$\displaystyle=\int_{D}[F(\omega)]^{q}\,d\omega=\sum_{k\in\mathbb{Z}^{d}}\int_{M_{k}}[F(\omega)]^{q}\,d\omega$
$\displaystyle\leq C\cdot\sum_{k\in\mathbb{Z}^{d}}(u_{k}\cdot
c_{k})^{q}=C\cdot\|c\|_{\ell_{u}^{q}}^{q}$ $\displaystyle\leq
C\cdot\|\Gamma\|^{q}\cdot\|f\|_{\mathcal{D}(\mathcal{Q}_{\Phi}^{(\delta,r)},\mathbf{L}^{p},\ell_{u}^{q})}^{q}<\infty.$

This proves the claim in case of $q<\infty$.

For $q=\infty$, the proof is slightly simpler: From Equation (4.17), and
denoting by $\tilde{C}>0$ the implied constant there, we get

$\displaystyle F(\omega)$ $\displaystyle\leq\tilde{C}\cdot
u_{k}\cdot\sum_{\ell\in k^{\ast}}d_{\ell}=\tilde{C}\cdot u_{k}\cdot
d_{k}^{\ast}$
$\displaystyle\leq\tilde{C}\cdot\|(d_{k}^{\ast})_{k\in\mathbb{Z}^{d}}\|_{\ell_{u}^{\infty}}\leq\tilde{C}\cdot\|\Gamma\|\cdot\|d\|_{\ell_{u}^{\infty}}$
$\displaystyle\leq\tilde{C}\cdot\|\Gamma\|\cdot\|f\|_{\mathcal{D}(\mathcal{Q}_{\Phi}^{(\delta,r)},\mathbf{L}^{p},\ell_{u}^{q})}<\infty,$

which proves the claim, since $q=\infty$ implies
$\|V_{\theta,\Phi}f\|_{\mathbf{L}_{\kappa}^{p,q}}=\mathop{\operatorname{ess~{}sup}}_{\omega\in
D}F(\omega)$. ∎

Now, we can finally show our claim that the coorbit space
$\operatorname{Co}(\Phi,\mathbf{L}_{\kappa}^{p,q})$ is canonically isomorphic
to the decomposition space
$\mathcal{D}(\mathcal{Q}_{\Phi}^{(\delta,r)},\mathbf{L}^{p},\ell_{u}^{q})$.

######  Corollary 4.15.

Let $\Phi:D\rightarrow\mathbb{R}^{d}$ and $\kappa:D\rightarrow\mathbb{R}^{+}$
be compatible, see Definition 2.4, and fix $\delta>0$, $r>\sqrt{d}/2$, and
$0<t<r-\sqrt{d}/2$. Finally, choose
$0\neq\theta\in\mathcal{D}(\mathbb{R}^{d})$ such that Equation (4.6) is
satisfied.

For $p,q\in[1,\infty]$, the bounded linear map

$\Psi\colon\operatorname{Co}(\Phi,\mathbf{L}_{\kappa}^{p,q})\to\mathcal{D}(\mathcal{Q}_{\Phi}^{(\delta,r)},\mathbf{L}^{p},\ell_{u}^{q}),\quad
f\mapsto\psi_{f}\quad\text{with}\quad\psi_{f}:Z(D)\to\mathbb{C},\quad g\mapsto
f(\overline{g})=\langle
f,\overline{g}\rangle_{(\mathcal{H}_{v}^{1})^{\urcorner},\mathcal{H}_{v}^{1}}.$

is an isomorphism of Banach spaces. Here, $u=(u_{k})_{k\in\mathbb{Z}^{d}}$,
$u_{k}:=\kappa_{\Phi}(\delta k)\cdot[w(\delta k)]^{\frac{1}{q}-\frac{1}{2}}$
for $k\in\mathbb{Z}^{d}$, as before.

###### Proof.

By Theorem 4.8 and the bounded inverse theorem, it suffices to show that
$\Psi$ is surjective. Thus, let
$f\in\mathcal{D}(\mathcal{Q}_{\Phi}^{(\delta,r)},\mathbf{L}^{p},\ell_{u}^{q})\subset
Z^{\prime}(D)$ be arbitrary, and let $F:=V_{\theta,\Phi}f$. By Lemma 4.14, we
have $F\in\mathbf{L}_{\kappa}^{p,q}$. Furthermore, Lemma 4.12 and Remark 4.13
show that $F=\|\theta\|^{-2}_{\mathbf{L}^{2}(\mathbb{R}^{d})}\cdot
K_{\theta,\Phi}(F)$, where $K_{\theta,\Phi}$ is as in the lemma.

Apply [22, Lemma 2.30] (with the reproducing kernel
$R=\|\theta\|^{-2}_{\mathbf{L}^{2}(\mathbb{R}^{d})}\cdot K_{\theta,\Phi}$) to
see that $F=V_{\theta,\Phi}f_{0}$ for some
$f_{0}\in\operatorname{Co}(\Phi,\mathbf{L}_{\kappa}^{p,q})$. With $f:=\Psi
f_{0}=\psi_{f_{0}}$, Theorem 4.8 shows
$f\in\mathcal{D}(\mathcal{Q}_{\Phi}^{(\delta,r)},\mathbf{L}^{p},\ell_{u}^{q})\subset
Z^{\prime}(D)$. Furthermore, we have

$\displaystyle(V_{\theta,\Phi}f)(y,\omega)$ $\displaystyle=\left\langle
f,\overline{g_{y,\omega}}\right\rangle_{Z^{\prime}(D),Z(D)}$
$\displaystyle=\left\langle\psi_{f_{0}},\overline{g_{y,\omega}}\right\rangle_{Z^{\prime}(D),Z(D)}$
$\displaystyle({\scriptstyle{\text{Def. of }\psi_{f_{0}}}})$
$\displaystyle=\left\langle
f_{0},g_{y,\omega}\right\rangle_{(\mathcal{H}_{v}^{1})^{\urcorner},\mathcal{H}_{v}^{1}}$
$\displaystyle=(V_{\theta,\Phi}f_{0})(y,\omega)=G(y,\omega)=(V_{\theta,\Phi}f)(y,\omega)$

for all $(y,\omega)\in\mathbb{R}^{d}\times D$.

But as shown in Lemma 4.12, this implies $f=\Psi
f_{0}\in\mathrm{image}(\Psi)$, so that $\Psi$ is indeed surjective and the
proof is complete. ∎

At last, we drop our special assumptions regarding $\theta$.

######  Theorem 4.16.

Let $\Phi:D\rightarrow\mathbb{R}^{d}$ and $\kappa:D\rightarrow\mathbb{R}^{+}$
be compatible, see Definition 2.4, and let
$0\neq\theta\in\mathcal{D}(\mathbb{R}^{d})$. Further, fix $\delta>0$ and
$r>\sqrt{d}/2$ and consider the $(\delta,r)$-fine frequency covering
$\mathcal{Q}_{\Phi}^{(\delta,r)}$ induced by $\Phi$, see Definition 1.1.

The map $\Psi$ defined in Corollary 4.3 restricts to an isomorphism of Banach
spaces

$\Psi:\operatorname{Co}(\mathcal{G}(\theta,\Phi),\mathbf{L}_{\kappa}^{p,q})\to\mathcal{D}(\mathcal{Q}_{\Phi}^{(\delta,r)},\mathbf{L}^{p},\ell_{u}^{q})\quad\text{
with }\quad u=(u_{k})_{k\in\mathbb{Z}^{d}}\text{ and
}u_{k}=\kappa(\Phi^{-1}(\delta k))\cdot[w(\delta
k)]^{\frac{1}{q}-\frac{1}{2}}\,\,.$

###### Proof.

By Theorem 2.5, the coorbit space
$\operatorname{Co}(\Phi,\mathbf{L}_{\kappa}^{p,q})=\operatorname{Co}(\mathcal{G}(\theta,\Phi),\mathbf{L}_{\kappa}^{p,q})$
is independent of the choice of $0\neq\theta\in\mathcal{D}(\mathbb{R}^{d})$.
Hence, the result follows immediately from Corollary 4.15. ∎

##  5\. Fundamentals of radial warping

The frequency coverings associated with well-known instances of decomposition
spaces, say Besov or $\alpha$-modulation spaces, exhibit symmetries with
respect to the modulus of the frequency position. The same can be said about
related constructions of frame decompositions. This observation has motivated
the study of _radial warping functions_ in [19]. Since such radially warped
time-frequency systems feature prominently in our embedding results presented
in Sections 6–9. Finally, we obtain conditions for equality of warped coorbit
spaces and embeddings between warped coorbit spaces with respect to radial
warping functions, in Section , we shortly recall the most important results
on radial warping presented in [19].

######  Definition 5.1 ([19, Definition 8.3]).

For a diffeomorphism $\varrho:\mathbb{R}\to\mathbb{R}$, with
$\varrho(\xi)=c\xi$ for all $\xi\in(-\varepsilon,\varepsilon)$ and some
$\varepsilon,c>0$, the associated radial warping function is given by

$\Phi_{\varrho}:\mathbb{R}^{d}\to\mathbb{R}^{d},\xi\mapsto\widetilde{\varrho}(|\xi|)\cdot\xi,\quad\text{with}\quad\widetilde{\varrho}(t):=\varrho(t)/t\quad\text{for
}t\in\mathbb{R}\setminus\\{0\\}\,,\quad\text{and}\quad\widetilde{\varrho}(0):=c\,.$
(5.1)

For a more concise notation and to stay consistent with [19] we denote by
$\varrho_{\ast}:=\varrho^{-1}$ the inverse of a bijection
$\varrho:\mathbb{R}\to\mathbb{R}$. If $\varrho$ satisfies some more structured
assumptions, specified in the following definition, then $\Phi_{\varrho}$ is a
$k$-admissible warping function in the sense of Definition 2.2, see below.

######  Definition 5.2 ([19, Definition 8.1]).

Let $k\in\mathbb{N}_{0}$. A function $\varrho:\mathbb{R}\to\mathbb{R}$ is
called a $k$-admissible radial component with control weight
$v:\mathbb{R}\to(0,\infty)$, if the following hold:

  1. (1)

$\varrho$ is a strictly increasing $\mathcal{C}^{k+1}$-diffeomorphism with
inverse $\varrho_{\ast}=\varrho^{-1}$.

  2. (2)

$\varrho$ is antisymmetric, that is, $\varrho(-\xi)=-\varrho(\xi)$ for all
$\xi\in\mathbb{R}$. In particular, $\varrho(0)=0$.

  3. (3)

There are $\varepsilon>0$ and $c>0$ with $\varrho(\xi)=c\xi$ for all
$\xi\in(-\varepsilon,\varepsilon)$.

  4. (4)

The weight $v$ is continuous, submultiplicative, and radially increasing.
Additionally, $\varrho_{\ast}^{\prime}$ and

$\widetilde{\varrho_{\ast}}:\mathbb{R}\to(0,\infty),\qquad\text{defined
by}\qquad\widetilde{\varrho_{\ast}}(\xi):=\varrho_{\ast}(\xi)/\xi\quad\text{for}\quad\xi\neq
0,\quad\text{and}\quad\widetilde{\varrho_{\ast}}(0):=c^{-1}.$ (5.2)

are $v$-moderate.

  5. (5)

There are constants $C_{0},C_{1}>0$ with

$C_{0}\cdot\widetilde{\varrho_{\ast}}(\xi)\leq\varrho_{\ast}^{\prime}(\xi)\leq
C_{1}\cdot(1+\xi)\cdot\widetilde{\varrho_{\ast}}(\xi)\qquad\forall\,\xi\in(0,\infty).$
(5.3)

  6. (6)

We have $|\varrho_{\ast}^{(\ell)}|\lesssim\varrho_{\ast}^{\prime}$, for all
$\ell\in\underline{k+1}$.

######  Remark 5.3.

Note that here, we use a simpler, equivalent form of Item (6) already
mentioned in [19], after Definition 8.1 therein. An analogous simplification
applies in the case of weakly admissible radial components, see Definition 5.5
below.

######  Theorem 5.4 ([19, Proposition 8.7, Corollary 8.8]).

Let $\varrho:\mathbb{R}\to\mathbb{R}$ be a $k$-admissible radial component,
for some $k\in\mathbb{N}$ with $k\geq d+1$, with control weight
$v:\mathbb{R}\\!\to\\!(0,\infty)$. Then the associated radial warping function
$\Phi_{\varrho}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is a $k$-admissible warping
function. Furthermore, there is a constant $C>0$, such that the control weight
$v_{0}$ for $\Phi_{\varrho}$ can be chosen as

$v_{0}:\mathbb{R}^{d}\to(0,\infty),\tau\mapsto C\cdot(1+|\tau|)\cdot
v(|\tau|),$ (5.4)

and the weight $w=\det(\mathrm{D}\Phi_{\varrho}^{-1})$ is given by

$w(\tau)=\varrho_{\ast}^{\prime}(|\tau|)\cdot[\widetilde{\varrho_{\ast}}(|\tau|)]^{d-1}\,.$
(5.5)

The technical requirement that $\varrho$ be linear in a neighborhood of the
origin (Condition (3) in Definition 5.2) is often unnatural, but for every
function $\varsigma$ that satisfies the following weaker assumption, an
_equivalent_ $k$-admissible radial component $\varrho$ can be constructed, see
also Remark 6.7 for a discussion of said equivalence.

######  Definition 5.5 ([19, Definition 8.12]).

Let $k\in\mathbb{N}_{0}$. A continuous function
$\varsigma:[0,\infty)\to[0,\infty)$ is called a weakly $k$-admissible radial
component with control weight $u:[0,\infty)\to(0,\infty)$, if it satisfies the
following conditions:

  1. (1)

$\varsigma$ is $\mathcal{C}^{k+1}$ on $(0,\infty)$, with
$\varsigma^{\prime}(\xi)>0$ for all $\xi\in(0,\infty)$. In particular,
$\varsigma_{\ast}=\varsigma^{-1}$ exists and is $\mathcal{C}^{k+1}$ on its
domain.

  2. (2)

$\varsigma(0)=0$ and $\varsigma(\xi)\to\infty$ as $\xi\to\infty$.

  3. (3)

The control weight $u$ is continuous and increasing with $u(\xi+\eta)\leq
u(\xi)\cdot u(\eta)$ for all $\xi,\eta\in[0,\infty)$. Furthermore, there are
$\delta>0$ and $C_{0},C_{1}>0$, such that the following properties hold for
all $\xi,\eta\notin B_{\delta}(0)$:

$\displaystyle
C_{0}\cdot\widetilde{\varsigma_{\ast}}(\xi)\leq\varsigma_{\ast}^{\prime}(\xi)\leq
C_{1}\cdot\varsigma_{\ast}(\xi),$ (5.6)
$\displaystyle\widetilde{\varsigma_{\ast}}(\xi)\leq\widetilde{\varsigma_{\ast}}(\eta)\cdot
u(|\xi-\eta|),$ (5.7)
$\displaystyle|\varsigma_{\ast}^{(\ell)}(\xi)|\lesssim\varsigma_{\ast}^{\prime}(\xi)$
$\displaystyle\qquad\forall\,\ell\in\underline{k+1}\,,$ (5.8)

where $\widetilde{\varsigma_{\ast}}(\eta)=\varsigma_{\ast}(\eta)/\eta$, for
all $\eta\in(0,\infty)$, analogous to $\widetilde{\varrho_{\ast}}$ in (5.2).

######  Theorem 5.6 ([19, Proposition 8.14]).

Let $k\in\mathbb{N}_{0}$, and let $\varsigma:[0,\infty)\to[0,\infty)$ be a
weakly $k$-admissible radial component with control weight
$u:[0,\infty)\to(0,\infty)$.

Fix some $\varepsilon>0$, some
$c\in\big{(}0,\varsigma(\varepsilon)/(2\varepsilon)\big{)}$, and an even
function $\Omega\in C_{c}^{\infty}(\mathbb{R})$ that satisfies $\Omega(\xi)=1$
for $x\in B_{\varepsilon}(0)$, $\Omega(\xi)=0$ for $x\not\in
B_{2\varepsilon}(0)$, and $\Omega^{\prime}(\xi)\leq 0$ for $\xi\in[0,\infty)$.
Then the function

$\varrho\colon\mathbb{R}\to\mathbb{R},\
\xi\mapsto\begin{cases}c\xi\cdot\Omega(\xi)+\mathop{\operatorname{sgn}}(\xi)\cdot(1-\Omega(\xi))\cdot\varsigma(|\xi|),&\text{if
}\xi\neq 0\,,\\\ 0,&\text{if }\xi=0\,,\end{cases}$ (5.9)

is a $k$-admissible radial component. There is a a constant $C\geq 1$, such
that

$v\colon\mathbb{R}\to(0,\infty),\ \xi\mapsto C\cdot u(|\xi|)\,$

can be chosen as control weight for $\varrho$.

######  Remark 5.7.

In [19], the radial component $\varrho$ constructed in Theorem 5.6 is referred
to as a slow start version of $\varsigma$. We will adopt this terminology in
the following sections. Although
$\Phi_{\varsigma}:\xi\mapsto\xi/|\xi|\cdot\varsigma(|\xi|)$ is not a warping
function in the sense of Definition 2.1, it is continuous and invertible, such
that we can define the induced $(\delta,r)$-fine frequency coverings
$\mathcal{Q}_{\Phi_{\varsigma}}^{(\delta,r)}:=\big{[}\,Q_{\Phi_{\varsigma},k}^{(\delta,r)}\,\big{]}_{k\in\mathbb{Z}^{d}}$
by $Q_{\Phi_{\varsigma},k}^{(\delta,r)}:=\Phi_{\varsigma}^{-1}(\delta\cdot
B_{r}(k))$, for all $k\in\mathbb{Z}^{d}$, as in Definition 1.1. The coverings
$\mathcal{Q}^{(\delta,r)}_{\Phi_{\varsigma}}$ and
$\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho}}$ coincide up to finitely many
covering elements which have nontrivial intersection with the ball
$B_{2\varepsilon}(0)$. It is thus easy to show that they are weakly
equivalent. Since all the elements of either covering are path-connected, they
are equivalent by Lemma 2.14 and, by Theorem 2.13, generate the same
decomposition spaces (in the sense of equivalent norms). Strictly speaking, it
must be verified that $\mathcal{Q}^{(\delta,r)}_{\Phi_{\varsigma}}$ is indeed
a decomposition covering. This, however, is a straightforward task, which only
requires the modification of the BAPU in Definition 3.7 for the finite number
of elements in $\mathcal{Q}^{(\delta,r)}_{\Phi_{\varsigma}}$ that differ from
those in $\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho}}$.

##  6\. Embedding relation between warped coorbit spaces

We begin by recalling some results from [36], which will be useful for the
verification of embedding relations between pairs of warped coorbit spaces in
this section and between Besov spaces and warped coorbit spaces in Section 7.
In Sections 8 and 9 we will further show that $\alpha$-modulation spaces and
certain spaces of dominating mixed smoothness coincide with certain warped
coorbit spaces. This will be accomplished by verifying the conditions of
Theorem 2.13.

######  Definition 6.1.

Let $\varnothing\neq D\subset\mathbb{R}^{d}$ be open and let
$\mathcal{Q}=(Q_{i})_{i\in I}=(T_{i}Q^{\prime}_{i}+b_{i})_{i\in I}$ and
$\mathcal{P}=(P_{j})_{j\in J}=(S_{j}P^{\prime}_{j}+c_{j})_{j\in J}$ be two
open, tight, semi-structured coverings of $D$, and $v=(v_{j})_{j\in J}$ a
$\mathcal{P}$-moderate weight. We say that $v$ is _relatively
$\mathcal{Q}$-moderate_, if

$\sup_{i\in I}\sup_{k,\ell\in\\{j\in J\,:\,P_{j}\cap
Q_{i}\neq\varnothing\\}}v_{k}/v_{\ell}<\infty,$

and $\mathcal{P}$ is _relatively $\mathcal{Q}$-moderate_, if

$\sup_{i\in I}\sup_{k,\ell\in\\{j\in J\,:\,P_{j}\cap
Q_{i}\neq\varnothing\\}}|\det(S_{k})|/|\det(S_{\ell})|<\infty.$

In the following theorem, we use the convention that the conjugate exponent
$p^{\prime}$ is defined for $p\in(0,1]$ to equal $p^{\prime}=\infty$.

######  Theorem 6.2.

Let $\varnothing\neq D\subset\mathbb{R}^{d}$ be open and let
$\mathcal{Q}=(Q_{i})_{i\in I}=(T_{i}Q^{\prime}_{i}+b_{i})_{i\in I}$ and
$\mathcal{P}=(P_{j})_{j\in J}=(S_{j}P^{\prime}_{j}+c_{j})_{j\in J}$ be two
open, tight, semi-structured coverings of $D$. Furthermore, let
$u=(u_{i})_{i\in I}$ be $\mathcal{Q}$-moderate and $v=(v_{j})_{j\in J}$ be
$\mathcal{P}$-moderate, and let $p_{1},p_{2},q_{1},q_{2}\in[1,\infty]$. Assume
that $\mathcal{P}$ is almost subordinate to $\mathcal{Q}$ and choose for each
$i\in I$ some $j_{i}\in J$ with $Q_{i}\cap P_{j_{i}}\neq\varnothing$. If
$\mathcal{P}$ and $v$ are relatively $\mathcal{Q}$-moderate. Then the
following hold:

  1. (1)

_[36, Theorem 7.2 (4)]_
$\mathcal{D}(\mathcal{Q},\mathbf{L}^{p_{1}},\ell^{q_{1}}_{u})\hookrightarrow\mathcal{D}(\mathcal{P},\mathbf{L}^{p_{2}},\ell^{q_{2}}_{v})$
if and only if

$p_{1}\leq p_{2}\quad\text{and}\quad
K_{t}:=\left\|\left(\frac{v_{j_{i}}}{u_{i}}\cdot|\det T_{i}|^{t}\cdot|\det
S_{j_{i}}|^{p_{1}^{-1}-p_{2}^{-1}-t}\right)_{i\in
I}\right\|_{\ell^{q_{2}\cdot(q_{1}/q_{2})^{\prime}}}<\infty,$

with $t=\max(0,q_{2}^{-1}-\min(p_{1}^{-1},1-p_{1}^{-1}))$.

  2. (2)

_[36, Theorem 7.4 (4)]_
$\mathcal{D}(\mathcal{P},\mathbf{L}^{p_{1}},\ell^{q_{1}}_{v})\hookrightarrow\mathcal{D}(\mathcal{Q},\mathbf{L}^{p_{2}},\ell^{q_{2}}_{u})$
if and only if

$p_{1}\leq p_{2}\quad\text{and}\quad
K_{\tilde{t}}:=\left\|\left(\frac{u_{i}}{v_{j_{i}}}\cdot|\det
T_{i}|^{\tilde{t}}\cdot|\det
S_{j_{i}}|^{p_{1}^{-1}-p_{2}^{-1}-\tilde{t}}\right)_{i\in
I}\right\|_{\ell^{q_{2}\cdot(q_{1}/q_{2})^{\prime}}}<\infty,$

with $\tilde{t}=\max(0,\max(p_{2}^{-1},1-p_{2}^{-1})-q_{1}^{-1})$.

  3. (3)

_[36, Corollary 7.3]_ If $\mathcal{Q}=\mathcal{P}$, then
$\mathcal{D}(\mathcal{Q},\mathbf{L}^{p_{1}},\ell^{q_{1}}_{u})\hookrightarrow\mathcal{D}(\mathcal{Q},\mathbf{L}^{p_{2}},\ell^{q_{2}}_{v})$
if and only if

$p_{1}\leq p_{2}\quad\text{and}\quad
K:=\left\|\left(\frac{v_{i}}{u_{i}}\cdot|\det
T_{i}|^{p_{1}^{-1}-p_{2}^{-1}}\right)_{i\in
I}\right\|_{\ell^{q_{2}\cdot(q_{1}/q_{2})^{\prime}}}<\infty.$

Interpreting warped coorbit spaces as decomposition spaces provides access to
Theorem 6.2 for proving embedding relations between spaces
$\operatorname{Co}(\Phi_{1},\mathbf{L}^{p_{1},q_{1}}_{\kappa_{1}})$ and
$\operatorname{Co}(\Phi_{2},\mathbf{L}^{p_{2},q_{2}}_{\kappa_{2}})$. In
general, for an embedding
$\operatorname{Co}(\Phi_{1},\mathbf{L}^{p_{1},q_{1}}_{\kappa_{1}})\hookrightarrow\operatorname{Co}(\Phi_{2},\mathbf{L}^{p_{2},q_{2}}_{\kappa_{2}})$,
Theorem 6.2 suggests that $p_{1}\leq p_{2}$ is necessary, such that we can
assume $p_{\delta}=p_{1}^{-1}-p_{2}^{-1}\geq 0$.

