# Finite width effects on the azimuthal asymmetry in proton-nucleus collisions
in the Color Glass Condensate

Pedro Agostini<EMAIL_ADDRESS>Tolga Altinoluk
<EMAIL_ADDRESS>Néstor Armesto<EMAIL_ADDRESS>Theoretical
Physics Division, National Centre for Nuclear Research, Pasteura 7, Warsaw
02-093, Poland  Instituto Galego de Física de Altas Enerxías IGFAE,
Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Galicia-
Spain

###### Abstract

We perform a numerical analysis of the two particle azimuthal correlations at
central rapidities generated in $p$A collisions within the framework of the
Color Glass Condensate. We extend the standard computations to include the
subeikonal corrections which stem from considering the finite longitudinal
width of the dense target. For practical reasons, we only consider the next-
to-next-to-eikonal corrections instead of using the all-order expressions for
the inclusive two gluon production cross section. We show that the subeikonal
terms that we account for in the two gluon yields contribute to both even and
odd harmonics, the latter being absent in the standard Color Glass Condensate
calculations performed at eikonal accuracy. Our analysis confirms the
vanishing of the subeikonal effects with increasing collision energy and when
going to forward rapidities, as expected.

††journal: Physics Letters B

##  1 Introduction

Experimental analyses of particle correlations performed in collision systems
from large to small: heavy ions, proton-nucleus ($p$A) and proton-proton
($pp$), showed an asymmetry of the azimuthal distribution of the produced
particles. Such asymmetry persists even for large rapidity difference of the
particles, for which it has been named the ridge. This finding is among the
key ones that demonstrate the existence of collective effects in small systems
[1, 2, 3, 4, 5]. These collective effects can be explained either by final
state effects, where an anisotropic hydrodynamical flow is generated due to
spatial asymmetries which result in momentum ones (see e.g. [6, 7]), or by
initial state effects, where azimuthal correlations of the produced particles
are seen as a result of the momentum correlations in the wave function of the
colliding particles (see the review [8]). Since the experiments involving
azimuthal anisotropies probe the low-$x$ region of the nuclear target, the
explanations based on the initial stage of the collision are usually performed
within the Color Glass Condensate (CGC) effective theory [9, 10].

In the present manuscript we focus on the azimuthal angle dependence of the
two-gluon spectrum at central rapidities generated in a $p$A collision within
the CGC. The two-particle azimuthal distribution of particles with transverse
momenta ${\bf k}_{1}$ and ${\bf k}_{2}$ and rapidities $\eta_{1}$, $\eta_{2}$
is quantified through its Fourier harmonics that we define as

$\displaystyle v_{n}^{2}(p_{\perp},p_{\perp}^{\rm
ref})=\frac{\int\frac{d^{2}N}{d\eta_{1}d^{2}{\bf k}_{1}d\eta_{2}d^{2}{\bf
k}_{2}}e^{in(\phi_{2}-\phi_{1})}d\phi_{1}d\phi_{2}}{\int\frac{d^{2}N}{d\eta_{1}d^{2}{\bf
k}_{1}d\eta_{2}d^{2}{\bf
k}_{2}}d\phi_{1}d\phi_{2}}\Bigg{|}_{\begin{subarray}{c}|{\bf
k}_{1}|=p_{\perp}\\\ |{\bf k}_{2}|=p_{\perp}^{\rm ref}\end{subarray}},$ (1)

where $\phi_{i}$ is the azimuthal angle of the $i^{\rm th}$ particle,
$d^{2}{\bf k}_{i}=|{\bf k}_{i}|d|{\bf k}_{i}|d\phi_{i}$ and
$\frac{d^{2}N}{d\eta_{1}d^{2}{\bf k}_{1}d\eta_{2}d^{2}{\bf k}_{2}}$ is the
two-particle spectrum. Note that other definitions exist but the exact
definition of the harmonics is not relevant for our purposes.

In high energy collisions, where the small-$x$ tail of the projectile or
target wave function is probed, the CGC offers a nonperturbative but weak
coupling description of the system. In this regime where parton densities in
the target are saturated, the projectile is assumed to be populated by almost
collinear gluons that scatter eikonally on a dense and Lorentz contracted
target. It is well known that in this scenario the scattering matrix for a
gluon moving in the plus light-cone direction is described by a Wilson line in
the adjoint representation,

$\displaystyle\mathcal{U}_{[x^{+},y^{+}]}({\bf
x})=\mathcal{P}^{+}\exp\left\\{-ig\int_{y^{+}}^{x^{+}}dz^{+}A^{-}(z^{+},{\bf
x})\right\\},$ (2)

which represents the path ordered propagation of a gluon, from $y^{+}$ to
$x^{+}$ at a fixed transverse position ${\bf x}$, in a classical field,
$A^{\mu}(x)\equiv T^{a}A^{\mu}_{a}(x)$, dominated by its minus component. In
standard CGC calculations, the reality of the adjoint representation implies
that the leading order two-gluon spectrum has an accidental ${\bf
k}_{i}\rightarrow-{\bf k}_{i}$ symmetry which prevents the generation of odd
azimuthal harmonics (see e.g. the discussion in [11]). This absence of odd
harmonics is, of course, not consistent with experimental results.

Among several proposed solutions to this problem [8], it has been seen
recently that relaxing the eikonal approximation by taking into account a
finite target width, the scattering amplitude is no longer real and therefore
the two-gluon azimuthal distribution entails odd harmonics [12, 13]. This
property has been proven in the case of a dilute target, within the glasma
graph approximation suitable for $pp$ collisions, where the scattering
amplitude is modified by a non-eikonal phase $-ik^{-}L^{+}$ containing the
target length along the plus light-cone direction $L^{+}$ and the minus light-
cone momentum of the produced gluon $k^{-}$.

In this manuscript our aim is to generalize the numerical study of finite
width target effects on two-particle azimuthal correlations performed in [13]
for $pp$ collisions to the case of $p$A collisions, where the target is
modelled as a dense gluonic ensemble with a finite (light-cone) width $L^{+}$.
The framework for including these corrections has been presented by us in a
recent work [14] where we have shown the equivalence to the standard approach
for jet quenching (see the reviews [15, 16, 17]) where the final state gluon
acquires its transverse momentum through a random walk inside the target
medium. In this case, in contrast to the eikonal one, the scattering matrix is
non-diagonal in transverse position and it is written in terms of the scalar
gluon propagator,

$\displaystyle\mathcal{G}_{k^{+}}(x^{+},{\bf x};y^{+},{\bf y})=\int_{{\bf
z}(y^{+})={\bf y}}^{{\bf z}(x^{+})={\bf x}}[\mathcal{D}{\bf
z}(z^{+})]\exp\left\\{\frac{ik^{+}}{2}\int_{y^{+}}^{x^{+}}dz^{+}\dot{{\bf
z}}^{2}\right\\}\mathcal{U}_{[x^{+},y^{+}]}({\bf z}),$ (3)

which represents the Brownian motion of a gluon with finite longitudinal
momentum111Note that the eikonal approximation is recovered taking
$k^{+}\to\infty$. $k^{+}$ inside a classical gauge field $A^{-}(z^{+},{\bf
z})$ from transverse position ${\bf y}$ at $y^{+}$ to ${\bf x}$ at $x^{+}$.
The method for computing multi-gluon production, based on the McLerran-
Venugopalan (MV) and Area Enhancement (AE) models, is given in [14, 18].
However, in this work we will restrict ourselves to the numerical analysis of
two-gluon production. As usual in CGC calculations, we assume that
correlations between the final particles reflect those of the produced gluons
and are not washed out by final state effects or hadronisation. Our aim here
is not to attempt any description of experimental data, as we assume that our
model does not contain the full ingredients required for it, but to examine
where the contribution from non-eikonal effects on odd harmonics is sizable
and where the approximations that we employ break.

This manuscript is structured as follows: In section 2 we review very briefly
the formalism developed in [14], with detailed formulae provided in A. In
section 3 we describe the numerical results, while our conclusions and outlook
are given in section 4.

##  2 Two-gluon production beyond the shockwave approximation

The building blocks of the multi-gluon spectrum in $p$A collisions within the
eikonal approximation are colour neutral objects known as multipoles. In
particular, in the case of two-gluon production the building blocks are the
dipole function, $\tfrac{1}{N_{c}^{2}-1}\langle{\rm
Tr}[\mathcal{U}\mathcal{U}^{\dagger}]\rangle$, the double dipole function,
$\tfrac{1}{(N_{c}^{2}-1)^{2}}\langle{\rm
Tr}[\mathcal{U}\mathcal{U}^{\dagger}]\,{\rm
Tr}[\mathcal{U}\mathcal{U}^{\dagger}]\rangle$, and the quadrupole function,
$\tfrac{1}{N_{c}^{2}-1}\langle{\rm
Tr}[\mathcal{U}\mathcal{U}^{\dagger}\mathcal{U}\mathcal{U}^{\dagger}]\rangle$,
where $N_{c}$ is the dimension of the colour gauge group and
$\langle\cdots\rangle$ denotes the average over all possible configurations of
colour charge density inside the target. Within the AE model [14, 18, 19, 20,
21], we can assume that the Wilson lines follow a Gaussian distribution and
therefore, using the Wick theorem, the multipoles can be written in terms of
dipole functions.

As discussed in section 1, if we relax the eikonal approximation by including
a finite width222Strictly speaking this is a relaxation of the shockwave
approximation, $L^{+}\to 0$, since a proper calculation beyond eikonality is
more involved: it should also take into account the $x^{-}$ dependence [22]
and the perpendicular component [23, 24] of the gluon field, classical quark
fields in the target wave function, quantum corrections to the classical
fields and non light-like components of the initial currents. for the target,
$L^{+}$, the scattering matrix – or equivalently the scalar gluon propagator
within a classical medium, $\mathcal{G}$ – becomes a more involved object than
a Wilson line. Thus, within the AE model, the non-eikonal multi-gluon spectrum
will be described by three building blocks that generalize the dipole function
for a finite width target. These objects are
$\tfrac{1}{N_{c}^{2}-1}\langle{\rm
Tr}[\mathcal{U}\mathcal{U}^{\dagger}]\rangle$,
$\tfrac{1}{N_{c}^{2}-1}\langle{\rm
Tr}[\mathcal{G}\mathcal{U}^{\dagger}]\rangle$ and
$\tfrac{1}{N_{c}^{2}-1}\langle{\rm
Tr}[\mathcal{G}\mathcal{G}^{\dagger}]\rangle$. As indicated in eq. 3, the
generalizations of the dipole function are given by path integrals. In [14] we
have computed these objects in the case of a small dipole, $|{\bf x}-{\bf
y}|\ll\Lambda_{\rm QCD}^{-1}$, where the correlator of two Wilson lines can be
written as a Gaussian function and the resulting harmonic oscillator-like path
integrals can be solved analytically. In A we summarize the results obtained
within this approach.

