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Journal of Function Spaces and Applications
Volume 2012, Article ID 523586, 41 pages
http://dx.doi.org/10.1155/2012/523586
Research Article

Homogeneous Besov Spaces on Stratified Lie Groups and Their Wavelet Characterization

1Lehrstuhl A für Mathematik, RWTH Aachen University, D-52056 Aachen, Germany
2Department of Mathematics and Computer Sciences, City University of New York (CUNY), Queensborough College, 222-05 56th Avenue Bayside, NY 11364, USA

Received 17 January 2012; Accepted 29 January 2012

Academic Editor: Hans G. Feichtinger

Copyright © 2012 Hartmut Führ and Azita Mayeli. 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 establish wavelet characterizations of homogeneous Besov spaces on stratified Lie groups, both in terms of continuous and discrete wavelet systems. We first introduce a notion of homogeneous Besov space ̇𝐵𝑠𝑝,𝑞 in terms of a Littlewood-Paley-type decomposition, in analogy to the well-known characterization of the Euclidean case. Such decompositions can be defined via the spectral measure of a suitably chosen sub-Laplacian. We prove that the scale of Besov spaces is independent of the precise choice of Littlewood-Paley decomposition. In particular, different sub-Laplacians yield the same Besov spaces. We then turn to wavelet characterizations, first via continuous wavelet transforms (which can be viewed as continuous-scale Littlewood-Paley decompositions), then via discretely indexed systems. We prove the existence of wavelet frames and associated atomic decomposition formulas for all homogeneous Besov spaces ̇𝐵𝑠𝑝,𝑞 with 1𝑝,𝑞< and 𝑠.