###  6.1. Embeddings between warped coorbit spaces generated by the same
warping function

For now, assume that $\Phi=\Phi_{1}=\Phi_{2}$ and $(\Phi,\kappa_{1})$ and
$(\Phi,\kappa_{2})$ form compatible pairs as per Definition 2.4. In
particular, $(\kappa_{j})_{\Phi}=\kappa_{j}\circ\Phi^{-1}$, $j\in\\{1,2\\}$,
is $\tilde{\kappa_{j}}$-moderate, where $\tilde{\kappa_{j}}$ is continuous,
submultiplicative and radially increasing. Recall from Lemma 3.2 and
Proposition 3.9 that the induced $(\delta,r)$-fine frequency covering
$\mathcal{Q}^{(\delta,r)}_{\Phi}$, with $\delta>0$ and $r>\sqrt{d}/2$ is tight
and semi-structured. Under these assumptions, Theorem 4.16 yields

$\operatorname{Co}(\Phi,\mathbf{L}^{p_{1},q_{1}}_{\kappa_{1}})=\mathcal{D}(\mathcal{Q}_{\Phi}^{(\delta,r)},\mathbf{L}^{p_{1}},\ell_{u^{(q_{1})}}^{q_{1}})\quad\text{and}\quad\operatorname{Co}(\Phi,\mathbf{L}^{p_{2},q_{2}}_{\kappa_{2}})=\mathcal{D}(\mathcal{Q}_{\Phi}^{(\delta,r)},\mathbf{L}^{p_{2}},\ell_{v^{(q_{2})}}^{q_{2}})$

up to canonical identifications, with the sequence elements of
$u^{(q_{1})},v^{(q_{2})}$ given by

$u^{(q_{1})}_{k}=(\kappa_{1})_{\Phi}(\delta k)\cdot[w(\delta
k)]^{\frac{1}{q_{1}}-\frac{1}{2}}\quad\text{and}\quad
v^{(q_{2})}_{k}=(\kappa_{2})_{\Phi}(\delta k)\cdot[w(\delta
k)]^{\frac{1}{q_{2}}-\frac{1}{2}},\quad k\in\mathbb{Z}^{d}.$

Finally, the linear maps $T_{k}$, $k\in\mathbb{Z}^{d}$, in Definition 2.7 (2)
take the form $T_{k}=A(\delta k)=D\Phi^{-1}(\delta k)$, such that $|\det
T_{k}|=w(\delta k)$, see Lemma 3.2.

Overall, we can apply Theorem 6.2 (3) to show that the relation
$\operatorname{Co}(\Phi,\mathbf{L}^{p_{1},q_{1}}_{\kappa_{1}})\hookrightarrow\operatorname{Co}(\Phi,\mathbf{L}^{p_{2},q_{2}}_{\kappa_{2}})$
holds if and only if

$p_{1}\leq p_{2}\quad\text{and}\quad K:=\left\|\Big{(}[w(\delta
k)]^{p_{\delta}-q_{\delta}}\cdot\frac{(\kappa_{2})_{\Phi}(\delta
k)}{(\kappa_{1})_{\Phi}(\delta
k)}\Big{)}_{k\in\mathbb{Z}^{d}}\right\|_{\ell^{q_{2}\cdot(q_{1}/q_{2})^{\prime}}}<\infty\quad\text{with
}q_{\delta}=q_{1}^{-1}-q_{2}^{-2}.$ (6.1)

If $q_{1}\leq q_{2}$, then $(q_{1}/q_{2})^{\prime}=\infty$ such that the above
simplifies to $(\kappa_{2})_{\Phi}\lesssim
w^{q_{\delta}-p_{\delta}}\cdot(\kappa_{1})_{\Phi}$, which is equivalent to

$\kappa_{2}\lesssim[w\circ\Phi]^{q_{\delta}-p_{\delta}}\cdot\kappa_{1}.$

If even $\kappa_{1}\asymp\kappa_{2}$, we simply obtain $[w(\delta
k)]^{p_{\delta}-q_{\delta}}\lesssim 1$. If on the other hand $q_{1}>q_{2}$,
then (6.1) depends heavily on the integrability of $w^{p_{\delta}-q_{\delta}}$
or the compensation of said term’s non-integrability by means of a weight
$\kappa_{1}$ with sufficient growth. For example, if $\kappa_{2}\equiv 1$,
then it is easy to verify that

$\begin{split}\text{\eqref{eq:SimplifiedWarpedEmbeddingCond} fails
if}\quad\kappa_{1}&\lesssim[w\circ\Phi]^{p_{\delta}-q_{\delta}}\cdot[1+|\Phi(\bullet)|]^{\frac{d}{q_{2}\cdot(q_{1}/q_{2})^{\prime}}},\\\
\text{but \eqref{eq:SimplifiedWarpedEmbeddingCond} holds
if}\quad\kappa_{1}&\gtrsim[w\circ\Phi]^{p_{\delta}-q_{\delta}}\cdot[1+|\Phi(\bullet)|]^{\frac{d+\varepsilon}{q_{2}\cdot(q_{1}/q_{2})^{\prime}}},\text{
for some }\varepsilon>0.\end{split}$

###  6.2. Embeddings between warped coorbit spaces generated by different
warping functions

Here, we only consider the case $p=p_{1}=p_{2}$ and $q=q_{1}=q_{2}$.
Specifically, Theorem 6.4 introduces a condition that yields equality of
warped coorbit spaces, whereas Theorem 6.9 studies embeddings between warped
coorbit spaces in the important special case of radial warping functions. As
preparation, the next lemma is concerned with subordinateness relations
between the $(\delta,r)$-coverings induced by $\Phi_{1}$ and $\Phi_{2}$.

######  Lemma 6.3.

Let $\Phi_{1},\Phi_{2}:D\rightarrow\mathbb{R}^{d}$ be two $0$-admissible
warping functions. If there exists a constant $C>0$, such that

$\|D\Phi_{2}(\Phi_{1}^{-1}(\tau))\cdot D\Phi_{1}^{-1}(\tau)\|\leq
C,\quad\text{for all }\tau\in\mathbb{R}^{d},$ (6.2)

then $\mathcal{Q}^{(\delta_{1},r_{1})}_{\Phi_{1}}$ is almost subordinate to
$\mathcal{Q}^{(\delta_{2},r_{2})}_{\Phi_{2}}$, for all
$\delta_{1},\delta_{2}>0$ and $r_{1},r_{2}>\sqrt{d}/2$.

Given a weight $\kappa:D\rightarrow\mathbb{R}^{+}$ such that
$\kappa\circ\Phi_{2}^{-1}$ is $\tilde{\kappa}$-moderate for some continuous,
submultiplicative, and radially increasing weight
$\tilde{\kappa}\colon\mathbb{R}^{d}\rightarrow[1,\infty)$, we additionally
have

$\kappa(\Phi_{1}^{-1}(\delta_{1}k))\asymp\kappa(\Phi_{2}^{-1}(\delta_{2}\ell)),\quad\text{for
all }k,\ell\in\mathbb{Z}^{d},\text{ with
}Q_{\Phi_{1},k}^{(\delta_{1},r_{1})}\cap
Q_{\Phi_{2},\ell}^{(\delta_{2},r_{2})}\neq\varnothing.$ (6.3)

In particular, if $\Phi_{1},\Phi_{2}$ are
$\mathcal{C}^{\infty}$-diffeomorphisms and $(d+1)$-admissible warping
functions, then compatibility of $\Phi_{2}$ and $\kappa$ in the sense of
Definition 2.4 implies that $\Phi_{1}$ and $\kappa$ are compatible as well.

###### Proof.

Our goal is to show that $\mathcal{Q}^{(\delta_{1},r_{1})}_{\Phi_{1}}$ is
almost subordinate to $\mathcal{Q}^{(\delta_{2},r_{2})}_{\Phi_{2}}$, for all
$\delta_{1},\delta_{2}>0$ and $r_{1},r_{2}>\sqrt{d}/2$. To this end, we begin
by showing that, for a given $\delta>0$ and $r>\sqrt{d}/{2}$, and provided
that (6.2) holds, there exists $R>\sqrt{d}/2$, such that
$\mathcal{Q}_{\Phi_{1}}^{(\delta,r)}$ is subordinate, and thereby almost
subordinate, to $\mathcal{Q}_{\Phi_{2}}^{(\delta,R)}$. Since almost
subordinateness is a transitive relation, and equivalent covers are almost
subordinate to one another, Lemma 3.2, which showed the equivalence of all
$\Phi$-induced $(\delta,r)$-fine coverings, can be applied to $\Phi_{1}$ and
to $\Phi_{2}$ in order to obtain the result for arbitrary
$\delta_{1},\delta_{2}>0$ and $r_{1},r_{2}>\sqrt{d}/2$.

Fix any $\delta>0$, $r>\sqrt{d}/2$. Since
$\mathcal{Q}_{\Phi_{2}}^{(\delta,r)}$ is a covering of $D$ by Lemma 3.2, there
exists, for every $k\in\mathbb{Z}^{d}$, a $j=j(k)\in\mathbb{Z}^{d}$, such that
$\Phi^{-1}_{1}(\delta k)\in\Phi^{-1}_{2}(\delta\cdot B_{r}(j))$, or
equivalently

$\Phi_{2}(\Phi^{-1}_{1}(\delta k))\in B_{\delta r}(\delta j).$ (6.4)

By the chain rule, (6.2) states that

$\|D(\Phi_{2}\circ\Phi_{1}^{-1})(\tau)\|=\|D\Phi_{2}(\Phi_{1}^{-1}(\tau))\cdot
D\Phi_{1}^{-1}(\tau)\|\leq C,$

for all $\tau\in\mathbb{R}^{d}$, i.e.,
$\Phi_{2}\circ\Phi_{1}^{-1}\colon\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ is
Lipschitz-continuous with Lipschitz constant no larger than $C$. If $\xi\in
Q^{(\delta,r)}_{\Phi_{1},k}$, then $\tau:=\Phi_{1}(\xi)\in B_{\delta r}(\delta
k)$, and thus

$\left|\Phi_{2}(\Phi^{-1}_{1}(\tau))-\Phi_{2}(\Phi^{-1}_{1}(\delta
k))\right|<C\delta r.$

Together with (6.4) and $R:=r+C\delta r/\delta=(C+1)r$, we obtain
$\Phi_{2}(\xi)=\Phi_{2}(\Phi^{-1}_{1}(\tau))\in\delta B_{R}(j)$ and thus
$\tau\in Q^{(\delta,R)}_{\Phi_{2},j}$. Since the choice of $\xi\in
Q^{(\delta,r)}_{\Phi_{1},k}$ was arbitrary, this implies

$Q^{(\delta,r)}_{\Phi_{1},k}\subset Q^{(\delta,R)}_{\Phi_{2},j}.$

Since further $R=(C+1)r$ is independent of the choice of $k\in\mathbb{Z}^{d}$,
$\mathcal{Q}_{\Phi_{1}}^{(\delta,r)}$ is subordinate to
$\mathcal{Q}_{\Phi_{2}}^{(\delta,R)}$, proving the first part of the lemma, as
discussed in the first paragraph of the proof.

To prove the second part, assume that $k,\ell\in\mathbb{Z}^{d}$ are such that
$Q_{\Phi_{1},k}^{(\delta_{1},r_{1})}\cap
Q_{\Phi_{2},\ell}^{(\delta_{2},r_{2})}\neq\varnothing$ and let $\xi\in
Q_{\Phi_{1},k}^{(\delta_{1},r_{1})}\cap
Q_{\Phi_{2},\ell}^{(\delta_{2},r_{2})}$. This implies

$|\Phi_{1}(\xi)-\delta_{1}k|<\delta_{1}r_{1}\quad\text{and}\quad|\Phi_{2}(\xi)-\delta_{2}\ell|<\delta_{2}r_{2}.$

Using the Lipschitz-continuity of $\Phi_{2}\circ\Phi_{1}^{-1}$, see above, and
the triangle inequality, we conclude that

$|\Phi_{2}(\xi)-\Phi_{2}(\Phi_{1}^{-1}(\delta_{1}k))|<C\delta_{1}r_{1}\quad\text{and}\quad|\delta_{2}\ell-\Phi_{2}(\Phi_{1}^{-1}(\delta_{1}k))|<\delta_{2}r_{2}+C\delta_{1}r_{1}.$

With this estimate and the $\tilde{\kappa}$-moderateness of
$\kappa\circ\Phi_{2}$, it follows that

$\frac{\kappa(\Phi^{-1}_{1}(\delta_{1}k))}{\kappa(\Phi^{-1}_{2}(\delta_{2}\ell))}=\frac{(\kappa\circ\Phi^{-1}_{2})((\Phi_{2}\circ\Phi^{-1}_{1})(\delta_{1}k))}{(\kappa\circ\Phi^{-1}_{2})(\delta_{2}\ell)}\leq\tilde{\kappa}((\delta_{2}r_{2}+C\delta_{1}r_{1})\cdot
e_{1}),$

and the lower bound
$\tfrac{\kappa(\Phi^{-1}_{1}(\delta_{1}k))}{\kappa(\Phi^{-1}_{2}(\delta_{2}k))}\geq[\tilde{\kappa}((\delta_{2}r_{2}+C\delta_{1}r_{1})\cdot
e_{1}]^{-1}$ is obtained analogously. Here, we used that $\tilde{\kappa}$ is
radially increasing.

The statement about compatibility of $\Phi_{1},\ \Phi_{2}$ and $\kappa$ now
follows by noting that the assumptions on $\kappa$ are exactly those in
Definition 2.4, whereas the calculation

$\frac{\kappa(\Phi^{-1}_{1}(\tau))}{\kappa(\Phi^{-1}_{1}(\sigma))}=\frac{\kappa(\Phi^{-1}_{2}(\Phi_{2}\circ\Phi^{-1}_{1}(\tau)))}{\kappa(\Phi^{-1}_{2}(\Phi_{2}\circ\Phi^{-1}_{1}(\sigma)))}\leq\tilde{\kappa}(|\Phi_{2}\circ\Phi^{-1}_{1}(\tau)-\Phi_{2}\circ\Phi^{-1}_{1}(\sigma)|\cdot
e_{1})\leq\tilde{\kappa}(C\cdot|\tau-\sigma|\cdot
e_{1})=\tilde{\kappa}(C\cdot(\tau-\sigma))$

shows that $\kappa\circ\Phi^{-1}_{1}$ is
$(\tilde{\kappa}(C\bullet))$-moderate, i.e., $\Phi_{1}$ and $\kappa$ satisfy
the conditions of Definition 2.4 as well. ∎

If (6.2) additionally holds with interchanged roles of $\Phi_{1}$ and
$\Phi_{2}$, then we obtain equality of warped coorbit spaces.

######  Theorem 6.4.

Let $\Phi_{1},\Phi_{2}\colon D\rightarrow\mathbb{R}^{d}$ and
$\kappa:D\rightarrow\mathbb{R}^{+}$ be such that $(\Phi_{1},\kappa)$ and
$(\Phi_{2},\kappa)$ form compatible pairs as per Definition 2.4, and suppose
that there is a constant $C>0$, such that

$\|D\Phi_{3-j}(\Phi_{j}^{-1}(\tau))\cdot D\Phi_{j}^{-1}(\tau)\|\leq
C,\quad\text{for all }\tau\in\mathbb{R}^{d}\text{ and }j\in\\{1,2\\}.$ (6.5)

Then we have, for all $p,q\in[1,\infty]$, the equality

$\operatorname{Co}(\Phi_{1},\mathbf{L}_{\kappa}^{p,q})=\operatorname{Co}(\Phi_{2},\mathbf{L}_{\kappa}^{p,q}).$
(6.6)

###### Proof.

By Theorem 4.16 we have, for arbitrary $\delta>0$, $r>\sqrt{d}/2$ and
$j\in\\{1,2\\}$, the equalities

$\operatorname{Co}(\Phi_{j},\mathbf{L}_{\kappa}^{p,q})=\mathcal{D}(\mathcal{Q}^{(\delta,r)}_{\Phi_{j}},\mathbf{L}^{p},\ell^{q}_{u_{j}}),\quad\text{with
}(u_{j})_{\ell}=\kappa(\Phi_{j}^{-1}(\delta\ell))\cdot[w_{j}(\delta\ell)]^{1/q-1/2},\
\ell\in\mathbb{Z}^{d}.$

Since (6.5) holds, we can apply Lemma 6.3 as is and with interchanged roles of
$\Phi_{1}$ and $\Phi_{2}$, to obtain equivalence of the coverings
$\mathcal{Q}^{(\delta,r)}_{\Phi_{1}}$ and
$\mathcal{Q}^{(\delta,r)}_{\Phi_{2}}$. By Theorem 2.13, we obtain the desired
equality (6.6), provided that $u_{1}\asymp u_{2}$. Lemma 6.3 further implies
that

$\kappa(\Phi_{1}^{-1}(\delta
k))\asymp\kappa(\Phi_{2}^{-1}(\delta\ell)),\quad\text{for all
}k,\ell\in\mathbb{Z}^{d},\ \text{with }Q^{(\delta,r)}_{\Phi_{1},k}\cap
Q^{(\delta,r)}_{\Phi_{2},\ell}\neq\varnothing.$

It remains to show that, likewise,

$w_{1}(\delta k)\asymp w_{2}(\delta\ell),\quad\text{for all
}k,\ell\in\mathbb{Z}^{d},\ \text{with }Q^{(\delta,r)}_{\Phi_{1},k}\cap
Q^{(\delta,r)}_{\Phi_{2},\ell}\neq\varnothing.$

To do so, first note that, for any $\delta>0$, $r>\sqrt{d}/2$ and
$j\in\\{1,2\\}$, we can estimate the Lebesgue measure $\mu$ of
$Q^{(\delta,r)}_{\Phi_{j},k}$ a follows:

$\mu(Q^{(\delta,r)}_{\Phi_{j},k})=\int_{D}{\mathds{1}}_{Q^{(\delta,r)}_{\Phi_{j},k}}(\xi)~{}d\xi=\int_{\mathbb{R}^{d}}|\det(D\Phi_{j}^{-1})(\tau)|\cdot{\mathds{1}}_{Q^{(\delta,r)}_{\Phi_{j},k}}(\Phi_{j}^{-1}(\tau))~{}d\tau=\int_{\delta\cdot
B_{r}(k)}w_{j}(\tau)~{}d\tau\asymp w_{j}(\delta k),$ (6.7)

for all $k\in\mathbb{Z}^{d}$. Here, we used that $w_{j}$ is $v_{j}$-moderate,
with respect to the control weight $v_{j}$ for the $(d+1)$-admissible warping
function $\Phi_{j}$, $j\in\\{1,2\\}$, such that the implied constants in (6.7)
depend on $\delta$, $r$, $d$, and $v_{j}$. Now let $R=(C+1)r$ be as in the
proof of Lemma 6.3 and $R_{2}=(C+1)R$, with $C>0$ as in (6.5). Then, as in the
proof of said Lemma, $Q^{(\delta,r)}_{\Phi_{1},k}\cap
Q^{(\delta,r)}_{\Phi_{2},\ell}\neq\varnothing$ implies
$Q^{(\delta,r)}_{\Phi_{1},k}\subset Q^{(\delta,R)}_{\Phi_{2},\ell}$ and,
likewise, $Q^{(\delta,R)}_{\Phi_{2},\ell}\subset
Q^{(\delta,R_{2})}_{\Phi_{1},k}$. By an application of (6.7) with $j=1$, we
thus find that

$w_{1}(\delta
k)\lesssim\mu(Q^{(\delta,r)}_{\Phi_{1},k})\leq\mu(Q^{(\delta,R)}_{\Phi_{2},\ell})\leq\mu(Q^{(\delta,R_{2})}_{\Phi_{1},k})\lesssim
w_{1}(\delta k).$

Applying (6.7) again with $j=2$, we obtain $w_{1}(\delta k)\asymp
w_{2}(\delta\ell)$ as desired. Therefore, we can apply Theorem 4.16 to show
the desired equality (6.6) of warped coorbit spaces. ∎

As we now show, the assumptions of both results above simplify greatly in the
case of radial warping functions $\Phi_{1}=\Phi_{\varrho_{1}}$,
$\Phi_{2}=\Phi_{\varrho_{2}}$ generated from the the radial components
$\varrho_{1},\varrho_{2}$.

######  Corollary 6.5.

Let $\varrho_{1},\varrho_{2}:\mathbb{R}\to\mathbb{R}$ be $0$-admissible radial
components, such that there is a constant $C>0$ with

$(\varrho_{2})^{\prime}(\xi)\leq C(\varrho_{1})^{\prime}(\xi),\quad\text{for
all }\xi\in\mathbb{R}^{+},$ (6.8)

then $\Phi_{\varrho_{1}},\Phi_{\varrho_{2}}$ satisfy the conditions of Lemma
6.3. In particular, the covering
$\mathcal{Q}^{(\delta_{1},r_{1})}_{\Phi_{\varrho_{1}}}$ is almost subordinate
to $\mathcal{Q}^{(\delta_{2},r_{2})}_{\Phi_{\varrho_{2}}}$, for every
$\delta_{1},\delta_{2}>0$ and $r_{1},r_{2}>\sqrt{d}/2$.

