Calderón–Zygmund theory on some Lie groups of exponential growth

Let $G=N⋊A$, where $N$ is a stratified Lie group and $A=R_{+}$ 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 et al., and provides a new approach in the development of the 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.