The space $H^1$ for nondoubling measures in terms of a grand maximal operator

Let $\mu$ be a Radon measure on $\mathbb{R} ^d$, which may be nondoubling. The only condition that $\mu$ must satisfy is the size condition $\mu(B(x,r))\leq C\,r^n$, for some fixed $0<n\leq d$. Recently, some spaces of type $BMO(\mu)$ and $H^1(\mu)$ were introduced by the author. These new spaces have properties similar to those of the classical spaces ${BMO}$ and $H^1$defined for doubling measures, and they have proved to be useful for studying the $L^p(\mu)$ boundedness of Calderón-Zygmund operators without assuming doubling conditions. In this paper a characterization of the new atomic Hardy space $H^1(\mu)$ in terms of a maximal operator $M_\Phi$ is given. It is shown that $f$ belongs to $H^1(\mu)$ if and only if $f\in L^1(\mu)$, $\int f\, d\mu=0$ and $M_\Phi f \in L^1(\mu)$, as in the usual doubling situation.