###### Proof.

By [19, Lemma 8.4], we have for
$\varrho\in\\{\varrho_{1},\varrho_{2},(\varrho_{1})_{\ast},(\varrho_{2})_{\ast}\\}$
and $\xi\in\mathbb{R}^{d}\setminus\\{0\\}$ that

$D\Phi_{\varrho}(\xi)=\widetilde{\varrho}(|\xi|)\cdot\pi_{\xi}^{\perp}+\varrho^{\prime}(|\xi|)\cdot\pi_{\xi},$

where $\pi_{\xi}$ is the orthogonal projection onto the span of $\xi$, and
$\pi_{\xi}^{\perp}=\operatorname{id}_{\mathbb{R}^{d}}-\pi_{\xi}$. It is easy
to see that

$(a\pi_{\xi}^{\perp}+b\pi_{\xi})\cdot(c\pi_{\xi}^{\perp}+d\pi_{\xi})=ac\pi_{\xi}^{\perp}+bd\pi_{\xi}\quad\text{and}\quad\|a\pi_{\xi}^{\perp}+b\pi_{\xi}\|=\max\\{a,b\\},$

for all $a,b,c,d\in\mathbb{R}^{+}$. Using
$\Phi_{\varrho_{1}}^{-1}=\Phi_{{\varrho_{1}}_{\ast}}$, we obtain for
$\tau\in\mathbb{R}^{d}\setminus\\{0\\}$

$\|D\Phi_{\varrho_{2}}(\Phi_{\varrho_{1}}^{-1}(\tau))\cdot
D\Phi_{\varrho_{1}}^{-1}(\tau)\|\leq\max\\{\
(\varrho_{1})_{\ast}^{\prime}(|\tau|)\cdot(\varrho_{2})^{\prime}((\varrho_{1})_{\ast}(|\tau|)),\
\widetilde{(\varrho_{1})_{\ast}}(|\tau|)\cdot\widetilde{\varrho_{2}}((\varrho_{1})_{\ast}(|\tau|))\
\\}.$

By assumption,
$(\varrho_{1})_{\ast}^{\prime}(|\tau|)\cdot(\varrho_{2})^{\prime}((\varrho_{1})_{\ast}(|\tau|))\leq
C(\varrho_{1})_{\ast}^{\prime}(|\tau|)\cdot(\varrho_{1})^{\prime}((\varrho_{1})_{\ast}(|\tau|))=C$.
Further, since $\varrho_{1}(0)=\varrho_{2}(0)=0$, Equation (6.8) implies
$\varrho_{2}\leq C\varrho_{1}$ on $\mathbb{R}^{+}$, and thus

$\widetilde{(\varrho_{1})_{\ast}}(|\tau|)\cdot\widetilde{\varrho_{2}}((\varrho_{1})_{\ast}(|\tau|))=\widetilde{(\varrho_{1})_{\ast}}(|\tau|)\cdot\frac{\varrho_{2}((\varrho_{1})_{\ast}(|\tau|))}{(\varrho_{1})_{\ast}(|\tau|)}\leq
C\widetilde{(\varrho_{1})_{\ast}}(|\tau|)\cdot\frac{\varrho_{1}((\varrho_{1})_{\ast}(|\tau|))}{(\varrho_{1})_{\ast}(|\tau|)}=C.$

By continuity, the estimate
$\|D\Phi_{\varrho_{2}}(\Phi_{\varrho_{1}}^{-1}(\tau))\cdot
D\Phi_{\varrho_{1}}^{-1}(\tau)\|\leq C$ extends from
$\tau\in\mathbb{R}^{d}\setminus\\{0\\}$ to all $\tau\in\mathbb{R}^{d}$. This
finishes the proof. ∎

######  Corollary 6.6.

Let $\varrho_{1},\varrho_{2}:\mathbb{R}\to\mathbb{R}$ and
$\kappa:\mathbb{R}^{d}\rightarrow\mathbb{R}^{+}$ be such that
$(\Phi_{\varrho_{1}},\kappa)$ and $(\Phi_{\varrho_{2}},\kappa)$ form
compatible pairs as per Definition 2.4. In particular, assume that
$\varrho_{1},\varrho_{2}$ are $(d+1)$-admissible radial components. If
$\varrho_{1}^{\prime}\asymp\varrho_{2}^{\prime}$ and $p,q\in[1,\infty]$, then
we have the equality

$\operatorname{Co}(\Phi_{\varrho_{1}},\mathbf{L}_{\kappa}^{p,q})=\operatorname{Co}(\Phi_{\varrho_{2}},\mathbf{L}_{\kappa}^{p,q}).$
(6.9)

###### Proof.

Use Corollary 6.5 twice to verify (6.5) in the assumptions of Theorem 6.4,
setting $\Phi_{1}=\Phi_{\varrho_{1}}$ and $\Phi_{2}=\Phi_{\varrho_{2}}$. Then
said theorem yields (6.9). ∎

######  Remark 6.7.

As a consequence of Corollary 6.6, we immediately obtain that different slow
start versions $\varrho_{1},\ \varrho_{2}$ of the same weakly $k$-admissible
radial component $\varsigma$ generate the same coorbit spaces.

If $\varrho_{1}^{\prime}$ and $\varrho_{2}^{\prime}$ are not proportional,
i.e., (6.8) _does not hold_ with the roles of $\varrho_{1}$ and $\varrho_{2}$
interchanged, then we require additional restrictions on $\varrho_{1}$ to
prove embeddings between the coorbit spaces associated with the radial warping
functions $\Phi_{\varrho_{1}}$ and $\Phi_{\varrho_{2}}$. The next result,
concerned with relative moderateness of the induced $(\delta,r)$-frequency
coverings, is the final step towards such embedding relations.

######  Proposition 6.8.

Let $\varrho_{1},\varrho_{2}:\mathbb{R}\to\mathbb{R}$ be $0$-admissible radial
components and $\mathcal{Q}_{\Phi_{\varrho_{1}}}^{(\delta,r)}$,
$\mathcal{Q}_{\Phi_{\varrho_{2}}}^{(\delta,r)}$ the respective induced
$(\delta,r)$-fine frequency coverings, for arbitrary $\delta>0$ and
$r>\sqrt{d}/2$. Then

$1+|\xi|\asymp
1+(\varrho_{j})_{\ast}(\delta|k|)\quad\forall\,j\in\\{1,2\\},\,k\in\mathbb{Z}^{d},\,\text{
and }\xi\in Q_{\Phi_{\varrho_{j}},k}^{(\delta,r)}\,.$ (6.10)

If, additionally, there exists a function
$C_{\varrho_{1}}:[1,\infty)\rightarrow[1,\infty)$, such that

$C_{\varrho_{1}}(\alpha)^{-1}\varrho_{1}^{\prime}(\xi)\leq\varrho_{1}^{\prime}(\alpha\xi)\leq
C_{\varrho_{1}}(\alpha)\varrho_{1}^{\prime}(\xi),\quad\text{for all
}\alpha\in[1,\infty),\ \xi\in\mathbb{R}^{+},$ (6.11)

then $\mathcal{Q}_{\Phi_{\varrho_{1}}}^{(\delta,r)}$ is relatively
$\mathcal{Q}_{\Phi_{\varrho_{2}}}^{(\delta,r)}$-moderate.

Specifically, if (6.11) holds, we have

$\widetilde{(\varrho_{1})_{\ast}}(\delta|k|)\asymp[\widetilde{\varrho_{1}}((\varrho_{2})_{\ast}(\delta|\ell|))]^{-1}\quad\forall\,k,\ell\in\mathbb{Z}^{d}\text{
with }Q_{\Phi_{\varrho_{1}},k}^{(\delta,r)}\cap
Q_{\Phi_{\varrho_{2}},\ell}^{(\delta,r)}\neq\varnothing\,,$ (6.12)

and

$w_{1}(\delta k)=\det(D\Phi_{\varrho_{1}}^{-1}(\delta
k))\asymp\frac{[\widetilde{\varrho_{1}}((\varrho_{2})_{\ast}(\delta|\ell|))]^{1-d}}{\varrho_{1}^{\prime}((\varrho_{2})_{\ast}(\delta|\ell|))}\quad\text{if}\quad
Q_{\Phi_{\varrho_{1}},k}^{(\delta,r)}\cap
Q_{\Phi_{\varrho_{2}},\ell}^{(\delta,r)}\neq\varnothing\,.$ (6.13)

In particular, $w_{1}$ is relatively
$\mathcal{Q}_{\Phi_{\varrho_{2}}}^{(\delta,r)}$-moderate.

###### Proof.

We first prove (6.10). Fix $j\in\\{1,2\\}$ and let
$v_{j}:\mathbb{R}\to\mathbb{R}^{+}$ be a control weight for $\varrho_{j}$. For
$\xi\in
Q_{\Phi_{\varrho_{j}},k}^{(\delta,r)}=\Phi_{\varrho_{j}}^{-1}\big{(}\delta\cdot
B_{r}(k)\big{)}$, we have
${\varrho_{j}}(|\xi|)=|\Phi_{\varrho_{j}}(\xi)|\in\delta|k|+[-\delta r,\delta
r]$. Since ${\varrho_{j}}$ is strictly increasing, this implies

$({\varrho_{j}})_{\ast}(\delta|k|-\delta
r)\leq|\xi|\leq({\varrho_{j}})_{\ast}(\delta|k|+\delta r)\,.$ (6.14)

Now, let us first consider the case $|k|\geq 2r+2$, so that $|k|\asymp|k|-r>0$
and $|k|+r\asymp|k|$. Since by definition of a $0$-admissible radial
component, $\widetilde{({\varrho_{j}})_{\ast}}$ is $v$-moderate, we see

$|\xi|\leq({\varrho_{j}})_{\ast}(\delta|k|+\delta
r)=\widetilde{({\varrho_{j}})_{\ast}}(\delta|k|+\delta
r)\cdot(\delta|k|+\delta r)\lesssim v_{j}(\delta
r)\cdot\widetilde{({\varrho_{j}})_{\ast}}(\delta|k|)\cdot\delta|k|=v_{j}(\delta
r)\cdot(\varrho_{j})_{\ast}(\delta|k|)\,,$

for $\xi\in Q_{\Phi_{\varrho_{j}},k}^{(\delta,r)}$. The lower bound
$[v_{j}(\delta r)]^{-1}({\varrho_{j}})_{\ast}(\delta|k|)\lesssim|\xi|$ is
derived similarly, such that overall
$|\xi|\asymp({\varrho_{j}})_{\ast}(\delta|k|)$, for $\xi\in
Q_{\Phi_{\varrho_{j}},k}^{(\delta,r)}$, and thus, whenever $|k|\geq 2r+2$, we
have $1+|\xi|\asymp 1+({\varrho_{j}})_{\ast}(\delta|k|)$.

To prove (6.10), it remains to consider the case $|k|\leq 2r+2$, for which
Equation (6.14) yields

$|\xi|\leq(\varrho_{j})_{\ast}(\delta|k|+\delta
r)\leq(\varrho_{j})_{\ast}(3\delta r+2\delta)\lesssim
1\quad\text{and}\quad(\varrho_{j})_{\ast}(\delta|k|)\leq(\varrho_{j})_{\ast}(2\delta
r+2\delta)\lesssim 1.$

Together, $1+|\xi|\asymp 1\asymp 1+(\varrho_{j})_{\ast}(\delta|k|)$ if $\xi\in
Q_{\Phi_{\varrho},k}^{(\delta,r)}$ and $|k|\leq 2r+2$, as desired.

To prove Equation (6.12), first note that $\varnothing\neq
Q_{\Phi_{\varrho_{1}},k}^{(\delta,r)}\cap
Q_{\Phi_{\varrho_{2}},\ell}^{(\delta,r)}\ni\xi$ implies, by (6.10), that
$1+({\varrho_{1}})_{\ast}(\delta|k|)\asymp 1+|\xi|\asymp
1+({\varrho_{2}})_{\ast}(\delta|\ell|)$, i.e., there exists $C_{0}\geq 1$,
such that

$C_{0}^{-1}\cdot(1+({\varrho_{2}})_{\ast}(\delta|\ell|))\leq
1+({\varrho_{1}})_{\ast}(\delta|k|)\leq
C_{0}\cdot(1+({\varrho_{2}})_{\ast}(\delta|\ell|)).$

Moreover, since $\varrho_{1}(0)=0$, (6.11) implies

$\alpha
C_{\varrho}(\alpha)^{-1}\varrho_{1}(\xi)\leq\varrho_{1}(\alpha\xi)\leq\alpha
C_{\varrho}(\alpha)\varrho_{1}(\xi),\quad\text{for all }\alpha\in[1,\infty),\
\xi\in\mathbb{R}^{+}.$ (6.15)

If $k=0$ or $\ell=0$, then Equations (6.12) and (6.13) are trivial: Both sides
are positive and there are only finitely many $k\in\mathbb{Z}^{d}$ with
$Q_{\Phi_{\varrho_{1}},k}^{(\delta,r)}\cap
Q_{\Phi_{\varrho_{2}},0}^{(\delta,r)}\neq\varnothing$. Likewise, there are
only finitely many $\ell\in\mathbb{Z}^{d}$ with
$Q_{\Phi_{\varrho_{1}},0}^{(\delta,r)}\cap
Q_{\Phi_{\varrho_{2}},\ell}^{(\delta,r)}\neq\varnothing$. Hence, we can assume
that $k\neq 0\neq\ell$. Then, (6.10) implies
$({\varrho_{1}})_{\ast}(\delta|k|)\asymp({\varrho_{2}})_{\ast}(\delta|\ell|)$,
and we obtain

$[\widetilde{\varrho_{1}}((\varrho_{2})_{\ast}(\delta|\ell|))]^{-1}=\frac{(\varrho_{2})_{\ast}(\delta|\ell|)}{\varrho_{1}((\varrho_{2})_{\ast}(\delta|\ell|))}\asymp\frac{(\varrho_{1})_{\ast}(\delta|k|)}{\delta|k|}=\widetilde{(\varrho_{1})_{\ast}}(\delta|k|),$

where we used (6.15) and
$(\varrho_{1})_{\ast}(\delta|k|)\asymp(\varrho_{2})_{\ast}(\delta|\ell|)$ to
deduce
$\varrho_{1}((\varrho_{2})_{\ast}(\delta|\ell|))\asymp\varrho_{1}((\varrho_{1})_{\ast}(\delta|k|))=\delta|k|$.

To prove the last part of the proposition, recall from (5.5) that
$\det(D\Phi_{\varrho_{1}}^{-1}(\delta
k))=(\varrho_{1})_{\ast}^{\prime}(\delta|k|)\cdot[\widetilde{(\varrho_{1})_{\ast}}(\delta|k|)]^{d-1}$
and recall that it suffices to consider the case $k\neq 0\neq l$. Because of
$(\varrho_{1})_{\ast}(\delta|k|)\asymp(\varrho_{2})_{\ast}(\delta|\ell|)$ we
get

$(\varrho_{1})_{\ast}^{\prime}(\delta|k|)=[\varrho_{1}^{\prime}((\varrho_{1})_{\ast}(\delta|k|))]^{-1}\asymp[\varrho_{1}^{\prime}((\varrho_{2})_{\ast}(\delta|\ell|))]^{-1},$

by an application of (6.11). Estimating the remaining term
$[\widetilde{(\varrho_{1})_{\ast}}(\delta|k|)]^{d-1}$ with (6.12) yields the
desired estimate (6.13).

With (6.13), we can derive that
$\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{1}}}$ is relatively
$\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{2}}}$-moderate. To see this, recall
from Lemma 3.2 that the elements $Q^{(\delta,r)}_{\Phi_{\varrho_{1}},k}$ of
the semi-structured coverning $\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{1}}}$
can be written as
$Q^{(\delta,r)}_{\Phi_{\varrho_{1}},k}=T_{k}Q_{k}^{\prime}+b_{k}$, with
$T_{k}=D\Phi_{\varrho_{1}}^{-1}(\delta k)$, and hence
$\det(T_{k})=w_{1}(\delta k)$, such that (6.13) yields relative
$\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{2}}}$-moderateness. ∎

We are now ready to prove embeddings
$\operatorname{Co}(\Phi_{\varrho_{1}},\mathbf{L}_{\kappa_{1}}^{p,q})\hookrightarrow\operatorname{Co}(\Phi_{\varrho_{2}},\mathbf{L}_{\kappa_{2}}^{p,q})$,
for appropriate weights $\kappa_{1},\kappa_{2}$.

######  Theorem 6.9.

Let $\varrho_{1},\varrho_{2}:\mathbb{R}\to\mathbb{R}$ and
$\kappa:\mathbb{R}^{d}\rightarrow\mathbb{R}^{+}$ be such that
$(\Phi_{\varrho_{1}},\kappa)$ and $(\Phi_{\varrho_{2}},\kappa)$ form
compatible pairs as per Definition 2.4. In particular, assume that
$\varrho_{1},\varrho_{2}$ are $(d+1)$-admissible radial components. If there
is a constant $C>0$, and a function
$C_{\varrho_{1}}:[1,\infty)\rightarrow[1,\infty)$, such that

$(\varrho_{2})^{\prime}(\xi)\leq
C(\varrho_{1})^{\prime}(\xi),\quad\text{and}\quad
C_{\varrho_{1}}(\alpha)^{-1}\varrho_{1}^{\prime}(\xi)\leq\varrho_{1}^{\prime}(\alpha\xi)\leq
C_{\varrho_{1}}(\alpha)\varrho_{1}^{\prime}(\xi),\quad\text{for all
}\alpha\in[1,\infty),\ \xi\in\mathbb{R}^{+},$

then, for all $p,q\in[1,\infty]$, we have the embeddings

$\operatorname{Co}(\Phi_{\varrho_{1}},\mathbf{L}_{\kappa_{\varrho_{1},\varrho_{2},-\tilde{t}}}^{p,q})\hookrightarrow\operatorname{Co}(\Phi_{\varrho_{2}},\mathbf{L}_{\kappa}^{p,q})\hookrightarrow\operatorname{Co}(\Phi_{\varrho_{1}},\mathbf{L}_{\kappa_{\varrho_{1},\varrho_{2},t}}^{p,q}),\
$ (6.16)

with $\tilde{t}=\max(0,\max(p^{-1},1-p^{-1})-q^{-1})\leq 1$, and
$t=\max(0,q^{-1}-\min(p^{-1},1-p^{-1}))\leq 1$. Here, for $t\in\mathbb{R}$,

$\kappa_{\varrho_{1},\varrho_{2},t}:=\kappa_{\varrho_{1},\varrho_{2},t}^{(q)}:=\kappa\cdot\left[\frac{w_{2}\circ\Phi_{\varrho_{2}}}{w_{1}\circ\Phi_{\varrho_{1}}}\right]^{\frac{1}{q}-\frac{1}{2}-t}\quad\text{and}\quad
w_{j}=\det(D\Phi^{-1}_{\varrho_{j}}),\quad j\in\\{1,2\\}.$

###### Proof.

We begin by showing that $(\Phi_{1},\kappa_{\varrho_{1},\varrho_{2},t})$ forms
a compatible pair in the sense of Definition 2.4, for any $t\in\mathbb{R}$. By
assumption, $(\Phi_{\varrho_{1}},\kappa)$ form a compatible pair, and
$w_{1}\circ\Phi_{\varrho_{1}}\circ\Phi_{\varrho_{1}}^{-1}=w_{1}$ is
$v_{1}$-moderate, with $v_{1}$ a (submultiplicative) control weight for the
$(d+1)$-admissible warping function $\Phi_{\varrho_{1}}$. Further, it is a
direct consequence of the definition of admissible warping functions,
Definition 2.2, that $(\Phi_{\varrho_{2}},w_{2}\circ\Phi_{\varrho_{2}})$ form
a compatible pair. By Corollary 6.5, we can apply Lemma 6.3, which yields
compatibility of $\Phi_{\varrho_{1}}$ and $w_{2}\circ\Phi_{\varrho_{2}}$.
Altogether, we conclude that
$\kappa_{\varrho_{1},\varrho_{2},t}\circ\Phi_{\varrho_{1}}^{-1}$ is moderate
with respect to the weight

$\tilde{\kappa}_{1}\cdot\left[v_{2}(C\bullet)\cdot
v_{1}\right]^{\left|\frac{1}{q}-\frac{1}{2}-t\right|},$

where $\tilde{\kappa}_{1}$ is the moderating weight for the compatible pair
$(\Phi_{\varrho_{1}},\kappa)$, see Definition 2.4, $v_{2}$ is a
submultiplicative, radially increasing control weight for
$\Phi_{\varrho_{2}}$, see Definition 2.2, and the proof of Lemma 6.3 shows
that $w_{2}\circ\Phi_{\varrho_{2}}\circ\Phi_{\varrho_{1}}^{-1}$ is moderate
with respect to $v_{2}(C\bullet)$.

Hence, by Theorem 4.16 we can identify
$\operatorname{Co}(\Phi_{\varrho_{1}},\mathbf{L}_{\kappa_{\varrho_{1},\varrho_{2},t}}^{p,q})$
and $\operatorname{Co}(\Phi_{\varrho_{2}},\mathbf{L}_{\kappa}^{p,q})$ with
decomposition spaces. Specifically, we obtain, for all $\delta>0$,
$r>\sqrt{d}/2$,

$\operatorname{Co}(\Phi_{\varrho_{2}},\mathbf{L}_{\kappa}^{p,q})=\mathcal{D}(\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{2}}},\mathbf{L}^{p},\ell^{q}_{u_{2}}),\quad\text{and}\quad\operatorname{Co}(\Phi_{\varrho_{1}},\mathbf{L}_{\kappa_{\varrho_{1},\varrho_{2},t}}^{p,q})=\mathcal{D}(\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{1}}},\mathbf{L}^{p},\ell^{q}_{u_{1}^{(t)}}),$

with

$(u_{2})_{\ell}=\kappa(\Phi_{\varrho_{2}}^{-1}(\delta\ell))\cdot[w_{2}(\delta\ell)]^{1/q-1/2}\
\ \text{and}\ \ (u_{1}^{(t)})_{k}=\kappa(\Phi_{\varrho_{1}}^{-1}(\delta
k))\cdot\left[w_{1}(\delta
k)\right]^{t}\cdot\left[w_{2}(\Phi_{\varrho_{2}}(\Phi_{\varrho_{1}}^{-1}(\delta
k)))\right]^{1/q-1/2-t},$

for all $\ell,k\in\mathbb{Z}^{d}$. For ease of notation, and without loss of
generality, we will subsequently choose the same values of $\delta>0$ and
$r>\sqrt{d}/2$ for both coverings.

In order to apply Theorem 6.2 to prove the desired embeddings, it remains to
verify that $\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{1}}}$ is almost
subordinate to $\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{2}}}$, and that
$\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{1}}}$ and $u_{1}^{(t)}$ are
relatively $\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{2}}}$-moderate. The first
property follows directly from Corollary 6.5, and an application of
Proposition 6.8 yields relative
$\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{2}}}$-moderateness of
$\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{1}}}$.

To show that $u_{1}:=u_{1}^{(t)}$ is relatively
$\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{2}}}$-moderate, note as a
consequence of Lemma 6.3 (applied with $\kappa$ and with
$w_{2}\circ\Phi_{\varrho_{2}}$ instead of $\kappa$) that if
$Q^{(\delta,r)}_{\Phi_{\varrho_{1}},k}\cap
Q^{(\delta,r)}_{\Phi_{\varrho_{2}},\ell}\neq\varnothing$, then

$\kappa(\Phi^{-1}_{\varrho_{1}}(\delta
k))\asymp\kappa(\Phi^{-1}_{\varrho_{2}}(\delta\ell))\quad\text{and}\quad
w_{2}(\delta\ell)\asymp
w_{2}((\Phi_{\varrho_{2}}\circ\Phi_{\varrho_{1}}^{-1}(\delta k)).$ (6.17)

Finally, Equation (6.13) in Proposition 6.8 shows that $w_{1}(\delta
k)=\det(D\Phi_{\varrho_{1}}^{-1}(\delta
k))\asymp\frac{[\widetilde{\varrho_{1}}((\varrho_{2})_{\ast}(\delta|\ell|))]^{1-d}}{\varrho_{1}^{\prime}((\varrho_{2})_{\ast}(\delta|\ell|))}$.
This clearly shows that $u_{1}^{(t)}$ is relatively
$\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{2}}}$-moderate.

