Calderón-Zygmund theory on some Lie groups of exponential growth

Abstract

Let $G=N⋊A$, where $N$ is a stratified Lie group and $A={\mathbb{R}}_{+}$ acts on $N$ via automorphic dilations. We prove that the group $G$ has the Calderón-Zygmund property, in the sense of Hebisch and Steger, with respect to a family of flow measures and metrics. This generalizes in various directions previous works by Hebisch and Steger and Martini, Ottazzi and Vallarino, and provides a new approach in the development of Calderón-Zygmund theory in Lie groups of exponential growth. We also prove a weak type $(1,1)$ estimate for the Hardy-Littlewood maximal operator naturally arising in this setting.