$\displaystyle\leq\int_{\mathbb{R}^{d}}\sum_{i\in\mathbb{Z}^{d}}{\mathds{1}}_{B_{\delta^{\prime}}({\scaleobj{0.65}{\Upsilon}}_{i})}({\scaleobj{0.65}{\Upsilon}})\cdot|g_{\tau,\tau_{0}}({\scaleobj{0.65}{\Upsilon}})|\,d{\scaleobj{0.65}{\Upsilon}}$
$\displaystyle({\scriptstyle{\text{Eq.
}\eqref{eq:uniform_covering_admissible}}})$
$\displaystyle\leq(1+4d)^{d}\cdot\|g_{\tau,\tau_{0}}\|_{\mathbf{L}^{1}}$
$\displaystyle({\scriptstyle{\text{since
}\frac{w({\scaleobj{0.65}{\Upsilon}}+\tau_{0})}{w(\tau_{0})}\leq
w_{0}({\scaleobj{0.65}{\Upsilon}})}})$
$\displaystyle\leq(1+4d)^{d}\cdot\|w_{0}\cdot\theta_{2}\cdot\mathbf{T}_{\tau}\theta_{1}\|_{\mathbf{L}^{1}}$
$\displaystyle({\scriptstyle{w_{0}\text{ is submultiplicative}}})$
$\displaystyle\leq\sqrt{w_{0}(\tau)}\cdot(1+4d)^{d}\cdot\|\theta_{2}\|_{\mathbf{L}^{2}_{\sqrt{w_{0}}}}\cdot\|\mathbf{T}_{\tau}\theta_{1}\|_{\mathbf{L}^{2}_{\sqrt{w_{0}}}}<\infty,$

the dominated convergence theorem justifies (4.32). ∎

To prepare for an estimate of
$\Square_{j_{i},\tau_{0},x}^{n}\,g_{i,\tau,\tau_{0}}$ itself, we consider the
partial derivatives of $g_{i,\tau,\tau_{0}}$.

######  Lemma 4.14.

Let $\Phi$ be a $k$-admissible warping function with control weight $v_{0}$,
let
$\theta_{1},\theta_{2}\in\mathbf{L}^{2}_{\sqrt{w_{0}}}\cap\mathcal{C}^{k}$,
and let $(\varphi_{i})_{i\in\mathbb{Z}^{d}}$ be a bounded partition of unity
as in Lemma 4.12, for some given $\delta^{\prime}>0$. For any fixed
$j\in\underline{d}$, $i\in\mathbb{Z}^{d}$ and
$\tau,\tau_{0}\in\mathbb{R}^{d}$, we have

$\frac{\partial^{n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{n}}\,g_{i,\tau,\tau_{0}}=\sum_{m=0}^{n}\sum_{\ell=0}^{m}\sum_{\alpha=0}^{\ell}\left(C_{n,m,\ell,\alpha}\cdot\frac{\frac{\partial^{n-m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{n-m}}\mathbf{T}_{-\tau_{0}}w}{w(\tau_{0})}\cdot\frac{\partial^{m-\ell}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m-\ell}}\varphi_{i}\cdot\frac{\partial^{\ell-\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\ell-\alpha}}\theta_{2}\cdot\frac{\partial^{\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\alpha}}\overline{\mathbf{T}_{\tau}\theta_{1}}\right),$
(4.33)

for some constants $C_{n,m,\ell,\alpha}\in\mathbb{N}$ and arbitrary $n\leq k$.
Here,
$g_{i,\tau,\tau_{0}}({\scaleobj{0.65}{\Upsilon}})=\frac{w({\scaleobj{0.65}{\Upsilon}}+\tau_{0})}{w(\tau_{0})}\varphi_{i}({\scaleobj{0.65}{\Upsilon}})\left(\theta_{2}\cdot\overline{\mathbf{T}_{\tau}\theta_{1}}\right)({\scaleobj{0.65}{\Upsilon}})$
is as in (4.30). Moreover,

$\left|\frac{\partial^{n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{n}}\,g_{i,\tau,\tau_{0}}\right|\leq
C_{n}\cdot
v_{0}^{d}\cdot\sum_{\ell=0}^{n}\sum_{\alpha=0}^{\ell}\left|\frac{\partial^{\ell-\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\ell-\alpha}}\theta_{2}\cdot\frac{\partial^{\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\alpha}}\overline{\mathbf{T}_{\tau}\theta_{1}}\right|,\text{
for all }n\leq k,$ (4.34)

for some constant $C_{n}=C_{n}(\Phi,\delta^{\prime},d)>0$.

###### Proof.

The first part of the proof is just a nested application of Leibniz’s rule:

$\begin{split}&\frac{\partial^{n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{n}}\,g_{i,\tau,\tau_{0}}=\frac{\partial^{n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{n}}\left(\frac{\mathbf{T}_{-\tau_{0}}w}{w(\tau_{0})}\cdot\varphi_{i}\cdot\theta_{2}\cdot\overline{\mathbf{T}_{\tau}\theta_{1}}\right)\\\
&=\frac{1}{w(\tau_{0})}\sum_{m=0}^{n}\left(\binom{n}{m}\frac{\partial^{n-m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{n-m}}\mathbf{T}_{-\tau_{0}}w\cdot\sum_{\ell=0}^{m}\left(\binom{m}{\ell}\frac{\partial^{m-\ell}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m-\ell}}\varphi_{i}\cdot\sum_{\alpha=0}^{\ell}\left(\binom{\ell}{\alpha}\frac{\partial^{\ell-\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\ell-\alpha}}\theta_{2}\cdot\frac{\partial^{\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\alpha}}\overline{\mathbf{T}_{\tau}\theta_{1}}\right)\right)\right)\\\
&=\frac{1}{w(\tau_{0})}\sum_{m=0}^{n}\sum_{\ell=0}^{m}\sum_{\alpha=0}^{\ell}\left(C_{n,m,\ell,\alpha}\cdot\frac{\partial^{n-m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{n-m}}\mathbf{T}_{-\tau_{0}}w\cdot\frac{\partial^{m-\ell}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m-\ell}}\varphi_{i}\cdot\frac{\partial^{\ell-\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\ell-\alpha}}\theta_{2}\cdot\frac{\partial^{\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\alpha}}\overline{\mathbf{T}_{\tau}\theta_{1}}\right),\end{split}$

with $C_{n,m,\ell,\alpha}=\binom{n}{m}\binom{m}{\ell}\binom{\ell}{\alpha}$.
This proves (4.33).

For the second assertion, we consider each term appearing in (4.33)
individually. Since all the involved sums are finite, there is a finite
constant $\widetilde{C}_{n}=\widetilde{C}_{n}(\delta^{\prime},d)>0$, such that

$\max_{n,m,\ell,\alpha}\left|C_{n,m,\ell,\alpha}\frac{\partial^{m-\ell}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m-\ell}}\varphi_{i}\right|\leq\max_{n,m,\ell,\alpha}\left(C_{n,m,\ell,\alpha}\cdot
D^{(\delta^{\prime})}_{(m-\ell)e_{j}}\right)\leq\widetilde{C}_{n}\,,$

where property (4) of $(\varphi_{i})_{i\in\mathbb{Z}^{d}}$ in Lemma 4.12 was
used, and $e_{j}$ is interpreted as a multi-index.

For the term
$[w(\tau_{0})]^{-1}\cdot\frac{\partial^{n-m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{n-m}}w({\scaleobj{0.65}{\Upsilon}}+\tau_{0})$
on the other hand, we apply the estimate given in Lemma 4.10, i.e.

$[w(\tau_{0})]^{-1}\cdot\left|\frac{\partial^{n-m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{n-m}}w({\scaleobj{0.65}{\Upsilon}}+\tau_{0})\right|\leq
D_{n-m}\cdot[v_{0}({\scaleobj{0.65}{\Upsilon}})]^{d}\leq\left(\max_{0\leq
m\leq
n}D_{n-m}\right)\cdot[v_{0}({\scaleobj{0.65}{\Upsilon}})]^{d}=:\widetilde{D}_{n}\cdot[v_{0}({\scaleobj{0.65}{\Upsilon}})]^{d},$

where $\widetilde{D}_{n}=\widetilde{D}_{n}(\Phi)$. Therefore, we see for a
suitable constant $C_{n}=C_{n}(\delta^{\prime},d,\Phi)>0$ that

$\begin{split}\left|\frac{\partial^{n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{n}}\,g_{i,\tau,\tau_{0}}\right|&\leq\widetilde{D}_{n}\widetilde{C}_{n}\cdot\sum_{m=0}^{n}\sum_{\ell=0}^{m}\sum_{\alpha=0}^{\ell}\left|v_{0}^{d}\cdot\frac{\partial^{\ell-\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\ell-\alpha}}\theta_{2}\cdot\frac{\partial^{\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\alpha}}\overline{\mathbf{T}_{\tau}\theta_{1}}\right|\\\
&\leq C_{n}\cdot
v_{0}^{d}\cdot\sum_{\ell=0}^{n}\sum_{\alpha=0}^{\ell}\left|\frac{\partial^{\ell-\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\ell-\alpha}}\theta_{2}\cdot\frac{\partial^{\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\alpha}}\overline{\mathbf{T}_{\tau}\theta_{1}}\right|.\qed\end{split}$

The next lemma provides an estimate of $|\Square_{j_{i},\tau_{0},x}^{n}\,g|$
in terms of the partial derivatives of $g$ and the weight function $v_{0}$
from Definition 4.2.

######  Lemma 4.15.

Let $\Phi$ be a $k$-admissible warping function with control weight $v_{0}$
and choose $\delta>0$ such that $\delta\cdot v_{0}(\delta/(4d)\cdot e_{1})\leq
1/\sqrt{d}$. Fix $j\in\underline{d}$ and
${\scaleobj{0.65}{\Upsilon}}_{0},\tau_{0}\in\mathbb{R}^{d}$ and let
$U_{j}^{({\scaleobj{0.65}{\Upsilon}}_{0},\tau_{0})}$ be as in Lemma 4.11(1).
If $g\in\mathcal{C}^{k}_{c}(B_{\delta/(4d)}({\scaleobj{0.65}{\Upsilon}}_{0}))$
and if $x\in\mathbb{R}^{d}\setminus\\{0\\}$ satisfies $x/|x|\in
U_{j}^{({\scaleobj{0.65}{\Upsilon}}_{0},\tau_{0})}$ then, with

$\left(\Square_{j,\tau_{0},x}\,g\right)=(2\pi
i|x|)^{-1}\frac{\partial}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}}\left[\frac{g(\bullet)}{\bigl{(}\phi_{\tau_{0}}(\bullet)\langle
x/|x|\rangle\bigr{)}_{j}}\right]$

as in (4.26), there exists $D_{n,\delta}=D_{n,\Phi,\delta}>0$, independent of
$j,x,{\scaleobj{0.65}{\Upsilon}}_{0},\tau_{0},g$, such that

$\left|\Square^{n}_{j,\tau_{0},x}\,g\right|\leq
D_{n,\delta}\cdot(2\pi|x|)^{-n}\cdot
v_{0}^{3n}\cdot\sum_{m=0}^{n}\left|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}^{m}_{j}}\,g\right|$

holds for all $0\leq n\leq k$, where $\Square^{n}_{j,\tau_{0},x}$ denotes the
$n$-fold application of $\Square_{j,\tau_{0},x}$.

###### Proof.

Step 1 (Preparation): Given $j\in\underline{d}$ and a strictly positive (or
strictly negative) function $h\in\mathcal{C}^{1}(U)$ defined on an open set
$\varnothing\neq U\subset\mathbb{R}^{d}$, we define the differential operator
$\blacksquare_{j,h}$ by
$\blacksquare_{j,h}\,\,g:=\frac{\partial}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}}\left(\frac{g}{h}\right)$.
Then the following identity can then be derived from the quotient rule by a
tedious, but straightforward induction:

$\blacksquare_{j,h}^{n}\,\,g=h^{-2n}\cdot\sum_{m=0}^{n}\Bigg{(}\frac{\partial^{m}g}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\cdot\sum_{\begin{subarray}{c}\alpha\in\mathbb{N}_{0}^{n}\\\
|\alpha|=n-m\end{subarray}}\bigg{(}C^{(m,\alpha)}\cdot\prod_{\ell=1}^{n}\frac{\partial^{\alpha_{\ell}}h}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\alpha_{\ell}}}\bigg{)}\Bigg{)},\text{
for all }g\in{\mathcal{C}}^{k}(U)\text{ and }n\in\mathbb{N}_{\leq k},$ (4.35)

for suitable constants $C^{(m,\alpha)}\in\mathbb{Z}$ that depend only on
$\alpha\in\mathbb{N}_{0}^{n}$ and on $m\in\\{0,\ldots,n\\}$. Furthermore, we
have the equality

$\Square^{n}_{i,\tau_{0},x}\,g({\scaleobj{0.65}{\Upsilon}})=(2\pi
i|x|)^{-n}\cdot\blacksquare_{j,\left(\phi_{\tau_{0}}(\cdot)\langle
x/|x|\rangle\right)_{j}}^{n}\,g({\scaleobj{0.65}{\Upsilon}}).$

Step 2 (Completing the proof): For $n=0$, the claim is clear. Hence, we can
assume $n\in\underline{k}$. With
$U_{j}^{({\scaleobj{0.65}{\Upsilon}}_{0},\tau_{0})}\subset S^{d-1}$ as in
Lemma 4.11(1), there is a $j\in\underline{d}$, such that $x/|x|=:\rho_{x}\in
U_{j}^{({\scaleobj{0.65}{\Upsilon}}_{0},\tau_{0})}$ and therefore,
$\Square_{j,\tau_{0},x}g$ is well-defined for arbitrary
$g\in\mathcal{C}_{c}^{k}(B_{\delta/(4d)}({\scaleobj{0.65}{\Upsilon}}_{0}))$ by
Lemma 4.11(2). Now, (4.35) provides

$\begin{split}\blacksquare_{j,\left(\phi_{\tau_{0}}(\cdot)\langle\rho_{x}\rangle\right)_{j}}^{n}\,g&=\bigl{(}\phi_{\tau_{0}}(\cdot)\langle\rho_{x}\rangle\bigr{)}_{j}^{-2n}\cdot\sum_{m=0}^{n}\Bigg{(}\frac{\partial^{m}g}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\cdot\sum_{\begin{subarray}{c}\alpha\in\mathbb{N}_{0}^{n}\\\
|\alpha|=n-m\end{subarray}}\bigg{(}C^{(m,\alpha)}\cdot\prod_{\ell=1}^{n}\frac{\partial^{\alpha_{\ell}}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\alpha_{\ell}}}\big{(}\phi_{\tau_{0}}(\cdot)\langle\rho_{x}\rangle\big{)}_{j}\bigg{)}\Bigg{)}.\end{split}$
(4.36)

We now estimate the magnitude of the innermost product by using (4.5):

$\left|\prod_{\ell=1}^{n}\frac{\partial^{\alpha_{\ell}}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\alpha_{\ell}}}\big{(}\phi_{\tau_{0}}({\scaleobj{0.65}{\Upsilon}})\langle\rho_{x}\rangle\big{)}_{j}\right|\overset{|\rho_{x}|=1}{\leq}\prod_{\ell=1}^{n}\left\|\frac{\partial^{\alpha_{\ell}}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\alpha_{\ell}}}\phi_{\tau_{0}}({\scaleobj{0.65}{\Upsilon}})\right\|\leq
v_{0}^{n}({\scaleobj{0.65}{\Upsilon}}).$

Insert this estimate into (4.36) to obtain

$\begin{split}\left|\blacksquare_{j,\left(\phi_{\tau_{0}}(\cdot)\langle\rho_{x}\rangle\right)_{j}}^{n}\,g({\scaleobj{0.65}{\Upsilon}})\right|&\leq[v_{0}({\scaleobj{0.65}{\Upsilon}})]^{n}\cdot\left|\left(\phi_{\tau_{0}}({\scaleobj{0.65}{\Upsilon}})\langle\rho_{x}\rangle\right)_{j}\right|^{-2n}\sum_{m=0}^{n}\Bigg{(}\left|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}g({\scaleobj{0.65}{\Upsilon}})\right|\cdot\sum_{\begin{subarray}{c}\alpha\in\mathbb{N}_{0}^{n}\\\
|\alpha|=n-m\end{subarray}}\bigl{|}C^{(m,\alpha)}\bigr{|}\Bigg{)}\\\
({\scriptstyle{\text{Lemma
}\ref{lem:phiIsReasonable}}})&\leq[v_{0}({\scaleobj{0.65}{\Upsilon}})]^{n}\cdot
C_{\delta}^{-2n}\cdot[v_{0}({\scaleobj{0.65}{\Upsilon}})]^{2n}\cdot\sum_{m=0}^{n}\Bigg{(}\left|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}^{m}_{j}}g({\scaleobj{0.65}{\Upsilon}})\right|\cdot\sum_{\begin{subarray}{c}\alpha\in\mathbb{N}_{0}^{n}\\\
|\alpha|=n-m\end{subarray}}\left|C^{(m,\alpha)}\right|\Bigg{)}\\\ &\leq
D_{n,\delta}\cdot[v_{0}({\scaleobj{0.65}{\Upsilon}})]^{3n}\cdot\sum_{m=0}^{n}\left|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}^{m}_{j}}g({\scaleobj{0.65}{\Upsilon}})\right|,\end{split}$

where
$D_{n,\delta}:=C_{\delta}^{-2n}\cdot\max_{m=0,\ldots,n}\left(\sum_{|\alpha|=n-m}|C^{(m,\alpha)}|\right)$
only depends on $n\leq k$, $\delta>0$ and $\Phi$. ∎

We are ready to prove Theorem 4.8, in particular we can now estimate the
integral appearing on the right-hand side of (4.7).

###### Proof of Theorem 4.8.

Recall from Lemma 4.9 that $w_{0}=v_{0}^{d}$. Furthermore, note by
sumbultiplicativity of $w_{1}$ that $w_{1}(0)=w_{1}(0+0)\leq[w_{1}(0)]^{2}$,
and hence $w_{1}(0)\geq 1$. This implies $w_{1}\geq 1$, since we see by
another application of submultiplicativity that $1\leq
w_{1}(0)=w_{1}({\scaleobj{0.65}{\Upsilon}}+(-{\scaleobj{0.65}{\Upsilon}}))\leq
w_{1}({\scaleobj{0.65}{\Upsilon}})\cdot
w_{1}(-{\scaleobj{0.65}{\Upsilon}})=[w_{1}({\scaleobj{0.65}{\Upsilon}})]^{2}$,
since
$w_{1}(-{\scaleobj{0.65}{\Upsilon}})=w_{1}({\scaleobj{0.65}{\Upsilon}})$. By
the same arguments, we see $v_{0}\geq 1$. Therefore, we see that (4.16)
implies
$\theta_{\ell}\in\mathbf{L}_{v_{0}^{d/2}}^{2}(\mathbb{R}^{d})=\mathbf{L}_{\sqrt{w_{0}}}^{2}(\mathbb{R}^{d})$,
i.e., $\theta_{1},\theta_{2}$ satisfy the conditions of Lemma 4.13.

In the following, we only consider the case $\ell=1$; the corresponding
estimates for $\ell=2$ can be obtained simply by swapping
$\theta_{1},\theta_{2}$; our assumptions, and the definition of $C_{\max}$,
are invariant under this operation.

A first estimate for the RHS of (4.7)—which is effective for $|x|\leq 1$ and
which can be obtained using the $v_{0}^{d}$-moderateness of $w$ (see Lemma
4.9) and the submultiplicativity of $w_{1},v_{0}$—reads as follows:

$\begin{split}&\left|\int_{\mathbb{R}^{d}}\frac{w({\scaleobj{0.65}{\Upsilon}}+\tau_{0})}{w(\tau_{0})}\left(\theta_{2}\cdot\overline{\mathbf{T}_{\tau}\theta_{1}}\right)\\!({\scaleobj{0.65}{\Upsilon}})\,e_{\tau_{0}}(x,{\scaleobj{0.65}{\Upsilon}})~{}d{\scaleobj{0.65}{\Upsilon}}\right|\\\
&\leq\int_{\mathbb{R}^{d}}v_{0}^{d}({\scaleobj{0.65}{\Upsilon}})\cdot\left|\theta_{2}({\scaleobj{0.65}{\Upsilon}})\right|\cdot\left|\theta_{1}({\scaleobj{0.65}{\Upsilon}}-\tau)\right|~{}d{\scaleobj{0.65}{\Upsilon}}\\\
&=w_{1}(\tau)^{-1}\cdot
w_{1}(\tau)\cdot\int_{\mathbb{R}^{d}}v_{0}^{d}({\scaleobj{0.65}{\Upsilon}})\cdot\left|\theta_{2}({\scaleobj{0.65}{\Upsilon}})\right|\cdot\left|\theta_{1}({\scaleobj{0.65}{\Upsilon}}-\tau)\right|~{}d{\scaleobj{0.65}{\Upsilon}}\\\
&\leq
w_{1}(\tau)^{-1}\int_{\mathbb{R}^{d}}\left|v_{0}^{d}({\scaleobj{0.65}{\Upsilon}})w_{1}({\scaleobj{0.65}{\Upsilon}})\theta_{2}({\scaleobj{0.65}{\Upsilon}})\right|\left|w_{1}(\tau-{\scaleobj{0.65}{\Upsilon}})\theta_{1}({\scaleobj{0.65}{\Upsilon}}-\tau)\right|~{}d{\scaleobj{0.65}{\Upsilon}}\\\
\text{\scriptsize{($w_{1}$ is radial) }}&\leq
w_{1}(\tau)^{-1}\cdot\|\theta_{1}\|_{\mathbf{L}^{2}_{w_{1}}}\cdot\|\theta_{2}\|_{\mathbf{L}^{2}_{v_{0}^{d}w_{1}}}\leq
C_{\max}\cdot[w_{1}(\tau)]^{-1}.\end{split}$ (4.37)

The last step used $v_{0}\geq 1$, such that
$\|\theta_{1}\|_{\mathbf{L}^{2}_{w_{1}}}\leq\|\theta_{1}\|_{\mathbf{L}^{2}_{w_{2}}}$
and likewise
$\|\theta_{1}\|_{\mathbf{L}^{2}_{v_{0}^{d}w_{1}}}\leq\|\theta_{1}\|_{\mathbf{L}^{2}_{w_{2}}}$.

To obtain an estimate which is effective for large $|x|$, we have to work
harder: We fix some $\delta=\delta(d,\Phi,v_{0})>0$, such that $\delta
v_{0}(\delta/(4d)\cdot e_{1})<1/\sqrt{d}$. Hence, we can apply Lemma 4.13 to
obtain

$\begin{split}\left|\int_{\mathbb{R}^{d}}\frac{w({\scaleobj{0.65}{\Upsilon}}+\tau_{0})}{w(\tau_{0})}\left(\theta_{2}\cdot\overline{\mathbf{T}_{\tau}\theta_{1}}\right)({\scaleobj{0.65}{\Upsilon}})\cdot
e_{\tau_{0}}(x,{\scaleobj{0.65}{\Upsilon}})~{}d{\scaleobj{0.65}{\Upsilon}}\right|&=\left|\sum_{i\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}\left(\Square_{j_{i},\tau_{0},x}^{k}\,\,g_{i,\tau,\tau_{0}}\right)({\scaleobj{0.65}{\Upsilon}})\cdot
e_{\tau_{0}}(x,{\scaleobj{0.65}{\Upsilon}})~{}d{\scaleobj{0.65}{\Upsilon}}\right|\\\
&\leq\int_{\mathbb{R}^{d}}\sum_{i\in\mathbb{Z}^{d}}\left|\left(\Square_{j_{i},\tau_{0},x}^{k}\,\,g_{i,\tau,\tau_{0}}\right)({\scaleobj{0.65}{\Upsilon}})\right|~{}d{\scaleobj{0.65}{\Upsilon}}\\\
&=\sum_{j\in\underline{d}}\int_{\mathbb{R}^{d}}\sum_{\begin{subarray}{c}i\in\mathbb{Z}^{d}\\\
\text{s.t.
}j_{i}=j\end{subarray}}\left|\left(\Square_{j,\tau_{0},x}^{k}\,\,g_{i,\tau,\tau_{0}}\right)({\scaleobj{0.65}{\Upsilon}})\right|~{}d{\scaleobj{0.65}{\Upsilon}},\end{split}$
(4.38)

for any $x\in\mathbb{R}^{d}\setminus\\{0\\}$,
$\tau,\tau_{0}\in\mathbb{R}^{d}$.

