Using Lemma 29, we obtain that the maps $H_{i}$ from (31) are of regularity
$\eta$.

###### Proposition 30

For all $i\in\\{1,...,m\\}$, the application $H_{i}$ belongs to
$\mathcal{H}^{\eta}_{C}(A,\mathbb{R})$.

Proof  Define

$F_{i}:=\phi_{i}^{-1}\circ T_{|\mathcal{M}}^{-1}\circ T\ \text{ and }\
\sigma_{i}:=\mathds{1}_{\\{T_{|\mathcal{M}}^{-1}\circ
T(\cdot)\in\phi_{i|B^{d}(0,\tau)}\\}}\zeta_{i}\circ F_{i}.$

We have $F_{i}\in\mathcal{H}^{\beta+1}_{C}(A,\mathbb{R}^{d})$ and as
$supp(\zeta_{i})\subset B^{d}(0,\tau)$, the application $\sigma_{i}$ belongs
to $\mathcal{H}^{\beta+1}_{C}(A,\mathbb{R}^{p})$. Furthermore, we have
$(\text{Id}-T)\in\mathcal{H}^{\beta+1}_{C}(A,\mathbb{R}^{p})$ and
$\tilde{\Gamma}^{-\eta+1,-2}_{\epsilon}(L_{i})\in\mathcal{H}_{C}^{\eta-1}(B^{d}(0,\tau),\mathbb{R}^{p})$,
so we get from the Faa di Bruno formula that
$H_{i}\in\mathcal{H}^{\eta-1}_{C}(A,\mathbb{R})$. Let
$i_{1},...,i_{\lfloor\eta\rfloor}\in\\{1,...,p\\}$ and write
$P(\\{i_{1},...,i_{\lfloor\eta\rfloor}\\})$ the set of subsets of
$\\{i_{1},...,i_{\lfloor\eta\rfloor}\\}$. For $x\in A$, we have

$\displaystyle\partial_{i_{1},...,i_{\lfloor\eta\rfloor}}^{\lfloor\eta\rfloor}H_{i}(x)$
$\displaystyle=\sum\limits_{S\in
P(\\{i_{1},...,i_{\lfloor\eta\rfloor}\\})}\Big{\langle}\partial^{|S|}_{i\in
S}(\tilde{\Gamma}^{-\eta+1,-2}_{\epsilon}(L_{i})\circ F_{i})(x)\ ,\
\partial^{\lfloor\eta\rfloor-|S|}_{i\notin
S}((\text{Id}-T)\sigma_{i})(x)\Big{\rangle}.$

For all $S\in P(\\{i_{1},...,i_{\lfloor\eta\rfloor}\\})$ with
$|S|<\lfloor\eta\rfloor$, we have that

$x\mapsto\Big{\langle}\partial^{|S|}_{i\in
S}(\tilde{\Gamma}^{-\eta+1,-2}_{\epsilon}(L_{i})\circ F_{i})(x)\ ,\
\partial^{\lfloor\eta\rfloor-|S|}_{i\notin
S}(\text{Id}-T)\sigma_{i}(x)\Big{\rangle}\in\mathcal{H}^{\eta-\lfloor\eta\rfloor}_{C}(A,\mathbb{R}).$

From the previous calculations, we know that

$\displaystyle|\partial_{i_{1},...,i_{\lfloor\eta\rfloor}}^{\lfloor\eta\rfloor}\tilde{\Gamma}^{-\eta+1,-2}_{\epsilon}(L_{i})(F_{i}(x))|$
$\displaystyle\leq CK_{X}\delta_{F_{i}(x)}^{i-(\lfloor\eta\rfloor+1-\eta)}$
$\displaystyle=CK_{X}\|T_{|\mathcal{M}}^{-1}\circ
T(x)-T(T_{|\mathcal{M}}^{-1}\circ T(x))\|^{-(\lfloor\eta\rfloor+1-\eta)}$
$\displaystyle=CK_{X}\|T_{|\mathcal{M}}^{-1}\circ
T(x)-T(x)\|^{-(\lfloor\eta\rfloor+1-\eta)}$ $\displaystyle\leq
C\|x-T(x)\|^{-(\lfloor\eta\rfloor+1-\eta)},$

so

$|\Big{\langle}\partial_{i_{1},...,i_{\lfloor\eta\rfloor}}^{\lfloor\eta\rfloor}(\tilde{\Gamma}^{-\eta+1,-2}_{\epsilon}(L_{i})\circ
F_{i})(x)\ ,(\text{Id}-T)\sigma_{i}(x)\Big{\rangle}|\leq
C\|x-T(x)\|^{\eta-\lfloor\eta\rfloor}\sigma_{i}(x).$ (36)

Let $x,y\in A$ with $T_{|\mathcal{M}}^{-1}\circ
T(x),T_{|\mathcal{M}}^{-1}\circ T(y)\in\phi_{i}(B^{d}(0,\tau))$ and
$\|x-T(x)\|\leq\|y-T(y)\|$, from (C.2) we get

