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

Abstract

Let $μ$ be a Radon measure on $R^d$, which may be non doubling. The only condition that $μ$ must satisfy is $μ(B(x,r))\leq C r^n$, for all $x,r$ and for some fixed $0<n\leq d$. Recently we introduced spaces of type $BMO(μ)$ and $H^1(μ)$ which proved to be useful to study the $L^p(μ)$ boundedness of Calderón-Zygmund operators without assuming doubling conditions. In this paper a characterization of the new atomic space $H^1(μ)$ in terms of a grand maximal operator $M_Φ$ is given. It is shown that $f$ belongs to $H^1(μ)$ iff $f\in L^1(μ)$, $\int f dμ=0$ and $M_Φ(f)\in L^1(μ)$, as in the usual doubling situation. The lack of any regularity condition on $μ$, 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.