The non-eikonal two-gluon yield depends, apart from the transverse momenta
${\bf k}_{i}$ ($i=1,2$), on the longitudinal momenta $k_{i}^{+}$ of the
produced particles, on the target longitudinal width $L^{+}$ (assuming that
the gauge field has support within a region $z^{+}\in[0,L^{+}]$ of the
longitudinal space) and on the saturation momentum $Q_{s}$, which stems from
the application of the MV [25, 26, 27] model to the target ensemble. Thus, it
is natural to define the non-eikonal parameter333Note that the definition in
this work is different than the one that we gave in [14] where the non-eikonal
parameter read  $\epsilon_{i}=\sqrt{\frac{iQ_{s}^{2}L^{+}}{2k_{i}^{+}}}.$
Since it is real valued and does not involve a square root, the definition in
eq. 4 is more convenient for a numerical or perturbative analysis.

$\epsilon_{i}=\frac{Q_{s}^{2}L^{+}}{2k_{i}^{+}}.$ (4)

When $k_{i}^{+}\to\infty$ the non-eikonal parameter vanishes and we recover
the eikonal result. Since the transverse momenta acquired by the produced
gluons are of order $Q_{s}$, we can interpret $2k_{i}^{+}/Q_{s}^{2}$ as their
correlation length, $\lambda_{i}^{+}$. Thus we can write $\epsilon_{i}\sim
L^{+}/\lambda_{i}^{+}$ so that the eikonal approximation is recovered when
$\lambda_{i}^{+}\gg L^{+}$ as we should expect. We also note that, as
discussed in [14], corrections to the AE model start to be sizable when
$\epsilon_{i}>1$.

If we work in the center-of-mass (CoM) frame, the non-eikonal parameter can be
written in terms of the collision CoM energy $\sqrt{s}$ and the
pseudorapidities $\eta_{i}$ of the produced gluons. This representation in
terms of measured quantities is more convenient for phenomenological analyses.
In order to do so we note that, in the CoM frame, the Lorentz-contracted
target width can be written as $L^{+}=2r_{A}/(\gamma\sqrt{2})$, where
$r_{A}\sim 5A^{1/3}\ {\rm GeV}^{-1}$ is the nuclear radius, $A$ the mass
number, $\gamma=\sqrt{s}/{(2m_{N})}$ the Lorentz factor and $m_{N}\sim 1$ GeV
the nucleon mass. On the other hand, the longitudinal momentum of the on-shell
produced gluons can be expressed in terms of their pseudorapidities as
$k_{i}^{+}=|{\bf k}_{i}|e^{\eta_{i}}/\sqrt{2}$. Thus we can write

$\epsilon_{i}\sim 10\frac{Q_{s}^{2}A^{1/3}}{|{\bf
k}_{i}|\sqrt{s}}e^{-\eta_{i}}.$ (5)

In fig. 1 we plot this parameter as a function of $\sqrt{s}$ and $\eta$ for
three values of the gluon transverse momentum $p_{\perp}\equiv|{\bf k}|$,
where we have chosen a target with $Q_{s}=1$ GeV and $A^{1/3}=6$. We see that,
for $p_{\perp}\gtrsim 1$ GeV, $\epsilon\lesssim 1$ in most of the kinematic
region probed at the Relativistic Heavy Ion Collider (RHIC),
$\sqrt{s}\in[20,200]$ GeV and $\eta\in[-4,4]$, and the Electron Ion Collider
(EIC)444While the EIC will study electroproduction on protons and nuclei
rather than $p$A collisions, we expect that the general conclusions extracted
here hold there, and an analysis similar to the one in this work can be done
in DIS processes., $\sqrt{s}\in[30,90]$ GeV and $\eta\in[-4,4]$. However, when
$p_{\perp}\sim 0.5$ GeV we have that $\epsilon\sim 1$ in a large region of the
kinematic space and the present analysis would not be justifiable. From fig. 1
we conclude that our analysis should be valid at momenta $p_{\perp}\gtrsim 1$
GeV and CoM energies $\sqrt{s}\gtrsim 50$ GeV.

Figure 1: Dependence of the non-eikonal parameter, eq. 5, on $p_{\perp}$,
$\eta$ and $\sqrt{s}$ with a target defined by $Q_{s}=1$ GeV and $A^{1/3}=6$.
We see that the analysis performed in this work is valid, i.e.,
$\epsilon\lesssim 1$, when $p_{\perp}\gtrsim 1$ GeV and $\sqrt{s}\gtrsim 50$
GeV.

The two-gluon spectrum given in appendix A has a complicated non linear
dependence on the parameters $\epsilon_{1}$ and $\epsilon_{2}$ which makes the
evaluation of the azimuthal harmonics, eq. 1, computationally expensive. In
order to make the numerical analysis simpler we have decided to expand
appendix A perturbatively in terms of $\epsilon_{i}$. This expansion is
justified as long as $\epsilon_{i}\ll 1$ which is the case in the kinematic
region probed in this work.

Thus we expand appendix A at the first non trivial order555The first order
term in the expansion in $\epsilon_{i}$ is known [28, 29] to give a vanishing
result for the single inclusive spectra; the same happens for the double
inclusive. This observation is only true for the non-eikonal contributions
that stem from finite size corrections, which are the ones considered in this
work., i.e., at second order in $\epsilon_{i}$. We refer to this approximation
as the Next-to-Next-to-Eikonal (NNE) order and it reads

$\displaystyle N_{2}(\epsilon_{1},\epsilon_{2})\big{|}_{\rm
NNE}=N_{2}(0,0)+\frac{\epsilon_{1}^{2}}{2}\frac{\partial^{2}N_{2}(\epsilon_{1},0)}{\partial^{2}\epsilon_{1}}\Bigg{|}_{\epsilon_{1}=0}+\frac{\epsilon_{2}^{2}}{2}\frac{\partial^{2}N_{2}(0,\epsilon_{2})}{\partial^{2}\epsilon_{2}}\Bigg{|}_{\epsilon_{2}=0}+\epsilon_{1}\epsilon_{2}\frac{\partial
N_{2}(\epsilon_{1},\epsilon_{2})}{\partial\epsilon_{1}\partial\epsilon_{2}}\Bigg{|}_{\epsilon_{1}=\epsilon_{2}=0},$
(6)

where we have introduced the shorthand notation
$N_{2}(\epsilon_{1},\epsilon_{2})\equiv\tfrac{d^{2}N}{dk_{1}^{+}d^{2}{\bf
k}_{1}dk_{2}^{+}d^{2}{\bf k}_{2}}$.

In order to double check the validity of the NNE expansion, in fig. 2 we plot
the relative difference of the single particle spectrum666Ideally we should
study the relative difference of the NNE expansion of
$N_{2}(\epsilon_{1},\epsilon_{2})$ with the full result. However, such
analysis is more involved numerically. We expect a similar convergence for the
case of two-gluon production since in appendix A the dominant term is the
single gluon spectrum squared while the other ones are suppressed by powers of
$N_{c}^{2}-1$. at NNE accuracy, $N_{1}(\epsilon)|_{\rm NNE}$, with the full
non-eikonal result at different values of $p_{\perp}$, $\sqrt{s}$ and $\eta$
with a target defined by $Q_{s}=1$ GeV and $A^{1/3}=6$. We observe a better
convergence than the one predicted by fig. 1 since the relative difference is
smaller than 20 % in all the kinematic region studied in this work.

Figure 2: Relative difference between the NNE expansion of the single particle
spectrum and its full non-eikonal result as a function of $p_{\perp}$,
$\sqrt{s}$ and $\eta$, at a fixed value of $Q_{s}=1$ GeV and $A^{1/3}=6$. The
oscillations in the difference in the plot on the right come from the
oscillatory nature of the full result in some kinematic regions.

##  3 Numerical results

In this section we present the numerical results for the azimuthal harmonics,
eq. 1, extracted from the double gluon spectrum at NNE accuracy, eq. 6, for
different values of $\epsilon_{1}$ and $\epsilon_{2}$. We fix $p_{\perp}^{\rm
ref}=1$ GeV in the whole analysis.

Before evaluating eq. 6, we have to deal with the infrared (IR) divergences
that appear in the factors of appendix A. In order to do so we introduce an IR
regulator, $m_{g}=0.4$ GeV, in the denominators by making the change $1/{\bf
q}_{i}^{2}\to 1/({\bf q}_{i}^{2}+m_{g}^{2})$ in appendix A. We have checked
that the result does not depend sizably under small variations of the
parameter $m_{g}$.

In fig. 3 we plot the azimuthal harmonics $v_{2}$ and $v_{3}$ computed from
eq. 6 within a range $0.5\ {\rm GeV}<p_{\perp}<2\ {\rm GeV}$ for three values
of $\sqrt{s}$ and at fixed pseudorapidities $\eta_{1}=0.1$ and $\eta_{2}=0.5$.
We have chosen a nucleus with $Q_{s}=1$ GeV and $A^{1/3}=6$. We also compare
the result with the one that we get in the eikonal approximation
($\epsilon_{1}=\epsilon_{2}=0$) where, because of the accidental symmetry,
$v_{3}=0$. We see that, in the non-eikonal case, we obtain a non vanishing odd
azimuthal harmonic $v_{3}$ whose value becomes smaller at higher CoM energies
as we should expect since we asymptotically approach the eikonal
approximation. Besides, we should note that the result when $\sqrt{s}=50$ GeV
has to be taken with care around the region with $p_{\perp}\lesssim 0.5$ GeV
where the expansion parameter is not small.

Figure 3: Dependence of the azimuthal harmonics, $v_{2}$ and $v_{3}$ on the
transverse momentum at three values of the CoM energy. The bands indicate the
estimated numerical error of the integration. For this plot we have fixed
$\eta_{1}=0.1$, $\eta_{2}=0.5$, $Q_{s}=1$ GeV and $A^{1/3}=6$. We also compare
the result with the one obtained in the eikonal approximation. In contrast to
the eikonal case, we obtain a non vanishing $v_{3}$ that decreases as we
increase the CoM energy of the collision. In the plot in the middle the point
corresponding to $v_{3}$ at $p_{\perp}=1.8$ GeV is not shown as its numerical
uncertainty is too large for it to be meaningful.

In fig. 4 we plot the dependence of the azimuthal harmonics with the CoM
energy at fixed $\eta_{1}=0.1$ and $\eta_{2}=0.5$, and the dependence with the
pseudorapidity of one of the gluons at fixed $\sqrt{s}=70$ GeV and
$\eta_{2}=1$. Both plots show results computed with at a fixed value of
$p_{\perp}=1$ GeV and compared with the eikonal result. We see that as we
increase the value of the pseudorapidity or the CoM energy, i.e., as we
asymptotically approach the eikonal limit, the value of $v_{3}$ decreases and
$v_{2}$ approaches $v_{2}^{\rm(eik)}$. We observe an unexpected behaviour of
$v_{2}$, approaching the eikonal value, for small $\eta_{1}$ (left plot) or
$\sqrt{s}$ (right plot) that we attribute to the increasing value of
$\epsilon_{i}$ which deteriorates the validity of the NNE expansion in these
kinematics.

