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Let $S$ be the Lie group $\mathrm{R}^n\ltimes \mathrm{R}^+$ endowed with theleft-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 familyof sets, called Calder\'on--Zygmund sets. In this paper, we first show theexistence of a dyadic grid in the group $S$, which has {nice} propertiessimilar to the classical Euclidean dyadic cubes. Using the properties of thedyadic grid we shall prove a Fefferman--Stein type inequality, involving thedyadic maximal Hardy--Littlewood function and the dyadic sharp dyadic function.As a consequence, we obtain a complex interpolation theorem involving the Hardyspace $H^1$ and the $BMO$ space introduced in [Collect. Math. 60(2009),277--295].

Dyadic Sets, Maximal Functions and Applications on $ax+b$ --Groups