This is joint work with M. Vallarino. In the setting of

the three-dimensional affine group $ax+b$, Vallarino has introduced

an atomic Hardy space $H^1$. It is inspired by Hebisch

and Steger's version of the Calder/'on-Zygmund decomposition on

this and similar groups of exponential growth. We prove that

the heat maximal operator maps $H^1$ into  $L^1$, but that this does not give

 a characterisation of $H^1$.