Weighted weak type endpoint estimates for the composition of Calderon-Zygmund operators

Abstract

Let $T_1$, $T_2$ be two Calder\'on-Zygmund operators and $T_{1,\,b}$ be the commutator of $T_1$ with symbol $b\in {\rm BMO}(\mathbb{R}^n)$. In this paper, the author prove that, the composite operator $T_1T_2$ satisfies the following estimate: for $\lambda>0$ and weight $w\in A_1(\mathbb{R}^n)$, \begin{eqnarray*}&&w\big(\{x\in\mathbb{R}^n:\,|T_{1} T_2f(x)|>\lambda\}\big)\\ &&\quad\lesssim [w]_{A_1}[w]_{A_{\infty}}\log ({\rm e}+[w]_{A_{\infty}}\big) \int_{\mathbb{R}^n}\frac{|f(x)|}{\lambda}\log \Big({\rm e}+\frac{|f(x)|}{\lambda}\Big)w(x)dx,\nonumber \end{eqnarray*} and the composite operator $T_{1,b}T_2$ satisfies that \begin{eqnarray*}&&w\big(\{x\in\mathbb{R}^n:\,|T_{1,b} T_2f(x)|>\lambda\}\big)\\ &&\quad\lesssim [w]_{A_1}[w]_{A_{\infty}}\log^2 ({\rm e}+[w]_{A_{\infty}}\big) \int_{\mathbb{R}^n}\frac{|f(x)|}{\lambda}\log^2 \Big({\rm e}+\frac{|f(x)|}{\lambda}\Big)w(x)dx. \end{eqnarray*}