the existence of a unique polynomial $P_{k}(z,z_{0},\rho,f)$ such that

(B.6)
$\inf_{p\in\mathcal{P}_{k}}\int_{\Omega(x_{0},\rho)}|f(z)-p(z)|^{q}\,{\rm
d}z=\int_{\Omega(x_{0},\rho)}|f(z)-P_{k}(z,z_{0},\rho,f)|^{q}\,{\rm d}z$

Indeed, $P_{k}(z,z_{0},\rho,f)$ is the kinetic Taylor expansion of $f$ at
$z_{0}$. Let $P\in\mathcal{P}_{k}$ and write

$P(z)=\sum_{j\in{\mathbb{N}}^{1+2d},|J|\leq
k}\frac{a_{j}(z_{0})}{j!}(z-z_{0})^{j}.$

We denote

$\displaystyle h(\\{a_{j}\\})=\|f-P\|_{L^{q}(\Omega(z_{0},\rho))},$

where $\Omega(z_{0},\rho)=Q_{R}(\tilde{z}_{0})\cap Q_{\rho}(z_{0})$ with
$Q_{R}(\tilde{z}_{0})$ as in the statement of Theorem 2.7. Note that $h$ is a
non-negative continuous real function of the coefficients of $P$. The infimum
of $h$ will be attained in a compact set containing the origin, so that the
existence of $P_{k}$ follows standardly. The uniqueness of $P_{k}$ follows by
uniform convexity of the Lebesgue spaces $L^{q}$. We will denote the
coefficients of $P_{k}(z,z_{0},\rho,f)$ with $a_{j}(z_{0},\rho)$. Note that
they are given by

(B.7)
$a_{j}(z_{0},\rho,f)=(\partial_{t}+v\cdot\nabla_{x})^{j_{0}}\partial_{x_{1}}^{j_{1}}\cdots\partial_{x_{d}}^{j_{d}}\partial_{v_{1}}^{j_{d+1}}\cdots\partial_{v_{d}}^{j_{2d}}P_{k}(z,z_{0},\rho,f)\big{|}_{z=z_{0}}.$

######  Lemma B.2.

For $f\in\mathcal{L}^{q,\lambda}_{k}(\Omega)$ there exists a constant
$c(q,\lambda)>0$ such that for any $z_{0}\in\Omega$ and
$0<\rho\leq\textrm{diam }\Omega$ and $l\in\mathbb{N}_{0}$ there holds

$\int_{\Omega(z_{0},\rho 2^{-(l-1)})}\big{|}P_{k}(z,z_{0},\rho
2^{-l},f)-P_{k}(z,z_{0},\rho 2^{-l-1},f)\big{|}^{q}\,{\rm d}z\leq
c2^{-l\lambda}\rho^{\lambda}[f]_{\mathcal{L}^{q,\lambda}_{k}}^{q}$

###### Proof.

For all $z\in\Omega\big{(}z_{0},\rho 2^{-(l-1)}\big{)}$ there holds

$\displaystyle|P_{k}(z,z_{0},\rho 2^{-l},f)-P_{k}(z,z_{0},\rho
2^{-l-1},f)|^{q}\leq 2^{q}|P_{k}(z,z_{0},\rho
2^{-l},f)-f(z)|^{q}+2^{q}|P_{k}(z,z_{0},\rho 2^{-(l-1)},f)-f(z)|^{q}$

Thus

$\displaystyle\int_{\Omega(z_{0},\rho 2^{-(l-1)})}|P_{k}(z,z_{0},\rho
2^{-l},f)-P_{k}(z,z_{0},\rho 2^{-l-1},f)|^{q}\,{\rm d}z$ $\displaystyle\leq
2^{q}[f]_{\mathcal{L}^{q,\lambda}_{k}}^{q}\big{(}2^{-l\lambda}\rho^{\lambda}+2^{(-l-1)\lambda}\rho^{\lambda}\big{)}$
$\displaystyle=2^{q}(1+2^{-\lambda})2^{-l\lambda}\rho^{\lambda}[f]_{\mathcal{L}^{q,\lambda}_{k}}^{q}.$

∎

######  Lemma B.3.

Suppose $f\in\mathcal{L}^{q,\lambda}_{k}(\Omega)$. Then for any
$z_{0},z_{1}\in\bar{\Omega}$ and for any multi-index $l$ such that $|L|=k$
with $|L|=2s\cdot l_{0}+(1+2s)|L_{1}|+|L_{2}|$ there holds

(B.8)
$\displaystyle\big{|}a_{l}(z_{0},2d_{\ell}(z_{0},z_{1}),f)-a_{l}(z_{1},2d_{\ell}(z_{0},z_{1}),f)\big{|}^{q}\leq
c2^{q+1+\lambda}[f]_{\mathcal{L}^{q,\lambda}_{k}}^{q}d_{\ell}(z_{0},z_{1})^{\lambda-
n-kq},$

where $d_{\ell}$ is the kinetic distance defined in 2.1.

###### Proof.

Let $z_{0},z_{1}\in\bar{\Omega}$. We write $\rho=d_{\ell}(z_{0},z_{1})$ and
$I_{\rho}=\Omega(z_{0},2\rho)\cap\Omega(z_{1},2\rho)$. Then we have

$\displaystyle|P_{k}(z,z_{0},2\rho,f)-P_{k}(z,z_{1},2\rho,f)|^{q}\leq
2^{q}|P_{k}(z,z_{0},2\rho,f)-f(z)|^{q}+2^{q}|P_{k}(z,z_{1},2\rho,f)-f(z)|^{q}.$

Integrating over $\Omega(z_{0},\rho)\subset I_{\rho}$ we obtain

(B.9) $\displaystyle\int_{\Omega(z_{0},\rho)}$
$\displaystyle|P_{k}(z,z_{0},2\rho,f)-P_{k}(z,z_{1},2\rho,f)|^{q}\,{\rm d}z$
$\displaystyle\leq
2^{q}\int_{\Omega(z_{0},\rho)}|P_{k}(z,z_{0},2\rho,f)-f(z)|^{q}\,{\rm
d}z+2^{q}\int_{\Omega(z_{0},\rho)}|P_{k}(z,z_{1},2\rho,f)-f(z)|^{q}\,{\rm d}z$
$\displaystyle\leq
2^{q+\lambda+1}\rho^{\lambda}[f]_{\mathcal{L}^{q,\lambda}_{k}}^{q}.$

On the other hand, by (B.7), and Lemma B.1 applied to
$P(z)=P_{k}(z,z_{0},2\rho,f)-P_{k}(z,z_{1},2\rho,f)$ and since the $k$-th
derivative of a polynomial of degree $k$ is constant, we have

(B.10) $\displaystyle\big{|}a_{l}\big{(}z_{0},2d_{\ell}(z_{0},z_{1}),f\big{)}$
$\displaystyle-a_{l}\big{(}z_{1},2d_{\ell}(z_{0},z_{1}),f\big{)}\big{|}^{q}$
$\displaystyle\leq
c\rho^{-(n+kq)}\int_{\Omega(z_{0},\rho)}\big{|}P_{k}(z,z_{0},2\rho,f)-P_{k}(z,z_{1},2\rho,f)\big{|}^{q}\,{\rm
d}z.$

Finally, the combination of (B.9) and (B.10) implies (B.8) and concludes the
proof. ∎

######  Lemma B.4.

Let $f\in\mathcal{L}_{k}^{q,\lambda}(\Omega)$. Then there exists a constant
$c$ such that for all $z_{0}\in\bar{\Omega},0<\rho\leq\textrm{diam }\Omega$,
$i\in{\mathbb{N}}$ and multi-index $l\in{\mathbb{N}}^{1+2d}$ with $|L|\leq k$
there holds

$\displaystyle\big{|}a_{l}(z_{0},\rho,f)-a_{l}(z_{0},\rho 2^{-i},f)\big{|}\leq
c[f]_{\mathcal{L}_{k}^{q,\lambda}}\sum_{m=0}^{i-1}2^{m\big{(}\frac{n+|L|q-\lambda}{q}\big{)}}\rho^{\frac{\lambda-n-|L|q}{q}}.$

