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WAVELET CHARACTERIZATIONS FOR ANISOTROPIC BESOV SPACES WITH 0<p<1

Published online by Cambridge University Press:  09 November 2004

Gustavo Garrigós
Affiliation:
Departamento de Matemáticas C-XV, Universidad Autónoma de Madrid, Ciudad Universitaria Cantoblanco, 28049 Madrid, Spain ([email protected])
Reinhard Hochmuth
Affiliation:
Institut für Angewandte Analysis, TU Bergakademie Freiberg, Agricolastra{\ss}e 1, 09596 Freiberg, Germany ([email protected])
Anita Tabacco
Affiliation:
Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy ([email protected])
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Abstract

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We present a wavelet characterization of anisotropic Besov spaces $B_{p,q}^{\bm{\alpha}}(\mathbb{R}^n)$, valid for the whole range $0\ltp,q\lt\infty$, 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 $p\lt1$. Among the consequences of our results, we characterize $B_{p,q}^{\bm{\alpha}}$ as a linear approximation space, and derive embeddings and interpolation formulae for $B_{p,q}^{\bm{\alpha}}$, which appear to be new in the literature when $p\lt1$.

AMS 2000 Mathematics subject classification: Primary 42B35; 42C40; 41A17

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2004