Figure 4: Left: dependence of the azimuthal harmonics on the pseudorapidity of
one of the gluons $\eta\equiv\eta_{1}$ at a fixed value of $\sqrt{s}=70$ GeV
and $\eta_{2}=1$. Right: dependence of the azimuthal harmonics on the CoM
energy at a fixed value of $\eta_{1}=0.1$ and $\eta_{2}=0.5$. The bands
indicate the estimated numerical error of the integration. In both plots we
have fixed $p_{\perp}=1$ GeV, $Q_{s}=1$ GeV and $A^{1/3}=6$.

##  4 Conclusions and outlook

In this work we have quantified the effect of the non-eikonal corrections that
come from considering a target with a finite longitudinal size, on the
azimuthal asymmetries of produced particles in $p$A collisions at central
rapidities. We generalise the CGC expressions in order to include these
corrections, as developed in [14], and check that in $p$A collisions odd
azimuthal harmonics, absent in standard eikonal computations, are indeed
present. For practical reasons we do not consider the full non-eikonal
expressions but truncate them to next-to-next-to eikonal accuracy, which is
the lowest order that provides non-trivial results. We check the validity of
such truncation and provide the final results for the second and third order
azimuthal harmonics. We observe the vanishing of the non-eikonal effects with
increasing collision energy and when going to forward rapidities, as expected
and observed in previous works for $pp$ collisions [13].

We note that, as usual in CGC calculations, we assume that correlations
between the final particles reflect those of the produced gluons and are not
washed out by final state effects or hadronisation. We do not to attempt any
description of experimental data, as we assume that our model does not contain
the full ingredients required for it.

The numerical analysis in this work is performed for $p$A collisions and
therefore should be applicable to the low energy regime of the Relativistic
Heavy Ion Collider. As an immediate outlook of this work, we plan to perform
this analysis in $e$A collisions in order to check that we get similar results
in Deep Inelastic Scattering at the energies of the Electron Ion Collider,
even if only part of the non-eikonal corrections are included in the framework
presented here.

Although conceptually more challenging, two other directions must be explored.
On the one hand, so far we have only included the corrections due to a finite
width of the target. The next step is to use the results of [22, 23] and
generalize this analysis in order to include $A_{\perp}$ and the dependence of
the gauge field on $x^{-}$. In order to do this we must modify the standard MV
model to include averages $\langle A^{\mu}(x)A^{\nu}(y)\rangle$ for all
components of the gauge field that depend on all four coordinates. On the
other hand, we must study the limitations to the AE model that are already
present at eikonal accuracy and are expected to be larger beyond the eikonal
approximation.

## Acknowledgements

PA and NA have received financial support from Xunta de Galicia (Centro
singular de investigación de Galicia accreditation 2019-2022), by European
Union ERDF, by the “María de Maeztu” Units of Excellence program
MDM-2016-0692, and by the Spanish Research State Agency under project
PID2020-119632GB-I00. TA is supported in part by the National Science Centre
(Poland) under the research grant no. 2018/31/D/ST2/00666 (SONATA 14). PA is
supported by grant ED481B-2022-050 by Xunta de Galicia. This work has been
performed in the framework of the European Research Council project
ERC-2018-ADG-835105 YoctoLHC and the MSCA RISE 823947 “Heavy ion collisions:
collectivity and precision in saturation physics” (HIEIC), and has received
funding from the European Union’s Horizon 2020 research and innovation
programme under grant agreement No. 824093.

##  Appendix A Double gluon production beyond the shockwave approximation

In this appendix we summarize the results of [14] that are relevant for the
present work. In [14], we have used the harmonic oscillator approximation in
order to solve the generalized dipole functions within a finite width
background field. The main results can be written777Note that there is a
redefinition $\epsilon_{i}\to\sqrt{i\epsilon_{i}}$ with respect to [14].

$\displaystyle d^{(0)}(x^{+},y^{+}|{\bf y},\bar{\bf
y})\equiv\frac{1}{N_{c}^{2}-1}\Big{\langle}{\rm
Tr}\Big{[}\mathcal{U}_{[x^{+},y^{+}]}({\bf
y})\mathcal{U}^{\dagger}_{[x^{+},y^{+}]}(\bar{\bf
y})\Big{]}\Big{\rangle}=\exp\Bigg{\\{}-\frac{\Delta^{+}}{L^{+}}\frac{Q_{s}^{2}}{4}({\bf
y}-\bar{\bf y})^{2}\Bigg{\\}},$ (7) $\displaystyle d^{(1)}(x^{+},y^{+}|{\bf
x},{\bf y},k_{i}^{+};\bar{\bf y})$
$\displaystyle\equiv\frac{1}{N_{c}^{2}-1}\Big{\langle}{\rm
Tr}\Big{[}\mathcal{G}_{k_{i}^{+}}(x^{+},{\bf x};y^{+},{\bf
y})\mathcal{U}^{\dagger}_{[x^{+},y^{+}]}(\bar{\bf y})\Big{]}\Big{\rangle}$
$\displaystyle=\frac{-Q_{s}^{2}}{4\pi\sqrt{i\epsilon_{i}}\sin\frac{\sqrt{i\epsilon_{i}}\Delta^{+}}{L^{+}}}\exp\Bigg{\\{}\frac{Q_{s}^{2}}{4\sqrt{i\epsilon_{i}}}\left[\frac{({\bf
y}-{\bf\bar{y}})^{2}+({\bf
x}-{\bf\bar{y}})^{2}}{\tan\frac{\sqrt{i\epsilon_{i}}\Delta^{+}}{L^{+}}}-2\frac{({\bf
y}-{\bf\bar{y}})\cdot({\bf
x}-{\bf\bar{y}})}{\sin\frac{\sqrt{i\epsilon_{i}}\Delta^{+}}{L^{+}}}\right]\Bigg{\\}}$
(8)

and

$\displaystyle d^{(2)}(x^{+},y^{+}|{\bf x},{\bf y},k_{1}^{+};\bar{\bf
x},\bar{\bf y},k_{2}^{+})\equiv\frac{1}{N_{c}^{2}-1}\Big{\langle}{\rm
Tr}\Big{[}\mathcal{G}_{k_{1}^{+}}(x^{+},{\bf x};y^{+},{\bf
y})\mathcal{G}_{k_{2}^{+}}(x^{+},\bar{{\bf x}};y^{+},\bar{{\bf
y}})\Big{]}\Big{\rangle}$ $\displaystyle=\hskip
0.0pt\frac{Q_{s}^{4}}{(4\pi)^{2}}\frac{\epsilon_{-}L^{+}}{\Delta^{+}\epsilon_{1}\epsilon_{2}\sin\frac{\Delta^{+}\epsilon_{-}}{L^{+}}}$
$\displaystyle\hskip 56.9055pt\times{\rm
exp}\Bigg{\\{}\frac{Q_{s}^{2}}{4\epsilon_{-}^{2}}\Bigg{(}\frac{\epsilon_{-}({\bf
r}_{0}^{2}+{\bf
r}_{N}^{2})}{\tan\frac{\Delta^{+}\epsilon_{-}}{L^{+}}}-2\frac{\epsilon_{-}{\bf
r}_{0}\cdot{\bf
r_{N}}}{\sin\frac{\Delta^{+}\epsilon_{-}}{L^{+}}}+\frac{L^{+}}{\Delta^{+}\epsilon_{1}\epsilon_{2}\epsilon_{+}^{4}}\left[2\epsilon_{1}\epsilon_{2}({\bf
r}_{N}-{\bf r}_{0})+\epsilon_{+}^{2}\epsilon_{-}^{2}({\bf b}_{N}-{\bf
b}_{0})\right]^{2}\Bigg{)}\Bigg{\\}}.$ (9)

These equations were computed for $x^{+}>y^{+}$ and
$\Delta^{+}=[\min(x^{+},L^{+})-\max(y^{+},0)]\Theta(x^{+})\Theta(L^{+}-y^{+})$
is the longitudinal width traversed by the emitted gluon inside the medium,
$\epsilon_{\pm}^{2}=i(\epsilon_{1}\pm\epsilon_{2})$, ${\bf r}_{0}={\bf
y}-{\bf\bar{y}}$, ${\bf r}_{N}={\bf x}-{\bf\bar{x}}$, ${\bf
b}_{0}=(k_{1}^{+}{\bf y}+k_{2}^{+}{\bf\bar{y}})/(k_{1}^{+}+k_{2}^{+})$ and
${\bf b}_{N}=(k_{1}^{+}{\bf x}+k_{2}^{+}{\bf\bar{x}})/(k_{1}^{+}+k_{2}^{+})$.

The double inclusive gluon spectrum can be written in terms of the generalized
dipole functions by using the Area Enhancement model which leads to the
following expression:

$\displaystyle\frac{d^{2}N}{d\eta_{1}d^{2}{\bf k}_{1}d\eta_{3}d^{2}{\bf
k}_{3}}=\frac{g^{4}}{4(2\pi)^{6}}\int_{{\bf q}_{1},{\bf q}_{2},{\bf
q}_{3},{\bf q}_{4}}$
$\displaystyle\Bigg{(}\Big{\langle}\Omega_{1}\Omega_{2}\Big{\rangle}_{p}\Big{\langle}\Omega_{3}\Omega_{4}\Big{\rangle}_{p}+\Big{\langle}\Omega_{1}\Omega_{3}\Big{\rangle}_{p}\Big{\langle}\Omega_{2}\Omega_{4}\Big{\rangle}_{p}+\Big{\langle}\Omega_{1}\Omega_{4}\Big{\rangle}_{p}\Big{\langle}\Omega_{2}\Omega_{3}\Big{\rangle}_{p}\Bigg{)}$
$\displaystyle\times$
$\displaystyle\Bigg{(}\Big{\langle}\Lambda_{1}\Lambda_{2}\Big{\rangle}_{T}\Big{\langle}\Lambda_{3}\Lambda_{4}\Big{\rangle}_{T}+\Big{\langle}\Lambda_{1}\Lambda_{3}\Big{\rangle}_{T}\Big{\langle}\Lambda_{2}\Lambda_{4}\Big{\rangle}_{T}+\Big{\langle}\Lambda_{1}\Lambda_{4}\Big{\rangle}_{T}\Big{\langle}\Lambda_{2}\Lambda_{3}\Big{\rangle}_{T}\Bigg{)},$
(10)

where

$\displaystyle\Big{\langle}\Omega_{i}\Omega_{j}\Big{\rangle}_{p}=\frac{\delta^{b_{i}b_{j}}}{N_{c}^{2}-1}\mu^{2}\big{[}(-1)^{i+1}\textbf{q}_{i},(-1)^{j+1}\textbf{q}_{j}\big{]},\quad\mu^{2}[{\bf
k},{\bf q}]=\pi B_{p}e^{-\frac{({\bf k}+{\bf q})^{2}}{4}B_{p}}$ (11)

is the projectile charge density correlator and $B_{p}$ is the gluonic
transverse area of the projectile which we have fixed to $B_{p}=4$ GeV-2 in
the whole manuscript. On the other hand,