###### Proof.

We have

$\displaystyle\big{|}a_{l}(z_{0},\rho,f)-a_{l}(z_{0},\rho
2^{-i},f)\big{|}\leq\sum_{m=0}^{i-1}\big{|}a_{l}(z_{0},\rho
2^{-m},f)-a_{l}(z_{0},\rho 2^{-m-1},f)\big{|}.$

Using the relation (B.7) and applying Lemma B.1 to $P_{k}(z,z_{0},\rho
2^{-m},f)-P_{k}(z,z_{0},\rho 2^{-m-1},f)$ we get

$\displaystyle\big{|}a_{l}(z_{0},\rho,f)-a_{l}(z_{0},\rho 2^{-i},f)\big{|}$
$\displaystyle\quad\leq
c\rho^{-\frac{n}{q}-|L|}\sum_{m=0}^{i-1}2^{(m+1)\big{(}\frac{n}{q}+|L|\big{)}}\Bigg{[}\int_{\Omega(z_{0},\rho
2^{-m-1})}|P_{k}(z,z_{0},\rho 2^{-m},f)-P_{k}(z,z_{0},\rho
2^{-m-1},f)|^{q}\,{\rm d}z\Bigg{]}^{\frac{1}{q}}.$

We conclude using Lemma B.2. ∎

Now we can prove the following useful lemma.

######  Lemma B.5.

Let $f\in\mathcal{L}_{k}^{q,\lambda}(\Omega)$ such that
$n+\tilde{k}q<\lambda\leq n+(\tilde{k}+1)q$ where $0\leq\tilde{k}\leq k$. Then
there exists functions $\\{g_{j}(z_{0})\\}$ for $j\in{\mathbb{N}}^{1+2d}$ with
$|J|\leq\tilde{k}$ such that for all $0<\rho\leq\textrm{diam
}\bar{\Omega},z_{0}\in\bar{\Omega}$ there holds

(B.11) $\big{|}a_{j}(z_{0},\rho,f)-g_{j}(z_{0})\big{|}\leq
c(\lambda,q,k,n,B)\rho^{\frac{\lambda-n-|J|q}{q}}[f]_{\mathcal{L}^{q,\lambda}_{k}}.$

As a consequence, there holds

(B.12) $\lim_{\rho\to 0}a_{j}(z_{0},\rho,f)=g_{j}(z_{0}),$

uniformly with respect to $z_{0}$.

###### Proof.

We show that the sequence $\\{a_{j}(z_{0},\rho 2^{-i},f)\\}$ converges in the
limit $i\to\infty$. Let $i_{1},i_{2}$ be two non-negative integers and assume
without loss in generality that $i_{2}>i_{1}$. With Lemma B.4 we obtain

$\displaystyle\big{|}a_{j}(z_{0},\rho 2^{-i_{2}},f)-a_{j}(z_{0},\rho
2^{-i_{1}},f)\big{|}\leq
c[f]_{\mathcal{L}_{k}^{q,\lambda}}\sum_{m=i_{1}}^{i_{2}-1}2^{m\big{(}\frac{n+|J|q-\lambda}{q}\big{)}}\rho^{\frac{\lambda-n-|J|q}{q}}.$

Since $|J|\leq p=\tilde{k}$ and $\lambda>n+\tilde{k}q$ the series
$\sum_{m=0}^{\infty}2^{m\big{(}\frac{n+|J|q-\lambda}{q}\big{)}}$ converges.
Thus $\\{a_{j}(z_{0},\rho 2^{-i},f)\\}$ is a Cauchy sequence and hence
converges as $i\to\infty$.

We now show that the limit is uniform in $\rho$. Let $\rho_{1}$ and $\rho_{2}$
be such that $0<\rho_{1}\leq\rho_{2}\leq\textrm{diam }\Omega$. With Lemma B.1
we get

$\displaystyle\big{|}a_{j}(z_{0},\rho_{1}2^{-i},f)-a_{j}(z_{0},\rho_{2}2^{-i},f)\big{|}^{q}$
$\displaystyle\leq
c\frac{2^{i(n+|J|q)}}{\rho_{1}^{n+|J|q}}\int_{\Omega(z_{0},\rho_{1}2^{-i})}\big{|}P_{k}(z,z_{0},\rho_{1}2^{-i},f)-P_{k}(z,z_{0},\rho_{2}2^{-i},f)\big{|}^{q}\,{\rm
d}z$ $\displaystyle\leq
c\frac{2^{i(n+|J|q)}}{\rho_{1}^{n+|J|q}}\Bigg{[}\int_{\Omega(z_{0},\rho_{1}2^{-i})}\big{|}P_{k}(z,z_{0},\rho_{1}2^{-i},f)-f(z)\big{|}^{q}\,{\rm
d}z$
$\displaystyle\qquad\qquad\qquad+\int_{\Omega(z_{0},\rho_{2}2^{-i})}\big{|}P_{k}(z,z_{0},\rho_{2}2^{-i},f)-f(z)\big{|}^{q}\,{\rm
d}z\Bigg{]}$ $\displaystyle\leq
c2^{q}\frac{\rho_{1}^{\lambda}+\rho_{2}^{\lambda}}{\rho_{1}^{n+|J|q}}2^{-i(\lambda-n-|J|q)}[f]_{\mathcal{L}^{q,\lambda}_{k}}\rightarrow
0,$

as $i\to\infty$ since $\lambda-n-|J|q>0$.

Thus for $z_{0}\in\bar{\Omega},0<\rho\leq\textrm{diam }(\Omega)$ and
$|J|\leq\tilde{k}$ we can take

(B.13) $g_{j}(z_{0})=\lim_{i\to\infty}a_{j}(z_{0},\rho 2^{-i},f).$

The sequence $g_{j}(z_{0})$ is well-defined in $\bar{\Omega}$. Since the
series $\sum_{m=0}^{\infty}2^{m\big{(}\frac{n+|J|q-\lambda}{q}\big{)}}$
converges, we deduce from Lemma B.4

(B.14) $\displaystyle\big{|}a_{j}(z_{0},\rho,f)-a_{j}(z_{0},\rho
2^{-i},f)\big{|}\leq
c[f]_{\mathcal{L}_{k}^{q,\lambda}}\rho^{\frac{\lambda-n-|J|q}{q}}.$

Combining (B.13) and (B.14) yields the result. ∎

###  B.3. The function $g_{j}(z_{0})$

We have the following theorem.

######  Theorem B.6.

Let $f\in\mathcal{L}^{q,\lambda}_{k}(\Omega)$ with $n+kq<\lambda$. Then the
functions $g_{j}(z_{0})$ with $|J|=k$ are Hölder continuous in $\bar{\Omega}$
and for any $z_{1},z_{2}\in\bar{\Omega}$ there holds

(B.15) $|g_{j}(z_{1})-g_{j}(z_{2})|\leq
c[f]_{\mathcal{L}^{q,\lambda}_{k}}d_{\ell}(z_{1},z_{2})^{\frac{\lambda-n-
kq}{q}}.$

###### Proof.

Take $z_{1},z_{2}\in\bar{\Omega}$ such that
$\rho=d_{\ell}(z_{1},z_{2})\leq\frac{\textrm{diam }\Omega}{2}$. Then

$\displaystyle|g_{j}(z_{1})-g_{j}(z_{2})|\leq|g_{j}(z_{1})-a_{j}(z_{1},2\rho)|+|g_{j}(z_{2})-a_{j}(z_{2},2\rho)|+|a_{j}(z_{1},2\rho)-a_{j}(z_{2},2\rho)|.$

On the one hand, by (B.11) we have

$\displaystyle|g_{j}(z_{1})-a_{j}(z_{1},2\rho)|\leq c2^{\frac{\lambda-n-
kq}{q}}\rho^{\frac{\lambda-n-kq}{q}}[f]_{\mathcal{L}^{q,\lambda}_{k}},$

and

$\displaystyle|g_{j}(z_{2})-a_{j}(z_{2},2\rho)|\leq c2^{\frac{\lambda-n-
kq}{q}}\rho^{\frac{\lambda-n-kq}{q}}[f]_{\mathcal{L}^{q,\lambda}_{k}}.$

On the other hand (B.8) implies

$\displaystyle|a_{j}(z_{1},2\rho)-a_{j}(z_{2},2\rho)|\leq
c2^{\frac{q+1+\lambda}{q}}\rho^{\frac{\lambda-n-
kq}{q}}[f]_{\mathcal{L}^{q,\lambda}_{k}}.$

This yields the result in case that
$d_{\ell}(z_{1},z_{2})\leq\frac{\textrm{diam }\Omega}{2}$.

