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

摘要

We work on RD-spaces𝒳, namely, spaces of homogeneous type in thesense of Coifman and Weiss with the additional property that a reverse doubling property holds in𝒳. An important example is the Carnot-Carathéodoryspace with doubling measure. By constructing an approximation of the identity with bounded support of Coifman type, we develop a theory of Besovand Triebel-Lizorkin spaces on the underlying spaces. In particular, thisincludes a theory of Hardy spacesHp(𝒳)and local Hardy spaceshp(𝒳)on RD-spaces, which appears to be new in this setting. Among other things, wegive frame characterization of these function spaces, study interpolation ofsuch spaces by the real method, and determine their dual spaces whenp≥1.The relations among homogeneous Besov spaces and Triebel-Lizorkin spaces,inhomogeneous Besov spaces and Triebel-Lizorkin spaces, Hardy spaces, andBMO are also presented. Moreover, we prove boundedness results on theseBesov and Triebel-Lizorkin spaces for classes of singular integral operators,which include non-isotropic smoothing operators of order zero in the sense ofNagel and Stein that appear in estimates for solutions of the Kohn-Laplacianon certain classes of model domains inℂN. Our theory applies in a widerange of settings.