# Classifying decomposition and wavelet coorbit spaces using coarse geometry

Hartmut Führ  Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen, Germany
<EMAIL_ADDRESS>René Koch  Lehrstuhl A für Mathematik, RWTH Aachen,
52056 Aachen, Germany<EMAIL_ADDRESS>
###### 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 fully 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.
These spaces can be understood as special instances 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 _shearlet dilation
groups_ , where this quasi-isometry condition can be characterized
algebraically. We close with the discussion of selected examples.

##  1 Introduction

This paper is concerned with the study of two related classes of function
spaces on euclidean spaces, wavelet coorbit spaces and decomposition spaces.
Both classes can be understood as rather sweeping generalizations of Besov
spaces, and their inception dates back to work by Feichtinger, Gröchenig and
Gröbner (e.g. [12, 13, 14, 15]) in the 1980s. We refer to [17] for more
background information on this development, and in particular for an
explanation of the relationship of these notions to the influential work of
Frazier and Jawerth [19].

The main thrust of the initial work was to provide a unified view onto a
variety of function spaces, specifically Besov spaces and modulation spaces,
using notions from abstract harmonic and Fourier analysis. Coorbit spaces are
defined using group representation theory [14, 15], and special instances of
coorbit spaces have been studied in a variety of settings and for various
classes of underlying locally compact groups in recent years; see e.g. [11,
10, 24, 25, 16, 18] for a small sample of the literature. The class of
admissible dilation groups underlying higher dimensional wavelet coorbit
spaces turns out to be a particularly rich source of examples [24], see

  * 1.

[22] for a classification of admissible dilation groups in dimension two up to
conjugacy;

  * 2.

[21] for the systematic construction and classification (up to conjugacy) of
abelian admissible dilation groups;

  * 3.

[11, 10] for the initial definition of shearlet dilation groups in dimension
two and higher;

  * 4.

[9] for the introduction of Toeplitz shearlet dilation groups in dimension
three and higher;

  * 5.

[25, 1] for the introduction and systematic construction of generalized
shearlet dilation groups in arbitrary dimensions (comprising the previous
examples);

  * 6.

[8] for a classification of admissible dilation groups in dimension three, up
to conjugacy.

We emphasize that the items on this list do not refer to single dilation
groups, but rather to classes of groups, most of which tend to become quite
large for higher dimensions.

By contrast, decomposition spaces are of a more Fourier analytic nature.
Whereas coorbit spaces depend on the choice of a particular group
representation, the construction of decomposition spaces departs from a
certain covering of the frequency domain. After their inception in [12], the
theory of decomposition spaces lay somewhat dormant for a while, but their
usefulness in capturing approximation-theoretic features of anisotropic
systems such as curvelets was realized later by Borup and Nielsen, see e.g.
[3, 32]. A further major boost to the subject was provided by the
transformative work of Voigtlaender [36, 37], which is also the foundation of
the current paper. At this point in time, large classes of function spaces
have been recognized as special instances of the decomposition space
formalism, as the following list shows:

  * 1.

$\alpha$-modulation spaces [28];

  * 2.

curvelet smoothness spaces [3];

  * 3.

shearlet smoothness spaces [32];

  * 4.

homogeneous and inhomogeneous anisotropic Besov spaces [4, 6];

  * 5.

wave packet smoothness spaces [5];

  * 6.

wavelet coorbit spaces associated to admissible dilation group, i.e., all
examples on the previous list [27]);

  * 7.

wavelet coorbit spaces associated to integrably admissible dilation groups,
see [26]. This class properly contains the class mentioned in the previous
item, but also the class of homogeneous anisotropic Besov spaces.

Again, most of the items on this list represent classes or families of
constructions rather than a single one, and these classes can become quite
large with increasing dimension.

While coorbit and decomposition spaces rest on somewhat distinct mathematical
foundations, the resulting spaces share an important common approximation-
theoretic interpretation. What is common both to coorbit and decomposition
spaces is the existence of a family of vectors $(\eta_{x})_{x\in\mathcal{X}}$
in a suitable Hilbert space $\mathcal{H}$ that acts as a continuous frame of
$\mathcal{H}$, i.e., guaranteeing a norm equivalence

$\|f\|_{\mathcal{H}}\asymp\|(\langle
f,\eta_{x}\rangle)_{x\in\mathcal{X}}\|_{L^{2}(\mathcal{X},d\mu)}~{},$ (1.1)

with respect to a suitably chosen measure $\mu$ on $\mathcal{X}$, as well as
an associated inversion formula

$f=\int_{\mathcal{X}}\langle f,\eta_{x}\rangle\widetilde{\eta}_{x}d\mu(x)~{},$

with a suitably chosen dual system
$(\widetilde{\eta}_{x})_{x\in\mathcal{X}}\subset\mathcal{H}$. The operator

$V_{\eta}:f\mapsto(\langle f,\eta_{x}\rangle)_{x\in\mathcal{X}}$

mapping $f$ to its expansion coefficients can be understood as a generalized
wavelet transform. Typically, the parameter space $\mathcal{X}$ provides an
interpretation of the elements $\eta_{x}$ of the continuous frame as basic
building blocks, and the inversion formula expresses $f$ as a continuous
expansion in these building blocks.

For concreteness, let us quickly sketch the case of generalized wavelet
systems: Given a dilation group $H<GL(\mathbb{R}^{d})$, we let
$\mathcal{X}=\mathbb{R}^{d}\rtimes H$ denote the subgroup of the full affine
group of $\mathbb{R}^{d}$ generated by the translations in $\mathbb{R}^{d}$
and $H$, and define the quasi-regular representation

$\pi:\mathbb{R}^{n}\rtimes
H\to\mathcal{U}(L^{2}(\mathbb{R}^{n}))~{},~{}\left(\pi(x,h)f\right)(y)=|{\mathrm{det}}(h)|^{-1/2}f(h^{-1}(y-x))~{},$

and for suitably chosen $\eta\in L^{2}(\mathbb{R}^{n})$, we let

$\eta_{(x,h)}=\pi(x,h)\eta~{}.$

The element $(x,h)\in\mathcal{X}$ corresponds to a wavelet $\eta_{(x,h)}$
centered at $x$ and scaled by $h\in H$. Very often, explicit parametrizations
of $H$ allow to interpret the scaling variable $h$ in $(x,h)$ (and thus
$\eta_{(x,h)}$) further, e.g. as a combination of isotropic scaling and
rotation in the case of the similitude group [2], or as a combination of
anisotropic scaling and shearing in the case of a shearlet dilation group
[10]. In this group-theoretic context, the norm equivalence (1.1) becomes a
norm equality, when $\eta$ is chosen as an admissible vector and $\mu$ is
chosen as the left Haar measure on $G$, and the inversion formula holds with
$\widetilde{\eta}_{(x,h)}=\eta_{(x,h)}$.

One can then proceed to define norms on elements of $L^{2}(\mathbb{R}^{n})$ by
imposing integrability conditions that are more stringent than the
$L^{2}$-norm entering in the Hilbert space norm equivalence (1.1), i.e., by
introducing weights $\nu$ and considering integrability exponents $p<2$,
leading to considering norms of the type

$\|V_{\eta}f\|_{L^{p}_{\nu}}~{}.$

Furthermore, discretization results allow to replace the (typically
continuously indexed) systems $(\eta_{x})_{x\in\mathcal{X}}$ by suitably
chosen discrete subsystems – in fact, (quasi-)Banach frames – corresponding to
discrete subsets $\mathcal{X}_{d}\subset\mathcal{X}$, while preserving norm
equivalences such as

$\|V_{\eta}f\|_{L^{p}_{\nu}}\asymp\|V_{\eta}f|_{\mathcal{X}_{d}}\|_{\ell^{p}_{\nu}}~{},$

see e.g. [12, 13, 14, 15]. For $p<2$ and constant weights this norm
equivalence attains additional relevance: Here the discretization results of
the just cited references allow to conclude, that vectors having $p$-summable
frame coefficients have a nontrivial non-linear approximation rate with
respect to the frame, with the decay rate increasing as $p$ decreases. In this
way, both coorbit and decomposition spaces associated to integrability
exponents $p,q<2$ have natural interpretations as spaces of sparse signals
with respect to the respective systems of building blocks. These spaces
therefore capture the approximation-theoretic properties of the building
blocks.

Much of the foundational work in [12, 13, 14, 15] was devoted to proving that
the definitions were consistent, i.e., essentially independent of various
design choices that enter into the construction of the systems. In the case of
wavelet coorbit spaces, these results were mostly related to the proper choice
of the analyzing wavelet $\eta$ entering the definition of the wavelet system
$(\eta_{(x,h)})_{(x,h)\in\mathcal{X}}=(\pi(x,h)\eta)_{(x,h)\in\mathcal{X}}$.

By contrast, the influence of the primary choices in the design of the system
of building blocks, i.e., the choice of dilation group in the case of
generalized wavelet systems, and the choice of covering in the case of
decomposition spaces, is much less understood. Generally speaking, one very
much expects that qualitatively different primary choices will result in
different approximation-theoretic properties of the building blocks; for
systems such as curvelets or shearlets, this expectation was the driving
factor for their inception. That said, the understanding what “qualitatively
different” actually means in this context is obviously an important part of
this discussion, and it is currently not fully developed.

Prior to the work of Voigtlaender, with few exceptions such as [32], most of
the work on coorbit or decomposition spaces tended to stay within the confines
provided by a fixed primary choice, i.e., tended to concentrate on, say,
$\alpha$-modulation spaces, or on shearlet coorbit spaces. Voigtlaender’s
publications since 2015, specifically [36, 27, 37], systematically develop
techniques that allow to cross these boundaries between different classes of
spaces, by providing sharp embedding results between decomposition spaces
associated to different coverings, or into classical smoothness spaces.

By contrast to the scope of the mentioned papers [27, 36, 37], the aims of
this paper are somewhat more modest and elementary. We address the following
fundamental question, both for decomposition spaces and for wavelet coorbit
spaces: When do two initially different primary design choices (e.g.,
different coverings or dilation groups) result in the same scales of
associated spaces? Our central contribution essentially amounts to rewriting
the pertinent criteria formulated in [37] by introducing a novel ingredient to
the discussion, namely coarse geometry, and demonstrating that this new
perspective can be used effectively.

###  1.1 Structure and overview of the paper

This paper is based on parts of the PhD thesis [30] by the second author, with
some results substantially expanded.

As mentioned at the end of the previous subsection, we address two main
questions:

  1. 1.

When do two coverings result in the same scale of decomposition spaces?

  2. 2.

When do two dilation groups possess the same scale of coorbit spaces?

As pointed out in the previous subsection, Question 2 is a special case of
Question 1, and both are essentially answered by results in [37]. However, as
with many results in this domain, the precise criteria are somewhat
unintuitive, and tend to be cumbersome to apply in concrete settings. It is
one of the main contributions of our paper to rewrite the criteria in terms of
suitable metrics, following an initial observation made in [12], and to
demonstrate the usefulness of this reformulation with the help of various
examples.

We now give an overview of the paper. The following summary glosses over
various technicalities (such as additional conditions on coverings, or matters
related to the question how coorbit spaces associated to different groups can
be understood to coincide, or the role of connectedness issues), that are
explained in more detail in the subsequent text.

Section 2 recalls the fundamentals of decomposition spaces and coorbit spaces,
including the definitions of the various spaces, and their basic properties.
Decomposition spaces, as used throughout the paper, rely on the notion of
_(structured) admissible covering_. We will concentrate on decomposition
spaces associated to weighted mixed $L^{p}$-norms, i.e., to inner norms
$\|\cdot\|_{L^{p}}$ and outer norms $\|\cdot\|_{\ell^{q}_{v}}$. The main
takeaway from subsection 2.1 is the definition of the notion of weak
equivalence of coverings, and its relevance for the associated decomposition
spaces. Lemmas 2.3 and 2.4 show that weak equivalence of two admissible
coverings is equivalent to the fact that the associated scales of
decomposition spaces coincide. They also formulate a rigidity result stating
that if two decomposition spaces associated to different coverings coincide
for some pair of integrability exponents $(p,q)\neq(2,2)$, then the full
scales of decomposition spaces agree. A complementary result that is of
particular relevance to wavelet coorbit spaces addresses the question when
dual coverings of different sets give rise to identical decomposition spaces,
see Theorem 2.5.

We then recount the relevant results of wavelet coorbit theory in higher
dimensions, in particular the notion of admissible dilation groups $H$ and
their unique open dual orbits $\mathcal{O}=H^{T}\xi$, for suitably chosen
$\xi\in\mathbb{R}^{d}$. The decomposition space description of the coorbit
spaces associated to the quasi-regular representation of
$\mathbb{R}^{d}\rtimes H$ is obtained via the so-called induced covering
arising from the dual action of $H$ on $\mathcal{O}$, see Theorem 2.13. We
then introduce the notion of coorbit equivalence for two dilation groups
$H_{1},H_{2}$, expressing when their coorbit spaces coincide; see Definition
2.16. We combine Theorem 2.13 with the results characterizing when two dual
coverings result in the same scale of decomposition spaces, and with Theorem
2.5, to obtain an important intermediate characterization of coorbit
equivalence in Theorem 2.18.

Section 3 then proceeds to introduce metric language to the discussion. We
first give a short review of the relevant notions from coarse geometry in
Subsection 3.1. A covering $\mathcal{P}$ of a frequency set $\mathcal{O}$
allows to introduce, in a very natural way, a related metric
$d_{\mathcal{P}}$, and it turns out that weak equivalence of $\mathcal{P}$ to
a second covering $\mathcal{Q}$ of the same set $\mathcal{O}$ is equivalent to
the fact that
$id_{\mathcal{O}}:(\mathcal{O},d_{\mathcal{P}})\to(\mathcal{O},d_{\mathcal{Q}})$
is a quasi-isometry, or equivalently, a coarse equivalence; see Theorem 3.22.

As already mentioned, it is important to point out that the definition of the
metric goes back to the original source [12], and Proposition 3.8 of the
mentioned paper is a relevant precursor of Theorem 3.22. However, weak
equivalence of coverings was initially only known to be a _sufficient_
criterion for the property that two coverings define the same scale of
decomposition spaces; the converse was proved later in [36].

Section 4 transfers the results obtained for decomposition spaces to the
coorbit setting. We begin by reviewing basic notions from metric group theory
in Subsection 4.1. Subsection 4.2 establishes that the canonical projection
map from the group onto the dual orbit is a quasi-isometry. Subsection 4.3
contains one of the main results of this paper, namely Theorem 4.17, which
combines the observations from Subsection 4.2 with the criteria from the
previous sections to formulate a metric criterion for coorbit equivalence: Two
admissible groups $H_{1}$ and $H_{2}$ can only be coorbit equivalent if their
dual orbits coincide, i.e., $\mathcal{O}=H_{1}^{T}\xi_{0}=H_{2}^{T}\xi_{0}$
for some suitable $\xi_{0}\in\mathbb{R}^{d}$. If that condition is fulfilled,
consider the two associated canonical projections
$p_{\xi_{0}}^{H_{1}}:H_{1}\ni h\mapsto h^{-T}\xi_{0}\in\mathcal{O}$, and
$p_{\xi_{0}}^{H_{2}}:H_{2}\to\mathcal{O}$ defined analogously. Let
$(p_{\xi_{0}}^{H_{2}})^{-1}:\mathcal{O}\to H_{2}$ denote any right inverse of
$p_{\xi_{0}}^{H_{2}}$. Then $H_{1}$ and $H_{2}$ are coorbit equivalent if and
only if $(p_{\xi_{0}}^{H_{2}})^{-1}\circ
p_{\xi_{0}}^{H_{1}}:(H_{1},d_{H_{1}})\to(H_{2},d_{H_{2}})$ is a quasi-isometry
with respect to any choice of word metrics $d_{H_{1}},d_{H_{2}}$ on
$H_{1},H_{2}$ associated to suitable relatively compact neighborhoods of
unity. Again, the quasi-isometry property is equivalent to coarse equivalence.
Thus coorbit equivalence has been translated to a concrete geometric group
theory problem.

Section 5 contains an illustration that this translation can be put to use for
the characterization of coorbit equivalence within a large example class,
namely that of generalized shearlet dilation groups. The main result of that
section is Theorem 5.9, which contains a concise characterization of coorbit
equivalence for shearlet dilation groups. The fact that this equivalence leads
to rather stringent conditions on the groups under consideration emphasizes
the richness of coorbit theory in higher dimensions. The results indicate that
in higher dimensions, coorbit equivalence of different groups is a fairly rare
phenomenon. While this observation may have been expected, it is the main
contribution of this paper to provide sharp criteria, methods for their
verification, and explicit classes of groups corroborating this expectation.

The closing section contains various examples that further illustrate the
usefulness of the approach developed in this paper.

##  2 Fundamentals of coorbit and decomposition spaces

###  2.1 Decomposition spaces

The starting point for the definition of decomposition spaces is the notion of
an _admissible covering_ $\mathcal{Q}=(Q_{i})_{i\in I}$ of some open set
$\mathcal{O}\subset\mathbb{R}^{d}$ (see [12]), which is a family of nonempty
sets $Q_{i}\subset\mathbb{R}^{d}$ such that

  1. i)

$\bigcup_{i\in I}Q_{i}=\mathcal{O}$ and

  2. ii)

$\sup_{i\in I}\sharp\\{j\in I:Q_{i}\cap Q_{j}\neq\emptyset\\}<\infty$.

Throughout this paper, we will concentrate on the class of _(tight) structured
admissible covering_ , see Definition 2.5 of [37]. This means that
$Q_{i}=T_{i}Q+b_{i}$ with $T_{i}\in\mathrm{GL}(\mathbb{R}^{d})$,
$b_{i}\in\mathbb{R}^{d}$ with an open, precompact set $Q$, and the involved
matrices fulfill

$\sup_{i,j\in I:Q_{j}\cap Q_{j}\neq\emptyset}\|T_{i}^{-1}T_{j}\|<\infty.$
(2.1)

The next ingredient in the definition of decomposition spaces is a special
partition of unity $\Phi=(\varphi_{i})_{i\in I}$ subordinate to $\mathcal{Q}$,
also called $\mathrm{L}^{p}$-BAPU (bounded admissible partition of unity),
with the following properties

  1. i)

$\varphi_{i}\in C_{c}^{\infty}(\mathcal{O})\quad\forall i\in I$,

  2. ii)

$\sum_{i\in I}\varphi_{i}(x)=1\quad\forall x\in\mathcal{O}$,

  3. iii)

$\varphi_{i}(x)=0$ for $x\in\mathbb{R}^{d}\setminus Q_{i}$ and $i\in I$,

  4. iv)

if $1\leq p\leq\infty$: $\sup_{i\in
I}\|\mathcal{F}^{-1}\varphi_{i}\|_{\mathrm{L}^{1}}<\infty$,

if $0<p<1$: $\sup_{i\in
I}|\det(T_{i})|^{\frac{1}{p}-1}\|\mathcal{F}^{-1}\varphi_{i}\|_{\mathrm{L}^{p}}<\infty$.

