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**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)].

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 |