Noncommutative analysis of Hermite expansions

TL;DR

This paper develops noncommutative harmonic analysis tools for Hermite operators, establishing maximal inequalities, convergence theorems, and multiplier results on noncommutative Lp-spaces, extending classical analysis to a noncommutative setting.

Contribution

It introduces noncommutative maximal inequalities, convergence theorems, and multiplier theorems for Hermite operators, utilizing a noncommutative Littlewood-Paley-Stein theory.

Findings

Established noncommutative maximal inequalities for Hermite Bochner-Riesz means

Proved pointwise convergence theorems for these means

Developed noncommutative multiplier theorems for Hermite operators

Abstract

This paper is devoted to the study of Hermite operators acting on noncommutative $L_{p}$-spaces. In the first part, we establish the noncommutative maximal inequalities for Bochner-Riesz means associated with Hermite operators and then obtain the corresponding pointwise convergence theorems. In particular, we develop a noncommutative Stein\textquoteright s theorem of Bochner-Riesz means for the Hermite operators. The second part of this paper deals with two multiplier theorems for Hermite operators. Our analysis on this part is based on a noncommutative analogue of the classical Littlewood-Paley-Stein theory associated with Hermite expansions.