Here, $\mathcal{F}$ denotes the usual Fourier transform of a function in
$\mathrm{L^{2}}(\mathbb{R}^{d})$ defined by

$\displaystyle\mathcal{F}f(\xi):=\int_{\mathbb{R}^{d}}f(x)e^{-2\pi i\langle
x,\xi\rangle}\mathrm{d}x$

for $\xi\in\mathbb{R}^{d}$. We also use the notation
$\widehat{f}:=\mathcal{F}(f)$. The definition of decomposition spaces requires
one last ingredient, namely a weight $(u_{i})_{i\in I}$ such that there exists
$C>0$ with $u_{i}\leq Cu_{j}$ for all $i,j\in I:Q_{i}\cap Q_{j}\neq\emptyset$.
A weight with this property is also called _$\mathcal{Q}$ -moderate_. The
interpretation of this property is that the value of $(u_{i})_{i\in I}$ is
comparable for indices corresponding to sets which are “close” to each other.
Finally, we define the (Fourier-side) decomposition space with respect to the
covering $\mathcal{Q}$ and the weight $(u_{i})_{i\in I}$ with integrability
exponents $0<p,q\leq\infty$ as

$\displaystyle\mathcal{D}(\mathcal{Q},\mathrm{L}^{p},\ell^{q}_{u}):=\\{f\in\mathcal{D}^{\prime}(\mathcal{O}):\|f\|_{\mathcal{D}(\mathcal{Q},\mathrm{L}^{p},\ell^{q}_{u})}<\infty\\}$
(2.2)

for

$\displaystyle\|f\|_{\mathcal{D}(\mathcal{Q},\mathrm{L}^{p},\ell^{q}_{u})}:=\left\|\left(u_{i}\cdot\|\mathcal{F}^{-1}(\varphi_{i}f)\|_{\mathrm{L}^{p}(\mathbb{R}^{d})}\right)_{i\in
I}\right\|_{\ell^{q}(I)}.$ (2.3)

As the notation suggests, the decomposition spaces are independent of the
precise choice of $\Phi$ [36, Corollary 3.4.11].

A crucial concept is the definition of the set of neighbors of a covering.

######  Definition 2.1 ([12] Definition 2.3)

For a covering $\mathcal{Q}=(Q_{i})_{i\in I}$ of $\mathcal{O}$ with
$Q_{i}\subset\mathcal{O}$ for all $i\in I$, we define the set of neighbors of
a subset $J\subset I$ as

$\displaystyle J^{*}:=\Set{i\in I}{\exists j\in J:\ Q_{i}\cap
Q_{j}\neq\emptyset}.$

By induction, we set $J^{0*}:=J$ and $J^{(n+1)*}=\left(J^{n*}\right)^{*}$ for
$n\in\mathbb{N}_{0}$. Moreover, we use the shorthand notations
$i^{k*}:=\Set{i}^{k*}$ and define $Q_{i}^{k*}:=\bigcup_{j\in i^{k*}}Q_{j}$ for
$i\in I$ and $k\in\mathbb{N}_{0}$.

In the remaining part of this subsection, we take a look at relations between
different coverings. The ultimate purpose of these relations is the
clarification when two coverings lead to the same scale of decomposition
spaces. While our exposition follows [36], most of the definitions hark back
to [12].

######  Definition 2.2 ([36] Definition 3.3.1.)

Let $\mathcal{Q}=(Q_{i})_{i\in I}$ and $\mathcal{P}=(P_{j})_{j\in J}$ be
families of subsets of $\mathbb{R}^{d}$.

  1. i)

We define the set of $\mathcal{P}$-neighbors of $i\in I$ by $J_{i}:=\Set{j\in
J}{Q_{i}\cap P_{j}\neq\emptyset}$. More generally, we call $J_{i}$ and $I_{j}$
intersection sets for the coverings $\mathcal{Q}$ and $\mathcal{P}$.

  2. ii)

We call $\mathcal{Q}$ weakly subordinate to $\mathcal{P}$ if
$N(\mathcal{Q},\mathcal{P}):=\sup_{i\in
I}\mathopen{\lvert}J_{i}\mathclose{\rvert}<\infty$.

The quantity $N(\mathcal{P},\mathcal{Q})$ is defined analogously, and we call
$\mathcal{Q}$ and $\mathcal{P}$ weakly equivalent if
$N(\mathcal{P},\mathcal{Q})<\infty$ and $N(\mathcal{Q},\mathcal{P})<\infty$.

  3. iii)

We call $\mathcal{Q}$ almost subordinate to $\mathcal{P}$ if there exists
$k\in\mathbb{N}_{0}$ such that for every $i\in I$ there exists an $j_{i}\in J$
with $Q_{i}\subset P_{j_{i}}^{k*}$. If $k=0$ is a valid choice, then we call
$\mathcal{Q}$ subordinate to $\mathcal{P}$.

  4. iv)

We call $\mathcal{Q}$ weakly equivalent to $\mathcal{P}$ if $\mathcal{Q}$ is
weakly subordinate to $\mathcal{P}$ and $\mathcal{P}$ is weakly subordinate to
$\mathcal{Q}$.

The relevance of these notions, in particular of weak equivalence, is spelled
out in the next two lemmas. The formulation of the next lemma is a special
case of the cited result.

######  Lemma 2.3 ([37] Theorem 6.9)

Let $\mathcal{Q}=(Q_{i})_{i\in I},\mathcal{P}=(P_{j})_{j\in J}$ be two
structured admissible coverings of the open set
$\mathcal{O}\subset\mathbb{R}^{d}$. If $\mathcal{Q}$ and $\mathcal{P}$ are not
weakly equivalent, then

$\mathcal{D}(\mathcal{Q},L^{p},\ell^{q}_{u_{1}})\neq\mathcal{D}(\mathcal{P},L^{p},\ell^{q}_{u_{2}})$

for all $\mathcal{Q}$-moderate weights $u_{1}:I\to(0,\infty)$, for all
$\mathcal{P}$-moderate weights $u_{2}:J\to(0,\infty)$ and all
$p,q\in(0,\infty]$ with $(p,q)\neq(2,2)$.

The exception $(p,q)\neq(2,2)$ is necessary to exclude trivial cases: In the
case of $(p,q)=(2,2)$ the associated decomposition spaces are just weighted
$\mathrm{L}^{2}$ spaces, by the Plancherel Theorem. In particular, if
$\mathcal{O}\subset\mathbb{R}^{d}$ is open and of full measure, and the weight
$v$ is constant, one finds that
$\mathcal{D}(\mathcal{P},L^{2},\ell^{2}_{v})=L^{2}(\mathbb{R}^{d})$, for all
admissible coverings $\mathcal{P}$.

Weak subordinateness and equivalence of coverings are important assumptions
for a multitude of sufficient criteria for embeddings of decomposition spaces
and their equality, as developed in [37]. The following statement is [37,
Lemma 6.11]. The statement about the range $0\leq p,q\leq\infty$ is justified
by the remark following the cited lemma.

###### Lemma 2.4

Let $1\leq p,q\leq\infty$ and let
$\emptyset\neq\mathcal{O}\subset\mathbb{R}^{d}$ be open. Further, let
$\mathcal{Q}=(Q_{i})_{i\in I},\mathcal{P}=(P_{j})_{j\in J}$ be two tight
structured admissible coverings of $\mathcal{O}$, and let $u_{1}$ be a
$\mathcal{Q}$-moderate weight and $u_{2}$ a $\mathcal{P}$-moderate weight.

If $\mathcal{Q}$ and $\mathcal{P}$ are weakly equivalent and there exists
$C>0$ such that $C^{-1}u_{1}(i)\leq u_{2}(j)\leq Cu_{1}(i)$ for all $i\in I$
and $j\in J$ with $Q_{i}\cap P_{j}\neq\emptyset$, then

$\displaystyle\mathcal{D}(\mathcal{Q},L^{p},\ell^{q}_{u_{1}})=\mathcal{D}(\mathcal{P},L^{p},\ell^{q}_{u_{2}})$

with equivalent norms.

If all sets in the coverings are connected, the conclusion holds for the range
$0\leq p,q\leq\infty$.

Under suitable assumptions, the two lemmas show that an equality

$\mathcal{D}(\mathcal{Q},L^{p_{1}},\ell^{q_{1}}_{u_{1}})=\mathcal{D}(\mathcal{P},L^{p_{2}},\ell^{q_{2}}_{u_{2}})$

with non-trivial exponents $(p_{1},q_{1})\neq(2,2)$ and/or
$(p_{2},q_{2})\neq(2,2)$ holds if and only if $\mathcal{Q}$ and $\mathcal{P}$
are weakly equivalent, with $(p_{1},q_{1})=(p_{2},q_{2})$ and the involved
weights are equivalent. Note that one such equality entails

$\mathcal{D}(\mathcal{Q},L^{p},\ell^{q}_{u_{1}})=\mathcal{D}(\mathcal{P},L^{p},\ell^{q}_{u_{2}})$

for _all_ exponents $0\leq p,q\leq\infty$ and all weights $u_{1}$, with
suitably chosen $u_{2}$.

If the coverings consist of open and connected sets, weak subordinateness
implies almost subordinateness (cf. [36] Corollary 3.3.4.). Consequently, we
will mainly be interested in studying necessary and sufficient conditions for
the weak subordinateness of coverings.

While the results so far compare decomposition spaces associated to different
coverings $\mathcal{Q},\mathcal{P}$ of the same set $\mathcal{O}$, the
following Theorem examines pairs of decomposition spaces associated to
coverings of different sets $\mathcal{O}$, $\mathcal{O}^{\prime}$. It is a
special case of [37, Theorem 6.9 $\frac{1}{2}$].

###### Theorem 2.5

Let $\emptyset\neq\mathcal{O},\mathcal{O}^{\prime}\subset\mathbb{R}^{d}$ open.
Let $\mathcal{Q}=(Q_{i})_{i\in I}$ denote an admissible covering of
$\mathcal{O}$, $\mathcal{P}=(P_{j})_{j\in J}$ denote an admissible covering of
$\mathcal{O}^{\prime}$. Assume that either
$\mathcal{O}^{\prime}\cap\partial\mathcal{O}\neq\emptyset$ or
$\mathcal{O}\cap\partial\mathcal{O}^{\prime}\neq\emptyset$ holds, and that
$\mathcal{O}\cap\mathcal{O}^{\prime}$ is unbounded. Let
$p_{1},p_{2},q_{1},q_{2}\in(0,\infty]$. Then

$\forall f\in
C_{c}^{\infty}(\mathcal{O}\cap\mathcal{O}^{\prime})~{}:~{}\|f\|_{D(\mathcal{Q},L^{p_{1}},\ell^{q_{1}}_{v})}\asymp\|f\|_{D(\mathcal{P},L^{p_{2}},\ell^{q_{2}}_{w})}$

can only hold in the trivial case, i.e., when
$(p_{1},q_{1})=(2,2)=(p_{2},q_{2})$ and $v_{i}\asymp w_{j}$ whenever
$Q_{i}\cap P_{j}\neq\emptyset$.

Note that the assumptions on $\mathcal{O},\mathcal{O}^{\prime}$ are fulfilled
if they are distinct open and dense subsets. Density and openness of
$\mathcal{O}$ imply $\partial\mathcal{O}=\mathbb{R}^{d}\setminus\mathcal{O}$,
and thus $\mathcal{O}^{\prime}\cap\partial\mathcal{O}=\emptyset$ can only
happen if $\mathcal{O}^{\prime}\subsetneq\mathcal{O}$. In that case however we
get
$\partial\mathcal{O}^{\prime}\cap\mathcal{O}=(\mathbb{R}^{d}\setminus\mathcal{O}^{\prime})\cap\mathcal{O}\neq\emptyset$.
Furthermore, $\mathcal{O}\cap\mathcal{O}^{\prime}$ is dense, hence unbounded.

###### Remark 2.6

The results in [37] exhibit an occasional subtle influence of the various
assumptions on the elements of the coverings $\mathcal{P}$ and $\mathcal{Q}$
on the comparison of associated decomposition spaces. An instance can be
witnessed in Lemma 2.4, where connectedness assumptions of the covering sets,
as well as tightness and structuredness of the cover, have an influence on the
range of summability parameters for which the conclusions of the lemma hold.
The connectedness assumption will play a more explicit role in the formulation
and proof of our metric characterization of weak equivalence in Theorem 3.22.

For further remarks on these issues, we refer to 3.23 below.

For the discussion of coorbit spaces associated to different dilation groups,
a certain class of weights will be of particular importance. Throughout the
paper, we will use $|A|$ to denote the Lebesgue measure of a Borel set
$A\subset\mathbb{R}^{d}$.

###### Definition 2.7

Let $\mathcal{Q}=(Q_{i})_{i\in I}$ denote a (tight) structured admissible
covering, and $\alpha\in\mathbb{R}$. The intrinsic weight with exponent
$\alpha$ is the family $(u_{i})_{i\in I}$ defined by

$u_{i}=|Q_{i}|^{\alpha}~{}.$

Note that for $Q_{i}=T_{i}Q+b_{i}$, one obtains $|Q_{i}|\asymp|\det(T_{i})|$.
Hence the intrinsic weight is indeed $\mathcal{Q}$-moderate, by the condition
(2.1) on structured coverings.

The following observation notes that intrinsic weights are robust under weak
equivalence.

###### Lemma 2.8

Let $\mathcal{Q}=(Q_{i})_{i\in I}$ and $\mathcal{P}=(P_{j})_{j\in J}$ denote
two tight structured admissible coverings consisting of connected sets. If
$\mathcal{Q}$ and $\mathcal{P}$ are weakly equivalent, then the associated
intrinsic weights are equivalent as well, i.e., there exists a constant $C>0$,
such that one has

$\forall i\in I\forall j\in J:Q_{i}\cap P_{j}\neq\emptyset\Rightarrow
C^{-1}\leq\frac{|P_{j}|}{|Q_{i}|}\leq C~{}.$

###### Proof 1

Let $x\in Q_{i}\cap P_{j}$ be given, for suitable $i\in I$ and $j\in J$. We
first note that if $\mathcal{Q}$ is weakly subordinate to $\mathcal{P}$, and
consists of connected sets, then it is almost subordinate to $\mathcal{P}$, by
[37, Corollary 2.13]. Hence there exists $n\in\mathbb{N}$ such that for each
$i\in I$ there exists $j_{0}\in J$ with

$Q_{i}\subset P_{j_{0}}^{n\ast}~{}.$

Let $C_{1},C_{2}\geq 1$ denote the constants such that

$\forall j,\ell\in J:P_{j}\cap
P_{\ell}\neq\emptyset\Rightarrow\frac{|P_{j}|}{|P_{\ell}|}\leq C_{1}~{},$

which results from condition (2.1), and

$C_{2}=\sup_{j\in J}\sharp\\{\ell\in J:P_{j}\cap P_{\ell}\neq\emptyset\\}~{}.$

Then $P_{j_{0}}^{n\ast}$ arises as the union of at most
$\sum_{k=0}^{n}C_{2}^{k}\leq C_{2}^{n+1}$ sets, each of measure at most
$C_{1}^{n}|P_{j_{0}}|$, i.e.,

$|Q_{i}|\leq|P_{j_{0}}^{n\ast}|\leq|P_{j_{0}}|C_{1}^{n}C_{2}^{n+1}~{}.$

But the fact that $x\in P_{j}\cap P_{j_{0}}^{n\ast}$ furthermore implies that

$\frac{|P_{j_{0}}|}{|P_{j}|}\leq C_{1}^{n+1}~{},$

whence finally

$|Q_{i}|\leq|P_{j}|C_{1}^{2n+1}C_{2}^{n+1}~{}.$

Here it is important to observe that both constants $C_{1}$ and $C_{2}$ are
independent of $i\in I$. Hence this estimate shows one direction of the
desired equivalence, and the other one follows by symmetry.

###  2.2 Admissible groups and induced coverings

For a closed matrix group $H\leq\mathrm{GL}(\mathbb{R}^{d})$, which we also
call dilation group in the following, we define the group
$G:=\mathbb{R}^{d}\rtimes H$, generated by dilations with elements of $H$ and
arbitrary translations, with the group law $(x,h)\circ(y,g):=(x+hy,hg)$. We
denote integration with respect to a left Haar measure on $H$ with
$\mathrm{d}h$, the associated left Haar measure on $G$ is then given by
$d(x,h)=\mathopen{\lvert}\det h\mathclose{\rvert}^{-1}\mathrm{d}x\mathrm{d}h$.
The Lebesgue spaces on $G$ are always defined through integration with respect
to a Haar measure. The group $G$ acts on the space
$\mathrm{L}^{2}(\mathbb{R}^{d})$ through the quasi-regular representation
$\pi$ defined by $[\pi(x,h)f](y):=\mathopen{\lvert}\det
h\mathclose{\rvert}^{-1/2}f(h^{{-1}}(y-x))$ for
$f\in\mathrm{L}^{2}(\mathbb{R}^{d})$. The generalized continuous wavelet
transform (with respect to $\psi\in\mathrm{L^{2}}(\mathbb{R}^{d})$) of $f$ is
then given as the function
$W_{\psi}f:G\to\mathbb{C}:(x,h)\mapsto\braket{f,\pi(x,h)\psi}$. We will
sometimes use the notation $W_{\psi}^{H}f$, in places where an explicit
reference to the choice of dilation group $H$ is required. Important
properties of the map $W_{\psi}:f\mapsto W_{\psi}f$ depend on $H$ and the
chosen $\psi$. If the quasi-regular representation is square-integrable, which
means that there exists $\psi\neq 0$ with $W_{\psi}\psi\in\mathrm{L}^{2}(G)$,
and irreducible, then we call $H$ admissible. In this case the map
$W_{\psi}:\mathrm{L}^{2}(\mathbb{R}^{d})\to\mathrm{L}^{2}(G)$ is a multiple of
an isometry, which gives rise to the (weak-sense) inversion formula

$f=\frac{1}{C_{\psi}}\int_{G}W_{\psi}f(x,h)\pi(x,h)\psi\mathrm{d}(x,h)~{},$
(2.4)

i.e., each $f\in\mathrm{L}^{2}(\mathbb{R}^{d})$ is a continuous superposition
of the wavelet system. According to results in [20], [23], the admissibility
of $H$ can be characterized by the dual action defined by
$\mathbb{R}^{d}\times H\to\mathbb{R}^{d},(h,\xi)\mapsto h^{-T}\xi$. In fact,
$H$ is admissible iff the dual action has a single open orbit
$\mathcal{O}:=H^{{-T}}\xi_{0}\subset\mathbb{R}^{d}$ of full measure for some
$\xi_{0}\in\mathbb{R}^{d}$ and additionally the isotropy group
$H_{\xi_{0}}:=\Set{h:p_{\xi_{0}}(h)=\xi_{0}}\subset H$ is compact; see e.g.
[23].

Every admissible group gives rise to an associated a covering. This is done
using the _dual action_ by picking a _well-spread_ family in $H$, i.e. a
family of elements $(h_{i})_{i\in I}\subset H$ with the properties

  1. i)

there exists a relatively compact neighborhood $U\subset H$ of the identity
such that $\bigcup_{i\in I}h_{i}U=H$ – we say $(h_{i})_{i\in I}$ is $U$-dense
in this case – and

  2. ii)

there exists a neighborhood $V\subset H$ of the identity such that $h_{i}V\cap
h_{j}V=\emptyset$ for $i\neq j$ – we say $(h_{i})_{i\in I}$ is $V$-separated
in this case.

The dual covering induced by $H$ is then given by the family
$\mathcal{Q}=(Q_{i})_{i\in I}$, where $Q_{i}=p_{\xi_{0}}(h_{i}U)$ for some
$\xi_{0}$ with $H^{-T}\xi_{0}=\mathcal{O}$. It can be shown that well-spread
families always exist, and that the induced covering is indeed a tight
structured admissible covering in the sense of decomposition space theory, for
which $\mathrm{L}^{p}$-BAPUs exist, according to Theorems 4.4.6 and 4.4.13 of
[36]. Furthermore, there always exist induced coverings consisting of open and
connected sets, an additional feature which can facilitate the investigations
in some cases. For ease of reference, we state this as a lemma.