$\displaystyle\Big{\langle}\Lambda_{\alpha}\Lambda_{\beta}\Big{\rangle}_{T}=\frac{\delta^{a_{\alpha}a_{\beta}}\delta^{b_{\alpha}b_{\beta}}}{N_{c}^{2}-1}\int_{\textbf{y}_{\alpha},\textbf{y}_{\beta},\textbf{x}_{\alpha},\textbf{x}_{\beta}}e^{i(-1)^{\alpha+1}(\textbf{q}_{\alpha}\cdot\textbf{y}_{\alpha}-\textbf{k}_{\alpha}\cdot\textbf{x}_{\alpha})+i(-1)^{\beta+1}(\textbf{q}_{\beta}\cdot\textbf{y}_{\beta}-\textbf{k}_{\beta}\cdot\textbf{x}_{\beta})}$
$\displaystyle\hskip 28.45274pt\times\Bigg{\\{}2\frac{{\bf
k}_{\alpha}^{\lambda_{\alpha}}{\bf k}_{\beta}^{\lambda_{\beta}}}{{\bf
k}_{\alpha}^{2}{\bf k}_{\beta}^{2}}\delta^{(2)}({\bf x}_{\alpha}-{\bf
y}_{\alpha})\delta^{(2)}({\bf x}_{\beta}-{\bf y}_{\beta})d^{(0)}(L^{+},0|{\bf
y}_{\alpha},{\bf y}_{\beta})$ $\displaystyle\hskip 28.45274pt-4\frac{{\bf
k}_{\alpha}^{\lambda_{\alpha}}{\bf q}_{\beta}^{\lambda_{\beta}}}{{\bf
k}_{\alpha}^{2}{\bf q}_{\beta}^{2}}\delta^{(2)}({\bf x}_{\alpha}-{\bf
y}_{\alpha})d^{(1)}(L^{+},0|{\bf y}_{\alpha};{\bf x}_{\beta},{\bf
y}_{\beta},(-1)^{\beta+1}k_{\beta}^{+})$ $\displaystyle\hskip
28.45274pt+2\frac{{\bf q}_{\alpha}^{\lambda_{\alpha}}{\bf
q}_{\beta}^{\lambda_{\beta}}}{{\bf q}_{\alpha}^{2}{\bf
q}_{\beta}^{2}}d^{(2)}(L^{+},0|{\bf x}_{\alpha},{\bf
y}_{\alpha},(-1)^{\alpha+1}k_{\alpha}^{+};{\bf x}_{\beta},{\bf
y}_{\beta},(-1)^{\beta}k_{\beta}^{+})$ $\displaystyle\hskip
28.45274pt+2\frac{L^{+}}{k_{\beta}^{+}}\frac{{\bf
k}_{\alpha}^{\lambda_{\alpha}}}{{\bf k}_{\alpha}^{2}}\delta^{(2)}({\bf
x}_{\alpha}-{\bf y}_{\alpha})\int_{0}^{1}df_{\beta}\Big{[}\partial_{{\bf
y}_{\alpha}^{\lambda_{\alpha}}}d^{(1)}(L^{+},f_{\beta}L^{+}|{\bf
y}_{\alpha};{\bf x}_{\beta},{\bf
y}_{\beta},(-1)^{\beta+1}k_{\beta}^{+})\Big{]}d^{(0)}(f_{\beta}L^{+},0|{\bf
y}_{\alpha},{\bf y}_{\beta})$ $\displaystyle\hskip
0.0pt-2\frac{L^{+}}{k_{\beta}^{+}}\frac{{\bf
q}_{\alpha}^{\lambda_{\alpha}}}{{\bf
q}_{\alpha}^{2}}\int_{0}^{1}df_{\beta}\int_{{\bf u}}\Big{[}\partial_{{\bf
y}_{\beta}^{\lambda_{\beta}}}d^{(2)}(L^{+},f_{\beta}L^{+}|{\bf
x}_{\alpha},{\bf u},(-1)^{\alpha+1}k_{\alpha}^{+};{\bf x}_{\beta},{\bf
y}_{\beta},(-1)^{\beta}k_{\beta}^{+})\Big{]}d^{(1)}(f_{\beta}L^{+},0|{\bf
y}_{\beta};{\bf u},{\bf y}_{\alpha},(-1)^{\alpha+1}k_{\alpha}^{+})$
$\displaystyle\hskip
14.22636pt+\frac{L^{+}}{k_{\alpha}^{+}}\frac{L^{+}}{k_{\beta}^{+}}\int_{0}^{1}df_{\beta}\int_{0}^{f_{\beta}}df_{\alpha}\int_{{\bf
u}}\Big{[}\partial_{{\bf
y}_{\beta}^{\lambda_{\beta}}}d^{(2)}(L^{+},f_{\beta}L^{+}|{\bf
x}_{\alpha},{\bf u},(-1)^{\alpha+1}k_{\alpha}^{+};{\bf x}_{\beta},{\bf
y}_{\beta},(-1)^{\beta}k_{\beta}^{+})\Big{]}$ $\displaystyle\hskip
85.35826pt\times\Big{[}\partial_{{\bf
y}_{\alpha}^{\lambda_{\alpha}}}d^{(1)}(f_{\beta}L^{+},f_{\alpha}L^{+}|{\bf
y}_{\beta};{\bf u},{\bf
y}_{\alpha},(-1)^{\alpha+1}k_{\alpha}^{+})\Big{]}d^{(0)}(f_{\alpha}L^{+},0|{\bf
y}_{\alpha},{\bf y}_{\beta})+(\alpha\leftrightarrow\beta)\Bigg{\\}}$ (12)

is the correlation function of the reduced matrix amplitude [14] which
contains the information about the target.

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dimensionality of an open ball by a superscript in the remainder of the proof,
e.g., $B_{r}^{(d)}(\xi)$ denotes the $d$-dimensional open ball of radius $r>0$
around $\xi\in\mathbb{R}^{d}$. It is easy to see that

$\Phi(Q^{(\delta,r)}_{\Phi,\ell})=B_{\delta r}^{(d)}(\delta\ell)\subset
B_{\delta r}^{(d_{1})}(\delta\ell_{1})\times\ldots\times B_{\delta
r}^{(d_{N})}(\delta\ell_{N})=\Phi(Q^{(\delta,r)}_{\Phi,\otimes,\ell}),\
\text{for all }\ell\in\mathbb{Z}^{d},$

such that $\mathcal{Q}^{(\delta,r)}_{\Phi}$ is subordinate to
$\mathcal{Q}^{(\delta,r)}_{\Phi,\otimes}$, in particular, it is almost
subordinate. On the other hand,

$\Phi(Q^{(\delta,r)}_{\Phi,\otimes,\ell})=B_{\delta
r}^{(d_{1})}(\delta\ell_{1})\times\ldots\times B_{\delta
r}^{(d_{N})}(\delta\ell_{N})\subset
B_{\delta\cdot\sqrt{d}r}^{(d)}(\delta\ell)=\Phi(Q^{(\delta,\sqrt{d}r)}_{\Phi,\ell}),\
\text{for all }\ell\in\mathbb{Z}^{d},$

i.e., $\mathcal{Q}^{(\delta,r)}_{\Phi,\otimes}$ is subordinate to
$\mathcal{Q}^{(\delta,\sqrt{d}r)}_{\Phi}$ and therefore almost subordinate to
$\mathcal{Q}^{(\delta,r)}_{\Phi}$, which is equivalent to
$\mathcal{Q}^{(\delta,\sqrt{d}r)}_{\Phi}$ by Lemma 3.2. ∎

We are now ready to show that the spaces of dominating smoothness of Besov
type can be realized as certain warped coorbit spaces.

######  Theorem 9.6.

Let $1\leq p,q\leq\infty$, $s\in\mathbb{R}^{d}$, and $\varrho_{1}$ any slow
start version of the weakly $k$-admissible radial component
$\varsigma_{1}=\ln(1+\bullet)$. If
$\Phi_{\varrho_{1}}:\mathbb{R}\rightarrow\mathbb{R}$ is the $1$-dimensional
radial warping function associated with $\varrho_{1}$ and

$\Phi:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d},\
\xi\mapsto(\Phi_{\varrho_{1}}(\xi_{1}),\ldots,\Phi_{\varrho_{1}}(\xi_{d})),$

then with

$\kappa:\mathbb{R}^{d}\to\mathbb{R}^{+},\xi=(\xi_{1},\dots,\xi_{d})\mapsto\prod_{i=1}^{d}(1+|\xi_{i}|)^{s_{i}+2^{-1}-q^{-1}}\,,$

we have

$S^{s}_{p,q}B(\mathbb{R}^{d})=\mathcal{D}(\mathcal{B}_{d},\mathbf{L}^{p},\ell_{v^{(s)}}^{q})=\operatorname{Co}(\Phi,\mathbf{L}^{p,q}_{\kappa})\,,$

up to canonical identifications. Here, $S^{s}_{p,q}B(\mathbb{R}^{d})$ and
$v^{(s)}$ are as in Definition 9.1,

###### Proof.

Combining Remark 9.2 and Proposition 9.4, we have already shown that
$S^{s}_{p,q}B(\mathbb{R}^{d})=\mathcal{D}(\mathcal{B}_{d},\mathbf{L}^{p},\ell_{v^{(s)}}^{q})$
up to canonical identifications.

It remains to prove that the decomposition space
$\mathcal{D}(\mathcal{B}_{d},\mathbf{L}^{p},\ell_{v^{(s)}}^{q})$ coincides
with the warped coorbit space
$\operatorname{Co}(\Phi,\mathbf{L}^{p,q}_{\kappa})$. We have already discussed
in Section 7, that the warping function
$\Phi_{\varrho_{1}}:\mathbb{R}\to\mathbb{R}$ is smooth and $d+1$-admissible
and that $Q_{\Phi_{\varrho_{1}},k}^{(\delta,r)}$ is equivalent to the
$1$-dimensional, inhomogeneous Besov covering $\mathcal{B}$, for any
$\delta>0$ and $r>\sqrt{d}/2$.

