## Metric cotype.(English)Zbl 1187.46014

In order to study the uniform structure of Banach spaces, it is important to have notions which are invariant under uniform homeomorphisms. M. Ribe proved in [Ark. Mat. 14, 237–244 (1976; Zbl 0336.46018)] that two uniformly homeomorphic Banach spaces are finitely representable in each other. Hence, it is important to have metric versions of local properties of Banach spaces. A metric version of the type (the Enflo type) was introduced as early as 1968 (and hence before the notion of type of a Banach space was explicitly defined) by P. Enflo [Ark. Mat. 8, 103–105 (1971; Zbl 0196.14002); Ark. Mat. 8, 107–109 (1971; Zbl 0196.14003)] and a slighly different one (called BMW type by the authors) by J. Bourgain, V. Milman and H. Wolfson [Trans. Am. Math. Soc. 294, 295–317 (1986; Zbl 0617.46024)]. These notions are sufficiently good to prove that every Banach space with BMW type $$p > 0$$ has Rademacher type $$p'$$ for all $$0 < p' < p$$ [Bourgain-Milman-Wolfson, op. cit.] and that every Banach space with Rademacher type $$p > 0$$ has Enflo type $$p'$$ for all $$0 < p' < p$$ [G. Pisier, Lect. Notes Math. 1206, 167–241 (1986; Zbl 0606.60008)].
However, no satisfactory metric version of the cotype had been found until now.
In the paper under review, the authors are able to introduce a notion of metric cotype which allows them to prove that, for every Banach space $$X$$, and $$q \geq 2$$, $$X$$ has metric cotype $$q$$ if and only if $$X$$ has Rademacher cotype $$q$$. Moreover, if $$C_q (X)$$ is its usual Rademacher cotype $$q$$ constant and $$\Gamma_q (X)$$ its metric cotype $$q$$ constant, then $$C_q (X) / 2\pi \leq \Gamma_q (X) \leq 90\, C_q (X)$$. The cases of Hilbert spaces and $$K$$-convex spaces, for which proofs are easier and give better constants, are treated separately.
Several applications are given. Among them, the authors prove: (1) a version of the Maurey-Pisier theorem (every Banach space without finite cotype contains the $$\ell_\infty^n$$’s uniformly), using a variant of their notion of metric cotype; (2) that for every Banach space $$X$$ with nontrivial type (the authors conjecture that this condition is not necessary), and every Banach space $$Y$$ which uniformly embeds, or more generally coarsely embeds, into $$X$$, one has $$q_Y \leq q_X$$; a result from which it follows that, for $$p, q > 0$$, $$L_p$$ embeds uniformly into $$L_q$$ if and only if $$p \leq q$$ or $$q \leq p \leq 2$$. Before this paper, it seems that it was only known that $$L_p$$ cannot be uniformly (resp., coarsely) embedded into $$L_2$$ if $$p > 2$$ [I. Aharoni, B. Maurey and B. S. Mityagin, Isr. J. Math. 52, 251–265 (1985; Zbl 0596.46010), resp., W. B. Johnson and N. L. Randrianarivony, Proc. Am. Math. Soc. 134, No. 4, 1045–1050 (2006; Zbl 1097.46051)].
Reviewer: Daniel Li (Lens)

### MSC:

 46B80 Nonlinear classification of Banach spaces; nonlinear quotients 46B85 Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science 46B07 Local theory of Banach spaces
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