Weighted Littlewood--Paley inequalities for heat flows in $\RCD$ spaces

We establish inequalities on vertical Littlewood--Paley square functions for heat flows in the weighted $L^2$ space over metric measure spaces satisfying the $\RCD^\ast(0,N)$ condition with $N\in [1,\infty)$ and the maximum volume growth assumption. In the noncompact setting, the later assumption can be removed by showing that the volume of the ball growths at least linearly. The estimates are sharp on the growth of the 2-heat weight and the 2-Muckenhoupt weight considered. The $p$-Muckenhoupt weight and the $p$-heat weight are also compared for all $p\in(1,\infty)$.