Classifying decomposition and wavelet coorbit spaces using coarse geometry

TL;DR

This paper introduces a coarse geometric method to determine when different coverings or dilation groups produce the same Besov-type decomposition and wavelet coorbit spaces, with applications to shearlet dilation groups.

Contribution

It develops a coarse geometric criterion based on quasi-isometries to compare coorbit spaces from different dilation groups, especially shearlet groups.

Findings

Quasi-isometry condition characterizes when two dilation groups have the same coorbit spaces.

The criterion can be checked algebraically for shearlet dilation groups.

Examples illustrate the application of the geometric approach.

Abstract

This paper is concerned with the study of Besov-type decomposition spaces, which are scales of spaces associated to suitably defined coverings of the euclidean space $\mathbb{R}^d$, or suitable open subsets thereof. A fundamental problem in this domain, that is currently not well understood, is deciding when two different coverings give rise to the same scale of decomposition spaces. In this paper, we establish a coarse geometric approach to this problem, and show how it specializes for the case of wavelet coorbit spaces associated to a particular class of matrix groups $H < GL(\mathbb{R}^d)$ acting via dilations. This class can be understood as a special case of decomposition spaces, and it turns out that the question whether two different dilation groups $H_1,H_2$ have the same coorbit spaces can be decided by investigating whether a suitably defined map $\phi: H_1 \to H_2$ is a quasi-isometry with respect to suitably defined word metrics. We then proceed to apply this criterion to a large class of dilation groups called {\em shearlet dilation groups}, where this quasi-isometry condition can be characterized algebraically. We close with the discussion of selected examples.