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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| Acceso en línea: | https://arxiv.org/abs/2212.07804 |
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| _version_ | 1866929226909745152 |
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| author | Kazaniecki, Krystian Müller, Paul F. X. |
| author_facet | Kazaniecki, Krystian Müller, Paul F. X. |
| contents | In the present paper we identify those filtered probability spaces $(Ω,\, \mathcal{F},\, \left(\mathcal{F}_n\right),\, \mathbb{P})$ that determine already the martingale type of a Banach space $X$. We isolate intrinsic conditions on the filtration $(\mathcal{F}_n)$ of purely atomic $σ$-algebras which determine that the upper $\ell^p$ estimates
\[
\|f\|_{L^p(Ω,\, X)}^p\leq C^p\left( \|\mathbb{E} f|\mathcal{F}_0\|^p_{L^p(Ω,\, X)}+\sum_{n=1}^{\infty} \|Δ_n f\|^p_{L^p(Ω,\, X)}\right),\qquad f\in L^p(Ω,X)\] imply that the Banach space $X$ is of martingale type $p$. Our paper complements \mbox{G. Pisier's} investigation \cite{Pisier1975} and continues the work by S. Geiss and second named author in \cite{Geiss2008}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_07804 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Martingale Type, the Gamlen-Gaudet Construction and a Greedy Algorithm Kazaniecki, Krystian Müller, Paul F. X. Functional Analysis 2020: 46B20, 46B09, 46B07 In the present paper we identify those filtered probability spaces $(Ω,\, \mathcal{F},\, \left(\mathcal{F}_n\right),\, \mathbb{P})$ that determine already the martingale type of a Banach space $X$. We isolate intrinsic conditions on the filtration $(\mathcal{F}_n)$ of purely atomic $σ$-algebras which determine that the upper $\ell^p$ estimates \[ \|f\|_{L^p(Ω,\, X)}^p\leq C^p\left( \|\mathbb{E} f|\mathcal{F}_0\|^p_{L^p(Ω,\, X)}+\sum_{n=1}^{\infty} \|Δ_n f\|^p_{L^p(Ω,\, X)}\right),\qquad f\in L^p(Ω,X)\] imply that the Banach space $X$ is of martingale type $p$. Our paper complements \mbox{G. Pisier's} investigation \cite{Pisier1975} and continues the work by S. Geiss and second named author in \cite{Geiss2008}. |
| title | Martingale Type, the Gamlen-Gaudet Construction and a Greedy Algorithm |
| topic | Functional Analysis 2020: 46B20, 46B09, 46B07 |
| url | https://arxiv.org/abs/2212.07804 |