On the asymptotics of coefficients of Rankin–Selberg L-functions

Let f and g be two different holomorphic cusp froms or Maass cusp forms for the full modular group \(SL(2,\mathbb{Z})\). We are interested in coefficients of Rankin–Selberg L-functions, and establish some bounds for

$$\begin{aligned}\sum_{n\leq x} \lambda_{{\rm sym}^if\times {\rm sym}^jg}(n),\quad \sum_{n\leq x}\lambda_f(n^i)\lambda_g(n^j), \\sum_{n\leq x} |\lambda_{{\rm sym}^if\times {\rm sym}^jg}(n)|, \quad \sum_{n\leq x}|\lambda_f(n^i)\lambda_g(n^j)|, \end{aligned}$$

and

$$\sum _{n\leq x} \max \bigl\{|\lambda_{{\rm sym}^if\times {\rm sym}^jg}(n)|^{2\varphi}, |\lambda_{{\rm sym}^if\times {\rm sym}^jg}(n+h)|^{2\varphi} \bigr\}, $$

where \(\varphi>0\) and h is a fixed positive integer.