Weighted vector-valued estimates for a non-standard Calder\'on-Zygmund operator

Abstract

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