######  Lemma 2.9 ([30] Corollary 2.5.9)

Let $H$ denote an admissible dilation group, with open dual orbit
$\mathcal{O}$. Then there always exists an induced covering of $\mathcal{O}$
by $H$ that is a tight structured admissible covering consisting of (path-)
connected open sets.

We call any induced covering that is a structured admissible covering
consisting of open and connected sets an induced connected covering of
$\mathcal{O}$ by $H$. Furthermore, two different induced coverings of the same
group are always weakly equivalent, see [30] Corollary 2.6.5.).

In Subsection 4.2, the next result will be crucial.

######  Lemma 2.10 (cf. [27] Lemma 18)

Let $(h_{i})_{i\in I}$ be a well-spread family in $H$ and let
$K_{1},K_{2}\subset\mathcal{O}$ be compact sets. Given $h\in H$, let

$\displaystyle I_{h}(K_{1},K_{2}):=\Set{i\in I}{h^{{-T}}K_{1}\cap
h_{i}^{{-T}}K_{2}\neq\emptyset}.$

There are $C_{1}=C_{1}(K_{1},K_{2})>0$ and
$C_{2}=C_{2}(K_{1},K_{2},(h_{i})_{i\in I})>0$ with
$\mathopen{\lvert}I_{h}\mathclose{\rvert}\leq C_{2}$ for all $h\in H$ and with
$\mathopen{}\mathclose{\left\lVert h^{T}h_{i}^{{-T}}\right\rVert}\leq C_{1}$
for all $i\in I_{h}$.

###  2.3 Coorbit theory and its connection to decomposition spaces

Coorbit spaces are defined in terms of the decay behavior of the generalized
wavelet transform. To give a precise definition, we introduce weighted mixed
$\mathrm{L}^{p}$-spaces on $G$, denoted by $\mathrm{L}^{p,q}_{v}(G)$. By
definition, this space is the set of functions

$\displaystyle\left\\{f:G\to\mathbb{C}:\int_{H}\left(\int_{\mathbb{R}^{d}}\left|f(x,h)\right|^{p}v(x,h)^{p}\mathrm{d}x\right)^{q/p}\frac{\mathrm{d}h}{|\det(h)|}<\infty\right\\},$

with natural (quasi-)norm $\|\cdot\|_{\mathrm{L}^{p,q}_{v}}$. This definition
is valid for $0<p,q<\infty$, for $p=\infty$ or $q=\infty$ the essential
supremum has to be taken at the appropriate place instead. The function
$v:G\to\mathbb{R}^{>0}$ is a measurable weight function that fulfills the
condition $v(ghk)\leq v_{0}(g)v(h)v_{0}(k)$ for some that is
submultiplicative, i.e. fulfills $v_{0}(gk)\leq v_{0}(g)v_{0}(k)$. If this
last condition on $v$ is satisfied, we call $v$ left- and right moderate with
respect to $v_{0}$. Thus, the expression
$\|W_{\psi}f\|_{\mathrm{L}^{p,q}_{v}}$ can be read as a measure of wavelet
coefficient decay of $f$. We consider weights which only depend on $H$. The
coorbit space $\mathrm{Co}\left(\mathrm{L}^{p,q}_{v}(\mathbb{R}^{d}\rtimes
H)\right)$ is then defined as the space

$\displaystyle\left\\{f\in\mathcal{(H}^{1}_{w})^{\neg}:W_{\psi}f\in
W(\mathrm{L}^{p,q}_{v}(\mathbb{R}^{d}\rtimes H))\right\\}$ (2.5)

for a suitable wavelet $\psi$ fulfilling various technical conditions, and
some control weight $w$ associated to $v$. The space
$(\mathcal{H}^{1}_{w})^{\neg}$ denotes the space of antilinear functionals on
$\mathcal{H}^{1}_{w}:=\left\\{f\in\mathrm{L}^{2}(\mathbb{R}^{d}):W_{\psi}f\in\mathrm{L}^{1}_{w}(G)\right\\}$
and $W(Y)$ for a function space $Y$ on $G$ denotes the Wiener amalgam space
defined by $W_{Q}(Y):=\\{f\in\mathrm{L}^{\infty}_{\text{loc}}(G)|M_{Q}f\in
Y\\}$ with quasi-norm $\|f\|_{W_{Q}(Y)}:=\|M_{Q}f\|_{Y}$ for $f\in W_{Q}(Y)$,
where the maximal function $M_{Q}f$ for some suitable unit neighborhood
$Q\subset G$ is $M_{Q}f:G\to[0,\infty],\ x\mapsto\operatorname{ess\ sup}_{y\in
xQ}|f(y)|$.

The appearance of the Wiener amalgam space in (2.5) is necessary to guarantee
consistently defined quasi-Banach spaces in the case
$\\{p,q\\}\cap(0,1)\neq\emptyset$, see [34] and [36]. In the classical coorbit
theory for Banach spaces, which was developed in [14, 15], the Wiener amalgam
space is replaced by $\mathrm{L}^{p,q}_{v}(G)$ and this change leads to the
same space for $p,q\geq 1$, see [34].

Many useful properties of these spaces are known and hold in the quasi-Banach
space case as well as in the Banach space case. The most prominent examples of
coorbit spaces associated to generalized wavelet transforms are the
homogeneous Besov spaces and the modulation spaces. However, each shearlet
group, a class of groups we introduce in the next subsection, gives rise to
its own scale of coorbit spaces, as well; see [31], [9] and [24].

###### Remark 2.11

The use of the reservoir space $(\mathcal{H}^{1}_{w})^{\neg}$ is a major
obstacle for the comparison of coorbit spaces associated to different dilation
groups. From a purely set-theoretic point of view, different dilation groups
necessarily result in disjoint coorbit spaces. However, this distinction is
somewhat beside the point, and it can be remedied by the following
observations; see also [14, Corollary 4.4]. By the Fischer-Riesz theorem, the
inclusion $\mathcal{H}^{1}_{w}\subset L^{2}(\mathbb{R}^{d})$ provides a
canonical embedding
$L^{2}(\mathbb{R}^{d})\subset(\mathcal{H}^{1}_{w})^{\neg}$, which allows to
identify $L^{2}(\mathbb{R}^{d})$ with a subspace of
$(\mathcal{H}^{1}_{w})^{\neg}$. Now the fact that the wavelet transform is
$L^{2}$-isometric (and the associated inversion formula) implies that the
range of this identification map is precisely given by
$Co(L^{2}(\mathbb{R}^{d}\rtimes H))$.

We next observe that $Co(L^{p}(\mathbb{R}^{d}\rtimes H))\subset
Co(L^{2}(\mathbb{R}^{d}\rtimes H))$ for $1\leq p\leq 2$. This follows from the
fact that the discrete sequence space $Y_{d}$, associated to
$Y=L^{p}(\mathbb{R}^{d}\times H)$ according to Definition 3.4 of [14], turns
out to be $Y_{d}=\ell^{p}\subset\ell^{2}$ (see [14, Lemma 3.5(e)]).
Furthermore, $Co(Y)\subset Co(Z)$ holds iff $Y_{d}\subset Z_{d}$, by [15,
Theorem 8.4], hence $Co(L^{p})\subset Co(L^{2})$.

Combining this embedding with the identification of
$Co(L^{2}(\mathbb{R}^{d}\rtimes H))$ with $L^{2}(\mathbb{R}^{d})$, we find the
following canonical identification for $1\leq p\leq 2$, valid for a suitably
chosen analyzing vector $\psi$:

$Co(L^{p}(\mathbb{R}^{d}\rtimes H))=\\{f\in
L^{2}(\mathbb{R}^{d}):W_{\psi}^{H}f\in L^{p}(\mathbb{R}^{d}\rtimes H)\\}~{},$

with the obvious norm. But now the new reservoir space $L^{2}(\mathbb{R}^{d})$
is independent of $H$.

We note that this identification extends to all weighted
$L^{p,q}_{v}(\mathbb{R}^{d}\rtimes H)$ for which the associated sequence space
fulfills $\ell^{p,q}_{v}\subset\ell^{2}$, in particular for $1\leq p,q\leq 2$
and constant weight $v$.

An alternative approach to the just-mentioned embeddings of coorbit spaces,
pointed out to us by the referee, is obtained by observing the following well-
known inclusion relations between Wiener amalgam spaces, valid for $0<p<2$,

$W(L^{p})\subset W(L^{2})\subset L^{2}~{},$

which via equation (2.5) translates to the inclusions of the associated
coorbit spaces, and has the advantage of directly generalizing to $p<1$.

The next definition will be useful for the transfer of weights from the
coorbit to the decomposition space setting.

######  Definition 2.12 ([36] Definition 4.5.3.)

For $q\in(0,\infty]$ and a weight $m:H\to(0,\infty)$, we define the weight
$m^{(q)}:H\to(0,\infty),\
h\mapsto\mathopen{\lvert}\det(h)\mathclose{\rvert}^{\frac{1}{2}-\frac{1}{q}}m(h)$.
Here, we set $\frac{1}{\infty}:=0$.

The connection between coorbit spaces and decomposition spaces is given by the
next theorem. For the Banach space case, we also refer to [27]. A recent
extension beyond the irreducible setting can be found in [26].

######  Theorem 2.13 ([36] Theorem 4.6.3)

Let $\mathcal{Q}$ be a covering of the dual orbit $\mathcal{O}$ induced by
$H$, $0<p,q\leq\infty$ and $u=(u_{i})_{i\in I}$ a suitable weight, then the
Fourier transform
$\mathcal{F}:\mathrm{Co}\left(\mathrm{L}^{p,q}_{v}(\mathbb{R}^{d}\rtimes
H)\right)\to\mathcal{D}(\mathcal{Q},\mathrm{L}^{p},\ell^{q}_{u})$ is an
isomorphism of (quasi-) Banach spaces. The weight $(u_{i})_{i\in I}$ can be
chosen as $u_{i}:=v^{(q)}(h_{i})$, where $(h_{i})_{i\in I}$ is the well-spread
family used in the construction of $\mathcal{Q}$. We call such a weight a
$\mathcal{Q}$-discretization of $v$.

###### Remark 2.14

In the following, we will mostly concentrate on constant weights, i.e. on the
study of coorbit spaces of the type $Co(L^{p,q}(G))$ corresponding to $v\equiv
1$. This has the important consequence that the $\mathcal{Q}$-discretization
$(u_{i})_{i\in I}$ obtained from a dual covering $\mathcal{Q}=(Q_{i})_{i\in
I}=(h_{i}^{-T}Q)_{i\in I}$ fulfills

$u_{i}=|{\mathrm{det}}(h_{i})|^{\frac{1}{2}-\frac{1}{q}}=|Q|^{\frac{1}{2}-\frac{1}{q}}|Q_{i}|^{\frac{1}{q}-\frac{1}{2}}\asymp|Q_{i}|^{\frac{1}{q}-\frac{1}{2}}~{}.$

In other words, the induced weight $(u_{i})_{i\in I}$ is intrinsic with
exponent $\alpha=\frac{1}{q}-\frac{1}{2}$.

###### Remark 2.15

Since the domain of the Fourier transform in the previous theorem consists of
spaces that depend on the choice of dilation groups, some clarification is in
order, see [27, Corollary 10] for more details: Given
$f\in(\mathcal{H}^{1}_{w})^{\neg}$, we let
$\mathcal{F}(f)\in\mathcal{D}^{\prime}(\mathcal{O})$ be defined as the map
$g\mapsto f(\mathcal{F}^{-1}\overline{g})$. Here $\mathcal{F}$ denotes the
Fourier transform defined on the Schwartz functions, and the definition makes
use of $\mathcal{D}(\mathcal{O})\subset\mathcal{S}(\mathcal{O})$, as well as
$\mathcal{F}^{-1}(\mathcal{D}(\mathcal{O}))\subset\mathcal{H}^{1}_{w}$ [24,
Theorem 2.1].

By the Fischer-Riesz theorem, the inclusion $\mathcal{H}^{1}_{w}\subset
L^{2}(\mathbb{R}^{d})$ provides a canonical embedding
$L^{2}(\mathbb{R}^{d})\subset(\mathcal{H}^{1}_{w})^{\neg}$. Hence for elements
of $L^{2}(\mathbb{R}^{d})$, one can rewrite the formula for $\mathcal{F}(f)$
as

$\mathcal{F}(f)(g)=\langle\widehat{f},\overline{g}\rangle_{L^{2}(\mathbb{R}^{d})}~{},$

where $\widehat{f}$ denotes the Plancherel transform of $f$. In particular,
for all coorbit spaces $Co(L^{p}(\mathbb{R}^{d}\rtimes H))$ having a canonical
embedding into $L^{2}(\mathbb{R}^{d})$ according to Remark 2.11, the Fourier
transform in the theorem coincides with the Plancherel transform, combined
with the obvious identification of the $L^{2}$-function $\widehat{f}$ with the
distribution on $\mathcal{D}(\mathcal{O})$ obtained by integration against
$\widehat{f}$. In this way, the Fourier transform $\mathcal{F}$ on
$Co(L^{p}(\mathbb{R}^{d}\rtimes H))$, as defined in Theorem 2.13, also becomes
independent of the choice of dilation group, whenever we canonically identify
the coorbit space with a subspace of $L^{2}(\mathbb{R}^{d})$.

We next formalize the property that two admissible dilation groups have the
same coorbit spaces. We already pointed out that a literal interpretation of
this property is not generally available, at least not for all possible
choices of coorbit space norms.

###### Definition 2.16

Let $H_{1},H_{2}\leq GL(\mathbb{R}^{d})$ denote admissible matrix groups. We
call $H_{1},H_{2}$ coorbit equivalent if for all $0<p,q\leq\infty$ and for all
$f\in L^{2}(\mathbb{R}^{d})$ we have

$\|f\|_{Co(L^{p,q}(\mathbb{R}^{d}\rtimes
H_{1}))}\asymp\|f\|_{Co(L^{p,q}(\mathbb{R}^{d}\rtimes H_{2}))}~{}.$

Here the norm equivalence is understood in the generalized sense that one side
is infinite iff the other side is.

###### Remark 2.17

An example of distinct groups that are coorbit equivalent can be found in
[27], Section 9: If $H=\mathbb{R}^{+}\times SO(d)$, for $d>1$, and $C\in
GL(\mathbb{R}^{d})$ is arbitrary, then $H$ and $C^{-1}HC$ are coorbit
equivalent, but typically distinct.

Now the decomposition space characterization from Theorem 2.13, together with
the results from the previous subsection characterizing when admissible
coverings lead to identical scales of decomposition spaces, provides a rather
stringent characterization of coorbit equivalence.

###### Theorem 2.18

Let $H_{1},H_{2}\leq GL(\mathbb{R}^{d})$ denote admissible matrix groups, and
let $\mathcal{O}_{1},\mathcal{O}_{2}$ denote the associated open dual orbits.
Then the following are equivalent:

  1. (a)

$H_{1}$ and $H_{2}$ are coorbit equivalent.

  2. (b)

For all $1\leq p,q\leq 2$: $Co(L^{p,q}(\mathbb{R}^{d}\rtimes
H_{1}))=Co(L^{p,q}(\mathbb{R}^{d}\rtimes H_{2}))$, as subspaces of
$L^{2}(\mathbb{R}^{d})$.

  3. (c)

There exists $1\leq p,q\leq 2$ with $(p,q)\neq(2,2)$, such that
$Co(L^{p,q}(\mathbb{R}^{d}\rtimes H_{1}))=Co(L^{p,q}(\mathbb{R}^{d}\rtimes
H_{2}))$, as subspaces of $L^{2}(\mathbb{R}^{d})$.

  4. (d)

$\mathcal{O}_{1}=\mathcal{O}_{2}$, and the coverings induced by $H_{1}$ and
$H_{2}$ on the common open orbit are weakly equivalent.

###### Proof 2

The implication $(a)\Rightarrow(b)$ is clear by the nature of the canonical
embedding of the coorbit spaces into $L^{2}(\mathbb{R}^{d})$ detailed in
Remark 2.11. $(b)\Rightarrow(c)$ is obvious.

For the direction $(c)\Rightarrow(d)$, recall that
$\mathcal{O}_{1},\mathcal{O}_{2}$ are both open and dense. Hence, following
the observations after Theorem 2.5, if these sets are distinct, then either
$\mathcal{O}_{1}\cap\partial\mathcal{O}_{2}\neq\emptyset$ or
$\mathcal{O}_{2}\cap\partial\mathcal{O}_{1}\neq\emptyset$ follows, and in
addition $\mathcal{O}_{1}\cap\mathcal{O}_{2}$ is unbounded. Moreover, the
space $\mathcal{F}^{-1}(C_{c}^{\infty}(\mathcal{O}_{1}\cap\mathcal{O}_{2}))$
is a subspace of $L^{2}(\mathbb{R}^{d})$, and the assumption on the coorbit
spaces combined with Theorem 2.13, yields that the decomposition space norms
$\|\cdot\|_{\mathcal{D}(\mathcal{Q},\mathrm{L}^{p},\ell^{q}_{v})}$ and
$\|\cdot\|_{\mathcal{D}(\mathcal{P},\mathrm{L}^{p},\ell^{q}_{u})}$ are
equivalent on $C_{c}^{\infty}(\mathcal{O}_{1}\cap\mathcal{O}_{2})$, whenever
$\mathcal{Q}$ and $\mathcal{P}$ are coverings induced by $H_{1},H_{2}$,
respectively. Here $u,v$ are the intrinsic weights of exponent
$\alpha=\frac{1}{q}-\frac{1}{2}$ associated to the coverings, see Remark 2.14.
Now the fact that $(p,q)\neq(2,2)$ implies via Theorem 2.5 that
$\mathcal{O}_{1}=\mathcal{O}_{2}$.

We observed in Remark 2.15 that the $L^{2}$-Fourier transform $\mathcal{F}$
induces isomorphisms

$Co(L^{p,q}(\mathbb{R}^{d}\rtimes
H_{1}))\to\mathcal{D}(\mathcal{P},L^{p},\ell^{q}_{u})~{},~{}Co(L^{p,q}(\mathbb{R}^{d}\rtimes
H_{2}))\to\mathcal{D}(\mathcal{P},L^{p},\ell^{q}_{v})$

and by assumption, the coorbit spaces coincide. But then

$\mathcal{D}(\mathcal{Q},L^{p},\ell^{q}_{u})=\mathcal{D}(\mathcal{P},L^{p},\ell^{q}_{v})~{}.$

Now Lemma 2.3 implies weak equivalence of the coverings.

Finally, $(d)\Rightarrow(a)$ follows combining Theorem 2.13 and Lemma 2.4.
Observe also that the induced weights are equivalent by Remark 2.14 and Lemma
2.8.

###### Remark 2.19

The unweighted case, which is in the focus of the theorem, covers all spaces
of the type $L^{p}(\mathbb{R}^{d}\rtimes H)$, for $1\leq p<2$, which are of
special importance for nonlinear approximation. Considering general, non-
constant weights on the dilation groups requires solving a technical issue,
that is currently not well-understood. In this general setting, the question
becomes whether any choice of weight $v_{1}$ given on one group $H_{1}$ can be
matched by an appropriate weight $v_{2}$ on $H_{2}$, such that the resulting
weights on the induced coverings are equivalent. In principle it seems
possible, using cross-sections, to construct $v_{2}$ from $v_{1}$. However, it
is currently unclear whether, starting from a weight $v_{1}$ that is moderate
with respect to a suitable sub-multiplicative function on $H_{1}$, the same
will automatically hold true for the newly constructed weight $v_{2}$ on
$H_{2}$.

##  3 A metric characterization of weak equivalence

This section contains the central observation of this paper, namely that the
question of weak equivalence of coverings, and thus of equality of the scales
of decomposition spaces associated to the coverings, can be reformulated using
the language of coarse geometry.

