Nonlinear exponential twists of $\rm GL_2\times\rm GL_2$ automorphic forms

Let $f$ and $g$ be holomorphic or Maass cusp forms for $\rm SL_2(\mathbb{Z})$ with normalized Fourier coefficients $\lambda_f(n)$ and $\lambda_g(n)$, respectively. In this paper, we prove nontrivial estimates for the sum $$ \sum_{n=1}^{\infty}\lambda_f(n) \lambda_g(n)e\left(t \varphi\left(\frac{n}{X}\right)\right)V\left(\frac{n}{X}\right), $$ where $e(x)=e^{2\pi ix}$, $V(x)\in \mathcal{C}_c^{\infty}(1,2)$, $t\geq 1$ is a large parameter and $\varphi(x)$ is some nonlinear real valued smooth function. Applications of these estimates include a subconvex bound for the Rankin-Selberg $L$-function $L(s,f\otimes g)$ in the $t$-aspect and an improved estimate for the nonlinear exponential twisted sum $$ \sum_{n\leq X}\lambda_f(n) \lambda_g(n)e\big(\alpha n^{\beta}\big), $$ for $12/25<\beta\leq 4/7$, $\beta\neq 1/2$ and $\alpha\in \mathbb{R}\backslash\{0\}$.