Analytic twists of GL2×GL2$\rm GL_2\times \rm GL_2$ automorphic forms

Let f and g be holomorphic or Maass cusp forms for SL2(Z)$\rm SL_2(\mathbb {Z})$ with normalized Fourier coefficients λf(n)$\lambda _f(n)$ and λg(n)$\lambda _g(n)$, respectively. In this paper, we prove nontrivial estimates for the sum ∑n=1∞λf(n)λg(n)etφnXVnX,$$\begin{equation*} \hspace*{8pc}\sum _{n=1}^{\infty }\lambda _f(n) \lambda _g(n)e{\left(t \varphi {\left(\frac{n}{X}\right)}\right)}V{\left(\frac{n}{X}\right)}, \end{equation*}$$where e(x)=e2πix$e(x)=\text{e}^{2\pi ix}$, V(x)∈Cc∞(1,2)$V(x)\in \mathcal {C}_c^{\infty }(1,2)$, t≥1$t\ge 1$ is a large parameter and φ(x)$\varphi (x)$ is some nonlinear real‐valued smooth function. Applications of these estimates include a subconvex bound for the Rankin–Selberg L‐function L(s,f⊗g)$L(s,f\otimes g)$ in the t‐aspect, an improved estimate for a nonlinear exponential twisted sum and the following asymptotic formula for the sum of the Fourier coefficients of certain GL5$\rm {GL}_5$ Eisenstein series ∑n≤Xλ1⊞(f×g)(n)=L(1,f×g)X+O(X23−1356+ε)$$\begin{equation*} \hspace*{6pc}\sum _{n \le X}\lambda _{1\boxplus (f\times g)}(n) =L(1,f\times g)X + O(X^{\frac{2}{3}-\frac{1}{356}+\varepsilon }) \end{equation*}$$for any ε>0$\varepsilon >0$.