$\displaystyle|\Big{\langle}$
$\displaystyle\partial_{i_{1},...,i_{\lfloor\eta\rfloor}}^{\lfloor\eta\rfloor}(\tilde{\Gamma}^{-\eta+1,-2}_{\epsilon}(L_{i})\circ
F_{i})(x)\ ,(\text{Id}-T)\sigma_{i}(x)\Big{\rangle}$
$\displaystyle-\Big{\langle}\partial_{i_{1},...,i_{\lfloor\eta\rfloor}}^{\lfloor\eta\rfloor}(\tilde{\Gamma}^{-\eta+1,-2}_{\epsilon}(L_{i})\circ
F_{i})(y)\ ,(\text{Id}-T)\sigma_{i}(y)\Big{\rangle}|$ $\displaystyle\leq$
$\displaystyle|\Big{\langle}\partial_{i_{1},...,i_{\lfloor\eta\rfloor}}^{\lfloor\eta\rfloor}(\tilde{\Gamma}^{-\eta+1,-2}_{\epsilon}(L_{i})\circ
F_{i})(x)-\partial_{i_{1},...,i_{\lfloor\eta\rfloor}}^{\lfloor\eta\rfloor}(\tilde{\Gamma}^{-\eta+1,-2}_{\epsilon}(L_{i})\circ
F_{i})(y)\ ,(\text{Id}-T)\sigma_{i}(x)\Big{\rangle}|$
$\displaystyle+|\Big{\langle}\partial_{i_{1},...,i_{\lfloor\eta\rfloor}}^{\lfloor\eta\rfloor}(\tilde{\Gamma}^{-\eta+1,-2}_{\epsilon}(L_{i})\circ
F_{i})(y)\
,(\text{Id}-T)\sigma_{i}(x)-(\text{Id}-T)\sigma_{i}(y)\Big{\rangle}|$
$\displaystyle\leq$ $\displaystyle
C\min(\|x-T(x)\|,\|y-T(y)\|)^{-1}\|x-y\|^{\eta-\lfloor\eta\rfloor}\|x-T(x)\|$
$\displaystyle+C\|T_{|\mathcal{M}}^{-1}\circ
T(y)-T(y)\|^{-(1-(\eta-\lfloor\eta\rfloor))}\|(\text{Id}-T)\sigma_{i}(x)-(\text{Id}-T)\sigma_{i}(y)\|$
$\displaystyle\leq$ $\displaystyle
C\|x-y\|^{\eta-\lfloor\eta\rfloor}+C\|T_{|\mathcal{M}}^{-1}\circ
T(y)-T(y)\|^{-(1-(\eta-\lfloor\eta\rfloor))}\|(\text{Id}-T)\sigma_{i}(x)-(\text{Id}-T)\sigma_{i}(y)\|.$

Furthermore,

$\displaystyle\|$
$\displaystyle(\text{Id}-T)\sigma_{i}(x)-(\text{Id}-T)\sigma_{i}(y)\|$
$\displaystyle=\|(\text{Id}-T)\sigma_{i}(x)-(\text{Id}-T)\sigma_{i}(y)\|^{1-(\eta-\lfloor\eta\rfloor)}\|(\text{Id}-T)\sigma_{i}(x)-(\text{Id}-T)\sigma_{i}(y)\|^{\eta-\lfloor\eta\rfloor}$
$\displaystyle\leq(\|(\text{Id}-T)(x)\|+\|(\text{Id}-T)(y)\|)^{1-(\eta-\lfloor\eta\rfloor)}\|x-y\|^{\eta-\lfloor\eta\rfloor}$
$\displaystyle\leq
2\|(\text{Id}-T)(y)\|^{1-(\eta-\lfloor\eta\rfloor)}\|x-y\|^{\eta-\lfloor\eta\rfloor}$
$\displaystyle\leq 2\|T_{|\mathcal{M}}^{-1}\circ
T(y)-T(y)\|^{1-(\eta-\lfloor\eta\rfloor)}\|x-y\|^{\eta-\lfloor\eta\rfloor}.$

Now if $T_{|\mathcal{M}}^{-1}\circ T(x)\in\phi_{i}(B^{d}(0,\tau))$ and
$T_{|\mathcal{M}}^{-1}\circ T(y)\notin\phi_{i}(B^{d}(0,\tau))$, we have from
(36),

$\displaystyle|\Big{\langle}\partial_{i_{1},...,i_{\lfloor\eta\rfloor}}^{\lfloor\eta\rfloor}(\tilde{\Gamma}^{-\eta+1,-2}_{\epsilon}(L_{i})\circ
F_{i})(x)\ ,(\text{Id}-T)\sigma_{i}(x)\Big{\rangle}|$ $\displaystyle\leq
C\|x-T(x)\|^{\eta-\lfloor\eta\rfloor}\sigma_{i}(x)$ $\displaystyle\leq
C\sigma_{i}(x)$ $\displaystyle=C(\sigma_{i}(x)-\sigma_{i}(y))$
$\displaystyle\leq C\|x-y\|.$

Therefore we have finally that $H_{i}\in\mathcal{H}^{\eta}_{C}(A,\mathbb{R})$.
Let us now prove that we can extend $H$ to the whole $\mathbb{R}^{p}$ space
using our extension result for maps defined on the neighborhood of a
submanifold (Proposition 25). Define $D:\mathcal{M}^{\star t}\rightarrow A$ by

$D(x)=\left\\{\begin{array}[]{ll}x&\text{if }x\in A\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\ \frac{x-T(x)}{\|x-T(x)\|}\|T_{|\mathcal{M}}^{-1}\circ
T(x)-T(x)\|+T(x)&\text{if }x\notin A.\end{array}\right.$

For $x\notin A$ we have

$\displaystyle\nabla D(x)=$ $\displaystyle\frac{\|T_{|\mathcal{M}}^{-1}\circ
T(x)-T(x)\|}{\|x-T(x)\|}\left(\nabla(\text{Id}-T)(x)-\frac{x-T(x)}{\|x-T(x)\|}\frac{(x-T(x))^{\top}}{\|x-T(x)\|}\nabla(\text{Id}-T)(x)\right)$
$\displaystyle+\frac{x-T(x)}{\|x-T(x)\|}\frac{(x-T(x))^{\top}}{\|x-T(x)\|}\nabla(T_{|\mathcal{M}}^{-1}\circ
T-T)(x)+\nabla T(x),$

so $\|\nabla D(x)\|\leq C$. Then using Proposition 25, for all
$i\in\\{1,...,m\\}$, $H_{i}$ can be extended to a map in
$\mathcal{H}^{\eta}_{C}(\mathbb{R}^{p},\mathbb{R})$.