###  3.1 Elementary notions and results from coarse geometry

In this section we collect the basic results from coarse geometry that are
needed for the following. Since the terminology employed in coarse geometry is
not entirely unified, and for reasons of self-containedness, we include proofs
of some basic results for which we could not find a handy reference.

###### Remark 3.1

In the following, we define different metrics on subsets of $\mathbb{R}^{d}$
and on matrix groups, but we will only be concerned with the coarse properties
of these metrics. In particular, the terms open, closed, connected component
and the continuity of maps between these spaces have to be understood with
respect to the standard topology on the respective sets and not with respect
to the topology that may be induced by these different metrics.

The pivotal role continuous maps play in the study of topological spaces is
taken over by coarse maps in the study of large scale properties of metric
spaces.

######  Definition 3.2 (cf. [35] Definition 1.8)

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces and let $f:X\to Y$ be a map.

  * 1.

We call $f$ metrically proper if the inverse image under $f$ of every bounded
subset in $Y$ is bounded in $X$, that is

$\displaystyle\sup\left\\{d_{Y}(y,y^{\prime}):\ y,y^{\prime}\in
A\right\\}<\infty$
$\displaystyle\quad\Rightarrow\sup\left\\{d_{X}(x,x^{\prime}):\
x,x^{\prime}\in f^{{-1}}(A)\right\\}<\infty$

for all sets $A\subset Y$.

  * 2.

We call $f$ uniformly bornologous if for every $R>0$ there exists $S>0$ such
that

$\displaystyle d_{X}(x,x^{\prime})<R\quad\Longrightarrow\quad
d_{Y}\left(f(x),f(x^{\prime})\right)<S$

for all $x,x^{\prime}$ in $X$.

  * 3.

A coarse map from $(X,d_{X})$ to $(Y,d_{Y})$ is a metrically proper and
uniformly bornologous map.

A simple argument shows that the composition of coarse maps is coarse as well.
In order to define when we want to consider coarse structures on different
spaces as essentially the same, we introduce the notion of closeness of maps.

######  Definition 3.3 (cf. [33] Definition 1.3.2)

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces and $f:X\to Y$ and $g:X\to Y$
be maps. We say $f$ and $g$ are _close_ if

$\displaystyle\sup\left\\{d_{Y}\left(f(x),g(x)\right):\ x\in
X\right\\}<\infty.$

######  Definition 3.4 (cf. [35] Page 6)

Two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$ are _coarsely equivalent_ if
there exist two coarse maps

$f:X\to Y\quad\text{and}\quad g:Y\to X$

such that $f\circ g$ is close to $\mathrm{id}_{Y}$ and $g\circ f$ is close to
$\mathrm{id}_{X}$. $f$ is then called a _metric coarse equivalence_.

As the name suggests, this relation is in fact an equivalence relation. An
important class of coarse maps are quasi-isometries.

######  Definition 3.5 (cf. [35] Remark 1.9)

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces and $f:X\to Y$ be a map. We
call $f$ large-scale Lipschitz if there are $L,C>0$ such that
$d_{Y}\left(f(x),f(x^{\prime})\right)\leq Ld_{X}\left(x,x^{\prime}\right)+C$
for all $x,x^{\prime}\in X$.

######  Definition 3.6 (cf. [33] Definition 1.3.4)

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces. A map $f:X\to Y$ is a quasi-
isometry if the following conditions are satisfied:

  1. i)

The map $f$ is a quasi-isometric embedding. This means there exist constants
$L,C>0$ such that

$\displaystyle L^{-1}d_{X}(x,x^{\prime})-C\leq
d_{Y}\left(f(x),f(x^{\prime})\right)\leq Ld_{X}(x,x^{\prime})+C$

for all $x,x^{\prime}\in X$.

  2. ii)

The map $f$ is coarsely surjective. This means that there exists $K>0$ such
that for every $y\in Y$ exists an $x\in X$ with $d_{Y}\left(f(x),y\right)\leq
K$.

###### Remark 3.7

It is easy to see that a surjective map is coarsely surjective. In the
concrete cases we consider, this will be the standard justification for this
property.

As for coarse maps, it is easy to see that the composition of quasi-isometric
maps is again quasi-isometric. The following lemma has a straightforward
proof.

###### Lemma 3.8

Every quasi-isometric embedding is coarse.

###### Definition 3.9

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces and $f:X\to Y$. We call a map
$h:Y\to X$ such that $f\circ h$ is close to $\mathrm{id}_{Y}$ a coarse right
inverse of $f$.

###### Example 3.10

The map $f:(\mathbb{R},|\cdot|)\to(\mathbb{R}^{2},|\cdot|_{1}),\
x\mapsto(x,0)^{T}$ illustrates that not every coarse map has a coarse right
inverse.

Quasi-isometries always have coarse right inverses. We include a proof, for
the reasons mentioned above.

###### Lemma 3.11

Every quasi-isometry has a coarse right inverse. Furthermore, any coarse right
inverse of a quasi-isometry is a quasi-isometry.

###### Proof 3

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces and $f:X\to Y$ be a quasi-
isometry. This entails the existence of some $K>0$ with the property that for
every $y\in Y$ exists an $x\in X$ with $d_{Y}\left(f(x),y\right)\leq K$.
Consequently, for every $y\in Y$, the set

$A_{y}:=\left\\{x\in X:\ d_{Y}\left(f(x),y\right)\leq K\right\\}$

is nonempty. The axiom of choice implies the existence of at least one coarse
right inverse of $f$ because $\prod_{y\in Y}A_{y}$ is nonempty.

Since $f$ is a quasi-isometry, there exist $L,C>0$ with

$\displaystyle L^{-1}d_{X}(x,x^{\prime})-C\leq
d_{Y}\left(f(x),f(x^{\prime})\right)\leq Ld_{X}(x,x^{\prime})+C$

for all $x,x^{\prime}\in X$. Moreover, let $h:Y\to X$ be any coarse right
inverse of $f$ such that $\sup_{y\in Y}d_{Y}\left((f\circ h)(y),y)\right)\leq
M$ for some $M>0$. By application of the triangle inequality twice, it follows
that

$\displaystyle L^{{-1}}d_{X}\left(h(y),h(y^{\prime})\right)-C$
$\displaystyle\leq d_{Y}\left((f\circ h)(y),(f\circ h)(y^{\prime})\right)$
$\displaystyle\leq d_{Y}\left((f\circ
h)(y),y\right)+d_{Y}\left(y^{\prime},y\right)$
$\displaystyle\qquad+d_{Y}\left(y^{\prime},(f\circ h)(y^{\prime})\right)$
$\displaystyle\leq 2M+d_{Y}\left(y^{\prime},y\right)$

for all $y,y^{\prime}\in Y$. This inequality implies

$\displaystyle d_{X}\left(h(y),h(y^{\prime})\right)\leq
Ld_{Y}\left(y^{\prime},y\right)+2LM+CL.$

We continue by using the reverse triangle inequality twice to get

$\displaystyle Ld_{X}\left(h(y),h(y^{\prime})\right)+C$ $\displaystyle\geq
d_{Y}\left((f\circ h)(y),(f\circ h)(y^{\prime})\right)$ $\displaystyle\geq
d_{Y}\left(y,(f\circ h)(y^{\prime})\right)-d_{Y}\left((f\circ h)(y),y\right)$
$\displaystyle\geq d_{Y}\left(y,y^{\prime}\right)-d_{Y}\left((f\circ
h)(y^{\prime}),y^{\prime}\right)$ $\displaystyle\qquad-d_{Y}\left((f\circ
h)(y),y\right)$ $\displaystyle\geq d_{Y}\left(y,y^{\prime}\right)-2M.$

This inequality implies

$\displaystyle d_{X}\left(h(y),h(y^{\prime})\right)\geq
L^{{-1}}d_{Y}\left(y^{\prime},y\right)-2L^{{-1}}M-CL^{{-1}}$

for all $y,y^{\prime}\in Y$. It follows that $h$ is a quasi-isometric
embedding.

In order to show that $h$ is a quasi-isometry, let $x\in X$ be arbitrary and
choose $y:=f(x)$. We conclude

$\displaystyle d_{X}\left(h(f(x)),x\right)$ $\displaystyle\leq
Ld_{Y}\left(f\circ(h\circ f)(x),f(x)\right)+CL$
$\displaystyle=Ld_{Y}\left((f\circ h)(f(x)),f(x)\right)+CL$
$\displaystyle=Ld_{Y}\left((f\circ h)(y),y\right)+CL$ $\displaystyle\leq
LM+CL$

and the inequality

$\displaystyle d_{X}\left(h(f(x)),x\right)\leq LM+CL$ (3.1)

shows that $h$ is coarsely surjective. In summary, this shows that $h$ is a
quasi-isometry.

###### Remark 3.12

The inequality (3.1) also shows that if $f:X\to Y$ is a quasi-isometry and
$h:Y\to X$ is any coarse right inverse of $f$, then $f$ is a coarse right
inverse of $h$. If we define a coarse left inverse analogously, then this
shows that every coarse right inverse of a quasi-isometry is also a coarse
left inverse of this quasi-isometry.

###### Theorem 3.13

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces. If there exists a quasi-
isometry $f:X\to Y$, then $X$ and $Y$ are coarsely equivalent.

###### Proof 4

According to Lemma 3.11, $f$ has a coarse right inverse $h$, which is also a
quasi-isometry. By Lemma 3.8, $f$ and $h$ are coarse maps. Since $h$ is a
coarse right inverse of $f$, the map $f\circ h$ is close to $\mathrm{id}_{Y}$.
From Remark 3.12 follows that $h\circ f$ is close to $\mathrm{id}_{X}$. This
shows that $f$ and $h$ induce a coarse equivalence of $X$ and $Y$.

The following definition allows to identify a general setting in which the
terms “coarse equivalence” and “quasi-isometry” are synonymous. See [7,
Definition 3.B.1].

###### Definition 3.14

Let $(X,d)$ denote a metric space. For $c>0$, $X$ is called $c$-large-scale
geodesic if there exist constants $a>0,b\geq 0$ such that for every pair of
points $x,x^{\prime}$ there exists a finite sequence
$x_{0}=x,x_{1},\ldots,x_{n}=x^{\prime}$ satisfying $d(x_{i-1},x_{i})\leq c$
and $n\leq ad(x,x^{\prime})+b$. $(X,d)$ is called large-scale geodesic if it
is $c$-large-scale geodesic for some $c>0$.

The following result is [7, Proposition 3.B.9].

###### Proposition 3.15

Let $(X,d)$ and $(Y,d^{\prime})$ denote large-scale geodesic spaces, and
$f:X\to Y$. Then $f$ is a coarse equivalence if and only if it is a quasi-
isometry.

###  3.2 Metrics induced by coverings

We start with a preparation for the definition of the metric associated to a
covering.

###### Definition 3.16

Let $\mathcal{O}\subset\mathbb{R}^{d}$ be open and let
$\mathcal{Q}=(Q_{i})_{i\in I}$ be a covering of $\mathcal{O}$. For
$x,y\in\mathcal{O}$, we say $x$ and $y$ are connected by a $\mathcal{Q}$-chain
(of length m) if there exist $Q_{1},\ldots,Q_{m}\in\mathcal{Q}$ such that
$x\in Q_{1}$, $y\in Q_{m}$ and $Q_{k}\cap Q_{k+1}\neq\emptyset$ for all
$k\in\\{1,\ldots,m-1\\}$.

We next define the metric, which is called $\mathcal{Q}$-chain distance in
[14, Definition 3.4].

###### Definition 3.17

Let $\mathcal{O}\subset\mathbb{R}^{d}$ be open and let
$\mathcal{Q}=(Q_{i})_{i\in I}$ be a covering of $\mathcal{O}$. Define the map
$d_{\mathcal{Q}}:\mathcal{O}\times\mathcal{O}\to\mathbb{N}_{0}\cup\\{\infty\\}$
by

$\displaystyle
d_{\mathcal{Q}}(x,y)=\begin{cases}\inf\Set{m\in\mathbb{N}}{\begin{array}[]{l}x,y\text{
are connected by a}\\\ \mathcal{Q}\text{-chain of length }m\end{array}},&x\neq
y\\\ 0,&x=y,\end{cases}$

where we set $\inf\emptyset=\infty$.

The above map in fact defines a metric (we allow the value $\infty$ for a
metric) on $\mathcal{O}$, without any further restrictions on the covering.

###### Remark 3.18

Let $\mathcal{O}\subset\mathbb{R}^{d}$ be open and let
$\mathcal{Q}=(Q_{i})_{i\in I}$ be a covering of $\mathcal{O}$. One could also
be tempted to define a function by considering the neighbors of a set (recall
Definition 2.1)

$\displaystyle\widetilde{d}:$
$\displaystyle\mathcal{O}\times\mathcal{O}\to\mathbb{N}_{0}\cup\\{\infty\\},$
$\displaystyle(x,y)\mapsto\begin{cases}\inf\Set{n\in\mathbb{N}}{\exists
Q\in\mathcal{Q}\text{ with }x,y\in Q^{n\ast}},&x\neq y\\\ 0&x=y.\end{cases}$

In general, this map does not satisfy the triangle inequality and is therefore
not a metric on $\mathcal{O}$. It suffices to consider a uniform covering of
$\mathcal{O}=\mathbb{R}$ by intervals of length $2$ as depicted in the diagram
below. Here, the intervals are represented by rectangles. For the points
$x,y,z\in\mathbb{R}$, we have $\widetilde{d}(x,y)=1$, $\widetilde{d}(y,z)=1$
but $\widetilde{d}(x,z)=3$.

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$x$$y$$z$

###  3.3 Characterizing weak equivalence

We will only consider coverings consisting of open connected sets. Here the
property $d(x,y)=\infty$ can be related to the partition of $\mathcal{O}$ into
connected components.

###### Lemma 3.19

Let $\mathcal{O}\subset\mathbb{R}^{d}$ be open and let
$\mathcal{Q}=(Q_{i})_{i\in I}$ be a covering of $\mathcal{O}$ comprised of
open connected subsets of $\mathcal{O}$. Then $d_{\mathcal{Q}}(x,y)<\infty$ if
and only if $x,y$ lie in the same connected component of $\mathcal{O}$.

###### Proof 5

We consider the equivalence relation $x\sim y:\Leftrightarrow d(x,y)<\infty$.
Fix $x$, and let $y\in\mathcal{O}$ with $x\sim y$. By definition there exists
a $\mathcal{Q}$-chain connecting $x$ and $y$. Since the union of two connected
sets with nontrivial intersection is connected again, we may thus conclude
that $x$ and $y$ are contained in a common connected subset of $\mathcal{O}$.
Hence the equivalence class of $x$ modulo $\sim$ is contained in the connected
component of $x$.

To see the converse direction, observe that for each $x\in\mathcal{O}$, there
exists $i\in I$ with $x\in Q_{i}$, and thus $d(x,y)\leq 1<\infty$ for all
$y\in Q_{i}$, which is an open set. This shows that all $\sim$-equivalence
classes are open, and consequently closed, since they partition $\mathcal{O}$.
Hence the connected component of $x$ is contained in the $\sim$-equivalence
class.

###### Remark 3.20

As we remarked above, we allow our metrics $d_{\mathcal{Q}}$ to assume the
value $\infty$, which constitutes a mild abuse of notation. For the purpose of
the subsequent arguments, this abuse can be justified as follows: We will only
be interested in the comparison of different coverings
$\mathcal{Q},\mathcal{P}$ of the same set $\mathcal{O}$, which amounts to
studying the coarse properties of the identity map
${\mathrm{id}}_{\mathcal{O}}:(\mathcal{O},d_{\mathcal{P}})\to(\mathcal{O},d_{\mathcal{Q}})$.
Since every connected component of $\mathcal{O}$ is mapped onto itself, one
has $d_{\mathcal{P}}(x,y)<\infty$ iff $d_{\mathcal{Q}}(x,y)<\infty$.
Furthermore, we will often restrict attention to the case that $\mathcal{O}$
has finitely many connected components.

Under these restrictions, one can extend the various properties of maps
between metric spaces (understood in the strict sense) to the setting of
generalized metrics (allowed to take the value $\infty$), by requiring that
for each connected component $\mathcal{O}_{0}\subset\mathcal{O}$, the
restriction
$id_{\mathcal{O}_{0}}:(\mathcal{O}_{0},d_{\mathcal{P}})\to(\mathcal{O}_{0},d_{\mathcal{Q}})$
has the required properties. Then all results from the previous section become
available for the slightly more general setting. Note in particular that
${\mathrm{id}}_{\mathcal{O}}:(\mathcal{O},d_{\mathcal{P}})\to(\mathcal{O},d_{\mathcal{Q}})$
is a quasi-isometry if and only if all restrictions to connected components
are, provided the number of connected components is finite. Here we simply
take minima resp. maxima of the finitely many constants involved in the
formulation of quasi-isometry statements for each component, to obtain
globally valid constants.

###### Remark 3.21

We note that each covering-induced metric $d_{\mathcal{P}}$ is $c$-large scale
geodesic, for $c=1$.

The property of weak equivalence of two admissible coverings has a
characterization in terms of the coarse structure of their associated metrics.
A result similar to $i)\Rightarrow ii)$ in the following Theorem can be found
in [12, Proposition 3.8 C)].

###### Theorem 3.22

Let $\mathcal{O}\subset\mathbb{R}^{d}$ be open and let
$\mathcal{Q}=(Q_{i})_{i\in I}$ and $\mathcal{P}=(P_{j})_{j\in J}$ be
admissible coverings of $\mathcal{O}$ comprised of open connected subsets of
$\mathcal{O}$. Then the following statements are equivalent:

  1. i)

The coverings $\mathcal{Q}$ and $\mathcal{P}$ are weakly equivalent.

  2. ii)

The map
$\mathrm{id}:(\mathcal{O},d_{\mathcal{Q}})\to(\mathcal{O},d_{\mathcal{P}}),\
x\mapsto x$ is a quasi-isometry.

If $\mathcal{O}$ has only finitely many connected components, $(i)$ and $(ii)$
are equivalent to

  1. (iii)

The map
$\mathrm{id}:(\mathcal{O},d_{\mathcal{Q}})\to(\mathcal{O},d_{\mathcal{P}}),\
x\mapsto x$ is a coarse equivalence.

###### Proof 6

$i)\Rightarrow ii)$: Assume that the coverings are weakly equivalent. Our goal
is to show the existence of $L>0$ such that

$\displaystyle L^{{-1}}d_{\mathcal{Q}}(x,y)\leq d_{\mathcal{P}}(x,y)\leq
Ld_{\mathcal{Q}}(x,y)$

for all $x,y\in\mathcal{O}$. It suffices to show the right inequality, the
left inequality follows by interchanging the roles of $\mathcal{P}$ and
$\mathcal{Q}$. Let $x,y\in\mathcal{O}$. We first consider the case that
$d_{\mathcal{Q}}(x,y)=1$, this implies the existence of a set
$Q\in\mathcal{Q}$ with $x,y\in Q$. The weak subordinateness of $\mathcal{Q}$
to $\mathcal{P}$ implies that there are $P_{1},\ldots,P_{K}\in\mathcal{P}$ for
some $K\leq N(\mathcal{Q},\mathcal{P})$ (see Definition 2.2) with the property
$Q\cap P_{j}\neq\emptyset$ for $j\in\\{1,\ldots,K\\}$,
$Q\subset\bigcup_{j=1}^{K}P_{j}$ and $Q\cap P=\emptyset$ for all
$P\in\mathcal{P}\setminus\\{P_{1},\ldots,P_{K}\\}$.