In case that $d_{l}(z_{1},z_{2})>\frac{\textrm{diam }\Omega}{2}$ we can
construct a polygon contained in $\bar{\Omega}$ with extremal points $z_{1}$
and $z_{2}$ and with sides of length smaller or equal to $\frac{\textrm{diam
}\Omega}{2}$, see Figure 1. The length of the sides can be bounded by
$\textrm{diam }\Omega$ uniformly with respect to $z_{1}$ and $z_{2}$. Thus to
conclude it suffices to apply (B.15) to all points at the end of the sides of
such a polygonal.

$\Omega$$z_{1}$$z_{2}$$z_{3}$$l_{1}$$l_{2}$ Figure 1. In case that
$d_{l}(z_{1},z_{2})>\frac{\textrm{diam }\Omega}{2}$ we construct a polygon
with side lengths $l_{1},l_{2}$ such that $l_{1},l_{2}\leq\frac{\textrm{diam
}\Omega}{2}$.

∎

For the sequel, we denote by $(0)$ the $d$-tuple $(0,\dots,0)$ and by $e_{i}$
the vector in ${\mathbb{R}}^{d}$ with the $i$-th coordinate equal to $1$ and
else $0$. We also note that any polynomial degree
$k\in{\mathbb{N}}+2s{\mathbb{N}}$ can be written as $k=2s\cdot
k_{0}+(1+2s)\cdot k_{1}+k_{2}$ with $k_{0},k_{1},k_{2}\in{\mathbb{N}}$.

######  Theorem B.7.

Let $f\in\mathcal{L}^{q,\lambda}_{k}(\Omega)$ with $k_{0},k_{1},k_{2}\geq 1$
and $n+kq<\lambda$. Then for any multi-index $j\in{\mathbb{N}}^{1+2d}$ such
that $|J|\leq k$ the function $g_{j}$ has a first partial derivative in
$\Omega$, and for any $z\in\Omega$ and $i=1,\dots,d$ there holds

(B.16)
$\displaystyle\mathcal{T}g_{j}(z)=g_{(j_{0}+1,J_{1},J_{2})}(z),\qquad\qquad
j_{0}\leq k_{0}-1,|J_{1}|\leq k_{1},|J_{2}|\leq k_{2}$
$\displaystyle\frac{\partial g_{j}(z)}{\partial
x_{i}}=g_{(j_{0},J_{1}+e_{i},J_{2})}(z),\qquad\qquad j_{0}\leq
k_{0},|J_{1}|\leq k_{1}-1,|J_{2}|\leq k_{2}$ $\displaystyle\frac{\partial
g_{j}(z)}{\partial v_{i}}=g_{(0,J_{1},J_{2}+e_{i})}(z),\qquad\qquad
j_{0}=0,|J_{1}|\leq k_{1},|J_{2}|\leq k_{2}-1.$

###### Proof.

For this proof we omit the dependency on $f$ in the coefficients
$a_{j}(z_{0},\rho,f)$ and $P_{k}(z,z_{0},\rho,f)$ and simply write
$a_{j}(z_{0},\rho)$ and $P_{k}(z,z_{0},\rho)$, respectively.

Step 1. We will start proving the first line. We consider
$j=(j_{0},J_{1},J_{2})$ for $j_{0}\leq k_{0}-1,|J_{1}|=k_{1},|J_{2}|=k_{2}$.
Theorem B.15 proves that $g_{(k_{0},J_{1},J_{2})}$ is Hölder continuous in a
classical sense for $|J_{1}|=k_{1},|J_{2}|=k_{2}$ and in particular
continuous. Thus we may assume that $g_{(j_{0}+\delta,J_{1},J_{2})}$ is
continuous in $\bar{\Omega}$ for $\delta=1,\dots,k_{0}-j_{0}$. Let
$z_{0}\in\Omega$ and $\rho$ be such that $B_{|\rho|}(z_{0})\subset\Omega$. By
(B.7) we have

(B.17)
$\displaystyle\frac{a_{j}\big{(}z_{0}+(\rho,(0),(0)),2|\rho|\big{)}-a_{j}(z_{0},2|\rho|)}{\rho}$
$\displaystyle=\frac{D^{j}\big{[}P_{k}\big{(}z,z_{0}+(\rho,(0),(0)),2|\rho|\big{)}-P_{k}(z,z_{0},2|\rho|)\big{]}}{\rho}$
$\displaystyle\qquad-\sum_{\delta=1}^{k_{0}-j_{0}}\frac{(-1)^{\delta}}{\delta!}\rho^{\delta-1}a_{j}\big{(}z_{0}+(\rho,(0),(0)),2|\rho|\big{)}.$

With Lemma B.1 and (B.9) we obtain

(B.18)
$\displaystyle\Bigg{|}\frac{D^{j}\big{[}P_{k}\big{(}z,z_{0}+(\rho,(0),(0)),2|\rho|\big{)}-P_{k}(z,z_{0},2|\rho|)\big{]}}{\rho}\Bigg{|}^{q}$
$\displaystyle\qquad\qquad\leq
c{|\rho|}^{-n-|J|q}\int_{\Omega(z_{0},|\rho|)}\Big{|}P_{k}\big{(}z,z_{0}+(\rho,(0),(0)),2|\rho|\big{)}-P_{k}(z,z_{0},2|\rho|)\Big{|}^{q}\,{\rm
d}z$ $\displaystyle\qquad\qquad\leq
c2^{q+\lambda+1}|\rho|^{\lambda-n-|J|q}[f]_{\mathcal{L}^{q,\lambda}_{k}}.$

Moreover, for $1\leq\delta\leq k_{0}-j_{0}$ there holds

(B.19)
$\displaystyle\Big{|}a_{(j_{0}+\delta,J_{1},J_{2})}\big{(}z_{0}+(\rho,(0),(0)),2|\rho|\big{)}-g_{(j_{0}+\delta,J_{1},J_{2})}(z_{0})\Big{|}$
$\displaystyle\qquad\qquad\leq\big{|}a_{(j_{0}+\delta,J_{1},J_{2})}\big{(}z_{0}+(\rho,(0),(0)),2|\rho|\big{)}-g_{(j_{0}+\delta,J_{1},J_{2})}\big{(}z_{0}+(\rho,(0),(0))\big{)}\big{|}$
$\displaystyle\qquad\qquad\quad+\big{|}g_{(j_{0}+\delta,J_{1},J_{2})}\big{(}z_{0}+(\rho,(0),(0))\big{)}-g_{(j_{0}+\delta,J_{1},J_{2})}(z_{0})\big{|}.$

Using (B.11) we can estimate the first term on the right hand side of (B.19)
by

(B.20)
$\displaystyle\Big{|}a_{(j_{0}+\delta,J_{1},J_{2})}\big{(}z_{0}+(\rho,(0),(0)),2|\rho|\big{)}-g_{(j_{0}+\delta,J_{1},J_{2})}$
$\displaystyle\big{(}z_{0}+(\rho,(0),(0))\big{)}\Big{|}$ $\displaystyle\leq
c2^{\frac{\lambda-n-(|J|+2s\delta)}{q}}|\rho|^{\frac{\lambda-n-(|J|+2s\delta)}{q}}[f]_{\mathcal{L}_{k}^{q,\lambda}}.$