Furthermore, Lemma 7.8 shows that if we set
$\kappa_{i}:\mathbb{R}^{d}\to\mathbb{R}^{+},\xi\mapsto(1+|\xi|)^{s_{i}+2^{-1}-q^{-1}}$,
then $\kappa_{i}\circ\Phi_{\varrho_{1}}$ is $\tilde{\kappa}_{i}$-moderate for
some continuous, submultiplicative, radially increasing weight
$\tilde{\kappa}_{i}:\mathbb{R}^{d}\to\mathbb{R}^{+}$. Noting that
$\kappa(\xi)=\prod_{i=1}^{d}\kappa_{i}(\xi_{i})$, we conclude that Lemma 9.5
shows that $(\Phi,\kappa)$ is a compatible pair, see Definition 2.4. Thus,
Theorem 4.16 shows that
$\operatorname{Co}(\Phi,\mathbf{L}_{\kappa}^{p,q})=\mathcal{D}(\mathcal{Q}_{\Phi}^{(\delta,r)},\mathbf{L}^{p},\ell_{u}^{q})$
up to canonical identifications, with $u_{k}=\kappa_{\Phi}(\delta
k)\cdot[w(\delta k)]^{q^{-1}-2^{-1}}$. Here, by Equation (9.3) in the proof of
Lemma 9.5, with $w_{1}=\det(\Phi^{-1}_{\varrho_{1}}(\bullet))$, the weight $w$
is given by $w(\tau)=\prod_{i=1}^{d}w_{1}(\tau_{i})$.

As a direct consequence of Equation (7.18) in the proof of Theorem 7.10, we
can use Proposition 6.8 and Lemma 7.3 to obtain weak equivalence of
$\mathcal{B}$ and $\mathcal{Q}_{\Phi_{\varrho_{1}}}^{(\delta,r)}$ and by
extension of $\mathcal{B}_{d}$ and the product covering
$\mathcal{Q}_{\Phi,\otimes}^{(\delta,r)}$ defined in Lemma 9.5. By that lemma,
$\mathcal{Q}_{\Phi,\otimes}^{(\delta,r)}$ is further equivalent to the usual
$\Phi$-induced $(\delta,r)$-fine frequency covering
$\mathcal{Q}_{\Phi}^{(\delta,r)}$, such that in total, $\mathcal{B}_{d}$ is
weakly equivalent to $\mathcal{Q}_{\Phi}^{(\delta,r)}$.

Hence, Theorem 2.13 shows that
$\mathcal{D}(\mathcal{Q}_{\Phi}^{(\delta,r)},\mathbf{L}^{p},\ell_{u}^{q})=\mathcal{D}(\mathcal{B}_{d},\mathbf{L}^{p},\ell_{v^{(s)}}^{q})$,
provided that

$u_{k}\asymp v_{\ell}^{(s)}\quad\text{for
all}\quad\ell\in\mathbb{N}_{0}^{d}\text{ and }k\in\mathbb{Z}^{d}\text{
satisfying }Q_{\Phi,k}^{(\delta,r)}\cap B_{d,\ell}\neq\varnothing\,.$ (9.4)

Using $Q_{\Phi,k}^{(\delta,r)}\subset Q_{\Phi,\otimes,k}^{(\delta,r)}$, we can
instead show that (9.4) holds for $\ell,k$ such that
$Q_{\Phi,\otimes,k}^{(\delta,r)}\cap B_{d,\ell}\neq\varnothing$. Since $\xi\in
Q_{\Phi,\otimes,k}^{(\delta,r)}\cap B_{d,\ell}$ is equivalent to $\xi_{i}\in
Q_{\Phi_{\varrho_{1}},k_{i}}\cap B_{\ell_{i}}$ and both $u$ and $v^{(s)}$ have
product structure, we can verify $u_{k}\asymp v_{\ell}^{(s)}$ componentwise.
However,

$(1+|\xi_{i}|)^{s_{i}+2^{-1}-q^{-1}}\cdot[w(\delta
k)]^{q^{-1}-2^{-1}}\asymp(1+|\xi_{i}|)^{s_{i}}\asymp 2^{s_{i}\ell_{i}}$

is a direct consequence of Equations (6.10) and (7.5), after noting that
$1+(\varrho_{1})_{\ast}(|\tau|)=(\varrho_{1})_{\ast}^{\prime}(|\tau|)=e^{|\tau|}$.
Hence, we can apply Theorem 2.13 to obtain

$\operatorname{Co}(\Phi,\mathbf{L}_{\kappa}^{p,q})=\mathcal{D}(\mathcal{Q}_{\Phi}^{(\delta,r)},\mathbf{L}^{p},\ell_{u}^{q})=\mathcal{D}(\mathcal{B}_{d},\mathbf{L}^{p},\ell_{v^{(s)}}^{q})=S^{s}_{p,q}B(\mathbb{R}^{d}).$

as desired. ∎

######  Remark 9.7.

Using Lemma 9.5 and the equality
$\operatorname{Co}(\Phi,\mathbf{L}_{\kappa}^{p,q})=S^{s}_{p,q}B(\mathbb{R}^{d})$,
it is straightforward to extend Definition 9.1 to obtain a larger family of
anisotropic smoothness spaces. In particular, choosing
$\alpha\in(-\infty,1]^{d}$ and with
$\varsigma_{{\alpha_{i}}}=(1+\bullet)^{(1-\alpha_{i})}$ if
$\alpha_{i}\in(-\infty,1)$, and $\varsigma_{1}=\ln(1+\bullet)$ otherwise, we
can define _anisotropic $\alpha$-modulation spaces_ as
$\operatorname{Co}(\Phi_{\alpha},\mathbf{L}_{\kappa}^{p,q})$ with an
appropriate choice of $\kappa$ and
$\Phi_{\varrho_{\alpha}}=\Phi_{\varrho_{{\alpha_{1}}}}\otimes\ldots\otimes\Phi_{\varrho_{{\alpha_{d}}}}$,
where each $\varrho_{{\alpha_{i}}}$ is a slow start version of
$\varsigma_{{\alpha_{i}}}$, and the radial warping function
$\Phi_{\varrho_{{\alpha_{i}}}}:\mathbb{R}\rightarrow\mathbb{R}$ is defined as
usual.

A rather straightforward adaptation of Proposition 8.5 using the
considerations in the proof of Theorem 9.6 yields embeddings between these
_anisotropic $\alpha$-modulation spaces_ and spaces of dominating mixed
smoothness.

##  10\. Conclusion

In this paper we have shown that warped coorbit spaces
$\operatorname{Co}(\Phi,\mathbf{L}^{p,q}_{\kappa})$ can be identified with
certain decomposition spaces. This has allowed us to derive embedding
relations between warped coorbit spaces and study the relations between these
spaces and semi-classical smoothness spaces. In particular, we have shown that
certain warped coorbit spaces coincide with $\alpha$-modulation spaces and
(Besov-type) spaces of dominating mixed smoothness. For Besov spaces spaces,
we have shown that such equality is only possible when considering functions
on the real line. For Besov spaces of functions on $\mathbb{R}^{d}$ with
$d>1$, we have proposed a warping function that achieves a certain _closeness_
relation between the associated warped coorbit spaces and Besov spaces.

##  Appendix A $\alpha$-coverings for general $\alpha\in\mathbb{R}$

In this appendix, we collect two secondary facts about $\alpha$-coverings. As
our first result, we show that any two $\alpha$-coverings, with respect to the
same $\alpha\in\mathbb{R}$, of $\mathbb{R}^{d}$ are weakly equivalent. This
generalizes [2, Lemma B.2], where the same observation is made for
$\alpha\in[0,1]$. As our second result, we show that there are no
$\alpha$-coverings for $\alpha>1$, such that $\alpha\in(-\infty,1]$, as
considered in Section 8, is indeed the most general setting. For ease of
reference, we recall the definition of an $\alpha$-covering, where, for now,
we only assume $\alpha\in\mathbb{R}$.

######  Definition A.1.

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

  1. (1)

With

$\quad\quad\quad
r_{Q}:=\sup\\{r>0\,:\,\exists\,\xi\in\mathbb{R}^{d}:B_{r}(\xi)\subset
Q\\}\quad\\!\\!\text{and}\quad\\!\\!R_{Q}:=\inf\\{R>0\,:\,\exists\,\xi\in\mathbb{R}^{d}:Q\subset
B_{R}(\xi)\\}\,,$ (A.1)

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

  2. (2)

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

Below, we will repeatedly use the following consequence of the definition of
$\alpha$-coverings: If $\mathcal{Q}=(Q_{i})_{i\in I}$ is an $\alpha$-covering,
then $r_{Q_{i}}^{d}\lesssim\mu(Q_{i})\lesssim R_{Q_{i}}^{d}\lesssim
r_{Q_{i}}^{d}$, and hence

$(1+|\xi|)^{\alpha d}\asymp\mu(Q_{i})\asymp R_{Q_{i}}^{d}\asymp
r_{Q_{i}}^{d}\qquad\forall\,i\in I\text{ and }\xi\in Q_{i}\,.$ (A.2)

The next result shows that the specific $\alpha$-covering chosen to define
$M^{s,\alpha}_{p,q}(\mathbb{R}^{d})$ is inconsequential, by Theorem 2.13.

######  Lemma A.2.

Let $\alpha\in\mathbb{R}$, and let $\mathcal{Q}$ and $\mathcal{P}$ be two
$\alpha$-coverings of $\mathbb{R}^{d}$. Then $\mathcal{Q}$ and $\mathcal{P}$
are weakly equivalent.

###### Proof.

Before we properly start the proof, we observe for any bounded set
$Q\subset\mathbb{R}^{d}$ that $|\xi-\eta|\leq 2R_{Q}$ for all $\xi,\eta\in Q$.
Indeed, for arbitrary $R>R_{Q}$, there is some $\zeta\in\mathbb{R}^{d}$ with
$Q\subset\zeta+B_{R}(0)$, and hence
$|\xi-\eta|\leq|\xi-\zeta|+|\zeta-\eta|\leq 2R$.

Now, write $\mathcal{Q}=(Q_{i})_{i\in I}$ and $\mathcal{P}=(P_{j})_{j\in J}$.
By symmetry, it suffices to show that there is some $N\in\mathbb{N}$ with
$|I_{j}|\leq N$ for all $j\in J$, with $I_{j}=\\{i\in I\,:\,Q_{i}\cap
P_{j}\neq\varnothing\\}$.