For $j_{i}=j$ (which implies $\rho_{x}\in U_{j_{i}}^{(i)}=U_{j}^{(i)}$) we
further see that

$\begin{split}&\left|\left(\Square_{j,\tau_{0},x}^{k}\,\,g_{i,\tau,\tau_{0}}\right)({\scaleobj{0.65}{\Upsilon}})\right|\\\
&\overset{\text{Lem.
\ref{lem:estimate_diffop_abs}}}{\leq}D_{k,\delta}\cdot(2\pi|x|)^{-k}\cdot
v_{0}^{3k}({\scaleobj{0.65}{\Upsilon}})\cdot\sum_{n=0}^{k}\left|\frac{\partial^{n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{n}}g_{i,\tau,\tau_{0}}({\scaleobj{0.65}{\Upsilon}})\right|\\\
&\overset{\text{Lem.
\ref{lem:partial_derivs_of_localized_gte}}}{\leq}D_{k,\delta}\cdot{\mathds{1}}_{B_{\delta/(4d)}({\scaleobj{0.65}{\Upsilon}}_{i})}({\scaleobj{0.65}{\Upsilon}})\cdot(2\pi|x|)^{-k}\cdot
v_{0}^{d+3k}({\scaleobj{0.65}{\Upsilon}})\cdot\sum_{n=0}^{k}C_{n}\sum_{\ell=0}^{n}\sum_{\alpha=0}^{\ell}\left|\frac{\partial^{\ell-\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\ell-\alpha}}\theta_{2}({\scaleobj{0.65}{\Upsilon}})\cdot\frac{\partial^{\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\alpha}}\overline{\theta_{1}({\scaleobj{0.65}{\Upsilon}}-\tau)}\right|.\end{split}$

Note that constants above are independent of $i\in\mathbb{Z}^{d}$. Next, using
the finite overlap property, (4.29), we get

$\sum_{\begin{subarray}{c}i\in\mathbb{Z}^{d}\\\ \text{s.t.
}j_{i}=j\end{subarray}}\left|\left(\Square_{j,\tau_{0},x}^{k}\,\,g_{i,\tau,\tau_{0}}\right)({\scaleobj{0.65}{\Upsilon}})\right|\leq\widetilde{C}\cdot(2\pi|x|)^{-k}v_{0}^{d+3k}({\scaleobj{0.65}{\Upsilon}})\sum_{n=0}^{k}\sum_{\ell=0}^{n}\sum_{\alpha=0}^{\ell}\left|\frac{\partial^{\ell-\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\ell-\alpha}}\theta_{2}({\scaleobj{0.65}{\Upsilon}})\cdot\frac{\partial^{\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\alpha}}\overline{\theta_{1}({\scaleobj{0.65}{\Upsilon}}-\tau)}\right|,$

where $\widetilde{C}:=(1+4d)^{d}\cdot
D_{k,\delta}\cdot\max_{n=0,\dots,k}C_{n}$. Insert this estimate into the final
line of (4.38), apply the Cauchy-Schwarz inequality, and recall that $w_{1}$
is submultiplicative and satisfies
$w_{1}(-{\scaleobj{0.65}{\Upsilon}})=w_{1}({\scaleobj{0.65}{\Upsilon}})$,
whence $1=[w_{1}(\tau)]^{-1}\cdot
w_{1}({\scaleobj{0.65}{\Upsilon}}+\tau-{\scaleobj{0.65}{\Upsilon}})\leq[w_{1}(\tau)]^{-1}\cdot
w_{1}({\scaleobj{0.65}{\Upsilon}})\cdot
w_{1}({\scaleobj{0.65}{\Upsilon}}-\tau),$ to obtain

$\begin{split}\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\sum_{j\in\underline{d}}\int_{\mathbb{R}^{d}}\sum_{\begin{subarray}{c}i\in\mathbb{Z}^{d}\\\
\text{s.t.
}j_{i}=j\end{subarray}}\left|\left(\Square_{j,\tau_{0},x}^{k}\,\,g_{i,\tau,\tau_{0}}\right)({\scaleobj{0.65}{\Upsilon}})\right|~{}d{\scaleobj{0.65}{\Upsilon}}$\mbox{}\hfil\\\
&\leq\widetilde{C}\cdot(2\pi|x|)^{-k}\sum_{j\in\underline{d}}\int_{\mathbb{R}^{d}}v_{0}^{d+3k}({\scaleobj{0.65}{\Upsilon}})\sum_{n=0}^{k}\sum_{\ell=0}^{n}\sum_{\alpha=0}^{\ell}\left|\frac{\partial^{\ell-\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\ell-\alpha}}\theta_{2}({\scaleobj{0.65}{\Upsilon}})\cdot\frac{\partial^{\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\alpha}}\overline{\theta_{1}({\scaleobj{0.65}{\Upsilon}}-\tau)}\right|~{}d{\scaleobj{0.65}{\Upsilon}}\\\
&\leq\widetilde{C}\cdot(2\pi|x|)^{-k}[w_{1}(\tau)]^{-1}\cdot\\\ &\hskip
40.0pt\sum_{j\in\underline{d}}\sum_{n=0}^{k}\sum_{\ell=0}^{n}\sum_{\alpha=0}^{\ell}\int_{\mathbb{R}^{d}}\left|v_{0}^{d+3k}({\scaleobj{0.65}{\Upsilon}})\cdot
w_{1}({\scaleobj{0.65}{\Upsilon}})\frac{\partial^{\ell-\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\ell-\alpha}}\theta_{2}({\scaleobj{0.65}{\Upsilon}})\cdot
w_{1}({\scaleobj{0.65}{\Upsilon}}-\tau)\frac{\partial^{\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\alpha}}\overline{\theta_{1}({\scaleobj{0.65}{\Upsilon}}-\tau)}\right|~{}d{\scaleobj{0.65}{\Upsilon}}\\\
&\leq\widetilde{C}\cdot(2\pi|x|)^{-k}w_{1}(\tau)^{-1}\sum_{j\in\underline{d}}\sum_{n=0}^{k}\sum_{\ell=0}^{n}\sum_{\alpha=0}^{\ell}\left\|\frac{\partial^{\ell-\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\ell-\alpha}}\theta_{2}\right\|_{\mathbf{L}^{2}_{w_{2}}}\cdot\left\|\frac{\partial^{\alpha}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\alpha}}\overline{\theta_{1}}\right\|_{\mathbf{L}^{2}_{w_{1}}}.\end{split}$

Since all the involved sums are finite, so is the total number of summands.
Moreover, the highest order partial derivatives that appear are
$\frac{\partial^{k}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{k}}\theta_{1}$
and
$\frac{\partial^{k}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{k}}\theta_{2}$,
for arbitrary $j\in\underline{d}$. Hence, a joint maximization over
$j\in\underline{d}$ and the partial derivatives of $\theta_{1},\theta_{2}$
yields

$\begin{split}\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\left|\int_{\mathbb{R}^{d}}\frac{w({\scaleobj{0.65}{\Upsilon}}+\tau_{0})}{w(\tau_{0})}\left(\theta_{2}\cdot\overline{\mathbf{T}_{\tau}\theta_{1}}\right)\\!({\scaleobj{0.65}{\Upsilon}})\,e_{\tau_{0}}(x,{\scaleobj{0.65}{\Upsilon}})~{}d{\scaleobj{0.65}{\Upsilon}}\right|$\mbox{}\hfil\\\
&\leq
C^{\prime}\cdot(2\pi|x|)^{-k}w_{1}(\tau)^{-1}\max_{j\in\underline{d}}\left\\{\left(\max_{n=0,\dots,k}\left\|\frac{\partial^{n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{n}}\theta_{1}\right\|_{\mathbf{L}^{2}_{w_{1}}}\right)\cdot\left(\max_{n=0,\dots,k}\left\|\frac{\partial^{n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{n}}\theta_{2}\right\|_{\mathbf{L}^{2}_{w_{2}}}\right)\right\\}\\\
&\leq C^{\prime}\cdot(2\pi|x|)^{-k}\cdot[w_{1}(\tau)]^{-1}\cdot
C_{\max},\end{split}$ (4.39)

for a suitable (large) constant
$C^{\prime}=C^{\prime}(d,k,\delta,\Phi,v_{0})>0$. Here, the last step used
again that $w_{1}\leq w_{2}$.

Now, define

$F(x):=\begin{cases}C_{\max},&\text{ if }|x|<1\\\ C^{\prime}\cdot
C_{\max}\cdot(2\pi|x|)^{-k},&\text{ else.}\end{cases}$

It is not hard to see $|F(x)|\leq C^{\prime\prime}\cdot
C_{\max}\cdot(1+|x|)^{-k}$ for some constant
$C^{\prime\prime}=C^{\prime\prime}(d,k,\Phi,v_{0})>0$, since we have
$\delta=\delta(d,v_{0},\Phi)$. Combining the inequalities (4.37) and (4.39),
we obtain

$\displaystyle\left|\int_{\mathbb{R}^{d}}\frac{w({\scaleobj{0.65}{\Upsilon}}+\tau_{0})}{w(\tau_{0})}\left(\theta_{2}\cdot\overline{\mathbf{T}_{\tau}\theta_{1}}\right)\\!({\scaleobj{0.65}{\Upsilon}})\,e_{\tau_{0}}(x,{\scaleobj{0.65}{\Upsilon}})~{}d{\scaleobj{0.65}{\Upsilon}}\right|$
$\displaystyle\leq[w_{1}(\tau)]^{-1}\cdot F(x)$ $\displaystyle\leq
C^{\prime\prime}\cdot
C_{\max}\cdot(1+|x|)^{-k}\cdot[w_{1}(\tau)]^{-1},\quad\forall\,x,\tau,\tau_{0}\in\mathbb{R}^{d}\,.\qed$

###  4.3 Proof of Theorem 4.4

Recall that $w=\det A$ is $w_{0}:=v_{0}^{d}$-moderate (Lemma 4.9) and
$v_{0},v_{1}\geq 1$ (see proof of Theorem 4.8), such that $w_{2}\geq
v_{0}^{d/2}=\sqrt{w_{0}}$ and
$\theta_{1},\theta_{2}\in\mathbf{L}_{\sqrt{w_{0}}}^{2}(\mathbb{R}^{d})$
follows. That $m$ is $\Phi$-compatible with dominating weight $m^{\Phi}$ is an
immediate consequence of the inequality (4.6), i.e.,

$m\bigl{(}(y,\xi),(z,\eta)\bigr{)}\leq(1+|y-z|)^{p}\cdot
v_{1}\bigl{(}\Phi(\xi)-\Phi(\eta)\bigr{)},\text{ for all
}y,z\in\mathbb{R}^{d}\text{ and }\xi,\eta\in D,$

and the choice of $p\in\mathbb{N}_{0}$ (in particular, $p=0$ if
$R_{\Phi}=\infty$).

Thus, Lemma 4.6 and Lemma 4.7 can be applied, showing that

$\|K_{\theta_{1},\theta_{2}}\|_{\mathcal{B}_{m}}\leq\max_{\ell\in\\{1,2\\}}\mathop{\operatorname{ess~{}sup}}_{\tau_{0}\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}M(x,\tau)L^{(\ell)}_{\tau_{0}}(x,\tau)~{}dx~{}d\tau,$

where $M(x,\tau)\\!=\sup_{y\in\mathbb{R}^{d},|y|\leq R|x|}(1+|y|)^{p}\cdot
v_{0}^{d/2}(\tau)\cdot v_{1}(\tau)$. Note that
$M(x,\tau)\\!\leq\\!C_{\Phi}\cdot(1+|x|)^{p}\cdot v_{0}^{d/2}(\tau)\cdot
v_{1}(\tau)$, where $C_{\Phi}:=\max\big{\\{}1,\sup_{\xi\in
D}\|\mathrm{D}\Phi(\xi)\|^{p}\big{\\}}$ if $p>0$ and $C_{\Phi}:=1$ otherwise.

Define
$w_{1}:\mathbb{R}^{d}\to\mathbb{R}^{+},{\scaleobj{0.65}{\Upsilon}}\mapsto(1+|{\scaleobj{0.65}{\Upsilon}}|)^{d+1}\cdot
v_{1}({\scaleobj{0.65}{\Upsilon}})\cdot[v_{0}({\scaleobj{0.65}{\Upsilon}})]^{d/2}.$
Since $v_{0},v_{1}$ are submultiplicative and satisfy
$v_{\ell}(-{\scaleobj{0.65}{\Upsilon}})=v_{\ell}({\scaleobj{0.65}{\Upsilon}})$
for $\ell\in\\{0,1\\}$ and ${\scaleobj{0.65}{\Upsilon}}\in\mathbb{R}^{d}$, it
is easy to see that $w_{1}$ satisfies the same two properties. Furthermore,
$w_{2}({\scaleobj{0.65}{\Upsilon}})=w_{1}({\scaleobj{0.65}{\Upsilon}})\cdot[v_{0}({\scaleobj{0.65}{\Upsilon}})]^{d+3(d+p+1)}$,
so that Theorem 4.8, with $k=d+p+1$, yields a constant $C=C(d,p,\Phi,v_{0})>0$
satisfying

$\begin{split}\|K_{\theta_{1},\theta_{2}}\|_{\mathcal{B}_{m}}&\leq\max_{\ell\in\\{1,2\\}}\mathop{\operatorname{ess~{}sup}}_{\tau_{0}\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}M(x,\tau)L^{(\ell)}_{\tau_{0}}(x,\tau)~{}dx~{}d\tau\\\
&\leq
C_{\Phi}\cdot\mathop{\operatorname{ess~{}sup}}_{\tau_{0}\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}(1+|x|)^{p}\cdot
v_{0}^{d/2}(\tau)\cdot
v_{1}(\tau)\cdot\max_{\ell\in\\{1,2\\}}L^{(\ell)}_{\tau_{0}}(x,\tau)~{}dx~{}d\tau\\\
\text{\scriptsize{(Thm. \ref{lem:NiceCrossGramianEstimate})}}&\leq
CC_{\Phi}C_{\max}\cdot\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}v_{0}^{d/2}(\tau)\cdot
v_{1}(\tau)\cdot[w_{1}(\tau)]^{-1}\cdot(1+|x|)^{-(d+1)}~{}d\tau~{}dx\\\ &\leq
CC_{\Phi}C_{\max}\cdot\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}(1+|\tau|)^{-(d+1)}(1+|x|)^{-(d+1)}~{}d\tau~{}dx\\\
&=:\widetilde{C}\cdot C_{\max}<\infty.\end{split}$

Here, the final constant $\widetilde{C}=\widetilde{C}(d,p,\Phi,v_{0})>0$ is
finite, simply because
$(1+|\cdot|)^{-(d+1)}\in\mathbf{L}^{1}(\mathbb{R}^{d})$. This concludes the
proof. ∎

##  5 The phase-space coverings induced by the warping function $\Phi$

To prepare for the estimation of
$\|{\mathrm{osc}}_{\mathcal{V}^{\delta},\Gamma}\|_{\mathcal{B}_{m}}$ we
construct families of coverings
$\mathcal{V}_{\Phi}^{\delta}=(V_{i}^{\delta})_{i\in I}$ of the phase space
$\Lambda$, induced by a given warping function $\Phi$ and study their
properties. In the next section, we will show that
$\|{\mathrm{osc}}_{\mathcal{V}^{\delta},\Gamma}\|_{\mathcal{B}_{m}}\to 0$ as
$\delta\to 0$, with ${\mathrm{osc}}_{\mathcal{V}^{\delta},\Gamma}$ as
introduced in Definition 2.16.

######  Definition 5.1.

Let $\Phi\colon D\rightarrow\mathbb{R}^{d}$ be a warping function. Define

$Q_{\Phi,\tau}^{(\delta,r)}:=\Phi^{-1}(\delta\cdot B_{r}(\tau)),\quad\text{for
all }r,\delta>0\text{ and }\tau\in\mathbb{R}^{d}.$ (5.1)

We call
$\mathcal{V}^{\delta}_{\Phi}=(V^{\delta}_{\ell,k})_{\ell,k\in\mathbb{Z}^{d}}$,
defined by

$V_{\ell,k}^{\delta}:=A^{-T}(\delta k/\sqrt{d})\left\langle\delta\cdot
B_{1}(\ell/\sqrt{d})\right\rangle\times Q_{k}^{\delta},\quad\text{with}\quad
Q_{k}^{\delta}:=Q_{\Phi,k/\sqrt{d}}^{(\delta,1)}=\Phi^{-1}\bigl{(}\delta\cdot
B_{1}(k/\sqrt{d})\bigr{)},$ (5.2)

the _$\Phi$ -induced $\delta$-fine (phase-space) covering_.

By allowing $r\neq 1$ in (5.1), it is possible to control the overlap of the
covering elements. In particular, any radius strictly larger than $1/2$
provides a covering. For proving the feasibility of discretization in coorbit
spaces, however, the above choice of $r=1$ in (5.2) is completely sufficient.

######  Proposition 5.2.

Let $\Phi$ be a $0$-admissible warping function with control weight $v_{0}$
(see Definition 4.2). Then the $\Phi$-induced $\delta$-fine phase-space
covering
$\mathcal{V}^{\delta}_{\Phi}=(V^{\delta}_{\ell,k})_{\ell,k\in\mathbb{Z}^{d}}$
is a topologically admissible cover of $\Lambda=\mathbb{R}^{d}\times D$ which
is also product-admissible as per Definition 2.4. More precisely, we have the
following properties:

  1. (1)

If $k,\ell,k_{0},\ell_{0}\in\mathbb{Z}^{d}$ satisfy $|k-k_{0}|>2\sqrt{d}$,
then $V^{\delta}_{\ell,k}\cap V^{\delta}_{\ell_{0},k_{0}}=\varnothing$.
Furthermore,

$\qquad\sup_{(\ell,k)\in\mathbb{Z}^{2d}}\\#\big{\\{}(\ell_{0},k_{0})\in\mathbb{Z}^{d}\times\mathbb{Z}^{d}~{}:~{}V^{\delta}_{\ell,k}\cap
V^{\delta}_{\ell_{0},k_{0}}\neq\varnothing\big{\\}}\leq(1+4d)^{d}\big{(}1+2\sqrt{d}\cdot\left(1+v_{0}(2\delta)\right)\big{)}^{d}.$

  2. (2)

We have
$[v_{0}(\delta)]^{-d}\leq\frac{\mu(V^{\delta}_{\ell,k})}{[\mu(B_{1}(0))]^{2}\cdot\delta^{2d}}\leq[v_{0}(\delta)]^{d}$
for all $k,\ell\in\mathbb{Z}^{d}$.

  3. (3)

We have
$\mu(V_{\ell,k}^{\delta})/\mu(V_{\ell_{0},k_{0}}^{\delta})\leq[v_{0}(\delta)]^{2d}$
for arbitrary $\ell,k,\ell_{0},k_{0}\in\mathbb{Z}^{d}$.

  4. (4)

For each fixed $\delta>0$, the weight $w_{\mathcal{V}_{\Phi}^{\delta}}$ as
given in Equation (2.8) satisfies

$\begin{split}\quad\bigl{(}w_{\mathcal{V}_{\Phi}^{\delta}}\bigr{)}_{\ell,k}&\asymp\min\big{\\{}w(\delta\cdot
k/\sqrt{d}),[w(\delta\cdot k/\sqrt{d})]^{-1}\big{\\}}\\\
&\asymp\min\big{\\{}w(\Phi(\xi)),[w(\Phi(\xi))]^{-1}\big{\\}}\gtrsim[v_{0}(\Phi(\xi))]^{-d},\quad\text{for
all }\ell,k\in\mathbb{Z}^{d}\text{ and }\xi\in Q^{\delta}_{k}.\end{split}$
(5.3)

In particular, there exists a constant $C=C(d,\delta,\Phi,v_{0})>0$ such that
$(w_{\mathcal{V}_{\Phi}^{\delta}})_{\ell,k}\big{/}(w_{\mathcal{V}_{\Phi}^{\delta}})_{\ell_{0},k_{0}}\leq
C$ for all $\ell,k,\ell_{0},k_{0}\in\mathbb{Z}^{d}$ with
$V^{\delta}_{\ell,k}\cap V^{\delta}_{\ell_{0},k_{0}}\neq\varnothing$.
Moreover, (2.9) holds with

$w_{\mathcal{V}_{\Phi}^{\delta}}^{c}:\quad\Lambda\rightarrow\mathbb{R}^{+},\quad(x,\xi)\mapsto\min\big{\\{}w(\Phi(\xi)),\,\,[w(\Phi(\xi))]^{-1}\big{\\}}.$

###### Proof.

Note that the family $\delta\cdot B_{1}(\ell/\sqrt{d})$,
$\ell\in\mathbb{Z}^{d}$ forms a covering of $\mathbb{R}^{d}$, since
$\frac{1}{\sqrt{d}}\big{(}\ell+[0,1)^{d}\big{)}\\!\subset\\!B_{1}(\ell/\sqrt{d})$.
Considering that $\Phi:D\to\mathbb{R}^{d}$ is a diffeomorphism and
$A^{-T}(\delta k/\sqrt{d})$, for any $k\in\mathbb{Z}^{d}$, is an invertible
matrix, it follows that $\mathcal{V}^{\delta}_{\Phi}$ indeed covers all of
$\Lambda$.

We first prove part (1). For $k,\ell\in\mathbb{Z}^{d}$, let

$J_{\ell,k}:=\left\\{\left(\ell_{0},k_{0}\right)\in\mathbb{Z}^{d}\times\mathbb{Z}^{d}\,:\,V_{\ell,k}^{\delta}\cap
V_{\ell_{0},k_{0}}^{\delta}\neq\varnothing\right\\}.$

If $V_{\ell,k}^{\delta}\cap V_{\ell_{0},k_{0}}^{\delta}\neq\varnothing$, then
in particular $Q_{k}^{\delta}\cap Q_{k_{0}}^{\delta}\neq\varnothing$.
Straightforward calculations show that the latter implies $|k_{0}-k|\leq
2\sqrt{d}$, and then $k_{0}\in k+\left\\{-2d,\dots,2d\right\\}^{d}$. Moreover,
if $(\ell_{0},k_{0})\in J_{\ell,k}$, then an easy calculation shows that there
exist $x_{1},x_{2}\in B_{1}(0)$ such that

$\ell_{0}=A^{T}\left(\delta k_{0}/\smash{\sqrt{d}}\right)\cdot
A^{-T}\left(\delta k/\smash{\sqrt{d}}\right)\left\langle\ell+\sqrt{d}\cdot
x_{1}\right\rangle-\sqrt{d}\cdot x_{2}.$

Property (4.5) shows that $A_{k,k_{0}}:=A^{T}\left(\delta
k_{0}/\smash{\sqrt{d}}\right)\cdot A^{-T}\left(\delta
k/\smash{\sqrt{d}}\right)=\phi_{\delta
k/\sqrt{d}}(\delta\cdot(k_{0}-k)/\sqrt{d})$ satisfies

$\left\|A_{k,k_{0}}\right\|\leq
v_{0}\left(\tfrac{\delta}{\sqrt{d}}(k_{0}-k)\right)\leq v_{0}(2\delta).$

Here, we used $|k_{0}-k|\leq 2\sqrt{d}$ and that $v_{0}$ is radially
increasing. Since $x_{1},x_{2}\in B_{1}(0)$, we thus have $\ell_{0}\in
A_{k,k_{0}}\ell+[-C_{1},C_{1}]^{d}$, where

$C_{1}:=\sqrt{d}\cdot\bigl{(}1+v_{0}(2\delta)\bigr{)}\geq\left|\sqrt{d}\cdot
A_{k,k_{0}}x_{1}-\sqrt{d}\cdot x_{2}\right|.$

Altogether, we have shown

$J_{\ell,k}\subset\bigcup_{k_{0}\in
k+\left\\{-2d,\dots,2d\right\\}^{d}}\left(\left[\mathbb{Z}^{d}\cap\left(A_{k,k_{0}}\ell+\left[-C_{1},C_{1}\right]^{d}\right)\right]\times\left\\{k_{0}\right\\}\right).$

But we have
$\\#\big{[}\mathbb{Z}^{d}\cap\big{(}A_{k,k_{0}}\ell+\left[-C_{1},C_{1}\right]^{d}\big{)}\big{]}\leq\left(1+2C_{1}\right)^{d}$
and hence

$|J_{\ell,k}|\leq\sum_{k_{0}\in
k+\\{-2d,\dots,2d\\}^{d}}(1+2C_{1})^{d}=(1+4d)^{d}(1+2C_{1})^{d},$

with $C_{1}=\sqrt{d}\cdot\left(1+v_{0}(2\delta)\right)$, completing the proof
of part (1). This also shows that $\mathcal{V}_{\Phi}^{\delta}$ is an
admissible covering. Since each $V_{\ell,k}^{\delta}$ is open and relatively
compact in $\Lambda$, we see that $\mathcal{V}_{\Phi}^{\delta}$ is
topologically admissible.