From (B.19) and (B.20) and since by induction hypothesis
$g_{(j_{0}+\delta,J_{1},J_{2})}$ are continuous for
$\delta=1,\dots,k_{0}-j_{0}$ we have

(B.21) $\lim_{\rho\to
0}a_{(j+\delta,J_{1},J_{2})}\big{(}z_{0}+(\rho,(0),(0)),2|\rho|\big{)}=g_{(j_{0}+\delta,J_{1},J_{2})}(z_{0})\qquad\delta=1,\dots,k_{0}-j_{0}.$

Thus from (B.17), (B.18) and (B.21) we deduce that

$\lim_{\rho\to
0}\frac{a_{j}\big{(}z_{0}+(\rho,(0),(0)),2|\rho|\big{)}-a_{j}(z_{0},2|\rho|)}{\rho}=g_{(j_{0}+1,J_{1},J_{2})}(z_{0}),$

uniformly in $z_{0}$. Thus if we can show that

(B.22) $\lim_{\rho\to
0}\frac{g_{j}\big{(}z_{0}+(\rho,(0),(0))\big{)}-g_{j}(z_{0})}{\rho}=\lim_{\rho\to
0}\frac{a_{j}\big{(}z_{0}+(\rho,(0),(0)),2|\rho|\big{)}-a_{j}(z_{0},2\rho)}{\rho},$

then we can conclude the proof of the first line of (B.16). We first notice
that by (B.11)

(B.23)
$\displaystyle\Bigg{|}\frac{g_{j}\big{(}z_{0}+(\rho,(0),(0))\big{)}-a_{j}\big{(}z_{0}+(\rho,(0),(0)),2|\rho|\big{)}}{\rho}\Bigg{|}\leq
c2^{\frac{\lambda-n-|J|q}{q}}|\rho|^{\frac{\lambda-n}{q}-|J|-1}[f]_{\mathcal{L}^{q,\lambda}_{k}},$

and

(B.24)
$\displaystyle\Big{|}\frac{g_{j}(z_{0})-a_{j}(z_{0},2|\rho|)}{\rho}\Big{|}\leq
c2^{\frac{\lambda-n-|J|q}{q}}|\rho|^{\frac{\lambda-n}{q}-|J|-1}[f]_{\mathcal{L}^{q,\lambda}_{k}}.$

Thus with the triangle inequality (B.23) and (B.24) imply (B.22), which in
turn implies the first line of(B.16).

Step 2. To prove the second statement in (B.16) we proceed as in Step 1. Now
we consider $j_{0}=1,\dots,k_{0},|J_{1}|\leq k_{1}-1$ and $|J_{2}|=k_{2}$. We
have shown that $g_{j}$ is continuous for
$j_{0}=1,\dots,k_{0},|J_{1}|=k_{1},|J_{2}|=k_{2}$. Assume then that
$g_{(j_{0},J_{1}+\delta e_{i},J_{2})}$ is continuous in $\bar{\Omega}$ for
$\delta=1,\dots,k_{1}-|J_{1}|$. We again have by (B.7)

(B.25)
$\displaystyle\frac{a_{j}\big{(}z_{0}+\rho(0,e_{i},(0)),2|\rho|\big{)}-a_{j}(z_{0},2|\rho|)}{\rho}$
$\displaystyle=\frac{D^{j}\big{[}P_{k}\big{(}z,z_{0}+\rho(0,e_{i},(0)),2|\rho|\big{)}-P_{k}(z,z_{0},2|\rho|)\big{]}}{\rho}$
$\displaystyle\quad-\sum_{\delta=1}^{k_{1}-|J_{1}|}\frac{(-1)^{\delta}}{\delta!}\rho^{\delta-1}a_{j}\big{(}z_{0}+\rho(0,e_{i},(0)),2|\rho|\big{)}.$

The proof is exactly the same if we replace $(\rho,(0),(0))$ with
$\rho(0,e_{i},(0))$, $k_{0}-j_{0}$ with $k_{1}-|J_{1}|$ and instead of
$2s\delta$ in the exponent of (B.20) we get $(1+2s)\delta$.

Step 3. To deduce the final statement in (B.16) the ideas are the same but the
statement only holds for $j_{0}=0$ since $\mathcal{T}$ and $D_{v}$ do not
commute. Therefore it was important to prove the first statement first, since
now we know that $g_{j}$ is continuous for $j_{0}=0,|J_{1}|\leq k_{1}$ and
$|J_{2}|=k_{2}$. We now assume that $g_{(j_{0},J_{1},J_{2}+\delta e_{i})}$ is
continuous in $\bar{\Omega}$ for $\delta=1,\dots,k_{2}-|J_{2}|$. Replacing
$(\rho,(0),(0))$ with $\rho(0,(0),e_{i})$, $k_{0}-j_{0}$ with $k_{2}-|J_{2}|$
and $2s\delta$ in the exponent of (B.20) with $\delta$, but otherwise
proceeding as above, we conclude.

Finally, combining the argument for the continuity of $g_{j}$ in all three
steps yields the improvement in ranges of $|J_{1}|$ and $|J_{2}|$ as stated in
the theorem. ∎

As a corollary of Theorem B.15 and B.16 we get

######  Theorem B.8.

Let $f\in\mathcal{L}^{q,\lambda}_{k}(\Omega)$ with $n+kq<\lambda$. Then the
function $g_{(0)}\in C_{\ell}^{\beta}(\bar{\Omega})$ where
$\beta=\frac{\lambda-n}{q}$ and there holds

$\mathcal{T}^{j_{0}}D_{x}^{J_{1}}D_{v}^{J_{2}}g_{(0)}(z)=g_{j}(z)\qquad\forall
z\in\Omega,~{}\forall|J|\leq k.$

Recall $j=(j_{0},J_{1},J_{2})\in{\mathbb{N}}^{1+2d}$ and $|J|=2s\cdot
j_{0}+(1+2s)\cdot|J_{1}|+|J_{2}|$.

######  Remark B.9.

For $f\in\mathcal{L}^{q,\lambda}_{k}(\Omega)$ with $n+(k+1)q<\lambda$ we
deduce from (B.15) that $g_{j}$ with $|J|=k$ are constant and thus by Theorem
B.8, $g_{(0)}$ is a polynomial of kinetic degree at most $k$.

###  B.4. Comparing the Hölder norm and the Campanato norm

######  Theorem B.10.

Let $f\in\mathcal{L}^{q,\lambda}_{k}(\Omega)$ with $n+kq<\lambda\leq
n+(k+1)q$. Then $f\in C_{\ell}^{\beta}(\Omega)$ where
$\beta=\frac{\lambda-n}{q}$ and there holds

(B.26) $[f]_{C_{\ell}^{\beta}}\leq c[f]_{\mathcal{L}^{q,\lambda}_{k}}.$

If $\lambda>n+(k+1)q$ then $f$ is a polynomial of kinetic degree at most $k$.

###### Proof.

Due to Theorem B.8 and Remark B.9 it suffices to show that
$f(z)=g_{(0)}(z)=\lim_{\rho\to 0}a_{(0)}(z,\rho)$ for almost every
$z\in\Omega$. Then (B.26) follows from (B.15) in Theorem B.15 and Taylor’s
formula.