We can now conclude the proof of Lemma 7 using Proposition 28 as

$\displaystyle\int_{\mathcal{M}}H_{i}(x)d\mu(x)\leq
C\sup\limits_{f\in\mathcal{H}^{\eta}_{1},\
f_{|\mathcal{M}^{\star}}=0}\int_{\mathcal{M}}f(x)d\mu(x).$

###  C.3 Proof of Lemma 8

Proof  Let $T$ be the $(\mathcal{M},\mathcal{M}^{\star})$-compatible map given
by Theorem 6. As
$T_{|_{\mathcal{M}}}^{-1}\in\mathcal{H}^{\beta+1}_{C}(\mathcal{M}^{\star},\mathcal{M}),$
we have that $T_{\\#\mu}$ admits a density
$f_{T}\in\mathcal{H}^{\beta}_{C}(\mathcal{M}^{\star},\mathbb{R})$ with respect
to the volume measure on $\mathcal{M}^{\star}$. For $t\geq C^{-1}$ the radius
of compatibility of the map $T$ (see Definition 4) define the $t$-envelope of
$f_{T}$ and $f_{\mu^{\star}}$ as:

$f_{T}^{t}(x)=\frac{f_{T}(T(x))\Theta(4\|x-T(x)\|^{2}/t^{2})|\text{ap}_{d}(\nabla
T(x))|}{\int_{\mathbb{R}^{p-d}}\Theta(4\|y\|^{2}/t^{2})d\lambda^{p-d}(y)}$

and

$f_{\mu^{\star}}^{t}(x)=\frac{f_{\mu^{\star}}(T(x))\Theta(4\|x-T(x)\|^{2}/t^{2})|\text{ap}_{d}(\nabla
T(x))|}{\int_{\mathbb{R}^{p-d}}\Theta(4\|y\|^{2}/t^{2})d\lambda^{p-d}(y)}$

for $\text{ap}_{d}$ the approximate jacobian defined in Definition 6 and
$\Theta\in\mathcal{H}_{C}^{\beta}(\mathbb{R},\mathbb{R}_{+})$ such that
$\Theta(x)=\Theta(-x)$, $\Theta(0)=1$ and $\Theta_{|(1,\infty)}=0$.

As $\nabla
T\in\mathcal{H}_{K}^{\beta}(\mathbb{R}^{p},L(\mathbb{R}^{p},\mathbb{R}^{p}))$
and $\text{ap}_{d}(\nabla T(x))$ is bounded below by $K_{T}^{-1}$ ($T$ verifes
point (iv) of Definition 4), then from the Faa di Bruno formula, we deduce
that $\text{ap}_{d}(\nabla T)$ belongs to
$\mathcal{H}_{C}^{\beta}(\mathcal{M}^{\star t},\mathbb{R})$. Let
$h\in\mathcal{H}_{K}^{\gamma}(\mathbb{R}^{p},\mathbb{R})$ and define
$\overline{h}:\mathcal{M}^{\star t}\rightarrow\mathbb{R}$ by

$\overline{h}(x)=h(T(x))\kappa(2\|x-T(x)\|^{2}/t^{2}),$

for $\kappa\in\mathcal{H}^{\gamma}_{C}(\mathbb{R},\mathbb{R})$ such that
$\kappa_{|(-\infty,1/2)}=1$ and $\kappa_{|(1,\infty)}=0$. Then
$\overline{h}\in\mathcal{H}^{\gamma}_{C}(\mathbb{R}^{p},\mathbb{R})$ and from
Proposition 21 we have,

$\displaystyle\int_{\mathcal{M}}$ $\displaystyle
h(T(x))d\mu(x)-\int_{\mathcal{M}^{\star}}h(x)d\mu^{\star}(x)=\int_{\mathcal{M}^{\star}}h(x)(f_{T}(x)-f_{\mu^{\star}}(x))d\lambda_{\mathcal{M}^{\star}}(x)$
$\displaystyle=\int_{\mathcal{M}^{\star}}\frac{h(x)(f_{T}(x)-f_{\mu^{\star}}(x))}{\int_{\mathbb{R}^{p-d}}\Theta(4\|y\|^{2}/t^{2})d\lambda^{p-d}(y)}\int_{T^{-1}(\\{x\\})}\Theta(4\|z-T(z)\|^{2}/t^{2})d\lambda^{p-d}(z)d\lambda_{\mathcal{M}^{\star}}(x)$
$\displaystyle=\int_{\mathcal{M}^{\star}}\int_{T^{-1}(\\{x\\})}\frac{h(T(z))(f_{T}(T(z))-f_{\mu^{\star}}(T(z)))\Theta(4\|z-T(z)\|^{2}/t^{2})}{\int_{\mathbb{R}^{p-d}}\Theta(4\|y\|^{2}/t^{2})d\lambda^{p-d}(y)}d\lambda^{p-d}(z)d\lambda_{\mathcal{M}^{\star}}(x)$
$\displaystyle=\int_{\mathbb{R}^{p}}h(T(x))(f_{T}^{t}(T(x))-f_{\mu^{\star}}^{t}(T(x)))d\lambda^{p}(x)$
$\displaystyle=\int_{\mathbb{R}^{p}}\overline{h}(x)(f_{T}^{t}(x)-f_{\mu^{\star}}^{t}(x))d\lambda^{p}(x).$

