Joint cubic moment of Eisenstein series and Hecke-Maass cusp forms

Let $\psi$ be a smooth compactly supported function on $\mathbb{X}=SL(2,\mathbb{Z})\mathrm{\backslash }\mathbb{H}$. In this paper, we are interested in the joint cubic moments of automorphic forms when the spectral parameters go to infinity. We show that the diagonal case for Eisenstein series ${\int }_{\mathbb{X}}\psi (z)E(z,1\mathrm{/}2+it{)}^{3}d\mu z={\mathcal{O}}_{\psi }(t^{-1\mathrm{/}3+\epsilon })$. In off-diagonal case we prove $\frac{1}{2\log t}{\int }_{\mathbb{X}}\psi (z)\mathrm{\mid }E(z,1\mathrm{/}2+it){\mathrm{\mid }}^{2}g(z)d\mu z=o(1)$ as long as $\min \{t,t_g\}\to \mathrm{\infty }$. Finally we show ${\int }_{\mathbb{X}}\psi (z)f^2(z)g(z)d\mu z=o(1)$ in the range $\mathrm{\mid }{t}_f-t_g\mathrm{\mid }\le t_f^{2\mathrm{/}3-\epsilon }$ where $f,g$ are two Hecke-Maass cusp forms.