We have now verified all the prerequisites for applying Theorem 6.2, Items (1)
and (2), with $\mathcal{P}=\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{1}}}$,
$\mathcal{Q}=\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{2}}}$ and the weights
$v=u_{1}^{(t)}$, $u=u_{2}$. To apply Theorem 6.2(1), we set
$t=\max(0,q^{-1}-\min(p^{-1},1-p^{-1})$. Select, for each
$\ell\in\mathbb{Z}^{d}$, some arbitrary $k_{\ell}\in\mathbb{Z}^{d}$ such that
$Q^{(\delta,r)}_{\Phi_{\varrho_{1}},k_{\ell}}\cap
Q^{(\delta,r)}_{\Phi_{\varrho_{2}},\ell}\neq\varnothing$. Then, recalling the
form of $\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{1}}}$ and
$\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{2}}}$ as semi-structured coverings,
given in Lemma 3.2, $K_{t}$ in Theorem 6.2(1) takes the form

$\begin{split}K_{t}&=\sup_{\ell\in\mathbb{Z}^{d}}\left|\frac{(u_{1}^{(t)})_{k_{\ell}}}{(u_{2})_{\ell}}\cdot[w_{2}(\delta{\ell})]^{t}\cdot[w_{1}(\delta
k_{\ell})]^{-t}\right|\\\
&=\sup_{\ell\in\mathbb{Z}^{d}}\left|\frac{\kappa(\Phi_{\varrho_{1}}^{-1}(\delta
k_{\ell}))}{\kappa(\Phi_{\varrho_{2}}^{-1}(\delta\ell))}\right|\cdot\left[\frac{w_{1}(\delta
k_{\ell})}{w_{1}(\delta
k_{\ell})}\right]^{t}\cdot\left[\frac{w_{2}(\Phi_{\varrho_{2}}(\Phi_{\varrho_{1}}^{-1}(\delta
k_{\ell})))}{w_{2}(\delta{\ell})}\right]^{1/q+1/2-t}.\end{split}$

The second factor above trivially equals $1$. The other two factors are
bounded by a constant, see Equation (6.17). In total, we obtain
$K_{t}<\infty$. Since $K_{t}$ is finite, we obtain the embedding
$\operatorname{Co}(\Phi_{\varrho_{2}},\mathbf{L}_{\kappa}^{p,q})\hookrightarrow\operatorname{Co}(\Phi_{\varrho_{1}},\mathbf{L}_{\kappa_{\varrho_{1},\varrho_{2},t}}^{p,q})$.
The second embedding is proven analogously, with an application of Theorem
6.2(2). Specifically, we set
$t=-\tilde{t}=-\max(0,\max(p^{-1},1-p^{-1})-q^{-1})$ and obtain

$K_{\tilde{t}}=\sup_{\ell\in\mathbb{Z}^{d}}\left|\frac{(u_{2})_{\ell}}{(u_{1}^{(-\tilde{t})})_{k_{\ell}}}\cdot[w_{2}(\delta{\ell})]^{\tilde{t}}\cdot[w_{1}(\delta
k_{\ell})]^{-\tilde{t}}\right|<\infty,$

such that Theorem 6.2(2) yields
$\operatorname{Co}(\Phi_{\varrho_{1}},\mathbf{L}_{\kappa_{\varrho_{1},\varrho_{2},-\tilde{t}}}^{p,q})\hookrightarrow\operatorname{Co}(\Phi_{\varrho_{2}},\mathbf{L}_{\kappa}^{p,q})$.
This finishes the proof. ∎

######  Remark 6.10.

Note that, in in the context of Theorem 6.9, the seemingly weaker assumption
that $\Phi_{\varrho_{1}}$ be a $\mathcal{C}^{\infty}$-diffeomorphism and a
$(d+1)$-admissible warping function is, in fact, equivalent to
$(\Phi_{\varrho_{1}},\kappa)$ being a compatible pair: Under the conditions of
the theorem, we can apply Lemma 6.3, which then yields compatibility of
$\Phi_{\varrho_{1}}$ and $\kappa$.

##  7\. Embeddings between Besov spaces and warped coorbit spaces

The homogeneous and inhomogeneous Besov spaces [1, 34], although they have
historically been defined in various ways, are prototypical decomposition
spaces. Specifically, they can be described as decomposition spaces with
respect to a radial, dyadic covering and exponential weights. In particular,
for the _inhomogeneous_ Besov spaces $B^{p,q}_{s}(\mathbb{R}^{d})$, we have
the equality
$B_{s}^{p,q}(\mathbb{R}^{d})=\mathcal{D}(\mathcal{B},\mathbf{L}^{p},\ell_{v^{s}}^{q})$,
with the weight $v^{s}=(v^{s}_{j})_{j\in\mathbb{N}_{0}}$, $v^{s}_{j}=2^{js}$
and the _inhomogeneous Besov covering_
$\mathcal{B}=(B_{j})_{j\in\mathbb{N}_{0}}$ of $\mathbb{R}^{d}$, given by

$B_{0}:=\\{\xi\in\mathbb{R}^{d}\,:\,|\xi|<2\\}\quad\text{and}\quad
B_{j}:=\\{\xi\in\mathbb{R}^{d}\,:\,2^{j-1}<|\xi|<2^{j+1}\\}\quad\text{for
}j\in\mathbb{N}\,,$ (7.1)

see also [14, Definition 2.2.1]. In particular, $v^{s}$ is
$\mathcal{B}$-moderate and the covering $\mathcal{B}$ is semi-structured and
tight [36, Lemma 9.10].

We now proceed to investigate how warped coorbit spaces relate to
inhomogeneous Besov spaces. In particular, we show that, for any admissible
radial component $\varrho$, the $\Phi_{\varrho}$-induced $(\delta,r)$-fine
frequency covering is almost subordinate to the standard inhomogeneous Besov
covering $\mathcal{B}$ just defined. However, we also show that the converse
can only be true in dimension $d=1$, even if $\Phi$ is not a radial warping
function. Nonetheless, we obtain general conditions for embeddings between
$\operatorname{Co}(\Phi_{\varrho},\mathbf{L}_{\kappa}^{p_{1},q_{1}})$ and
$B_{s}^{p_{2},q_{2}}(\mathbb{R}^{d})$, for specific weights $\kappa$ depending
on $s$. Specifically, if $\varrho_{1}$ is any slow start version of the weakly
admissible radial component $\varsigma_{1}=\ln(1+\bullet)$, we show equality
of $\operatorname{Co}(\Phi_{\varrho},\mathbf{L}_{\kappa}^{p,q})$ and
$B_{s}^{p,q}(\mathbb{R}^{d})$ in dimension $d=1$, for a certain weight
$\kappa$. In dimension $d\geq 2$, we obtain a _closeness_ relation, in the
sense that $B_{s}^{p,q}(\mathbb{R}^{d})$ can be nested between two warped
coorbit spaces with respect to $\Phi_{\varrho_{1}}$ that only differ by a
minor modification of the weight $\kappa$.

The following result shows that the covering
$\mathcal{Q}_{\Phi_{\varrho}}^{(\delta,r)}$ induced by $\Phi_{\varrho}$ is
almost subordinate to, and even relatively moderate with respect to, the
inhomogeneous Besov covering.

######  Proposition 7.1.

Let $\varrho:\mathbb{R}\to\mathbb{R}$ be a $0$-admissible radial component and
let $\mathcal{B}$ be the inhomogeneous Besov covering. Then the
$(\delta,r)$-fine frequency covering
$\mathcal{Q}_{\Phi_{\varrho}}^{(\delta,r)}$ induced by $\Phi_{\varrho}$ is
almost subordinate to $\mathcal{B}$, for arbitrary $\delta>0$, $r>\sqrt{d}/2$.
Specifically, we have the properties

$\Big{|}j-\log_{2}\big{(}1+\varrho_{\ast}(\delta|k|)\big{)}\Big{|}\lesssim
1\quad\text{if}\quad B_{j}\cap
Q_{\Phi_{\varrho},k}^{(\delta,r)}\neq\varnothing\,,$ (7.2)

as well as

$\widetilde{\varrho_{\ast}}(\delta|k|)\asymp\frac{2^{j}}{\varrho(2^{j})}\quad\forall\,j\in\mathbb{N}_{0}\text{
and }k\in\mathbb{Z}^{d}\text{ with }B_{j}\cap
Q_{\Phi_{\varrho},k}^{(\delta,r)}\neq\varnothing\,.$ (7.3)

Furthermore, if $\varrho$ is $C^{2}$, and there are $R,C>0$ such that

$\left|\frac{\varrho_{\ast}(\gamma)\cdot\varrho_{\ast}^{\prime\prime}(\gamma)}{[\varrho_{\ast}^{\prime}(\gamma)]^{2}}\right|\leq
C<\infty\quad\forall\,\gamma\in[R,\infty)\,,$ (7.4)

then $\mathcal{Q}_{\Phi_{\varrho}}^{(\delta,r)}$ is relatively
$\mathcal{B}$-moderate, with

$w(\delta
k)\asymp\varrho_{\ast}^{\prime}\big{(}\varrho(2^{j})\big{)}\cdot\left(2^{j}\big{/}\varrho(2^{j})\right)^{d-1}\quad\text{whenever
}j\in\mathbb{N}_{0}\text{ and }k\in\mathbb{Z}^{d}\text{ with }B_{j}\cap
Q_{\Phi_{\varrho},k}^{(\delta,r)}\neq\varnothing\,.$ (7.5)

######  Remark 7.2.

If Condition (7.4) is satisfied, and if the weight
$\kappa:\mathbb{R}^{d}\to\mathbb{R}^{+}$ is given by
$\kappa(\xi)=(1+|\xi|)^{s}$ for some $s\in\mathbb{R}$, then the associated
weight $u^{(q)}$ defined in (4.2) is relatively $\mathcal{B}$-moderate.
Indeed, if $\varnothing\neq B_{j}\cap
Q_{\Phi_{\varrho},k}^{(\delta,r)}\ni\xi$, then $1+|\xi|\asymp 2^{j}$, so that
Equation (6.10) shows

$\kappa\big{(}\Phi_{\varrho}^{-1}(\delta
k)\big{)}=\kappa\big{(}\Phi_{\varrho_{\ast}}(\delta
k)\big{)}=(1+|\Phi_{\varrho_{\ast}}(\delta
k)|)^{s}=\big{(}1+\varrho_{\ast}(\delta|k|)\big{)}^{s}\asymp(1+|\xi|)^{s}\asymp
2^{sj}\,.$

In combination with Equation (7.5), we thus see

$u_{k}^{(q)}=\kappa\big{(}\Phi_{\varrho}^{-1}(\delta k)\big{)}\cdot[w(\delta
k)]^{\frac{1}{q}-\frac{1}{2}}\asymp
2^{sj}\cdot\left[\varrho_{\ast}^{\prime}\big{(}\varrho(2^{j})\big{)}\cdot\left(2^{j}\big{/}\varrho(2^{j})\right)^{d-1}\right]^{\frac{1}{q}-\frac{1}{2}}\quad\text{if
}B_{j}\cap Q_{\Phi_{\varrho},k}^{(\delta,r)}\neq\varnothing\,,$ (7.6)

which implies in particular that $u^{(q)}$ is relatively
$\mathcal{B}$-moderate.

###### Proof.

Let $v:\mathbb{R}\to\mathbb{R}^{+}$ be a control weight for $\varrho$. To
prove Equation (7.2), fix $k\in\mathbb{Z}^{d}$, and let $j\in\mathbb{N}_{0}$
be arbitrary. If $\varnothing\neq B_{j}\cap
Q_{\Phi_{\varrho},k}^{(\delta,r)}\ni\xi$, then $1+|\xi|\asymp 2^{j}$ follows
directly from the definition of $B_{j}$. In combination with (6.10), we thus
see that there are $0<c_{0}\leq 1\leq c_{1}$ (independent of $j,k,\xi$) such
that $c_{0}\cdot\big{(}1+\varrho_{\ast}(\delta|k|)\big{)}\leq 2^{j}\leq
c_{1}\cdot\big{(}1+\varrho_{\ast}(\delta|k|)\big{)}$. Hence,
$\big{|}j-\log_{2}(1+\varrho_{\ast}(\delta|k|))\big{|}\leq\max\\{\log_{2}(c_{1}),\log_{2}(c_{0}^{-1})\\}$,
such that (7.2) is satisfied.

In particular, there are no more than
$1+2\max\\{\log_{2}(c_{1}),\log_{2}(c_{0}^{-1})\\}$ nonnegative integers
$j\in\mathbb{N}_{0}$ which satisfy this estimate. We thus see that

$\sup_{k\in\mathbb{Z}^{d}}|\\{j\in\mathbb{N}_{0}\,:\,B_{j}\cap
Q_{\Phi_{\varrho},k}^{(\delta,r)}\neq\varnothing\\}|\leq
1+2\max\\{\log_{2}(c_{1}),\log_{2}(c_{0}^{-1})\\}<\infty\,,$

and $\mathcal{Q}_{\Phi_{\varrho}}^{(\delta,r)}$ is weakly subordinate to
$\mathcal{B}$. Since the elements of
$\mathcal{Q}_{\Phi_{\varrho}}^{(\delta,r)}$ are path-connected, Lemma 2.14
shows that $\mathcal{Q}_{\Phi_{\varrho}}^{(\delta,r)}$ is almost subordinate
to $\mathcal{B}$.

Our proof of Equation (7.3) distinguishes two cases: Since
$\mathcal{Q}_{\Phi_{\varrho}}^{(\delta,r)}$ is weakly subordinate to
$\mathcal{B}$, the set
$\\{(j,k)\in\mathbb{N}_{0}\times\mathbb{Z}^{d}\,:\,|k|\leq 2r\text{ and
}B_{j}\cap Q_{\Phi_{\varrho},k}^{(\delta,r)}\neq\varnothing\\}$ is finite.
Since $\widetilde{\varrho_{\ast}}$ and
$2^{\bullet}\big{/}\varrho(2^{\bullet})$ are positive functions,
$\widetilde{\varrho_{\ast}}(\delta|k|)\asymp 2^{j}\big{/}\varrho(2^{j})$ for
all $(j,k)$ in the above set. Note that Equation (6.14) implies that the set
$\\{k\in\mathbb{Z}^{d}\,:\,B_{0}\cap
Q_{\Phi_{\varrho},k}^{(\delta,r)}\neq\varnothing\\}$ is likewise finite,
yielding $\widetilde{\varrho_{\ast}}(\delta|k|)\asymp
2^{0}\big{/}\varrho(2^{0})$ for all such $k$.

Finally, we consider the case $|k|>2r$ and $j\neq 0$. To this end, note that
the map $h:\mathbb{R}\to\mathbb{R},\tau\mapsto\ln(\varrho(e^{\tau}))$ is well-
defined and strictly increasing, since $\varrho$ is strictly increasing with
$\varrho(0)=0$. Setting $\gamma:=\varrho(e^{\tau})>0$, we see that the
derivative of $h$ satisfies

$0\leq
h^{\prime}(\tau)=\frac{1}{\varrho(e^{\tau})}\cdot\varrho^{\prime}(e^{\tau})\cdot
e^{\tau}=\frac{1}{\gamma}\cdot\frac{1}{\varrho_{\ast}^{\prime}(\gamma)}\cdot\varrho_{\ast}(\gamma)=\frac{\widetilde{\varrho_{\ast}}(\gamma)}{\varrho_{\ast}^{\prime}(\gamma)}\leq
C_{0}^{-1}\,,$

where we used $\varrho^{\prime}=[\varrho_{\ast}^{\prime}\circ\varrho]^{-1}$ by
the inverse function theorem and, since $\varrho$ is $0$-admissible, that
$\varrho_{\ast}^{\prime}(\gamma)\geq
C_{0}\cdot\widetilde{\varrho_{\ast}}(\gamma)$, cf. Equation (5.3). The above
estimate implies in particular that $|h(\tau)-h(\sigma)|\leq
C_{0}^{-1}\cdot|\tau-\sigma|$ for all $\tau,\sigma\in\mathbb{R}$. Let $\xi\in
B_{j}\cap Q^{\delta,r}_{\Phi_{\varrho},k}$ and set $\tau:=\ln(|\xi|)$ and
$\sigma:=\ln(2^{j})$, such that $|\tau-\sigma|\leq\ln(2)$, since
$2^{-1}\leq|\xi|/2^{j}\leq 2$. Therefore,

$C_{0}^{-1}\ln(2)\geq|h(\tau)-h(\sigma)|=\big{|}\ln\big{(}\varrho(|\xi|)\big{)}-\ln\big{(}\varrho(2^{j})\big{)}\big{|}=\ln\bigg{(}\max\bigg{\\{}\frac{\varrho(|\xi|)}{\varrho(2^{j})},\frac{\varrho(2^{j})}{\varrho(|\xi|)}\bigg{\\}}\bigg{)}\,,$
(7.7)

and hence $\varrho(2^{j})\asymp\varrho(|\xi|)\asymp\delta|k|$, where the
latter equivalence follows from Equation (6.14), in the proof of Proposition
6.8, with $|k|\geq 2r$. Since $k,j\neq 0$ and $|\xi|\geq 2^{j-1}\geq 1$, we
have $1+\varrho_{\ast}(\delta|k|)\asymp\varrho_{\ast}(\delta|k|)$ and
$1+|\xi|\asymp|\xi|$. Therefore, Equation (6.10) implies
$\varrho_{\ast}(\delta|k|)\asymp|\xi|\asymp 2^{j}$. Together with the prior
estimate, we obtain (7.3).

To prove the last part of the proposition, let us assume that (7.4) is
satisfied. Note that $\varrho_{\ast}^{\prime}$ is a _positive_ , continuous
function, since $\varrho$ is a strictly increasing diffeomorphism. Therefore,
the left-hand side of (7.4) is a continuous function, so that (7.4) is in fact
satisfied for all $\gamma\in[0,\infty)$, after possibly enlarging the constant
$C$. In the following, we will assume that $C$ is chosen in this way.

Now, define
$f:\mathbb{R}\to\mathbb{R}^{+},t\mapsto\varrho_{\ast}^{\prime}(\varrho(t))$
and $g:\mathbb{R}\to\mathbb{R},\tau\mapsto\ln(f(e^{\tau}))$. Setting
$\gamma:=\varrho(e^{\tau})>0$, we obtain, with (7.4),

$|g^{\prime}(\tau)|=\left|\frac{1}{f(e^{\tau})}\cdot f^{\prime}(e^{\tau})\cdot
e^{\tau}\right|=\left|\frac{e^{\tau}\cdot\varrho_{\ast}^{\prime\prime}(\varrho(e^{\tau}))\cdot\varrho^{\prime}(e^{\tau})}{\varrho_{\ast}^{\prime}(\varrho(e^{\tau}))}\right|=\left|\frac{\varrho_{\ast}(\gamma)\cdot\varrho_{\ast}^{\prime\prime}(\gamma)}{[\varrho_{\ast}^{\prime}(\gamma)]^{2}}\right|\leq
C<\infty,\quad\text{for all }\tau\in\mathbb{R},$

where we used $\varrho^{\prime}=[\varrho_{\ast}^{\prime}\circ\varrho]^{-1}$.
We conclude that

$|g(\tau)-g(\sigma)|\leq C\cdot|\tau-\sigma|,\quad\text{for all
}\tau,\sigma\in\mathbb{R}.$ (7.8)

Now, let $j\in\mathbb{N}$ and $k\in\mathbb{Z}^{d}$ satisfy $\varnothing\neq
B_{j}\cap Q_{\Phi_{\varrho},k}^{(\delta,r)}\ni\xi$, and recall from (6.14)
that $\big{|}\,\varrho(|\xi|)-\delta|k|\,\big{|}\leq\delta r$. Since
$\varrho_{\ast}^{\prime}$ is $v$-moderate, and since $v$ is radially
increasing, this implies

$\varrho_{\ast}^{\prime}(\delta|k|)=\varrho_{\ast}^{\prime}\big{(}\delta|k|-\varrho(|\xi|)+\varrho(|\xi|)\big{)}\leq
v(\delta r)\cdot\varrho_{\ast}^{\prime}(\varrho(|\xi|))=v(\delta r)\cdot
f(|\xi|)\,.$

Likewise, $f(|\xi|)\leq v(\delta r)\cdot\varrho_{\ast}^{\prime}(\delta|k|)$.
Now, $f(|\xi|)\asymp f(2^{j})$ and consequently
$\varrho_{\ast}^{\prime}(\delta|k|)\asymp\varrho_{\ast}^{\prime}(\varrho(2^{j}))$,
for all $j\in\mathbb{N}$, $k\in\mathbb{Z}^{d}$ with $\varnothing\neq B_{j}\cap
Q_{\Phi_{\varrho},k}^{(\delta,r)}$ is shown analogous to (7.7), with an
application of (7.8). Combining this with Equations (5.5) and (7.3), we thus
see

$w(\delta
k)=\varrho_{\ast}^{\prime}(\delta|k|)\cdot[\widetilde{\varrho_{\ast}}(\delta|k|)]^{d-1}\asymp\varrho_{\ast}^{\prime}(\varrho(2^{j}))\cdot\left(2^{j}\big{/}\varrho(2^{j})\right)^{d-1}$

for $j\in\mathbb{N}$ and $k\in\mathbb{Z}^{d}$ with $B_{j}\cap
Q_{\Phi_{\varrho},k}^{(\delta,r)}\neq\varnothing$, proving (7.5) for positive
$j\in\mathbb{N}$. However, if $j=0$, then we recall that there are only
finitely many $k\in\mathbb{Z}^{d}$ such that $B_{0}\cap
Q_{\Phi_{\varrho},k}^{(\delta,r)}\neq\varnothing$. Since both sides of (7.5)
are positive, equivalence is trivial. ∎

We just saw that the covering $\mathcal{Q}_{\Phi_{\varrho}}^{(\delta,r)}$ is
always almost subordinate to the inhomogeneous Besov covering $\mathcal{B}$.
Our next result provides a straightforward, sufficient condition for the
converse to be true, but only in the one-dimensional case $d=1$. In Theorem
7.10, we will construct an example of an admissible radial component which
satisfies this condition. The situation is radically different in higher
dimensions: In Theorem 7.4 below, we show that $\mathcal{B}$ is never weakly
subordinate to $\mathcal{Q}_{\Phi}^{(\delta,r)}$, if $d\geq 2$. This holds
regardless of $\Phi$ being radial or not.

######  Lemma 7.3.

Let $d=1$, let $r>1/2$, let $\varrho:\mathbb{R}\to\mathbb{R}$ be a
$0$-admissible radial component, and assume that $\varrho$ is $C^{2}$ and
satisfies Equation (7.4). If

$2^{j}\lesssim\varrho_{\ast}^{\prime}(\varrho(2^{j}))=[\varrho^{\prime}(2^{j})]^{-1}\qquad\forall\,j\in\mathbb{N}_{0}\,,$
(7.9)

then the inhomogeneous Besov covering
$\mathcal{B}=(B_{j})_{j\in\mathbb{N}_{0}}$ defined in (7.1) is weakly
subordinate to $\mathcal{Q}_{\Phi_{\varrho}}^{(\delta,r)}$.

###### Proof.

Proposition 7.1 shows that there is some $N\in\mathbb{N}$ such that, for each
$k\in\mathbb{Z}$, there is some $j_{k}\in\mathbb{N}_{0}$ satisfying
$Q_{\Phi_{\varrho},k}^{(\delta,r)}\subset B_{j_{k}^{N\ast}}$. For
$j\in\mathbb{N}_{0}$, set

$I_{j}:=\big{\\{}k\in\mathbb{Z}\,:\,Q_{\Phi_{\varrho},k}^{(\delta,r)}\cap
B_{j}\neq\varnothing\big{\\}}\,.$

Now, if $k\in I_{j}$, then $\varnothing\neq B_{j}\cap
Q_{\Phi_{\varrho},k}^{(\delta,r)}\subset B_{j}\cap B_{j_{k}^{N\ast}}$, and
hence $j_{k}\in j^{(N+1)\ast}$, which implies $j_{k}^{N\ast}\subset
j^{(2N+1)\ast}$, and finally

$Q_{\Phi_{\varrho},k}^{(\delta,r)}\subset B_{j_{k}^{N\ast}}\subset
B_{j^{(2N+1)\ast}}\qquad\forall\,j\in\mathbb{N}_{0}\text{ and }k\in I_{j}\,.$
(7.10)

Since $\mathcal{Q}_{\Phi_{\varrho}}^{(\delta,r)}$ is an admissible covering,
there is some $K\in\mathbb{N}$ such that
$\sum_{k\in\mathbb{Z}}{\mathds{1}}_{Q_{\Phi_{\varrho},k}^{(\delta,r)}}\leq K$.
In view of Equation (7.10), this shows $\sum_{k\in
I_{j}}{\mathds{1}}_{Q_{\Phi_{\varrho},k}^{(\delta,r)}}\leq
K\cdot{\mathds{1}}_{B_{j^{(2N+1)\ast}}}$. Thus, noting that the factor
$(2^{j}/\varrho(2^{j}))^{d-1}$ in Equation (7.5) vanishes for $d=1$, we obtain

$\begin{split}|I_{j}|\cdot\varrho_{\ast}^{\prime}(\varrho(2^{j}))&\overset{\eqref{eq:RadialWarpingCoveringRelativeModerateness}}{\asymp}\sum_{k\in
I_{j}}w(\delta k)\overset{\text{ Lem.
}\ref{lem:DecompInducedCoveringIsNice}}{\asymp}\sum_{k\in
I_{j}}\mu\big{(}Q_{\Phi_{\varrho},k}^{(\delta,r)}\big{)}\\\
&\overset{\text{(MCT)}}{=}\int_{\mathbb{R}}\sum_{k\in
I_{j}}{\mathds{1}}_{Q_{\Phi_{\varrho},k}^{(\delta,r)}}(\xi)\,d\xi\leq
K\cdot\mu\big{(}B_{j^{(2N+1)\ast}}\big{)}\lesssim 2^{j}\,,\end{split}$

where (MCT) indicates an application of the monotone convergence theorem. Now,
apply (7.9) to show that $\mathcal{B}$ is weakly subordinate to
$\mathcal{Q}_{\Phi_{\varrho}}^{(\delta,r)}$. Specifially, we obtain
$|I_{j}|\lesssim 2^{j}\big{/}\varrho_{\ast}^{\prime}(\varrho(2^{j}))\lesssim
1$. ∎

The next result shows that Lemma 7.3 cannot be extended to dimensions $d\geq
2$.

