Characterization of the atomic space $H^1$ for non doubling measures in terms of a grand maximal operator

Let $\mu$ be a Radon measure on $R^d$, which may be non doubling. The only condition that $\mu$ must satisfy is $\mu(B(x,r))\leq C r^n$, for all $x,r$ and for some fixed $0<n\leq d$. Recently we introduced spaces of type $BMO(\mu)$ and $H^1(\mu)$ which proved to be useful to study the $L^p(\mu)$ boundedness of Calder\'on-Zygmund operators without assuming doubling conditions. In this paper a characterization of the new atomic space $H^1(\mu)$ in terms of a grand maximal operator $M_\Phi$ is given. It is shown that $f$ belongs to $H^1(\mu)$ iff $f\in L^1(\mu)$, $\int f d\mu=0$ and $M_\Phi(f)\in L^1(\mu)$, as in the usual doubling situation. The lack of any regularity condition on $\mu$, apart from the size condition stated above, is one of the main difficulties that appears when one tries to extend the classical arguments to the present situation.