Maximal weak Orlicz types and the strong maximal on von Neumann algebras
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917646599979008 |
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| author | Pérez, Adrián M. González Parcet, Javier García, Jorge Pérez |
| author_facet | Pérez, Adrián M. González Parcet, Javier García, Jorge Pérez |
| contents | 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 $(Φ, Φ)$ for a family of functions including $Φ(t) = t \, \log^{2+\varepsilon} t$, 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 $(Φ, Φ)$, then $Φ(s) \not\in o(s \, \log^2 s)$. 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 $p \to 1^{+}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_12061 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximal weak Orlicz types and the strong maximal on von Neumann algebras Pérez, Adrián M. González Parcet, Javier García, Jorge Pérez Operator Algebras Classical Analysis and ODEs Functional Analysis 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 $(Φ, Φ)$ for a family of functions including $Φ(t) = t \, \log^{2+\varepsilon} t$, 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 $(Φ, Φ)$, then $Φ(s) \not\in o(s \, \log^2 s)$. 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 $p \to 1^{+}$. |
| title | Maximal weak Orlicz types and the strong maximal on von Neumann algebras |
| topic | Operator Algebras Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2404.12061 |