A classification of anisotropic Besov spaces

TL;DR

This paper classifies anisotropic Besov spaces associated with dilation matrices, establishing when different matrices induce the same function space scales by analyzing their Jordan normal forms and relating these spaces to decomposition spaces.

Contribution

It provides a complete classification of dilation matrices based on the Besov spaces they induce, linking algebraic properties to functional space equivalences.

Findings

Homogeneous and inhomogeneous Besov spaces differ in their dependence on dilation matrices.

Two matrices induce the same homogeneous Besov space scale if and only if they do so for the inhomogeneous scale.

Classification is achieved via Jordan normal forms of the matrices.

Abstract

We study (homogeneous and inhomogeneous) anisotropic Besov spaces associated to expansive dilation matrices $A \in {\rm GL}(d,\mathbb{R})$, 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 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. Furthermore, the question whether $A,B$ induce the same scale of homogeneous spaces is closely related to the question whether they induce the same scale of Hardy spaces; the latter question had been previously studied by Bownik. We give a complete characterization of the different types of equivalence in terms of the Jordan normal forms of $A,B$.