Since $f\in L^{q}(\Omega)$ there holds for almost every $z_{0}\in\Omega$

(B.27) $\lim_{\rho\to
0}\frac{1}{|\Omega(z_{0},\rho)|}\int_{\Omega(z_{0},\rho)}|f(z)-f(z_{0})|^{q}\,{\rm
d}z=0.$

Now let $z_{0}\in\Omega$ be such that (B.27) holds. Then for almost every
$z\in\Omega$ we have

$\displaystyle\big{|}a_{(0)}(z_{0},\rho)-f(z_{0})\big{|}^{q}\leq
c\Big{(}\big{|}P_{k}(z,z_{0},\rho)-a_{(0)}(z_{0},\rho)\big{|}^{q}+\big{|}P_{k}(z,z_{0},\rho)-f(z)\big{|}^{q}+|f(z)-f(z_{0})|^{q}\Big{)}.$

Integrating this inequality over $\Omega(z_{0},\rho)$ yields

(B.28) $\displaystyle\big{|}a_{(0)}(z_{0},\rho)-f(z_{0})\big{|}^{q}$
$\displaystyle\leq\frac{c}{A_{1}\rho^{n}}\int_{\Omega(z_{0},\rho)}\big{|}P_{k}(z,z_{0},\rho)-a_{(0)}(z_{0},\rho)\big{|}^{q}\,{\rm
d}z$
$\displaystyle\quad+\frac{c}{A_{1}\rho^{n}}\int_{\Omega(z_{0},\rho)}\big{|}P_{k}(z,z_{0},\rho)-f(z)\big{|}^{q}\,{\rm
d}z+\frac{c}{A_{1}\rho^{n}}\int_{\Omega(z_{0},\rho)}|f(z)-f(z_{0})|^{q}\,{\rm
d}z.$

By definition of $\mathcal{L}^{q,\lambda}_{k}$ we have

$\frac{c}{R^{-n}|Q_{R}(\tilde{z}_{0})|\rho^{n}}\int_{\Omega(z_{0},\rho)}\big{|}P_{k}(z,z_{0},\rho)-f(z)\big{|}^{q}\,{\rm
d}z\leq
c\frac{\rho^{\lambda-n}}{R^{-n}|Q_{R}(\tilde{z}_{0})|}[f]_{\mathcal{L}^{q,\lambda}_{k}}\xrightarrow[\rho\to
0]{}0.$

Due to (B.27) the last integral in (B.28) vanishes as well in the limit
$\rho\to 0$. Finally there holds

$\displaystyle\frac{c}{R^{-n}|Q_{R}(\tilde{z}_{0})|\rho^{n}}\int_{\Omega(z_{0},\rho)}\big{|}P_{k}(z,z_{0},\rho)-a_{(0)}(z_{0},\rho)\big{|}^{q}\,{\rm
d}z\leq c(n,q,k)\sum_{\begin{subarray}{c}j\in{\mathbb{N}}^{1+2d},\\\ |J|\leq
k\end{subarray}}\big{|}a_{j}(z_{0},\rho)\big{|}^{q}\rho^{|J|q}.$

Due to (B.12) this integral vanishes in the limit $\rho\to 0$, so that (B.28)
gives for almost every $z_{0}\in\Omega$

$\lim_{\rho\to 0}a_{(0)}(z_{0},\rho)=f(z_{0}).$

Equivalently, there holds $f(z)=g_{(0)}(z)$ almost everywhere in $\Omega$. ∎

###### Proof of Theorem 2.7.

If $f\in\mathcal{L}_{k}^{p,\lambda}(\Omega)$, then Theorem B.10 yields $f\in
C_{\ell}^{\beta}(\Omega)$ and the Hölder semi-norm is bounded above by the
Campanato semi-norm (B.26).

Conversely, let $f\in C_{\ell}^{\beta}(\bar{\Omega})$ and
$P\in\mathcal{P}_{k}$ where $k=\rm{deg}_{\rm{kin}}P<\beta$. For $z\in
Q_{r}(z_{0})\cap\Omega$ we have

$|f(z)-P(z)|\leq[f]_{C_{\ell}^{\beta}}r^{\beta}.$

Thus for $\beta=\frac{\lambda-n}{p}$ there holds

$\displaystyle\frac{1}{r^{\lambda}}\int_{Q_{r}(z_{0})\cap\Omega}|f(z)-P(z)|^{p}\,{\rm
d}z\leq
C[f]^{p}_{C_{\ell}^{\beta}}r^{p\beta-\lambda+n}=C[f]^{p}_{C_{\ell}^{\beta}}.$

∎

##  Appendix C Interpolation Inequality for Hölder spaces

For the sake of completeness, we prove Lemma 2.12 following the arguments of
Imbert-Silvestre [19, Proposition 2.10].

###### Proof of Lemma 2.12.

It suffices to prove the statement for $\beta_{3}$ sufficiently close to
$\beta_{1}$. Thus we assume that there exists only one element
$\bar{\beta}\in{\mathbb{N}}+2s{\mathbb{N}}$ such that
$\bar{\beta}\in[\beta_{1},\beta_{3})$. We know that if
$p^{i}_{z}\in\mathcal{P}_{\beta_{i}}$ is the polynomial expansion of $f$ at
$z$ of order less than $\beta_{i}$ for all $i\in\\{1,\dots,3\\}$, then for all
$z\circ\xi\in Q_{1}$

(C.1)
$\big{|}f(z\circ\xi)-p^{i}_{z}(\xi)\big{|}\leq[f]_{C_{\ell}^{\beta_{i}}}\|\xi\|^{\beta_{i}},\quad
i=1,2,3.$

The polynomials $p_{z}^{i}$ are of increasingly higher order. We assume that
the difference of degree of homogeneity of $p_{z}^{1}$ and $p_{z}^{3}$ is at
most one, so that $p_{z}^{2}$ coincides with either $p_{z}^{1}$ or
$p_{z}^{3}$, depending on whether $\bar{\beta}\geq\beta_{2}$ or
$\bar{\beta}<\beta_{2}$. If there is no $\bar{\beta}$ then all three
polynomials coincide. Let us first assume therefore that there is exactly one
$\bar{\beta}$. We have by subtracting (C.1) for $i=1,3$ from each other

(C.2)
$|p_{z}^{3}(\xi)-p_{z}^{1}(\xi)|\leq[f]_{C_{\ell}^{\beta_{1}}}\|\xi\|^{\beta_{1}}+[f]_{C_{\ell}^{\beta_{3}}}\|\xi\|^{\beta_{3}}.$

For any $R\in(0,1]$ and $z\in Q_{1}$ we pick $\xi_{1}\in Q_{1}$ such that
$\|\xi_{1}\|\leq R$ and whenever $d_{\ell}(\xi_{1},\xi)<cR$, then $\|\xi\|\leq
R$ and $z\circ\xi\in Q_{1}$ with some universal constant $c$. From (C.2) we
then have

$\sup_{\xi:d_{\ell}(\xi_{1},\xi)\leq
cR}|p_{z}^{3}(\xi)-p_{z}^{1}(\xi)|\leq[f]_{C_{\ell}^{\beta_{1}}}R^{\beta_{1}}+[f]_{C_{\ell}^{\beta_{3}}}R^{\beta_{3}}.$

Since $p_{z}^{3}-p_{z}^{1}$ is homogeneous of degree $\bar{\beta}$ we get by
scaling

$\sup_{\xi:d_{l}((\xi_{1})_{R^{-1}},\xi)\leq
c}|p_{z}^{3}(\xi)-p_{z}^{1}(\xi)|\leq[f]_{C_{\ell}^{\beta_{1}}}R^{\beta_{1}-\bar{\beta}}+[f]_{C_{\ell}^{\beta_{3}}}R^{\beta_{3}-\bar{\beta}}.$

Using the triangle inequality from [19, Prop. 2.2] we can assure that whenever
$|\xi|\leq 1$ then $d_{\ell}\big{(}(\xi_{1})_{R^{-1}},\xi\big{)}\leq C$ for
some universal constant $C$. Since all norms on the space of polynomials are
equivalent, we have

$\displaystyle\|p_{z}^{3}-p_{z}^{1}\|$ $\displaystyle=\sup_{\xi:\|\xi\|\leq
1}|p_{z}^{3}(\xi)-p_{z}^{1}(\xi)|\leq
C\sup_{\xi:d_{\ell}\big{(}(\xi_{1})_{R^{-1}},\xi\big{)}\leq
c}|p_{z}^{3}(\xi)-p_{z}^{1}(\xi)|$ $\displaystyle\leq
C[f]_{C_{\ell}^{\beta_{1}}}R^{\beta_{1}-\bar{\beta}}+C[f]_{C_{\ell}^{\beta_{3}}}R^{\beta_{3}-\bar{\beta}}.$

For

$R=\Bigg{(}\frac{[f]_{C_{\ell}^{\beta_{1}}}}{[f]_{C_{\ell}^{\beta_{3}}}}\Bigg{)}^{\frac{1}{\beta_{3}-\beta_{1}}}$

we obtain

$\|p_{z}^{3}-p_{z}^{1}\|\leq
C[f]_{C_{\ell}^{\beta_{1}}}^{\bar{\theta}}[f]_{C_{\ell}^{\beta_{3}}}^{1-\bar{\theta}}+[f]_{C_{\ell}^{\beta_{1}}},$

where $\bar{\beta}=\bar{\theta}\beta_{1}+(1-\bar{\theta})\beta_{3}$.