We proceed to prove item (2). By the change of variables formula, cf. Equation
(3.4), we get

$\mu(Q_{k}^{\delta})=\int_{D}{\mathds{1}}_{\delta\cdot
B_{1}(k/\sqrt{d})}(\Phi(\xi))~{}d\xi=\int_{\delta\cdot
B_{1}(k/\sqrt{d})}w(\tau)~{}d\tau.$

Recall that $w$ is $v_{0}^{d}$-moderate by Lemma 4.9, where $v_{0}$ is
submultiplicative and radially increasing. Therefore,

$[v_{0}(\delta)]^{-d}\leq\frac{w(\tau)}{w(\delta\cdot
k/\sqrt{d})}\leq[v_{0}(\delta)]^{d},\text{ for all
}\tau\in\delta\cdot\overline{B_{1}}(k/\sqrt{d})$

In combination, the two preceding displayed equations show that

$\mu(Q_{k}^{\delta})\in\mu(B_{1}(0))\cdot\delta^{d}\cdot w(\delta\cdot
k/\sqrt{d})\cdot\left[[v_{0}(\delta)]^{-d},[v_{0}(\delta)]^{d}\right].$ (5.4)

Moreover,

$\mu\left(A^{-T}(\delta k/\sqrt{d})\langle\delta\cdot
B_{1}(\ell/\sqrt{d})\rangle\right)=\left|\det\left(A^{-T}(\delta\cdot
k/\sqrt{d})\right)\right|\cdot\mu\left(\delta\cdot
B_{1}(\ell/\sqrt{d})\right)=\frac{\mu(B_{1}(0))\cdot\delta^{d}}{w(\delta\cdot
k/\sqrt{d})}.$ (5.5)

Since $\mu(V_{\ell,k}^{\delta})=\mu\big{(}A^{-T}(\delta
k/\sqrt{d})\langle\delta\cdot
B_{1}(\ell/\sqrt{d})\rangle\big{)}\cdot\mu(Q_{k}^{\delta})$, this proves part
(2). Finally, part (3) is a direct consequence of part (2).

It remains to prove part (4). Since $\mathcal{V}^{\delta}_{\Phi}$ is a
covering of $\Lambda=\mathbb{R}^{d}\times D$ with countable index set and with
each set $V_{\ell,k}^{\delta}$ being a Cartesian product of open sets, this
will then imply that $\mathcal{V}^{\delta}_{\Phi}$ is product-admissible.
First note that $\min\bigl{\\{}1,\mu(V^{\delta}_{\ell,k})\bigr{\\}}\asymp 1$
as a function in $\ell,k\in\mathbb{Z}^{d}$ and that
$V_{\ell,k}^{\delta}=V_{1,(\ell,k)}^{\delta}\times V_{2,(\ell,k)}^{\delta}$
with $V^{\delta}_{1,(\ell,k)}=A^{-T}(\delta k/\sqrt{d})\langle\delta\cdot
B_{1}(\ell/\sqrt{d})\rangle$ and $V^{\delta}_{2,(\ell,k)}=Q^{\delta}_{k}$.
Hence, by (5.5) and (5.4), we have

$\min\big{\\{}\mu(V^{\delta}_{1,(\ell,k)}),\,\,\mu(V^{\delta}_{2,(\ell,k)})\big{\\}}\asymp\min\big{\\{}w(\delta
k/\sqrt{d}),[w(\delta k/\sqrt{d})]^{-1}\big{\\}},\,\text{as a function in
}\ell,k\in\mathbb{Z}^{d}.$

Together, this yields the first estimate in (5.3). The other two estimates in
(5.3) are simple consequences of $w$ being $v_{0}^{d}$-moderate (and thus
$w^{-1}$ is as well) and of the identity $\Phi(Q^{\delta}_{k})=\delta\cdot
B_{1}(k/\sqrt{d})$. Note that (5.3) implies (2.9) with the stated choice of
$w^{c}_{\mathcal{V}^{\delta}_{\Phi}}$.

To prove that
$(w_{\mathcal{V}_{\Phi}^{\delta}})_{\ell,k}/(w_{\mathcal{V}_{\Phi}^{\delta}})_{\ell_{0},k_{0}}\lesssim
1$ if $V_{\ell,k}^{\delta}\cap V_{\ell_{0},k_{0}}^{\delta}\neq\varnothing$,
first note that since $w$ is $v_{0}^{d}$-moderate and $v_{0}$ is radially
increasing (and hence radial). Note that taking reciprocal values, as well as
pointwise minima/maxima preserve moderateness relations, see Remark 2.6, such
that

$\frac{\min\bigl{\\{}w(\delta k/\sqrt{d}),[w(\delta
k/\sqrt{d})]^{-1}\bigr{\\}}}{\min\bigl{\\{}w(\delta k_{0}/\sqrt{d}),[w(\delta
k_{0}/\sqrt{d})]^{-1}\bigr{\\}}}\leq\bigl{[}v_{0}(\delta(k-k_{0})/\sqrt{d})\bigr{]}^{d}\qquad\forall\,k,k_{0}\in\mathbb{Z}^{d}.$

Furthermore, part (1) of the proposition shows that if
$V_{\ell,k}^{\delta}\cap V_{\ell_{0},k_{0}}^{\delta}\neq\varnothing$, then
$|k-k_{0}|\leq 2\sqrt{d}$. Combining these observations with Equation (5.3)
and with the fact that $v_{0}$ is radially increasing, we see
$(w_{\mathcal{V}_{\Phi}^{\delta}})_{\ell,k}/(w_{\mathcal{V}_{\Phi}^{\delta}})_{\ell_{0},k_{0}}\leq
C_{1}\cdot v_{0}(2\delta)^{d}\leq C_{2}$ for suitable constants $C_{1},C_{2}$
depending (only) on $d,\delta,\Phi,v_{0}$. ∎

The next lemma is concerned with the sets $\mathbf{V}_{\lambda}=\bigcup_{i\in
I\text{ s.t. }\lambda\in V_{i}}V_{i}$ defining the oscillation
${\mathrm{osc}}_{\mathcal{V},\Gamma}$, see Definition 2.16. For the induced
coverings $\mathcal{V}^{\delta}_{\Phi}$, the set
$\mathbf{V}_{\lambda}^{\delta}$ can once more be estimated by a convenient
product set. Moreover, the lemma implies that if
$\lambda=(z,\eta)\in\mathbf{V}_{\lambda_{0}}^{\delta}$, with
$\lambda_{0}=(y,\omega)$, then

$|A^{T}(\Phi(\omega))\langle z-y\rangle|\leq
C_{\delta}\qquad\text{and}\qquad|\Phi(\eta)-\Phi(\omega)|\leq
C_{\delta},\quad\text{with}\quad C_{\delta}>0\text{ independent of
}\lambda,\lambda_{0}.$

In particular, this holds if there exists $(\ell,k)\in\mathbb{Z}^{2d}$ such
that $\lambda,\lambda_{0}\in V_{\ell,k}^{\delta}$. These estimates will be
crucial for estimating
$\|{\mathrm{osc}}_{\mathcal{V}^{\delta},\Gamma}\|_{\mathcal{B}_{m}}$.

######  Lemma 5.3.

Let $\Phi$ be a warping function, and let $\mathcal{V}^{\delta}_{\Phi}$ be the
$\Phi$-induced $\delta$-fine covering. For all $(y,\omega)\in\Lambda$ and all
$\delta>0$, we have

$\mathbf{V}_{(y,\omega)}^{\delta}=\bigcup_{\begin{subarray}{c}(\ell,k)\text{
s.t.}\\\ (y,\omega)\in
V_{\ell,k}^{\delta}\end{subarray}}V_{\ell,k}^{\delta}\subset(y+\mathbf{P}_{\omega}^{\delta})\times\mathbf{Q}_{\omega}^{\delta},\vspace{-0.3cm}$
(5.6)

where

$\mathbf{Q}_{\omega}^{\delta}:=\Phi^{-1}\bigl{(}\Phi(\omega)+B_{2\delta}(0)\bigr{)}\qquad\text{
and }\qquad\mathbf{P}_{\omega}^{\delta}:=v_{0}(\delta)\cdot
A^{-T}(\Phi(\omega))\left\langle B_{2\delta}(0)\right\rangle.$ (5.7)

###### Proof.

Let $(\ell,k)\in\mathbb{Z}^{d}\times\mathbb{Z}^{d}$ be such that
$(y,\omega)\in V_{\ell,k}^{\delta}$. Then $\delta
k/\sqrt{d}\in\Phi(\omega)+\delta\cdot B_{1}(0)$ and by extension of that
argument, $Q_{k}^{\delta}\subset\Phi^{-1}(\Phi(\omega)+2\delta
B_{1}(0))=\mathbf{Q}_{\omega}^{\delta}$, which proves the first part of the
claim.

Next, for $(x,\xi)\in V_{\ell,k}^{\delta}$, we have $|A^{T}(\delta
k/\sqrt{d})\left\langle x-y\right\rangle|<2\delta$, since $x,y\in
A^{-T}(\delta k/\sqrt{d})\left\langle\delta
B_{1}(\ell/\sqrt{d})\right\rangle$. Hence,

$\displaystyle\left|A^{T}(\Phi(\omega))\left\langle x-y\right\rangle\right|$
$\displaystyle=\left|A^{T}(\Phi(\omega))A^{-T}(\delta k/\sqrt{d})A^{T}(\delta
k/\sqrt{d})\left\langle x-y\right\rangle\right|$
$\displaystyle<2\delta\cdot\big{\|}A^{T}(\Phi(\omega))A^{-T}(\delta
k/\sqrt{d})\big{\|}=2\delta\cdot\big{\|}\phi_{\delta
k/\sqrt{d}}\bigl{(}\Phi(\omega)-\delta k/\sqrt{d}\bigr{)}\big{\|}$
$\displaystyle({\scriptstyle\text{cf.~{}Eq.~{}\eqref{eq:PhiHigherDerivativeEstimate}}})$
$\displaystyle\leq 2\delta\cdot v_{0}(\Phi(\omega)-\delta k/\sqrt{d})\leq
2\delta\cdot v_{0}(\delta),$

which shows $x-y\in A^{-T}(\Phi(\omega))\left\langle 2\delta
v_{0}(\delta)B_{1}(0)\right\rangle$, and thus $x\in
y+\mathbf{P}_{\omega}^{\delta}$, as desired. ∎

######  Proposition 5.4.

Let $\Phi$ be a $0$-admissible warping function with control weight $v_{0}$.
Let further $m_{0}:\Lambda\times\Lambda\to\mathbb{R}^{+}$ be continuous and
symmetric, with $1\leq m_{0}(\lambda,\rho)\leq C^{(0)}\cdot
m_{0}(\lambda,{\nu})\cdot m_{0}({\nu},\rho)$, for all
$\lambda,\rho,{\nu}\in\Lambda$ and some $C^{(0)}\geq 1$, satisfy

$m_{0}((y,\xi),(z,\eta))\leq(1+|y-z|)^{p}\cdot\zeta_{1}\bigl{(}\Phi(\xi)-\Phi(\eta)\bigr{)}\quad\forall\,(y,\xi),(z,\eta)\in\Lambda.$

Here, $p=0$ if $R_{\Phi}=\sup_{\xi\in D}\|\mathrm{D}\Phi(\xi)\|=\infty$ and
$p\in\mathbb{N}_{0}$ otherwise, and
$\zeta_{1}:\mathbb{R}^{d}\to\mathbb{R}^{+}$ is a continuous function with
$\zeta_{1}(-\tau)=\zeta_{1}(\tau)$ for all $\tau\in\mathbb{R}^{d}$. Define,
for some arbitrary, fixed ${\nu}\in\Lambda$,

$\begin{split}u:\quad\Lambda\to\mathbb{R}^{+},&\quad\lambda\mapsto
m_{0}(\lambda,{\nu})\qquad\text{and}\\\
v:\quad\Lambda\to\mathbb{R}^{+},&\quad(y,\xi)\mapsto
u(y,\xi)\cdot\max\big{\\{}w(\Phi(\xi)),[w(\Phi(\xi))]^{-1}\big{\\}}\end{split}$

and let $m_{v}$ be as in Equation (2.10). Then $u$ is
$\mathcal{V}_{\Phi}^{\delta}$-moderate, for any $\delta>0$, and $m_{0}$ and
$m_{v}$ are $\Phi$-convolution-dominated by
$(1+|\bullet|)^{p}\cdot\zeta_{1}(\bullet)$ and
$m_{v}^{\Phi}:=(1+|\bullet|)^{p}\cdot\zeta_{2}(\bullet)$, where
$\zeta_{2}=v_{0}^{d}\cdot\zeta_{1}$. In particular, items (1)-(3) of
Assumption 2.11 are satisfied.

###### Proof.

Proposition 5.2 provides product-admissibility of
$\mathcal{V}^{\delta}_{\Phi}$, such that item (1) of Assumption 2.11 is
satisfied. Item (3) is a direct consequence of the symmetry of $m_{0}$:

$m_{0}(\lambda,\rho)\leq C^{(0)}\cdot m(\lambda,{\nu})\cdot
m({\nu},\rho)=C^{(0)}\cdot u(\lambda)\cdot u(\rho).$

To show $\mathcal{V}^{\delta}_{\Phi}$-moderateness of $u$ (which coincides
with item (2) of Assumption 2.11), observe that

$\frac{u(\lambda)}{u(\rho)}\leq
C^{(0)}\frac{m_{0}(\lambda,\rho)m_{0}(\rho,{\nu})}{m_{0}(\rho,{\nu})}=C^{(0)}m_{0}(\lambda,\rho).$
(5.8)

If $\lambda=(y,\xi)$ and $\rho=(z,\eta)$ are both contained in
$V^{\delta}_{\ell,k}$, for some $\ell,k\in\mathbb{Z}^{d}$, then
$|\Phi(\xi)-\Phi(\eta)|<\delta$, and

$|y-z|\leq\delta\cdot\|A^{-T}(\delta k/\sqrt{\delta})\|\leq\delta\cdot
R_{\Phi},\quad\text{if }R_{\Phi}<\infty.$

Hence, and $\frac{u(y,\xi)}{u(z,\eta)}\leq C^{(0)}m_{0}((y,\xi),(z,\eta))\leq
C^{(0)}(1+\delta R_{\Phi})^{p}\cdot\zeta_{1}(\delta),$ as desired. If
$R_{\Phi}=\infty$, then $\frac{u(y,\xi)}{u(z,\eta)}\leq
C^{(0)}m_{0}((y,\xi),(z,\eta))\leq C^{(0)}\zeta_{1}(\delta)$ instead.

That $m_{0}$ is $\Phi$-convolution-dominated by
$(1+|\bullet|)^{p}\cdot\zeta_{1}(\bullet)$ is immediate. To prove that $m_{v}$
is $\Phi$-convolution-dominated by $m_{v}^{\Phi}$, observe

$\frac{\max\big{\\{}w(\tau),[w(\tau)]^{-1}\big{\\}}}{\max\big{\\{}w({\scaleobj{0.65}{\Upsilon}}),[w({\scaleobj{0.65}{\Upsilon}})]^{-1}\big{\\}}}\leq\max\left\\{\frac{w(\tau)}{w({\scaleobj{0.65}{\Upsilon}})},\frac{w({\scaleobj{0.65}{\Upsilon}})}{w(\tau)}\right\\}\leq
v_{0}^{d}(\tau-{\scaleobj{0.65}{\Upsilon}}),\text{ for all
}\tau,{\scaleobj{0.65}{\Upsilon}}\in\mathbb{R}^{d}.$

Combine the above with (5.8), such that

$\frac{v(y,\Phi(\xi))}{v(z,\Phi(\eta))}\leq C^{(0)}\cdot
m_{0}((y,\Phi(\xi)),(z,\Phi(\eta)))\cdot v_{0}^{d}(\Phi(\xi)-\Phi(\eta)).\qed$

##  6 Controlling the $\mathcal{B}_{m}$-norm of the oscillation

In this section, we employ the $\Phi$-induced $\delta$-fine phase-space
coverings $\mathcal{V}^{\delta}_{\Phi}$, constructed in the previous section,
to derive conditions concerning the prototype function $\theta$ which ensure
that
$\|{\mathrm{osc}}_{\mathcal{V}^{\delta}_{\Phi},\Gamma}\|_{\mathcal{B}_{m}}<\infty$
with
$\|{\mathrm{osc}}_{\mathcal{V}^{\delta}_{\Phi},\Gamma}\|_{\mathcal{B}_{m}}\rightarrow
0$ as $\delta\rightarrow 0$. We will obtain the following result.

######  Theorem 6.1.

Let $\Phi$ be a $(d+p+1)$-admissible warping function with control weight
$v_{0}$, where $p=0$ if $R_{\Phi}=\sup_{\xi\in
D}\|\mathrm{D}\Phi(\xi)\|=\infty$ and $p\in\mathbb{N}_{0}$ otherwise. Let
furthermore $m:\Lambda\times\Lambda\to\mathbb{R}^{+}$ be a symmetric weight
that satisfies

$m((y,\xi),(z,\eta))\leq(1+|y-z|)^{p}\cdot v_{1}(\Phi(\xi)-\Phi(\eta)),\text{
for all }y,z\in\mathbb{R}^{d}\text{ and }\xi,\eta\in D,\
\tau,{\scaleobj{0.65}{\Upsilon}}\in\mathbb{R}^{d},$ (6.1)

for some continuous and submultiplicative weight
$v_{1}:\mathbb{R}^{d}\to\mathbb{R}^{+}$ satisfying
$v_{1}({\scaleobj{0.65}{\Upsilon}})=v_{1}(-{\scaleobj{0.65}{\Upsilon}})$ for
all ${\scaleobj{0.65}{\Upsilon}}\in\mathbb{R}^{d}$.

Finally, with

$w_{2}:\mathbb{R}^{d}\to\mathbb{R}^{+},{\scaleobj{0.65}{\Upsilon}}\mapsto(1+|{\scaleobj{0.65}{\Upsilon}}|)^{d+1}\cdot
v_{1}({\scaleobj{0.65}{\Upsilon}})\cdot[v_{0}({\scaleobj{0.65}{\Upsilon}})]^{9d/2+3p+3},$

assume that $\theta\in\mathcal{C}^{d+p+1}(\mathbb{R}^{d})$ and

$v_{0}^{n}\cdot\frac{\partial^{(d+p+1)-n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{(d+p+1)-n}}\theta\in\mathbf{L}^{2}_{w_{2}}(\mathbb{R}^{d}),\qquad\text{
for all }i\in\underline{d},\,\,\,0\leq n\leq d+p+1.$

Then, with
$\Gamma:\Lambda\times\Lambda\to\mathbb{C},\bigl{(}(y,\omega),(z,\eta)\bigr{)}\mapsto
e^{-2\pi i\langle y-z,\omega\rangle},$ and
$\mathcal{V}_{\Phi}^{\delta}=(V_{\ell,k}^{\delta})_{\ell,k\in\mathbb{Z}^{d}}$
the $\Phi$-induced $\delta$-fine covering:

$\|{\mathrm{osc}}_{\mathcal{V}^{\delta}_{\Phi},\Gamma}\|_{\mathcal{B}_{m}}<\infty\text{
for all }\delta>0\quad\text{ and
}\quad\|{\mathrm{osc}}_{\mathcal{V}^{\delta}_{\Phi},\Gamma}\|_{\mathcal{B}_{m}}\overset{\delta\rightarrow
0}{\rightarrow}0.$ (6.2)

######  Remark 6.2.

The conditions of Theorem 6.1 are largely the same as those for Theorem 4.4.
The only difference is the appearance of an additional factor $v_{0}^{n}$, for
certain $n\in\mathbb{N}_{0}$, in the conditions on $\theta$. Since $v_{0}\geq
v_{0}(0)$, the conditions of Theorem 6.1 imply those of Theorem 4.4.

To prove Theorem 6.1, we study the second component of the oscillation, i.e.,
$g_{\lambda}-\Gamma(\lambda,\rho)g_{\rho},$ for
$\rho\in\mathcal{V}^{\delta}_{\lambda}$. If we can bound certain weighted
$\mathbf{L}^{2}$-norms of this difference and its derivatives uniformly in
$\lambda\in\Lambda$ and $\rho\in\mathcal{V}^{\delta}_{\lambda}$, then we can
show that
${\mathrm{osc}}_{\mathcal{V}^{\delta}_{\Phi},\Gamma}\in\mathcal{B}_{m}$ by a
slight variation on Theorem 4.4. In fact, the estimates we obtain converge to
$0$ for $\delta\to 0$, such that we naturally obtain the second part of
Equation 6.2 as well.

###  6.1 Local behavior of the oscillating component

In order to rely on the machinery we already developed in Section 4, it will
be useful to rewrite $g_{\lambda}-\Gamma(\lambda,\rho)g_{\rho}$ as the warping
of a function $\tilde{\theta}_{\lambda,\rho}\in\mathbf{L}^{2}_{\sqrt{w_{0}}}$
(dependent on $\lambda,\rho\in\Lambda$) derived from the prototype $\theta$.

