07-2   Berichtsreihe des Mathematischen Seminars der Universität Kiel

## Besov and Triebel-Lizorkin Spaces on Metric Measure Spaces Modeled on Carnot-Caratheodory Spaces

In this paper, we work on RD-spaces \$\cx\$, i.\,e., spaces of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in \$\cx.\$ An important class of RD-spaces is provided by Carnot-Carath\'eodory spaces with a doubling measure. By constructing an approximation of the identity with bounded support of Coifman type, we develop a theory of Besov and Triebel-Lizorkin spaces on the underlying spaces. In particular, this includes a theory of Hardy spaces \$\hx\$ and local Hardy spaces \$\shx\$ on RD-spaces, which appears to be new in this setting. Moreover, we prove boundedness results on these Besov and Triebel-Lizorkin spaces for classes of singular integral operators, which include non-isotropic smoothing operators of order zero in the sense of Nagel and Stein that appear in estimates for solutions of the Kohn-Laplacian on certain classes of model domains in \$\cc^N\$. Our theory applies in a wide range of settings, for instance, to Ahlfors \$n\$-regular metric measure spaces, Lie groups of polynomial volume growth, compact Carnot-Carath\'eodory (also called sub-Riemannian) manifolds and to boundaries of certain unbounded model domains of polynomial type in \$\cc^N\$ appearing in the work of Nagel and Stein.

Mathematics Subject Classification (1991): Primary 42B35; Secondary 46E35, 42B25, 42B20, 42B30, 43A99

Keywords: space of homogeneous type, RD-space, Carnot-Caratheodory space, Ahlfors \$n\$-regular metric measure space, Lie group of polynomial volume growth, unbounded model domain of polynomial type, Besov space, Triebel-Lizorkin space, Hardy space, BMO, test function, approximation of the identity, Calderon reproducing formula, Littlewood-Paley function, singular integral operator, frame, interpolation, dual space