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

Abstract

Let $\mu$ be a Radon measure on $R^d$, which may be non doubling. The only condition that $\mu$ must satisfy is $\mu (B(x,r))\le C{r}^n$, for all $x,r$ and for some fixed $0<n\le d$. 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 Calderón-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 $f\in L^1(\mu )$, $\int fd\mu =0$ 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.

arXiv comment47 pages

References in corpus (1)

• Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators in nonhomogeneous spaces

Cited by in corpus (1)

• Littlewood-Paley theory and the T(1) theorem with non doubling measures