######  Proposition 6.3.

For $D\subset\mathbb{R}^{d}$ open, let $\Lambda=\mathbb{R}^{d}\times D$, and
define the phase function $\Gamma$ via

$\Gamma:\Lambda\times\Lambda\to\mathbb{C},((y,\omega),(z,\eta))\mapsto
e^{-2\pi i\langle y-z,\omega\rangle}.$ (6.3)

Let $\Phi:D\to\mathbb{R}^{d}$ be a warping function, assume
$\theta\in\mathbf{L}^{2}_{\sqrt{w_{0}}}(\mathbb{R}^{d})$ and denote
$(g_{y,\omega})_{(y,\omega)\in\Lambda}=\mathcal{G}(\theta,\Phi)$ as usual.
Then the identity

$\widehat{g_{y,\omega}}-\Gamma((y,\omega),(z,\eta))\widehat{g_{z,\eta}}=e^{-2\pi
i\langle
y,\cdot\rangle}\cdot\left(w(\Phi(\omega))^{-1/2}\cdot\left(\mathbf{T}_{\Phi(\omega)}\tilde{\theta}_{(y,\omega),(z,\eta)}\right)\circ\Phi\right),$
(6.4)

holds for all $(y,\omega),(z,\eta)\in\Lambda$, with

$\tilde{\theta}_{(y,\omega),(z,\eta)}:=\left(\theta-\sqrt{\frac{w(\Phi(\omega))}{w(\Phi(\eta))}}\cdot\mathbf{\operatorname{E}}_{\Phi(\omega),A^{T}(\Phi(\omega))\langle
y-z\rangle}\left(\mathbf{T}_{\Phi(\eta)-\Phi(\omega)}\theta\right)\right)\in\mathbf{L}^{2}_{\sqrt{w_{0}}}.$
(6.5)

The operator $\mathbf{\operatorname{E}}_{\tau,\varepsilon}$ in Equation 6.5 is
a multiplication operator defined by

$\mathbf{\operatorname{E}}_{\tau,\varepsilon}f:=e^{2\pi i\langle
A^{-T}(\tau)\langle\varepsilon\rangle,\Phi^{-1}(\cdot+\tau)-\Phi^{-1}(\tau)\rangle}\cdot
f\quad\text{ for all }\quad f:\mathbb{R}^{d}\to\mathbb{C}\text{ and
}\tau,\varepsilon\in\mathbb{R}^{d}.$ (6.6)

###### Proof.

To see that
$\tilde{\theta}_{(y,\omega),(z,\eta)}\in\mathbf{L}^{2}_{\sqrt{w_{0}}}$, note
that $\mathbf{T}_{\Phi(\eta)-\Phi(\omega)}$ and
$\mathbf{\operatorname{E}}_{\tau,\varepsilon}$ are bounded operators on
$\mathbf{L}^{2}_{\sqrt{w_{0}}}$ and that
$\sqrt{w(\Phi(\omega))/w(\Phi(\eta))}$ is finite for all $\omega,\eta\in D$.
Here, boundedness of $\mathbf{T}_{x}$ on $\mathbf{L}^{2}_{\sqrt{w_{0}}}$ is a
consequence of (3.5), since $w_{0}$ is submultiplicative. To prove (6.4), note
that, by definition,

$\begin{split}\left(\widehat{g_{y,\omega}}-\Gamma((y,\omega),(z,\eta))\widehat{g_{z,\eta}}\right)(\xi)&=e^{-2\pi
i\langle y,\xi\rangle}g_{\omega}(\xi)-e^{-2\pi i\langle
y-z,\omega\rangle}e^{-2\pi i\langle z,\xi\rangle}g_{\eta}(\xi)\\\ &=e^{-2\pi
i\langle y,\xi\rangle}\left(g_{\omega}-e^{-2\pi i\langle
y-z,\omega-\cdot\rangle}g_{\eta}\right)(\xi),\end{split}$

and furthermore

$\displaystyle g_{\omega}-e^{-2\pi i\langle y-z,\omega-\cdot\rangle}g_{\eta}$
$\displaystyle=w(\Phi(\omega))^{-1/2}\left(\mathbf{T}_{\Phi(\omega)}\theta\right)\circ\Phi-e^{-2\pi
i\langle
y-z,\omega-\cdot\rangle}w(\Phi(\eta))^{-1/2}\left(\mathbf{T}_{\Phi(\eta)}\theta\right)\circ\Phi$
$\displaystyle=w(\Phi(\omega))^{-1/2}\left(\mathbf{T}_{\Phi(\omega)}\theta-\sqrt{\frac{w(\Phi(\omega))}{w(\Phi(\eta))}}\cdot
e^{2\pi i\langle
y-z,\Phi^{-1}(\cdot)-\omega\rangle}\mathbf{T}_{\Phi(\eta)}\theta\right)\circ\Phi$
$\displaystyle=w(\Phi(\omega))^{-1/2}\left(\mathbf{T}_{\Phi(\omega)}\left(\theta-\sqrt{\frac{w(\Phi(\omega))}{w(\Phi(\eta))}}\cdot
e^{2\pi i\langle
y-z,\Phi^{-1}(\cdot+\Phi(\omega))-\omega\rangle}\mathbf{T}_{\Phi(\eta)-\Phi(\omega)}\theta\right)\right)\circ\Phi$
$\displaystyle=w(\Phi(\omega))^{-1/2}\left(\mathbf{T}_{\Phi(\omega)}\tilde{\theta}_{(y,\omega),(z,\eta)}\right)\circ\Phi.\qed$

Now that we can express $g_{\lambda}-\Gamma(\lambda,\rho)g_{\rho}$ through
$\tilde{\theta}(\lambda,\rho)$, we aim to derive conditions on $\theta$, such
that Lemma 4.8 can be applied with
$\theta_{1}=\theta,\theta_{2}=\tilde{\theta}(\lambda,\rho)$. In particular, we
investigate the (uniform) continuity of the map
$(\tau,\varepsilon)\mapsto\mathbf{\operatorname{E}}_{\tau,\varepsilon}$, in
the next lemma. Here, $\mathbf{\operatorname{E}}_{\tau,\varepsilon}$ is
considered as an operator on $\mathbf{L}^{q}_{\tilde{w}}(\mathbb{R}^{d})$, for
suitable weights $\tilde{w}$.

######  Lemma 6.4.

Let $q\in[1,\infty)$ and let $\tilde{w}:\mathbb{R}^{d}\to\mathbb{R}^{+}$ be a
continuous weight function. Furthermore, assume that $\Phi$ is a
$k$-admissible warping function with control weight $v_{0}$.

The operator
$\mathbf{\operatorname{E}}_{\tau,\varepsilon}~{}:~{}\mathbf{L}^{q}_{\tilde{w}}(\mathbb{R}^{d})\to\mathbf{L}^{q}_{\tilde{w}}(\mathbb{R}^{d})$,
$\tau,\varepsilon\in\mathbb{R}^{d}$, given by (6.6), is well-defined and has
the following properties:

  * (1)

If $\vartheta\in\mathbf{L}^{q}_{\tilde{w}}(\mathbb{R}^{d})$ with
$\mathop{\operatorname{supp}}(\vartheta)\subset\overline{B_{\delta}}(0)$ for
some $\delta>0$, then

$\|\vartheta-\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta\|_{\mathbf{L}^{q}_{\tilde{w}}}\leq\sqrt{2[1-\cos\left(\pi\cdot\min\\{1,2|\varepsilon|\delta
v_{0}(\delta)\\}\right)]}\cdot\|\vartheta\|_{\mathbf{L}^{q}_{\tilde{w}}}.$
(6.7)

  * (2)

If $\vartheta\in\mathbf{L}^{q}_{\tilde{w}}(\mathbb{R}^{d})$, then
$\sup\limits_{\tau\in\mathbb{R}^{d}}\|\vartheta-\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta\|_{\mathbf{L}^{q}_{\tilde{w}}}\overset{\varepsilon\rightarrow
0}{\rightarrow}0.$

  * (3)

If $\vartheta\in\mathcal{C}^{m}(\mathbb{R}^{d})$ for some $0\leq m\leq k+1$,
and if $j\in\underline{d}$ with

$v_{0}^{n}\cdot\frac{\partial^{m-n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m-n}}\vartheta\in\mathbf{L}^{q}_{\tilde{w}}(\mathbb{R}^{d})\text{
for all }0\leq n\leq m,$ (6.8)

then
$\frac{\partial^{\ell}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\ell}}\vartheta\in\mathbf{L}_{\tilde{w}}^{p}(\mathbb{R}^{d})$
for $0\leq\ell\leq m$,
$\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}(\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta)\in\mathbf{L}^{q}_{\tilde{w}}(\mathbb{R}^{d})$
for all $\tau,\varepsilon\in\mathbb{R}^{d}$, and

$\sup\limits_{\tau\in\mathbb{R}^{d}}\left\|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}(\vartheta-\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta)\right\|_{\mathbf{L}^{q}_{\tilde{w}}}\overset{\varepsilon\rightarrow
0}{\rightarrow}0.$

Furthermore, for each $\varepsilon_{0}>0$, there is a constant
$C_{m,\varepsilon_{0}}>0$ satisfying for all
$|\varepsilon|\leq\varepsilon_{0}$ that

$\sup_{\tau\in\mathbb{R}^{d}}\left\|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}(\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta)\right\|_{\mathbf{L}^{q}_{\tilde{w}}}\leq\left\|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\vartheta\right\|_{\mathbf{L}^{q}_{\tilde{w}}}+C_{m,\varepsilon_{0}}\cdot|\varepsilon|\cdot\sum_{m=1}^{m}\left\|v_{0}^{n}\cdot\frac{\partial^{m-n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m-n}}\vartheta\right\|_{\mathbf{L}^{q}_{\tilde{w}}}<\infty.$
(6.9)

###### Proof.

Assumption (6.8) implies
$\frac{\partial^{\ell}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\ell}}\vartheta\in\mathbf{L}^{q}_{\tilde{w}}(\mathbb{R}^{d})$
for all $0\leq\ell\leq n$, since $v_{0}$ is radially increasing. Now, to prove
(1), note for arbitrary ${\scaleobj{0.65}{\Upsilon}}\in\mathbb{R}^{d}$ that

$|\vartheta({\scaleobj{0.65}{\Upsilon}})-(\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta)({\scaleobj{0.65}{\Upsilon}})|=\left|1-e^{2\pi
i\langle\varepsilon,A^{-1}(\tau)(\Phi^{-1}({\scaleobj{0.65}{\Upsilon}}+\tau)-\Phi^{-1}(\tau))\rangle}\right|\cdot|\vartheta({\scaleobj{0.65}{\Upsilon}})|,$

where $\mathop{\operatorname{supp}}\vartheta\subset\overline{B_{\delta}}(0)$,
such that it suffices to estimate this expression for
$|{\scaleobj{0.65}{\Upsilon}}|\leq\delta$. We begin by expressing the
difference $\Phi^{-1}({\scaleobj{0.65}{\Upsilon}}+\tau)-\Phi^{-1}(\tau)$
through the Jacobian $A=\mathrm{D}\Phi^{-1}$ of $\Phi^{-1}$ by using the
directional derivative. This furnishes the following estimate:

$\begin{split}|A^{-1}(\tau)(\Phi^{-1}({\scaleobj{0.65}{\Upsilon}}+\tau)-\Phi^{-1}(\tau))|&=\left|\int_{0}^{1}A^{-1}(\tau)A(\tau+r{\scaleobj{0.65}{\Upsilon}})\langle{\scaleobj{0.65}{\Upsilon}}\rangle~{}dr\right|\\\
&\leq|{\scaleobj{0.65}{\Upsilon}}|\cdot\max\limits_{r\in[0,1]}\|A^{-1}(\tau)A(\tau+r{\scaleobj{0.65}{\Upsilon}})\|\overset{{\scaleobj{0.65}{\Upsilon}}\in\overline{B_{\delta}}(0)}{\leq}\delta\cdot
v_{0}(\delta),\end{split}$

where we used (4.5) in the last step. Therefore,
$|\langle\varepsilon,A^{-1}(\tau)(\Phi^{-1}({\scaleobj{0.65}{\Upsilon}}+\tau)-\Phi^{-1}(\tau))\rangle|\leq|\varepsilon|\cdot\delta\cdot
v_{0}(\delta).$

Next, a simple calculation shows that $|1-e^{\pi ir}|=\sqrt{2[1-\cos(\pi
r)]}$, which is an even function that is increasing on $[0,1]$ and converges
to $0$ for $r\rightarrow 0$. Thus, we obtain

$|(\vartheta-\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta)({\scaleobj{0.65}{\Upsilon}})|\leq\sqrt{2\left[1-\cos(\pi\cdot
2|\varepsilon|\delta
v_{0}(\delta))\right]}\cdot|\vartheta({\scaleobj{0.65}{\Upsilon}})|$

for all $0\leq|\varepsilon|\leq\frac{1}{2\delta v_{0}(\delta)}$. For
$|\varepsilon|>(2\delta v_{0}(\delta))^{-1}$ apply the trivial estimate
$|1-e^{\pi ir}|\leq 2=\sqrt{2[1-\cos(\pi)]}$ instead. This easily yields
(6.7), in fact for any solid Banach space $X$, and not only for
$\mathbf{L}^{q}_{\tilde{w}}$.

To prove (2), note that for a given
$\vartheta\in\mathbf{L}^{q}_{\tilde{w}}(\mathbb{R}^{d})$, we have
$\|\vartheta-\vartheta_{n}\|_{\mathbf{L}^{q}_{\tilde{w}}}\to 0$ as
$n\to\infty$ for the sequence
$\vartheta_{n}=\vartheta\cdot{\mathds{1}}_{\overline{B_{n}}(0)}$, by the
dominated convergence theorem. Furthermore, for every $n\in\mathbb{N}$,

$\begin{split}\sup\limits_{\tau\in\mathbb{R}^{d}}\|\vartheta-\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta\|_{\mathbf{L}^{q}_{\tilde{w}}}&\leq\|\vartheta-\vartheta_{n}\|_{\mathbf{L}^{q}_{\tilde{w}}}+\sup\limits_{\tau\in\mathbb{R}^{d}}\left(\|\vartheta_{n}-\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta_{n}\|_{\mathbf{L}^{q}_{\tilde{w}}}+\|\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta_{n}-\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta\|_{\mathbf{L}^{q}_{\tilde{w}}}\right)\\\
&=2\|\vartheta-\vartheta_{n}\|_{\mathbf{L}^{q}_{\tilde{w}}}+\sup\limits_{\tau\in\mathbb{R}^{d}}\|\vartheta_{n}-\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta_{n}\|_{\mathbf{L}^{q}_{\tilde{w}}}.\end{split}$

For any $n\in\mathbb{N}$ and any $\varepsilon_{0}>0$, we can choose
$\varepsilon_{n}>0$ such that

$3\|\vartheta\|_{\mathbf{L}^{q}_{\tilde{w}}}\cdot\sqrt{2[1-\cos\left(\pi\cdot\min\\{1,2|\varepsilon_{n}|nv_{0}(n)\right)\\}}<\varepsilon_{0}.$

By (6.7), we obtain
$\|\vartheta_{n}-\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta_{n}\|_{\mathbf{L}^{q}_{\tilde{w}}}<\varepsilon_{0}/3$,
for all $\varepsilon\in\overline{B_{\varepsilon_{n}}}(0)$,
$\tau\in\mathbb{R}^{d}$. If additionally, $n\in\mathbb{N}$ is such that
$\|\vartheta-\vartheta_{n}\|_{\mathbf{L}^{q}_{\tilde{w}}}<\varepsilon_{0}/3$,
then
$\|\vartheta-\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta\|_{\mathbf{L}^{q}_{\tilde{w}}}<\varepsilon_{0}$.
Since $\varepsilon_{0}>0$ was arbitrary, we obtain

$\forall\ \varepsilon_{0}>0\ \exists\ n\in\mathbb{N}\text{ and
}\varepsilon_{n}>0,\text{ such that
}\varepsilon\in\overline{B_{\varepsilon_{n}}}(0)\text{ implies
}\sup_{\tau\in\mathbb{R}^{d}}\|\vartheta-\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta\|_{\mathbf{L}^{q}_{\tilde{w}}}<\varepsilon_{0}.$

To prove (3), we first note that for $m=0$, all claims in this part are easy
consequences of the definitions and of item (2). Therefore, we can assume
$m\in\underline{k+1}$. Apply Leibniz’s rule to obtain

$\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}(\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta)({\scaleobj{0.65}{\Upsilon}})=\sum_{n=0}^{m}\left(\binom{m}{n}\frac{\partial^{n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{n}}\left(e^{2\pi
i\langle
A^{-T}(\tau)\langle\varepsilon\rangle,\Phi^{-1}(\cdot+\tau)-\Phi^{-1}(\tau)\rangle}\right)({\scaleobj{0.65}{\Upsilon}})\cdot\frac{\partial^{m-n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m-n}}\vartheta({\scaleobj{0.65}{\Upsilon}})\right).$
(6.10)

Moreover, Faa Di Bruno’s formula [28, Corollary 2.10]—a form of the chain rule
for higher derivatives—yields for $n\in\underline{m}$ that

$\frac{\partial^{n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{n}}\left(e^{2\pi
i\langle
A^{-T}(\tau)\langle\varepsilon\rangle,\Phi^{-1}(\cdot+\tau)-\Phi^{-1}(\tau)\rangle}\right)({\scaleobj{0.65}{\Upsilon}})=e^{2\pi
i\langle
A^{-T}(\tau)\langle\varepsilon\rangle,\Phi^{-1}({\scaleobj{0.65}{\Upsilon}}+\tau)-\Phi^{-1}(\tau)\rangle}\cdot
P_{n,\tau,\varepsilon}({\scaleobj{0.65}{\Upsilon}})=\mathbf{\operatorname{E}}_{\tau,\varepsilon}P_{n,\tau,\varepsilon}({\scaleobj{0.65}{\Upsilon}}),$

where

$\begin{split}P_{n,\tau,\varepsilon}({\scaleobj{0.65}{\Upsilon}})&=\sum_{\ell=1}^{n}\left((2\pi
i)^{\ell}\cdot\sum_{\sigma\in(\underline{n-\ell+1})^{\ell}}\left(C_{\sigma}\cdot\prod_{i=1}^{\ell}\frac{\partial^{\sigma_{i}}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\sigma_{i}}}\left\langle
A^{-T}(\tau)\langle\varepsilon\rangle,\Phi^{-1}({\scaleobj{0.65}{\Upsilon}}+\tau)-\Phi^{-1}(\tau)\right\rangle\right)\right)\\\
&=\sum_{\ell=1}^{n}\left((2\pi
i)^{\ell}\cdot\sum_{\sigma\in(\underline{n-\ell+1})^{\ell}}\left(C_{\sigma}\cdot\prod_{i=1}^{\ell}\frac{\partial^{\sigma_{i}}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\sigma_{i}}}\left\langle\varepsilon,A^{-1}(\tau)\langle\Phi^{-1}({\scaleobj{0.65}{\Upsilon}}+\tau)\rangle\right\rangle\right)\right),\end{split}$

for suitable constants $C_{\sigma}\geq 0$. For the second equality, note that
$\sigma_{i}\geq 1$ for all $i$, so that the term $\langle
A^{-T}(\tau)\langle\varepsilon\rangle,\Phi^{-1}(\tau)\rangle$—which is
constant with respect to ${\scaleobj{0.65}{\Upsilon}}$—can be ignored. In
fact, the main statement of Faa Di Bruno’s formula is exactly which
$C_{\sigma}$ are nonzero and what value they attain, see also Lemma 8.6, but
these details are not required here. Similar to (4.25), we have that

$\begin{split}\frac{\partial^{\sigma_{i}}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\sigma_{i}}}\left\langle\varepsilon,A^{-1}(\tau)\langle\Phi^{-1}({\scaleobj{0.65}{\Upsilon}}+\tau)\rangle\right\rangle&=\frac{\partial^{\sigma_{i}-1}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\sigma_{i}-1}}\left\langle\varepsilon,A^{-1}(\tau)A({\scaleobj{0.65}{\Upsilon}}+\tau)\langle
e_{i}\rangle\right\rangle\\\
&=\frac{\partial^{\sigma_{i}-1}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\sigma_{i}-1}}\left(\left[A^{-1}(\tau)A({\scaleobj{0.65}{\Upsilon}}+\tau)\right]^{T}\varepsilon\right)_{i}=\left(\frac{\partial^{\sigma_{i}-1}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\sigma_{i}-1}}\phi_{\tau}({\scaleobj{0.65}{\Upsilon}})\langle\varepsilon\rangle\right)_{i}\,\,,\end{split}$
(6.11)

where $\phi_{\tau}=\left[A^{-1}(\tau)A(\cdot+\tau)\right]^{T}$ is as in (4.4).
By (4.5), we can estimate

$\left|\left(\frac{\partial^{\sigma_{i}-1}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\sigma_{i}-1}}\phi_{\tau}({\scaleobj{0.65}{\Upsilon}})\langle\varepsilon\rangle\right)_{i}\right|\leq\left\|\frac{\partial^{\sigma_{i}-1}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{\sigma_{i}-1}}\phi_{\tau}({\scaleobj{0.65}{\Upsilon}})\right\|\cdot|\varepsilon|\leq
v_{0}({\scaleobj{0.65}{\Upsilon}})\cdot|\varepsilon|\quad\text{ and inserting
this into \eqref{eq:PMTEest},}$
$|P_{n,\tau,\varepsilon}({\scaleobj{0.65}{\Upsilon}})|\leq\sum_{\ell=1}^{n}\left((2\pi\cdot
v_{0}({\scaleobj{0.65}{\Upsilon}})\cdot|\varepsilon|)^{\ell}\cdot\sum_{\sigma\in(\underline{n-\ell+1})^{\ell}}C_{\sigma}\right)\leq\tilde{C}\cdot|\varepsilon|\cdot\sum_{\ell=1}^{n}\left(v_{0}({\scaleobj{0.65}{\Upsilon}})^{\ell}\cdot|\varepsilon|^{\ell-1}\right),$

for a suitably large $\tilde{C}=\tilde{C}(n)>0$. Since we only consider
$n\in\underline{m}$, we can in fact choose the same constant $\tilde{C}$ for
all values of $n$. Moreover, $v_{0}^{\ell}\leq v_{0}^{n}$ for all $\ell\leq
n$.

By assembling all the pieces and by separating the term $n=0$ in (6.10), we
thus get

$\begin{split}\left|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}(\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta)({\scaleobj{0.65}{\Upsilon}})-\mathbf{\operatorname{E}}_{\tau,\varepsilon}\left(\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\vartheta\right)({\scaleobj{0.65}{\Upsilon}})\right|&\leq\sum_{n=1}^{m}\binom{m}{n}\left|(\mathbf{\operatorname{E}}_{\tau,\varepsilon}P_{n,\tau,\varepsilon})({\scaleobj{0.65}{\Upsilon}})\cdot\left(\frac{\partial^{m-n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m-n}}\vartheta\right)({\scaleobj{0.65}{\Upsilon}})\right|\\\
&=\sum_{n=1}^{m}\binom{m}{n}\left|P_{n,\tau,\varepsilon}({\scaleobj{0.65}{\Upsilon}})\cdot\left(\frac{\partial^{m-n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m-n}}\vartheta\right)({\scaleobj{0.65}{\Upsilon}})\right|\\\
&\leq|\varepsilon|\cdot\sum_{n=1}^{m}\left(\left|\frac{\partial^{m-n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m-n}}\vartheta({\scaleobj{0.65}{\Upsilon}})\right|\cdot\sum_{\ell=1}^{n}\left(\tilde{C}\binom{m}{n}v_{0}({\scaleobj{0.65}{\Upsilon}})^{\ell}\cdot|\varepsilon|^{\ell-1}\right)\right)\\\
&\leq|\varepsilon|\cdot\sum_{n=1}^{m}\left(\left|\left(v_{0}^{n}\cdot\frac{\partial^{m-n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m-n}}\vartheta({\scaleobj{0.65}{\Upsilon}})\right)\right|\cdot\sum_{\ell=1}^{n}\left(\tilde{C}\binom{m}{n}\cdot|\varepsilon|^{\ell-1}\right)\right).\end{split}$

Let $0\leq
C_{m,\varepsilon}:=\max_{n\in\underline{m}}\left(\sum_{\ell=1}^{n}\binom{m}{n}\cdot\tilde{C}|\varepsilon|^{\ell-1}\right)<\infty$
to obtain the estimate

$\left|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}(\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta)({\scaleobj{0.65}{\Upsilon}})-\mathbf{\operatorname{E}}_{\tau,\varepsilon}\left(\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\vartheta\right)({\scaleobj{0.65}{\Upsilon}})\right|\leq
C_{m,\varepsilon}\cdot|\varepsilon|\cdot\sum_{n=1}^{m}\left|v_{0}^{n}\cdot\frac{\partial^{m-n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m-n}}\vartheta({\scaleobj{0.65}{\Upsilon}})\right|.$

Since $\mathbf{L}^{q}_{\tilde{w}}$ is solid, we conclude

$\left\|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}(\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta)-\mathbf{\operatorname{E}}_{\tau,\varepsilon}\left(\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\vartheta\right)\right\|_{\mathbf{L}^{q}_{\tilde{w}}}\leq
C_{m,\varepsilon}\cdot|\varepsilon|\cdot\sum_{n=1}^{m}\left\|v_{0}^{n}\cdot\frac{\partial^{m-n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m-n}}\vartheta\right\|_{\mathbf{L}^{q}_{\tilde{w}}}<\infty.$
(6.12)

Finally, with $C_{m,\varepsilon}\leq C_{m,\varepsilon_{0}}$ for
$|\varepsilon|\leq\varepsilon_{0}$, we obtain

$\displaystyle\sup_{\tau\in\mathbb{R}^{d}}\left\|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}(\vartheta-\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta)\right\|_{\mathbf{L}^{q}_{\tilde{w}}}$
$\displaystyle\leq\sup_{\tau\in\mathbb{R}^{d}}\left(\left\|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\vartheta-\mathbf{\operatorname{E}}_{\tau,\varepsilon}\left(\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\vartheta\right)\right\|_{\mathbf{L}^{q}_{\tilde{w}}}+\left\|\mathbf{\operatorname{E}}_{\tau,\varepsilon}\left(\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\vartheta\right)-\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}(\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta)\right\|_{\mathbf{L}^{q}_{\tilde{w}}}\right)\xrightarrow[|\varepsilon|\rightarrow
0]{}0,$

as a consequence of part (2), and (6.12).