Define the set

$\displaystyle C_{x}:=\Set{z\in Q}{\begin{array}[]{l}x,z\text{ are connected
by a }\mathcal{P}\text{-chain }\\\ (P_{\ell}^{*})_{\ell=1}^{m}\text{ of length
}m\leq K\mbox{ with }P_{\ell}^{*}\cap Q\neq\emptyset\\\ \text{and
}P_{\ell}^{*}\neq P_{\ell^{\prime}}^{*}\text{ for
}\ell\neq\ell^{\prime}\end{array}}.$

As in the proof of the previous lemma, the set $C_{x}$ is relatively open in
$Q$. Next, we show that it is also relatively closed in $Q$. Let $z_{1}$ be an
element of $\overline{C_{x}}^{Q}$. Because $\mathcal{P}$ is a covering of
$\mathcal{O}$, there is $P\in\mathcal{P}$ with $z_{1}\in P$. The set $Q\cap P$
is a relatively open neighborhood of $z_{1}$ in $Q$, this implies the
existence of some $z\in(Q\cap P)\cap C_{x}$. Hence, there is a
$\mathcal{P}$-chain $P^{\prime}_{1},\ldots,P^{\prime}_{m}$ of length $m\leq K$
that connects $x$ and $z$.

If $P=P^{\prime}_{j}$ for some $j\in\\{1,\ldots,m\\}$, then
$P^{\prime}_{1},\ldots,P^{\prime}_{j}$ is a $\mathcal{P}$-chain of length
$j\leq K$, with $P^{\prime}_{j}\cap Q\neq\emptyset$, that connects $x$, and
$z_{1}\in C_{x}$.

If $P\neq P^{\prime}_{j}$ for all $j\in\\{1,\ldots,m\\}$, the inclusion
$\\{P_{1}^{\prime},\ldots,P_{m}^{\prime}\\}\subset\\{P_{1},\ldots,P_{K}\\}$
together with $P_{\ell}^{\prime}\neq P_{\ell^{\prime}}^{\prime}$ for
$\ell\neq\ell^{\prime}$ implies via the pigeonhole principle that $m<K$. But
then $P^{\prime}_{1},\ldots,P^{\prime}_{m},P$ is a $\mathcal{P}$-chain of
length $m+1\leq K$ that connects $x$ and $z_{1}$. Here, $z\in
P_{m}^{\prime}\cap P\neq\emptyset$. In every case, $z_{1}\in C_{x}$ and
$C_{x}$ is relatively closed in $Q$. Since $Q$ is connected, we conclude
$C_{x}=Q$ and $d_{\mathcal{P}}(x,y)\leq K\leq N(\mathcal{Q},\mathcal{P})$.

Next, we consider the general case $d_{\mathcal{Q}}(x,y)=m\in\mathbb{N}$. In
this case, there exists a $\mathcal{Q}$-chain $Q_{1},\ldots,Q_{m}$ of length
$m$ that connects $x$ and $y$. The set $Q:=\bigcup_{i=1}^{m}Q_{i}$ is
connected, open and has nonempty intersection with at most
$mN(\mathcal{Q},\mathcal{P})$ elements of the covering $\mathcal{P}$ (every
$Q_{i}$ has nonempty intersection with at most $N(\mathcal{Q},\mathcal{P})$
elements of the covering $\mathcal{P}$ for $i\in\\{1,\ldots,m\\}$). Hence, an
adaptation of the argument for the case $m=1$ leads to the inequality

$\displaystyle d_{\mathcal{P}}(x,y)\leq mN(\mathcal{Q},\mathcal{P})\leq
N(\mathcal{Q},\mathcal{P})d_{\mathcal{Q}}(x,y).$

Interchanging of the roles of the coverings leads to

$\displaystyle N(\mathcal{P},\mathcal{Q})^{{-1}}d_{\mathcal{Q}}(x,y)\leq
d_{\mathcal{P}}(x,y)\leq N(\mathcal{Q},\mathcal{P})d_{\mathcal{Q}}(x,y).$

This shows that ${\mathrm{id}}$ is a quasi-isometric embedding. Since it is
also bijective, it is therefore a quasi-isometry.

$ii)\Rightarrow i)$: Assume that $\mathcal{Q}$ and $\mathcal{P}$ are not
weakly equivalent. Without loss of generality, assume that $\mathcal{Q}$ is
not weakly subordinate to $\mathcal{P}$. This implies the existence of a
sequence $(Q_{i_{n}})_{n\in\mathbb{N}}$ in $\mathcal{Q}$ such that

$\displaystyle\mathopen{}\mathclose{\left\lvert
A_{n}\right\rvert}>N_{\mathcal{P}}^{n+1}$

and
$\mathopen{\lvert}A_{n}\mathclose{\rvert}<\mathopen{\lvert}A_{n+1}\mathclose{\rvert}$
for all $n\in\mathbb{N}$, where

$A_{n}:=\Set{j\in J}{Q_{i_{n}}\cap P_{j}\neq\emptyset}$

and $N_{\mathcal{P}}$ is the admissibility constant for the covering
$\mathcal{P}$.

The next step is to show that for every $x\in Q_{i_{n}}$ and all $j_{x}\in J$
with $x\in P_{j_{x}}$, the set $Q_{i_{n}}\setminus P_{j_{x}}^{n*}$ is nonempty
and that $d_{\mathcal{P}}(x,y)\geq n$ for $x\in Q_{i_{n}}$ and $y\in
Q_{i_{n}}\setminus P_{j_{x}}^{n*}$ (cf. Definition 2.1).

Let $x\in Q_{i_{n}}$ be arbitrary for some $n\in\mathbb{N}$. Since
$\mathcal{P}$ is a covering of $\mathcal{O}$, there is at least one $j_{x}\in
J$ such that $x\in P_{j_{x}}$. If $Q_{i_{n}}\setminus
P_{j_{x}}^{n*}=\emptyset$, then $Q_{i_{n}}\subset P_{j_{x}}^{n*}$ and for all
$j\in J$ with $P_{j}\cap Q_{i_{n}}\neq\emptyset$, we have $P_{j}\cap
P_{j_{x}}^{n*}\neq\emptyset$, which implies $j\in j_{x}^{(n+1)*}$. Hence,

$\displaystyle\mathopen{\lvert}A_{n}\mathclose{\rvert}\leq\mathopen{\lvert}j_{x}^{(n+1)*}\mathclose{\rvert}\leq
N_{\mathcal{P}}^{n+1},$

which is a contradiction to the properties of the set $A_{n}$. It follows
$Q_{i_{n}}\setminus P_{j_{x}}^{n*}\neq\emptyset$.

Let $x\in Q_{i_{n}}$ and $y\in Q_{i_{n}}\setminus P_{j_{x}}^{n*}$. If
$d_{\mathcal{P}}(x,y)<n$, then there exists a $\mathcal{P}$-chain
$P_{1},\ldots,P_{m}$ that connects $x$ and $y$ with $m\leq n-1$. The
definition of a $\mathcal{P}$-chain implies $y\in P_{m}\subset P_{1}^{m*}$ and
$x\in P_{j_{x}}\cap P_{1}$ implies $P_{1}\subset P_{j_{x}}^{*}$. In
conclusion, we have $y\in P_{j_{x}}^{(m+1)*}\subset P_{j_{x}}^{n*}$, which is
a contradiction to the choice of $y$. Hence, $d_{\mathcal{P}}(x,y)\geq n$ for
all $x\in Q_{i_{n}}$ and all $y\in Q_{i_{n}}\setminus P_{j_{x}}^{n*}$ if $x\in
P_{j_{x}}$.

Now, choose $x_{n}\in Q_{i_{n}}$ and $y_{n}\in Q_{i_{n}}\setminus
P_{j_{x_{n}}}^{n*}\neq\emptyset$, where $x_{n}\in P_{j_{x_{n}}}$. The prior
part shows $d_{\mathcal{P}}(x_{n},y_{n})\geq n\xrightarrow{n\to\infty}\infty$,
but $d_{\mathcal{Q}}(x_{n},y_{n})\leq 1$ since $x_{n},y_{n}\in
Q_{i_{n}}\in\mathcal{Q}$. All in all, there cannot exist constants $L,C>0$
such that

$\displaystyle d_{\mathcal{P}}(x,y)\leq Ld_{\mathcal{Q}}(x,y)+C$

for all $x,y\in\mathcal{O}$. It follows that
$\mathrm{id}:(\mathcal{O},d_{\mathcal{Q}})\to(\mathcal{O},d_{\mathcal{P}})$ is
not a quasi-isometry.

The equivalence $ii)\Leftrightarrow iii)$ follows from Proposition 3.15 and
Remarks 3.20, 3.21.

###### Remark 3.23

Given the role of the connectedness assumption on the elements of the
coverings in the various results leading up to the theorem, it seems
reasonable to expect that they cannot be dropped from the theorem. In fact, it
is not hard to observe that, without some structural assumptions on the
coverings, Theorem 3.22 cannot hold. To see this, one can pick an admissible
covering $\mathcal{P}$, and construct a second admissible covering
$\mathcal{Q}$ from $\mathcal{P}$ by replacing suitable pairs of sets from
$\mathcal{P}$ by their unions. If the distances of the pairs of sets making up
$\mathcal{Q}$, when measured in the $\mathcal{P}$-chain distance
$d_{\mathcal{P}}$, are unbounded, $d_{\mathcal{Q}}$ will not be quasi-
equivalent to $d_{\mathcal{P}}$. On the other hand, one easily sees that the
coverings must be weakly equivalent.

However, it is currently open whether such a counterexample can be constructed
with both $\mathcal{P}$ and $\mathcal{Q}$ fulfilling the other standing
assumptions on decomposition coverings, namely tightness and structuredness,
stipulated at the beginning of Subsection 2.1. Hence it is not clear whether
the connectedness assumption is the main ingredient making the proof of
Theorem 3.22 work.

##  4 A metric characterization of coorbit equivalence

###  4.1 Word metrics on locally compact groups

Throughout this section, $H$ denotes an admissible dilation group, and
$H_{0}<H$ denotes the connected component of the identity element in $H$.
Since $H$ is a Lie group, $H_{0}$ is an open subgroup. We define a metric
(which is allowed to take the value $\infty$) by picking a word metric on
$H_{0}$, and suitably extending it to $H$.

###### Definition 4.1

Let $H$ be a locally compact group and let $W\subset H$ be a unit
neighborhood. Define the map $d_{W}:H\times
H\to\mathbb{N}_{0}\cup\\{\infty\\}$ in the following way

$\displaystyle d_{W}(x,y)=\begin{cases}\inf\Set{m\in\mathbb{N}}{x^{{-1}}y\in
W^{m}}&x\neq y\\\ 0&x=y,\end{cases}$

where we again set $\inf\emptyset=\infty$.

The expression word metric is motivated by the fact that $d_{W}(x,y)=m$
entails that the minimal number of elements (letters) $w_{1},\ldots,w_{m}\in
W$ that make the equation $y=xw_{1}\cdots w_{m}$ true is precisely $m$. The
following fact is well-known, and we omit the proof.

###### Lemma 4.2

Let $H$ be a locally compact group and let $W\subset H$ be a symmetric
($W=W^{{-1}}$) unit neighborhood. Then $d_{W}$ is a metric on $H$, that is
left invariant, i.e., $d_{W}(x,y)=d_{W}(zx,zy)$ for all $x,y,z\in H$.

As for the metric in the last section, we can impose a connectedness
restriction on $W$ in such a way that precisely the elements in the same
connected component have finite distance with respect to $d_{W}$. The proof is
essentially identical to the previous one, and therefore omitted.

###### Lemma 4.3

Let $H$ be a locally compact group and let $W\subset H$ be a symmetric unit
neighborhood with $W\subset H_{0}$, where $H_{0}$ denotes the connected
component of the neutral element in $H$. Then $d_{W}(x,y)<\infty$ if and only
if $x,y$ lie in the same connected component of $H$.

We next recall that the metric spaces $(H,d_{W})$ and $(H,d_{V})$ are coarsely
equivalent for relatively compact $W,V\subset H_{0}$. Here we refer to [7,
Corollary 4.A.6]

###### Lemma 4.4

Let $H$ be a locally compact group and let $V,W\subset H$ be relatively
compact, symmetric unit neighborhoods with $V,W\subset H_{0}$. Then

$\displaystyle\mathrm{id}_{W}^{V}:(H,d_{W})\to(H,d_{V}),h\mapsto h$

is a quasi-isometry and the spaces $(H,d_{W})$ and $(H,d_{V})$ are coarsely
equivalent.

In the terminology of [7], $U$-well-spread families are metric lattices with
respect to a properly chosen word metric. The following lemma is a further
example of a folklore result for which we could not locate a handy reference.

###### Lemma 4.5

Let $H$ be a locally compact group, let $W\subset H$ be a relatively compact,
symmetric unit neighborhood with $W\subset H_{0}$. Furthermore, let $X\subset
H$ be a $U$-well-spread family for some relatively compact unit neighborhood
$U\subset H_{0}$. Then $(H,d_{W})$ and $(X,d_{W}|_{X\times X})$ are coarsely
equivalent.

###### Proof 7

The identity

$\displaystyle d_{W}|_{X\times X}(x,y)=d_{W}(x,y)$

for all $x,y\in X$ shows that $g:X\to H,x\mapsto x$ is a quasi-isometric
embedding. It is coarsely surjective because for every $y\in H$ exists an
$x_{y}\in X$ such that $y\in x_{y}U$, which implies $x_{y}^{{-1}}y\in U$.

Since $W$ is symmetric, $\bigcup_{n\in\mathbb{N}}W^{n}$ is easily seen to be
an open subgroup of $H_{0}$, and connectedness of $H_{0}$ then entails
$H_{0}=\bigcup_{n\in\mathbb{N}}W^{n}$. In fact, the right hand side is an open
covering by increasing sets. Since $\overline{U}\subset H_{0}$ is compact,
this entails the existence of $n\in\mathbb{N}$ with $U\subset W^{n}$. Hence,

$\displaystyle d_{W}(x_{y},y)\leq n.$

It follows that $g$ is a quasi-isometry and that

$(H,d_{W})\text{ and }(X,d_{W}|_{X\times X})$

are coarsely equivalent, according to Theorem 3.13.

The last result of this section will be useful, but has a straightforward
proof, which is therefore omitted.

###### Lemma 4.6

Let $H,H^{\prime}$ be locally compact groups, let $W\subset H$ be a relatively
compact, symmetric unit neighborhood with $W\subset H_{0}$ and let $V\subset
H^{\prime}$ be a relatively compact, symmetric unit neighborhood with
$V\subset H_{0}^{\prime}$. Assume that $\phi:H\to H^{\prime}$ is a topological
group isomorphism, i.e. it is a continuous isomorphism with continuous
inverse. Then $\phi$ is a quasi-isometry from $(H,d_{W})$ to
$(H^{\prime},d_{V})$. In particular, $(H,d_{W})$ and $(H^{\prime},d_{V})$ are
coarsely equivalent.

###  4.2 Coarse equivalence of the dual orbit and the dilation group

In this section, we investigate the relation between the coarse structure
induced on the dual orbit $\mathcal{O}\subset\mathbb{R}^{d}$ of an admissible
group $H\subset\mathrm{GL}(\mathbb{R}^{d})$ by means of an induced covering on
the one hand, and the coarse structure induced by a relatively compact unit
neighborhood on the group $H$ on the other. It is the main purpose of this
subsection to prove that the orbit map

$p_{\xi}:H\to\mathcal{O},h\mapsto h^{{-T}}\xi$

is a quasi-isometry, when $H$ is endowed with a suitable metric, and
$\mathcal{O}$ is endowed with the covering-induced metric.

The next result collects various topological properties of the orbit map.

###### Lemma 4.7

For each $\xi\in\mathcal{O}$, the orbit map is continuous, surjective, closed,
open, and proper, i.e. $p_{\xi}^{{-1}}(K)\subset H$ is compact for compact
$K\subset\mathcal{O}$.

Furthermore, the orbit $\mathcal{O}$ has finitely many components.

###### Proof 8

We refer to [21] and [36, Corollary 4.1.3] for all properties except
closedness of $p_{\xi}$. But this property follows from properness: Let
$A\subset H$ be closed, and let $(h_{n})_{n\in\mathbb{N}}\subset A$ with
$p_{\xi}(h_{n})\to\eta\in\mathcal{O}$. Then
$K=\\{p_{\xi}(h_{n}):n\in\mathbb{N}\\}\cup\\{\eta\\}\subset\mathcal{O}$ is
compact, and then the same follows (by properness) for $p_{\xi}^{-1}(K)\subset
H$. This set contains the sequence $(h_{n})_{n\in\mathbb{N}}$. Hence, possibly
after passing to a subsequence, $h_{n}\to h\in H$, and then $h\in A$ by
closedness of $A$. But this finally entails
$\eta=\lim_{n\to\infty}p_{\xi}(h_{n})=p_{\xi}(h)\in p_{\xi}(A)$, as desired.

Recall from Section 3, that we are primarily interested in induced coverings
consisting of connected sets. In order to properly relate the metric arising
from the induced covering with a word metric on $H$, the condition
$H_{\xi}\subset H_{0}$ will be important, where $H_{0}\subset H$ denotes the
connected component of the identity element in $H$. The following lemma makes
some basic observations following from this assumption.

###### Lemma 4.8

Assume that the isotropy group $H_{\xi}=\Set{h\in H}{p_{\xi}(h)=\xi}$ is a
subset of the identity component $H_{0}\subset H$.

  1. (i)

The orbit map $p_{\xi}$ induces a bijection between the sets of connected
components of $H$ and those of $\mathcal{O}$.

  2. (ii)

For every connected set $B\subset\mathcal{O}$, there is a connected set
$A\subset H$ such that

$\displaystyle p_{\xi}^{{-1}}(B)\subset A.$

###### Proof 9

We start the proof of (i) by noting that $H_{0}$ is a normal subgroup of $H$.
Then the factor group $H/H_{0}$ acts on the set of connected components of
$\mathcal{O}$ in a canonical way. To see this, note that if
$C\subset\mathcal{O}$ is a connected component, then $H_{0}^{T}C$ is a
connected open set with open complement (the union of the remaining
$H_{0}^{T}$-orbits), which intersects $C$ nontrivially. This implies
$H_{0}^{T}C\subset C$ by connectedness of the left-hand side, and
$H_{0}^{T}C=C$ by the remaining properties of $H_{0}^{T}C$. As a consequence,
if $\mathcal{C}$ denotes the set of connected components of $\mathcal{O}$, the
map $(H/H_{0})\times\mathcal{C}\ni(hH_{0},C)\mapsto h^{-T}C\in\mathcal{C}$ is
a well-defined action of the quotient group on $\mathcal{C}$. Furthermore, it
is clearly transitive, since the action of $H$ on $\mathcal{O}$ is transitive.

Finally, the condition $H_{\xi}\subset H_{0}$ implies that the action of
$H/H_{0}$ is free. To see this, it suffices to show that the stabilizer of
$C=H_{0}^{T}\xi$ is trivial, since the action is transitive. For any $h\in H$
satisfying $h^{-T}H_{0}^{T}C=C$ we get $\xi\in hH_{0}^{T}\xi$, and thus
$h^{-T}h_{0}^{-T}\xi=\xi$, with suitably chosen $h_{0}\in H_{0}$. Hence
$hh_{0}\in H_{\xi}\subset H_{0}$, which implies $h\in H_{0}$. This finishes
the proof of (i).

For the proof of (ii), we first note that by assumption on $B$, $B$ is
contained in a connected component of $\mathcal{O}$. By part (i), the inverse
image of this connected component is a connected component $A\subset H$.

Without the condition $H_{\xi}\subset H_{0}$, the orbit map is not a quasi-
isometry or even a coarse map between the group $H$ and the associated dual
orbit.

