Wavelet characterizations for anisotropic Besov spaces with 0 < p < 1

authored by: Gustavo Garrigós, Reinhard Hochmuth, Anita Tabacco
Abstract: We present a wavelet characterization of anisotropic Besov spaces B _p,q^α(ℝⁿ), valid for the whole range 0 < p, q < ∞, and in terms of multi-resolution analyses with dilation adapted to the anisotropy of the space. Our proofs combine classical techniques based on Bernstein and Jackson-type inequalities, and nonlinear methods for the cases p < 1. Among the consequences of our results, we characterize B_p,q^α as a linear approximation space, and derive embeddings and interpolation formulae for B_p,q ^α, which appear to be new in the literature when p < 1.
External Organisation(s): Universidad Autónoma de Madrid
TU Bergakademie Freiberg - University of Resources
Politecnico di Torino (POLITO)
Type: Article
Journal: Proceedings of the Edinburgh Mathematical Society
Volume: 47
Pages: 573-595
No. of pages: 23
ISSN: 0013-0915
Publication date: 10.2004
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: General Mathematics
Electronic version(s): https://doi.org/10.1017/S001309150300107X (Access: Open)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{408ba73c439643508c30effd2423c60f,
title = "Wavelet characterizations for anisotropic Besov spaces with 0 < p < 1",
abstract = "We present a wavelet characterization of anisotropic Besov spaces B p,qα(ℝn), valid for the whole range 0 < p, q < ∞, and in terms of multi-resolution analyses with dilation adapted to the anisotropy of the space. Our proofs combine classical techniques based on Bernstein and Jackson-type inequalities, and nonlinear methods for the cases p < 1. Among the consequences of our results, we characterize Bp,qα as a linear approximation space, and derive embeddings and interpolation formulae for Bp,q α, which appear to be new in the literature when p < 1.",
keywords = "Approximation and interpolation spaces, Jackson and Bernstein inequalities, Multilevel decomposition",
author = "Gustavo Garrig{\'o}s and Reinhard Hochmuth and Anita Tabacco",
note = "Funding Information: Acknowledgements. Work partially supported by the European Community Human Potential Programme, contracts HPRN-CT-2002-00286 {\textquoteleft}Breaking Complexity{\textquoteright} and HPRN-CT-2001-00273 {\textquoteleft}HARP{\textquoteright}. G.G. was also supported by {\textquoteleft}Programa Ram{\'o}n y Cajal{\textquoteright} and grant BMF2001-0189, MCyT (Spain). The authors thank an anonymous referee whose careful reading and suggestions led to a much improved version of this paper.",
year = "2004",
month = oct,
doi = "10.1017/S001309150300107X",
language = "English",
volume = "47",
pages = "573--595",
journal = "Proceedings of the Edinburgh Mathematical Society",
issn = "0013-0915",
publisher = "Cambridge University Press",
number = "3",
}