######  Theorem 7.4.

Let $2\leq d\in\mathbb{N}$, $D\subset\mathbb{R}^{d}$ open, and $\Phi\colon
D\rightarrow\mathbb{R}^{d}$ a diffeomorphism. The $(\delta,r)$-fine frequency
covering $\mathcal{Q}^{(\delta,r)}_{\Phi}$ induced by $\Phi$, see Definition
1.1, is not weakly equivalent to $\mathcal{B}$, for any choice of $\delta>0$
and $r>\sqrt{d}/2$.

Before we prove Theorem 7.4, we require some auxiliary results, estimating the
number of $n$-th order neighbors for $\mathcal{Q}^{(\delta,r)}_{\Phi}$ and
$\mathcal{B}$, see also Remark 2.8.

######  Lemma 7.5.

Let $2\leq d\in\mathbb{N}$, $D\subset\mathbb{R}^{d}$ open, and $\Phi\colon
D\rightarrow\mathbb{R}^{d}$ a diffeomorphism. Then, for any $\delta>0$ and
$r>\sqrt{d}/2$, the $(\delta,r)$-fine frequency covering
$\mathcal{Q}^{(\delta,r)}_{\Phi}=(Q^{(\delta,r)}_{\Phi,k})_{k\in\mathbb{Z}^{d}}$
induced by $\Phi$ satisfies

$k^{n\ast}\supset\\{\ell\in\mathbb{Z}^{d}~{}:~{}\|k-\ell\|_{\ell^{\infty}}\leq
n\\},\quad\text{such that }|k^{n\ast}|\geq(1+2n)^{d},\quad\text{for all
}k\in\mathbb{Z}^{d},\ n\in\mathbb{N}_{0}.$ (7.11)

Moreover, the inhomogeneous Besov covering
$\mathcal{B}=(B_{j})_{j\in\mathbb{N}_{0}}$ satisfies

$j^{n\ast}\subset\\{\ell\in\mathbb{N}_{0}~{}:~{}|j-\ell|\leq
n\\},\quad\text{such that }|j^{n\ast}|\leq 1+2n,\quad\text{for all
}j,n\in\mathbb{N}_{0}.$ (7.12)

###### Proof.

We first prove the upper bound for $|j^{n\ast}|$, i.e., Equation (7.12). If
$\xi\in B_{j}\cap B_{\ell}\neq\varnothing$, with $j,\ell\in\mathbb{N}$, then
$2^{\ell-1}<|\xi|<2^{j+1}$, and thus $\ell-1<j+1$. implying $\ell\leq j+1$.
Repeat the argument with interchanged roles of $j,\ell$ to obtain
$|j-\ell|\leq 1$. Further, if $j=0$ instead, then $2^{\ell-1}<|\xi|<2=2^{1}$,
similarly yielding $|0-\ell|\leq 1$. Overall,

$j^{\ast}\subset\\{\ell\in\mathbb{N}_{0}~{}:~{}|j-\ell|\leq
1\\},\quad\text{such that }|j^{\ast}|\leq 3,\quad\text{for all
}j\in\mathbb{N}_{0}.$

Equation (7.12) now follows by a straightforward induction.

The proof of Equation (7.11) is slightly more involved. Note that
$r>\sqrt{d}/2$ implies

$P_{k}\subset B_{r}(k),\quad\text{where}\quad P_{k}=k+[-1/2,1/2]^{d}.$

Therefore $P_{k}\cap P_{\ell}\neq\varnothing$ implies
$Q^{(\delta,r)}_{\Phi,k}\cap
Q^{(\delta,r)}_{\Phi,\ell}=\Phi^{-1}(\delta\cdot(B_{r}(k)\cap
B_{r}(\ell)))\neq\varnothing$. Hence, it suffices to prove (7.11) for
$\mathcal{P}=(P_{k})_{k\in\mathbb{Z}^{d}}$ in place of
$\mathcal{Q}^{(\delta,r)}_{\Phi}$.

Now define, for $k,\ell\in\mathbb{Z}^{d}$, $\varepsilon:=k-\ell$ and note that
$\|k-\ell\|_{\ell^{\infty}}\leq 1$ implies $\varepsilon\in[-1,1]^{d}$. In that
case, we clearly have $k-\varepsilon/2=\ell+\varepsilon/2\in P_{k}\cap
P_{\ell}$. Hence,

$\ell\in k^{\ast},\quad\text{whenever }k,\ell\in\mathbb{Z}^{d}\text{ with
}\|k-\ell\|_{\ell^{\infty}}\leq 1.$

Finally, assume that the first part of Equation (7.11) holds for some
$n\in\mathbb{N}_{0}$ and let $k,\ell\in\mathbb{Z}^{d}$ with
$\|k-\ell\|_{\ell^{\infty}}=n+1$. There exists a $\tilde{k}\in\mathbb{Z}^{d}$
with $\|k-\tilde{k}\|_{\ell^{\infty}}\leq n$ and
$\|\tilde{k}-\ell\|_{\ell^{\infty}}=1$. Hence, $\tilde{k}\in k^{n\ast}$ and
$\ell\in\tilde{k}^{\ast}$, such that $\ell\in k^{(n+1)\ast}$ follows. This
proves the first part of Equation (7.11) by induction. To prove the second
part, simply note that
$|\\{\ell\in\mathbb{Z}^{d}~{}:~{}\|k-\ell\|_{\ell^{\infty}}\leq
n\\}|=(1+2n)^{d}$. ∎

Our second auxiliary result provides an estimate for neighbor sets of
arbitrary order, for two equivalent coverings.

######  Lemma 7.6.

Let $\mathcal{Q}=(Q_{i})_{i\in I}$ and $\mathcal{P}=(P_{j})_{j\in J}$ be two
equivalent, admissible coverings of a common set $\mathcal{O}$, such that
$Q_{i}$, $i\in I$, are nonempty. Then there are constants $M,N\in\mathbb{N}$
such that

$|i^{n\ast}|\leq M\cdot|j^{N(n+1)\ast}|,$ (7.13)

for all $i\in I,j\in J$ such that $Q_{i}\cap P_{j}\neq\varnothing$, and
arbitrary $n\in\mathbb{N}_{0}$.

###### Proof.

Recall that $Q_{L}$, for some $L\subset I$, denotes the set
$Q_{L}:=\bigcup_{i\in L}Q_{i}$ and $P_{L}$ for some $L\subset J$ is defined
analogously. Since $\mathcal{Q}$ is almost subordinate to $\mathcal{P}$, there
exists, by [36, Lemma 2.11 (3)], an $N\in\mathbb{N}$, such that $Q_{i}\subset
P_{j^{N\ast}}$, for all $i\in I,j\in J$ such that $Q_{i}\cap
P_{j}\neq\varnothing$. A straightforward induction shows that, in fact,
$Q_{i^{n\ast}}\subset P_{j^{N(n+1)\ast}}$ holds for all $n\in\mathbb{N}_{0}$,
whenever $Q_{i}\cap P_{j}\neq\varnothing$. Denote by $I_{j}$, $j\in J$, the
sets

$I_{j}=\\{i\in I~{}:~{}Q_{i}\cap P_{j}\neq\varnothing\\}.$

Since $\mathcal{P}$ is almost subordinate to $\mathcal{Q}$, it is, in
particular, weakly subordinate to $\mathcal{Q}$ ([36, Lemma 2.11 (1)], see
also Remark 2.12), such that $M:=\sup_{j\in J}|I_{j}|<\infty$.

Now, let $i\in I$ and $j\in J$ be arbitrary, with $P_{j}\cap
Q_{i}\neq\varnothing$. Since $Q_{i^{n\ast}}\subset P_{j^{N(n+1)\ast}}$ holds,
we also have $Q_{k}\subset P_{j^{N(n+1)\ast}}$ for all $k\in i^{n\ast}$, i.e.,
there is some $\ell\in j^{N(n+1)\ast}$ with $Q_{k}\cap
P_{\ell}\neq\varnothing$. We conclude that $i^{n\ast}\subset\bigcup_{\ell\in
j^{N(n+1)\ast}}I_{\ell}$. Therefore,

$|i^{n\ast}|\leq\Big{|}\bigcup_{\ell\in
j^{N(n+1)\ast}}I_{\ell}\Big{|}\leq\sum_{\ell\in j^{N(n+1)\ast}}|I_{\ell}|\leq
M\cdot|j^{N(n+1)\ast}|.\hfill\qed$

###### Proof of Theorem 7.4.

If $\mathcal{B}$ and $\mathcal{Q}^{(\delta,r)}_{\Phi}$ are weakly equivalent,
then they are equivalent, since their elements are path-connected non-empty.
Applying Lemma 7.6 with $\mathcal{Q}=\mathcal{Q}^{(\delta,r)}_{\Phi}$ and
$\mathcal{P}=\mathcal{B}$ and inserting the estimates in Lemma 7.5, we obtain,
for all $n\in\mathbb{N}_{0}$,

$(1+2n)^{d}\leq|k^{n\ast}|\leq M\cdot|j^{N(n+1)\ast}|\leq M\cdot(1+2N(n+1)),$

for some constants $M,N\in\mathbb{N}$, and all $k\in\mathbb{Z}^{d}$,
$j\in\mathbb{N}_{0}$ with $\mathcal{Q}^{(\delta,r)}_{\Phi,k}\cap
B_{j}\neq\varnothing$. Since $d\geq 2$, this is a contradiction (let
$n\rightarrow\infty$). ∎

As a consequence of Theorem 7.4, we obtain the following negative result on
equality between warped coorbit spaces and Besov spaces.

######  Theorem 7.7.

Let $d\geq 2$, $p_{1},p_{2},q_{1},q_{2}\in[1,\infty]$, and let
$\Phi\colon\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$,
$\kappa\colon\mathbb{R}^{d}\rightarrow\mathbb{R}^{+}$ be compatible in the
sense of Definition 2.4. If
$\operatorname{Co}(\Phi,\mathbf{L}^{p_{1},q_{1}}_{\kappa})$ coincides, up to
canonical identifications, with an (inhomogeneous) Besov space
$B^{p_{2},q_{2}}_{s}(\mathbb{R}^{d})$, then $p_{1}=p_{2}=q_{1}=q_{2}=2$.

###### Proof.

By Theorem 4.16, we have

$\operatorname{Co}(\Phi,\mathbf{L}^{p_{1}.q_{1}}_{\kappa})=\mathcal{D}(\mathcal{Q}^{(\delta,r)}_{\Phi},\mathbf{L}^{p_{1}},\ell^{q_{1}}_{u}),$

where
$\mathcal{Q}^{(\delta,r)}_{\Phi}=(Q^{(\delta,r)}_{\Phi,k})_{k\in\mathbb{Z}^{d}}$,
with arbitrary $\delta>0$ and $r>\sqrt{d}/2$, is the $\Phi$-induced
$(\delta,r)$-fine frequency covering and $u=(u_{k})_{k\in\mathbb{Z}^{d}}$ is a
suitable $\mathcal{Q}^{(\delta,r)}_{\Phi}$-moderate weight. Hence, the
assumptions of the theorem imply

$\mathcal{D}(\mathcal{Q}^{(\delta,r)}_{\Phi},\mathbf{L}^{p_{1}},\ell^{q_{1}}_{u})=\operatorname{Co}(\Phi,\mathbf{L}^{p_{1}.q_{1}}_{\kappa})=B^{p_{2},q_{2}}_{s}(\mathbb{R}^{d})=\mathcal{D}(\mathcal{B},\mathbf{L}^{p_{2}},\ell^{q_{2}}_{v}),$

where $v=(2^{js})_{j\in\mathbb{N}_{0}}$. Since
$\mathcal{Q}^{(\delta,r)}_{\Phi}$ and $\mathcal{B}$ are not weakly equivalent
by Theorem 7.4, [36, Theorem 6.9, Items (1) and (3)] shows that
$p_{1}=p_{2}=2$. Finally, [36, Theorem 6.9, Item (4)] shows that
$q_{1}=q_{2}=2$ and the proof is finished. ∎

Now that we have determined the limits of identifying warped coorbit spaces
with Besov spaces, we return to what is possible in the setting of radial
warping functions $\Phi_{\varrho}$. The next lemma is the final step for
preparing our embedding result, stated in Proposition 7.9 below. We show that
the weight $\kappa(\xi)=(1+|\xi|)^{s}$, together with any radial warping
function $\Phi_{\varrho}$ derived from a $0$-admissible radial component
$\varrho$, form a compatible pair, see Definition 2.4.

######  Lemma 7.8.

Let $\varrho:\mathbb{R}\to\mathbb{R}$ be a $0$-admissible radial component
with control weight $v\colon\mathbb{R}\rightarrow\mathbb{R}^{+}$, and let
$\kappa:\mathbb{R}^{d}\to\mathbb{R}^{+},\xi\mapsto(1+|\xi|)^{s}$ for some
$s\in\mathbb{R}$. Then, there exists a constant $C>0$, such that
$\kappa\circ\Phi_{\varrho}^{-1}$ is $\tilde{\kappa}$-moderate, with
$\tilde{\kappa}=\left(1+C\cdot(1+|\bullet|)\cdot v(\bullet)\right)^{|s|}$. In
particular, if $\varrho\in\mathcal{C}^{\infty}(\mathbb{R})$ is
$(d+1)$-admissible, then $\Phi_{\varrho}$ and $\kappa$ are compatible in the
sense of Definition 2.4.

###### Proof.

It is sufficient to show the result for $\kappa,\tilde{\kappa}$ with $s=1$,
since all relevant properties are retained if real powers $s\in\mathbb{R}$ of
the moderate weight, respectively nonnegative powers $|s|$ of the moderating
weight are taken.

Let $v:\mathbb{R}\to\mathbb{R}^{+}$ denote a control weight for the radial
component $\varrho$. In particular, this implies that
$\widetilde{\varrho_{\ast}}$ is $v$-moderate, see Definition 5.2. We will show
that $\kappa_{\Phi_{\varrho}}=\kappa\circ\Phi_{\varrho}^{-1}$ is
$\tilde{\kappa}$-moderate for the weight

$\tilde{\kappa}\colon\mathbb{R}^{d}\to\mathbb{R}^{+},\ \xi\mapsto
1+C\cdot(1+|\xi|)\cdot v(|\xi|)\,,\quad\text{with}\quad
C:=\max\\{2,\max_{\tau_{0}\in[-1,1]}\widetilde{\varrho_{\ast}}(\tau_{0})\\},$

which is easily seen to be radially increasing, continuous, and
submultiplicative.

Observe that
$\kappa_{\Phi_{\varrho}}(\xi)=1+|\Phi_{\varrho_{\ast}}(\xi)|=1+\varrho_{\ast}(|\xi|)$,
for all $\xi\in\mathbb{R}^{d}$. Since $\varrho_{\ast}$ is increasing, and
since $\widetilde{\varrho_{\ast}}$ is $v$-moderate, we have

$\kappa_{\Phi_{\varrho}}(\xi+\tau)=1+\varrho_{\ast}(|\xi+\tau|)\leq
1+\varrho_{\ast}(|\xi|+|\tau|)=1+\widetilde{\varrho_{\ast}}(|\xi|+|\tau|)\cdot(|\xi|+|\tau|)\leq
1+\big{(}v(|\xi|)\cdot\widetilde{\varrho_{\ast}}(|\tau|)\big{)}\cdot(|\xi|+|\tau|)=:\circledast\,.$

We now distinguish three cases:

Case 1: If $|\tau|\leq 1$, then

$\circledast\leq
1+\max_{\tau_{0}\in[-1,1]}\widetilde{\varrho_{\ast}}(\tau_{0})\cdot(1+|\xi|)\cdot
v(|\xi|)\leq\tilde{\kappa}(\xi)\leq\tilde{\kappa}(\xi)\cdot(1+\varrho_{\ast}(|\tau|))=\tilde{\kappa}(\xi)\cdot\kappa_{\Phi_{\varrho}}(\tau)\,.$

Case 2: If $|\xi|\leq|\tau|$ and $|\tau|>1$, then

$\circledast\leq 1+v(|\xi|)\cdot\widetilde{\varrho_{\ast}}(|\tau|)\cdot
2|\tau|=1+2v(|\xi|)\cdot\varrho_{\ast}(|\tau|)\leq\tilde{\kappa}(\xi)\cdot(1+\varrho_{\ast}(|\tau|))=\tilde{\kappa}(\xi)\cdot\kappa_{\Phi_{\varrho}}(\tau)\,.$

Case 3: If $1<|\tau|\leq|\xi|$, then
$\widetilde{\varrho_{\ast}}(|\tau|)=\varrho_{\ast}(|\tau|)/|\tau|\leq\varrho_{\ast}(|\tau|)$,
and

$\circledast\leq 1+2|\xi|\cdot
v(|\xi|)\cdot\varrho_{\ast}(|\tau|)\leq\tilde{\kappa}(\xi)\cdot(1+\varrho_{\ast}(|\tau|))=\tilde{\kappa}(\xi)\cdot\kappa_{\Phi_{\varrho}}(\tau)\,.\qed$

Finally, we can characterize the existence of embeddings of the warped coorbit
spaces $\operatorname{Co}(\Phi_{\varrho},\mathbf{L}_{\kappa}^{p,q})$, with the
weight $\kappa=(1+|\bullet|)^{s}$ into inhomogeneous Besov spaces, and vice
versa.

######  Proposition 7.9.

Let $\varrho:\mathbb{R}\to\mathbb{R}$ be $\mathcal{C}^{\infty}$ and a
$(d+1)$-admissible radial component, with

$\left|\tfrac{\varrho_{\ast}(\gamma)\cdot\varrho_{\ast}^{\prime\prime}(\gamma)}{[\varrho_{\ast}^{\prime}(\gamma)]^{2}}\right|\leq
C<\infty\quad\text{for some }R>0\ \text{and}\ \gamma\in[R,\infty)\,.$

Furthermore, let
$\kappa:\mathbb{R}^{d}\to\mathbb{R}^{+},\xi\mapsto(1+|\xi|)^{s_{1}}$ for some
$s_{1}\in\mathbb{R}$.

Then the warped coorbit spaces
$\operatorname{Co}(\Phi_{\varrho},\mathbf{L}_{\kappa}^{p_{1},q_{1}})$ are
well-defined and independent of the choice of
$0\neq\theta\in\mathcal{D}(\mathbb{R}^{d})\setminus\\{0\\}$, for arbitrary
$p_{1},q_{1}\in[1,\infty]$. We further have the following equivalences, for
$p_{1},p_{2}\in[1,\infty]$:

  1. (1)

$\operatorname{Co}(\Phi_{\varrho},\mathbf{L}_{\kappa}^{p_{1},q_{1}})\hookrightarrow
B_{s_{2}}^{p_{2},q_{2}}(\mathbb{R}^{d})$ holds if and only if

$p_{1}\leq p_{2}\quad\text{and
}\left\|\left(2^{j(s_{2}-s_{1}+d\tilde{t})}\cdot\left[\varrho_{\ast}^{\prime}(\varrho(2^{j}))\cdot\big{(}2^{j}\big{/}\varrho(2^{j})\big{)}^{d-1}\right]^{p_{1}^{-1}-p_{2}^{-1}-\tilde{t}+2^{-1}-q_{1}^{-1}}\right)_{j\in\mathbb{N}_{0}}\right\|_{\ell^{q_{2}\cdot(q_{1}/q_{2})^{\prime}}}<\infty\,,$
(7.14)

where $\tilde{t}=\max(0,\max(p_{2}^{-1},1-p_{2}^{-1})-q_{1}^{-1})$.

  2. (2)

$B_{s_{2}}^{p_{2},q_{2}}(\mathbb{R}^{d})\hookrightarrow\operatorname{Co}(\Phi_{\varrho},\mathbf{L}_{\kappa}^{p_{1},q_{1}})$
holds if and only if

$p_{2}\leq p_{1}\quad\text{and
}\left\|\left(2^{j(s_{1}-s_{2}+dt)}\cdot\left[\varrho_{\ast}^{\prime}(\varrho(2^{j}))\cdot\big{(}2^{j}\big{/}\varrho(2^{j})\big{)}^{d-1}\right]^{p_{2}^{-1}-p_{1}^{-1}-t+q_{1}^{-1}-2^{-1}}\right)_{j\in\mathbb{N}_{0}}\right\|_{\ell^{q_{1}\cdot(q_{2}/q_{1})^{\prime}}}<\infty\,,$
(7.15)

where $t:=\max(0,q_{1}^{-1}-\min(p_{2}^{-1},1-p_{2}^{-1}))$.

###### Proof.

By [36, Lemma 9.10], $\mathcal{B}$ is a tight semi-structured covering of
$\mathbb{R}^{d}$. Further, Theorem 5.4 shows that $\Phi_{\varrho}$ is a
$(d+1)$-admissible warping function, such that Theorem 2.5 shows that the
warped coorbit spaces
$\operatorname{Co}(\Phi_{\varrho},\mathbf{L}_{\kappa}^{p_{1},q_{1}})$ are
well-defined and independent of the choice of
$0\neq\theta\in\mathcal{D}(\mathbb{R}^{d})$. Lemmas 3.2 and 3.5 show that
$\mathcal{Q}_{\Phi_{\varrho}}^{(\delta,r)}$ is semi-structured and tight.
Finally, by Lemma 7.8, the tuple $(\Phi_{\varrho},\kappa)$ is a compatible
pair as per Definition 2.4, such that the assumptions of Theorem 4.16 are
satisfied, and
$\operatorname{Co}(\Phi_{\varrho},\mathbf{L}_{\kappa}^{p_{1},q_{1}})=\mathcal{D}(\mathcal{Q}_{\Phi_{\varrho}}^{(\delta,r)},\mathbf{L}^{p_{1}},\ell_{u^{(q_{1})}}^{q_{1}})$,
with $\delta>0$ and $r>\sqrt{d}/2$ arbitrary, up to canonical identifications.
The weight $u^{(q_{1})}$ is defined as in (4.2), i.e.,

$u_{k}^{(q_{1})}=\kappa_{\Phi_{\varrho}}(\delta k)\cdot[w(\delta
k)]^{\frac{1}{q_{1}}-\frac{1}{2}}.$

Since, by assumption, there exist $C,R>0$, such that
$\left|\tfrac{\varrho_{\ast}(\gamma)\cdot\varrho_{\ast}^{\prime\prime}(\gamma)}{[\varrho_{\ast}^{\prime}(\gamma)]^{2}}\right|\leq
C$, for all $\gamma\in[R,\infty)$, Proposition 7.1 shows that
$\mathcal{Q}_{\Phi_{\varrho}}^{(\delta,r)}$ is almost subordinate to, and
relatively moderate with respect to, $\mathcal{B}$ and, as discussed in Remark
7.2, $u^{(q_{1})}$ is relatively $\mathcal{B}$-moderate as well. Hence, the
assumptions of Theorem 6.2, Items (1) and (2), are satisfied.

