Homogeneous Besov Spaces on Stratified Lie Groups and Their Wavelet Characterization. (23rd May 2012)

• Record Type: Journal Article
• Authors: Führ, Hartmut; Mayeli, Azita
• Other Names: Feichtinger Hans G. Academic Editor.
• Is Part Of: Journal of function spaces and applications. Volume 2012(2012)
• Publication Date: 2012-05-23
• Subjects: Function spaces -- Periodicals; 515.73
• DOI: 10.1155/2012/523586
• Languages: English
• ISSNs: 0972-6802

Abstract

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 spaceB ˙ p, q s 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 spacesB ˙ p, q s with1 ≤ p, q < ∞ ands ∈ ℝ .