07-1   Berichtsreihe des Mathematischen Seminars der Universität Kiel

## Littlewood-Paley Characterizations for Hardy Spaces

Let $(\cx, d, \mu)$ be a space of homogeneous type in the sense of Coifman and Weiss. Assuming that $\mu$ satisfies certain estimates from below and there exists a suitable Calder\'on reproducing formula in $L^2(\cx)$, the authors establish a Lusin-area characterization for the atomic Hardy spaces $\hax$ of Coifman and Weiss for $p\in (p_0,1]$, where $p_0=3Dn/(n+\ez_1)$ depends on the = dimension" $n$ of $\cx$ and the regularity" $\ez_1$ of the Calder\'on reproducing formula. Using this characterization, the authors further obtain a=20 Littlewood-Paley $g_\lambda^\ast$-function characterization for $\hx$ when $\lz>n+2n/p$ and the boundedness of Calder\'on-Zygmund operators on $\hx$. The results apply, for instance, to Ahlfors $n$-regular metric measure spaces, Lie groups of polynomial volume growth and boundaries of some unbounded model domains of polynomial type in $\cc^N$.

Mathematics Subject Classification (1991): primary 42B25; secondary 42B35; 42B20; 43A99

Keywords: space of homogeneous type, Calderon reproducing formula, space of test function, Littlewood-Paley function, Hardy space, atom, singular integral, dual space.