A classification of anisotropic Besov spaces

Abstract

We study (homogeneous and inhomogeneous) anisotropic Besov spaces associated to expansive dilation matrices A∈GL(d,R), with the goal of clarifying when two such matrices induce the same scale of Besov spaces. For this purpose, we first establish that anisotropic Besov spaces have an alternative description as decomposition spaces. This result allows to relate properties of function spaces to combinatorial properties of the underlying coverings. This principle is applied to the question of classifying dilation matrices. It turns out that the scales of homogeneous and inhomogeneous Besov spaces differ in the way they depend on the dilation matrix: Two matrices A,B that induce the same scale of homogeneous Besov spaces also induce the same scale of inhomogeneous spaces, but the converse of this statement is generally false. We give a complete characterization of the different types of equivalence in terms of the Jordan normal forms of A,B.