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

Let $T_1$, $T_2$ be two Calder'on-Zygmund operators and $T_{1,b}$ be the commutator of $T_1$ with symbol $b\in \mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n)$. In this paper, the author prove that, the composite operator $T_1{T}_2$ 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*}