On the best constants of the noncommutative Littlewood-Paley-Stein inequalities

Let $1<p<\mathrm{\infty }$. Let $\{T_t{\}}_{t>0}$ be a noncommutative symmetric diffusion semigroup on a semifinite von Neumann algebra $\mathcal{M}$, and let $\{P_t{\}}_{t>0}$ be its associated subordinated Poisson semigroup. The celebrated noncommutative Littlewood-Paley-Stein inequality asserts that for any $x\in L_p(\mathcal{M})$, \begin{equation*} α_p^{-1}|x|{p}\le |x|{p,P}\le β_p |x|_{p}, \end{equation*} where $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{p,P}$ is the $L_p(\mathcal{M})$-norm of square functions associated with $\{P_t{\}}_{t>0}$, and ${\alpha }_p,{\beta }_p$ are the best constants only depending on $p$. We show that as $p\to \mathrm{\infty }$, $${\beta }_p\lesssim p,$$ and $p$ is the optimal possible order of ${\beta }_p$ as well. We also obtain some lower and upper bounds of ${\alpha }_p$ and ${\beta }_p$ in the other cases.