To see this, fix $j\in J$ and $\eta\in P_{j}$. For any $i\in I_{j}$, there is
some $\xi_{i}\in Q_{i}\cap P_{j}$. For arbitrary $\xi\in Q_{i}$, this implies

$\displaystyle|\xi-\eta|$ $\displaystyle\leq|\xi-\xi_{i}|+|\xi_{i}-\eta|\leq
2(R_{Q_{i}}+R_{P_{j}})$
$\displaystyle({\scriptstyle{\text{Eq.~{}}\eqref{eq:AlphaCoveringAsymptotic}\text{
and }\xi_{i}\in Q_{i}\cap P_{j}}})$
$\displaystyle\lesssim(1+|\xi_{i}|)^{\alpha}$
$\displaystyle({\scriptstyle{\xi_{i}\in P_{j}\text{ and }\eta\in P_{j}}})$
$\displaystyle\lesssim(1+|\eta|)^{\alpha}\,.$

Therefore, $Q_{i}\subset B_{C(1+|\eta|)^{\alpha}}(\eta)$ for a constant
$C=C(d,\alpha,\mathcal{Q},\mathcal{P})$ independent of $i,j$. Since we also
have $\sum_{i\in I_{j}}{\mathds{1}}_{Q_{i}}\lesssim 1$ by the admissibility of
$\mathcal{Q}$, we see $\sum_{i\in
I_{j}}{\mathds{1}}_{Q_{i}}\lesssim{\mathds{1}}_{B_{C(1+|\eta|)^{\alpha}}(\eta)}$,
and hence

$\displaystyle(1+|\eta|)^{\alpha d}$
$\displaystyle\gtrsim\mu(B_{C(1+|\eta|)^{\alpha}}(\eta))\gtrsim\int\sum_{i\in
I_{j}}{\mathds{1}}_{Q_{i}}\,dx=\sum_{i\in I_{j}}\mu(Q_{i})$
$\displaystyle({\scriptstyle{\text{Eq. }\eqref{eq:AlphaCoveringAsymptotic}}})$
$\displaystyle\gtrsim\sum_{i\in I_{j}}(1+|\xi_{i}|)^{\alpha d}$
$\displaystyle({\scriptstyle{\text{Eq.
}\eqref{eq:AlphaCoveringAsymptotic}\text{ and }\xi_{i},\eta\in P_{j}}})$
$\displaystyle\gtrsim\sum_{i\in I_{j}}(1+|\eta|)^{\alpha
d}=|I_{j}|\cdot(1+|\eta|)^{\alpha d}\,,$

which implies $|I_{j}|\lesssim 1$, as desired. ∎

######  Lemma A.3.

Let $\alpha>1$ and $d\in\mathbb{N}$ be arbitrary. Then there does not exist an
$\alpha$-covering of $\mathbb{R}^{d}$.

###### Proof.

Assume towards a contradiction that $\mathcal{Q}=(Q_{i})_{i\in I}$ is such a
covering. As seen in Equation (A.2), there is a constant $c>0$ with
$r_{Q_{i}}\geq 2c\cdot(1+|\eta|)^{\alpha}$ for all $i\in I$ and $\eta\in
Q_{i}$.

By definition of $r_{Q}$, there is for each $i\in I$ some
$\xi_{i}\in\mathbb{R}^{d}$ such that

$\xi_{i}+B_{c\,(1+|\eta|)^{\alpha}}(0)\subset\xi_{i}+B_{r_{Q_{i}}/2}\,(0)\subset
Q_{i}\qquad\forall\,i\in I\text{ and }\eta\in Q_{i}\,.$

Next, since $\mathcal{Q}$ covers all of $\mathbb{R}^{d}$, there is for each
$n\in\mathbb{N}$ some $i_{n}\in I$ satisfying $\eta_{n}:=(n,0,\dots,0)\in
Q_{i_{n}}$. Therefore,

$\xi_{i_{n}}+B_{c\,\cdot
n^{\alpha}}(0)\subset\xi_{i_{n}}+B_{c\,(1+|\eta_{n}|)^{\alpha}}\,(0)\subset
Q_{i_{n}}\,.$

It is not hard to see that there is some $\zeta_{n}\in B_{c\cdot
n^{\alpha}}(0)$ satisfying
$|\xi_{i_{n}}+\zeta_{n}|=|\xi_{i_{n}}|+\frac{c}{2}\cdot n^{\alpha}\gtrsim
n^{\alpha}$. However, since $\xi_{i_{n}}+\zeta_{n}\in Q_{i_{n}}$, $\eta_{n}\in
Q_{i_{n}}$, and $\alpha d\neq 0$, Equation (A.2) implies

$n\asymp 1+|\eta_{n}|\asymp 1+|\xi_{i_{n}}+\zeta_{n}|\gtrsim n^{\alpha}\,.$

Since this holds for all $n\in\mathbb{N}$, with the implicit constant being
independent of $n$, and since $\alpha>1$, this yields the desired
contradiction. ∎

##  Appendix B Products of semi-structured coverings

For the sake of completeness, we prove that the product of several (tight)
semi-structured coverings is again (tight and) semi-structured, that
subordinateness relations are inherited by the product covering and
moderateness relations between coverings and weights are inherited by the
product covering and a separable product of the weights.

######  Lemma B.1.

For each $\ell=1,\dots,N$, let $\mathcal{Q}_{\ell}=(Q_{i}^{(\ell)})_{i\in
I_{\ell}}=(T_{i}^{(\ell)}P_{i}^{(\ell)}+b_{i}^{(\ell)})_{i\in I_{\ell}}$ be a
semi-structured covering of an open set
$\mathcal{O}_{\ell}\subset\mathbb{R}^{d_{\ell}}$. Let
$I:=I_{1}\times\cdots\times I_{N}$ and $d:=d_{1}+\dots+d_{N}$, and for
$i=(i_{1},\dots,i_{N})\in I$ set

$T_{i}:=\mathrm{diag}\big{(}T_{i_{1}}^{(1)},\dots,T_{i_{N}}^{(N)}\big{)}\in\mathrm{GL}(\mathbb{R}^{d})\,,\quad
b_{i}:=\big{(}b_{i_{1}}^{(1)},\dots,b_{i_{N}}^{(N)}\big{)}\in\mathbb{R}^{d}\,,\quad
Q_{i}^{\prime}:=P_{i_{1}}^{(1)}\times\cdots\times
P_{i_{N}}^{(N)}\subset\mathbb{R}^{d}\,.$

Then the following hold:

  1. (1)

The covering
$\mathcal{Q}_{1}\otimes\cdots\otimes\mathcal{Q}_{N}:=\big{(}Q_{i_{1}}^{(1)}\times\cdots\times
Q_{i_{N}}^{(N)}\big{)}_{i\in
I}=\big{(}T_{i}\,Q_{i}^{\prime}+b_{i}\big{)}_{i\in I}$ is a semi-structured
covering of
$\mathcal{O}:=\mathcal{O}_{1}\times\cdots\times\mathcal{O}_{N}\subset\mathbb{R}^{d}$.
If the $\mathcal{Q}_{\ell}$, $\ell=1,\dots,N$, are tight, then so is
$\mathcal{Q}_{1}\otimes\cdots\otimes\mathcal{Q}_{N}$.

  2. (2)

If $v^{(\ell)}$ is a $\mathcal{Q}_{\ell}$-moderate weight, for
$\ell=1,\dots,N$, then the weight $v=(v_{i})_{i\in I}$, with
$v_{i}=\prod_{\ell=1}^{N}v_{i_{\ell}}^{(\ell)}$ is
$\mathcal{Q}_{1}\otimes\cdots\otimes\mathcal{Q}_{N}$-moderate.

  3. (3)

If $\mathcal{Q}_{\ell}=(Q_{i}^{(\ell)})_{i\in I_{\ell}}$ and
$\mathcal{P}_{\ell}=(P_{j}^{(\ell)})_{j\in J_{\ell}}$, for $\ell=1,\dots,N$,
are admissible coverings of $\mathcal{O}_{\ell}\subset\mathbb{R}^{d_{\ell}}$,
and if, for all $\ell$, $\mathcal{Q}_{\ell}$ is weakly subordinate to
$\mathcal{P}_{\ell}$, then
$\mathcal{Q}_{1}\otimes\cdots\otimes\mathcal{Q}_{N}$ is weakly subordinate to
$\mathcal{P}_{1}\otimes\cdots\otimes\mathcal{P}_{N}$. If the
$\mathcal{Q}_{\ell}$ are even almost subordinate to $\mathcal{P}_{\ell}$, then
$\mathcal{Q}_{1}\otimes\cdots\otimes\mathcal{Q}_{N}$ is almost subordinate to
$\mathcal{P}_{1}\otimes\cdots\otimes\mathcal{P}_{N}$.

###### Proof.

For brevity, let
$\mathcal{Q}:=\mathcal{Q}_{1}\otimes\cdots\otimes\mathcal{Q}_{N}$. Clearly,
$\mathcal{Q}$ is indeed a covering of $\mathcal{O}$. To prove the semi-
structuredness of $\mathcal{Q}$, first note that for each $\ell=1,\dots,N$,
there is some $R_{\ell}>0$ such that
$Q_{i}^{(\ell)}\subset\overline{B_{R_{\ell}}(0)}$ for all $i\in I_{\ell}$. If
we set $R:=\sqrt{N}\cdot\max\\{R_{\ell}\,:\,\ell=1,\dots,N\\}$, then this
implies $Q_{i}^{\prime}\subset\overline{B_{R}(0)}$ for all $i\in I$.

Next, note for $i=(i_{1},\dots,i_{N})\in I$ and $j=(j_{1},\dots,j_{N})\in I$
that

$i\in j^{\ast}\quad\Longleftrightarrow\quad Q_{i}\cap
Q_{j}\neq\varnothing\quad\Longleftrightarrow\quad\forall\,\ell=1,\dots,N\,:Q_{i_{\ell}}\cap
Q_{j_{\ell}}\neq\varnothing\quad\Longleftrightarrow\quad
i\in\prod_{\ell=1}^{N}j_{\ell}^{\ast}\,.$

On the one hand, this shows $M_{\mathcal{Q}}=\sup_{i\in
I}|i^{\ast}|=\sup_{i\in
I}\prod_{\ell=1}^{N}|i_{\ell}^{\ast}|\leq\prod_{\ell=1}^{N}M_{\mathcal{Q}_{\ell}}<\infty$,
so that $\mathcal{Q}$ is admissible. On the other hand, we see that if $i\in
j^{\ast}$, then

$\displaystyle\|T_{i}^{-1}T_{j}\|$
$\displaystyle=\Big{\|}\mathrm{diag}\big{(}(T_{i_{1}}^{(1)})^{-1}\,T_{j_{1}}^{(1)},\dots,(T_{i_{N}}^{(N)})^{-1}\,T_{j_{N}}^{(N)}\big{)}\Big{\|}$
$\displaystyle\leq\max\left\\{\sup_{\begin{subarray}{c}m,n\in I_{\ell}\text{
and }\\\ Q_{m}^{(\ell)}\cap
Q_{n}^{(\ell)}\neq\varnothing\end{subarray}}\|(T_{m}^{(\ell)})^{-1}T_{n}^{(\ell)}\|\,:\,\ell=1,\dots,N\right\\}\leq\max\\{C_{\mathcal{Q}_{\ell}}\,:\,\ell=1,\dots,N\\}\,,$

where $C_{\mathcal{Q}_{\ell}}$ is the constant $C>0$ in Definition 2.7(2), for
$\mathcal{Q}=\mathcal{Q}_{\ell}$. Hence, $\mathcal{Q}$ is a semi-structured
covering of $\mathcal{O}$.

