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

Abstract:

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 $B\!M\!O(\mu )$ 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.