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Martingales with values in uniformly convex spaces

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Abstract

Using the techniques of martingale inequalities in the case of Banach space valued martingales, we give a new proof of a theorem of Enflo: every super-reflexive space admits an equivalent uniformly convex norm. Letr be a number in ]2, ∞[; we prove moreover that if a Banach spaceX is uniformly convex (resp. ifδ x(ɛ)/ɛ r whenɛ → 0) thenX admits for someq<∞ (resp. for someq<r) an equivalent norm for which the corresponding modulus of convexity satisfiesδ(ɛ)/ɛ q → ∞ whenɛ → 0. These results have dual analogues concerning the modulus of smoothness. Our method is to study some inequalities for martingales with values in super-reflexive or uniformly convex spaces which are characteristic of the geometry of these spaces up to isomorphism.

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  1. Centre de Mathématiques, Ecole Polytechnique, 17 rue Descartes, 75005, Paris, France

    Gilles Pisier

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  1. Gilles Pisier
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Pisier, G. Martingales with values in uniformly convex spaces. Israel J. Math. 20, 326–350 (1975). https://doi.org/10.1007/BF02760337

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  • DOI: https://doi.org/10.1007/BF02760337

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