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

Abstract

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