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*}
Categories: math.CA