On the best constants of the noncommutative Littlewood-Paley-Stein inequalities
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913576714764288 |
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| author | Wei, Zhenguo Zhang, Hao |
| author_facet | Wei, Zhenguo Zhang, Hao |
| contents | 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 $\|\cdot\|_{p,P}$ is the $L_p(\mathcal{M})$-norm of square functions associated with $\{P_t\}_{t>0}$, and $α_p, β_p$ are the best constants only depending on $p$.
We show that as $p\to \infty$,
$$ β_p\lesssim p, $$
and $p$ is the optimal possible order of $β_p$ as well. We also obtain some lower and upper bounds of $α_p$ and $β_p$ in the other cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_08731 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the best constants of the noncommutative Littlewood-Paley-Stein inequalities Wei, Zhenguo Zhang, Hao Operator Algebras Functional Analysis 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 $\|\cdot\|_{p,P}$ is the $L_p(\mathcal{M})$-norm of square functions associated with $\{P_t\}_{t>0}$, and $α_p, β_p$ are the best constants only depending on $p$. We show that as $p\to \infty$, $$ β_p\lesssim p, $$ and $p$ is the optimal possible order of $β_p$ as well. We also obtain some lower and upper bounds of $α_p$ and $β_p$ in the other cases. |
| title | On the best constants of the noncommutative Littlewood-Paley-Stein inequalities |
| topic | Operator Algebras Functional Analysis |
| url | https://arxiv.org/abs/2401.08731 |