To prove (6.9) (and thus also
$\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}(\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta)\in\mathbf{L}^{q}_{\tilde{w}}$),
observe
$\|\mathbf{\operatorname{E}}_{\tau,\varepsilon}f\|_{\mathbf{L}^{q}_{\tilde{w}}}=\|f\|_{\mathbf{L}^{q}_{\tilde{w}}}$
for all $f\in\mathbf{L}^{q}_{\tilde{w}}(\mathbb{R}^{d})$. By Equation (6.12)
the triangle inequality for norms yields

$\displaystyle\sup_{\tau\in\mathbb{R}^{d}}\left\|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}(\mathbf{\operatorname{E}}_{\tau,\varepsilon}\vartheta)\right\|_{\mathbf{L}^{q}_{\tilde{w}}}$
$\displaystyle\leq\sup_{\tau\in\mathbb{R}^{d}}\left\|\mathbf{\operatorname{E}}_{\tau,\varepsilon}\left(\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\vartheta\right)\right\|_{\mathbf{L}^{q}_{\tilde{w}}}+C_{m,\varepsilon}\cdot|\varepsilon|\cdot\sum_{n=1}^{m}\left\|v_{0}^{n}\cdot\frac{\partial^{m-n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m-n}}\vartheta\right\|_{\mathbf{L}^{q}_{\tilde{w}}}$
$\displaystyle=\left\|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\vartheta\right\|_{\mathbf{L}^{q}_{\tilde{w}}}+C_{m,\varepsilon}\cdot|\varepsilon|\cdot\sum_{n=1}^{m}\left\|v_{0}^{n}\cdot\frac{\partial^{m-n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m-n}}\vartheta\right\|_{\mathbf{L}^{q}_{\tilde{w}}}.$

This proves (6.9), since $C_{m,\varepsilon}\leq C_{m,\varepsilon_{0}}$. ∎

We now show that $\tilde{\theta}_{(y,\omega),(z,\eta)}$ uniformly converges to
$0$ as $\delta\\!\to\\!\\!0$, for $(y,\omega)\\!\in\\!\Lambda$ and
$(z,\eta)\\!\in\\!(y+\mathbf{P}^{\delta}_{\omega})\times\mathbf{Q}^{\delta}_{\omega}$.
Recall that
$(y+\mathbf{P}^{\delta}_{\omega})\times\mathbf{Q}^{\delta}_{\omega}$ was
introduced in Lemma 5.3 as a simple superset to
$\mathcal{V}^{\delta}_{(y,\omega)}=\bigcup_{V_{\ell,k}\ni(y,\omega)}V_{\ell,k}$,
appearing in the oscillation. The considered notion of convergence is in terms
of the $\mathbf{L}^{q}_{\tilde{w}}$-norm of certain derivatives of
$\tilde{\theta}_{(y,\omega),(z,\eta)}$. With Lemma 6.4, obtaining the desired
estimates for $\tilde{\theta}((y,\omega),(z,\eta))$ amounts to little more
than an application of the triangle inequality and a somewhat elaborate
three-$\varepsilon$-argument.

######  Lemma 6.5.

Let $q\in[1,\infty)$ and let $\tilde{w}:\mathbb{R}^{d}\to\mathbb{R}^{+}$ be a
continuous, submultiplicative weight function. Furthermore, assume that $\Phi$
is a $k$-admissible warping function with control weight $v_{0}$. If

$\theta\in\mathcal{C}^{m}(\mathbb{R}^{d})\text{ for some }0\leq m\leq
k+1,\quad\text{ and }\quad
v_{0}^{n}\cdot\frac{\partial^{m-n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m-n}}\theta\in\mathbf{L}^{q}_{\tilde{w}}(\mathbb{R}^{d})\text{
for all }0\leq n\leq m,\ j\in\underline{d},$

then

$\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\tilde{\theta}_{(y,\omega),(z,\eta)}=\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\left(\theta-\sqrt{\frac{w(\Phi(\omega))}{w(\Phi(\eta))}}\mathbf{\operatorname{E}}_{\Phi(\omega),A^{T}(\Phi(\omega))\langle
y-z\rangle}\left(\mathbf{T}_{\Phi(\eta)-\Phi(\omega)}\theta\right)\right)\in\mathbf{L}^{q}_{\tilde{w}}(\mathbb{R}^{d})$
(6.13)

for all $(y,\omega),(z,\eta)\in\Lambda$, and $j\in\underline{d}$. Furthermore,
with

$F_{j,m}(\delta;\theta,p,\tilde{w}):=\sup_{(y,\omega)\in\Lambda}\,\,\sup_{z\in(y+\mathbf{P}_{\omega}^{\delta}),\eta\in\mathbf{Q}_{\omega}^{\delta}}\left\|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\tilde{\theta}_{(y,\omega),(z,\eta)}\right\|_{\mathbf{L}^{q}_{\tilde{w}}},$
(6.14)

where $\mathbf{Q}_{\omega}^{\delta}$ and $\mathbf{P}_{\omega}^{\delta}$ are as
in Lemma 5.3, we have

$F_{j,m}(\delta;\theta,p,\tilde{w})<\infty\text{ for all }\delta>0,\quad\text{
and }\quad F_{j,m}(\delta;\theta,p,\tilde{w})\xrightarrow[\delta\to 0]{}0.$
(6.15)

###### Proof.

Since $v_{0}$ and $\tilde{w}$ are submultiplicative, so is
$v_{0}^{n}\tilde{w}$, and
$\mathbf{L}^{q}_{v_{0}^{n}\tilde{w}}(\mathbb{R}^{d})$ is translation-
invariant, see (3.5). Hence, since
$\frac{\partial^{m-n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m-n}}\theta\in\mathbf{L}^{q}_{v_{0}^{n}\tilde{w}}(\mathbb{R}^{d}),$
$0\leq n\leq m$ and $i\in\underline{d}$, the same holds arbitrary translates.
Thus, Lemma 6.4(3) shows
$\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\theta,\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\mathbf{\operatorname{E}}_{\tau,\varepsilon}(\mathbf{T}_{\tau_{0}}\theta)\in\mathbf{L}^{q}_{\tilde{w}}(\mathbb{R}^{d})$
for all $\tau_{0},\tau,\varepsilon\in\mathbb{R}^{d}$. This establishes (6.13),
since $\frac{w(\Phi(\omega))}{w(\Phi(\eta))}<\infty$.

Fix $\delta>0$ and $(y,\omega)\in\Lambda$ and
$(z,\eta)\in(y+\mathbf{P}_{\omega}^{\delta})\times\mathbf{Q}_{\omega}^{\delta}$.
For brevity, set $\tau:=\Phi(\omega)-\Phi(\eta)$ and
$\varepsilon:=A^{T}(\Phi(\omega))\langle y-z\rangle$, noting that $\tau\in
B_{2\delta}(0)$ and $\varepsilon\in
A^{T}(\Phi(\omega))\langle\mathbf{P}_{\omega}^{\delta}\rangle=B_{2\delta
v_{0}(\delta)}(0)=:B_{\varepsilon_{\delta}}(0).$ In particular,
$\varepsilon_{\delta}\leq\varepsilon_{\delta_{0}}$, for all
$\delta\leq\delta_{0}$, and $\varepsilon_{\delta}\rightarrow 0$ as
$\delta\rightarrow 0$. Recall the definition of
$\tilde{\theta}_{(y,\omega),(z,\eta)}$ ((6.5)), and apply the triangle
inequality twice to obtain the estimate

$\begin{split}\left\|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\tilde{\theta}_{(y,\omega),(z,\eta)}\right\|_{\mathbf{L}^{q}_{\tilde{w}}}&\leq\\!\left|1\\!-\\!\sqrt{\frac{w(\Phi(\omega))}{w(\Phi(\eta))}}\right|\cdot\left\|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\theta\right\|_{\mathbf{L}^{q}_{\tilde{w}}}+\sqrt{\frac{w(\Phi(\omega))}{w(\Phi(\eta))}}\cdot\\!\left\|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\\!\left(\theta\\!-\\!\mathbf{\operatorname{E}}_{\Phi(\omega),\varepsilon}\theta\right)\right\|_{\mathbf{L}^{q}_{\tilde{w}}}\\\
&\quad+\sqrt{\frac{w(\Phi(\omega))}{w(\Phi(\eta))}}\cdot\\!\left\|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\mathbf{\operatorname{E}}_{\Phi(\omega),\varepsilon}(\theta\\!-\\!\mathbf{T}_{\Phi(\eta)-\Phi(\omega)}\theta)\right\|_{\mathbf{L}^{q}_{\tilde{w}}}\\!.\end{split}$
(6.16)

Next, Lemma 6.4(3) yields

$\displaystyle E_{\delta}$
$\displaystyle:=\sup_{|\varepsilon|\leq\varepsilon_{\delta}}\,\,\sup_{\omega\in
D}\left\|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\\!\left(\theta-\mathbf{\operatorname{E}}_{\Phi(\omega),\varepsilon}\theta\right)\right\|_{\mathbf{L}^{q}_{\tilde{w}}}\leq\infty,\
\text{for all}\ \delta>0,\text{ with }E_{\delta}\rightarrow 0\ \text{ as
}\delta\rightarrow 0,\text{ and}$ (6.17) $\displaystyle F_{\delta}$
$\displaystyle:=\sup_{|\varepsilon|\leq\varepsilon_{\delta}}\,\,\sup_{\omega\in
D}\left\|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\mathbf{\operatorname{E}}_{\Phi(\omega),\varepsilon}(\theta\\!-\\!\mathbf{T}_{-\tau}\theta)\right\|_{\mathbf{L}^{q}_{\tilde{w}}}$
$\displaystyle\leq\left\|\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\theta-\mathbf{T}_{-\tau}\left(\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\theta\right)\right\|_{\mathbf{L}^{q}_{\tilde{w}}}+C_{m,\varepsilon_{\delta_{0}}}\cdot\varepsilon_{\delta}\cdot\sum_{n=1}^{m}\left\|v_{0}^{n}\cdot\frac{\partial^{m-n}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m-n}}(\theta\\!-\\!\mathbf{T}_{-\tau}\theta)\right\|_{\mathbf{L}^{q}_{\tilde{w}}}.$
(6.18)

Note that the first term of the right-hand side of (6.18) converges to $0$ for
$\delta\rightarrow 0$, since $|\tau|\leq 2\delta$ and translation is
continuous in $\mathbf{L}^{q}_{\tilde{w}}$, since $\tilde{w}$ is continuous
and submultiplicative. Furthermore, the sum over $n$ in the right-hand side of
(6.18) is finite, since $\mathbf{L}^{q}_{\tilde{w}}$ is translation-invariant
and hence, all summands are finite by assumption. Therefore, $F_{\delta}$
vanishes for $\delta\rightarrow 0$. In fact, since
$|\varepsilon|\leq|\varepsilon_{\delta}|$ and $w$ is $v_{0}^{d}$-moderate with
radially increasing $v_{0}$ (cf. Lemma 4.9),
$\frac{w(\Phi(\omega))}{w(\Phi(\eta))}\leq v_{0}^{d}(\varepsilon_{\delta})$,
which settles the desired convergence of the second and third term in (6.16).

To settle convergence of the first term, we need to show that
$\frac{w(\Phi(\omega))}{w(\Phi(\eta))}\overset{\delta\rightarrow
0}{\rightarrow}1$, uniformly with respect to $\omega\in
D,\eta\in\mathbf{Q}_{\omega}^{\delta}$. To this end, note that

$\frac{w(\Phi(\omega))}{w(\Phi(\eta))}=\frac{w(\Phi(\eta))+\int_{0}^{1}\frac{d}{dt}\big{|}_{t=s}\left[w(\Phi(\eta)+s\tau)\right]~{}ds}{w(\Phi(\eta))}\leq
1+\frac{\sup_{{\scaleobj{0.65}{\Upsilon}}\in
B_{2\delta}(\Phi(\eta))}\nabla_{\tau}w({\scaleobj{0.65}{\Upsilon}})}{w(\Phi(\eta))},$

where $\nabla_{\tau}$ denotes the derivative in direction
$\tau\in\mathbb{R}^{d}$. We now use Jacobi’s formula

$\frac{d}{dt}\det A(t)=\det
A(t)\cdot\mathop{\operatorname{trace}}([A(t)]^{-1}\cdot A^{\prime}(t)),$

valid for the derivative of the determinant of any differentiable function
$M:I\subset\mathbb{R}\to\mathrm{GL}(\mathbb{R}^{d})$ (see [70, Section 8.3,
Equation (2)]), to obtain

$\begin{split}\nabla_{\tau}w({\scaleobj{0.65}{\Upsilon}})&=\sum_{j\in\underline{d}}\tau_{j}\frac{\partial}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}}\det(A({\scaleobj{0.65}{\Upsilon}}))=\det(A({\scaleobj{0.65}{\Upsilon}}))\cdot\sum_{j\in\underline{d}}\tau_{j}\cdot\mathop{\operatorname{trace}}\left(A^{-1}({\scaleobj{0.65}{\Upsilon}})\frac{\partial}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}}A({\scaleobj{0.65}{\Upsilon}})\right)\\\
&=w({\scaleobj{0.65}{\Upsilon}})\cdot\sum_{j\in\underline{d}}\tau_{j}\cdot\mathop{\operatorname{trace}}\left(\left(\frac{\partial}{\partial\eta_{i}}\bigg{|}_{\eta=0}A^{-1}({\scaleobj{0.65}{\Upsilon}})A({\scaleobj{0.65}{\Upsilon}}+\eta)\right)^{T}\right)\\\
&=w({\scaleobj{0.65}{\Upsilon}})\cdot\sum_{j\in\underline{d}}\tau_{j}\cdot\mathop{\operatorname{trace}}\left(\frac{\partial}{\partial\eta_{i}}\bigg{|}_{\eta=0}\phi_{{\scaleobj{0.65}{\Upsilon}}}(\eta)\right),\end{split}$

with $\phi_{{\scaleobj{0.65}{\Upsilon}}}$ as in (4.4). Note that
$\phi_{{\scaleobj{0.65}{\Upsilon}}}(0)=\mathop{\operatorname{id}}$ for all
${\scaleobj{0.65}{\Upsilon}}\in\mathbb{R}^{d}$, so that (4.5) yields
$\|(\partial_{i}\phi_{{\scaleobj{0.65}{\Upsilon}}})(0)\|\leq v_{0}(0)$.
Additionally, the trace of a matrix $M\in\mathbb{R}^{d\times d}$ can be
(coarsely) estimated by $|\mathop{\operatorname{trace}}(M)|\leq d\|M\|$, such
that

$\left|\nabla_{\tau}w({\scaleobj{0.65}{\Upsilon}})\right|\leq d\cdot
w({\scaleobj{0.65}{\Upsilon}})\cdot\sum_{j\in\underline{d}}v_{0}(0)\cdot|\tau_{j}|\leq
d\cdot w({\scaleobj{0.65}{\Upsilon}})\cdot\|\tau\|_{1}\cdot v_{0}(0)\leq
d^{3/2}\cdot w({\scaleobj{0.65}{\Upsilon}})\cdot|\tau|\cdot v_{0}(0).$

Therefore, with $|\tau|\leq 2\delta$ and $v_{0}^{d}$-moderateness of $w$,

$\left|1-\frac{w(\Phi(\omega))}{w(\Phi(\eta))}\right|\leq|\tau|\cdot
d^{3/2}\cdot
v_{0}(0)\cdot\max_{r\in[0,1]}\frac{w(\Phi(\eta)+r\tau)}{w(\Phi(\eta))}\leq
2\delta\cdot d^{3/2}\cdot v_{0}^{d}(2\delta)\cdot
v_{0}(0)=:C^{\delta}<\infty.$ (6.19)

The final estimate is independent of $\omega\in D$, and of
$\eta\in\mathbf{Q}_{\omega}^{\delta}$, and $C^{\delta}\rightarrow 0$ as
$\delta\rightarrow 0$. ∎

We are now ready to prove Theorem 6.1.

###  6.2 Proof of Theorem 6.1

Recall that, by Remark 2.17,
${\mathrm{osc}}_{\mathcal{V}^{\delta}_{\Phi},\Gamma}$ is continuous. Using
Proposition 6.3 and Parseval’s formula, we can rewrite the oscillation at
$((y,\omega),(z,\eta))\in\Lambda\times\Lambda$, as follows:

$\displaystyle{\mathrm{osc}}_{\mathcal{V}^{\delta}_{\Phi},\Gamma}((y,\omega),(z,\eta))$
$\displaystyle=\sup_{(z_{0},\eta_{0})\in\mathcal{V}_{(z,\eta)}^{\delta}}\left|\left\langle\widehat{g_{y,\omega}},\widehat{g_{z,\eta}}-\Gamma((z,\eta),(z_{0},\eta_{0}))\cdot\widehat{g_{z_{0},\eta_{0}}}\right\rangle\right|$
$\displaystyle=\sup_{(z_{0},\eta_{0})\in\mathcal{V}_{(z,\eta)}^{\delta}}\left|K_{\theta,\tilde{\theta}_{(z,\eta),(z_{0},\eta_{0})},\Phi}((y,\omega),(z,\eta))\right|.$
(6.20)

Based on (6.20), Lemma 4.7 provides

$\left|K_{\theta,\tilde{\theta}_{(z,\eta),(z_{0},\eta_{0})},\Phi}((y,\omega),(z,\eta))\right|=\sqrt{\frac{w(\Phi(\eta))}{w(\Phi(\omega))}}\cdot
L_{\Phi(\eta)}[\theta,\tilde{\theta}_{(z,\eta),(z_{0},\eta_{0})}](A^{T}(\Phi(\eta))\langle
z-y\rangle,\Phi(\omega)-\Phi(\eta)),$

where $L_{\Phi(\eta)}$ is as in (4.7). If we define
$\mathcal{L}_{\tau_{0}}:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{+}_{0}$,
$\tau_{0}\in\mathbb{R}^{d}$, by

$\mathcal{L}_{\tau_{0}}(x,\tau):=\sup_{z\in\mathbb{R}^{d}}\sup_{(z_{0},\eta_{0})\in\mathcal{V}^{\delta}_{z,\Phi^{-1}(\tau_{0})}}L_{\tau_{0}}[\theta,\tilde{\theta}_{(z,\Phi^{-1}(\tau_{0})),(z_{0},\eta_{0})}](x,\tau),$
(6.21)

then, for all $(y,\omega),(z,\eta)\in\Lambda$,

${\mathrm{osc}}_{\mathcal{V}^{\delta}_{\Phi},\Gamma}((y,\omega),(z,\eta))\leq\sqrt{\frac{w(\Phi(\eta))}{w(\Phi(\omega))}}\mathcal{L}_{\Phi(\eta)}(A^{T}(\Phi(\eta))\langle
z-y\rangle,\Phi(\omega)-\Phi(\eta)).$

Via a tedious, but straightforward derivation involving several changes of
variable in a manner similar to the proof of Lemma 4.7, we obtain in
particular that

$\|{\mathrm{osc}}_{\mathcal{V}^{\delta}_{\Phi},\Gamma}\|_{\mathcal{B}_{m}}\leq\mathop{\operatorname{ess~{}sup}}_{\tau_{0}\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}M(x,\tau)\mathcal{L}_{\tau_{0}}(x,\tau)~{}dx~{}d\tau,$
(6.22)

where $M$ is defined as in (4.2) and we used that $m$ is $\Phi$-compatible
with the (symmetric) dominating weight $m^{\Phi}(x,\tau)=(1+|x|)^{p}\cdot
v_{1}(\tau)$.

By Lemma 6.5, with $\tilde{w}\\!=\\!w_{2}$, all functions
$\tilde{\theta}_{(z,\eta),(z_{0},\eta_{0})}$ with
$(z,\eta)\\!\\!\in\\!\Lambda$ and
$(z_{0},\eta_{0})\\!\\!\in\\!\mathcal{V}^{\delta}_{z,\eta}\\!\subset\\!(z+\mathbf{P}^{\delta}_{\eta})\times\mathbf{Q}^{\delta}_{\eta}$
satisfy the conditions of Theorem 4.8, as does $\theta$. Hence, for any
$z,\tau_{0}\in\mathbb{R}^{d}$ and
$(z_{0},\eta_{0})\in\mathcal{V}^{\delta}_{z,\Phi^{-1}(\tau_{0})}$, Theorem 4.8
yields

$L_{\tau_{0}}[\theta,\tilde{\theta}_{(z,\Phi^{-1}(\tau_{0})),(z_{0},\eta_{0})}](x,\tau)\leq
C\cdot C_{\textrm{max}}\cdot(1+|x|)^{-(d+p+1)}\cdot
v_{0}^{4d+p+1}(\tau)\cdot[w_{2}(\tau)]^{-1},$ (6.23)

with

$\begin{split}C_{\textrm{max}}=&\,\,C_{\textrm{max}}\left(d+p+1,\theta,\tilde{\theta}_{(z,\Phi^{-1}(\tau_{0})),(z_{0},\eta_{0})}\right)\\\
=&\max_{\begin{subarray}{c}j\in\underline{d}\\\ 0\leq m\leq
d+p+1\end{subarray}}\bigg{\|}\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\theta\bigg{\|}_{\mathbf{L}^{2}_{w_{2}}(\mathbb{R}^{d})}\cdot\max_{\begin{subarray}{c}j\in\underline{d}\\\
0\leq m\leq
d+p+1\end{subarray}}\bigg{\|}\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\tilde{\theta}_{(z,\Phi^{-1}(\tau_{0})),(z_{0},\eta_{0})}\bigg{\|}_{\mathbf{L}^{2}_{w_{2}}(\mathbb{R}^{d})}\\\
({\scriptstyle\text{Lem.
\ref{lem:convergence_of_tildetheta}}})\,\leq&\max_{\begin{subarray}{c}j\in\underline{d}\\\
0\leq m\leq
d+p+1\end{subarray}}\bigg{\|}\frac{\partial^{m}}{\partial{\scaleobj{0.65}{\Upsilon}}_{j}^{m}}\theta\bigg{\|}_{\mathbf{L}^{2}_{w_{2}}(\mathbb{R}^{d})}\cdot\max_{\begin{subarray}{c}j\in\underline{d}\\\
0\leq m\leq
d+p+1\end{subarray}}F_{j,m}(\delta;\theta,2,w_{2})=:D_{\textrm{max}}^{\delta}<\infty.\end{split}$

Note that the estimate for $D_{\textrm{max}}^{\delta}$ is independent of
$\tau_{0}\in\mathbb{R}^{D}$, $z\in\mathbb{R}^{d}$,
$(z_{0},\eta_{0})\in\mathcal{V}^{\delta}_{z,\Phi^{-1}(\tau_{0})}$, such that
taking $D_{\textrm{max}}^{\delta}$ instead of $C_{\textrm{max}}$ in (6.23)
produces a valid upper estimate for $\mathcal{L}_{\tau_{0}}(x,\tau)$.
Moreover, note that Lemma 6.5 implies $D_{\textrm{max}}^{\delta}\rightarrow 0$
as $\delta\rightarrow 0$.

