Banach space valued Pisier and Riesz type inequalities on discrete cube

Abstract

This is an attempt to build Banach space valued theory for certain singular integrals on Hamming cube. Of course all estimates below are dimension independent, and we tried to find ultimate sharp assumptions on the Banach space for a corresponding operators to be bounded. In certain cases we succeeded, although there are still many open questions, some of them are listed in the last Section. Using the approach of \cite{IVHV} and also quantum random variables approach of \cite{ELP} we generalize several theorems of Pisier \cite{P} and Hyt\"onen-Naor \cite{HN}. We also improve the constant in $L^1$-Poincar\'e inequality on Hamming cube, the previous results are due to Talagrand and Ben Efraim--Lust-Piquard.