Shifted Convolution Sum for $GL(3)\times GL(2)$ with Weighted Average

In this paper, we will prove the non-trivial bound for the weighted average version of shifted convolution sum for $GL(3)\times GL(2)$, i.e. for any $\epsilon >0$ and $X^{1\mathrm{/}4+\delta }\le H\le X$ with $\delta >0$, $$\frac{1}{H}\sum _{h=1}^{\mathrm{\infty }}{\lambda }_f(h)V\left(\frac{h}{H}\right)\sum _{n=1}^{\mathrm{\infty }}{\lambda }_{\pi }(1,n){\lambda }_g(n+h)W\left(\frac{n}{X}\right)\ll X^{1-\delta +\epsilon }$$ where $V,W$ are smooth compactly supported funtions, ${\lambda }_f(n),{\lambda }_g(n)$ and ${\lambda }_{\pi }(1,n)$ are the normalized n-th Fourier coefficients of $SL(2,\mathbb{Z})$ Hecke-Maass cusp forms $f,g$ and $SL(3,\mathbb{Z})$ Hecke-Maass cusp form $\pi$, respectively.