We further see that $v$ is $\mathcal{Q}$-moderate, since if $Q_{i}\cap
Q_{j}\neq\varnothing$, then

$\frac{v_{i}}{v_{j}}=\prod_{\ell=1}^{N}\frac{v_{i_{\ell}}^{(\ell)}}{v_{j_{\ell}}^{(\ell)}}\leq\prod_{\ell=1}^{N}C_{v^{(\ell)},\mathcal{Q}_{\ell}}<\infty\,.$

If additionally, all the coverings $\mathcal{Q}_{\ell}$ are tight, then there
is for each $\ell=1,\dots,N$ some $\varepsilon_{\ell}>0$ and for each $i\in
I_{\ell}$ some $c_{i}^{(\ell)}\in\mathbb{R}^{d_{\ell}}$ satisfying
$B_{\varepsilon_{\ell}}(c_{i}^{(\ell)})\subset Q_{i}^{(\ell)}$. Set
$c_{i}:=(c_{i_{1}}^{(1)},\dots,c_{i_{N}}^{(N)})\in\mathbb{R}^{d}$ for
$i=(i_{1},\dots,i_{N})\in I$ and furthermore
$\varepsilon:=\min\\{\varepsilon_{\ell}\,:\,\ell=1,\dots,N\\}$. It follows
that $B_{\varepsilon}(c_{i})\subset Q_{i}^{\prime}$ for all $i\in I$, i.e.,
$\mathcal{Q}$ is tight.

Finally, we prove the last part of the lemma. If we define
$J_{i}^{(\ell)}:=\\{j\in J_{\ell}\,:\,P_{j}^{(\ell)}\cap
Q_{i}^{(\ell)}\neq\varnothing\\}$ for $i\in I_{\ell}$, then there is some
$M\in\mathbb{N}$ such that $|J_{i}^{(\ell)}|\leq M$ for all $i\in I_{\ell}$
and $\ell=1,\dots,N$, since $\mathcal{Q}_{\ell}$ is weakly subordinate to
$\mathcal{P}_{\ell}$.

Now, set $I:=I_{1}\times\cdots\times I_{N}$ and $J:=J_{1}\times\cdots\times
J_{N}$. Furthermore, let $Q_{i}:=Q_{i_{1}}^{(1)}\times\cdots\times
Q_{i_{N}}^{(N)}$ and $P_{j}:=P_{j_{1}}^{(1)}\times\cdots\times
P_{j_{N}}^{(N)}$ for $i=(i_{1},\dots,i_{N})\in I$ and
$j=(j_{1},\dots,j_{N})\in J$. Then

$\begin{split}J_{i}:=\\{j\in J\,:\,P_{j}\cap
Q_{i}\neq\varnothing\\}&=\big{\\{}j=(j_{1},\dots,j_{N})\in J\,:\,\text{for all
}\ell=1,\dots,N\,:\,P_{j_{\ell}}^{(\ell)}\cap
Q_{i_{\ell}}^{(\ell)}\neq\varnothing\big{\\}}\\\
&=J_{i_{1}}^{(1)}\times\cdots\times J_{i_{N}}^{(N)}\,,\end{split}$

and hence $|J_{i}|=|J_{i_{1}}^{(1)}|\cdots|J_{i_{N}}^{(N)}|\leq M^{N}$ for
arbitrary $i=(i_{1},\dots,i_{N})\in I$. Hence,
$\mathcal{Q}_{1}\otimes\cdots\otimes\mathcal{Q}_{N}$ is weakly subordinate to
$\mathcal{P}_{1}\otimes\cdots\otimes\mathcal{P}_{N}$.

If $\mathcal{Q}_{\ell}$ is almost subordinate to $\mathcal{P}_{\ell}$,
$\ell=1,\dots,N$, i.e., there is $M_{\ell}\in\mathbb{N}$ such that for all
$i_{\ell}\in I_{\ell}$ there is a $j_{\ell}=j_{i,\ell}\in J_{\ell}$, such that
$Q^{(\ell)}_{i}\subset(P^{(\ell)}_{j})^{N_{\ell}\ast}$, then with
$M=\max_{\ell}M_{\ell}$,
$Q^{(\ell)}_{(i_{1},\cdot,i_{N})}\subset(P^{(\ell)}_{(j_{1},\ldots,j_{N})})^{M\ast}$
follows. ∎

##  Appendix C The structured $\Phi$-induced $(\delta,r)$-fine frequency
covering

In this final appendix, we demonstrate that every warping function $\Phi$
induces a _structured_ $(\delta,r)$-fine frequency covering, for
$r\in(0,\vartheta_{0})$ and $\delta\in(0,\Delta_{r})$, with $\vartheta_{0}>0$
as in Lemma 3.4 and a certain $\Delta_{r}>0$ .

######  Definition C.1.

A semi-structured covering $\mathcal{Q}=(Q_{i})_{i\in I}$ of $\mathcal{O}$,
with $Q_{i}=T_{i}Q_{i}^{\prime}+b_{i}$, $i\in I$, as in Definition 2.7(2) is
called _structured_ , if the operators $T_{i}$ can be chosen such that
$Q_{i}^{\prime}=Q$, for all $i\in I$, and there is an open set
$P\subset\mathbb{R}^{d}$ with $\overline{P}\subset Q$ and

$\mathcal{O}=\bigcup_{i\in I}(T_{i}P+b_{i}).$

It should be noted that every structured admissible covering $\mathcal{Q}$ is
a decomposition covering, that is, there is a $\mathcal{Q}$-BAPU, see [3,
Proposition 1]. While the covering $\mathcal{Q}_{\Phi}^{(\delta,r)}$ is in
general not structured, we can define an equivalent family of structured
coverings using Lemma 3.4. Thus, for the purpose of investigating the
associated decomposition spaces, it does not matter which of the two families
of coverings is considered, see Theorem 2.13. However, the structured covering
may simplify the derivation of certain results in future work.

######  Definition C.2.

Let $\Phi:D\to\mathbb{R}^{d}$ be a diffeomorphism. For $\delta,r>0$, we define

$\mathcal{S}_{\Phi}^{(\delta,r)}:=\left(S_{\Phi,k}^{(\delta,r)}\right)_{k\in\mathbb{Z}^{d}}:=\Big{(}\Phi^{-1}(\delta
k)+D\Phi^{-1}(\delta k)\langle B_{r}(0)\rangle\Big{)}_{k\in\mathbb{Z}^{d}}\,.$

The family $\mathcal{S}_{\Phi}^{(\delta,r)}$ is called the _structured
$(\delta,r)$-fine frequency covering induced by $\Phi$_.

We proceed to show that $\mathcal{Q}_{\Phi}^{(\varepsilon,\rho)}$ and
$\mathcal{S}_{\Phi}^{(\delta,r)}$ are equivalent, for suitable choices of the
parameters.

######  Lemma C.3.

Let $\Phi:D\to\mathbb{R}^{d}$ be a $1$-admissible warping function with
control weight $v_{0}$ and set $\vartheta_{0}:=(2dv_{0}(1))^{-1}$. If
$r\in(0,\vartheta_{0}/4)$ and $\delta\in\big{(}0,\min\\{1,4r\\}(2\sqrt{d}\cdot
v_{0}(1/2))^{-1}\big{)}$, then $\mathcal{S}_{\Phi}^{(\delta,r)}$ is a
structured admissible covering of $D$.

Moreover, $\mathcal{S}_{\Phi}^{(\delta,r)}$ is equivalent to
$\mathcal{S}_{\Phi}^{(\delta^{\prime},r^{\prime})}$ for any
$r^{\prime}\in(0,\vartheta_{0}/4)$ and
$\delta^{\prime}\in\big{(}0,\min\\{1,4r^{\prime}\\}(2\sqrt{d}\cdot
v_{0}(1/2))^{-1}\big{)}$, and to the coverings
$\mathcal{Q}_{\Phi}^{(\varepsilon,\rho)}$, for any $\varepsilon>0$,
$\rho>\sqrt{d}/2$.

###### Proof.

Let $r\in(0,\vartheta_{0}/4)$ be arbitrary, such that
$\vartheta:=4r\in(0,\vartheta_{0})$ and note that $\vartheta_{0}$ is as in
Lemma 3.4. Hence, Equation (3.3) of said lemma, with $\xi=\Phi^{-1}(\delta
k)$, yields

$\begin{split}S_{\Phi,k}^{(\delta,r)}&=\xi+D\Phi^{-1}(\Phi(\xi))\langle
B_{\vartheta/4}(0)\rangle\subset\Phi^{-1}\big{(}\Phi(\xi)+B_{\vartheta}(0)\big{)}\\\
&=\Phi^{-1}\big{(}\delta k+B_{4r}(0)\big{)}=\Phi^{-1}\big{(}\delta\cdot
B_{4r/\delta}(k)\big{)}\\\ &\subset\Phi^{-1}\big{(}\delta\cdot
B_{\rho^{\ast}}(k)\big{)}=Q_{\Phi,k}^{(\delta,\rho^{\ast})}\subset
D\,,\qquad\text{ whenever }\qquad\rho^{\ast}\geq 4r/\delta.\end{split}$ (C.1)

In addition to showing that the family $\mathcal{S}_{\Phi}^{(\delta,r)}$ is
subordinate (and thus almost subordinate) to the covering
$\mathcal{Q}_{\Phi}^{(\delta,\rho^{\ast})}$ with arbitrary
$\rho^{\ast}>\max\\{\frac{\sqrt{d}}{2},\frac{4r}{\delta}\\}$, this also
implies for all $k\in\mathbb{Z}^{d}$ that $S_{\Phi,k}^{(\delta,r)}\subset D$.
Furthermore, Equation (C.1) also shows that $\mathcal{S}_{\Phi}^{(\delta,r)}$
is admissible, since

$\left|\left\\{\ell\in\mathbb{Z}^{d}\,:\,S_{\Phi,\ell}^{(\delta,r)}\cap
S_{\Phi,k}^{(\delta,r)}\neq\varnothing\right\\}\right|\leq\left|\left\\{\ell\in\mathbb{Z}^{d}\,:\,Q_{\Phi,\ell}^{(\delta,\rho^{\ast})}\cap
Q_{\Phi,k}^{(\delta,\rho^{\ast})}\neq\varnothing\right\\}\right|\leq
N_{\mathcal{Q}_{\Phi}^{(\delta,\rho^{\ast})}}<\infty,$

for all $k\in\mathbb{Z}^{d}$, by admissibility of
$\mathcal{Q}_{\Phi}^{(\delta,\rho^{\ast})}$ (see Lemma 3.2).

