Flag Hardy space theory on Heisenberg groups and applications

2021/02/15 by Chen, Peng, Cowling, Michael G., Lee, Ming-Yi +2
#Analysis of PDEs (math.AP) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2102.07371

Abstract

We establish a complete theory of the flag Hardy space on the Heisenberg group \mathbb Hⁿ 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 L¹(\mathbb Hⁿ) 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.

Related

• Flag Hardy spaces and partial differential equations
• Flag-like singular integrals and associated Hardy spaces on a kind of nilpotent Lie groups of step two
• Boundedness of singular integrals on the flag Hardy spaces on Heisenberg group
• Characterizations of product Hardy spaces on stratified groups by singular integrals and maximal functions
• Hardy spaces and quasiconformal maps in the Heisenberg group