Proving
$\|{\mathrm{osc}}_{\mathcal{V}^{\delta}_{\Phi},\Gamma}\|_{\mathcal{B}_{m}}<\infty$
is now analogous to the proof of Theorem 4.4, and
$\|{\mathrm{osc}}_{\mathcal{V}^{\delta}_{\Phi},\Gamma}\|_{\mathcal{B}_{m}}\rightarrow
0$ as $\delta\rightarrow 0$ follows directly from
$D_{\textrm{max}}^{\delta}\rightarrow 0$. ∎

##  7 Coorbit space theory of warped time-frequency systems

We have now developed explicit sufficient conditions that ensure
$K_{\theta,\Phi},{\mathrm{osc}}_{\mathcal{V}^{\delta}_{\Phi},\Gamma}\in\mathcal{B}_{m}$
and hence, by Eq. (2.23),
$\mathrm{M}_{\mathcal{V}_{\Phi}^{\delta}}K_{\theta,\Phi}\in\mathcal{B}_{m}$,
since $\mathcal{B}_{m}$ is solid. These are the crucial ingredients for
applying coorbit theory in the setting of warped time-frequency
representations.

######  Theorem 7.1.

Let $\Phi$ be a $(d+p+1)$-admissible warping function with control weight
$v_{0}$, where $p=0$ if $R_{\Phi}=\sup_{\xi\in D}\mathrm{D}\Phi(\xi)\|=\infty$
and $p\in\mathbb{N}_{0}$ otherwise. Let furthermore
$m_{0}:\Lambda\times\Lambda\to\mathbb{R}^{+}$ be a symmetric weight that
satisfies $1\leq m_{0}(\lambda,\rho)\leq
C^{(0)}m_{0}(\lambda,{\nu})m_{0}({\nu},\rho)$ for all
$\lambda,\rho,{\nu}\in\Lambda$ and

$m_{0}((y,\xi),(z,\eta))\leq(1+|y-z|)^{p}\cdot
v_{1}(\Phi(\xi)-\Phi(\eta)),\text{ for all }y,z\in\mathbb{R}^{d}\text{ and
}\xi,\eta\in D,\ \tau,{\scaleobj{0.65}{\Upsilon}}\in\mathbb{R}^{d},$ (7.1)

for some continuous and submultiplicative weight
$v_{1}:\mathbb{R}^{d}\to\mathbb{R}^{+}$ with
$v_{1}({\scaleobj{0.65}{\Upsilon}})=v_{1}(-{\scaleobj{0.65}{\Upsilon}})$ for
all ${\scaleobj{0.65}{\Upsilon}}\in\mathbb{R}^{d}$.

Then there exist nonzero
$\theta\in\mathbf{L}^{2}_{v_{0}^{d/2}}(\mathbb{R}^{d})$, such that for any
rich, solid Banach space
$Y\hookrightarrow\mathbf{L}_{\mathrm{loc}}^{1}(\Lambda)$ with
$\mathcal{B}_{m_{0}}(Y)\hookrightarrow Y$,

  1. 1.

$\operatorname{Co}(\mathcal{G}(\theta,\Phi),Y)$ is a well-defined Banach
function space.

  2. 2.

There is a $\delta_{0}=\delta_{0}(\theta,\Phi,m)>0$ independent of $Y$, such
that

$(g_{y_{\ell,k},\omega_{\ell,k}})_{\ell,k\in\mathbb{Z}^{d}}\subset\mathcal{G}(\theta,\Phi)$

is a Banach frame decomposition for
$\operatorname{Co}(\mathcal{G}(\theta,\Phi),Y)$, whenever the points
$\left((y_{\ell,k},\omega_{\ell,k})\right)_{\ell,k\in\mathbb{Z}^{d}}\subset\Lambda$
satisfy $(y_{\ell,k},\omega_{\ell,k})\in V^{\delta}_{\ell,k}$, where
$\mathcal{V}_{\Phi}^{\delta}=(V_{\ell,k}^{\delta})_{\ell,k\in\mathbb{Z}^{d}}$
is the $\Phi$-induced $\delta$-fine covering and $\delta\leq\delta_{0}$.

In particular, items (1) and (2) above hold for
$Y=\mathbf{L}^{p,q}_{\kappa}(\Lambda)$, with $1\leq p,q\leq\infty$ and any
weight $\kappa:\Lambda\rightarrow[1,\infty)$ that satisfies
$m_{\kappa}\lesssim m_{0}$.

###### Proof.

By Propositions 5.2 and 5.4, the $\Phi$-induced $\delta$-fine covering
$\mathcal{V}_{\Phi}^{\delta}$ is a topologically admissible, product-
admissible covering that satisfies items (1)-(3) of Assumption 2.11 and item
(1) of Assumption 2.19. Moreover, item (6) of Assumption 2.11 is satisfied, by
the assumptions of this theorem.

Next, choose $\theta\in\mathbf{L}^{2}_{\sqrt{w_{0}}}(\mathbb{R}^{d})$, such
that $\|\theta\|_{\mathbf{L}^{2}(\mathbb{R}^{d})}=1$ and the assumptions of
Theorem 6.1 are satisfied with $m=m_{v}$ defined by

$\displaystyle
m_{v}((y,\omega),(z,\eta))=\max\left\\{\frac{v((y,\omega))}{v((z,\eta))},\frac{v((z,\eta))}{v((y,\omega))}\right\\},$
with $\displaystyle
v((y,\omega)):=m_{0}((y,\omega),(x,\xi))\cdot\max\\{w(\Phi(\omega)),[w(\Phi(\omega))]^{-1}\\},$

for all $(y,\omega),(z,\eta)\in\Lambda$ and some fixed, arbitrary
$(x,\xi)\in\Lambda$. This is always possible, since any function
$\theta\in\mathcal{C}^{\infty}_{c}(\mathbb{R}^{d})\subset\mathbf{L}^{2}(\mathbb{R}^{d})$
with unit $\mathbf{L}^{2}$-norm satisfies these assumptions. In particular,
the assumptions of Theorem 6.1 are also satisfied for $m=m_{0}\leq m_{v}$. By
Proposition 3.4, the map $(y,\omega)\mapsto g_{y,\omega}$ is continuous and by
Corollary 3.6, the warped time-frequency system $\mathcal{G}(\theta,\Phi)$ is
a tight Parseval frame, such that item (4) of Assumption 2.11 is satisfied. In
particular, by Eq. (3.7),
$\sup_{(y,\omega)\in\Lambda}\|g_{y,\omega}\|_{2}\leq\|\theta\|_{\mathbf{L}^{2}_{\sqrt{w_{0}}}}<\infty$.
Hence, with
$w^{c}_{\mathcal{V}_{\Phi}^{\delta}}=\max\\{w(\Phi(\omega)),[w(\Phi(\omega))]^{-1}\\}$
as in Proposition 5.2 and $u(\lambda):=m_{0}(\lambda,(x,\xi))$ with the same
choice of $(x,\xi)\in\Lambda$ as above, item (5) of Assumption 2.11 is
satisfied as well.

Moreover, by choice of $\theta$, and with $\Gamma$ as in Theorem 6.1, we have

$\|K_{\theta,\Phi}\|_{\mathcal{B}_{m_{v}}}<\infty\quad\text{and}\quad\|\mathrm{M}_{\mathcal{V}_{\Phi}^{\delta}}K_{\theta,\Phi}\|_{\mathcal{B}_{m_{0}}}\leq\|K_{\theta,\Phi}\|_{\mathcal{B}_{m_{0}}}+\|{\mathrm{osc}}_{\mathcal{V}^{\delta}_{\Phi},\Gamma}\|_{\mathcal{B}_{m_{0}}}<\infty,$

showing that the final item (7) of Assumption 2.11 is satisfied. Hence,
Assumption 2.11 is fully satisfied and we can apply Theorem 2.14 to show that
$\operatorname{Co}(\mathcal{G}(\theta,\Phi),Y)$ is a well-defined Banach
function space.

Finally, note that $\Gamma$ as in Theorem 6.1 is continuous, to verify that
item (2) of Assumption 2.19 is satisfied. By the same theorem, we can choose
$\delta_{0}>0$, such that

$\|{\mathrm{osc}}_{\mathcal{V},\Gamma}\|_{\mathcal{B}_{m_{v}}}\cdot(2\|K_{\Psi}\|_{\mathcal{B}_{m_{v}}}+\|{\mathrm{osc}}_{\mathcal{V}^{\delta}_{\Phi},\Gamma}\|_{\mathcal{B}_{m_{v}}})<1$

for all $\delta\leq\delta_{0}$, proving the second assertion. The proof is
completed by observing that the statement about weighted, mixed-norm Lebesgue
spaces is a direct consequence of (2.14). ∎

By definition, the coorbit space
$\operatorname{Co}(\mathcal{G}(\theta,\Phi),Y)$ depends on both the prototype
function $\theta$ and the warping function $\Phi$. The dependence on the
warping function $\Phi$ is an essential consequence of (sufficiently)
different warping functions inducing time-frequency representations with
vastly different properties. As teasered in the introduction, relations
between coorbit spaces associated to different warping functions will be
studied in the framework of decomposition spaces in a follow-up contribution.
Here, we will show that, the dependence on the generating prototype $\theta$
can be weakened, i.e., under certain conditions on $\theta_{1},\theta_{2}$,
the coorbit spaces $\operatorname{Co}(\mathcal{G}(\theta_{1},\Phi),Y)$ and
$\operatorname{Co}(\mathcal{G}(\theta_{2},\Phi),Y)$ are equal, similar to
modulation spaces for the STFT. Before we do so, however, we show that the
mixed kernel associated with two warped time-frequency systems inherits the
membership in $\mathcal{B}_{m}$ (or $\mathcal{A}_{m}$) from the kernels of the
individual systems.

######  Lemma 7.2.

Let $X\in\\{\mathcal{A}_{m},\mathcal{B}_{m}\\}$, with a symmetric weight $m$
satisfying $m(\lambda,\rho)\leq C^{(0)}m(\lambda,{\nu})m({\nu},\rho)$, for
some $C^{(0)}$ and all $\lambda,\rho,{\nu}\in\Lambda$. If
$\theta_{1},\theta_{2}\in\mathbf{L}^{2}_{\sqrt{w_{0}}}\cap\mathbf{L}^{2}(\mathbb{R}^{d})$
are nonzero and such that $K_{\theta_{1},\Phi},K_{\theta_{2},\Phi}\in X$, then

$K_{\theta_{1},\theta_{2},\Phi}:=K_{\mathcal{G}(\theta_{1},\Phi),\mathcal{G}(\theta_{2},\Phi)}\in
X.$ (7.2)

###### Proof.

We first consider the case $\langle\theta_{1},\theta_{2}\rangle\neq 0$. In
that case, the orthogonality relations, Theorem 3.5, applied to the kernel
$K_{\theta_{1},\Phi}\cdot K_{\theta_{2},\Phi}$ yield, for all
$(y,\omega),(z,\eta)\in\Lambda$,

$\begin{split}K_{\theta_{1},\Phi}\cdot
K_{\theta_{2},\Phi}((y,\omega),(z,\eta))&=\int_{\Lambda}K_{\theta_{1},\Phi}((y,\omega),(x,\xi))K_{\theta_{2},\Phi}((x,\xi),(z,\eta))~{}d(x,\xi)\\\
&=\int_{\Lambda}\overline{\langle
g^{(1)}_{y,\omega},g^{(1)}_{x,\xi}\rangle}\langle
g^{(2)}_{z,\eta},g^{(2)}_{x,\xi}\rangle~{}d(x,\xi)\\\
({\scriptstyle{\text{Def. of
}V_{\bullet,\Phi}}})&=\int_{\Lambda}V_{\theta_{2},\Phi}g^{(2)}_{z,\eta}(x,\xi)\overline{V_{\theta_{1},\Phi}g^{(1)}_{y,\omega}(x,\xi)}~{}d(x,\xi)=\langle
V_{\theta_{2},\Phi}g^{(2)}_{z,\eta},V_{\theta_{1},\Phi}g^{(1)}_{y,\omega}\rangle\\\
({\scriptstyle{\text{orth.~{}rel.}}})&=\langle
g^{(2)}_{z,\eta},g^{(1)}_{y,\omega}\rangle\langle\theta_{1},\theta_{2}\rangle=\langle\theta_{1},\theta_{2}\rangle\cdot
K_{\theta_{1},\theta_{2},\Phi}((y,\omega),(z,\eta)).\end{split}$

Since, under the conditions on $m$, $\mathcal{A}_{m},\mathcal{B}_{m}$ are
algebrae, this establishes (7.2).

If $\langle\theta_{1},\theta_{2}\rangle=0$, then we need an auxiliary function
$\theta_{3}$, which may be any function in
$\mathbf{L}^{2}_{\sqrt{w_{0}}}\cap\mathbf{L}^{2}(\mathbb{R}^{d})$ such that
$K_{\theta_{3},\Phi}\in X$ and that is neither orthogonal to $\theta_{1}$ nor
to $\theta_{2}$. For example, $\theta_{3}$ could satisfy the conditions of
Theorem. 4.4. By the first part of the proof, we obtain

$(K_{\theta_{1},\Phi}\cdot K_{\theta_{3},\Phi})\cdot(K_{\theta_{3},\Phi}\cdot
K_{\theta_{2},\Phi})=\langle\theta_{1},\theta_{3}\rangle\overline{\langle\theta_{2},\theta_{3}\rangle}\cdot\
K_{\theta_{1},\theta_{3},\Phi}\cdot K_{\theta_{3},\theta_{2},\Phi}.$

Now, apply the argument in the first part of the proof twice to obtain that

$K_{\theta_{1},\theta_{2},\Phi}=C^{-1}(K_{\theta_{1},\Phi}\cdot
K_{\theta_{3},\Phi})\cdot(K_{\theta_{3},\Phi}\cdot
K_{\theta_{2},\Phi})\qquad\text{with}\qquad
C=\|\theta_{3}\|^{2}\langle\theta_{1},\theta_{3}\rangle\overline{\langle\theta_{2},\theta_{3}\rangle}.\qed$

######  Remark 7.3.

If $\theta_{1},\theta_{2}$ satisfy the conditions of Theorem 4.4, then the
assumptions of Lemma 7.2 can be verified by applying that theorem. However,
since Theorem 4.4 only provides _sufficient_ conditions, there might be
$\theta_{1},\theta_{2}$ with
$K_{\theta_{1},\Phi},K_{\theta_{2},\Phi}\in\mathcal{B}_{m}$ that do not
satisfy those conditions, for which Lemma 7.2 remains valid.

######  Theorem 7.4.

Assume that $\Phi$, $m_{0}$ and both
$\theta_{1}\in\mathbf{L}^{2}_{\sqrt{w_{0}}}$ and
$\theta_{2}\in\mathbf{L}^{2}_{\sqrt{w_{0}}}$ jointly satisfy the conditions of
Theorem 6.1.

Then, for any rich, solid Banach space
$Y\hookrightarrow\mathbf{L}_{\mathrm{loc}}^{1}(\Lambda)$ with
$\mathcal{B}_{m_{0}}(Y)\hookrightarrow Y$, we have

$\operatorname{Co}(\mathcal{G}(\theta_{1},\Phi),Y)=\operatorname{Co}(\mathcal{G}(\theta_{2},\Phi),Y).$

In particular, the statement holds for $Y=\mathbf{L}^{p,q}_{\kappa}(\mu)$,
with $1\leq p,q\leq\infty$ and any weight
$\kappa:\Lambda\rightarrow[1,\infty)$ that satisfies $m_{\kappa}\lesssim
m_{0}$.

###### Proof.

The same derivations as in the proof of Theorem 7.1 show that Assumptions 2.11
and 2.19 are fully satisfied and consequently, by Theorem 2.14,
$\operatorname{Co}(\mathcal{G}(\theta_{1},\Phi),Y)$ and
$\operatorname{Co}(\mathcal{G}(\theta_{2},\Phi),Y)$ are well-defined Banach
spaces. By Lemma 7.2, the mixed kernel $K_{\theta_{1},\theta_{2}}$ is
contained in $\mathcal{B}_{m_{v}}\subset\mathcal{B}_{m_{0}}$, with $v$ as in
the proof of Theorem 7.1. Hence, we can apply Proposition 2.15 to obtain the
desired result. The statement about weighted, mixed-norm Lebesgue spaces is,
once more, a direct consequence of (2.14). ∎

##  8 Radial warping

In this section, we consider warped time-frequency representations for which
the warping of frequency space depends only on the modulus in the frequency
domain, i.e., we study maps of the form

$\Phi_{\varrho}:\mathbb{R}^{d}\to\mathbb{R}^{d},\xi\mapsto\xi/|\xi|\cdot\varrho(|\xi|)\,,$

which we call the radial warping function associated to the radial component
$\varrho:[0,\infty)\to[0,\infty)$. More precisely, we will provide conditions
on the radial component $\varrho$ which ensure that $\Phi_{\varrho}$ is a
($k$-admissible) warping function, as introduced in Definitions 3.1 and 4.2.
In particular, we will see that if $\varrho$ is a strictly increasing
$\mathcal{C}^{k+1}$ diffeomorphism which is also linear on a neighborhood of
the origin, then $\Phi_{\varrho}$ is a $\mathcal{C}^{k+1}$ diffeomorphism,
with inverse $\Phi_{\varrho}^{-1}=\Phi_{\varrho^{-1}}$. Finally, under
additional “moderateness assumptions” on the derivatives of $\varrho^{-1}$, we
will show that the diffeomorphism $\Phi_{\varrho}$ is a $k$-admissible warping
function. These claims will be established in Section 8.1.

Section 8.2 is concerned with circumventing the somewhat unnatural restriction
that $\varrho$ is linear in a neighborhood of the origin. Using the so-called
slow-start construction, one can associate to a “sufficiently well-behaved”
function $\varsigma:[0,\infty)\to[0,\infty)$ a $k$-admissible radial component
$\varrho:[0,\infty)\to[0,\infty)$, which equals $\varsigma$ on outside an
arbitrarily small neighborhood of the origin.

Finally, we discuss several examples of radial warping functions in Section
8.3.

###  8.1 General properties of radial warping functions

To enable a more compact notation, we will from now on denote by
$\varrho_{\ast}:=\varrho^{-1}$ the inverse of a bijection
$\varrho:\mathbb{R}\to\mathbb{R}$.

######  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 $\varrho(\xi)=c\cdot\xi$ for all
$\xi\in(-\varepsilon,\varepsilon)$.

  4. 4.

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

$\widetilde{\varrho_{\ast}}:\mathbb{R}\to(0,\infty)\qquad\text{is defined
by}\qquad\widetilde{\varrho_{\ast}}(\xi):=\varrho_{\ast}(\xi)/\xi\quad\text{for}\quad\xi\neq
0,\quad\text{and}\quad\widetilde{\varrho_{\ast}}(0):=c^{-1}.$ (8.1)

are $v$-moderate.

  5. 5.

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

$C_{0}\cdot\widetilde{\varrho_{\ast}}(\xi)\leq\varrho_{\ast}^{\prime}(\xi)\leq
C_{1}\cdot(1+\xi)\cdot\widetilde{\varrho_{\ast}}(\xi)\qquad\forall\,\xi\in(0,\infty).$
(8.2)

  6. 6.

We have

$|\varrho_{\ast}^{(\ell)}(\xi)|\leq
v(\xi-\eta)\cdot\varrho_{\ast}^{\prime}(\eta)\qquad\forall\,\eta,\xi\in[0,\infty)\text{
and }\ell\in\underline{k+1}\,.$ (8.3)

Note that the property (8.3) can equivalently be exchanged by the simpler
$|\varrho_{\ast}^{(\ell)}|\leq C\varrho_{\ast}^{\prime}$, for all
$\ell\in\underline{k+1}$ (using that $\varrho_{\ast}^{\prime}$ is $v$-moderate
and $v$ is submultiplicative), at the cost of introducing a multiplicative
constant $Cv(0)$ on the right-hand side of (8.3).

######  Remark 8.2.

(1) An important consequence of these assumptions is that there exists a
constant $C_{2}=C_{2}(C_{1},v)>0$ with

$|\varrho_{\ast}^{(\ell)}(\xi)|\leq
C_{2}\cdot(1+\xi)\cdot\widetilde{\varrho_{\ast}}(\xi)\qquad\forall\,\xi\in(0,\infty)\text{
and }\ell\in\\{0\\}\cup\underline{k+1}\,.$ (8.4)

Indeed, for $\ell=0$ (8.4) is always satisfied as long as $C_{2}\geq 1$, since
$\varrho_{\ast}$ is increasing with $\varrho_{\ast}(0)=0$, whence
$|\varrho_{\ast}^{(0)}(\xi)|=\varrho_{\ast}(\xi)=\xi\cdot\widetilde{\varrho_{\ast}}(\xi)\leq(1+\xi)\cdot\widetilde{\varrho_{\ast}}(\xi)$
for $\xi\in(0,\infty)$. Thus, it remains to verify Equation (8.4) for
$\ell\in\underline{k+1}$. But for this case, applying (8.3) with $\eta=\xi$,
we see that

$|\varrho_{\ast}^{(\ell)}(\xi)|\leq
v(\xi-\xi)\cdot\varrho_{\ast}^{\prime}(\xi)=v(0)\cdot\varrho_{\ast}^{\prime}(\xi)\,,$

so that (8.2) yields $|\varrho_{\ast}^{(\ell)}(\xi)|\leq
v(0)\cdot\varrho_{\ast}^{\prime}(\xi)\leq C_{1}\cdot
v(0)\cdot(1+\xi)\cdot\widetilde{\varrho_{\ast}}(\xi)$. Setting
$C_{2}:=\max\\{1,C_{1}\cdot v(0)\\}$, we have thus established (8.4).

