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On the best constants of the noncommutative Littlewood-Paley-Stein inequalities
Published 16 Jan 2024 in math.OA and math.FA | (2401.08731v2)
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*} \alpha_p{-1}|x|{p}\le |x|{p,P}\le \beta_p |x|{p}, \end{equation*} where $|\cdot|{p,P}$ 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.
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