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

Abstract

Let $1<p<\infty$. 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 $\Vert \cdot {\Vert }_{p,P}$ 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 $p\to \infty$, $${\beta }_p\lesssim p,$$ 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.