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

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