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Abstract and Applied Analysis
Volume 2014, Article ID 265378, 13 pages
http://dx.doi.org/10.1155/2014/265378
Research Article

Some Function Spaces via Orthonormal Bases on Spaces of Homogeneous Type

1Department of Mathematics, Sichuan University, Chengdu 610064, China
2Department of Mathematics, Macquarie University, NSW 2109, Australia
3Department of Mathematics, China University of Mining & Technology Beijing, Beijing 100083, China

Received 20 June 2014; Accepted 26 August 2014; Published 16 December 2014

Academic Editor: Daoyi Xu

Copyright © 2014 Chuang Chen 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

Let be a space of homogeneous type in the sense of Coifman and Weiss, where the quasi-metric d may have no regularity and the measure μ satisfies only the doubling property. Adapting the recently developed randomized dyadic structures of X and applying orthonormal bases of constructed recently by Auscher and Hytönen, we develop the Besov and Triebel-Lizorkin spaces on such a general setting. In this paper, we establish the wavelet characterizations and provide the dualities for these spaces. The results in this paper extend earlier related results with additional assumptions on the quasi-metric d and the measure μ to the full generality of the theory of these function spaces.