A shifted convolution sum for \(GL(3) \times GL(2)\) with weighted average

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