Let

$\tilde{\Gamma}^{0,-2}_{\epsilon}(\overline{h})(x)=\sum\limits_{j=0}^{\log(\lfloor\epsilon^{-1}\rfloor)}\sum\limits_{l=1}^{2^{d}}\sum\limits_{w\in\mathbb{Z}^{p}}(1+j)^{-2}S(j,l,w)\alpha_{\overline{h}}(j,l,w)\psi_{jlw}(x)$

for
$S(j,l,w)=\frac{\alpha_{\overline{h}}(j,l,w)(\alpha_{f_{T}^{t}}(j,l,w)-\alpha_{f_{\mu^{\star}}^{t}}(j,l,w))}{\alpha_{\overline{h}}(j,l,w)(\alpha_{f_{T}^{t}}(j,l,w)-\alpha_{f_{\mu^{\star}}^{t}}(j,l,w))|}$
. Then using Proposition 26, we have

$\int_{\mathbb{R}^{p}}\overline{h}(x)(f_{T}^{t}(x)-f_{\mu^{\star}}^{t}(x))d\lambda^{p}(x)\leq
C\log(\epsilon^{-1})^{2}\int_{\mathbb{R}^{p}}\tilde{\Gamma}^{0,-2}_{\epsilon}(\overline{h})(f_{T}^{t}(x)-f_{\mu^{\star}}^{t}(x))d\lambda^{p}(x)+C\epsilon.$

Applying Proposition 19 for
$h_{1}=\tilde{\Gamma}^{0,-2}_{\epsilon}(\overline{h})$,
$h_{2}=f_{T}^{t}-f_{\mu^{\star}}^{t}$, $s_{1}=\gamma$, $b_{1}=2$,
$s_{2}=\beta$, $b_{2}=0$, $\tau=\gamma-\eta$, $t=1$,
$r=\frac{\gamma}{\beta+\gamma}$ and $q=\frac{\beta+\eta}{\beta+\gamma}$ we get

$\displaystyle\int_{\mathbb{R}^{p}}$
$\displaystyle\overline{h}^{\epsilon}(x)(f_{T}^{t}(x)-f_{\mu^{\star}}^{t}(x))d\lambda^{p}(x)$
$\displaystyle\leq\Big{\langle}\tilde{\Gamma}^{\gamma-\eta,-2}_{\epsilon}(\overline{h}),f_{T}^{t}-f_{\mu^{\star}}^{t}\Big{\rangle}_{L^{2}(\mathbb{R}^{p})}^{\frac{\beta+\gamma}{\beta+\eta}}\Big{\langle}\tilde{\Gamma}^{\gamma,-2}_{\epsilon}(\overline{h}),\Gamma^{\beta}(f_{T}^{t}-f_{\mu^{\star}}^{t})\Big{\rangle}_{L^{2}(\mathbb{R}^{p})}^{\frac{\eta-\gamma}{\beta+\eta}}$

and

$\displaystyle\sum\limits_{j=0}^{\infty}\sum\limits_{l=1}^{2^{p}}\sum\limits_{w\in\mathbb{Z}^{p}}$
$\displaystyle
2^{j(\beta+\gamma)}|\alpha_{\tilde{\Gamma}^{0,-2}_{\epsilon}(\overline{h})}(j,l,w)||\alpha_{f_{T}}(j,l,w)-\alpha_{f_{\mu^{\star}}}(j,l,w)|$
$\displaystyle\leq\sum\limits_{j=0}^{\infty}\sum\limits_{l=1}^{2^{d}}\sum\limits_{w\in\mathbb{Z}^{p}}(1+j)^{-2}C2^{-jp}\mathds{1}_{\\{supp(\psi_{jlw}\cap
B^{p}(0,K))\neq\varnothing\\}}$ $\displaystyle\leq
C\sum\limits_{j=0}^{\infty}(1+j)^{-2}\leq C.$

Define $F_{h}:\mathcal{M}^{\star}\rightarrow\mathbb{R}$ by

$F_{h}(x)=\int_{T^{-1}(\\{x\\})}\tilde{\Gamma}^{\gamma-\eta,-2}_{\epsilon}(\overline{h})(z)\
\frac{\Theta(4\|z-T(z)\|^{2}/t^{2})}{\int_{\mathbb{R}^{p-d}}\Theta(4\|y\|^{2}/t^{2})d\lambda^{p-d}(y)}d\lambda_{T^{-1}(\\{x\\})}(z).$

Then

$\displaystyle\int_{\mathbb{R}^{p}}\tilde{\Gamma}^{\gamma-\eta,-2}_{\epsilon}(\overline{h})(x)(f_{T}^{t}(x)-f_{\mu^{\star}}^{t}(x))d\lambda^{p}(x)$
$\displaystyle=\int_{\mathcal{M}^{\star}}\int_{T^{-1}(\\{x\\})}\tilde{\Gamma}^{\gamma-\eta,-2}_{\epsilon}(\overline{h})(z)(f_{T}^{t}(z)-f_{\mu^{\star}}^{t}(z))|\text{ap}_{d}(\nabla
T(z))|^{-1}d\lambda^{p-d}(z)d\lambda_{\mathcal{M}^{\star}}(x)$
$\displaystyle=\int_{\mathcal{M}^{\star}}\int_{T^{-1}(\\{x\\})}\tilde{\Gamma}^{\gamma-\eta,-2}_{\epsilon}(\overline{h})(z)\frac{(f_{T}(T(z))-f_{\mu^{\star}}(T(z)))\Theta(4\frac{\|z-T(z)\|^{2}}{t^{2}})}{\int_{\mathbb{R}^{p-d}}\Theta(4\|y\|^{2}/t^{2})d\lambda^{p-d}(y)}d\lambda^{p-d}(z)d\lambda_{\mathcal{M}^{\star}}(x)$
$\displaystyle=\int_{\mathcal{M}^{\star}}F_{h}(x)(f_{T}(x)-f_{\mu^{\star}}(x))d\lambda_{\mathcal{M}^{\star}}(x).$

