Remarks on non linear type and Pisier’s inequality. (English) Zbl 1033.46013
For a function \(f\) on the discrete cube \({\mathbb D}_n = \{-1,1\}^n\) with values in a normed space \(X\), let \(F(\omega,\omega')= \sum_{i=1}^n \omega_i'\partial_i f(\omega)\) with \(\partial_i f(\omega) = (f(\omega)-f(S_i \omega))/2\), where \(S_i(\omega)\) has the same coordinates as \(\omega\) except for a sign change in the \(i\)th coordinate. G. Pisier proved in [ Probability and analysis, Lect. Notes Math. 1206, 167–241 (1986; Zbl 0606.60008)] that, for any \(1\leq p\leq\infty\),
\[
\left( {\mathbb E} \| f - {\mathbb E} f \| ^p \right)^{1/p} \leq 2 e \log n \left( {\mathbb E} \| F\| ^p \right)^{1/p},
\]
where \({\mathbb E}\) stands for the expectation with respect to the uniform probability on \({\mathbb D}_n\). For \(1\leq p <\infty\), the logarithmic factor is necessary in general as was shown by M. Talagrand [Geom. Funct. Anal. 3, 295–314 (1993; Zbl 0806.46035)], while R. Wagner [ Lect. Notes Math. 1745, 263–268 (2000; Zbl 0981.46021)] proved that it can be removed for \(p=\infty\).
The authors show that if \(X\) is a UMD Banach space, then the logarithmic factor can also be removed for all \(1\leq p<\infty\) at the expense of a constant depending on \(X\) but not on \(n\). They apply this result and their methods to study relations between different notions of linear and non-linear type. In particular, they show that Enflo type \(p\) and Rademacher type \(p\) are equivalent concepts for UMD spaces. Moreover, they prove that for UMD spaces both Rademacher type \(p\) and Enflo type \(p\) imply the non-linear type \(p\) of Bourgain, Milman and Wolfson.
The authors show that if \(X\) is a UMD Banach space, then the logarithmic factor can also be removed for all \(1\leq p<\infty\) at the expense of a constant depending on \(X\) but not on \(n\). They apply this result and their methods to study relations between different notions of linear and non-linear type. In particular, they show that Enflo type \(p\) and Rademacher type \(p\) are equivalent concepts for UMD spaces. Moreover, they prove that for UMD spaces both Rademacher type \(p\) and Enflo type \(p\) imply the non-linear type \(p\) of Bourgain, Milman and Wolfson.
Reviewer: Aicke Hinrichs (Jena)
MSC:
| 46B09 | Probabilistic methods in Banach space theory |
| 60B11 | Probability theory on linear topological spaces |
| 46B20 | Geometry and structure of normed linear spaces |
References:
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| [13] | Pisier, G., Sur les espaces qui ne contiennent pas de l 1n uniformement, Seminaire Maurey-Schwartz (1973- 1974), Espaces Lp, applications radonifiantes et geometrie des espaces de Banach.Exp. No.7, 19 pp. (errata, p. E.1) Centre de Math., Eacole Polytech., Paris 1974. |
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