Abstract and Applied Analysis
Volume 2008 (2008), Article ID 893409, 250 pages
doi:10.1155/2008/893409
Research Article

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

Yongsheng Han,1 Detlef Müller,2 and Dachun Yang3

1Department of Mathematics, Auburn University, Auburn, AL 36849-5310, USA
2Mathematisches Seminar, Christian-Albrechts-Universität Kiel, Ludewig-Meyn Strasse 4, 24098 Kiel, Germany
3School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

Received 13 February 2008; Accepted 23 May 2008

Academic Editor: Stephen Clark

Copyright © 2008 Yongsheng Han et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 Hp(𝒳) and local Hardy spaces hp(𝒳) 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≥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 ℂN. Our theory applies in a wide range of settings.