(2) To indicate that being an admissible radial component is a nontrivial
restriction on $\varrho$, we observe that condition (8.2) entails certain
growth restrictions on the function $\varrho_{\ast}=\varrho^{-1}$. Indeed, for
arbitrary $\varepsilon>0$ and $\xi\geq 1/\varepsilon$, Equation (8.2) shows
$\varrho_{\ast}^{\prime}(\xi)\leq
C_{1}\cdot(1+\xi)\cdot\varrho_{\ast}(\xi)/\xi\leq(1+\varepsilon)C_{1}\cdot\varrho_{\ast}(\xi)$.
This implies

$\displaystyle\frac{d}{d\xi}\left(e^{-(1+\varepsilon)C_{1}\xi}\cdot\varrho_{\ast}(\xi)\right)$
$\displaystyle=-(1+\varepsilon)C_{1}\cdot
e^{-(1+\varepsilon)C_{1}\xi}\cdot\varrho_{\ast}(\xi)+e^{-(1+\varepsilon)C_{1}\xi}\cdot\varrho_{\ast}^{\prime}(\xi)$
$\displaystyle\leq-(1+\varepsilon)C_{1}\cdot
e^{-(1+\varepsilon)C_{1}\xi}\cdot\varrho_{\ast}(\xi)+(1+\varepsilon)C_{1}\cdot
e^{-(1+\varepsilon)C_{1}\xi}\cdot\varrho_{\ast}(\xi)=0$

for all $\xi\geq 1/\varepsilon$. For any $\xi\geq a\geq 1/\varepsilon$, this
implies $e^{-(1+\varepsilon)C_{1}\xi}\cdot\varrho_{\ast}(\xi)\leq
e^{-(1+\varepsilon)C_{1}a}\cdot\varrho_{\ast}(a)$, and hence

$\varrho_{\ast}(\xi)\leq\frac{\varrho_{\ast}(a)}{e^{(1+\varepsilon)C_{1}a}}\cdot
e^{(1+\varepsilon)C_{1}\xi}\qquad\forall\,\xi\geq
a\geq\varepsilon^{-1}\,,\text{ for any }\varepsilon>0\,.$ (8.5)

Likewise, the lower bound in (8.2) implies

$\frac{d}{d\xi}\left(\xi^{-C_{0}}\cdot\varrho_{\ast}(\xi)\right)=(-C_{0})\xi^{-C_{0}-1}\cdot\varrho_{\ast}(\xi)+\xi^{-C_{0}}\cdot\varrho_{\ast}^{\prime}(\xi)\geq(-C_{0})\xi^{-C_{0}}\cdot\frac{\varrho_{\ast}(\xi)}{\xi}+C_{0}\cdot\xi^{-C_{0}}\cdot\frac{\varrho_{\ast}(\xi)}{\xi}=0$

for all $\xi\in(0,\infty)$. Thus, for $\xi\geq a>0$, we get
$\xi^{-C_{0}}\cdot\varrho_{\ast}(\xi)\geq a^{-C_{0}}\cdot\varrho_{\ast}(a)$,
and thus

$\varrho_{\ast}(\xi)\geq\frac{\varrho_{\ast}(a)}{a^{C_{0}}}\cdot\xi^{C_{0}}\qquad\forall\,\xi\geq
a>0\,.$ (8.6)

In words, Equations (8.5) and (8.6) show that _the inverse of an admissible
radial component $\varrho$ can grow at most exponentially, and has to grow at
least like a positive (not necessarily integer) power of $\xi$._

We define (for a larger class of radial components) the radial warping
function associated with $\varrho$.

######  Definition 8.3.

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

$\Phi_{\varrho}:\mathbb{R}^{d}\to\mathbb{R}^{d},\xi\mapsto\widetilde{\varrho}(|\xi|)\cdot\xi,\quad\text{with}\quad\widetilde{\varrho}(t):=\varrho(t)/t\quad\text{for
}t\in\mathbb{R}\setminus\\{0\\}\,,\quad\text{and}\quad\widetilde{\varrho}(0):=c\,.$
(8.7)

Clearly, if $\varrho\in\mathcal{C}^{k}(\mathbb{R})$, then
$\widetilde{\varrho}\in\mathcal{C}^{k}(\mathbb{R})$. Our goal in this section
is to show that $\Phi_{\varrho}$ is a $k$-admissible warping function as per
Definition 4.2, provided that $\varrho$ is a $k$-admissible radial component.
To this end, we first show that the inverse $\Phi_{\varrho}^{-1}$ of
$\Phi_{\varrho}$ is given by $\Phi_{\varrho}^{-1}=\Phi_{\varrho^{-1}}$, and
provide a convenient expression of the Jacobian
$\mathrm{D}\Phi_{\varrho}^{-1}$. The following notation will be helpful for
that purpose: For $\xi\in\mathbb{R}^{d}\setminus\\{0\\}$, we define

$\xi_{\circ}:=\xi/|\xi|,\qquad\pi_{\xi}:\mathbb{R}^{d}\to\mathbb{R}^{d},\tau\mapsto\langle\tau,\xi_{\circ}\rangle\cdot\xi_{\circ}\,,\qquad\text{and}\qquad\pi_{\xi}^{\perp}:=\mathrm{id}_{\mathbb{R}^{d}}-\pi_{\xi}\,,$
(8.8)

so that $\pi_{\xi}$ is the orthogonal projection on the space spanned by
$\xi$, while $\pi_{\xi}^{\perp}$ is the orthogonal projection on the
orthogonal complement of this space. With these notations, the derivative of
$\Phi_{\varrho}$ and $\Phi_{\varrho}^{-1}$ can be described as follows:

######  Lemma 8.4.

Let $\varrho:\mathbb{R}\to\mathbb{R}$ be a $\mathcal{C}^{k}$-diffeomorphism
with $\varrho(t)=ct$ for all $t\in(-\varepsilon,\varepsilon)$ and suitable
$\varepsilon,c>0$. Then $\Phi_{\varrho}$ is $\mathcal{C}^{k}$, and for
$\xi\in\mathbb{R}^{d}\setminus\\{0\\}$, we have

$\mathrm{D}\Phi_{\varrho}(\xi)=\widetilde{\varrho}(|\xi|)\cdot\pi_{\xi}^{\perp}+\varrho^{\prime}(|\xi|)\cdot\pi_{\xi}\,,\qquad\text{and}\qquad[\mathrm{D}\Phi_{\varrho}(\xi)]^{-1}=[\widetilde{\varrho}(|\xi|)]^{-1}\cdot\pi_{\xi}^{\perp}+[\varrho^{\prime}(|\xi|)]^{-1}\cdot\pi_{\xi}\,.$
(8.9)

Furthermore, $\Phi_{\varrho}$ is a $\mathcal{C}^{k}$-diffeomorphism, with
inverse $\Phi_{\varrho}^{-1}=\Phi_{\varrho_{\ast}}$ and satisfies
$\varrho_{\ast}(t)=t/c$ for $t\in(-c\varepsilon,c\varepsilon)$.

Finally, if $\varrho$ is a $0$-admissible radial component, then we have

$\|[\mathrm{D}\Phi_{\varrho_{\ast}}(\xi)]^{-1}\|\lesssim
1/\widetilde{\varrho_{\ast}}(|\xi|)\qquad\forall\,\xi\in\mathbb{R}^{d},\quad\text{with
}\widetilde{\varrho_{\ast}}\text{ as in }\eqref{eq:PsiTildeDefinition},$
(8.10)

where the implied constant only depends on the constant $C_{0}$ in (8.2).

###### Proof.

Recall that $\widetilde{\varrho}\in\mathcal{C}^{k}(\mathbb{R})$, with
$\widetilde{\varrho}\equiv c$ on $(-\varepsilon,\varepsilon)$, and hence
$\Phi_{\varrho}\in\mathcal{C}^{k}(\mathbb{R}^{d})$.

Now, a direct computation using the identity $\partial_{j}|\xi|=\xi_{j}/|\xi|$
shows for $\xi\in\mathbb{R}^{d}\setminus\\{0\\}$ that

$\displaystyle\partial_{j}(\Phi_{\varrho})_{i}(\xi)=\partial_{j}\left(\xi_{i}\cdot\frac{\varrho(|\xi|)}{|\xi|}\right)$
$\displaystyle=\widetilde{\varrho}(|\xi|)\cdot\delta_{i,j}+\frac{\varrho^{\prime}(|\xi|)-\widetilde{\varrho}(|\xi|)}{|\xi|^{2}}\cdot\xi_{i}\xi_{j}\,.$

In vector notation, and with $\xi_{\circ}=\xi/|\xi|$ as in (8.8), this means

$\mathrm{D}\Phi_{\varrho}(\xi)=\widetilde{\varrho}(|\xi|)\cdot\mathrm{id}+\left(\varrho^{\prime}(|\xi|)-\widetilde{\varrho}(|\xi|)\right)\cdot\xi_{\circ}\xi_{\circ}^{T}\,.$

Now, recall that $\xi_{\circ}\xi_{\circ}^{T}$ is the matrix representing the
linear map $\pi_{\xi}$, and that $\mathrm{id}=\pi_{\xi}+\pi_{\xi}^{\perp}$.
Inserting these identities into the previous displayed equation establishes
the claimed formula for $\mathrm{D}\Phi_{\varrho}(\xi)$. In particular, each
$\eta\in\mathbb{R}^{d}$ with $\eta\perp\xi$ is mapped to
$\widetilde{\varrho}(|\xi|)\cdot\eta$ by $\mathrm{D}\Phi_{\varrho}(\xi)$,
while each $\eta\in\mathrm{span}(\xi)$ is mapped to
$\varrho^{\prime}(|\xi|)\cdot\eta$. Since
$\mathbb{R}^{d}=\xi^{\perp}\oplus\mathrm{span}(\xi)$, the stated formula for
$[\mathrm{D}\Phi_{\varrho}(\xi)]^{-1}$ follows.

Linearity of $\varrho_{\ast}(t)=t/c$ for $t\in(-c\varepsilon,c\varepsilon)$ is
clear, such that $\Phi_{\varrho_{\ast}}$ is a radial warping functions as per
Definition 8.3. Note $|\Phi_{\varrho}(\xi)|=\varrho(|\xi|)$ for
$\xi\in\mathbb{R}^{d}\setminus\\{0\\}$, such that
$\varrho_{\ast}(|\Phi_{\varrho}(\xi)|)=\varrho_{\ast}(\varrho(|\xi|))=|\xi|$
and $\Phi_{\varrho}(\xi)/|\Phi_{\varrho}(\xi)|=\xi/|\xi|$. Together, this
implies $\Phi_{\varrho_{\ast}}(\Phi_{\varrho}(\xi))=\xi\,,$ for all
$\xi\in\mathbb{R}^{d}\setminus\\{0\\}$ and thus, by continuity, for $\xi=0$ as
well. Repeating this argument after interchanging $\varrho_{\ast}$ and
$\varrho$ yields $\Phi_{\varrho}\circ\Phi_{\varrho_{\ast}}=\mathrm{id}$.

To prove (8.10), consider
$\xi\\!\in\\!\mathbb{R}^{d}\setminus\\{0\\}\vphantom{\sum_{j}}$ and observe
that
$\|[\mathrm{D}\Phi_{\varrho_{\ast}}(\xi)]^{-1}\|=\max\big{\\{}[\widetilde{\varrho_{\ast}}(|\xi|)]^{-1},[\varrho_{\ast}^{\prime}(|\xi|)]^{-1}\big{\\}}$,
by (8.9). Applying the lower inequality in (8.2), we get

$\|[\mathrm{D}\Phi_{\varrho_{\ast}}(\xi)]^{-1}\|\leq\max\\{1,C_{0}^{-1}\\}\cdot
1/\widetilde{\varrho_{\ast}}(|\xi|)\,.$

For $\xi=0$ the result follows by continuity. ∎

To prove that $\Phi_{\varrho}$ is $k$-admissible if $\varrho$ is a
$k$-admissible radial component, we need to combine Lemma 8.4 with the
following crucial estimate:

######  Lemma 8.5.

For arbitrary $\xi,\eta\in\mathbb{R}^{d}\setminus\\{0\\}$, and with
$\pi_{\xi}$ and $\eta_{\circ}$ as in (8.8), we have

$\max\\{|\xi|,|\eta|\\}\cdot|\pi_{\xi}^{\perp}(\eta_{\circ})|\leq\sqrt{2}\cdot|\xi-\eta|\,.$

###### Proof.

By the definition of the Euclidean norm in terms of the inner product,

$|\xi-\eta|^{2}=\langle\xi-\eta,\xi-\eta\rangle=|\xi|^{2}-2\langle\xi,\eta\rangle+|\eta|^{2}\geq(|\xi|^{2}+|\eta|^{2})\cdot(1-|\langle\xi_{\circ},\eta_{\circ}\rangle|)\quad\forall\,\xi,\eta\in\mathbb{R}^{d}\setminus\\{0\\}\,,$
(8.11)

where we used $2ab\leq a^{2}+b^{2}$, for $a,b\in\mathbb{R}$ (with $a=|\xi|$,
$b=|\eta|$).

Next, we note

$|\pi_{\xi}^{\perp}(\eta_{\circ})|^{2}=|\eta_{\circ}|^{2}-|\pi_{\xi}(\eta_{\circ})|^{2}=1-|\langle\xi_{\circ},\eta_{\circ}\rangle\cdot\xi_{\circ}|^{2}=1-|\langle\xi_{\circ},\eta_{\circ}\rangle|^{2}\,.$
(8.12)

Observe $|\langle\xi_{\circ},\eta_{\circ}\rangle|\in[0,1]$, and furthermore
$1-a^{2}=(1+a)\cdot(1-a)\leq 2\cdot(1-a)$, for $a\leq 1$, to furnish the final
estimate

$\displaystyle\max\\{|\xi|,|\eta|\\}\cdot|\pi_{\xi}^{\perp}(\eta_{\circ})|$
$\displaystyle\leq\Big{[}(|\xi|^{2}+|\eta|^{2})\cdot|\pi_{\xi}^{\perp}(\eta_{\circ})|^{2}\Big{]}^{1/2}\overset{\eqref{eq:PiPerpEst}}{=}\Big{[}(|\xi|^{2}+|\eta|^{2})\cdot(1-|\langle\xi_{\circ},\eta_{\circ}\rangle|^{2})\Big{]}^{1/2}$
$\displaystyle\leq\Big{[}(|\xi|^{2}+|\eta|^{2})\cdot
2\cdot(1-|\langle\xi_{\circ},\eta_{\circ}\rangle|)\Big{]}^{1/2}\overset{\eqref{eq:NormDifferenceInnerProductConnection}}{\leq}\sqrt{2}\cdot|\xi-\eta|\,.$

∎

To verify Property (4.5) of Definition 4.2, we need to control certain
derivatives of the (matrix-valued) function

$\phi_{\tau}\left({\scaleobj{0.65}{\Upsilon}}\right)=\left(A^{-1}(\tau)\cdot
A({\scaleobj{0.65}{\Upsilon}}+\tau)\right)^{T}\quad\text{with}\quad
A(\tau)=\mathrm{D}\Phi_{\varrho}^{-1}(\tau)$ (8.13)

from (4.4). To this end, we will frequently use _Faa di Bruno’s formula_ , a
chain rule for higher derivatives. Precisely, we will use the following form
of the formula, which is a slightly simplified (but less precise) version of
[28, Corollary 2.10]. Note that, for a nonnegative multiindex $\alpha$, i.e.,
$\alpha\in\mathbb{N}_{0}^{d}$, we denote the sum of its components by
$|\alpha|\geq 0$ and by $\alpha=0$ we refer to the unique multiindex with
$|\alpha|=0$.

######  Lemma 8.6.

For $\alpha\in\mathbb{N}_{0}^{d}\setminus\\{0\\}$ and
$n\in\underline{|\alpha|}$, set

$\Gamma_{\alpha,n}:=\left\\{\gamma=(\gamma_{1},\dots,\gamma_{n})\in\Big{[}\mathbb{N}_{0}^{d}\setminus\\{0\\}\Big{]}^{n}\,:\,\smash{\sum_{j=1}^{n}}\vphantom{\sum}\gamma_{j}=\alpha\right\\}\,.$

Furthermore, set
$\Gamma:=\bigcup_{\alpha\in\mathbb{N}_{0}^{d}\setminus\\{0\\}}\bigcup_{n=1}^{|\alpha|}\Gamma_{\alpha,n}$.

Then, for each $\gamma\in\Gamma$, there is a constant
$C_{\gamma}\in\mathbb{R}$ such that for any open sets $U\subset\mathbb{R}^{d}$
and $V\subset\mathbb{R}$, and any $\mathcal{C}^{k}$ functions
$f:V\to\mathbb{R}$ and $g:U\to V$, the following holds for any
$\alpha\in\mathbb{N}_{0}^{d}$ with $|\alpha|\in\underline{k}$:

$\partial^{\alpha}(f\circ
g)(x)=\sum_{n=1}^{|\alpha|}\left[f^{(n)}(g(x))\cdot\sum_{\gamma\in\Gamma_{\alpha,n}}\bigg{(}C_{\gamma}\cdot\prod_{j=1}^{n}(\partial^{\gamma_{j}}g)(x)\bigg{)}\right]\qquad\forall\,x\in
U\,,$

where $f^{(n)}$ denotes the $n$-th derivative of $f$.

######  Remark.

From the statement of [28, Corollary 2.10], it might appear that the constants
$C_{\gamma}$ also depend on $\alpha,n,d$, in addition to $\gamma$. But these
parameters are determined by $\gamma$: On the one hand, we have
$\gamma\in[\mathbb{N}_{0}^{d}]^{n}$, which uniquely determines $n$ and $d$. On
the other hand, $\alpha=\sum_{j=1}^{n}\gamma_{j}$ for
$\gamma\in\Gamma_{\alpha,n}$.

With these preparations, we can now prove that the radial warping function
$\Phi_{\varrho}$ associated to a $k$-admissible radial component $\varrho$ is
indeed a $k$-admissible warping function. Most significantly, the following
proposition proves that Property (4.5), stated in Definition 4.2, is satisfed.

######  Proposition 8.7.

Let $\varrho:\mathbb{R}\to\mathbb{R}$ be a $k$-admissible radial component
with control weight $v:\mathbb{R}\\!\to\\!(0,\infty)$, and with
$c,\varepsilon,C_{0},C_{1}$ as in Definition 8.1.

Then there is a constant $C=C(c,\varepsilon,C_{0},C_{1},v,k)\geq 1$ such that
with

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

the function $\Phi_{\varrho}$ satisfies (4.5) for all
$\alpha\in\mathbb{N}_{0}^{d}$ with $|\alpha|\leq k$.

###### Proof.

It is easy to see that $v_{0}$ is submultiplicative and radially increasing as
the product of submultiplicative and radially increasing weights $C$,
$(1+|\bullet|)$ and $v(|\bullet|)$.

The remainder of the proof is divided into five steps. As a preparation for
these, recall from Lemma 8.4 that
$\Phi_{\varrho}^{-1}=\Phi_{\varrho_{\ast}}=(\bullet)\cdot\widetilde{\varrho_{\ast}}(|\bullet|)$,
with $\varrho_{\ast}=\varrho^{-1}$ and $\widetilde{\varrho_{\ast}}$ as defined
in (8.1). By Lemma 8.4,
$\widetilde{\varrho_{\ast}}\in\mathcal{C}^{k+1}(\mathbb{R})$. Our main goal is
to estimate the derivatives of $\Phi_{\varrho_{\ast}}$.

Step 1 - Estimate the derivatives of $\widetilde{\varrho_{\ast}}$: A trivial
induction shows $\frac{d^{\ell}}{dt^{\ell}}t^{-1}=(-1)^{\ell}\cdot\ell!\cdot
t^{-(1+\ell)}$. With this, Leibniz’s rule shows for any $n\in\underline{k+1}$
and any $t\in[c\varepsilon,\infty)$ that

$\displaystyle\left|\widetilde{\varrho_{\ast}}^{(n)}(t)-\frac{\varrho_{\ast}^{(n)}(t)}{t}\right|=\left|\sum_{\ell=1}^{n}\binom{n}{\ell}\cdot\frac{d^{\ell}t^{-1}}{dt^{\ell}}\cdot\varrho_{\ast}^{(n-\ell)}(t)\right|$
$\displaystyle\leq
C^{(1)}\cdot\sum_{\ell=1}^{n}t^{-(1+\ell)}\cdot|\varrho_{\ast}^{(n-\ell)}(t)|$
$\displaystyle\overset{\eqref{eq:AdmissibilityHigherDerivativeConsequence}}{\leq}C^{(2)}\cdot\sum_{\ell=1}^{n}t^{-\ell+1}\cdot\left(\frac{1+t}{t}\right)^{2}\cdot\frac{\widetilde{\varrho_{\ast}}(t)}{1+t}$
$\displaystyle({\scriptstyle{t^{-1}\leq(c\varepsilon)^{-1}\text{ and
}(1+t)/t=t^{-1}+1\leq(c\varepsilon)^{-1}+1}})$ $\displaystyle\leq
C^{(3)}\cdot\widetilde{\varrho_{\ast}}(t)/(1+t)\,,$ (8.14)

where $C^{(1)}=C^{(1)}(k)\geq 1$ is a suitable constant,
$C^{(2)}=C^{(1)}C_{2}$, with $C_{2}$ as in (8.4), and $C^{(3)}$ is given by
$C^{(3)}=C^{(2)}\max\\{1,(c\varepsilon)^{-(k+1)}\\}((c\varepsilon)^{-1}+1)^{2}$.

In particular, $|\widetilde{\varrho_{\ast}}^{(n)}(t)|\leq
C^{(3)}\cdot\widetilde{\varrho_{\ast}}(t)/(1+t)+|\varrho_{\ast}^{(n)}(t)/t|,$
such that, with $(1+t)/t\leq(c\varepsilon)^{-1}+1$, (8.4) also yields

$|\widetilde{\varrho_{\ast}}^{(n)}(t)|\leq
C^{(3)}\cdot\frac{\widetilde{\varrho_{\ast}}(t)}{1+t}+\left|\frac{\varrho_{\ast}^{(n)}(t)}{t}\right|\overset{\eqref{eq:AdmissibilityFirstDerivative},\eqref{eq:AdmissibilityHigherDerivative}}{\leq}C^{(3)}\cdot(C_{0}^{-1}+v(0))\cdot\frac{\varrho^{\prime}(t)}{t},$
(8.15)

and, with $C^{(4)}=2C^{(3)}$,

$|\widetilde{\varrho_{\ast}}^{(n)}(t)|\leq
C^{(3)}\cdot\widetilde{\varrho_{\ast}}(t)+C_{2}\cdot((c\varepsilon)^{-1}+1)\widetilde{\varrho_{\ast}}(t)\leq
C^{(4)}\cdot\widetilde{\varrho_{\ast}}(t),\qquad,$ (8.16)

for all $t\in[c\varepsilon,\infty)$ and $n\in\underline{k+1}$.

Step 2 - Estimate the partial derivatives of the function
$\vartheta:\mathbb{R}^{d}\setminus\\{0\\}\to(0,\infty),\tau\mapsto|\tau|\,$:
Note that a straightforward induction shows for
$h:(0,\infty)\to(0,\infty),t\mapsto t^{1/2}$ that

$h^{(n)}(t)=c_{n}\cdot t^{-n+1/2}\quad\text{with}\quad
c_{n}=2^{-n}\cdot(-1)^{n-1}\cdot\prod_{j=1}^{n-1}(2j-1)\qquad\forall\,n\in\mathbb{N}\,.$
(8.17)

Next, it is not hard to see for arbitrary $\alpha\in\mathbb{N}_{0}^{d}$ that

$\partial^{\alpha}|\tau|^{2}=\begin{cases}|\tau|^{2},&\text{if }\alpha=0,\\\
2\tau_{j},&\text{if }\alpha=e_{j}\text{ for some }j\in\underline{d},\\\
2,&\text{if }\alpha=2e_{j}\text{ for some }j\in\underline{d},\\\
0,&\text{otherwise},\end{cases}$

which implies $\big{|}\,\partial^{\alpha}|\tau|^{2}\,\big{|}\leq
2\cdot|\tau|^{2-|\alpha|}$ for all $\tau\in\mathbb{R}^{d}\setminus\\{0\\}$ and
$\alpha\in\mathbb{N}_{0}^{d}$.