Therefore we can estimate $f-p_{z}^{2}$. Assume first
$\beta_{2}\leq\bar{\beta}$. Then $p_{z}^{2}=p_{z}^{1}$ and

$|f(z\circ\xi)-p_{z}^{2}(\xi)|\leq\begin{cases}[f]_{C_{\ell}^{\beta_{1}}}\|\xi\|^{\beta_{1}},\\\
[f]_{C_{\ell}^{\beta_{3}}}\|\xi\|^{\beta_{3}}+\left([f]_{C_{\ell}^{\beta_{1}}}^{\bar{\theta}}[f]_{C_{\ell}^{\beta_{3}}}^{1-\bar{\theta}}+[f]_{C_{\ell}^{\beta_{1}}}\right)\|\xi\|^{\bar{\beta}}.\end{cases}$

Now if $\|\xi\|\geq R$ then

$[f]_{C_{\ell}^{\beta_{1}}}\|\xi\|^{\beta_{1}}\leq[f]_{C_{\ell}^{\beta_{1}}}^{\theta}[f]_{C_{\ell}^{\beta_{3}}}^{1-\theta}\|\xi\|^{\beta_{2}}.$

Else if $\|\xi\|<R$

$[f]_{C_{\ell}^{\beta_{3}}}\|\xi\|^{\beta_{3}}+\Big{(}[f]_{C_{\ell}^{\beta_{1}}}^{\bar{\theta}}[f]_{C_{\ell}^{\beta_{3}}}^{1-\bar{\theta}}+[f]_{C_{\ell}^{\beta_{1}}}\Big{)}\|\xi\|^{\bar{\beta}}\leq[f]_{C_{\ell}^{\beta_{1}}}^{\theta}[f]_{C_{\ell}^{\beta_{3}}}^{1-\theta}\|\xi\|^{\beta_{2}}+[f]_{C_{\ell}^{\beta_{1}}}\|\xi\|^{\bar{\beta}}.$

Thus we conclude
$|f(z\circ\xi)-p_{z}^{2}(\xi)|\leq[f]_{C_{\ell}^{\beta_{1}}}^{\theta}[f]_{C_{\ell}^{\beta_{3}}}^{1-\theta}\|\xi\|^{\beta_{2}}+[f]_{C_{\ell}^{\beta_{1}}}\|\xi\|^{\beta_{2}}$.

In case that $\bar{\beta}<\beta_{2}$

$|f(z\circ\xi)-p_{z}^{2}(\xi)|\leq\begin{cases}[f]_{C_{\ell}^{\beta_{1}}}\|\xi\|^{\beta_{1}}+\Big{(}[f]_{C_{\ell}^{\beta_{1}}}^{\bar{\theta}}[f]_{C_{\ell}^{\beta_{3}}}^{1-\bar{\theta}}+[f]_{C_{\ell}^{\beta_{1}}}\Big{)}\|\xi\|^{\bar{\beta}},\\\
[f]_{C_{\ell}^{\beta_{3}}}\|\xi\|^{\beta_{3}}.\end{cases}$

and we conclude as above.

In case that no $\bar{\beta}$ exists, then all polynomials coincide and we get

$|f(z\circ\xi)-p_{z}^{2}(\xi)|\leq[f]_{C_{\ell}^{\beta_{1}}}^{\theta}[f]_{C_{\ell}^{\beta_{3}}}^{1-\theta}\|\xi\|^{\beta_{2}}.$

∎

##  Appendix D Proof of Bouchut’s Proposition

For the sake of self-containment, we recall the proof of Proposition 3.4 from
[3, Proposition 1.1].

###### Proof of Proposition 3.4.

We denote by $\hat{f}(\eta,k,v)$ the Fourier-transform of a solution $f$ of
(3.7) in time $t$ and space $x$. Then $\hat{f}$ solves

$i(\eta+v\cdot k)\hat{f}=\hat{S}.$

We introduce a smoothing sequence $\rho_{1}\in
C_{c}^{\infty}({\mathbb{R}}^{d})$ in velocity such that

(D.1)
$\rho_{\varepsilon}(v)=\frac{1}{\varepsilon^{d}}\rho_{1}\Big{(}\frac{v}{\varepsilon}\Big{)},\qquad\int\rho_{1}\,{\rm
d}v=1,\qquad\int v^{\alpha}\rho_{1}=0\textrm{ for }1\leq|\alpha|<|\beta|.$

For fixed $(\eta,k)$ we decompose

(D.2)
$\hat{f}(\eta,k,v)=\left(\rho_{\varepsilon}\ast_{v}\hat{f}\right)(\eta,k,v)+\left(\hat{f}-\left(\rho_{\varepsilon}\ast_{v}\hat{f}\right)\right)(\eta,k,v),$

where $\ast_{v}$ denotes the convolution in velocity $v$. Then by the
properties of $\rho$ (D.1) we can bound $|1-\hat{\rho}_{\varepsilon}|\leq
C_{d,\beta}|\varepsilon v|^{\beta}$ so that

(D.3)
$\Bigg{\|}\left(\hat{f}-\left(\rho_{\varepsilon}\ast_{v}\hat{f}\right)\right)(\eta,k,\cdot)\Bigg{\|}_{L^{2}({\mathbb{R}}^{d})}\leq
C_{d,\beta}\varepsilon^{\beta}\big{\|}|D_{v}|^{\beta}\hat{f}(\eta,k,\cdot)\big{\|}_{L^{2}({\mathbb{R}}^{d})}.$

For the first term in (D.2) we introduce $\lambda>0$ such that

$\big{(}\lambda+i(\eta+v\cdot
k)\big{)}\hat{f}(\eta,k,v)=\lambda\hat{f}(\eta,k,v)+\hat{S}(\eta,kv).$

Equivalently,

$\hat{f}(\eta,k,v)=\frac{\lambda\hat{f}(\eta,k,v)+\hat{S}(\eta,k,v)}{\lambda+i(\eta+v\cdot
k)},$

which yields

$\left(\rho_{\varepsilon}\ast_{v}\hat{f}\right)(\eta,k,v)=\int\frac{\lambda\hat{f}(\eta,k,\xi)+\hat{S}(\eta,k,\xi)}{\lambda+i(\eta+\xi\cdot
k)}\rho_{\varepsilon}(v-\xi)\,{\rm d}\xi.$

Then we bound

$\displaystyle\Big{|}$
$\displaystyle\left(\rho_{\varepsilon}\ast_{v}\hat{f}\right)(\eta,k,v)\Big{|}$
$\displaystyle\leq\Big{(}\big{\|}\hat{f}(\eta,k,\cdot)|\rho_{\varepsilon}(v-\cdot)|^{\frac{1}{2}}\big{\|}_{L^{2}({\mathbb{R}}^{d})}+\lambda^{-1}\big{\|}\hat{S}(\eta,k,\cdot)|\rho_{\varepsilon}(v-\cdot)|^{\frac{1}{2}}\big{\|}_{L^{2}({\mathbb{R}}^{d})}\Big{)}\Bigg{(}\int\frac{|\rho_{\varepsilon}(v-\xi)|}{|1+i(\eta+\xi\cdot
k)\lambda^{-1}|^{2}}\,{\rm d}\xi\Bigg{)}^{\frac{1}{2}}.$

