Flag Hardy space theory on Heisenberg groups and applications

TL;DR

This paper develops a comprehensive theory of flag Hardy spaces on the Heisenberg group, including characterizations and applications to singular integrals, overcoming challenges posed by noncommutativity and lack of classical tools.

Contribution

It introduces new techniques to define and analyze flag Hardy spaces on the Heisenberg group, with characterizations and applications to boundedness of singular integrals.

Findings

Established atomic, area, square, maximal function, and singular integral characterizations.

Proved boundedness of singular integrals from flag Hardy space to L^1.

Decomposed flag BMO space using singular integrals.

Abstract

We establish a complete theory of the flag Hardy space on the Heisenberg group $\mathbb H^{n}$ with characterisations via atomic decompositions, area functions, square functions, maximal functions and singular integrals. We introduce several new techniques to overcome the difficulties caused by the noncommutative Heisenberg group multiplication, and the lack of a suitable Fourier transformation and Cauchy--Riemann type equations. Applications include the boundedness from the flag Hardy space to $L^1(\mathbb H^n)$ of various singular integral operators that arise in complex analysis, a sharp boundedness result on the flag Hardy space of the Marcinkiewicz-type multipliers introduced by M\"uller, Ricci and Stein, and the decomposition of flag BMO space via singular integrals.