Finally, together with Lemma 3.2, which specifies that the operator $T_{k}$ in
the semi-structured representation of $Q_{\Phi_{\varrho},k}^{(\delta,r)}$ is
given by $T_{k}=D\Phi_{\varrho}^{-1}(\delta k)$, we obtain

$|\det T_{k}|=w(\delta
k)\asymp\varrho_{\ast}^{\prime}(\varrho(2^{j}))\cdot\big{(}2^{j}\big{/}\varrho(2^{j})\big{)}^{d-1}=:\gamma_{j}\quad\text{and}\quad
u_{k}^{(q_{1})}\asymp 2^{s_{1}j}\cdot\gamma_{j}^{q_{1}^{-1}-2^{-1}}\,,$

if $k\in\mathbb{Z}^{d}$ and $j\in\mathbb{N}_{0}$ satisfy $B_{j}\cap
Q_{\Phi_{\varrho},k}^{(\delta,r)}\neq\varnothing$. The equivalences (7.14) and
(7.15) are now direct consequences of Theorem 6.2, Items (2) and (1),
respectively. ∎

Closing the discussion of Besov embeddings, we show that for $d=1$, and
choosing the radial warping function derived from a certain weakly
$k$-admissible radial component, $B^{p,q}_{s}(\mathbb{R})$ is recovered as a
warped coorbit space. For $d\geq 2$, $B^{p,q}_{s}(\mathbb{R}^{d})$ is nested
between two warped coorbit spaces which only differ by a minimal change of the
weight $\kappa$.

######  Theorem 7.10.

The function $\varsigma_{1}:[0,\infty)\to[0,\infty),\xi\mapsto\ln(1+\xi)$ is a
weakly $k$-admissible radial component for any $k\in\mathbb{N}_{0}$. Assume
that $1\leq p,q\leq\infty$ and let
$\varrho_{1}:\mathbb{R}\rightarrow\mathbb{R}$ be any _slow-start_ version of
$\varsigma_{1}$, see Theorem 5.6. Further, define for $s\in\mathbb{R}$ and
$d\in\mathbb{N}$,

$\kappa^{(s,q)}:=\kappa^{(d,s,q)}\colon\mathbb{R}^{d}\rightarrow\mathbb{R}^{+},\quad\xi\mapsto(1+|\xi|)^{s+d(2^{-1}-q^{-1})}.$

We have

$\operatorname{Co}(\Phi_{\varrho_{1}},\mathbf{L}^{p,q}_{\kappa^{(s+\varepsilon,q)}})\hookrightarrow
B_{s}^{p,q}(\mathbb{R}^{d})\hookrightarrow\operatorname{Co}(\Phi_{\varrho_{1}},\mathbf{L}^{p,q}_{\kappa^{(s-\varepsilon,q)}})\qquad\forall\,\varepsilon>0\,.$
(7.16)

If and only if $d=1$ or $p=q=2$, the equality

$B_{s}^{p,q}(\mathbb{R}^{d})=\operatorname{Co}(\Phi_{\varrho_{1}},\mathbf{L}^{p,q}_{\kappa^{(s,q)}})\,$
(7.17)

holds. If $d\geq 2$ and $p\neq 2$, then neither of the two embeddings implied
by (7.17) is valid. If $d\geq 2$, and $p=2$, then
$B_{s}^{p,q}(\mathbb{R}^{d})\hookrightarrow\operatorname{Co}(\Phi_{\varrho_{1}},\mathbf{L}^{p,q}_{\kappa^{(s,q)}})$
if and only if $q\leq 2$ and, likewise,
$\operatorname{Co}(\Phi_{\varrho_{1}},\mathbf{L}^{p,q}_{\kappa^{(s,q)}})\hookrightarrow
B_{s}^{p,q}(\mathbb{R}^{d})$ if and only if $q\geq 2$.

###### Proof.

It was already shown in [19, Example 8.16] that $\varsigma_{1}$ is weakly
$k$-admissible, for any $k\in\mathbb{N}_{0}$. Hence, we conclude that
$\Phi_{\varrho}$ is $(d+1)$-admissible for any $d\in\mathbb{N}$, by Theorems
5.6 and 5.4. Before we can apply Proposition 7.9, it remains to show that
there exist $C,R>0$, such that
$\left|\tfrac{\varrho_{\ast}(\gamma)\cdot\varrho_{\ast}^{\prime\prime}(\gamma)}{[\varrho_{\ast}^{\prime}(\gamma)]^{2}}\right|\leq
C$, for all $\gamma\in[R,\infty)$. However, by construction, see Theorem 5.6,
there is some $R>0$ such that $\varrho_{1}(\xi)=\varsigma_{1}(\xi)$ for all
$\xi\in[R,\infty)$ and, possibly after increasing $R$,
$(\varrho_{1})_{\ast}(\xi)=(\varsigma_{1})_{\ast}(\xi)=e^{\xi}-1$ as well.
Consequently, we obtain the desired bound

$\Big{|}\frac{(\varrho_{1})_{\ast}(\gamma)\cdot(\varrho_{1})_{\ast}^{\prime\prime}(\gamma)}{[(\varrho_{1})_{\ast}^{\prime}(\gamma)]^{2}}\Big{|}=\frac{(e^{\gamma}-1)e^{\gamma}}{e^{2\gamma}}\leq
1\qquad\forall\,\gamma\in[R,\infty)\,.$

Given that $(\varrho_{1})_{\ast}^{\prime}(\xi)=e^{|\xi|}$ for all
$\xi\in\mathbb{R}\setminus(-R,R)$, a short calculation show that

$2^{j}\asymp(\varrho_{1})_{\ast}^{\prime}(\varrho_{1}(2^{j}))\qquad\forall\,j\in\mathbb{N}_{0}\,.$
(7.18)

We are now ready to apply Proposition 7.9, where due to the chosen weight
$\kappa^{(s,q)}$ we have $s_{1}+d(2^{-1}-q^{-1})$ in place of $s_{1}$. By
Equation (7.18), and additionally noting that
$\varrho_{1}(2^{j})\asymp\varsigma_{1}(2^{j})\asymp 1+j$, for all
$j\in\mathbb{N}_{0}$, we can rewrite the norm expression in Equation (7.14),
with $p_{1}=p_{2}=p$ and $q_{1}=q_{2}=q$, as

$\begin{split}K_{\tilde{t}}&=\left\|\left(2^{j(s_{2}-(s_{1}+d(2^{-1}-q^{-1}))+d\tilde{t})}\cdot\left[(\varrho_{1})_{\ast}^{\prime}(\varrho_{1}(2^{j}))\cdot\big{(}2^{j}\big{/}\varrho_{1}(2^{j})\big{)}^{d-1}\right]^{2^{-1}-q^{-1}-\tilde{t}}\right)_{j\in\mathbb{N}_{0}}\right\|_{\ell^{\infty}}\\\
&\asymp\left\|\left(2^{j(s_{2}-s_{1})}\big{/}(1+j)^{(d-1)(-\tilde{t}+2^{-1}-q^{-1})}\right)_{j\in\mathbb{N}_{0}}\right\|_{\ell^{\infty}}=:(*),\end{split}$
(7.19)

with $\tilde{t}=\max(0,\max(p^{-1},1-p^{-1})-q^{-1})$. Similarly, the norm
expression in Equation (7.15) simplifies to

$\begin{split}K_{t}&=\left\|\left(2^{j[(s_{1}+d(2^{-1}-q^{-1}))-s_{2}+dt]}\cdot\left[(\varrho_{1})_{\ast}^{\prime}(\varrho_{1}(2^{j}))\cdot\big{(}2^{j}\big{/}\varrho_{1}(2^{j})\big{)}^{d-1}\right]^{q^{-1}-2^{-1}-t}\right)_{j\in\mathbb{N}_{0}}\right\|_{\ell^{\infty}}\\\
&\asymp\left\|\left(2^{j(s_{1}-s_{2})}\big{/}(1+j)^{(d-1)(q^{-1}-2^{-1}-t)}\right)_{j\in\mathbb{N}_{0}}\right\|_{\ell^{\infty}}=:(\star),\end{split}$
(7.20)

with $t:=\max(0,q^{-1}-\min(p^{-1},1-p^{-1}))$. In the setting of this
Theorem, we consider $s_{2}=s$ and
$s_{1}\in\\{s-\varepsilon,s,s+\varepsilon\\}$. It is easily seen that
$(\ast)<\infty$ if $s=s_{2}<s_{1}=s+\varepsilon$ and likewise $(\star)<\infty$
if $s=s_{2}>s_{1}=s-\varepsilon$, such that the closeness relation (7.16)
follows from Proposition 7.9.

If $s_{1}=s_{2}=s$, then

$(\ast)=\left\|\left((1+j)^{(d-1)(\tilde{t}-2^{-1}+q^{-1})}\right)_{j\in\mathbb{N}_{0}}\right\|_{\ell^{\infty}},\quad\text{and}\quad(\star)=\left\|\left((1+j)^{(d-1)(t-q^{-1}+2^{-1})}\right)_{j\in\mathbb{N}_{0}}\right\|_{\ell^{\infty}}$

which are finite if and only if $0\geq(d-1)(\tilde{t}-2^{-1}+q^{-1})$,
respectively $0\geq(d-1)(t-q^{-1}+2^{-1})$. Both inequalities are trivially
satisfied if $d=1$, such that the equality (7.17) holds. If $d>1$, then
$d-1>0$, but a simple check of all possible configurations reveals that
$\tilde{t}-2^{-1}+q^{-1}$ is positive, unless $p=2$ and $q\geq 2$, when it
equals $0$. Likewise, $t-q^{-1}+2^{-1}>0$, unless $p=2$ and $q\leq 2$, when it
equals $0$. Hence, Proposition 7.9 yields the equality (7.17) for $d=1$ and
$p=q=2$. The above discussion also settles the remaining cases for $d\geq 2$,
as stated in the theorem. This completes the proof. ∎

######  Remark 7.11.

It is easy to see that the statement of Theorem 7.10 remains true, if we
substitute $\kappa=\kappa^{s,q}$ given in the theorem by

$\kappa^{(1,s,q)}:=\kappa^{(1,s,d,q)}\colon\mathbb{R}^{d}\rightarrow\mathbb{R}^{+},\
\xi\mapsto(1+|\xi|)^{s+d(2^{-1}-q^{-1})}\cdot(1+|\ln(1+|\xi|)|)^{(1-d)(2^{-1}-q^{-1})},$

a weight that we will encounter again in our study of embeddings between
warped coorbit spaces and $\alpha$-modulation spaces in Section 8.

To see that $\Phi_{\varrho_{1}}$ and $\kappa^{(1,s,q)}$ are compatible, recall
that $\Phi_{\varrho_{1}}$ and $(1+|\bullet|)^{s}$ are compatible by Lemma 7.8,
and that
$(1+|\bullet|)^{d(2^{-1}-q^{-1})}\cdot(1+|\ln(1+|\bullet|)|)^{(1-d)(2^{-1}-q^{-1})}$
is just $[w\circ\Phi_{\varrho_{1}}]^{2^{-1}-q^{-1}}$, which is clearly
compatible with $\Phi_{\varrho_{1}}$, in disguise: By construction of
$\varrho_{1}$, we have, for all $\xi\in\mathbb{R}^{d}\setminus
B_{2\varepsilon}(0)$, $|\Phi_{\varrho_{1}}(\xi)|=\ln(1+|\xi|)$. Furthermore,
with $\delta=\Phi_{\varrho_{1}}(2\varepsilon)>0$, we have, for all
$\tau\in\mathbb{R}^{d}\setminus B_{\delta}(0)$,that
$w(\tau)=e^{d|\tau|}\cdot|\tau|^{1-d}\asymp e^{d|\tau|}\cdot(1+|\tau|)^{1-d}$.
Hence,
$w(\Phi_{\varrho_{1}}(\xi))\asymp(1+|\bullet|)^{d}\cdot(1+|\ln(1+|\bullet|)|)^{(1-d)}$
for all $\xi\in\mathbb{R}^{d}\setminus B_{2\varepsilon}(0)$. On the other
hand, if $\xi\in\overline{B_{2\varepsilon}(0)}$, then
$w(\Phi_{\varrho_{1}}(\xi))\asymp
1\asymp(1+|\bullet|)^{d}\cdot(1+|\ln(1+|\bullet|)|)^{(1-d)}$, such that both
weights are equivalent. In particular,
$(1+|\bullet|)^{d}\cdot(1+|\ln(1+|\bullet|)|)^{(1-d)}$ is
$Cv_{0}^{d}$-moderate, where $C\geq 1$ and $v_{0}$ is a control weight for the
$k$-admissible warping function $\Phi_{\varrho_{1}}$.

Now, by exactly the same steps as in the proof of Theorem 7.10, the
expressions (7.14) and (7.15) are equivalent to the even simpler

$\left\|\left(2^{j(s_{2}-s_{1})}\cdot(1+j)^{(d-1)\tilde{t}}\right)_{j\in\mathbb{N}_{0}}\right\|_{\ell^{\infty}},\quad\text{and}\quad\left\|\left(2^{j(s_{1}-s_{2})}\cdot(1+j)^{(d-1)t}\right)_{j\in\mathbb{N}_{0}}\right\|_{\ell^{\infty}},\quad\text{respectively.}$

These conditions are, in turn, satisfied for the exact same values of $s_{1},\
s_{2}$ and $\tilde{t},\ t$ as the analogous conditions given in (7.19) and
(7.20).

######  Remark 7.12.

Note that it can be shown that shearlet-type coverings as discussed in [23,
39] are weakly subordinate to the covering
$\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho}}$. In fact, the _connected_ shearlet
covering (see [39, Definition 3.1]) is almost subordinate to
$\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho}}$ and also relatively moderate with
respect to $\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho}}$. Hence, one can also
completely characterize the embedding relations between
$\operatorname{Co}(\Phi_{\rho},L^{p,q}_{\kappa_{\gamma}})$ and the shearlet
smoothness spaces discussed in [23, 39]. We leave the detailed derivation to
the interested reader.

We will now turn our attention to _$\alpha$ -modulation spaces_ and (Besov-
type) _spaces of dominating mixed smoothness_ [27, 28, 40], showing that these
spaces coincide with certain warped coorbit spaces. In either case, we note
that the discretization results in [19] imply that warped time-frequency
systems provide atomic decompositions for the respective spaces that differ
from previous constructions, see, e.g., [2, 33, 6] for $\alpha$-modulation
spaces and [40, Section 2.4] for spaces of dominating mixed smoothness.

##  8\. Warped coorbit spaces as ($\alpha$-)modulation spaces

The $\alpha$-modulation spaces [15, 2] are a family of decomposition spaces
defined with respect to coverings that are in some sense geometrically
intermediate between the uniform covering and the dyadic, (inhomogeneous)
Besov covering. We will show that the radial warping obtained from
$\varsigma_{\beta}(t)=(1+t)^{1/\beta}-1$ for $\beta\geq 1$ recovers
$\alpha$-modulation spaces with $\alpha=1-\beta^{-1}\in[0,1)$. The following
definition is a slightly extended version of [2, Definitions 2.1 and 2.4].

######  Definition 8.1.

Let $\mathcal{Q}=(Q_{i})_{i\in I}$ be an admissible covering of
$\mathbb{R}^{d}$, and let $\alpha\in(-\infty,1]$. The family $\mathcal{Q}$ is
called an $\alpha$-covering of $\mathbb{R}^{d}$, if the following hold:

  1. (1)

With

$\quad\quad\quad r_{Q}:=\sup\\{r>0\,:\,\exists\
\xi\in\mathbb{R}^{d}:B_{r}(\xi)\subset
Q\\}\quad\\!\\!\text{and}\quad\\!\\!R_{Q}:=\inf\\{R>0\,:\,\exists\
\xi\in\mathbb{R}^{d}:Q\subset B_{R}(\xi)\\}\,,$ (8.1)

we have $\sup_{i\in I}R_{Q_{i}}/r_{Q_{i}}<\infty$.

  2. (2)

We have $\mu(Q_{i})\asymp(1+|\xi|)^{\alpha\cdot d}$ for all $i\in I$ and
$\xi\in Q_{i}$, where the implied constant does not depend on $i,\xi$, and
where $\mu$ denotes the Lebesgue measure as usual.

If $1\leq p,q\leq\infty$, $\mathcal{Q}^{(\alpha)}$ is an $\alpha$-covering,
and $u^{s}=(u^{s}_{i})_{i\in I}$, with
$u^{s}_{i}=[\mu(Q_{i}^{(\alpha)})]^{s(d\alpha)^{-1}}$ for some
$s\in\mathbb{R}$, then the _$\alpha$ -modulation space_
$M^{s,\alpha}_{p,q}(\mathbb{R}^{d})$ is defined by

$M_{p,q}^{s,\alpha}(\mathbb{R}^{d}):=\mathcal{D}(\mathcal{Q}^{(\alpha)},\mathbf{L}^{p},\ell_{u^{s}}^{q}).$
(8.2)

######  Remark 8.2.

In Appendix A, we show that the resulting function space is independent of the
choice of the $\alpha$-covering $\mathcal{Q}^{(\alpha)}$, see also [2, Lemma
B.2].

Note that prior work [15, 2] was concerned specifically with the case
$\alpha\in[0,1]$. We show in Appendix A that $\alpha\in(-\infty,1]$ is indeed
the most general setting, and the natural generalization of Defintion 8.1 to
$\alpha>1$ yields an empty set of $\alpha$-coverings for any such $\alpha$. To
justfiy the notion of $\alpha$-coverings, $\alpha\in[0,1]$, as _intermediate_
between the uniform and dyadic coverings, consider Item (2) in Definition 8.1
and observe that the uniform covering $\mathcal{Q}=(Q+k)_{k\in\mathbb{Z}^{d}}$
with a suitable base set $Q$ satisfies $\mu(Q+k)\asymp 1=(1+|\xi|)^{0\cdot d}$
for $\xi\in Q+k$, whereas the dyadic covering $\mathcal{B}$ consisting of the
annuli $B_{j}=B_{2^{j+1}}(0)\setminus\overline{B_{2^{j-1}}(0)}$ for
$j\in\mathbb{N}$, and a ball $B_{0}=B_{2}(0)$, satisfies $\mu(B_{j})\asymp
2^{dj}\asymp(1+|\xi|)^{1\cdot d}$ for $\xi\in B_{j}$.

In the following, we will consider warping functions
$\Phi_{\varrho_{\alpha}}$, $\alpha\in(-\infty,1)$, derived from a slow start
version of

$\varsigma_{\alpha}\colon[0,\infty)\rightarrow[0,\infty),\quad\xi\mapsto(1+\xi)^{1-\alpha}-1.$
(8.3)

A straightforward derivation shows that $\varsigma_{\alpha}$ is indeed a
weakly $k$-admissible radial component, for any $k\in\mathbb{N}_{0}$. Hence,
any slow start version $\varrho_{\alpha}$ of $\varsigma_{\alpha}$, generates a
$k$-admissible warping function $\Phi_{\varrho_{\alpha}}$, see Theorem 5.6 and
the ensuing remark. We now proceed to show that
$\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{\alpha}}}$ is an $\alpha$-covering,
such that
$\operatorname{Co}(\Phi_{\varrho_{\alpha}},\mathbf{L}^{p,q}_{\kappa})=\mathcal{D}(\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{\alpha}}},\mathbf{L}^{p},\ell_{u^{s}}^{q})=M_{p,q}^{s,\alpha}(\mathbb{R}^{d})$,
for an appropriate choice of $\kappa=\kappa^{(\alpha,s,d,q)}$.

######  Lemma 8.3.

Let $\alpha\in(-\infty,1)$ and $\varrho_{\alpha}$ be an arbitrary slow start
version of $\varsigma_{\alpha}$, as in Theorem 5.6. Then, for any $\delta>0$
and $r\geq\sqrt{d}/2$, $\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{\alpha}}}$ is
an $\alpha$-covering.

###### Proof.

Since $\varrho_{\alpha}$ is a slow start version of $\varsigma_{\alpha}$,
there is a $R>0$, such that with $\beta=(1-\alpha)^{-1}\in\mathbb{R}^{+}$, we
have

$(\varrho_{\alpha})_{\ast}(\xi)=(\varsigma_{\alpha})_{\ast}(\xi)=(1+\xi)^{\beta}-1\quad\text{and}\quad(\varrho_{\alpha})_{\ast}^{\prime}(\xi)=\beta\cdot(1+\xi)^{\beta-1},\quad\text{for
all }\xi\geq R,$

as well as $\widetilde{(\varrho_{\alpha})_{\ast}}\asymp
1\asymp(\varrho_{\alpha})_{\ast}^{\prime}$ in $[0,R)$. Using $\xi\asymp 1+\xi$
for $\xi>R$, we see that in fact

$\widetilde{(\varrho_{\alpha})_{\ast}}(\xi)\asymp(1+\xi)^{\beta-1}\asymp(\varrho_{\alpha})_{\ast}^{\prime}(\xi)$

on all of $[0,\infty)$ and with the trivial equality
$\frac{x^{p}}{y^{p}}=\frac{y^{-p}}{x^{-p}}$ for $x,y,p\in\mathbb{R}$, we find
that

$v\,\colon\,\mathbb{R}\rightarrow\mathbb{R}^{+},\ \xi\mapsto
C\cdot(1+|\xi|)^{|\beta-1|},$

is a control weight for $\varrho_{\alpha}$ as per Definition 5.2, for an
appropriate constant $C>0$ depending on $d$ and $\beta$.

