Marcinkiewicz Multipliers and Lipschitz Spaces on Heisenberg Groups

Han, Yanchang ; Han, Yongsheng ; Li, Ji ; Tan, Chaoqiang

Canadian journal of mathematics, Tome 71 (2019) no. 3, pp. 607-627

Résumé

The Marcinkiewicz multipliers are $L^{p}$ bounded for $1<p<\infty$ on the Heisenberg group $\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$ (Müller, Ricci, and Stein). This is surprising in the sense that these multipliers are invariant under a two parameter group of dilations on $\mathbb{C}^{n}\times \mathbb{R}$, while there is no two parameter group of automorphic dilations on $\mathbb{H}^{n}$. The purpose of this paper is to establish a theory of the flag Lipschitz space on the Heisenberg group $\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$ that is, in a sense, intermediate between that of the classical Lipschitz space on the Heisenberg group $\mathbb{H}^{n}$ and the product Lipschitz space on $\mathbb{C}^{n}\times \mathbb{R}$. We characterize this flag Lipschitz space via the Littlewood–Paley theory and prove that flag singular integral operators, which include the Marcinkiewicz multipliers, are bounded on these flag Lipschitz spaces.

Détail

DOI : 10.4153/CJM-2018-003-0

Mots-clés : Heisenberg group, Marcinkiewicz multiplier, flag singular integral, flag Lipschitz space, reproducing formula, discrete Littlewood–Paley analysis