Now, we can finally apply Faa di Bruno’s formula (see Lemma 8.6) to get for
arbitrary $\alpha\in\mathbb{N}_{0}^{d}\setminus\\{0\\}$ and
$\tau\in\mathbb{R}^{d}\setminus\\{0\\}$ that

$\big{|}\partial^{\alpha}\vartheta(\tau)\big{|}=\big{|}\partial^{\alpha}|\tau|\big{|}=\big{|}\partial^{\alpha}(h\circ|\bullet|^{2})(\tau)\big{|}=\left|\sum_{n=1}^{|\alpha|}h^{(n)}(|\tau|^{2})\cdot\sum_{\gamma\in\Gamma_{\alpha,n}}\bigg{(}C_{\gamma}\cdot\prod_{j=1}^{n}(\partial^{\gamma_{j}}|\bullet|^{2})(\tau)\bigg{)}\right|\,.$

But we have $\sum_{j=1}^{n}\gamma_{j}=\alpha$ for
$\gamma\in\Gamma_{\alpha,n}$, and hence,

$\left|\smash{\prod_{j=1}^{n}}\vphantom{\prod}(\partial^{\gamma_{j}}|\bullet|^{2})(\tau)\right|\leq
2^{n}\cdot\prod_{j=1}^{n}|\tau|^{2-|\gamma_{j}|}\leq
2^{|\alpha|}\cdot|\tau|^{2n-|\alpha|}.$

Furthermore, (8.17) shows $|h^{(n)}(|\tau|^{2})|\lesssim|\tau|^{1-2n}$, where
the implied constant only depends on $n\leq|\alpha|$. Overall, this yields

$\big{|}\partial^{\alpha}\vartheta(\tau)\big{|}=\big{|}\partial^{\alpha}|\tau|\big{|}\leq
C^{(5)}_{\alpha}\cdot|\tau|^{1-|\alpha|}\qquad\forall\,\tau\in\mathbb{R}^{d}\setminus\\{0\\}\text{
and }\alpha\in\mathbb{N}_{0}^{d},$ (8.18)

for some constants $C^{(5)}_{\alpha}=C^{(5)}_{\alpha}(d)$. The estimate is
trivial in case of $\alpha=0$.

Step 3 - Estimate the partial derivatives of
$\zeta:\mathbb{R}^{d}\setminus\\{0\\}\to\mathbb{R},\tau\mapsto\widetilde{\varrho_{\ast}}(|\tau|)$:
Note that this map is just the composition of $\widetilde{\varrho_{\ast}}$
with the map $\vartheta$ from the preceding step. Thus, Faa di Bruno’s formula
(see Lemma 8.6) shows for any $\alpha\in\mathbb{N}_{0}^{d}\setminus\\{0\\}$
with $|\alpha|\leq k+1$ and $\tau\in\mathbb{R}^{d}\setminus\\{0\\}$ that

$\partial^{\alpha}\zeta(\tau)=\sum_{n=1}^{|\alpha|}\left[\widetilde{\varrho_{\ast}}^{(n)}(|\tau|)\cdot\sum_{\gamma\in\Gamma_{\alpha,n}}\left(C_{\gamma}\cdot\smash{\prod_{j=1}^{n}}\vphantom{\prod}\,(\partial^{\gamma_{j}}\vartheta)(\tau)\right)\right]\,.$
(8.19)

In the previous step, we saw $|(\partial^{\gamma_{j}}\vartheta)(\tau)|\leq
C_{\gamma_{j}}^{(5)}\cdot|\tau|^{1-|\gamma_{j}|}$. Since
$\sum_{j=1}^{n}\gamma_{j}=\alpha$ for
$\gamma=(\gamma_{1},\dots,\gamma_{n})\in\Gamma_{\alpha,n}$, there thus exists
a constant $C_{\gamma}^{(6)}=C_{\gamma}^{(6)}(d,\gamma)$ such that

$\left|\prod_{j=1}^{n}(\partial^{\gamma_{j}}\vartheta)(\tau)\right|\leq
C_{\gamma}^{(6)}\cdot|\tau|^{n-|\alpha|}\qquad\forall\,\tau\in\mathbb{R}^{d}\setminus\\{0\\},\quad\alpha\in\mathbb{N}_{0}^{d}\setminus\\{0\\},\quad
n\in\underline{|\alpha|},\quad\text{and}\quad\gamma\in\Gamma_{\alpha,n}\,.$
(8.20)

Now, let us focus on the case $|\tau|\geq c\varepsilon$. Then, if
$n\in\underline{|\alpha|-1}$, the estimate (8.16) yields with
$|\tau|^{n-|\alpha|}\leq(c\varepsilon)^{n+1-|\alpha|}\cdot|\tau|^{-1}\leq(c\varepsilon)^{n+1-|\alpha|}(1+(c\varepsilon)^{-1})/(1+|\tau|)$
that

$\left|\widetilde{\varrho_{\ast}}^{(n)}(|\tau|)\cdot\sum_{\gamma\in\Gamma_{\alpha,n}}\\!\\!\\!\left(C_{\gamma}\cdot\smash{\prod_{j=1}^{n}}\vphantom{\prod}\,(\partial^{\gamma_{j}}\vartheta)(\tau)\right)\right|\leq
C^{(7)}\cdot\frac{\widetilde{\varrho_{\ast}}(|\tau|)}{1+|\tau|}\quad\forall\,\tau\in\mathbb{R}^{d}\setminus\overline{B_{c\varepsilon}}(0)\text{
and }n\in\underline{|\alpha|-1}\,.$ (8.21)

Here, $\alpha\in\mathbb{N}_{0}^{d}$ with $|\alpha|\in\underline{k+1}$ and
$C^{(7)}=C^{(7)}(C_{2},k,d,c,\varepsilon)$, with $C_{2}=C_{2}(C_{1},v)$ as in
(8.4).

Overall, by combining (8.19)–(8.21), we get

$\displaystyle|\partial^{\alpha}\zeta(\tau)|=\Big{|}\partial^{\alpha}\big{(}\widetilde{\varrho_{\ast}}(|\tau|)\big{)}\Big{|}$
$\displaystyle\leq
C^{(8)}\cdot\left(\Big{|}\widetilde{\varrho_{\ast}}^{(|\alpha|)}(|\tau|)\Big{|}+\frac{\widetilde{\varrho_{\ast}}(|\tau|)}{1+|\tau|}\right)$
(8.22)
$\displaystyle\overset{\eqref{eq:PsiTildeDerivativeEstimateEasyCase}}{\leq}C^{(9)}\cdot\widetilde{\varrho_{\ast}}(|\tau|)\quad\forall\,\alpha\in\mathbb{N}_{0}^{d}\,\text{
with }\,|\alpha|\leq k+1\,\text{ and }\,\tau\in\mathbb{R}^{d}\setminus
B_{c\varepsilon}(0)\,,$ (8.23)

with $C^{(8)}=C^{(8)}(C_{2},d,c,\varepsilon,k)$ and
$C^{(9)}=C^{(9)}(C_{2},d,c,\varepsilon,k)$. The case $\alpha=0$ is trivial.

Step 4 - Estimate $\partial^{\alpha}\phi_{\tau}({\scaleobj{0.65}{\Upsilon}})$
for $0\leq|\alpha|\leq k$ and
${\scaleobj{0.65}{\Upsilon}}\in\mathbb{R}^{d}\setminus
B_{c\varepsilon}(-\tau)$: Recall from (8.13) the definition of
$\phi_{\tau}({\scaleobj{0.65}{\Upsilon}})=\left(A^{-1}(\tau)\cdot
A(\tau+{\scaleobj{0.65}{\Upsilon}})\right)^{T}$. Since $\|M\|=\|M^{T}\|$ for
all $M\in\mathbb{R}^{d\times d}$, and since
$\partial^{\alpha}[M(\tau)]^{T}=[\partial^{\alpha}M(\tau)]^{T}$ for any
sufficiently smooth matrix-valued function
$M:\mathbb{R}^{d}\to\mathbb{R}^{d\times d}$, it is sufficient to estimate
$\|\partial^{\alpha}\varphi_{\tau}({\scaleobj{0.65}{\Upsilon}})\|$ with
$\varphi_{\tau}({\scaleobj{0.65}{\Upsilon}}):=A^{-1}(\tau)\cdot
A(\tau+{\scaleobj{0.65}{\Upsilon}})$, where
$A(\tau)=\mathrm{D}\Phi_{\varrho_{\ast}}(\tau)$.

Furthermore, $\|M\|\leq\sum_{j=1}^{d}|M_{\bullet,j}|$ for all
$M\in\mathbb{R}^{d\times d}$, such that it is sufficient to estimate the
columns of $M$ individually. In the following, we denote, for
$\alpha\in\mathbb{N}_{0}^{d}$ and
${\scaleobj{0.65}{\Upsilon}}\in\mathbb{R}^{d}$,
$\partial^{\alpha}_{\scaleobj{0.65}{\Upsilon}}=\frac{\partial^{|\alpha|}}{\partial^{\alpha_{1}}_{{\scaleobj{0.65}{\Upsilon}}_{1}}\,\cdots\,\partial^{\alpha_{d}}_{{\scaleobj{0.65}{\Upsilon}}_{d}}}.$
Let us fix $\tau\in\mathbb{R}^{d}\setminus\\{0\\}$. Then,
$\partial^{\alpha}\varphi_{\tau}({\scaleobj{0.65}{\Upsilon}})=A^{-1}(\tau)\cdot(\partial^{\alpha}A)(\tau+{\scaleobj{0.65}{\Upsilon}})$.
We see that the $j$-th column of
$\partial^{\alpha}\varphi_{\tau}({\scaleobj{0.65}{\Upsilon}})$ is simply

$\left[\partial^{\alpha}\varphi_{\tau}({\scaleobj{0.65}{\Upsilon}})\right]_{\bullet,j}=A^{-1}(\tau)\cdot\partial_{\scaleobj{0.65}{\Upsilon}}^{\alpha}[A(\tau+{\scaleobj{0.65}{\Upsilon}})]_{\bullet,j}=A^{-1}(\tau)\cdot(\partial_{\scaleobj{0.65}{\Upsilon}}^{\alpha+e_{j}}\Phi_{\varrho_{\ast}})(\tau+{\scaleobj{0.65}{\Upsilon}})\,.$
(8.24)

Now fix $j\in\underline{d}$, and set $\sigma:=\alpha+e_{j}$ for brevity. Note
$\sigma\in\mathbb{N}_{0}^{d}\setminus\\{0\\}$ with
$|\sigma|\in\underline{k+1}$.

By definition of $\Phi_{\varrho_{\ast}}$, the $i$-th entry of
$\Phi_{\varrho_{\ast}}(\tau)$ is
$[\Phi_{\varrho_{\ast}}(\tau)]_{i}=\tau_{i}\cdot\widetilde{\varrho_{\ast}}(|\tau|)$.
Let $\alpha_{+}$, for $\alpha\in\mathbb{Z}_{0}^{d}$, be the elementwise
positive part, i.e., $(\alpha_{+})_{i}=\max\\{0,\alpha_{i}\\}$,
$i\in\underline{d}$. The Leibniz rule, with $\partial^{\beta}\tau_{i}=0$ for
$\beta\notin\\{0,e_{i}\\}$ and $\partial_{i}\tau_{i}=1$, yields

$[\partial^{\sigma}\Phi_{\varrho_{\ast}}(\tau)]_{i}=\tau_{i}\cdot\big{[}\partial^{\sigma}(\widetilde{\varrho_{\ast}}(|\tau|))\big{]}+\sigma_{i}\cdot\partial_{\tau}^{(\sigma-
e_{i})_{+}}\big{(}\widetilde{\varrho_{\ast}}(|\tau|)\big{)}\qquad\forall\,i\in\underline{d}\,,$

or in other words,

$\partial^{\sigma}\Phi_{\varrho_{\ast}}(\tau)=\big{(}\partial_{\tau}^{\sigma}\big{[}\widetilde{\varrho_{\ast}}(|\tau|)\big{]}\big{)}\cdot\tau+v_{\sigma,\tau}\,,\quad\text{with}\quad
v_{\sigma,\tau}:=\left[\sigma_{i}\cdot\partial_{\tau}^{(\sigma-
e_{i})_{+}}\big{(}\widetilde{\varrho_{\ast}}(|\tau|)\big{)}\right]_{i=1,\dots,d}\,.$
(8.25)

Now, by (8.23), we have $|v_{\sigma,\tau}|\leq
C^{(10)}\cdot\widetilde{\varrho_{\ast}}(|\tau|)$, for all
$\tau\in\mathbb{R}^{d}\setminus B_{c\varepsilon}(0)$, with

$C^{(10)}=C^{(10)}(C^{(9)},k):=C^{(9)}(k+1)\geq C^{(9)}|\sigma|.$

Furthermore, Lemma 8.4 provides the estimate
$\|A^{-1}(\tau)\|=\|[\mathrm{D}\Phi_{\varrho_{\ast}}(\tau)]^{-1}\|\leq
C_{0}/\widetilde{\varrho_{\ast}}(|\tau|)$ with $C_{0}$ as in (8.2). Since
$\widetilde{\varrho_{\ast}}$ is $v$-moderate and $v$ is radially increasing,
this implies

$\begin{split}|A^{-1}(\tau)\cdot
v_{\sigma,\tau+{\scaleobj{0.65}{\Upsilon}}}|&\leq
C^{(10)}C_{0}\cdot\frac{\widetilde{\varrho_{\ast}}(|\tau+{\scaleobj{0.65}{\Upsilon}}|)}{\widetilde{\varrho_{\ast}}(|\tau|)}\\\
&\leq C^{(10)}C_{0}\cdot v(|{\scaleobj{0.65}{\Upsilon}}|)\leq
C^{(10)}C_{0}\cdot
v_{0}({\scaleobj{0.65}{\Upsilon}})\qquad\forall\,{\scaleobj{0.65}{\Upsilon}}\in\mathbb{R}^{d}\setminus
B_{c\varepsilon}(-\tau)\,.\end{split}$ (8.26)

Thus, in view of (8.25), it remains to estimate
$\big{(}\partial_{(\tau+{\scaleobj{0.65}{\Upsilon}})}^{\sigma}\big{[}\widetilde{\varrho_{\ast}}(|\tau+{\scaleobj{0.65}{\Upsilon}}|)\big{]}\big{)}\cdot
A^{-1}(\tau)\langle\tau+{\scaleobj{0.65}{\Upsilon}}\rangle$ for
$\tau+{\scaleobj{0.65}{\Upsilon}}\in\mathbb{R}^{d}\setminus
B_{c\varepsilon}(0)$.

Lemma 8.4 implies

$A^{-1}(\tau)=[\mathrm{D}\Phi_{\varrho_{\ast}}(\tau)]^{-1}=[\widetilde{\varrho_{\ast}}(|\tau|)]^{-1}\cdot\pi_{\tau}^{\perp}+[\varrho_{\ast}^{\prime}(|\tau|)]^{-1}\cdot\pi_{\tau}\,.$
(8.27)

Now, Lemma 8.5, together with (8.23) and $v$-moderateness of $\varrho_{\ast}$
for radially increasing $v$, shows

$\begin{split}&\big{|}\partial_{{\scaleobj{0.65}{\Upsilon}}}^{\sigma}\big{(}\widetilde{\varrho_{\ast}}(|\tau+{\scaleobj{0.65}{\Upsilon}}|)\big{)}\big{|}\cdot\big{[}\widetilde{\varrho_{\ast}}(|\tau|)\big{]}^{-1}\cdot|\pi_{\tau}^{\perp}(\tau+{\scaleobj{0.65}{\Upsilon}})|\\\
&\leq\sqrt{2}\cdot
C^{(9)}\cdot\big{[}\widetilde{\varrho_{\ast}}(|\tau|)\big{]}^{-1}\cdot\widetilde{\varrho_{\ast}}(|\tau+{\scaleobj{0.65}{\Upsilon}}|)\cdot|{\scaleobj{0.65}{\Upsilon}}|\\\
&\leq 2C^{(9)}\cdot
v(|{\scaleobj{0.65}{\Upsilon}}+\tau|-|\tau|)\cdot|{\scaleobj{0.65}{\Upsilon}}|\\\
({\scriptstyle{v_{0}\geq(1+|\bullet|)\cdot v(|\bullet|)}})&\leq 2C^{(9)}\cdot
v_{0}({\scaleobj{0.65}{\Upsilon}})\,.\end{split}$ (8.28)

Finally, with the elementary estimate
$|\pi_{\tau}({\scaleobj{0.65}{\Upsilon}}+\tau)|\leq|{\scaleobj{0.65}{\Upsilon}}+\tau|$

$\begin{split}&\big{|}\partial_{{\scaleobj{0.65}{\Upsilon}}}^{\sigma}\big{(}\widetilde{\varrho_{\ast}}(|{\scaleobj{0.65}{\Upsilon}}+\tau|)\big{)}\big{|}\cdot\left[\varrho_{\ast}^{\prime}(|\tau|)\right]^{-1}\cdot|\pi_{\tau}({\scaleobj{0.65}{\Upsilon}}+\tau)|\\\
({\scriptstyle{\eqref{eq:PsiTildeEuclideanTotalCase1}\text{ and
}\eqref{eq:PsiTildeDerivativeEstimateEasyCase2}}})&\leq
C^{(8)}C^{(3)}\cdot(C_{0}^{-1}+v(0))\cdot|{\scaleobj{0.65}{\Upsilon}}+\tau|\cdot\frac{\varrho_{\ast}^{\prime}(|{\scaleobj{0.65}{\Upsilon}}+\tau|)}{|{\scaleobj{0.65}{\Upsilon}}+\tau|\cdot\varrho_{\ast}^{\prime}(|\tau|)}\\\
&\leq C^{(11)}\cdot v(|{\scaleobj{0.65}{\Upsilon}}|)\leq C^{(11)}\cdot
v_{0}({\scaleobj{0.65}{\Upsilon}})\,,\end{split}$ (8.29)

with $C^{(11)}=C^{(8)}C^{(3)}\cdot(C_{0}^{-1}+v(0))$. Overall, combining
(8.24)–(8.29), we finally see

$\displaystyle\|\partial^{\alpha}\phi_{\tau}({\scaleobj{0.65}{\Upsilon}})\|=\|\partial^{\alpha}\varphi_{\tau}({\scaleobj{0.65}{\Upsilon}})\|$
$\displaystyle\leq
d\cdot\max_{j\in\underline{d}}\big{|}\big{[}\partial^{\alpha}\varphi_{\tau}({\scaleobj{0.65}{\Upsilon}})\big{]}_{\bullet,j}\big{|}$
$\displaystyle({\scriptstyle{\eqref{eq:RadialWarpingProductRulePreparation}}})$
$\displaystyle\leq
d\cdot\max_{j\in\underline{d}}\big{|}A^{-1}(\tau)\cdot(\partial^{\alpha+e_{j}}\Phi_{\varrho_{\ast}})(\tau+{\scaleobj{0.65}{\Upsilon}})\big{|}$
$\displaystyle({\scriptstyle{\eqref{eq:PhiProductRuleApplication}}})$
$\displaystyle\leq
d\cdot\max_{j\in\underline{d}}\left(\big{|}\partial_{{\scaleobj{0.65}{\Upsilon}}+\tau}^{\alpha+e_{j}}\big{(}\widetilde{\varrho_{\ast}}(|{\scaleobj{0.65}{\Upsilon}}+\tau|)\big{)}\big{|}\cdot|A^{-1}(\tau)\,({\scaleobj{0.65}{\Upsilon}}+\tau)|+|A^{-1}(\tau)\,v_{\alpha+e_{j},\tau+{\scaleobj{0.65}{\Upsilon}}}|\right)$
$\displaystyle({\scriptstyle{\eqref{eq:RadialWarpingAdmissibleMainStepTerm1}\text{--}\eqref{eq:RadialWarpingAdmissibleMainStepTerm3}}})$
$\displaystyle\leq C^{(12)}\cdot
v_{0}({\scaleobj{0.65}{\Upsilon}})\quad\text{for all
}{\scaleobj{0.65}{\Upsilon}}\in\mathbb{R}^{d}\setminus
B_{c\varepsilon}(-\tau)\text{ and }|\alpha|\leq k\,,$

with a suitable constant $C^{(12)}=C^{(12)}(C_{0},C_{2},d,c,\varepsilon,k)$.

Step 5 - Estimate $\partial^{\alpha}\phi_{\tau}({\scaleobj{0.65}{\Upsilon}})$
for $0\leq|\alpha|\leq k$ and ${\scaleobj{0.65}{\Upsilon}}\in
B_{c\varepsilon}(-\tau)$: By Lemma 8.4, $\varrho_{\ast}(t)=t/c$, and thus
$\widetilde{\varrho_{\ast}}(t)=c^{-1}$ for $t\in(-c\varepsilon,c\varepsilon)$.
Hence, $\Phi_{\varrho_{\ast}}(\tau)=c^{-1}\cdot\tau$ for all $\tau\in
B_{c\varepsilon}(0)$, so that
$A(\tau)=\mathrm{D}\Phi_{\varrho_{\ast}}(\tau)=c^{-1}\cdot\mathrm{id}_{\mathbb{R}^{d}}$
for $\tau\in B_{c\varepsilon}(0)$.

Hence,
$\phi_{\tau}({\scaleobj{0.65}{\Upsilon}})=A^{T}(\tau+{\scaleobj{0.65}{\Upsilon}})\cdot
A^{-T}(\tau)=c^{-1}\cdot A^{-T}(\tau)$, whence
$\|\partial^{\alpha}\phi_{\tau}({\scaleobj{0.65}{\Upsilon}})\|=0\leq
v_{0}({\scaleobj{0.65}{\Upsilon}})$ for ${\scaleobj{0.65}{\Upsilon}}\in
B_{c\varepsilon}(-\tau)$ and $\alpha\in\mathbb{N}_{0}^{d}$ with
$|\alpha|\in\underline{k}$. For $\alpha=0$, Eq. (8.10) in Lemma 8.4 shows

$\displaystyle\|\phi_{\tau}({\scaleobj{0.65}{\Upsilon}})\|=c^{-1}\cdot\|A^{-T}(\tau)\|=c^{-1}\cdot\|[\mathrm{D}\Phi_{\varrho_{\ast}}(\tau)]^{-1}\|\leq
C^{(10)}C_{0}\cdot c^{-1}/\,\widetilde{\varrho_{\ast}}(|\tau|)\,,$

with the constant $C^{(10)}C_{0}$ from Step 4. But since
$\widetilde{\varrho_{\ast}}$ is $v$-moderate, we have
$c^{-1}\\!=\\!\widetilde{\varrho_{\ast}}(0)\leq\widetilde{\varrho_{\ast}}(|\tau|)\cdot
v(|\tau|)\,,$ and finally $|\tau|\leq
c\varepsilon+|{\scaleobj{0.65}{\Upsilon}}|$, such that $v(|\tau|)\leq
v(c\varepsilon)\cdot v(|{\scaleobj{0.65}{\Upsilon}}|)$. Altogether,
$\|\phi_{\tau}({\scaleobj{0.65}{\Upsilon}})\|\leq C^{(10)}C_{0}\cdot
v(c\varepsilon)\cdot v(|{\scaleobj{0.65}{\Upsilon}}|)$ for all
$\tau\in\mathbb{R}^{d}$ and ${\scaleobj{0.65}{\Upsilon}}\in
B_{c\varepsilon}(-\tau)$. ∎

That every radial warping function associated to a $k$-admissible radial
component $\varrho$ is indeed a $k$-admissible warping function is now a
straightforward corollary.

######  Corollary 8.8.

Let $\varrho:\mathbb{R}\to\mathbb{R}$ be a $k$-admissible radial component,
for some $k\in\mathbb{N}$ with $k\geq d+1$. The associated radial warping
function $\Phi_{\varrho}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is a $k$-admissible
warping function, with control weight $v_{0}$ chosen as in Proposition 8.7.
Furthermore, the weight $w=\det(\mathrm{D}\Phi_{\varrho}^{-1})$ is given by

$w(\tau)=\varrho_{\ast}^{\prime}(|\tau|)\cdot[\widetilde{\varrho_{\ast}}(|\tau|)]^{d-1}\,.$
(8.30)

###### Proof.

Lemma 8.4 shows that $\Phi_{\varrho}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is a