Banach space valued Pisier and Riesz type inequalities on discrete cube

TL;DR

This paper develops a Banach space valued theory for singular integrals on the Hamming cube, establishing dimension-independent estimates and generalizing key inequalities with improved constants.

Contribution

It introduces new Banach space conditions for boundedness of operators on the Hamming cube and extends Pisier and Hytönen-Naor theorems using quantum and probabilistic approaches.

Findings

Dimension-independent estimates for singular integrals.

Generalization of Pisier and Hytönen-Naor inequalities.

Improved constants in $L^1$-Poincaré inequality.

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.