Triple Correlations of Multiplicative Functions

In this paper, we find asymptotic formula for the following sum with explicit error term: $$M_{x}(g_{1}, g_{2}, g_3)=\frac{1}{x}\sum_{n\le x}g_{1}(F_1(n))g_{2}(F_2(n))g_{3} (F_3(n)),$$ where $F_1(x), F_2(x)$ and $F_3(x)$ are polynomials with integer coefficients and $g_1,g_2,g_3$ are multilpicative functions with modulus less than or equal to $1.$ Moreover, under some assumption on $g_1,g_2,$ we prove that as $x\rightarrow \infty,$ $$\frac{1}{x}\sum\limits_{n\le x}g_1(n+3)g_2(n+2)\mu(n+1)=o(1)$$ and assuming $2$-point Chowla type conjecture we show that as $x\rightarrow \infty,$ $$\frac{1}{x}\sum\limits_{n\le x}g_1(n+3)\mu(n+2)\mu(n+1)=o(1).$$