The last integral is estimated using $|\rho_{\varepsilon}(v)|\leq
C_{d,\beta}\varepsilon^{-d}\chi_{|v|\leq\varepsilon}$, and decomposing
$\xi=\tilde{\xi}\frac{k}{|k|}+\xi^{\perp}$ with $\xi^{\perp}\cdot k=0$, so
that

$\displaystyle\int\frac{|\rho_{\varepsilon}(v-\xi)|}{|1+i(\eta+\xi\cdot
k)\lambda^{-1}|^{2}}\,{\rm d}\xi\leq
C_{d,\beta}\frac{1}{\varepsilon}\int\frac{\chi_{|\frac{v\cdot
k}{|k|}-\tilde{\xi}|<\varepsilon}}{|1+i(\eta+\tilde{\xi}\cdot
k)\lambda^{-1}|^{2}}\,{\rm d}\tilde{\xi}\leq
C_{d,\beta}\frac{\lambda}{\varepsilon|k|}.$

Thus

$\displaystyle\Big{\|}\left(\rho_{\varepsilon}\ast_{v}\hat{f}\right)(\eta,k,\cdot)\Big{\|}_{L^{2}({\mathbb{R}}^{d})}\leq
C_{d,\beta}\Big{(}\frac{\lambda}{\varepsilon|k|}\Big{)}^{\frac{1}{2}}\Big{(}\big{\|}\hat{f}(\eta,k,\cdot)\big{\|}_{L^{2}({\mathbb{R}}^{d})}+\lambda^{-1}\big{\|}\hat{S}(\eta,k,\cdot)\big{\|}_{L^{2}({\mathbb{R}}^{d})}\Big{)}.$

Choosing

$\lambda=\frac{\big{\|}\hat{S}(\eta,k,\cdot)\big{\|}_{L^{2}({\mathbb{R}}^{d})}}{\big{\|}\hat{f}(\eta,k,\cdot)\big{\|}_{L^{2}({\mathbb{R}}^{d})}}$

yields

(D.4)
$\Big{\|}\left(\rho_{\varepsilon}\ast_{v}\hat{f}\right)(\eta,k,\cdot)\Big{\|}_{L^{2}({\mathbb{R}}^{d})}\leq\frac{C_{d,\beta}}{\sqrt{\varepsilon|k|}}\big{\|}\hat{f}(\eta,k,\cdot)\big{\|}_{L^{2}({\mathbb{R}}^{d})}^{\frac{1}{2}}\big{\|}\hat{S}(\eta,k,\cdot)\big{\|}_{L^{2}({\mathbb{R}}^{d})}^{\frac{1}{2}}.$

Combining (D.2) with (D.3) and (D.4) yields

$\big{\|}\hat{f}(\eta,k,\cdot)\big{\|}_{L^{2}({\mathbb{R}}^{d})}\leq\frac{C_{d,\beta}}{\sqrt{\varepsilon|k|}}\big{\|}\hat{f}(\eta,k,\cdot)\big{\|}_{L^{2}({\mathbb{R}}^{d})}^{\frac{1}{2}}\big{\|}\hat{S}(\eta,k,\cdot)\big{\|}_{L^{2}({\mathbb{R}}^{d})}^{\frac{1}{2}}+C_{d,\beta}\varepsilon^{\beta}\big{\|}|D_{v}|^{\beta}\hat{f}(\eta,k,\cdot)\big{\|}_{L^{2}({\mathbb{R}}^{d})}.$

We finally optimise $\varepsilon$ so that

$\big{\|}\hat{f}(\eta,k,\cdot)\big{\|}_{L^{2}({\mathbb{R}}^{d})}\leq\Bigg{(}\frac{1}{|k|}\big{\|}\hat{f}(\eta,k,\cdot)\big{\|}_{L^{2}({\mathbb{R}}^{d})}\big{\|}\hat{S}(\eta,k,\cdot)\big{\|}_{L^{2}({\mathbb{R}}^{d})}\Bigg{)}^{\frac{\beta}{1+2\beta}}\big{\|}|D_{v}|^{\beta}\hat{f}(\eta,k,\cdot)\big{\|}^{\frac{1}{1+2\beta}}_{L^{2}({\mathbb{R}}^{d})}.$

Dividing by
$\big{\|}\hat{f}(\eta,k,\cdot)\big{\|}_{L^{2}({\mathbb{R}}^{d})}^{\frac{\beta}{1+2\beta}}$
yields

$\big{\|}\hat{f}(\eta,k,\cdot)\big{\|}_{L^{2}({\mathbb{R}}^{d})}\leq\Bigg{(}\frac{1}{|k|}\big{\|}\hat{S}(\eta,k,\cdot)\big{\|}_{L^{2}({\mathbb{R}}^{d})}\Bigg{)}^{\frac{\beta}{1+\beta}}\big{\|}|D_{v}|^{\beta}\hat{f}(\eta,k,\cdot)\big{\|}^{\frac{1}{1+\beta}}_{L^{2}({\mathbb{R}}^{d})},$

which concludes the proof of (3.8) after integrating over $(\eta,k)$. ∎

Acknowledgements. We thank Clément Mouhot for his continuous support and
intuitive discussions on the subject. This work was supported by the Cambridge
International & Newnham College Scholarship from the Cambridge Trust.

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# Reversal of quantised Hall drifts at non-interacting and interacting
topological boundaries

Zijie Zhu  Marius Gächter  Anne-Sophie Walter  Konrad Viebahn
<EMAIL_ADDRESS>Tilman Esslinger  Institute for Quantum Electronics &
Quantum Center, ETH Zurich, 8093 Zurich, Switzerland

###### Abstract

The transport properties of gapless edge modes at boundaries between
topologically distinct domains are of fundamental and technological
importance. Therefore, it is crucial to gain a better understanding of
topological edge states and their response to interparticle interactions.
Here, we experimentally study long-distance quantised Hall drifts in a
harmonically confined topological pump of non-interacting and interacting
ultracold fermionic atoms. We find that quantised drifts halt and reverse
their direction when the atoms reach a critical slope of the confining
potential, revealing the presence of a topological boundary. The drift
reversal corresponds to a band transfer between a band with Chern number
$C=+1$ and a band with $C=-1$ via a gapless edge mode, in agreement with the
bulk-edge correspondence for non-interacting particles. We establish that a
non-zero repulsive Hubbard interaction leads to the emergence of an additional
edge in the system, relying on a purely interaction-induced mechanism, in
which pairs of fermions are split.

####

The existence of individual edge modes at topological boundaries plays a
crucial role in quantum Hall physics. More specifically, a non-trivial
topology in the bulk of a material ensures that its edge modes are gapless and
chiral. Gaplessness is related to the bulk-edge correspondence, stating that
the number of topological edge modes is equal to the difference in Chern
number across an interface [1]. Consequently, a gapless mode should allow an
adiabatic transfer from one band to another, resulting in a reflection of
transverse bulk currents in the opposite direction if the two bands feature
opposite Chern numbers. However, the coherence time in most electronic
materials is not sufficient to observe this effect, and edges are generally
probed spectroscopically [1, 2, 3, 4]. Moreover, studies of edge physics in
engineered quantum systems, such as ultracold atoms and photonics, have so far
been focussed on chirality [5, 6, 7, 8, 9] and localisation [10, 11, 12, 13].
A boundary reflection has not been detected [14, 15], or it was disregarded
[16, 17], and to our knowledge it has never been studied for variable
interaction strength. Here, we observe the reversal of quantised bulk drifts
due to harmonic trapping in a topological Thouless pump, the temporal analogue
of the quantum Hall effect [18, 19, 20]. The reflection is a fundamental
manifestation of confined topological matter and directly shows the gapless
nature of topological edge modes. Going beyond the non-interacting regime, we
discover the emergence of a second edge for repulsive Hubbard $U$.