###### Corollary 4.9

Let $W\subset H$ be a relatively compact, symmetric unit neighborhood with
$W\subset H_{0}$. Let $\mathcal{Q}$ be an induced connected covering of
$\mathcal{O}$ by $H$. If $H_{\xi}\not\subset H_{0}$, then $p_{\xi}$ is not a
metrically proper map from $(H,d_{W})$ to $(\mathcal{O},d_{\mathcal{Q}})$.

###### Proof 10

If $H_{\xi}\not\subset H_{0}$, there exist $h_{1},h_{2}\in H_{\xi}$ that are
members of different connected components in $H$. This implies

$\displaystyle
d_{\mathcal{Q}}(p_{\xi}(h_{1}),p_{\xi}(h_{2}))=0\quad\text{and}\quad
d_{W}(h_{1},h_{2})=\infty$

because of $p_{\xi}(h_{1})=p_{\xi}(h_{2})$ and Lemma 4.3. This means the
inverse image of the bounded set $A:=\\{p_{\xi}(h_{1})\\}$ in
$(\mathcal{O},d_{\mathcal{Q}})$ under $p_{\xi}$ is not bounded in $(H,d_{W})$
since $p^{{-1}}_{\xi}(A)\supset\\{h_{1},h_{2}\\}$.

In the remaining part of this section, we show that the condition
$H_{\xi}\subset H_{0}$ ensures that the orbit map $p_{\xi}$ is indeed a quasi-
isometry for suitable $\xi\in\mathcal{O}$. As preparation for this, we cite
the next result.

######  Lemma 4.10 ([36] Lemma 4.1.4.)

Let $K_{1},K_{2}\subset\mathcal{O}$ be compact sets and define

$\displaystyle L:=L(K_{1},K_{2}):=p_{\xi}^{{-1}}(K_{1})\cdot
H_{\xi}\cdot\left(p_{\xi}^{{-1}}(K_{2})\right)^{{-1}}.$

Then $L\subset H$ is compact and for arbitrary $g,h\in H$ satisfying

$h^{{-T}}K_{1}\cap g^{{-T}}K_{2}\neq\emptyset,$

we have $g\in hL$.

We will need the following property of the set $L$ from the above lemma.

###### Lemma 4.11

If $H_{\xi}\subset H_{0}$ and $K_{1},K_{2}\subset\mathcal{O}$ are compact sets
contained in the same connected component of $\mathcal{O}$, then
$L(K_{1},K_{2})\subset H_{0}$ with the set $L(K_{1},K_{2})$ from Lemma 4.10.

###### Proof 11

If one of the sets $K_{1},K_{2}$ is empty, the claim is obvious. Consider the
case of nonempty $K_{1},K_{2}$ in the following. There is a connected
component $B\subset\mathcal{O}$ with $K_{1},K_{2}\subset B$. By Lemma 4.8,
there is a connected component $A\subset H$ such that

$\displaystyle p_{\xi}^{{-1}}(K_{1}),p_{\xi}^{{-1}}(K_{2})\subset
p_{\xi}^{{-1}}(B)\subset A.$

This implies

$\displaystyle L(K_{1},K_{2})\subset A\cdot H_{\xi}\cdot A^{{-1}}\subset
A\cdot H_{0}\cdot A^{{-1}}=:M.$

Since $K_{1},K_{2}$ are nonempty, the set $A$ is nonempty as well. Hence,
there is $a\in A$, which further implies $e=a\cdot e\cdot a^{{-1}}\in M$. As
$M$ is a connected set as product of connected sets, and contains $e$, we
infer $M\subset H_{0}$.

In the following statements and arguments, we will occasionally make the
assumption that $\xi\in Q$, where the set $Q$ is used in the construction of
an induced connected covering $(h_{i}^{{-T}}Q)_{i\in I}$ of $\mathcal{O}$ by
$H$. Observe that this can be achieved by relabeling the covering, e.g. by
picking $i_{0}\in I$ with $\xi\in h_{i_{0}}^{{-T}}Q=:Q^{\prime}$, and
considering the system $(g_{i}^{{-T}}Q^{\prime})_{i\in
I}=(h_{i}^{{-T}}Q)_{i\in I}$, where $g_{i}=h_{i}h_{i_{0}}^{-1}$. Here it is
easy to verify that $(g_{i})_{i\in I}$ is well-spread if $(h_{i})_{i\in I}$
is.

The following inequality is the first step towards showing that the orbit map
is a quasi-isometry.

###### Lemma 4.12

Assume $H_{\xi}\subset H_{0}$. Let $W\subset H$ be a relatively compact,
symmetric unit neighborhood with $W\subset H_{0}$. Let
$\mathcal{Q}=(h_{i}^{{-T}}Q)_{i\in I}$ be an induced connected covering of
$\mathcal{O}$ by $H$ with $\xi\in Q$. Then the following inequality holds

$\displaystyle d_{W}(g,h)\leq Ld_{\mathcal{Q}}(p_{\xi}(g),p_{\xi}(h))+C$

for suitable $L,C>0$ for all $g,h\in H$.

###### Proof 12

Let $g,h\in H$ be arbitrary. If
$d_{\mathcal{Q}}(p_{\xi}(g),p_{\xi}(h))=\infty$, there is nothing to show. If
$d_{\mathcal{Q}}(p_{\xi}(g),p_{\xi}(h))=0$, then

$p_{\xi}(g)=p_{\xi}(h)$

and $g^{{-1}}h\in H_{\xi}$. The set $H_{\xi}\subset H_{0}$ is compact,
according to the admissibility of $H$, so that there exists a
$C_{1}\in\mathbb{N}$ with $H_{\xi}\subset W^{C_{1}}$ (cf. the proof of Lemma
4.5 for a similar argument), which implies $d_{W}(g,h)\leq C_{1}$.

If $d_{\mathcal{Q}}(p_{\xi}(g),p_{\xi}(h))=m\in\mathbb{N}$, there exists a
$\mathcal{Q}$-chain from $p_{\xi}(g)$ to $p_{\xi}(h)$, which means there exist
$h_{i_{1}},\ldots,h_{i_{m}}$ in $(h_{i})_{i\in I}$ such that $p_{\xi}(g)\in
h_{i_{1}}^{{-T}}Q$, $p_{\xi}(h)\in h_{i_{m}}^{{-T}}Q$ and

$\displaystyle h_{i_{j}}^{{-T}}Q\cap h_{i_{j+1}}^{{-T}}Q\neq\emptyset$

for all $j\in\\{1,\ldots,m-1\\}$. According to Lemma 4.10, this implies

$\displaystyle g^{{-1}}h_{i_{1}}$ $\displaystyle\in
L(\\{\xi\\},\overline{Q}),$ $\displaystyle h_{i_{j}}^{{-1}}h_{i_{j+1}}$
$\displaystyle\in L(\overline{Q},\overline{Q})\quad\text{for }j\in\\{1,\ldots
m-1\\}$ $\displaystyle h_{i_{m}}^{{-1}}h$ $\displaystyle\in
L(\overline{Q},\\{\xi\\}),$

which allows us to deduce

$\displaystyle g^{{-1}}h$
$\displaystyle=g^{{-1}}h_{i_{1}}\left(\prod_{j=1}^{m-1}h_{i_{j}}^{{-1}}h_{i_{j+1}}\right)h_{i_{m}}^{{-1}}h$
$\displaystyle\in
L(\\{\xi\\},\overline{Q})L(\overline{Q},\overline{Q})^{m-1}L(\overline{Q},\\{\xi\\}).$

Because $\xi\in\overline{Q}$ and $\overline{Q}$ is a connected subset of
$\mathcal{O}$, Lemma 4.11 implies

$\displaystyle S:=L(\\{\xi\\},\overline{Q})\cup
L(\overline{Q},\overline{Q})\cup L(\overline{Q},\\{\xi\\})\subset H_{0}$

and $S$ is compact, according to Lemma 4.10. Connectedness of $H_{0}$ and
$W\subset H_{0}$ symmetric, open imply that
$H_{0}=\bigcup_{n\in\mathbb{N}}W^{n}$, hence compactness of $S$ implies
$S\subset W^{L_{1}}$ for suitable $L_{1}$, which implies

$\displaystyle g^{{-1}}h\in S^{m+1}\subset W^{L_{1}(m+1)}.$

We conclude

$\displaystyle d_{W}(g,h)\leq
L_{1}(m+1)=L_{1}d_{\mathcal{Q}}(p_{\xi}(g),p_{\xi}(h))+L_{1}.$

In summary, for $L:=L_{1}$ and $C:=L_{1}+C_{1}$, we have

$\displaystyle d_{W}(g,h)\leq Ld_{\mathcal{Q}}(p_{\xi}(g),p_{\xi}(h))+C$

for all $g,h\in H$.

In the next result, we show the remaining inequality for a special case.

###### Lemma 4.13

Let $W\subset H$ be a relatively compact, symmetric unit neighborhood with
$W\subset H_{0}$. Let $\mathcal{Q}=(h_{i}^{{-T}}Q)_{i\in I}$ be an induced
connected covering of $\mathcal{O}$ by $H$ with $\xi\in Q$ for some well-
spread family $(h_{i})_{i\in I}$ in $H$ and open, relatively compact
$Q\subset\mathcal{O}$. Then there exists a constant $C>0$ such that for all
$g,h\in H$ with $d_{W}(g,h)\leq 1$, the inequality

$\displaystyle d_{\mathcal{Q}}(p_{\xi}(g),p_{\xi}(h))\leq C$

holds.

###### Proof 13

If $g,h\in H$ with $d_{W}(g,h)\leq 1$, then $h=gw$ for some $w\in W$ (if
$d_{W}(g,h)=0$, then $w=e$). The set $W\subset H_{0}$ is relatively compact
therefore we can choose a relatively compact, connected set $V\subset H_{0}$
with $W\subset V$ (this is possible by choosing a relatively compact,
connected unit neighborhood $U\subset H$ and using that
$W\subset\bigcup_{m=1}^{\infty}(U^{\circ})^{m}$, then $V:=U^{m}$, for a
suitable $m$, is a valid choice). We have

$\overline{p_{\xi}(V)}\subset p_{\xi}(\overline{V})\subset\mathcal{O}$

because the orbit map is closed according to Lemma 4.7, which shows that
$K_{1}:=\overline{p_{\xi}(V)}$ is a compact and connected subset of
$\mathcal{O}$. If we set $K_{2}:=\overline{Q}$, then Lemma 2.10 implies that
the set

$\displaystyle I_{g}(K_{1},K_{2}):=\Set{i\in I}{g^{{-T}}K_{1}\cap
h_{i}^{{-T}}K_{2}\neq\emptyset}$

has at most $C_{2}=C_{2}(K_{1},K_{2},(h_{i})_{i\in I})$ elements. Crucial for
this proof is that $C_{2}$ does not depend on $g$. For arbitrary $x\in
g^{{-T}}K_{1}=:K$, consider the set

$\displaystyle C_{x}:=\Set{z\in K}{\begin{array}[]{l}x,z\text{ are connected
by a }\mathcal{Q}\text{-chain }\\\ (Q_{\ell})_{\ell=1}^{m}\text{ of length
}m\leq C_{2}\mbox{ with }Q_{\ell}\cap K\neq\emptyset\\\ \text{and
}Q_{\ell}\neq Q_{\ell^{\prime}}\text{ for }\ell\neq\ell^{\prime}\end{array}}.$

As in the proof of Theorem 3.22, we see that $C_{x}=K$ because $K$ is
connected as well. Since $x\in K$ was arbitrary, we have

$\displaystyle d_{\mathcal{Q}}(x,y)\leq C_{2}$

for all $x,y\in K$. In particular, it holds

$\displaystyle p_{\xi}(g)=g^{{-T}}\xi\in g^{{-T}}W^{{-T}}\xi\subset
g^{{-T}}\overline{p_{\xi}(V)}=g^{{-T}}K_{1}=K$

and

$\displaystyle p_{\xi}(gw)=g^{{-T}}w^{{-T}}\xi\in g^{{-T}}W^{{-T}}\xi\subset
g^{{-T}}\overline{p_{\xi}(V)}=g^{{-T}}K_{1}=K,$

which implies

$\displaystyle d_{\mathcal{Q}}(p_{\xi}(g),p_{\xi}(h))\leq C_{2}.$

As $C_{2}$ is independent of $g,h$ with $d_{W}(g,h)\leq 1$, this concludes the
proof.

This special case implies the general case.

###### Corollary 4.14

Let $W\subset H$ be a relatively compact, symmetric unit neighborhood with
$W\subset H_{0}$. Let $\mathcal{Q}=(h_{i}^{{-T}}Q)_{i\in I}$ be an induced
connected covering of $\mathcal{O}$ by $H$ with $\xi\in Q$, for some open,
relatively compact $Q\subset\mathcal{O}$. There exists a constant $C>0$ such
that for all $g,h\in H$, the inequality

$\displaystyle d_{\mathcal{Q}}(p_{\xi}(g),p_{\xi}(h))\leq Cd_{W}(g,h)$

holds.

###### Proof 14

Let $g,h$ be arbitrary elements in $H$. If $d_{W}(g,h)=\infty$, there is
nothing to show. If $d_{W}(g,h)=0$, then $g=h$ and also the left side of the
inequality vanishes. If $d_{W}(g,h)=m\in\mathbb{N}$, we can write

$\displaystyle h=g\prod_{k=1}^{m}w_{k}$

for some $w_{k}\in W$. Define $h_{n}:=g\prod_{k=1}^{n}w_{k}$ for
$n\in\\{0,\ldots,m\\}$. This entails

$\displaystyle d_{W}(h_{n},h_{n+1})=d_{W}(e,w_{n+1})\leq 1$

for all $n\in\\{0,\ldots,m-1\\}$, where we used the left invariance of the
metric $d_{W}$. Repeated application of the triangle inequality and Lemma 4.13
yield

$\displaystyle d_{\mathcal{Q}}(p_{\xi}(g),p_{\xi}(h))$
$\displaystyle\leq\sum_{n=0}^{m-1}d_{\mathcal{Q}}(p_{\xi}(h_{n}),p_{\xi}(h_{n+1}))$
$\displaystyle\leq Cm$ $\displaystyle=Cd_{W}(g,h).$

###### Remark 4.15

Note that we did not assume $H_{\xi}\subset H_{0}$ in Lemma 4.13 and Corollary
4.14, which means that even without this assumption, the orbit map is
uniformly bornologous.

The previous results combine to give the following important observation.

###### Theorem 4.16

Assume $H_{\xi}\subset H_{0}$. Let $W\subset H$ be a relatively compact,
symmetric unit neighborhood with $W\subset H_{0}$. Furthermore, let
$\mathcal{Q}=(h_{i}^{{-T}}Q)_{i\in I}$ be an induced connected covering of
$\mathcal{O}$ by $H$ with $\xi\in Q$, for some open, relatively compact
$Q\subset\mathcal{O}$. Then

$p_{\xi}:(H,d_{W})\to(\mathcal{O},d_{\mathcal{Q}}),h\mapsto h^{{-T}}\xi$

is a quasi-isometry. In particular $(H,d_{W})$ and
$(\mathcal{O},d_{\mathcal{Q}})$ are coarsely equivalent.

###### Proof 15

Lemma 4.12 and Corollary 4.14 show that

$p_{\xi}:(H,d_{W})\to(\mathcal{O},d_{\mathcal{Q}})$

is a quasi-isometric embedding. Since it is surjective, it is a quasi-
isometry. The coarse equivalence follows with Theorem 3.13.

###  4.3 Characterizing coorbit equivalence

Now we can state the central theorem characterizing whether two admissible
groups with the same dual orbit $\mathcal{O}$ induce weakly equivalent
coverings of $\mathcal{O}$, without the need to explicitly determine such
coverings.

The basic idea underlying the theorem is conveyed by the following commutative
diagram, and an appeal to Theorems 3.22 and 4.16.

${(\mathcal{O},d_{\mathcal{Q}})}$${(\mathcal{O},d_{\mathcal{P}})}$${(H_{1},d_{W})}$${(H_{2},d_{V})}$$\scriptstyle{\mathrm{id}_{\mathcal{Q}}^{\mathcal{P}}}$$\scriptstyle{\left(p_{\xi_{2}}^{H_{2}}\right)^{{-1}}}$$\scriptstyle{\phi}$$\scriptstyle{p_{\xi_{1}}^{H_{1}}}$

###### Theorem 4.17

Let $H_{1},H_{2}\subset\mathrm{GL}(\mathbb{R}^{d})$ be two admissible dilation
groups with dual orbit $\mathcal{O}=H_{1}^{-T}\xi_{1}$ and
$\mathcal{O}^{\prime}=H_{2}^{-T}\xi_{2}$. Assume that the isotropy groups are
contained in the respective connected components of $H_{1}$ and $H_{2}$. Let
$W\subset(H_{1})_{0},V\subset(H_{2})_{0}$ denote relatively compact, symmetric
neighborhoods, let $\mathcal{Q}=(h_{i}^{-T}Q)_{i\in I}$ and
$\mathcal{P}=(g_{j}^{-T}P)_{j\in J}$ denote induced, connected coverings of
$\mathcal{O}$ resp. $\mathcal{O}^{\prime}$, for suitable choices of well-
spread families $(h_{i})_{i\in I}\subset H_{1}$, $(g_{j})_{j\in J}\subset
H_{2}$.

Denote the orbit maps by

$\displaystyle
p_{\xi_{1}}^{H_{1}}:(H_{1},d_{W})\to(\mathcal{O},d_{\mathcal{Q}}),h\mapsto
h^{{-T}}\xi_{1}$

and

$\displaystyle
p_{\xi_{2}}^{H_{2}}:(H_{2},d_{V})\to(\mathcal{O},d_{\mathcal{P}}),h\mapsto
h^{{-T}}\xi_{2}.$

Let $\left(p_{\xi_{2}}^{H_{2}}\right)^{{-1}}$ denote an arbitrary right
inverse of $p_{\xi_{2}}^{H_{2}}$ (which exists since $p_{\xi_{2}}^{H_{2}}$ is
surjective). Then the following statements are equivalent:

  1. i)

$H_{1}$ and $H_{2}$ are coorbit equivalent.

  2. ii)

$\mathcal{O}=\mathcal{O}^{\prime}$, and the map

$\phi:=\left(p_{\xi_{2}}^{H_{2}}\right)^{{-1}}\circ
p_{\xi_{1}}^{H_{1}}:(H_{1},d_{W})\to(H_{2},d_{V})$

is a coarse equivalence.

  3. iii)

$\mathcal{O}=\mathcal{O}^{\prime}$, and the map

$\phi:=\left(p_{\xi_{2}}^{H_{2}}\right)^{{-1}}\circ
p_{\xi_{1}}^{H_{1}}:(H_{1},d_{W})\to(H_{2},d_{V})$

is a quasi-isometry.

###### Proof 16

First, we note that any right inverse of $p_{\xi_{2}}^{H_{2}}$ is also a
coarse right inverse in the sense of Definition 3.9. Since our assumptions and
Theorem 4.16 imply that
$p_{\xi_{2}}^{H_{2}}(H_{2},d_{V})\to(\mathcal{O}^{\prime},d_{\mathcal{P}})$ is
a quasi-isometry, Lemma 3.11 shows that

$\left(p_{\xi_{2}}^{H_{2}}\right)^{{-1}}:(\mathcal{O},d_{\mathcal{P}})\to(H_{2},d_{V})$

is also a quasi-isometry.

$i)\Rightarrow iii)$: By Theorem 2.18, $H_{1}$ and $H_{2}$ are coorbit
equivalent iff $\mathcal{O}=\mathcal{O}^{\prime}$, and the dual coverings
induced by $H_{1}$ and $H_{2}$ are weakly equivalent.

