Extension of Pisier’s inequality, going beyond Enflo’s conjecture, I

Banach space valued Poincaré inequality and singular integrals on discrete cube are the topics of this talk.

For Banach space valued functions Poincaré inequality is usually replaced by Pisier’s inequality. It is interesting to understand precisely for which Banach spaces X Pisier inequality on Hamming cube is dimension free. This has been done in a recent paper of Ivanisvili-Van Handel-Volberg (IVHV). There is a whole scale of related inequalities filling the gap between Pisier’s inequality and singular integral inequality on Hamming cube. For those inequalities the description of class of X is not known, the reason is related to the following fact: we are used to the “fact” that singular integrals on X-valued functions have to be bounded in Lp(X) if X is UMD. But on discrete cube this is not true anymore. However, we will show a wide class of spaces for which those inequalities hold. The proofs are the mixture of the formula of IVHV and quantum random variables technique à la Francoise Lust-Piquard.