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

Let $\psi$ be a smooth compactly supported function on $\mathbb{X} = SL(2,\mathbb{Z})\backslash\mathbb{H}$. 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 $\int_{\mathbb{X}}\psi(z)E(z,1/2+it)^{3} d\mu z = \mathcal{O}_{\psi}(t^{-1/3+\varepsilon})$. In off-diagonal case we prove $\frac{1}{2\log t}\int_{\mathbb{X}}\psi(z)|E(z,1/2+it)|^{2}g(z)d\mu z = o(1)$ as long as $\min\{t , t_{g}\} \rightarrow \infty$. Finally we show $\int_{\mathbb{X}}\psi(z)f^{2}(z)g(z)d\mu z = o(1)$ in the range $|t_{f} - t_{g}| \leq t_{f}^{2/3-\varepsilon}$ where $f,g$ are two Hecke-Maass cusp forms.