If $\rho>\sqrt{d}/2$ is arbitrary, then, by performing the same derivations as
in the proof of Lemma 3.2, we can show that

$Q_{\Phi,k}^{(\delta,\rho)}=\Phi^{-1}(\delta\cdot
B_{\rho}(k))\subset\Phi^{-1}(\delta k)+A(\delta k)\langle B_{\delta\rho\cdot
v_{0}(\delta\rho)}(0)\rangle=S_{\Phi,k}^{(\delta,\delta\rho\cdot
v_{0}(\delta\rho))}\,.$ (C.2)

Now, for fixed $r\in(0,\vartheta_{0}/4)$ and
$0<\delta<\frac{2}{\sqrt{d}}\frac{\min\\{1,4r\\}}{4\cdot v_{0}(1/2)}$ as per
our assumption, we can choose $\rho_{0}>\sqrt{d}/2$, such that

$\delta\rho_{0}<\frac{\min\\{1,4r\\}}{4\cdot v_{0}(1/2)}\leq 1/4.$

Hence, using that $v_{0}$ is radially increasing, we can estimate
$\delta\rho_{0}$ by either $v_{0}(1/2)^{-1}r$ or simply $1/4$, such that
$\delta\rho_{0}\cdot v_{0}(2\delta\rho_{0})<\frac{rv_{0}(1/2)}{v_{0}(1/2)}\leq
r$. Therefore, Equation (C.2) can be applied with $\rho=\rho_{0}$, yielding

$Q_{\Phi,k}^{(\delta,\rho_{0})}\subset S_{\Phi,k}^{(\delta,\delta\rho_{0}\cdot
v_{0}(2\delta\rho_{0}))}\subset S_{\Phi,k}^{(\delta,r)}\overset{\text{Equation
~{}}\eqref{eq:StructuredFrequencyCoveringSubordinateToFrequencyCovering}}{\subset}D\qquad\forall\,k\in\mathbb{Z}^{d}\,.$

On the one hand, this proves that $\mathcal{Q}_{\Phi}^{(\delta,\rho_{0})}$ is
subordinate to $\mathcal{S}_{\Phi}^{(\delta,r)}$. On the other hand, since
$\rho_{0}>\sqrt{d}/2$, Lemma 3.1 shows
$D=\bigcup_{k\in\mathbb{Z}^{d}}Q_{\Phi,k}^{(\delta,\rho_{0})}\subset\bigcup_{k\in\mathbb{Z}^{d}}S_{\Phi,k}^{(\delta,r)}\subset
D$, so that $\mathcal{S}_{\Phi}^{(\delta,r)}$ covers all of $D$.

Moreover, since all coverings $\mathcal{Q}_{\Phi}^{(\varepsilon,\rho)}$ with
$\rho>\sqrt{d}/2$ and $\varepsilon>0$ are equivalent, since
$\mathcal{Q}_{\Phi}^{(\delta,\rho_{0})}$ is subordinate to
$\mathcal{S}_{\Phi}^{(\delta,r)}$, and since we saw above that
$\mathcal{S}_{\Phi}^{(\delta,r)}$ is subordinate to
$\mathcal{Q}_{\Phi}^{(\delta,\rho^{\ast})}$ for
$\rho^{\ast}>\max\\{\frac{\sqrt{d}}{2},\frac{4r}{\delta}\\}$, we see that the
coverings $\mathcal{S}_{\Phi}^{(\delta,r)}$ and
$\mathcal{Q}_{\Phi}^{(\varepsilon,\rho)}$ are equivalent, as long as
$\rho>\sqrt{d}/2$, $r\in(0,\vartheta_{0}/4)$, and
$0<\delta<\frac{\min\\{1,4r\\}}{2\sqrt{d}\cdot v_{0}(1/2)}$.

It remains to show that $\mathcal{S}_{\Phi}^{(\delta,r)}$ is indeed a
structured covering. But setting $T_{k}=A(\delta k)$ and
$b_{k}=\Phi^{-1}(\delta k)$, as well as $Q:=B_{r}(0)$, we clearly have
$S_{\Phi,k}^{(\delta,r)}=T_{k}Q+b_{k}$ for all $k\in\mathbb{Z}^{d}$. Finally,
if we fix $\rho^{\ast}>\max\\{\frac{\sqrt{d}}{2},\frac{4r}{\delta}\\}$, and if
$S_{\Phi,k}^{(\delta,r)}\cap S_{\Phi,\ell}^{(\delta,r)}\neq\varnothing$, then
Equation (C.1) shows $\varnothing\neq Q_{\Phi,k}^{(\delta,\rho^{\ast})}\cap
Q_{\Phi,\ell}^{(\delta,\rho^{\ast})}$, and hence $\|T_{k}^{-1}T_{\ell}\|\leq
C$ for an absolute constant $C=C(\Phi,\delta,r)>0$, see Lemma 3.2.

Finally, if we choose $\tilde{r}\in(0,r)\subset(0,\vartheta_{0}/4)$ such that
we still have $\delta<\frac{\min\\{1,4\tilde{r}\\}}{2\sqrt{d}\cdot
v_{0}(1/2)}$, then it is clear that $P:=B_{\tilde{r}}(0)$ satisfies
$\overline{P}\subset Q$, and the preceding part of the proof shows that
$(T_{k}P+b_{k})_{k\in\mathbb{Z}^{d}}$ still covers $D$. ∎

Using the equivalence between the coverings
$\mathcal{Q}_{\Phi}^{(\delta,\varrho)}$ and $\mathcal{S}_{\Phi}^{(\delta,r)}$,
we can now show that the associated decomposition spaces also coincide.

######  Corollary C.4.

With notation as in Lemma C.3, let $r\in(0,\vartheta_{0}/4)$, $s>\sqrt{d}/2$,
and $0<\delta<\frac{\min\\{1,4r\\}}{2\sqrt{d}\cdot v_{0}(1/2)}$.

If $u=(u_{k})_{k\in\mathbb{Z}^{d}}$ is
$\mathcal{Q}_{\Phi}^{(\delta,s)}$-moderate, then $u$ is also
$\mathcal{S}_{\Phi}^{(\delta,r)}$-moderate, and we have

$\mathcal{D}(\mathcal{Q}_{\Phi}^{(\delta,s)},\mathbf{L}^{p},\ell_{u}^{q})=\mathcal{D}(\mathcal{S}_{\Phi}^{(\delta,r)},\mathbf{L}^{p},\ell_{u}^{q})\qquad\forall\,p,q\in[1,\infty]\,.$

###### Proof.

Directly from the definition of the coverings
$\mathcal{Q}_{\Phi}^{(\delta,s)}$ and $\mathcal{S}_{\Phi}^{(\delta,r)}$, we
see

$\Phi^{-1}(\delta k)\in S_{\Phi,k}^{(\delta,r)}\cap
Q_{\Phi,k}^{(\delta,s)}\neq\varnothing\qquad\forall\,k\in\mathbb{Z}^{d}\,.$

Note that this derivation crucially uses that the _sampling density_
$\delta>0$ of both coverings coincides.

In Lemma C.3, we saw that the coverings $\mathcal{Q}_{\Phi}^{(\delta,s)}$ and
$\mathcal{S}_{\Phi}^{(\delta,s)}$ are equivalent. In particular, there is a
fixed $N\in\mathbb{N}$ such that for each $k\in\mathbb{Z}^{d}$, there is some
$j_{k}\in\mathbb{Z}^{d}$ satisfying $S_{\Phi,k}^{(\delta,r)}\subset
Q_{\Phi,j_{k}^{N\ast}}^{(\delta,s)}$. But then

$\Phi^{-1}(\delta k)\in S_{\Phi,k}^{(\delta,r)}\cap
Q_{\Phi,k}^{(\delta,s)}\subset Q_{\Phi,j_{k}^{N\ast}}^{(\delta,s)}\cap
Q_{\Phi,k}^{(\delta,s)}\,,$

so that $k\in j_{k}^{(N+1)\ast}$, and hence $j_{k}\in k^{(N+1)\ast}$. But this
implies $j_{k}^{N\ast}\subset k^{(2N+1)\ast}$, and hence

$S_{\Phi,k}^{(\delta,r)}\subset Q_{\Phi,j_{k}^{N\ast}}^{(\delta,s)}\subset
Q_{\Phi,k^{(2N+1)\ast}}^{(\delta,s)}\qquad\forall\,k\in\mathbb{Z}^{d}\,.$
(C.3)

The inclusion (C.3) yields the $\mathcal{S}_{\Phi}^{(\delta,r)}$-moderateness
of $u$, since if $\varnothing\neq S_{\Phi,k}^{(\delta,r)}\cap
S_{\Phi,\ell}^{(\delta,r)}$, then $\varnothing\neq
Q_{\Phi,k^{(2N+1)\ast}}^{(\delta,s)}\cap
Q_{\Phi,\ell^{(2N+1)\ast}}^{(\delta,s)}$, and thus $\ell\in k^{(4N+3)\ast}$,
so that $u_{\ell}\asymp u_{k}$. Here, we used that $v_{i}\leq C^{\ell}\cdot
v_{j}$ for $j\in i^{\ell\ast}$, whenever $\ell\in\mathbb{N}$ and
$v=(v_{i})_{i\in I}$ is $\mathcal{Q}$-moderate, i.e., $v_{i}\leq C\cdot v_{j}$
if $Q_{i}\cap Q_{j}\neq\varnothing$. This inequality can be verified by a
straightforward induction.

Likewise, the inclusion (C.3) also implies that $u_{k}\asymp u_{\ell}$ if
$\varnothing\neq Q_{\Phi,k}^{(\delta,s)}\cap S_{\Phi,\ell}^{(\delta,r)}$,
since then $\varnothing\neq Q_{\Phi,k}^{(\delta,s)}\cap
Q_{\Phi,\ell^{(2N+1)\ast}}^{(\delta,s)}$, which implies
$k\in\ell^{(2N+2)\ast}$, and hence $u_{k}\asymp u_{\ell}$.

Overall, since the coverings $\mathcal{Q}_{\Phi}^{(\delta,s)}$ and
$\mathcal{S}_{\Phi}^{(\delta,r)}$ are equivalent, and since $u_{k}\asymp
u_{\ell}$ if $Q_{\Phi,k}^{(\delta,s)}\cap
S_{\Phi,\ell}^{(\delta,r)}\neq\varnothing$, Theorem 2.13 shows for arbitrary
$p,q\in[1,\infty]$ that
$\mathcal{D}(\mathcal{Q}_{\Phi}^{(\delta,s)},\mathbf{L}^{p},\ell_{u}^{q})=\mathcal{D}(\mathcal{S}_{\Phi}^{(\delta,r)},\mathbf{L}^{p},\ell_{u}^{q})$,
with equivalent norms. ∎

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