Martingale Type, the Gamlen-Gaudet Construction and a Greedy Algorithm

2022/12/15 by Krystian Kazaniecki, Kazaniecki, Krystian, Paul F. X. Müller +1
Mathematics · Social Sciences · #2020: 46B20 #46B07 #46B09 #Advanced Mathematical Identities #FOS: Mathematics #Functional Analysis (math.FA) #Historical Geography and Cartography

paper · pdf · doi:10.48550/arxiv.2212.07804

openalex publication_date 2022/12/15 · openalex created_date 2022/12/28 · openalex updated_date 2026/07/28

Abstract

In the present paper we identify those filtered probability spaces (Ω, F, (F_n), ℙ) that determine already the martingale type of a Banach space X. We isolate intrinsic conditions on the filtration (F_n) of purely atomic σ-algebras which determine that the upper ℓ^p estimates ‖f‖_{L^p(Ω, X)}^p≤ C^p( ‖𝔼 f|F₀‖^p_{L^p(Ω, X)}+∑_n=1^∞ ‖Δ_n f‖^p_{L^p(Ω, X)}), f∈ L^p(Ω,X) imply that the Banach space X is of martingale type p. Our paper complements G. Pisier's investigation \citePisier1975 and continues the work by S. Geiss and second named author in \citeGeiss2008.

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