We can now write

$\phi=\left(p_{\xi_{2}}^{H_{2}}\right)^{{-1}}\circ\mathrm{id}_{\mathcal{Q}}^{\mathcal{P}}\circ
p_{\xi_{1}}^{H_{1}}:(H_{1},d_{W})\to(H_{2},d_{V})~{},$

as a composition of quasi-isometries:
$\mathrm{id}_{\mathcal{Q}}^{\mathcal{P}}$ is a quasi-isometry
$(\mathcal{O},d_{\mathcal{Q}})\to(\mathcal{O},d_{\mathcal{P}})$ by Theorem
3.22, and the other two mappings are quasi-isometries by Theorem 4.16. Hence
$\phi$ is a quasi-isometry as well.

$iii)\Rightarrow i)$: Our assumptions imply that $p_{\xi_{1}}^{H_{1}}$ and
$p_{\xi_{2}}^{H_{2}}$ are quasi-isometries. Furthermore, the same argument as
for $p_{\xi_{2}}^{H_{2}}$ shows that there exists a quasi-isometric right
inverse $\left(p_{\xi_{1}}^{H_{1}}\right)^{{-1}}$ of $p_{\xi_{1}}^{H_{1}}$. We
conclude that the map

$\displaystyle
p_{\xi_{2}}^{H_{2}}\circ\phi\circ\left(p_{\xi_{1}}^{H_{1}}\right)^{{-1}}$
$\displaystyle=p_{\xi_{2}}^{H_{2}}\circ\left(\left(p_{\xi_{2}}^{H_{2}}\right)^{{-1}}\circ\mathrm{id}_{\mathcal{Q}}^{\mathcal{P}}\circ
p_{\xi_{1}}^{H_{1}}\right)\circ\left(p_{\xi_{1}}^{H_{1}}\right)^{{-1}}$
$\displaystyle=\left(p_{\xi_{2}}^{H_{2}}\circ\left(p_{\xi_{2}}^{H_{2}}\right)^{{-1}}\right)\circ\mathrm{id}_{\mathcal{Q}}^{\mathcal{P}}\circ\left(p_{\xi_{1}}^{H_{1}}\circ\left(p_{\xi_{1}}^{H_{1}}\right)^{{-1}}\right)$
$\displaystyle=\mathrm{id}_{\mathcal{Q}}^{\mathcal{P}}$

is a quasi-isometry as composition of quasi-isometric maps. Theorem 3.22
implies that the coverings $\mathcal{Q}$ and $\mathcal{P}$ are weakly
equivalent. This suffices to establish the coorbit equivalence of the groups
$H_{1}$ and $H_{2}$, by Theorem 3.22.

In order to prove the equivalence $ii)\Leftrightarrow iii)$, we first observe
that $H_{1},H_{2}$ both have finitely many connected components, by combining
Lemmas 4.7 and 4.8.

With that fact established for $H_{1}$ and $H_{2}$, $ii)\Leftrightarrow iii)$
follows from Proposition 3.15 and Remark 3.20, since the word metrics are
large scale geodesic.

###### Remark 4.18

The main appeal of the theorem is that it replaces the technicalities involved
with computing and comparing coverings induced by different dilation groups by
a problem in geometric group theory. Whether this new formalism actually
simplifies the task still depends on the groups at hand; the results of
Section 5 present a class of groups where this is arguably the case.

At least the theorem opens the door to the systematic application of geometric
group theory techniques to the problem of deciding coorbit equivalence. For
example, the machinery of coarse invariants of groups can be brought to bear
to derive necessary criteria for coorbit equivalence. E.g., if $H_{1}$ and
$H_{2}$ are coorbit equivalent and unimodular, and $H_{1}$ is amenable, then
$H_{2}$ must be amenable as well, by Corollary 4.F.9 of [7].

It is important to realize, though, that the theorem refers to coarse
properties of a very specific map between the groups $H_{1}$ and $H_{2}$,
which is induced by the respective dual actions. In particular, Section 5 will
provide a rich class of examples of pairs of groups $H_{1}$ and $H_{2}$ that
are topologically isomorphic (and thus quasi-isometric), without being coorbit
equivalent; see Remark 5.8 below.

##  5 Characterizing coorbit equivalence of shearlet dilation groups

Shearlet groups in arbitrary dimensions, and their associated coorbit spaces,
were first introduced in [11]. Later on, _Toeplitz shearlets_ , a variation on
the shearlet construction in dimensions $\geq 3$, were introduced in [9]. The
next definition provides the details. The initial definition of Toeplitz
shearlets only allowed the inclusion of isotropic dilation, corresponding to
the case $\delta=0$ in part ii). The availability of anisotropic scaling for
this class of groups was stated later in [1].

######  Definition 5.1 ([1] Example 17. and Example 18.)

  1. i)

For $\lambda=(\lambda_{1},\ldots,\lambda_{d-1})\in\mathbb{R}^{d-1}$, we define
the standard shearlet group in $d$-dimensions $H^{\lambda}$ as the set

$\displaystyle\Set{\epsilon\,\mathrm{diag}\left(a,a^{\lambda_{1}},\ldots,a^{\lambda_{d-1}}\right)\begin{pmatrix}1&s_{1}&\ldots&s_{d-1}\\\
&1&0\ldots&0\\\ &&\ddots&0\\\ &&&1\end{pmatrix}}{\begin{array}[]{l}a>0,\\\
s_{i}\in\mathbb{R},\\\ \epsilon\in\Set{\pm 1}\end{array}}.$

  2. ii)

For $\delta\in\mathbb{R}$, we define the Toeplitz shearlet group in
$d$-dimensions $H^{\delta}$ as the set

$\displaystyle\Set{\epsilon\,\mathrm{diag}\left(a,a^{1-\delta},\ldots,a^{1-(d-1)\delta}\right)\cdot
T(1,s_{1},\ldots,s_{d-1})}{\begin{array}[]{l}a>0,\\\ s_{i}\in\mathbb{R},\\\
\epsilon\in\Set{\pm 1}\end{array}},$

where the matrix $T(1,s_{1},\ldots,s_{d-1})$ is defined by

$\displaystyle
T(1,s_{1},\ldots,s_{d-1}):=\begin{pmatrix}1&s_{1}&s_{2}&\ldots&\ldots&s_{d-1}\\\
&1&s_{1}&s_{2}&\ldots&s_{d-2}\\\ &&\ddots&\ddots&\ddots&\\\
&&&1&s_{1}&s_{2}\\\ &&&&1&s_{1}\\\ &&&&&1\end{pmatrix}.$

###### Remark 5.2

The two constructions are special cases of the class of _generalized shearlet
dilation groups_ introduced in [25] and further studied in [1]; see Definition
5.3 below. Note that each class actually defines a whole family of groups,
parametrized by different choices for the diagonal.

In dimension two the distinction between the standard and Toeplitz shearlet
dilation groups is moot. In dimension three, one can show that standard and
Toeplitz shearlet are the only possible generalized shearlet dilation groups
(cf. [1], Remark 19). With increasing dimension, the set of generalized
shearlet dilation groups besides the standard and Toeplitz case quickly
becomes untractable.

Prior to the thesis [30], the simple question whether standard and Toeplitz
shearlets have distinct coorbit spaces was not under consideration. [30]
established that, for any dimension $d\geq 3$, two distinct groups from the
families in Definition 5.1 are not coorbit equivalent. For dimension three,
this was done by computing and comparing induced coverings, see Chapter 3 of
[30] for details. The arguments employed to establish this result nicely
illustrate the computational burden imposed by the use of induced coverings,
and motivated the introduction of coarse geometry as a tool in Chapter 5 of
[30]. The comparison of standard and Toeplitz shearlet groups in dimensions
$>3$ was then performed using the new toolbox. Our treatment of general
shearlet dilation groups is a generalization of the arguments from [30,
Section 5.5].

###  5.1 Shearlet dilation groups: definition and basic properties

We now recall basic notation regarding matrix groups and their Lie algebras.
$\mathfrak{gl}(\mathbb{R}^{d})$ denotes the set of all real $d\times
d$-matrices. We let

$\exp:\mathfrak{gl}(\mathbb{R}^{d})\to\mathrm{GL}(\mathbb{R}^{d})$

be the exponential map defined by

$\exp(A):=\sum_{k=0}^{\infty}\frac{A^{k}}{k!}~{}.$

Furthermore, we denote with
$T(\mathbb{R}^{d})\subset\mathrm{GL}(\mathbb{R}^{d})$ the group of upper
triangular $d\times d$-matrices with one on their diagonals. By definition,
the Lie algebra of a closed subgroup $H\subset\mathrm{GL}(\mathbb{R}^{d})$ is
the set $\mathfrak{h}$ of all matrices $Y$ in $\mathfrak{gl}(\mathbb{R}^{d})$
such that $\exp(tY)\in H$ for all $t\in\mathbb{R}$.

######  Definition 5.3 ([1] Definition 1.)

Let $H\subset\mathrm{GL}(\mathbb{R}^{d})$ be a closed, admissible dilation
group. The group $H$ is called generalized shearlet dilation group if there
exist two closed subgroups

$S,D\subset\mathrm{GL}(\mathbb{R}^{d})$

such that

  1. i)

$S$ is a connected abelian subgroup of $T(\mathbb{R}^{d})$,

  2. ii)

$D=\Set{\exp(rY)}{r\in\mathbb{R}}$ is a one-parameter group, where
$Y\in\mathfrak{gl}(\mathbb{R}^{d})$ is a diagonal matrix and

  3. iii)

every $h\in H$ has a unique representation as $h=\pm ds$ for some $d\in D$ and
$s\in S$.

$S$ is called the shearing subgroup of $H$, $D$ is called the scaling subgroup
of $H$, and $Y$ is called the infinitesimal generator of $D$.

The article [1] describes a systematic algebraic construction method for
shearlet groups in arbitrary dimension $d\geq 2$ that proceeds by first
constructing a shearing group $S$ from an arbitrary nilpotent associative
algebra of dimension $d-1$, and then determining conditions on the diagonal
complement $D$ from certain structure constants of the nilpotent algebra.

We denote the canonical basis of $\mathbb{R}^{d}$ with $e_{1},\ldots,e_{d}$
and the identity matrix in $\mathrm{GL}(\mathbb{R}^{d})$ with $I_{d}$. The
next result contains information about the structure of shearing subgroups.

######  Lemma 5.4 ([1] Lemma 5. and Lemma 6.)

Let $S$ be the shearing subgroup of a generalized shearlet dilation group
$H\subset\mathrm{GL}(\mathbb{R}^{d})$. Then the following statements hold:

  1. i)

There exists a unique basis $X_{2},\ldots,X_{d}$ of the Lie algebra
$\mathfrak{s}$ of $S$ with $X_{i}^{T}e_{1}=e_{i}$ for $2\leq i\leq d$, called
the canonical basis of $\mathfrak{s}$.

  2. ii)

We have $S=\Set{I_{d}+X}{X\in\mathfrak{s}}$.

  3. iii)

Let $\mathfrak{s}_{k}={\mathrm{span}}\\{X_{j}:j\geq k\\}$, for
$k\in\\{2,\ldots,d\\}$. These are associative matrix algebras satisfying
$\mathfrak{s}_{k}\mathfrak{s}_{\ell}\subset\mathfrak{s}_{k+\ell-1}$, where we
write $\mathfrak{s}_{m}=\\{0\\}$ for $m>d$.

  4. iv)

$H$ is the inner semidirect product of the normal subgroup $S$ with the closed
subgroup $D\cup-D$.

Every generalized shearlet dilation group $H$ is admissible, and the next
result shows that all of them share the same dual orbit. This ensures that one
of the basic conditions for coorbit equivalence is already fulfilled.

######  Lemma 5.5 ([1] Proposition 11.)

Let $S\subset T(\mathbb{R}^{d})$ be a shearing subgroup and $D$ be a
compatible scaling subgroup such that

$H=DS\cup(-DS)$

is a generalized shearlet dilation group. Then the unique open dual orbit of
$H$ is given by $\mathcal{O}=\mathbb{R}^{*}\times\mathbb{R}^{d-1}$ and the
isotropy group of $\xi\in\mathcal{O}$ with respect to the dual action is given
by $H_{\xi}=\Set{I_{d}}$.

Note that in particular, $H_{\xi}\subset H_{0}$, where $H_{0}=DS$ denotes the
connected component of the neutral element. Furthermore, the orbit map is
bijective, thus it has a unique (right) inverse. Throughout this section, we
will use the representative $\xi_{0}=(1,0,\ldots,0)$. As a consequence of the
bijectivity of $p_{\xi_{0}}$, each element of $H$ is uniquely determined by
its first line.

Note that if $H=DS\cup-DS$ is a shearlet dilation group, one can replace
$D=\exp(rY)$ be $D^{\prime}=\mathbb{R}^{+}\cdot I_{d}$ to obtain the
associated abelian shearlet dilation group $A=D^{\prime}S\cup-D^{\prime}S$.
The structure of abelian dilation groups was elucidated in [21], and the
following lemma notes some useful properties arising from this structure.

###### Lemma 5.6

  1. (a)

Let $A$ denote an abelian admissible dilation group. The Lie algebra
$\mathfrak{a}$ of $A$ coincides with the associative algebra
$\mathcal{A}={\mathrm{span}}(A)$ generated by $A$, and $A\subset\mathcal{A}$
is the group of invertible elements in $\mathcal{A}$. Furthermore, the dual
action of $A$ on the open dual orbit is free.

  2. (b)

Let $A,A^{\prime}\subset GL(\mathbb{R}^{d})$ be abelian admissible dilation
groups with $\mathcal{A}={\mathrm{span}}(A)$ and
$\mathcal{A}^{\prime}={\mathrm{span}}(A^{\prime})$. Let
$\alpha:\mathcal{A}\to\mathcal{A}^{\prime}$ denote an algebra isomorphism.
Then there exists $C\in GL(\mathbb{R}^{d})$ such that $\alpha(X)=C^{-1}XC$
holds, for all $X\in\mathcal{A}$.

###### Proof 17

The first statement of part (a) is noted in [21, Theorem 11], whereas the
second statement is due to [21, Lemma 6].

Note that a consequence of the freeness of the operation implies
$\dim(A)=\dim(\mathcal{A})=d$. For part (b) let
$\psi:\mathcal{A}\to\mathbb{R}^{d}$, $\psi(X)=X^{T}\xi_{0}$, and
$\psi^{\prime}:\mathcal{A}^{\prime}\to\mathbb{R}^{d}$ be defined analogously.
Then $\psi,\psi^{\prime}$ are vector space isomorphisms: They are linear maps,
whose images contain the (open) dual orbits, by part (a). Hence they are onto,
and therefore bijective, since the dimensions coincide.

Now let $B\in GL(\mathbb{R}^{d})$,
$B=\psi\circ\alpha^{-1}\circ(\psi^{\prime})^{-1}$, and $X\in\mathcal{A}$.
Given any $y\in\mathbb{R}^{d}$, one has $y=\psi^{\prime}(Y)=Y^{T}\xi_{0}$ with
a unique $Y\in\mathcal{A}^{\prime}$, and we can compute for arbitrary
$X\in\mathcal{A}$

$\displaystyle B^{-1}X^{T}By$ $\displaystyle=$
$\displaystyle\left((\psi^{\prime}\circ\alpha\circ\psi^{-1})\circ
X^{T}\circ(\psi\circ\alpha^{-1}\circ(\psi^{\prime})^{-1})\right)(y)$
$\displaystyle=$
$\displaystyle\left((\psi^{\prime}\circ\alpha\circ\psi^{-1})\circ
X^{T}\right)(\psi(\alpha^{-1}(Y)))~{}.$

The construction of $\psi$ and the fact that $\mathcal{A}$ is commutative
implies $\psi(XZ)=X^{T}\psi(Z)$ for all $X,Z\in\mathcal{A}$, hence we can
continue via

$\displaystyle B^{-1}X^{T}By$ $\displaystyle=$
$\displaystyle(\psi^{\prime}\circ\alpha\circ\psi^{-1})(\psi(X\alpha^{-1}(Y)))$
$\displaystyle=$ $\displaystyle(\psi^{\prime}\circ\alpha)(X\alpha^{-1}(Y))$
$\displaystyle=$
$\displaystyle\psi^{\prime}(\alpha(X)Y)=\alpha(X)^{T}Y^{T}\xi_{0}=\alpha(X)^{T}y~{},$

where the third equation used that $\alpha$ is an algebra homomorphism. Hence
$C=B^{-T}$ has the desired properties.

###### Remark 5.7

The associative algebra structure of $\mathfrak{s}$ is also the key to the
systematic construction of shearing subgroups. For the following facts, we
refer to [1], in particular to Lemma 13 and Remark 14 therein. Every nilpotent
commutative associative algebra $\mathcal{N}$ of dimension $d-1$ gives rise to
the Lie algebra $\mathfrak{s}$ of a shearing subgroup of a generalized
shearlet dilation group. The canonical basis of $\mathfrak{s}$ is constructed
by picking a Jordan-Hölder basis in $\mathcal{N}$, and applying a suitable
linear map $\mathcal{N}\to\mathfrak{gl}(d,\mathbb{R})$. Letting
$S=\exp(\mathfrak{s})$ and $D=\mathbb{R}^{+}\cdot I_{d}$ then results in an
abelian shearlet dilation group $H=DS\cup-DS$. Furthermore, all candidates of
infinitesimal generators for $D$ can be found by solving a system of linear
equations derived from $\mathfrak{s}$, see [1, Lemma 9].

It is useful to note that two different shearing subgroups $S_{1},S_{2}$ are
related to each other by conjugation iff their Lie algebras are isomorphic as
associative algebras. Here the necessity is clear; and the sufficiency follows
by Lemma 5.6 (b). Thus there is a one-to-one correspondence between conjugacy
classes of shearing subgroups (or alternatively: of abelian shearlet dilation
groups) in dimension $d$ and of isomorphism classes of nilpotent commutative
associative algebras of dimension $d-1$. This illustrates the increasing
richness of this class of dilation groups, as $d$ grows. The main theorem of
this section, which characterizes coorbit equivalence within the class of
shearlet dilation groups, exhibits the relevance of the conjugacy problem also
for coorbit equivalence.

###### Remark 5.8

Given a shearlet dilation group $H=DS\cup(-DS)$ with $D=\exp(\mathbb{R}Y)$ and
canonical basis $X_{2},\ldots,X_{d}$ of $S$, the entries of the infinitesimal
generator $Y$ can be related to certain algebraic relation. More precisely, a
necessary requirement for $H$ to be an admissible dilation group is that the
first diagonal entry of $Y$ does not vanish, by [1, Proposition 7]. $Y$ can
then be normalized to $1$ as first entry, i.e.,
$Y=\mathrm{diag}(1,\lambda_{2},\ldots,\lambda_{d})$, and the remaining entries
of $Y$ occur in the structure relations

$\forall
i,j=2,\ldots,d~{}:~{}[Y,X_{i}]=(1-\lambda_{i})X_{i}~{},~{}[X_{i},X_{j}]=0~{},$

characterizing the Lie algebra $\mathfrak{h}$ up to isomorphism. It turns out
that this extends to the Lie group level, yielding that $H$ is topologically
isomorphic to $H^{\prime}=D^{\prime}S^{\prime}\cup(-D^{\prime}S^{\prime})$ iff
$D=D^{\prime}$; and this clearly entails that $H$ and $H^{\prime}$ are
coarsely isometric.