In order to show that $\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{\alpha}}}$
satisfies Definition 8.1(1), note that

$\begin{split}\left|\Phi_{\varrho_{\alpha}}^{-1}(\xi)-\Phi_{\varrho_{\alpha}}^{-1}(\omega)\right|&=\left|\int_{0}^{1}D\Phi_{\varrho_{\alpha}}^{-1}(\omega+s(\xi-\omega))\langle\xi-\omega\rangle~{}ds\right|\\\
&\leq\int_{0}^{1}\left|D\Phi_{\varrho_{\alpha}}^{-1}(\omega+s(\xi-\omega))\langle\xi-\omega\rangle\right|~{}ds.\end{split}$

By [19, Lemma 8.4], we have

$D\Phi_{\varrho_{\alpha}}^{-1}(\omega+s(\xi-\omega))=\widetilde{(\varrho_{\alpha})_{\ast}}(|\omega+s(\xi-\omega)|)\cdot\pi^{\bot}_{\omega+s(\xi-\omega)}+(\varrho_{\alpha})_{\ast}^{\prime}(|\omega+s(\xi-\omega)|)\cdot\pi_{\omega+s(\xi-\omega)}$

In particular, an application of Parseval’s identity for orthonormal bases
yields

$\begin{split}\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\left|D\Phi_{\varrho_{\alpha}}^{-1}(\omega+s(\xi-\omega))\langle\xi-\omega\rangle\right|^{2}$\mbox{}\hfil\\\
&=|\widetilde{(\varrho_{\alpha})_{\ast}}(|\omega+s(\xi-\omega)|)\cdot\pi^{\bot}_{\omega+s(\xi-\omega)}(\xi-\omega)|^{2}+|(\varrho_{\alpha})_{\ast}^{\prime}(|\omega+s(\xi-\omega)|)\cdot\pi_{\omega+s(\xi-\omega)}(\xi-\omega)|^{2}\\\
&\asymp(\varrho_{\alpha})_{\ast}^{\prime}(|\omega+s(\xi-\omega)|)^{2}\left(|\pi^{\bot}_{\omega+s(\xi-\omega)}(\xi-\omega)|^{2}+|\pi_{\omega+s(\xi-\omega)}(\xi-\omega)|^{2}\right)\\\
&=|(\varrho_{\alpha})_{\ast}^{\prime}(|\omega+s(\xi-\omega)|)|^{2}\cdot|\xi-\omega|^{2},\end{split}$

where we used
$(\varrho_{\alpha})_{\ast}^{\prime}\asymp\widetilde{(\varrho_{\alpha})_{\ast}}$
and $(\varrho_{\alpha})_{\ast}^{\prime}>0$. Altogether, we obtain

$\left|\Phi_{\varrho_{\alpha}}^{-1}(\xi)-\Phi_{\varrho_{\alpha}}^{-1}(\omega)\right|\asymp\int_{0}^{1}|(\varrho_{\alpha})_{\ast}^{\prime}(|\omega+s(\xi-\omega)|)|\cdot|\xi-\omega|~{}ds.$

Now set $\omega=\delta k$ and let
$\xi\in\delta\cdot(\overline{B_{r}(k)}\setminus B_{r}(k))$, i.e., $|\delta
k-\xi|=\delta r$. Since $s(\omega-\xi)\in\overline{B_{\delta r}(0)}$, and
$(\varrho_{\alpha})_{\ast}^{\prime}$ is $v$-moderate with radially increasing
weight $v$, we further have

$[v(\delta r)]^{-1}\cdot(\varrho_{\alpha})_{\ast}^{\prime}(|\delta
k|)\leq(\varrho_{\alpha})_{\ast}^{\prime}(|\delta k+s(\xi-\delta k)|)\leq
v(\delta r)\cdot(\varrho_{\alpha})_{\ast}^{\prime}(|\delta k|),$

such that

$\left|\Phi_{\varrho_{\alpha}}^{-1}(\xi)-\Phi_{\varrho_{\alpha}}^{-1}(\delta
k)\right|\asymp(\varrho_{\alpha})_{\ast}^{\prime}(|\delta k|)\asymp(1+|\delta
k|)^{\beta-1}.$

We conclude that $R_{Q^{(\delta,r)}_{\Phi_{\varrho_{\alpha}},k}}$ and
$r_{Q^{(\delta,r)}_{\Phi_{\varrho_{\alpha}},k}}$, defined as in Equation (8.1)
satisfy

$R_{Q^{(\delta,r)}_{\Phi_{\varrho_{\alpha}},k}}\asymp
r_{Q^{(\delta,r)}_{\Phi_{\varrho_{\alpha}},k}}\asymp(1+|\delta
k|)^{\beta-1}\quad\text{and}\quad\sup_{k\in\mathbb{Z}^{d}}R_{Q^{(\delta,r)}_{\Phi_{\varrho_{\alpha}},k}}/r_{Q^{(\delta,r)}_{\Phi_{\varrho_{\alpha}},k}}<\infty.$
(8.4)

It remains to prove that
$\mu(Q^{(\delta,r)}_{\Phi_{\varrho_{\alpha}},k})\asymp(1+|\tau|)^{d\alpha}$,
for all $\tau\in Q^{(\delta,r)}_{\Phi_{\varrho_{\alpha}},k}$. For
$\tau=\Phi_{\varrho_{\alpha}}^{-1}(\delta k)$, we have $|\delta
k|=|\Phi_{\varrho_{\alpha}}(\tau)|\asymp(1+|\tau|)^{1/\beta}-1$, such that

$(1+|\delta
k|)^{\beta-1}\asymp(1+|\tau|)^{(\beta-1)/\beta}\asymp(1+|\tau|)^{\alpha}.$

With Equation (8.4), we conclude that
$\mu(Q^{(\delta,r)}_{\Phi_{\varrho_{\alpha}},k})\asymp(1+|\tau|)^{d\alpha}$
for this value of $\tau$. On the other hand, if
$\tau=\Phi_{\varrho_{\alpha}}^{-1}(\xi)$, with $\xi\in\delta\cdot B_{r}(\delta
k)$, then by the same argument,

$(1+|\tau|)^{(\beta-1)/\beta}\asymp(1+|\xi|)^{\beta-1}\overset{(\ast)}{\asymp}(1+|\delta
k|)^{\beta-1},$

where the proportionality constants for $(\ast)$ can be chosen as $(1+\delta
r)^{-|\beta-1|}$ and $(1+\delta r)^{|\beta-1|}$, using that
$(1+|\cdot|)^{\beta-1}$ is $(1+|\cdot|)^{|\beta-1|}$-moderate. Overall,
$\mu(Q^{(\delta,r)}_{\Phi_{\varrho_{\alpha}},k})\asymp(1+|\tau|)^{d\alpha}$
for all $\tau\in Q^{(\delta,r)}_{\Phi_{\varrho_{\alpha}},k}$, completing the
proof. ∎

We are now ready to show equality between $\alpha$-modulation spaces and
certain coorbit spaces associated to $\Phi_{\varrho_{\alpha}}$.

######  Proposition 8.4.

Let $\alpha\in(-\infty,1)$, $1\leq p,q\leq\infty$ and choose any slow start
version $\varrho_{\alpha}$ of $\varsigma_{\alpha}$. Denote
$w_{\alpha}=\det(D\Phi_{\varrho_{\alpha}}^{-1}(\bullet))$ and define the
weight
$\kappa:=\kappa^{(\alpha,s,d,q)}\,\colon\,\mathbb{R}^{d}\rightarrow[0,\infty)$
by

$\kappa:=\kappa^{(\alpha,s,d,q)}\,\colon\,\mathbb{R}^{d}\rightarrow[0,\infty),\
\xi\mapsto(1+|\xi|)^{\gamma},\quad\text{with
}\gamma=s-d\alpha(q^{-1}-2^{-1}).$ (8.5)

Then, up to canonical identifications,

$\operatorname{Co}(\Phi_{\varrho_{\alpha}},\mathbf{L}^{p,q}_{\kappa})=\mathcal{D}(\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{\alpha}}},\mathbf{L}^{p},\ell_{u}^{q})=M_{p,q}^{s,\alpha}(\mathbb{R}^{d})\,,$
(8.6)

where $u=(u_{k})_{k\in\mathbb{Z}^{d}}$, with
$u_{k}=\kappa(\Phi^{-1}_{\varrho_{\alpha}}(\delta k))\cdot[w_{\alpha}(\delta
k)]^{q^{-1}-2^{-1}}$.

###### Proof.

In Lemma 8.3, we have shown that
$\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{\alpha}}}$ is an $\alpha$-cover,
such that the second equality in (8.6) follows from the definition of the
$\alpha$-modulation space $M_{p,q}^{s,\alpha}(\mathbb{R}^{d})$ as
decomposition space if we show that

$u_{k}=\kappa(\Phi^{-1}_{\varrho_{\alpha}}(\delta k))\cdot[w_{\alpha}(\delta
k)]^{q^{-1}-2^{-1}}\asymp[\mu(Q^{(\delta,r)}_{\Phi_{\varrho_{\alpha}},k})]^{s(d\alpha)^{-1}}$

as a function in $k\in\mathbb{Z}^{d}$. In the proof of Lemma 8.3, we have
shown that $\mu(Q^{(\delta,r)}_{\Phi_{\varrho_{\alpha}},k})\asymp(1+|\delta
k|)^{d(\beta-1)}\asymp(1+|\delta k|)^{\frac{d\alpha}{1-\alpha}}$, with
$\beta=(1-\alpha)^{-1}$. Furthermore, $\kappa(\Phi^{-1}_{\alpha}(\delta
k))=(1+|\Phi_{\varrho_{\alpha}}^{-1}(\delta k)|)^{\gamma}$ and, by Theorem
5.4, $w(\delta k)=(\varrho_{\alpha})_{\ast}^{\prime}(|\delta
k|)\cdot[\widetilde{(\varrho_{\alpha})_{\ast}}(|\delta
k|)]^{d-1}\asymp(1+|\delta k|)^{d(\beta-1)}=(1+|\delta
k|)^{\frac{d\alpha}{1-\alpha}}$. Finally, since there exists some $R>0$, such
that
$(1+|\Phi_{\varrho_{\alpha}}^{-1}(\xi)|)^{\gamma}=(1+|\xi|)^{\gamma/(1-\alpha)}$
for all $\xi\in\mathbb{R}^{d}\setminus B_{R}(0)$, we have, in fact
$(1+|\Phi_{\varrho_{\alpha}}^{-1}(\xi)|)^{\gamma}\asymp(1+|\xi|)^{\gamma/(1-\alpha)}$
for all $\xi\in\mathbb{R}^{d}$. Putting all these estimates together, we
obtain

$u_{k}\asymp(1+|\delta k|)^{\frac{s}{1-\alpha}}=\left((1+|\delta
k|)^{\frac{d\alpha}{1-\alpha}}\right)^{\frac{s}{d\alpha}}\asymp[\mu(Q^{(\delta,r)}_{\Phi_{\varrho_{\alpha}},k})]^{s(d\alpha)^{-1}}.$

Hence, we have shown that
$\mathcal{D}(\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{\alpha}}},\mathbf{L}^{p},\ell_{u}^{q})=M_{p,q}^{s,\alpha}(\mathbb{R}^{d})$.

To show the first equality in (8.6), we want to apply Theorem 4.16. To do so,
we only need to show that
$(\Phi_{\varrho_{\alpha}},\kappa_{\Phi_{\varrho_{\alpha}}})$ form a compatible
pair as per Defintion 2.4. $\Phi_{\alpha}$ is a
$\mathcal{C}^{\infty}$-diffeomorphism and a $(d+1)$-admissible warping
function by construction, see also Theorem 5.4. As discussed above, the
transplanted weight $\kappa_{\Phi_{\varrho_{\alpha}}}$ satisfies
$\kappa_{\Phi_{\varrho_{\alpha}}}(\xi)\asymp(1+|\xi|)^{\gamma/(1-\alpha)}$,
for all $\xi\in\mathbb{R}^{d}$, and is clearly moderate with respect to the
continuous, radial and submultiplicative weight $\xi\mapsto
C(1+|\xi|)^{|\gamma/(1-\alpha)|}$, for any sufficiently large $C\geq 1$.
Hence, $\Phi_{\varrho_{\alpha}}$ and $\kappa_{\Phi_{\varrho_{\alpha}}}$ are
compatible and Theorem 4.16 yields the desired equality
$\operatorname{Co}(\Phi_{\varrho_{\alpha}},\mathbf{L}^{p,q}_{\kappa})=\mathcal{D}(\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{\alpha}}},\mathbf{L}^{p},\ell_{u}^{q})=M_{p,q}^{s,\alpha}(\mathbb{R}^{d})$.
∎

After Definition 8.1, we remarked the the inhomogeneous, dyadic Besov covering
$\mathcal{B}$ is an $\alpha$-covering for $\alpha=1$. However, the annulus-
based geometry of $\mathcal{B}$ is fundamentally different from the the
coverings $\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{\alpha}}}$,
$\alpha\in[0,1)$, which consist of convex sets. Warped coorbit spaces provide
an interesting alternative with radial warping based on the weakly
$k$-admissible radial component $\varsigma_{1}=\ln(1+\bullet)$, which we have
related to Besov spaces in the previous subsection. We obtain the following
result.

######  Proposition 8.5.

Let $\alpha\in(-\infty,1)$, $1\leq p,q\leq\infty$ and let $\varrho_{1}$ be any
slow start version of $\varsigma_{1}=\ln(1+\bullet)$. Define
$\Phi_{1}:=\Phi_{\varrho_{1}}$ and let $w_{1}:=\det(\mathrm{D}\Phi_{1}^{-1})$
be the associated weight. Fix $\varepsilon>0$ and let
$\tilde{T}:=\tilde{T}_{d,\alpha}:=\tilde{t}d(1-\alpha)$, with
$\tilde{t}=-\max(0,\max(p^{-1},1-p^{-1})-q^{-1})$ and
$T:=T_{d,\alpha}:=td(1-\alpha)\geq 0$, with
$t=\max(0,q^{-1}-\min(p^{-1},1-p^{-1}))$. Then we have the embedding

$M_{p,q}^{s+\tilde{T},\alpha}(\mathbb{R}^{d})\hookrightarrow\operatorname{Co}(\Phi_{\varrho_{1}},\mathbf{L}^{p,q}_{\kappa})\hookrightarrow
M_{p,q}^{s-T,\alpha}(\mathbb{R}^{d})$ (8.7)

with the weight
$\kappa:=\kappa^{(1,s,d,q)}\,\colon\,\mathbb{R}^{d}\rightarrow[0,\infty)$
given by

$\kappa^{(1,s,d,q)}\,\colon\,\mathbb{R}^{d}\rightarrow[0,\infty),\
\xi\mapsto(1+|\xi|)^{s+d(2^{-1}-q^{-1})}\cdot(1+\ln(1+|\xi|))^{(1-d)(2^{-1}-q^{-1})}.$
(8.8)

###### Proof.

Let, as before, $\varrho_{\alpha}$ be a slow start version of
$\varsigma_{\alpha}$, see Equation (8.3). Let further,
$w_{\alpha}=\det(D\Phi_{\varrho_{\alpha}}(\bullet))$ denote the weight
associated with $\Phi_{\varrho_{\alpha}}$. Proposition 8.4 shows that
$M_{p,q}^{s,\alpha}(\mathbb{R}^{d})=\operatorname{Co}(\Phi_{\varrho_{\alpha}},\mathbf{L}^{p,q}_{\kappa})$,
where
$\kappa(\xi)=\kappa^{(\alpha,s,d,q)}(\xi)=(1+|\xi|)^{s+d\alpha(2^{-1}-q^{-1})}$
is as in (8.5). In particular, $\Phi_{\varrho_{\alpha}}$ and
$\kappa^{(\alpha,s,d,q)}$ are compatible. Compatibility of
$\Phi_{\varrho_{1}}$ and $\kappa^{(1,s,d,q)}$ was already proven in Remark
7.11, where we also saw that
$w_{1}\circ\Phi_{\varrho_{1}}\asymp(1+|\bullet|)^{d}(1+|1+|\ln(1+|\bullet|)|)^{1-d}$,
such that
$\kappa^{(1,s,d,q)}\asymp(1+|\bullet|)^{s}\cdot[w_{1}\circ\Phi_{\varrho_{1}}]^{2^{-1}-q^{-1}}$.
Similarly we can show that
$w_{\alpha}\circ\Phi_{\varrho_{\alpha}}\asymp(1+|\bullet|)^{d\alpha}$ and thus
$\kappa^{(\alpha,s,d,q)}\asymp(1+|\bullet|)^{s}\cdot[w_{\alpha}\circ\Phi_{\varrho_{\alpha}}]^{2^{-1}-q^{-1}}$.

Moreover, with the above derivations, and setting $\kappa=\kappa^{(1,s,d,q)}$,
we can already derive the following equivalence, which holds for any
$t\in\mathbb{R}$ and will be used later in the proof.

$\begin{split}\kappa\cdot\left[\frac{w_{1}\circ\Phi_{\varrho_{1}}}{w_{\alpha}\circ\Phi_{\varrho_{\alpha}}}\right]^{q^{-1}-2^{-1}-t}&\asymp(1+|\bullet|)^{s}\cdot[w_{1}\circ\Phi_{\varrho_{1}}]^{-t}\cdot[w_{\alpha}\circ\Phi_{\varrho_{\alpha}}]^{t+2^{-1}-{q}^{-1}}\\\
&\asymp(1+|\xi|)^{s+td(\alpha-1)}\cdot[w_{\alpha}\circ\Phi_{\varrho_{\alpha}}]^{{2}^{-1}-{q}^{-1}}\cdot(1+|\ln(1+|\xi||)^{t(d-1)}\\\
&\asymp\kappa^{(\alpha,s-td(1-\alpha),d,q)}(\xi)\cdot(1+|\ln(1+|\xi|)|)^{t(d-1)}.\end{split}$
(8.9)

We will prove (8.7) by means of Theorem 6.9. Hence, we must first show that
its assumptions are met. By Definition of $\varrho_{\alpha},\ \varrho_{1}$,
there exists some $R>0$, such that
$(\varrho_{\alpha})^{\prime}(\xi)=(1-\alpha)(1+|\xi|)^{-\alpha}$ and
$(\varrho_{1})^{\prime}(\xi)=(1+|\xi|)^{-1}$ holds for all
$\xi\in\mathbb{R}\setminus[-R,R]$. The existence of $C_{\alpha}>0$, such that
$(\varrho_{1})^{\prime}(\xi)\leq C_{\alpha}(\varrho_{\alpha})^{\prime}(\xi)$
for all such $\xi$ is clear. After possibly increasing $C_{\alpha}$, this
inequality extends to all $\xi\in\mathbb{R}$, since $[-R,R]$ is compact and
$(\varrho_{1})^{\prime},(\varrho_{\alpha})^{\prime}>0$ are continuous. In
particular, we can use Corollary 6.5 to see that the conditions of Lemma 6.3
are satisfied, such that $\Phi_{\varrho_{\alpha}}$ and $\kappa^{(1,s,d,q)}$
are compatible. Finally, we have, for all $a\in[1,\infty)$,
$\xi\in\mathbb{R}$, and $\alpha\in[0,1)$, that
$(1+|a\xi|)^{-\alpha}\leq(1+|\xi|)^{-\alpha}\leq
a^{\alpha}\cdot(a(1+|\xi|))^{-\alpha}\leq C(a)(1+|a\xi|))^{-\alpha}$, with
$C(a)<\infty$. For $\alpha\in(-\infty,0)$, analogous steps yield a similar
proportionality relation. Hence, all assumptions of Theorem 6.9 are met, with
$\kappa=\kappa^{(1,s,d,q)}$.

Note that $t=\max(0,q^{-1}-\min(p^{-1},1-p^{-1}))\in[0,1]$, such that Equation
(8.9) yields, with $T:=T_{d,\alpha}:=td(1-\alpha)\geq 0$,

$\kappa_{\varrho_{\alpha},\varrho_{1},t}=\kappa\cdot\left[\frac{w_{1}\circ\Phi_{\varrho_{1}}}{w_{\alpha}\circ\Phi_{\varrho_{\alpha}}}\right]^{\frac{1}{q}-\frac{1}{2}-t}\gtrsim\kappa^{(\alpha,s-T,d,q)}.$

Likewise, if we set
$t=-\tilde{t}=\max(0,\max(p^{-1},1-p^{-1})-q^{-1})\in[-1,0]$ and
$\tilde{T}:=\tilde{T}_{d,\alpha}:=\tilde{t}d(1-\alpha)\geq 0$, then Equation
(8.9) yields

$\kappa_{\varrho_{\alpha},\varrho_{1},-\tilde{t}}=\kappa\cdot\left[\frac{w_{1}\circ\Phi_{\varrho_{1}}}{w_{\alpha}\circ\Phi_{\varrho_{\alpha}}}\right]^{\frac{1}{q}-\frac{1}{2}+\tilde{t}}\lesssim\kappa^{(\alpha,s+\tilde{T},d,q)}.$

The discussion in Section 6.1, in particular Equation (6.1), shows that the
estimates above imply

$\operatorname{Co}(\Phi_{\varrho_{\alpha}},\mathbf{L}^{p,q}_{\kappa^{(\alpha,s+\tilde{T},d,q)}})\hookrightarrow\operatorname{Co}(\Phi_{\varrho_{\alpha}},\mathbf{L}^{p,q}_{\kappa_{\varrho_{\alpha},\varrho_{1},-\tilde{t}}})\quad\text{and}\quad\operatorname{Co}(\Phi_{\varrho_{\alpha}},\mathbf{L}^{p,q}_{\kappa_{\varrho_{\alpha},\varrho_{1},t}})\hookrightarrow\operatorname{Co}(\Phi_{\varrho_{\alpha}},\mathbf{L}^{p,q}_{\kappa^{(\alpha,s-T,d,q)}}).$

Moreover, by Theorem 6.9, we obtain

$\operatorname{Co}(\Phi_{\varrho_{\alpha}},\mathbf{L}^{p,q}_{\kappa_{\varrho_{\alpha},\varrho_{1},-\tilde{t}}})\hookrightarrow\operatorname{Co}(\Phi_{\varrho_{1}},\mathbf{L}^{p,q}_{\kappa^{(1,s,d,q)}})\hookrightarrow\operatorname{Co}(\Phi_{\varrho_{\alpha}},\mathbf{L}^{p,q}_{\kappa_{\varrho_{\alpha},\varrho_{1},t}}),$

and Proposition 8.4 yields
$M_{p,q}^{s+\tilde{T},\alpha}(\mathbb{R}^{d})=\operatorname{Co}(\Phi_{\varrho_{\alpha}},\mathbf{L}^{p,q}_{\kappa^{(\alpha,s+\tilde{T},d,q)}})$
and
$\operatorname{Co}(\Phi_{\varrho_{\alpha}},\mathbf{L}^{p,q}_{\kappa^{(\alpha,s-T,d,q)}})=M_{p,q}^{s-T,\alpha}(\mathbb{R}^{d})$.
Overall, we have shown that

$M_{p,q}^{s+\tilde{T},\alpha}(\mathbb{R}^{d})\hookrightarrow\operatorname{Co}(\Phi_{\varrho_{1}},\mathbf{L}^{p,q}_{\kappa^{(1,s,d,q)}})\hookrightarrow
M_{p,q}^{s-T,\alpha}(\mathbb{R}^{d}),$

as desired.

∎

######  Remark 8.6.

Combining our results from this section and Section 7, we have shown that any
$\alpha$-covering with $\alpha\in(-\infty,1)$ is almost subordinate to
$\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{1}}}$, which is in turn almost
subordinate to the inhomogeneous Besov covering $\mathcal{B}$, which also is
an $\alpha$-covering for $\alpha=1$. In dimension $d=1$,
$\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{1}}}$ and $\mathcal{B}$ are weakly
equivalent, but for $d>1$, Theorem 7.4 shows that $\mathcal{B}$ is never
almost subordinate to $\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{1}}}$, i.e.,
it is not an $\alpha$-covering for $\alpha=1$. Likewise, it can be shown that
$\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{1}}}$ is not almost subordinate to
any $\alpha$-covering with $\alpha\in(-\infty,1)$, in particular, it is not an
$\alpha$-covering at all.

To see this, fix some $\delta>0$ and $r>\sqrt{d}/2$ and denote by
$K_{\ell}=\\{k\in\mathbb{Z}^{d}~{}:~{}Q^{(\delta,r)}_{\Phi_{\varrho_{\alpha}},k}\cap
Q^{(\delta,r)}_{\Phi_{\varrho_{1}},\ell}\neq\varnothing\\}$,
$\ell\in\mathbb{Z}^{d}$. Now, consider
$Q^{(\delta,r)}_{\Phi_{\varrho_{1}},ne_{1}}$, for $n\in\mathbb{N}$ and the
first standard unit vector $e_{1}\in\mathbb{R}^{d}$. For any sufficiently
large $n$, we have

$Q^{(\delta,r)}_{\Phi_{\varrho_{1}},ne_{1}}=\Phi_{\varrho_{1}}^{-1}(\delta\cdot
B_{r}(ne_{1}))\supset\\{ce_{1}~{}:~{}c+1\in e^{|\delta\cdot
B_{r}(ne_{1})|}\\}=\\{ce_{1}~{}:~{}c+1\in e^{(\delta(n-r),\delta(n+r))}\\}.$

Furthermore, for any sufficiently large $m\in\mathbb{N}$, we have

$Q^{(\delta,r)}_{\Phi_{\varrho_{\alpha}},me_{1}}\ni\Phi_{\varrho_{\alpha}}^{-1}(\delta
me_{1})=((1+\delta m)^{\frac{1}{1-\alpha}}-1)e_{1}.$

Hence, if $(1+\delta m)^{\frac{1}{1-\alpha}}\in
e^{(\delta(n-r),\delta(n+r))}$, or equivalently $\ln(1+\delta
m)\in(1-\alpha)\cdot(\delta(n-r),\delta(n+r))$, then $me_{1}\in K_{ne_{1}}$.
In other words, for any sufficiently large $n\in\mathbb{N}$,
$\\{m\in\mathbb{N}~{}:~{}\ln(1+\delta
m)\in(1-\alpha)\cdot(\delta(n-r),\delta(n+r))\\}\subset K_{ne_{1}}$, but
clearly

$|\\{m\in\mathbb{N}~{}:~{}\ln(1+\delta
m)\in(1-\alpha)\cdot(\delta(n-r),\delta(n+r))\\}|\overset{n\rightarrow\infty}{\longrightarrow}\infty.$

Hence, $\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{1}}}$ is not weakly
subordinate to $\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{\alpha}}}$. Since the
elements of both coverings are path-connected, it is thus not almost
subordinate to $\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{\alpha}}}$. Since all
$\alpha$-coverings are equivalent and $\alpha\in(-\infty,1)$ was arbitrary, we
conclude that $\mathcal{Q}^{(\delta,r)}_{\Phi_{\varrho_{1}}}$ is not almost
subordinate to any $\alpha$-covering.

