TAILIEUCHUNG - Đề tài " Metric cotype "

We introduce the notion of cotype of a metric space, and prove that for Banach spaces it coincides with the classical notion of Rademacher cotype. This yields a concrete version of Ribe’s theorem, settling a long standing open problem in the nonlinear theory of Banach spaces. We apply our results to several problems in metric geometry. Namely, we use metric cotype in the study of uniform and coarse embeddings, settling in particular the problem of classifying when Lp coarsely or uniformly embeds into Lq . We also prove a nonlinear analog of the Maurey-Pisier theorem, and use it to. | Annals of Mathematics Metric cotype By Manor Mendel and Assaf Naor Annals of Mathematics 168 2008 247-298 Metric cotype By Manor Mendel and Assaf Naor Abstract We introduce the notion of cotype of a metric space and prove that for Banach spaces it coincides with the classical notion of Rademacher cotype. This yields a concrete version of Ribe s theorem settling a long standing open problem in the nonlinear theory of Banach spaces. We apply our results to several problems in metric geometry. Namely we use metric cotype in the study of uniform and coarse embeddings settling in particular the problem of classifying when Lp coarsely or uniformly embeds into Lq. We also prove a nonlinear analog of the Maurey-Pisier theorem and use it to answer a question posed by Arora Lovasz Newman Rabani Rabinovich and Vempala and to obtain quantitative bounds in a metric Ramsey theorem due to Matousek. 1. Introduction In 1976 Ribe 62 see also 63 27 9 6 proved that if X and Y are uniformly homeomorphic Banach spaces then X is finitely representable in Y and vice versa X is said to be finitely representable in Y if there exists a constant K 0 such that any finite dimensional subspace of X is K-isomorphic to a subspace of Y . This theorem suggests that local properties of Banach spaces . properties whose definition involves statements about finitely many vectors have a purely metric characterization. Finding explicit manifestations of this phenomenon for specific local properties of Banach spaces such as type cotype and super-reflexivity has long been a major driving force in the biLipschitz theory of metric spaces see Bourgain s paper 8 for a discussion of this research program . Indeed as will become clear below the search for concrete versions of Ribe s theorem has fueled some of the field s most important achievements. The notions of type and cotype of Banach spaces are the basis of a deep and rich theory which encompasses diverse aspects of the local theory of Banach spaces. We

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