Observe that this characterization imposes no restrictions on $S$ and
$S^{\prime}$, respectively. By contrast, Theorem 5.9 establishes the necessary
criterion for coorbit equivalence that $S$ and $S^{\prime}$ are related by
conjugacy, and by Remark 5.7, this entails that the Lie algebras
$\mathfrak{s},\mathfrak{s}^{\prime}$ must be isomorphic as associative
algebras. In particular, by taking $H,H^{\prime}$ as standard resp. Toeplitz
shearlet dilation groups in dimension $d\geq 3$, with identical diagonal
groups $D=D^{\prime}$, we obtain pairs of dilation groups that are coarsely
isometric, but not coorbit equivalent. See also Remark 5.10 below.

To our knowledge, a classification of semidirect products of the type
$\mathbb{R}^{m}\rtimes\exp(\mathbb{R}Y)$ up to quasi-isometry is not
available. A subclass of these groups, for which a rich theory has been
developed, is given by the so-called Heintze groups [29]. Even here, a
classification up to quasi-isometry does not seem to be in sight. These
observations make clear that coarse geometric methods generally do not provide
readily applicable solutions for the problem of classifying dilation groups up
to coorbit equivalence. They also emphasize that coorbit equivalence of two
dilation groups depends on the coarse geometric properties of a very specific
map relating the dual actions of these groups.

We can now formulate a general characterization of coorbit equivalence for
shearlet groups. The following theorem generalizes the characterization for
two-dimensional shearlet groups obtained in [36], and the results for standard
and Toeplitz shearlet groups in [30].

###### Theorem 5.9

Let $H=DS\cup-
DS,H^{\prime}=D^{\prime}S^{\prime}\cup-D^{\prime}S^{\prime}\subset
GL(\mathbb{R}^{d})$ denote two shearlet dilation groups. Then the following
properties are equivalent:

  1. (a)

$H$ and $H^{\prime}$ are coorbit equivalent.

  2. (b)

$D=D^{\prime}$, and there exists a matrix $C\in GL(\mathbb{R}^{d})$ with the
following properties: $C^{-1}SC=S^{\prime}$, and the conjugation actions of
$D$ and $C$ on $S$ commute, i.e.

$\forall s\in S\forall d\in D~{}:~{}d^{-1}C^{-1}sCd=C^{-1}d^{-1}sdC~{}.$ (5.1)

###### Remark 5.10

We can apply the theorem to the two families of shearlet dilation groups
introduced in Definition 5.3. We first observe that the theorem explicitly
states that different elements in the same family (standard or Toeplitz),
i.e., pairs of shearlet dilation groups that differ only in the scaling
subgroup, are not coorbit equivalent.

The comparison of elements from different families leads to the question of
conjugacy, which in turn is closely related to the associative structures of
the respective Lie algebras. Let $\mathfrak{s}_{s}$ denote the Lie algebra of
the shearing subgroup of the standard shearlet dilation group in dimension
$d\geq 3$, and let $\mathfrak{s}_{T}$ denote its Toeplitz counterpart. Then
any two elements $X,Y\in\mathfrak{s}_{s}$ fulfill $XY=0$. By contrast,
$\mathfrak{s}_{T}$ is generated as an associative algebra by the element
$X_{2}$ of its canonical basis, showing that
$\mathfrak{s}_{T}\cong\mathbb{R}[X]/(X^{d-1})$. In particular,
$\mathfrak{s}_{s}$ and $\mathfrak{s}_{T}$ are not isomorphic as associative
algebras, and hence not conjugate.

In summary, any pair of distinct groups from Definition 5.3 has distinct
coorbit spaces. This fact was first established in [30].

###  5.2 Proof of Theorem 5.9

Throughout this subsection, we will denote $H=DS\cup(-DS)$,
$H^{\prime}=D^{\prime}S^{\prime}\cup(-D^{\prime}S^{\prime})$, with associated
Lie algebras
$\mathfrak{h},\mathfrak{h^{\prime}},\mathfrak{s},\mathfrak{s}^{\prime}$, and
canonical bases $Y,X_{2},\ldots,X_{d}\in\mathfrak{h}$ and
$Y^{\prime},X_{2}^{\prime},\ldots,X_{d}^{\prime}\in\mathfrak{h}^{\prime}$. By
Proposition 7 of [1], we can normalize $Y,Y^{\prime}$ so that

$Y={\mathrm{diag}}(1,\lambda_{2},\ldots,\lambda_{d})~{},~{}Y^{\prime}={\mathrm{diag}}(1,\lambda_{2}^{\prime},\ldots,\lambda_{d}^{\prime})~{}.$

We use
$\xi_{0}=(1,0,\ldots,0)\in\mathcal{O}=\mathbb{R}^{*}\times\mathbb{R}^{d-1}$.
We parametrize group elements of $H$ and $H^{\prime}$ as follows: Given
$r\in\mathbb{R}$ and $t=(t_{2},\ldots,t_{d})\in\mathbb{R}^{d-1}$, we let

$\displaystyle h(r,t)$ $\displaystyle=$
$\displaystyle\exp(-rY)\left(I_{d}+\sum_{j=2}^{d}t_{j}X_{j}\right)^{-1}\in
H~{},~{}$ (5.2) $\displaystyle h^{\prime}(r,t)$ $\displaystyle=$
$\displaystyle\exp(-rY^{\prime})\left(I_{d}+\sum_{j=2}^{d}t_{j}X_{j}^{\prime}\right)^{-1}\in
H^{\prime}~{}.$

Note that $D=\\{h(r,0):r\in\mathbb{R}\\}$ and
$S=\\{h(0,t):t\in\mathbb{R}^{d-1}\\}$, and analogous statements hold for
$D^{\prime},S^{\prime}$.

We can then compute

$p_{\xi_{0}}^{H}(h(r,t))=\exp(rY)(I_{d}+\sum_{j=2}^{d}t_{j}X_{j}^{T})\left(\begin{array}[]{c}1\\\
0\\\ \vdots\\\ 0\end{array}\right)=\left(\begin{array}[]{c}e^{r}\\\
e^{\lambda_{2}r}t_{2}\\\ \vdots\\\
e^{\lambda_{d}r}t_{d}\end{array}\right)~{}.$

The analogous calculation for $H^{\prime}$ yields

$p_{\xi_{0}}^{H^{\prime}}(h^{\prime}(r,t))=\left(\begin{array}[]{c}e^{r}\\\
e^{\lambda_{2}^{\prime}r}t_{2}\\\ \vdots\\\
e^{\lambda_{d}^{\prime}r}t_{d}\end{array}\right)$

Throughout this section, let $\phi=(p_{\xi_{0}}^{H^{\prime}})^{-1}\circ
p_{\xi_{0}}^{H}:H\to H^{\prime}$. Then a comparison of the formulae for
$p_{\xi_{0}}^{H}(h(r,t)),p_{\xi_{0}}^{H^{\prime}}(h^{\prime}(r,t))$ yields

$\phi(h(r,t))=h^{\prime}(r,t^{\prime})~{},~{}t^{\prime}=(\exp(r(\lambda_{2}-\lambda_{2}^{\prime}))t_{2},\ldots,\exp(r(\lambda_{d}-\lambda_{d}^{\prime}))t_{d})~{}.$
(5.3)

Setting $r=0$ in this formula shows that the restriction $\phi|_{S}$ in fact
maps $S$ bijectively onto $S^{\prime}$, and that it is independent of the
choices of $Y,Y^{\prime}$.

We can now make the first step towards the proof of the theorem, which
consists in comparing the diagonals of $H$ and $H^{\prime}$.

###### Lemma 5.11

If $H$ and $H^{\prime}$ are coorbit equivalent, then $Y=Y^{\prime}$, and thus
$D=D^{\prime}$.

###### Proof 18

Towards a proof by contradiction, assume that $H$ and $H^{\prime}$ are coorbit
equivalent, yet there exists $2\leq i\leq d$ such that
$\lambda_{i}\neq\lambda_{i}^{\prime}$. Let $\phi:H\to H^{\prime}$ denote the
map from Equation (5.3). Let $e_{i}\in\mathbb{R}^{d-1}$ denote the vector with
entry one at the $i-1$st position, and zeros elsewhere. Define the sequence
$(h_{n})_{n\in\mathbb{N}}$ as $h_{n}=h(1,e_{i})^{n}$, and
$h_{n}^{\prime}=\phi(h_{n})$. If $d:H\times H\to\mathbb{R}_{0}^{+}$ denotes a
word metric on $H$, we get by left-invariance that
$d(h_{n},h_{n+1})=d(e_{H},h)$. We need to prove that $\phi$ is not a quasi-
isometry with respect to a suitable word metric $d^{\prime}$ on $H^{\prime}$.
For this it is now sufficient to show that
$(d^{\prime}(h_{n}^{\prime},h_{n+1}^{\prime}))_{n\in\mathbb{N}}\subset\mathbb{R}^{+}$
is unbounded, or equivalently, that the sequence
$(h_{n+1}^{\prime})^{-1}h_{n}^{\prime}\in H^{\prime}$, $n\in\mathbb{N}$, is
not contained in a relatively compact set. This will be shown by concentrating
on suitably chosen matrix entries of these products.

We start out by noting that $h(1,e_{i})=h(1,0)h(0,e_{i})$. For the computation
of higher powers of this group element, we introduce
$T^{\prime}=\\{0\\}^{i}\times\mathbb{R}^{d-i-1}$. Then
$t=t_{i}e_{i}+t^{\prime}$, with $t^{\prime}\in T^{\prime}$, is a vector with
$i-2$ leading zeroes, followed by $t_{i}$, followed by the remaining entries
of $t^{\prime}$.

With this notation, we get

$h(-1,0)h(0,t_{i}e_{i}+t^{\prime})^{-1}h(1,0)=h(0,e^{1-\lambda_{i}}t_{i}e_{i}+t^{\prime\prime})^{-1}~{},$

with suitably chosen $t^{\prime\prime}\in T^{\prime}$ depending linearly on
$t^{\prime}$. To see this, it is sufficient to note that both sides of the
equation are elements of $H$ whose first lines can be made to coincide by
properly choosing $t^{\prime\prime}\in T^{\prime}$, and this first line
uniquely determines the elements of $H$ by Lemma 5.5 and the comments
thereafter. Since conjugation by $h(-1,0)$ is a group isomorphism, we also get

$h(-1,0)h(0,t_{i}e_{i}+t^{\prime})h(1,0)=h(0,e^{1-\lambda_{i}}t_{i}e_{i}+t^{\prime\prime})~{},$
(5.4)

and thus for all $n\in\mathbb{N}$:

$h(n,0)h(0,e_{i})h(-n,0)=h(0,e^{-n(1-\lambda_{i})}e_{i})~{}.$

Furthermore, the fact that
$X_{i}X_{j}\in{\mathrm{span}}\\{X_{k}:k>\max(i,j)\\}$ yields in particular,
for all $p,q\in\mathbb{R}$ and $t_{1}^{\prime},t_{2}^{\prime}\in T^{\prime}$

$h(0,pe_{i}+t_{1}^{\prime})h(0,qe_{i}+t_{2}^{\prime})=h(0,(p+q)e_{i}+t_{3}^{\prime})~{},$

with suitably chosen $t_{3}^{\prime}\in T^{\prime}$. In addition, utilizing
the Neumann series for the inverse of a matrix allows to compute for every
$p\in\mathbb{R}$ and every $t_{1}^{\prime}\in T^{\prime}$ that

$h(0,pe_{i}+t_{1}^{\prime})^{-1}=h(0,-pe_{i}+t_{2}^{\prime})$

with suitably chosen $t_{2}^{\prime}\in T^{\prime}$. We observe that an
analogous formula holds for $H^{\prime}$ as well.

Let us now first consider the case $\lambda_{i}\neq 1$. Then the above
formulae allow to show inductively that

$h^{n}=h(n,0)h\left(0,\frac{e^{n(1-\lambda_{i})}-1}{e^{1-\lambda_{i}}-1}e_{i}+t_{n}^{\prime}\right)~{},$
(5.5)

with suitably chosen $t_{n}^{\prime}\in T^{\prime}$. Plugging this into (5.3)
yields the following fairly explicit formula

$h_{n}^{\prime}=\phi(h_{n})=h^{\prime}(n,\exp(n(\lambda_{i}-\lambda_{i}^{\prime}))\cdot\frac{e^{n(1-\lambda_{i})}-1}{e^{1-\lambda_{i}}-1}e_{i}+s_{n}^{\prime})~{},$

with suitably chosen $s_{n}^{\prime}\in T^{\prime}$. In order to prove that
the sequence of matrices $(h_{n}^{\prime})^{-1}h_{n+1}^{\prime}$ is not
bounded in $H^{\prime}$, we compute

$\displaystyle(h_{n}^{\prime})^{-1}h_{n+1}^{\prime}$ $\displaystyle=$
$\displaystyle
h^{\prime}\left(0,\exp(n(\lambda_{i}-\lambda_{i}^{\prime}))\cdot\frac{e^{n(1-\lambda_{i})}-1}{e^{1-\lambda_{i}}-1}e_{i}+s_{n}^{\prime}\right)^{-1}h^{\prime}(1,0)\cdot$
$\displaystyle
h^{\prime}\left(0,\exp((n+1)(\lambda_{i}-\lambda_{i}^{\prime}))\cdot\frac{e^{(n+1)(1-\lambda_{i})}-1}{e^{1-\lambda_{i}}-1}e_{i}+s_{n+1}^{\prime}\right)$
$\displaystyle=$ $\displaystyle h^{\prime}(1,0)h(0,r_{n}e_{i}+u_{n})$

where $u_{n}\in T^{\prime}$, and

$\displaystyle r_{n}$ $\displaystyle=$
$\displaystyle-e^{1-\lambda_{i}^{\prime}}\exp(n(\lambda_{i}-\lambda_{i}^{\prime}))\cdot\frac{e^{n(1-\lambda_{i})}-1}{e^{1-\lambda_{i}}-1}+\exp((n+1)(\lambda_{i}-\lambda_{i}^{\prime}))\cdot\frac{e^{(n+1)(1-\lambda_{i})}-1}{e^{1-\lambda_{i}}-1}$
$\displaystyle=$
$\displaystyle\frac{\exp(n(\lambda_{i}-\lambda_{i}^{\prime}))}{e^{1-\lambda_{i}}-1}\left(-e^{1-\lambda_{i}^{\prime}+n(1-\lambda_{i})}+e^{1-\lambda_{i}^{\prime}}+e^{\lambda_{i}-\lambda_{i}^{\prime}+(n+1)(1-\lambda_{i})}-e^{\lambda_{i}-\lambda_{i}^{\prime}}\right)$
$\displaystyle=$
$\displaystyle\frac{\exp(n(\lambda_{i}-\lambda_{i}^{\prime}))}{e^{1-\lambda_{i}}-1}\left(e^{1-\lambda_{i}^{\prime}}-e^{\lambda_{i}-\lambda_{i}^{\prime}}\right)~{}.$

Now the assumptions $\lambda_{i}^{\prime}<\lambda_{i}$ and $\lambda_{i}\neq 1$
allow to conclude directly that $|r_{n}|\to\infty$, and thus the desired
conclusion can be drawn. In the case $\lambda_{i}^{\prime}>\lambda_{i}$ we can
conclude similarly, by considering the sequence $h_{n}=h^{-n}$. Hence the case
$\lambda_{i}\neq 1$ is settled. However, in the case $\lambda_{i}=1$, the
assumption $\lambda_{i}^{\prime}\neq\lambda_{i}$ yields that
$\lambda_{i}^{\prime}\neq 1$, and exchanging the roles of $H$, $H^{\prime}$ in
the above argument allows again to conclude that $H$ and $H^{\prime}$ are not
coorbit equivalent.

The next result looks at the restriction of $\phi$ to the shearing subgroup.
Throughout the following proof, we will call a map $\alpha:U\to V$ defined
between different vector spaces $U,V$ polynomial if the coordinates of
$\alpha(X)$ with respect to any fixed basis of $V$ depend polynomially on the
coordinates of $X$, taken with respect to any fixed basis of $U$. Clearly,
this notion is independent of the choice of bases.

###### Lemma 5.12

If $H$ and $H^{\prime}$ are coorbit equivalent, there exists a matrix $C\in
GL(\mathbb{R}^{d})$ satisfying the following two properties:

  1. (i)

The map $\phi$ from Equation (5.3) fulfills $\phi(s)=C^{-1}sC$, for all $s\in
S$.

  2. (ii)

The conjugation actions of $D$ and $C$ commute.

###### Proof 19

For the proof of (i), first note that equation (5.3) implies that $\phi|_{S}$
maps into $S^{\prime}$. We will first show that $\phi|_{S}:S\to S^{\prime}$ is
in fact a group homomorphism.

Let $\mathcal{A}={\mathrm{span}}(\\{I_{d}\\}\cup\mathfrak{s})$ and
$\mathcal{A}^{\prime}={\mathrm{span}}(\\{I_{d}\\}\cup\mathfrak{s}^{\prime})$.
These are the Lie algebras of the abelian shearlet dilation groups
$A,A^{\prime}$ associated to $H,H^{\prime}$. Note that
$p_{\xi_{0}}^{A}=\psi\circ\iota$, where $\iota$ is the inversion map $h\mapsto
h^{-1}$, and $\psi:\mathcal{A}\to\mathbb{R}^{d}$ is the linear bijective map
defined in the proof of Lemma 5.6. In the same way, we obtain
$p_{\xi_{0}}^{A^{\prime}}=\psi^{\prime}\circ\iota$, with analogously defined
linear isomorphism $\psi^{\prime}$.

It follows that $\phi_{A,A^{\prime}}=(p_{\xi_{0}}^{A^{\prime}})^{-1}\circ
p_{\xi_{0}}^{A}=\iota\circ(\psi^{\prime})^{-1}\circ\psi\circ\iota$. The same
then holds for the restriction of this map to $S$. But by the remark
immediately following (5.3), this restriction coincides with $\phi|_{S}:S\to
S^{\prime}$, the restriction of the map $\phi$ associated to $H$ and
$H^{\prime}$.

Now fix $s\in S$, and define the map

$\alpha:\mathfrak{s}\to
H^{\prime}~{},~{}\alpha(X)=\phi((I_{d}+X)s)^{-1}\phi(I_{d}+X)~{}.$

Then $\alpha$ is a polynomial map. To see this, observe that inversion is
polynomial on the set of unipotent matrices, since the inverse of a unipotent
map can be computed by finitely many terms of a Neumann series. Since $s$,
$I_{d}+X$ and $\phi((I_{d}+X)s)$ are all unipotent, this finally yields that
$\alpha$ is indeed a composition of polynomial maps, hence polynomial.

Assuming that $H,H^{\prime}$ are coorbit equivalent, we have that $\phi$ is a
quasi-isometry with respect to suitable word metrics $d_{H},d_{H^{\prime}}$ on
$H,H^{\prime}$ respectively, say with constants $a>0,b\geq 0$. Then left-
invariance of the metrics provides

$\displaystyle d_{H^{\prime}}(\alpha(X),e_{H^{\prime}})$ $\displaystyle=$
$\displaystyle
d_{H^{\prime}}(\phi((I_{d}+X)s)^{-1}\phi(I_{d}+X),e_{H^{\prime}})$
$\displaystyle=$ $\displaystyle
d_{H^{\prime}}(\phi(I_{d}+X),\phi((I_{d}+X)s)\leq
ad_{H}((I_{d}+X)),(I_{d}+X)s)+b$ $\displaystyle\leq$ $\displaystyle
ad_{H}(e_{H},s)+b~{},$

which is independent of $X\in\mathfrak{s}$. Since the metric $d_{H}$ is
proper, it follows that $\alpha:\mathfrak{s}\to\mathbb{R}^{d\times d}$ is a