We say that a family $S$ of cubes $Q$ in $\mathbb{R}^n$ is $\eta$-sparse if there is $0<\eta<1$ such that for every cube $Q\in S$ there is a subset $E_Q\subset Q$ such that $|E_Q|\geq\eta |Q|$ and the sets $E_Q$ are pairwise disjoint.
We define the sparse operator by $$A_Sf(x)=\sum_{Q\in S}f_Q \chi_Q(x)$$ when $f_Q=\frac{1}{|Q|}\int_Q f$. I need to prove that $$\int_{\{x\in\mathbb{R}^n:|A_Sf(x)|>\lambda\}}w(x)dx\leq\frac{C}{\lambda}\|f\|_{L^p(w)}$$ when the weight $w$ is locally nonegative integrable function such that $\frac{1}{Q}\int_Q w(x)dx\leq A \inf_{x\in Q}w(x)$ for each cube Q.
I'm trying to prove it for days now, untill now I proved $$\int_{\{x\in\mathbb{R}^n:|A_Sf(x)|>\lambda\}}w(x)dx\leq \frac{1}{\lambda}\int_{\mathbb{R}^n}A_Sf(x)w(x)dx=\frac{1}{\lambda}\sum_{Q\in S}f_Q w_Q|Q|\leq\frac{A}{\eta\lambda}\sum_{Q\in S}\int_{E_Q}f_Qw(x)dx\leq\frac{A}{\eta\lambda}\int_{\mathbb{R}^n}f_Qw(x)dx.$$ At this point I'm stuck, I tried to use the maximal operator but its still dont work. Does anyone know how to figure it out?