Flag Hardy space theory on Heisenberg groups and applications
Peng Chen, Michael G. Cowling, Ming-Yi Lee, Ji Li, Alessandro Ottazzi
TL;DR
This work develops a comprehensive flag Hardy space theory on the Heisenberg group $\mathbb{H}^{\nu}$, establishing a complete array of equivalent characterizations for the endpoint space $\mathsf{H}^1_{F}(\mathbb{H}^{\nu})$ via atomic, area, Lusin-area, square-function, maximal-function, and Riesz-transform formulations. The authors introduce novel geometric and analytic tools—flag tilings with shards, Journé-type covering lemmas, a two-Laplacian Sobolev framework, and a Calderón reproducing formula in a noncommutative setting—to overcome obstacles from noncommutativity and the absence of a standard Fourier transform. They prove that $\mathsf{H}^1_{F}(\mathbb{H}^{\nu})$ embeds as a proper subspace of the one-parameter $\mathsf{H}^1_{FSCG}(\mathbb{H}^{\nu})$, while still supporting endpoint boundedness for flag singular integrals (e.g., Cauchy–Szegő) and for Marcinkiewicz-type multipliers $\mathrm{M}(\mathcal{L}_{(1)}, i\mathcal{T})$ under sharp regularity assumptions. The dual space is identified as $\mathsf{BMO}_{F}(\mathbb{H}^{\nu})$, with a decomposition in terms of flag Riesz components and singular-integral operators applied to bounded functions. Collectively, the results extend multiparameter harmonic analysis to nilpotent Lie groups with implicit flag structures, enabling robust $\mathsf{L}^p$ and $\mathsf{H}^p$ theory for flag-singular integrals on complex-analytic domains and related CR-geometry problems.
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 Müller, Ricci and Stein, and the decomposition of flag BMO space via singular integrals.