Characterizations of operator-valued Hardy spaces and applications to harmonic analysis on quantum tori

TL;DR

This paper characterizes operator-valued Hardy spaces, showing the Poisson kernel can be replaced by other test functions, and applies this to quantum tori, impacting harmonic analysis and related function spaces.

Contribution

It provides a generalization of Hardy space characterizations on quantum tori, extending previous definitions and enabling new analysis tools.

Findings

Poisson kernel can be replaced by any reasonable test function in Hardy space definitions

Characterization of Hardy spaces on quantum tori established

Application to Triebel-Lizorkin spaces on quantum tori

Abstract

This paper deals with the operator-valued Hardy spaces introduced and studied by Tao Mei. Our principal result shows that the Poisson kernel in Mei's definition of these spaces can be replaced by any reasonable test function. As an application, we get a general characterization of Hardy spaces on quantum tori. The latter characterization plays a key role in our recent study of Triebel-Lizorkin spaces on quantum tori.