Spaces $H^1$ and $BMO$ on $ax+b$ - groups

Let $S$ be the group $\RR^d\ltimes \RR^+$ endowed with the Riemannian symmetric space metric $d$ and the right Haar measure $\rho$. The space $(S,d,\rho)$ is a Lie group of exponential growth. In this paper we define an Hardy space $H^1$ and a $BMO$ space in this context. We prove that the functions in $BMO$ satisfy the John--Nirenberg inequality and that $BMO$ may be identified with the dual space of $H^1$. We then prove that singular integral operators whose kernels satisfy a suitable integral H\"ormander condition are bounded from $H^1$ to $L^1$ and from $L^{\infty}$ to $BMO$. We also study the real interpolation between $H^1$, $BMO$ and the $L^p$ spaces.