Overall, it seems natural to consider $\mathcal{Q}^{(\delta,r)}_{\Phi_{1}}$ as
an intermediate between $\alpha$-coverings with $\alpha\rightarrow 1$ and the
inhomogeneous Besov covering $\mathcal{B}$. Interestingly, the embedding
relations in Proposition 8.5 exactly mirror those obtained between
$\alpha$-modulation and Besov spaces: Specifically, with
$\tilde{T}=\tilde{T}_{d,\alpha}$ and $T:=T_{d,\alpha}$ as in the proposition,
we obtain by [36, Theorem 9.13, Lemma 9.15, Corollary 9.16],

$M_{p,q}^{s+\tilde{T},\alpha}(\mathbb{R}^{d})\hookrightarrow
B_{s}^{p,q}(\mathbb{R}^{d})\hookrightarrow
M_{p,q}^{s-T,\alpha}(\mathbb{R}^{d}).$

##  9\. Warped coorbit spaces as spaces of dominating mixed smoothness

In Section 7, we have seen that—at least in dimension $d=1$—the radial warping
generated from $\varsigma=\log(1+|\cdot|)$ induces a frequency covering
equivalent to the inhomogeneous Besov covering, so that the associated coorbit
spaces are inhomogeneous Besov spaces. In the following, we will show, for
arbitrary dimension $d\in\mathbb{N}$, that certain warped coorbit spaces
obtained by $d$-dimensional _tensorization_ of a $1$-dimensional warping
function, i.e., $\Psi=\Psi_{\varrho}\otimes\ldots\otimes\Psi_{\varrho}$ (where
$\Phi_{\varrho}$ is symmetric and thus radial), are identical to (Besov-type)
_spaces of dominating mixed smoothness_ [27, 28, 40]. The latter are defined
as follows.

######  Definition 9.1 (Definition 1.2 [40]).

Let $1\leq p,q\leq\infty$ and $s\in\mathbb{R}^{d}$ and choose a
$\mathcal{B}$-BAPU
$\\{\varphi_{j}\\}_{j\in\mathbb{N}_{0}}\subset\mathcal{D}(\mathbb{R})$ for the
$1$-dimensional, inhomogenous Besov covering
$\mathcal{B}=\\{B_{i}\\}_{i\in\mathbb{N}_{0}}$, such that additionally, there
are $c_{k}>0$, $k\in\mathbb{N}_{0}$, with

$2^{j}k\cdot|\varphi_{j}^{(k)}(t)|\leq c_{k},\quad\text{for all
}j,k\in\mathbb{N}_{0},\ t\in\mathbb{R}.$ (9.1)

Define $\mathcal{B}_{d}=\\{B_{d,\ell}\\}_{\ell\in\mathbb{N}_{0}^{d}}$, where
$B_{d,\ell}=B_{\ell_{1}}\times\ldots\times B_{\ell_{d}}$, and
$v^{(s)}=\\{v^{(s)}_{\ell}\\}_{\ell\in\mathbb{N}_{0}^{d}}$, where
$v^{(s)}_{\ell}=2^{\langle\ell,s\rangle}$. Then the space
$S^{s}_{p,q}B(\mathbb{R}^{d})$ is given by the collection

$S^{s}_{p,q}B(\mathbb{R}^{d}):=\left\\{f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})~{}\colon~{}\|f\|_{\mathcal{D}(\mathcal{B}_{d},\mathbf{L}^{p},\ell_{v^{(s)}}^{q})}<\infty\right\\},$
(9.2)

equipped with the usual decomposition space norm
$f\mapsto\|f\|_{\mathcal{D}(\mathcal{B}_{d},\mathbf{L}^{p},\ell_{v^{(s)}}^{q})}$,
defined with respect to the $\mathcal{B}_{d}$-BAPU
$\\{{\psi_{\ell}}\\}_{\ell\in\mathbb{N}_{0}^{d}}\subset\mathcal{D}(\mathbb{R}^{d})$,
where
$\psi_{\ell}=\varphi_{\ell_{1}}(\bullet_{1})\cdot\ldots\cdot\varphi_{\ell_{d}}(\bullet_{d})$.

######  Remark 9.2.

The decomposition space norm in Definition 9.1 was used only for notational
convenience and it remains to be shown that $S^{s}_{p,q}B(\mathbb{R}^{d})$
does in fact coincide with
$\mathcal{D}(\mathcal{B}_{d},\mathbf{L}^{p},\ell_{v^{(s)}}^{q})$. Strictly
speaking, we also have to prove the claim in Definition 9.1, that
$\\{{\psi_{\ell}}\\}_{\ell\in\mathbb{N}_{0}^{d}}$ is a $\mathcal{B}_{d}$-BAPU.
The required support and summation properties are, however, obvious. Observe
further that (9.1) is equivalent to requiring that
$\\{\varphi_{j}\\}_{j\in\mathbb{N}_{0}}$ is a regular partition of unity
subordinate to $\mathcal{B}$ in the sense of [36, Definition 8.1], and as an
immediate consequence, $\\{{\psi_{\ell}}\\}_{\ell\in\mathbb{N}_{0}^{d}}$ is a
regular partition of unity subordinate to $\mathcal{B}_{d}$. Hence,
$\\{{\psi_{\ell}}\\}_{\ell\in\mathbb{N}_{0}^{d}}$ is in fact a
$\mathcal{B}_{d}$-BAPU (by [36, Theorem 8.2]) and $\mathcal{B}_{d}$ a
decomposition covering. Likewise, it is easy to see that
$\mathcal{B}_{d}$-moderateness of $v^{(s)}$ is implied by
$\mathcal{B}$-moderateness of the $1$-dimensional Besov weight $v^{s}$ with
$v^{s}_{j}=2^{js}$.

It can be seen for $d=1$ that the space $S^{s}_{p,q}B(\mathbb{R})$ of
dominating mixed smoothness coincides with the inhomogeneous Besov space
$B^{s}_{p,q}(\mathbb{R})$. For larger values of $d$,
$S^{s}_{p,q}B(\mathbb{R}^{d})$ is, in some sense, an anisotropic alternative
to Besov spaces, penalizing large values in the Fourier-domain with a weight
that does not depend only on the modulus of the frequency position
$\xi\in\mathbb{R}^{d}$, but the modulus of its individual components
$\xi_{1},\ldots,\xi_{d}\in\mathbb{R}$.

We now proceed to prove that $S^{s}_{p,q}B(\mathbb{R}^{d})$ coincides with the
decomposition space
$\mathcal{D}(\mathcal{B}_{d},\mathbf{L}^{p},\ell_{v^{(s)}}^{q})$. For this
purpose, we need an auxilliary result on the covering $\mathcal{B}_{d}$.

######  Proposition 9.3.

Let $\mathcal{B}=\\{B_{j}\\}_{j\in\mathbb{N}_{0}}$ denote the $1$-dimensional,
inhomogenous Besov covering, see (7.1). The covering
$\mathcal{B}_{d}=\\{B_{d,\ell}\\}_{\ell\in\mathbb{N}_{0}^{d}}$, where
$B_{d,\ell}=B_{\ell_{1}}\times\ldots\times B_{\ell_{d}}$, of $\mathbb{R}^{d}$
is semi-structured with

$B_{d,\ell}=S_{l}B_{d,\ell}^{\prime}\,,\quad\text{with}\quad
S_{l}=\mathrm{diag}(2^{\ell_{1}},\ldots,2^{\ell_{d}})\quad\text{and}\quad
B_{d,\ell}^{\prime}=B_{\ell_{1}}^{\prime}\times\cdots\times
B_{\ell_{d}}^{\prime}.$

Here, $B_{j}^{\prime}=(-2,-1/2)\cup(1/2,2)$ for $j\in\mathbb{N}$, and
$B_{0}^{\prime}=(-2,2)$. Furthermore, for all $1\leq p\leq\infty$, there
exists some $N\in\mathbb{N}_{0}$, such that the weight
$w^{(N)}:=w^{(N,p)}:=(w_{\ell}^{(N,p)})_{\ell\in\mathbb{N}_{0}^{d}}$, with

$w_{\ell}^{(N,p)}=|\det S_{\ell}|^{1/p}\cdot\Big{[}\inf_{\xi\in
B_{d,\ell^{\ast}}}(1+|\xi|)\Big{]}^{-N}\quad\text{for}\quad\ell\in\mathbb{N}_{0}^{d}\,,$

satisfies
$w^{(N)}\in\ell^{1}(\mathbb{N}_{0}^{d})\subset\ell^{q}(\mathbb{N}_{0}^{d})$.

###### Proof.

It is well-known that the inhomogeneous Besov covering
$\mathcal{B}=(B_{j})_{j\in\mathbb{N}_{0}}=(T_{j}B_{j}^{\prime})_{j\in\mathbb{N}_{0}}$
is a semi-structured covering of $\mathbb{R}$, with $T_{j}=2^{j}$ for
$j\in\mathbb{N}_{0}$, $B_{0}^{\prime}=(-2,2)$ and
$B_{j}^{\prime}=(-2,-1/2)\cup(1/2,2)$ for $j\in\mathbb{N}$. Hence, the
elements of $\mathcal{B}_{d}$ can be represented as
$B_{d,\ell}=S_{\ell}B_{d,\ell}^{\prime}$, with
$S_{\ell}=\mathrm{diag}(2^{\ell_{1}},\dots,2^{\ell_{d}})$ and
$B_{d,\ell}^{\prime}=B_{\ell_{1}}^{\prime}\times\cdots\times
B_{\ell_{d}}^{\prime}$ for $\ell\in\mathbb{N}_{0}^{d}$. In particular, $|\det
S_{\ell}|=2^{\|\ell\|_{1}}$ and $\|S_{\ell}^{-1}\|\leq 1$. That
$\mathcal{B}_{d}$ is semi-structured now follows easily.

In Section 7 we already noted that $1+|\xi|\asymp 2^{j}$ for all $\xi\in
B_{j}$, $j\in\mathbb{N}_{0}$, and it is easily seen that even $1+|\xi|\asymp
2^{j}$ for all $\xi\in B_{j^{\ast}}$. Since
$B_{d,\ell^{\ast}}=B_{\ell_{1}^{\ast}}\times\cdots\times B_{\ell_{d}^{\ast}}$,
we obtain

$1+|\xi|\asymp 1+\|\xi\|_{\infty}\asymp
2^{\|\ell\|_{\ell^{\infty}}}\quad\text{for}\quad\xi\in B_{d,\ell^{\ast}}\,.$

Therefore, if $N>d/p$

$w_{\ell}^{(N,p)}\lesssim 2^{\|\ell\|_{1}/p}\cdot
2^{-N\|\ell\|_{\ell^{\infty}}}\lesssim
2^{(\frac{d}{p}-N)\cdot\|\ell\|_{\ell^{\infty}}}\lesssim(1+\|\ell\|_{\ell^{\infty}})^{-(d+1)}\quad\text{for}\quad\ell\in\mathbb{N}_{0}^{d}\,,$

such that $w^{(N)}\in\ell^{1}(\mathbb{N}_{0}^{d})$. In the last step, we used
the elementary estimate $2^{x}=e^{x\cdot\ln 2}=\sum_{k=0}^{\infty}(x\cdot\ln
2)^{k}/k!\gtrsim(1+x)^{d+1}$, valid for $x\geq 0$. ∎

######  Corollary 9.4.

For $1\leq p,q\leq\infty$ and $s\in\mathbb{R}^{d}$ and with $\mathcal{B}_{d}$
and $\\{{\psi_{\ell}}\\}_{\ell\in\mathbb{N}_{0}^{d}}$ the
$\mathcal{B}_{d}$-BAPU as in Definition 9.1, then every
$g\in\mathcal{D}(\mathcal{B}_{d},\mathbf{L}^{p},\ell_{v^{(s)}}^{q})$ extends
to a tempered distribution $\tilde{g}\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$
given by

$\tilde{g}:\mathcal{S}(\mathbb{R}^{d})\rightarrow\mathbb{C},\
f\mapsto\sum_{\ell\in\mathbb{Z}^{d}}\langle\mathcal{F}^{-1}(\hat{g}\cdot\psi_{\ell}),f\rangle_{\mathcal{S}^{\prime},\mathcal{S}}.$

In particular,
$\langle\tilde{g},f\rangle_{\mathcal{S}^{\prime},\mathcal{S}}=\langle
g,f\rangle_{Z^{\prime},Z}$, for all $f\in Z(D)$ and
$g\in\mathcal{D}(\mathcal{B}_{d},\mathbf{L}^{p},\ell_{v^{(s)}}^{q})$.

###### Proof.

This is a direct application of [36, Theorem 8.3], provided that all its
assumptions are satisfied. As discussed in Remark 9.2,
$\\{{\psi_{\ell}}\\}_{\ell\in\mathbb{N}_{0}^{d}}$ is a regular partition of
unity subordinate to $\mathcal{B}_{d}$, such that $\mathcal{B}_{d}$ itself is
a regular covering in the sense of [36, Definition 8.1]. Note that
$\max\\{1,\|S^{-1}_{\ell}\|\\}=1$ for all $\ell\in\mathbb{N}^{d}_{0}$, as
observed in the proof of Proposition 9.3, such that the weight $w^{(N,p)}$ in
said proposition coincides with the weight
$w^{(N)}\in\mathbb{C}^{\mathbb{N}_{0}^{d}}$ defined in [36, Theorem 8.3].
Therefore Proposition 9.3 shows the existence of $N\in\mathbb{N}_{0}$, such
that $w^{(N)}$ is summable and all prerequisites of [36, Theorem 8.3] are
satisfied. This completes the proof. ∎

In the above proof, and in Remark 9.2, we did not derive the desired
properties of the product weight $v^{(s)}$ in full detail. Although these
derivations are rather straightforward, we provide, for the convenience of the
reader, a full proof in Appendix B, together with general results on products
of semi-structured coverings. There, we also show that the inheritance of
almost subordinateness, which will be used in Theorem 9.6, is a simple
consequence of the product construction.

The warping function that generates spaces of dominating mixed smoothness will
turn out to be a $d$-dimensional _tensor product_ , the factors of which are
given by the $1$-dimensional warping function $\Phi_{\varrho_{1}}$ considered
in our treatment of Besov spaces in Section 7. Therefore, the following lemma,
which considers general _separable_ warping functions $\Phi$, as they were
referred to previously [19], will be helpful.

######  Lemma 9.5.

For $i=1,\dots,N$ let $\Phi_{i}:D_{i}\to\mathbb{R}^{d_{i}}$ be a
$k$-admissible warping function with control weight $v_{i}$. Let
$d:=d_{1}+\dots+d_{N}$ and $D:=D_{1}\times\cdots\times D_{N}$. Then

$\Phi:=\Phi_{1}\otimes\cdots\otimes\Phi_{N}\colon D=D_{1}\times\cdots\times
D_{N}\to\mathbb{R}^{d},\
(\xi_{1},\dots,\xi_{N})\mapsto\big{(}\Phi_{1}(\xi_{1}),\dots,\Phi_{N}(\xi_{N})\big{)}$

is a $k$-admissible warping function with control weight
$v:\mathbb{R}^{d}\to\mathbb{R}^{+},\tau\mapsto\max\\{v_{1}(|\tau|e_{1}^{(d_{1})}),\dots,v_{N}(|\tau|e_{1}^{(d_{N})})\\}$,
where the superscripts indicate the dimensionality of the standard unit
vectors. Moreover, the $\Phi$-induced $(\delta,r)$-fine product covering
$\mathcal{Q}^{(\delta,r)}_{\Phi,\otimes}=\\{Q^{(\delta,r)}_{\Phi,\otimes,\ell}\\}_{\ell\in\mathbb{Z}^{d}}$,
where
$Q^{(\delta,r)}_{\Phi,\otimes,\ell}=Q^{(\delta,r)}_{\Phi_{1},\ell_{1}}\times\ldots\times
Q^{(\delta,r)}_{\Phi_{N},\ell_{N}}$, with
$\ell=(\ell_{1},\ldots,\ell_{N})\in\mathbb{Z}^{d}$ and
$\ell_{i}\in\mathbb{Z}^{d_{i}}$, $i=1,\ldots,N$, is equivalent to the usual
$\Phi$-induced $(\delta,r)$-fine frequency covering
$\mathcal{Q}^{(\delta,r)}_{\Phi}$, for all $\delta>0$, $r>\sqrt{d}/2$.

If furthermore for each $i=1,\dots,N$ we are given a weight
$\kappa_{i}:D_{i}\to\mathbb{R}^{+}$ such that $\kappa_{i}\circ\Phi_{i}^{-1}$
is $\tilde{\kappa}_{i}$-moderate, for some continuous, radially increasing,
submultiplicative weight $\tilde{\kappa}_{i}$, then setting

$\kappa:D\to\mathbb{R}^{+},(\xi_{1},\dots,\xi_{N})\mapsto\prod_{i=1}^{N}\kappa_{i}(\xi_{i})\quad\text{and}\quad\tilde{\kappa}:\mathbb{R}^{d}\to\mathbb{R}^{+},\xi\mapsto\prod_{i=1}^{N}\tilde{\kappa}_{i}(|\xi|)\,,$

the weight $\kappa_{\Phi}=\kappa\circ\Phi^{-1}$ is $\tilde{\kappa}$-moderate,
and $\tilde{\kappa}$ is continuous, radially increasing, and
submultiplicative. Hence, if $\Phi_{i}$ is a
$\mathcal{C}^{\infty}$-diffeomorphism, for all $i=1,\dots,N$, then
$(\Phi,\kappa)$ forms a compatible pair as per Definition 2.4.

###### Proof.

Across the proof, we will identify $\mathbb{R}^{d}$ with
$\mathbb{R}^{d_{1}}\times\cdots\times\mathbb{R}^{d_{N}}$, and likewise
$\mathbb{N}_{0}^{d}$ with
$\mathbb{N}_{0}^{d_{1}}\times\cdots\times\mathbb{N}_{0}^{d_{N}}$.

Since each $\Phi_{i}$ is a $C^{k+1}$-diffeomorphism, so is $\Phi$. The inverse
of $\Phi$ is given by

$\Phi^{-1}:\mathbb{R}^{d}\to
D,(\tau_{1},\dots,\tau_{N})\mapsto\big{(}\Phi_{1}^{-1}(\tau_{1}),\dots,\Phi_{N}^{-1}(\tau_{N})\big{)}\,.$

Therefore, if we set $A_{i}:=D\Phi_{i}^{-1}$, then

$A(\tau_{1},\dots,\tau_{N}):=D\Phi^{-1}(\tau_{1},\dots,\tau_{N})=\mathrm{diag}\big{(}A_{1}(\tau_{1}),\dots,A_{N}(\tau_{N})\big{)}$

is a block-diagonal matrix, so that

$w(\tau)=\det(A(\tau_{1},\dots,\tau_{N}))=\prod_{i=1}^{N}\det(A_{i}(\tau_{i}))=\prod_{i=1}^{N}w_{i}(\tau_{i})>0\,,$
(9.3)

where $w$ and $w_{i}$ are the weights associated with the warping functions
$\Phi$ and $\Phi_{i}$, respectively.

Furthermore, if we set
$\phi_{\tau}^{(i)}({\scaleobj{0.65}{\Upsilon}}):=A_{i}^{T}({\scaleobj{0.65}{\Upsilon}}+\tau)\cdot
A_{i}^{-T}(\tau)$ for $\tau,{\scaleobj{0.65}{\Upsilon}}\in\mathbb{R}^{d_{i}}$,
then for $\tau=(\tau_{1},\dots,\tau_{N})\in\mathbb{R}^{d}$ and
${\scaleobj{0.65}{\Upsilon}}=({\scaleobj{0.65}{\Upsilon}}_{1},\dots,{\scaleobj{0.65}{\Upsilon}}_{N})\in\mathbb{R}^{d}$,
we have

$\phi_{\tau}({\scaleobj{0.65}{\Upsilon}}):=A^{T}({\scaleobj{0.65}{\Upsilon}}+\tau)\cdot
A^{-T}(\tau)=\mathrm{diag}\big{(}\phi_{\tau_{1}}^{(1)}({\scaleobj{0.65}{\Upsilon}}_{1}),\dots,\phi_{\tau_{N}}^{(N)}({\scaleobj{0.65}{\Upsilon}}_{N})\big{)}\,.$

Therefore, for any $\alpha=(\alpha_{1},\dots,\alpha_{N})\in\mathbb{N}_{0}^{d}$
with $|\alpha|\leq k$, we have

$\displaystyle\|\partial^{\alpha}\phi_{\tau}({\scaleobj{0.65}{\Upsilon}})\|$
$\displaystyle=\big{\|}\mathrm{diag}\big{(}\partial^{\alpha_{1}}\phi_{\tau_{1}}^{(1)}({\scaleobj{0.65}{\Upsilon}}_{1}),\dots,\partial^{\alpha_{N}}\phi_{\tau_{N}}^{(N)}({\scaleobj{0.65}{\Upsilon}}_{N})\big{)}\big{\|}=\max\big{\\{}\|\partial^{\alpha_{i}}\phi_{\tau_{i}}^{(i)}({\scaleobj{0.65}{\Upsilon}}_{i})\|\,:\,i=1,\dots,N\big{\\}}$
$\displaystyle\overset{(\lozenge)}{\leq}\max\big{\\{}v_{i}(|{\scaleobj{0.65}{\Upsilon}}_{i}|e_{1}^{(d_{i})})\,:\,i=1,\dots,N\big{\\}}\overset{(\ast)}{\leq}\max\big{\\{}v_{i}(|{\scaleobj{0.65}{\Upsilon}}|e_{1}^{(d_{i})})\,:\,i=1,\dots,N\big{\\}}=:v({\scaleobj{0.65}{\Upsilon}})\,.$

Here, the step marked with $(\lozenge)$ used that each $\Phi_{i}$ is
$k$-admissible with control weight $v_{i}$, and the step marked with $(\ast)$
used that $v_{i}$ is radially increasing and that
$|{\scaleobj{0.65}{\Upsilon}}_{i}|\leq|{\scaleobj{0.65}{\Upsilon}}|$.

It is clear that $v$ is continuous and radially increasing. Furthermore,

$v_{i}(|\tau+{\scaleobj{0.65}{\Upsilon}}|e_{1}^{(d_{i})})\leq
v_{i}(|\tau|e_{1}^{(d_{i})}+|{\scaleobj{0.65}{\Upsilon}}|e_{1}^{(d_{i})})\leq
v_{i}(|\tau|e_{1}^{(d_{i})})\cdot
v_{i}(|{\scaleobj{0.65}{\Upsilon}}|e_{1}^{(d_{i})})\leq v(\tau)\cdot
v({\scaleobj{0.65}{\Upsilon}})$

for all $i=1,\dots,N$, which implies that $v$ is submultiplicative. In a
similar manner, one can show that $\tilde{\kappa}$ is continuous, radially
increasing, and submultiplicative.

Finally, to see that $\kappa_{\Phi}$ is $\tilde{\kappa}$-moderate, note for
$\tau=(\tau_{1},\dots,\tau_{N})\in\mathbb{R}^{d}$ and
${\scaleobj{0.65}{\Upsilon}}=({\scaleobj{0.65}{\Upsilon}}_{1},\dots,{\scaleobj{0.65}{\Upsilon}}_{N})\in\mathbb{R}^{d}$
that

$\displaystyle(\kappa_{\Phi})(\tau+{\scaleobj{0.65}{\Upsilon}})$
$\displaystyle=\prod_{i=1}^{N}(\kappa_{i}\circ\Phi_{i}^{-1})(\tau_{i}+{\scaleobj{0.65}{\Upsilon}}_{i})\leq\prod_{i=1}^{N}\left[(\kappa_{i}\circ\Phi_{i}^{-1})(\tau_{i})\cdot\tilde{\kappa}_{i}({\scaleobj{0.65}{\Upsilon}}_{i})\right]$
$\displaystyle\leq\Big{(}\prod_{i=1}^{N}\tilde{\kappa}_{i}(|{\scaleobj{0.65}{\Upsilon}}|)\Big{)}\cdot(\kappa_{\Phi})(\tau)=\tilde{\kappa}({\scaleobj{0.65}{\Upsilon}})\cdot(\kappa_{\Phi})(\tau)\,.$

To finish the proof, we will now show that
$\mathcal{Q}^{(\delta,r)}_{\Phi,\otimes}$ is equivalent to
$\mathcal{Q}^{(\delta,r)}_{\Phi}$. For clarity, we will indicate the