As $T$ verifies point iii) of Definition 4, we have
$F_{h}\in\mathcal{H}^{\eta}_{C}(\mathcal{M}^{\star},\mathbb{R})$ so it can be
extended using Proposition 25 to a map in
$\mathcal{H}^{\eta}_{C}(\mathbb{R}^{p},\mathbb{R})$. Then, we can conclude
that

$\displaystyle\int_{\mathcal{M}^{\star}}h(x)(f_{T}(x)-f_{\mu^{\star}}(x))d\lambda_{\mathcal{M}^{\star}}(x)\leq
C\sup\limits_{h\in\mathcal{H}^{\eta}_{1}}\left(\int_{\mathcal{M}^{\star}}h(x)(f_{T}(x)-f_{\mu^{\star}}(x))d\lambda_{\mathcal{M}^{\star}}(x)\right)^{\frac{\beta+\gamma}{\beta+\eta}}.$

###  C.4 Proof of Theorem 9

Proof  Let $C_{\eta}$ the constant given by Theorem 6 such that if

$d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})\leq C_{\eta}^{-1},$

then there exists a map $T$ being
$(\mathcal{M},\mathcal{M}^{\star})$-compatible.

Suppose first that
$d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})>C_{\eta}^{-1}$, then

$d_{\mathcal{H}^{\gamma}_{1}}(\mu,\mu^{\star})\leq 2K\leq
2KC_{\eta}d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})\leq
Cd_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})^{\frac{\beta+\gamma}{\beta+\eta}}.$

Suppose now that $d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})\leq
C_{\eta}^{-1}$. Using Corollary 20 we have for all $\epsilon\in(0,1)$

$d_{\mathcal{H}^{\gamma}_{1}}(\mu,\mu^{\star})\leq
C\log(\epsilon^{-1})^{2}d_{\mathcal{H}^{1}_{1}}(\mu,\mu^{\star})^{\frac{\eta-\gamma}{\eta-1}}d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})^{\frac{\gamma-1}{\eta-1}}+\epsilon.$
(37)

Suppose first that $\eta\leq\beta+1$. Using lemmas 7 and 8 we have

$\displaystyle d_{\mathcal{H}^{1}_{1}}(\mu,\mu^{\star})$ $\displaystyle\leq
C\log(\epsilon^{-1})^{4}\left(\sup\limits_{\begin{subarray}{c}h\in\mathcal{H}^{\eta}_{1}\\\
h_{|\mathcal{M}^{\star}}=0\end{subarray}}\left(\int_{\mathcal{M}}h(x)d\mu(x)\right)^{\frac{\beta+1}{\beta+\eta}}+d_{\mathcal{H}^{\eta}_{1}}(T_{\\#\mu},\mu^{\star})^{\frac{\beta+1}{\beta+\eta}}\right)+\epsilon.$

On one hand we have

$\sup\limits_{\begin{subarray}{c}h\in\mathcal{H}^{\eta}_{1}\\\
h_{|\mathcal{M}^{\star}}=0\end{subarray}}\int_{\mathcal{M}}h(x)d\mu(x)\leq\sup\limits_{h\in\mathcal{H}^{\eta}_{1}}\int_{\mathcal{M}}h(x)d\mu(x)-\int_{\mathcal{M}^{\star}}h(x)d\mu^{\star}(x)=d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star}),$

on the other hand

$d_{\mathcal{H}^{\eta}_{1}}(T_{\\#\mu},\mu^{\star})=\sup\limits_{h\in\mathcal{H}^{\eta}_{1}}\int_{\mathcal{M}}h(T(x))d\mu(x)-\int_{\mathcal{M}^{\star}}h(T(x))d\mu^{\star}(x)\leq
Cd_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star}),$

as any map $h\circ T\in\mathcal{H}^{\eta}_{C}(\mathcal{M}^{\star
t/4},\mathbb{R})$ can be extended to a map in
$\mathcal{H}^{\eta}_{C}(\mathbb{R}^{p},\mathbb{R})$ using Proposition 25.
Therefore, we have

$d_{\mathcal{H}^{1}_{1}}(\mu,\mu^{\star})\leq
C\log(\epsilon^{-1})^{4}d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})^{\frac{\beta+1}{\beta+\eta}}+\epsilon.$

Then plugging this in (37) we obtain

$\displaystyle d_{\mathcal{H}^{\gamma}_{1}}(\mu,\mu^{\star})$
$\displaystyle\leq
C\log(\epsilon^{-1})^{6}d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})^{\frac{\beta+1}{\beta+\eta}\frac{\eta-\gamma}{\eta-1}+\frac{\gamma-1}{\eta-1}}+\epsilon$
$\displaystyle=C\log(\epsilon^{-1})^{6}d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})^{\frac{\beta+\gamma}{\beta+\eta}}+\epsilon$

so taking
$\epsilon=d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})^{\frac{\beta+\gamma}{\beta+\eta}}$,
we get the result.

