Pisier type inequalities for $K$-convex spaces

TL;DR

This paper extends Pisier-type inequalities to K-convex spaces, providing new characterizations involving second order Riesz transforms, thereby deepening understanding of the geometric and functional properties of these spaces.

Contribution

It introduces a new necessary and sufficient condition for K-convexity based on the boundedness of second order Riesz transforms, generalizing prior theorems.

Findings

New characterization of K-convex spaces via Riesz transforms

Extension of Pisier-type inequalities to broader classes of spaces

Deepened understanding of the geometric structure of K-convex spaces

Abstract

We generalize several theorems of Hyt\"onen-Naor \cite{HN} using the approach from \cite{IVHV}. In particular, we give yet another necessary and sufficient condition (see (3.2)) to be a $K$-convex space, where the sufficiency was proved by Naor--Schechtman \cite{NS}. This condition is in terms of the boundedness of the second order Riesz transforms $\{\Delta^{-1} D_i\}_{i=1}^n$ in $L^p(\Omega_n, X)$.