Littlewood--Paley--Stein inequalities on $\textup{RCD}(K,\infty)$ spaces

The $L^p$ boundedness on vertical Littlewood--Paley square functions for heat flows on $\textup{RCD}(K,\infty)$ spaces with $K\in\mathbb{R}$ is proved. With regards to the proof, for $1<p\leq 2$, Stein's analytical method is applied, while for $2<p<\infty$, the probabilistic approach in the sense of Ba\~{n}uelos--Bogdan--Luks introduced recently is employed.