Now if $\eta>\beta+1$, using lemmas 7 and 8 we have again

$\displaystyle d_{\mathcal{H}^{1}_{1}}(\mu,\mu^{\star})$ $\displaystyle\leq
d_{\mathcal{H}^{1}_{1}}(\mu,T_{\\#\mu})+d_{\mathcal{H}^{1}_{1}}(T_{\\#\mu},\mu^{\star})$
$\displaystyle\leq
C\log(\epsilon^{-1})^{4}d_{\mathcal{H}^{\beta+1}_{1}}(\mu,\mu^{\star})^{\frac{\beta+1}{2\beta+1}}+\epsilon.$

And using again Corollary 20 we have

$d_{\mathcal{H}^{\beta+1}_{1}}(\mu,\mu^{\star})\leq
C\log(\epsilon^{-1})^{2}d_{\mathcal{H}^{1}_{1}}(\mu,\mu^{\star})^{\frac{\eta-(\beta+1)}{\eta-1}}d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})^{\frac{\beta}{\eta-1}}+\epsilon$

so

$\displaystyle d_{\mathcal{H}^{1}_{1}}(\mu,\mu^{\star})\leq
C\log(\epsilon^{-1})^{6}d_{\mathcal{H}^{1}_{1}}(\mu,\mu^{\star})^{\frac{\eta-(\beta+1)}{\eta-1}\frac{\beta+1}{2\beta+1}}d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})^{\frac{\beta}{\eta-1}\frac{\beta+1}{2\beta+1}}+\epsilon.$

Then, taking
$\epsilon=d_{\mathcal{H}^{1}_{1}}(\mu,\mu^{\star})d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})$,
we have

$\displaystyle d_{\mathcal{H}^{1}_{1}}(\mu,\mu^{\star})\leq
C\log(d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})^{-1})^{6}d_{\mathcal{H}^{1}_{1}}(\mu,\mu^{\star})^{\frac{\eta-(\beta+1)}{\eta-1}\frac{\beta+1}{2\beta+1}}d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})^{\frac{\beta}{\eta-1}\frac{\beta+1}{2\beta+1}},$

which gives

$\displaystyle
d_{\mathcal{H}^{1}_{1}}(\mu,\mu^{\star})^{(\eta-1)(2\beta+1)-(\eta-1-\beta)(\beta+1)}\leq$
$\displaystyle
C\log(d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})^{-1})^{6(\eta-1)(2\beta+1)}d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})^{\beta(\beta+1)}.$

As $(\eta-1)(2\beta+1)-(\eta-1-\beta)(\beta+1))=\beta(\eta+\beta)$ we obtain

$\displaystyle d_{\mathcal{H}^{1}_{1}}(\mu,\mu^{\star})\leq$ $\displaystyle
C\log(d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})^{-1})^{6\frac{(\eta-1)(2\beta+1)}{\beta(\eta+\beta)}}d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})^{\frac{\beta+1}{\beta+\eta}}.$

Then plugging this result in (37), we obtain for
$\epsilon=d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})$,

$\displaystyle d_{\mathcal{H}^{\gamma}_{1}}(\mu,\mu^{\star})$
$\displaystyle\leq
C\log(d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})^{-1})^{C_{2}}d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})^{\frac{\beta+1}{\beta+\eta}\frac{\eta-\gamma}{\eta-1}+\frac{\gamma-1}{\eta-1}}$
$\displaystyle=C\log(d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})^{-1})^{C_{2}}d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})^{\frac{\beta+\gamma}{\beta+\eta}}.$

###  C.5 Proof of Proposition 10

Proof  As explained at the beginning of the proof of Theorem 9, we can suppose
that the distance $d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})$ is small
enough so that there exists a $(\mathcal{M},\mathcal{M}^{\star})$-compatible
map $T$. We have

$\displaystyle d_{\mathcal{H}^{\gamma}_{1}}(\mu,\mu^{\star})$
$\displaystyle\leq
d_{\mathcal{H}^{\gamma}_{1}}(\mu,T_{\\#\mu})+d_{\mathcal{H}^{\gamma}_{1}}(T_{\\#\mu},\mu^{\star})$

Using Lemma 8 we obtain

$\displaystyle d_{\mathcal{H}^{\gamma}_{1}}(T_{\\#\mu},\mu^{\star})$
$\displaystyle\leq
C\log(\epsilon^{-1})^{2}d_{\mathcal{H}^{\eta}_{1}}(T_{\\#\mu},\mu^{\star})^{\frac{\beta+\gamma}{\beta+\eta}}+\epsilon$
$\displaystyle\leq
Cd_{\mathcal{H}^{\eta}_{1}}(T_{\\#\mu},\mu^{\star})^{\frac{\gamma}{\eta}}$

for
$\epsilon=d_{\mathcal{H}^{\eta}_{1}}(T_{\\#\mu},\mu^{\star})^{\frac{\gamma}{\eta}}$.
On the other hand,

$\displaystyle d_{\mathcal{H}^{\gamma}_{1}}(\mu,T_{\\#\mu})$
$\displaystyle=\sup\limits_{h\in\mathcal{H}^{\gamma}_{1}}\int_{\mathcal{M}}(h(x)-h(T(x)))f_{\mu}(x)d\lambda_{\mathcal{M}}(x)$
$\displaystyle\leq\int_{\mathcal{M}}\|x-T(x)\|^{\gamma}f_{\mu}(x)d\lambda_{\mathcal{M}}(x)$
$\displaystyle\leq\left(\int_{\mathcal{M}}\|x-T(x)\|^{\eta}f_{\mu}(x)d\lambda_{\mathcal{M}}(x)\right)^{\frac{\gamma}{\eta}}$

using Jensen’s inequality. The function $H:\mathcal{M}^{\star
t}\rightarrow\mathbb{R}$ defined by

$H(x)=\|x-T(x)\|^{\eta},$

belongs to $\mathcal{H}^{\eta}_{C}(\mathcal{M}^{\star t},\mathbb{R})$ and can
therefore be extend to a map in
$\mathcal{H}^{\eta}_{C}(\mathbb{R}^{p},\mathbb{R})$ using Proposition 25.

###  C.6 Proof of Proposition 12

Proof  Let $s\in(0,r_{\mathcal{M}^{\star}})$, we are going to show that
$W_{1}(\mu,\mu^{\star})$ is bounded below which will give us that
$d_{\mathcal{H}^{\eta}_{1}}(\mu,\mu^{\star})$ is also bounded below by Lemma
5.

Suppose first that $\mathbb{H}(\mathcal{M},\mathcal{M}^{\star})>s$ and for
$y\in\mathcal{M}^{\star}$ such that $d(y,\mathcal{M})\geq s$, define
$D^{y}_{s}:\mathbb{R}^{p}\rightarrow\mathbb{R}$ as

$D_{s}^{y}(x)=(s-\|x-y\|)\vee 0.$

As $D_{s}$ is $1$-Lipschitz, we have

$\displaystyle W_{1}(\mu,\mu^{\star})$
$\displaystyle\geq\left(\int_{\mathcal{M}^{\star}}D_{s}^{y}(x)f_{\mu^{\star}}(x)d\lambda_{\mathcal{M}^{\star}}(x)-\int_{\mathcal{M}}D_{s}^{y}(x)f_{\mu}(x)d\lambda_{\mathcal{M}}(x)\right)$
$\displaystyle=\int_{\mathcal{M}^{\star}\cap
B^{p}(y,s)}(s-\|x-y\|)f_{\mu^{\star}}(x)d\lambda_{\mathcal{M}^{\star}}$
$\displaystyle\geq Cs^{d}K^{-1}$

which gives the result. Now if
$\mathbb{H}(\mathcal{M},\mathcal{M}^{\star})\leq s$, by Lemma 22 we have that
for $s>0$ small enough, $\pi$ the canonical projection onto
$\mathcal{M}^{\star}$, is a diffeomorphism from $\mathcal{M}$ to
$\mathcal{M}^{\star}$. Le be $z\in\mathcal{M}$ such that $f_{\mu}(z)\leq s$.
Define $D_{s}^{z}:\mathbb{R}^{p}\rightarrow\mathbb{R}$ as

$D_{s}(x)=(\frac{s}{2}\wedge(s-\|x-\pi(z)\|))\vee 0.$

As $D_{s}$ is $1$-Lipschitz, we have

$\displaystyle W_{1}(\mu,\mu^{\star})$
$\displaystyle\geq\int_{\mathcal{M}^{\star}}D_{s}^{z}(x)f_{\mu^{\star}}(x)d\lambda_{\mathcal{M}^{\star}}(x)-\int_{\mathcal{M}}D_{s}^{z}(x)f_{\mu}(x)d\lambda_{\mathcal{M}}(x)$

and

$\displaystyle\int_{\mathcal{M}^{\star}}D_{s}^{z}(x)f_{\mu^{\star}}(x)d\lambda_{\mathcal{M}^{\star}}(x)$
$\displaystyle=\int_{\mathcal{M}^{\star}\cap
B^{p}(\pi(z),s)}\frac{s}{2}\wedge(s-\|x-\pi(z)\|)f_{\mu^{\star}}(x)d\lambda_{\mathcal{M}^{\star}}(x)$
$\displaystyle\geq\int_{\mathcal{M}^{\star}\cap
B^{p}(\pi(z),s/2)}\frac{s}{2}f_{\mu^{\star}}(x)d\lambda_{\mathcal{M}^{\star}}(x)$
$\displaystyle\geq Cs^{d+1}K^{-1}.$

On the other hand,

$\displaystyle\int_{\mathcal{M}}D_{s}^{z}(x)f_{\mu}(x)d\lambda_{\mathcal{M}}(x)$
$\displaystyle=\int_{\mathcal{M}\cap
B^{p}(\pi(z),s)}\frac{s}{2}\wedge(s-\|x-\pi(z)\|)f_{\mu}(x)d\lambda_{\mathcal{M}}(x)$
$\displaystyle\leq\int_{\mathcal{M}\cap
B^{p}(\pi(z),s)}sf_{\mu}(x)d\lambda_{\mathcal{M}}(x)$
$\displaystyle=s\int_{\varphi_{x}(\mathcal{M}\cap
B^{p}(\pi(z),s))}f_{\mu}(\varphi_{x}^{-1}(u))d\varphi_{x\\#\lambda_{\mathcal{M}}}(u)$
$\displaystyle\leq
s\int_{B^{d}(\varphi_{x}(z),2s)}f_{\mu}(\varphi_{x}^{-1}(u))K^{d}d\lambda^{d}(u)$
$\displaystyle\leq
Cs\int_{B^{d}(\varphi_{x}(\pi(z)),2s)}(f_{\mu}(z)+K\|\varphi_{x}^{-1}(z)-u\|))d\lambda^{d}(u)$
$\displaystyle\leq
Cs\int_{B^{d}(\varphi_{x}(\pi(z)),2s)}(1+2K)sd\lambda^{d}(u)$
$\displaystyle\leq Cs^{d+2}.$

Therefore we have

$\displaystyle W_{1}(\mu,\mu^{\star})$ $\displaystyle\geq Cs^{d+1}(1-C_{2}s),$

so for $s$ small enough we get the result.

###  C.7 Additional proofs from Section 3.1

####  C.7.1 Proof of Proposition 2

Proof  Using (6), we are looking for the smallest $q\in(0,1)$ such that
$d_{\mathcal{B}^{s-\frac{q^{\star}}{q}\tau}_{\infty,\infty}}(f,f^{\star})$ is
finite. Taking $f,f^{\star}\in\mathcal{B}^{\beta,2}_{\infty,\infty}$ compactly
supported in $B^{p}(0,K)$ we have

$\displaystyle
d_{\mathcal{B}^{s-\frac{q^{\star}}{q}\tau}_{\infty,\infty}}(f,f^{\star})$
$\displaystyle=\sup\limits_{h\in\mathcal{B}^{s-\frac{q^{\star}}{q}\tau}_{\infty,\infty}(1)}\sum\limits_{j=0}^{\infty}\sum\limits_{l=1}^{2^{p}}\sum\limits_{w\in\mathbb{Z}^{p}}\alpha_{h}(j,l,w)(\alpha_{f}(j,l,w)-\alpha_{f^{\star}}(j,l,w))$
$\displaystyle=\sum\limits_{j=0}^{\infty}\sum\limits_{l=1}^{2^{p}}\sum\limits_{w\in\mathbb{Z}^{p}}2^{-j(s-\frac{q^{\star}}{q}\tau+p/2)}\alpha_{h}(j,l,w)|\alpha_{f}(j,l,w)-\alpha_{f^{\star}}(j,l,w)|$
$\displaystyle\leq\sum\limits_{j=0}^{\infty}\sum\limits_{l=1}^{2^{p}}\sum\limits_{w\in\mathbb{Z}^{p}}2^{-j(s-\frac{q^{\star}}{q}\tau+p/2)}2^{-j(\beta+p/2)}(1+j)^{-2}\|f-f^{\star}\|_{\mathcal{B}^{\beta,2}_{\infty,\infty}}\mathds{1}_{\\{supp(\psi_{jlw})\cap
B^{p}(0,K)\neq\varnothing\\}}$ $\displaystyle\leq
C\|f-f^{\star}\|_{\mathcal{B}^{\beta,2}_{\infty,\infty}}\sum\limits_{j=0}^{\infty}2^{-j(s-\frac{q^{\star}}{q}\tau+\beta)}(1+j)^{-2}.$

The last quantity is finite iff $\beta\geq\frac{q^{\star}}{q}\tau-s$ which
gives $\frac{1}{q}\leq\frac{\beta+s}{\beta+s+\tau}$.

####  C.7.2 Proof of Theorem 3

Proof  Using lemma 1 we have that
$f-f^{\star}\in\mathcal{B}^{\beta}_{\infty,\infty}(C)$ and
$h\in\mathcal{B}^{\gamma}_{\infty,\infty}(C)$, then for $\epsilon\in(0,1)$ we
have

$\displaystyle\int_{\mathbb{R}^{p}}$ $\displaystyle
h(x)(f(x)-f^{\star}(x))d\lambda^{p}(x)$ $\displaystyle=$
$\displaystyle\sum\limits_{j=0}^{\infty}\sum\limits_{l=1}^{2^{p}}\sum\limits_{w\in\mathbb{Z}^{p}}\alpha_{h}(j,l,w)(\alpha_{f}(j,l,w)-\alpha_{f^{\star}}(j,l,w))$
$\displaystyle\leq$
$\displaystyle\sum\limits_{j=0}^{\log_{2}(\epsilon^{-1})}\sum\limits_{l=1}^{2^{p}}\sum\limits_{w\in\mathbb{Z}^{p}}\alpha_{h}(j,l,w)(\alpha_{f}(j,l,w)-\alpha_{f^{\star}}(j,l,w))$
$\displaystyle+\sum\limits_{j=\log_{2}(\epsilon^{-1})+1}^{\infty}\sum\limits_{l=1}^{2^{p}}\sum\limits_{w\in\mathbb{Z}^{p}}2^{-j(\gamma+\beta+p)}\mathds{1}_{\\{supp(\psi_{jlw})\cap
B^{p}(0,K)\neq\varnothing\\}}$ $\displaystyle\leq$
$\displaystyle\log_{2}(\epsilon^{-1})^{2}\sum\limits_{j=0}^{\log_{2}(\epsilon^{-1})}\sum\limits_{l=1}^{2^{p}}\sum\limits_{w\in\mathbb{Z}^{p}}(1+j)^{-2}|\alpha_{h}(j,l,w)(\alpha_{f}(j,l,w)-\alpha_{f^{\star}}(j,l,w))|$
$\displaystyle+C\epsilon^{\gamma+\beta}$ $\displaystyle\leq$ $\displaystyle
C\log_{2}(\epsilon^{-1})^{2}d_{\mathcal{B}^{\gamma}_{\infty,\infty}(1)}(\Gamma^{0,-2}(f),\Gamma^{0,-2}(f^{\star}))+C\epsilon^{\gamma+\beta}.$

As $f,f^{\star}\in\mathcal{B}^{\beta,2}_{\infty,\infty}(C)$, we can now use
Proposition 2 and obtain

$\displaystyle
d_{\mathcal{B}^{\gamma}_{\infty,\infty}(1)}(\Gamma^{0,-2}(f),\Gamma^{0,-2}(f^{\star}))$
$\displaystyle\leq
Cd_{\mathcal{B}^{\alpha}_{\infty,\infty}(1)}(\Gamma^{0,-2}(f),\Gamma^{0,-2}(f^{\star}))^{\frac{\beta+\gamma}{\beta+\alpha}}$
$\displaystyle=d_{\mathcal{B}^{\gamma,2}_{\infty,\infty}(1)}(f,f^{\star})^{\frac{\beta+\gamma}{\beta+\alpha}}$
$\displaystyle\leq
Cd_{\mathcal{H}^{\gamma}_{C}}(f,f^{\star})^{\frac{\beta+\gamma}{\beta+\alpha}},$

where we used Lemma 1 for the last inequality. Putting everything together and
taking $\epsilon=d_{\mathcal{H}^{\alpha}_{1}}(f,f^{\star})\wedge 1/2$ we
obtain the result.