paper, the author considers the weighted vector-valued estimate for the operator defined by $$T_{A}f(x)=p.v.{\int }_{{\mathbb{R}}^n}\frac{\Omega (x-y)}{|x-y{|}^{n+1}}(A(x)-A(y)-\nabla A(y))f(y)\mathrm{d}y,$$ and the corresponding maximal operator $T_A^{\ast }$, where $\Omega$ is homogeneous of degree zero, has vanishing moment of order one, $A$ is a function in ${\mathbb{R}}^n$ such that $\nabla A\in \mathrm{BMO}({\mathbb{R}}^n)$. By a pointwise estimate for $\Vert \{T_A{f}_k(x)\}{\Vert }_{l^q}$ and the weighted $L^p$ estimates for the sparse operator $${\mathcal{A}}_{\mathcal{S},L(\log L{)}^{\beta }}f(x)=\sum _{Q\in \mathcal{S}}\Vert f{\Vert }_{L(\log L{)}^{\beta },Q}{\chi }_Q(x),$$ the author establishes some weak and endpoint quantitative weighted vector-valued estimates for $T_A$ and $T_A^{\ast }$.

arXiv comment21 pages