تقرير
Maximal weak Orlicz types and the strong maximal on von Neumann algebras
| العنوان: | Maximal weak Orlicz types and the strong maximal on von Neumann algebras |
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| المؤلفون: | Pérez, Adrián M. González, Parcet, Javier, García, Jorge Pérez |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Operator Algebras, Mathematics - Classical Analysis and ODEs, Mathematics - Functional Analysis |
| الوصف: | Let $\mathbf{E}_n: \mathcal{M} \to \mathcal{M}_n$ and $\mathbf{E}_m: \mathcal{N} \to \mathcal{N}_m$ be two sequences of conditional expectations on finite von Neumann algebras. The optimal weak Orlicz type of the associated strong maximal operator $\mathcal{E} = (\mathbf{E}_n\otimes \mathbf{E}_m)_{n,m}$ is not yet known. In a recent work of Jose Conde and the two first-named authors, it was show that $\mathcal{E}$ has weak type $(\Phi, \Phi)$ for a family of functions including $\Phi(t) = t \, \log^{2+\varepsilon} t$, for every $\varepsilon > 0$. In this article, we prove that the weak Orlicz type of $\mathcal{E}$ cannot be lowered below $L \log^2 L$, meaning that if $\mathcal{E}$ is of weak type $(\Phi, \Phi)$, then $\Phi(s) \not\in o(s \, \log^2 s)$. Our proof is based on interpolation. Namely, we use recent techniques of Cadilhac/Ricard to formulate a Marcinkiewicz type theorem for maximal weak Orlicz types. Then, we show that a weak Orlicz type lower than $L \log^2 L$ would imply a $p$-operator constant for $\mathcal{E}$ smaller than the known optimum as $p \to 1^{+}$. Comment: 16 pages; Minor changes, acknowledgment added |
| نوع الوثيقة: | Working Paper |
| الوصول الحر: | http://arxiv.org/abs/2404.12061Test |
| رقم الانضمام: | edsarx.2404.12061 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |