A Theory of Besov and Triebel-Lizorkin Spaces on Metric Measure Spaces Modeled on Carnot-Carathéodory Spaces

Abstract

We work on RD-spaces $𝒳$, namely, spaces of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in $𝒳$. An important example is the Carnot-Carathéodory space with 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 $H^p\left(𝒳\right)$ and local Hardy spaces $h^p\left(𝒳\right)$ on RD-spaces, which appears to be new in this setting. Among other things, we give frame characterization of these function spaces, study interpolation of such spaces by the real method, and determine their dual spaces when $p\ge 1$. The relations among homogeneous Besov spaces and Triebel-Lizorkin spaces, inhomogeneous Besov spaces and Triebel-Lizorkin spaces, Hardy spaces, and BMO are also presented. 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 ${ℂ}^N$. Our theory applies in a wide range of settings.