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

Abstract

Let $T_1$, $T_2$ be two Calderón-Zygmund operators and $T_{1,b}$ be the commutator of $T_1$ with symbol $b\in \mathrm{BMO}({\mathbb{R}}^n)$. 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)|>λ\}\big)\\ &&\quad\lesssim [w]_{A_1}[w]_{A_{\infty}}\log ({\rm e}+[w]_{A_{\infty}}\big) \int_{\mathbb{R}^n}\frac{|f(x)|}λ\log \Big({\rm e}+\frac{|f(x)|}λ\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)|>λ\}\big)\\ &&\quad\lesssim [w]_{A_1}[w]_{A_{\infty}}\log^2 ({\rm e}+[w]_{A_{\infty}}\big) \int_{\mathbb{R}^n}\frac{|f(x)|}λ\log^2 \Big({\rm e}+\frac{|f(x)|}λ\Big)w(x)dx. \end{eqnarray*}

rewrite Lemma 3.1 and modify its proof. Corrected some misprints