  Let $S$ be the Lie group $\mathrm{R}^n\ltimes \mathrm{R}^+$ endowed with the
left-invariant Riemannian symmetric space structure and the right Haar measure
$\rho$, which is a Lie group of exponential growth. Hebisch and Steger in
[Math. Z. 245(2003), 37--61] proved that any integrable function on $(S,\rho)$
admits a Calder\'on--Zygmund decomposition which involves a particular family
of sets, called Calder\'on--Zygmund sets. In this paper, we first show the
existence of a dyadic grid in the group $S$, which has {nice} properties
similar to the classical Euclidean dyadic cubes. Using the properties of the
dyadic grid we shall prove a Fefferman--Stein type inequality, involving the
dyadic maximal Hardy--Littlewood function and the dyadic sharp dyadic function.
As a consequence, we obtain a complex interpolation theorem involving the Hardy
space $H^1$ and the $BMO$ space introduced in [Collect. Math. 60(2009),
277--295].
