Hyperbolic summation involving certain arithmetic functions and the integer part function

Let f be an arithmetic function satisfying certain conditions. In this paper, we give an asymptotic formula for the sum $$\sum_{n_1 n_2 \cdots n_r \leq x} f\left(\left\lfloor \frac{x}{n_1 n_2 \cdots n_r} \right\rfloor\right), \quad r \geq 2.$$, where $\lfloor . \rfloor$ denotes the integer part function.