Figure 1: Reflection of quantised Hall drifts off a topological interface. (A)
Topological Thouless pumping in the presence of confining potentials. In the
non-interacting case (top), a harmonic trap gives rise to topological trivial
($C=0$) and non-trivial ($C\neq 0$) regions, separated by a topological
interface. The atoms exhibit a quantised drift until they are reflected at the
interface. With repulsive on-site interactions (bottom), the reflection
happens closer to the centre, accompanied by atoms still drifting in the
original direction. (B) Using Floquet theory, the 1D Rice-Mele pump can be
mapped to a 2D Harper-Hofstadter-Hatsugai model with a linear gradient along
the synthetic dimension $n$ which represents the photon number. The magnetic
flux per plaquette is $\Phi=1/2$ in units of the magnetic flux quantum [21].
The gradient along $n$ leads to a transverse Hall drift along $x$ (red arrows)
due to the nontrivial topology of the bands. (C) Schematic spectrum of the
mapped 2D Hofstadter model in a semi-infinite geometry. The lowest two bands
have $C=\pm 1$, respectively. The linear gradient induces Bloch oscillations
in the synthetic reciprocal space (dashed arrows). A gapless edge mode (solid
arrow) appears at the topological interface. The reflection of the Hall drift
can be understood as atoms being transported from the lower band ($C=1$) to
the higher band ($C=-1$) via the topological edge mode.

####

The experiments are performed with ultracold fermionic potassium-40 atoms,
which are loaded into the potential of a generalised optical lattice formed by
a combination of standing and running waves of wavelength
$\lambda=$1064\text{\,}\mathrm{n}\mathrm{m}$$ [22]. This creates an array of
decoupled, one-dimensional tubes. Along the tube direction, the periodically
modulated Rice-Mele-Hubbard Hamiltonian with harmonic confinement is realised,

$\displaystyle\hat{H}(\tau)$ $\displaystyle=$
$\displaystyle-\sum_{j,\sigma}\left[t+(-1)^{j}\delta(\tau)\right]\left(\hat{c}_{j\sigma}^{\dagger}\hat{c}_{j+1\sigma}+\text{h.c.}\right)$
$\displaystyle+\,\Delta(\tau)\sum_{j,\sigma}(-1)^{j}\hat{c}_{j\sigma}^{\dagger}\hat{c}_{j\sigma}+U\sum_{j}\hat{c}_{j\uparrow}^{\dagger}\hat{c}_{j\uparrow}\hat{c}_{j\downarrow}^{\dagger}\hat{c}_{j\downarrow}~{}$
$\displaystyle+\,\sum_{j,\sigma}V_{j}\hat{c}_{j\sigma}^{\dagger}\hat{c}_{j\sigma}~{},$

where $\hat{c}_{j\sigma}$ is the fermionic annihilation operator for spin
$\sigma\in\\{\uparrow,\downarrow\\}$ on site $j$, and $t$ denotes the average
tunnelling. An adiabatic modulation of bond dimerisation
$\delta(\tau)=\delta_{0}\cos(2\pi\tau/T)$ and sublattice offset
$\Delta(\tau)=\Delta_{0}\sin(2\pi\tau/T)$ traces a closed trajectory in the
$\delta$–$\Delta$ plane around the origin, referred to as critical point.
Therefore, an insulator or homogeneously filled band at $U=0$ describes a
topological pump [20, 18] with $T$ being the pump period. Experimentally, the
topological pump manifests itself as a quantised drift of the atom position by
one unit cell per pump cycle [23, 24, 25, 26, 27].

####

The harmonic confinement is characterised by the trap frequency $\nu$,
entering Eq.  ‣ Reversal of quantised Hall drifts at non-interacting and
interacting topological boundaries as $V_{j}=\frac{1}{2}m(2\pi\nu
aj)^{2}\equiv V_{0}j^{2}$ ($a=\lambda/2$, lattice spacing; $m$, atomic mass).
Due to the confinement, the atoms are initially localised at the centre of the
trap. Topological pumping then leads to a quantised drift of atoms against the
confining potential (Fig. 1A). Our measurements show that the quantised drift
changes its direction at a certain distance from the trap centre. We will
demonstrate that this happens when the gradient of the confinement overcomes
the band gap and a boundary between topological and trivial regions emerges.
For repulsive interactions, we observe another reflection, closer to the trap
centre, while a part of the atoms keeps drifting in the original direction
(Fig. 1A).

####

In the following, we develop a description of the reflection in terms of
gapless edge modes and the bulk-edge correspondence within the framework of
the Harper-Hofstadter-Hatsugai (HHH) model with one real ($x$) and one
synthetic ($n$) dimension. The model features bulk Chern bands with $C=+1$ and
$C=-1$. An exact mapping between the non-interacting 1D Rice-Mele Hamiltonian
(Eq. 1) and the two-dimensional (2D) HHH model can be obtained using Floquet
theory, illustrated in Fig. 1B (for derivation see, e.g., refs. [19] and
111see supplemental materials, which include refs. [28, 29, 30, 31, 32, 33]).
A linear gradient along the synthetic dimension $n$ appears in the mapping
since the state with $n$ photons acquires an energy of $-n\hbar\omega$, where
$\omega=2\pi/T$ is the pump frequency. The gradient along $n$ or,
equivalently, an external force causes Bloch oscillations along the synthetic
reciprocal dimension $k_{n}$ which, in turn, lead to a Hall drift or
‘anomalous velocity’ along the transverse real direction $x$ [14, 15]. The
bulk Hall drift along $x$ corresponds exactly to the quantised displacement
measured in the topological pump. The trap induces a boundary between
topological ($C_{\text{centre}}=1$) and trivial ($C_{\text{right}}=0$) regions
and a single gapless edge mode emerges, according to the bulk-edge
correspondence: $C_{\text{centre}}-C_{\text{right}}=1$. The edge modes
connects two bands of opposite Chern invariant, as shown in Fig. 1C. Thus, a
Bloch oscillation transfers the atoms from the ground to the first excited
band via that edge mode. Since the first excited band has Chern number $-1$
the atoms are now moving ‘backwards’, resulting in a reversal of the quantised
Hall drift.

Figure 2: Measuring the reversal of a quantised Hall drift. (A) The time-trace
of atomic in-situ images shows a quantised drift along $x$ before the atoms
are reflected at the topological boundary. Each density image is averaged over
three individual measurements with the parameters $V_{0}=0.0148(9)t$,
$\Delta_{0}=2.7(1)t$, and $T=$3\text{\,}\mathrm{m}\mathrm{s}$=12.8(2)\hbar/t$.
The red dashed line indicates the topological boundary
$x_{\rm{edge}}/\left(2a\right)\approx\frac{1}{2}\Delta_{0}/V_{0}$. The white
dashed lines are linear fits to the atom drift, yielding slopes of
$1.00(1)\times 2a/T$ before, and $-0.99(3)\times 2a/T$ after the reflection.
Cloud positions, averaged over the transverse direction, are fitted using
Gaussians. The experimental Chern marker (lower panel, points) is determined
by the velocity of the right-moving cloud at different positions. The
theoretical Chern marker (line) is calculated in a staggered potential
$V_{j}=V_{0}\left(-1\right)^{j}j$ which has the same local tilt
$\Delta_{\rm{ext}}(j)=V_{0}\left|j-\frac{1}{2}\right|$ as the harmonic trap
[21]. In a local density approximation picture, the local tilt
$\Delta_{\rm{ext}}$ shifts the $\delta$–$\Delta$ pump trajectory upwards.
Depending on whether or not the trajectory encloses the critical point, the
pump is rendered topological or trivial. (B) Measured band populations as a
function of time $\tau$. Each density image is averaged over six individual
measurements with the parameters $V_{0}=0.0191(6)t$, $\Delta_{0}=3.2(2)t$ and
$T=$3\text{\,}\mathrm{m}\mathrm{s}$=10.7(2)\hbar/t$. The total atom number
remains constant, within error bars, throughout the experiment. Due to the
underlying honeycomb lattice geometry in the $x$–$z$ plane, the first
Brillouin zone has a diamond shape [21]. The band population is inverted when
the bulk current is reflected off the topological interface, manifesting the
gapless nature of the topological edge mode. (C) Hybridisation of the edge
modes at the topological interface due to tunnelling. Bloch oscillations along
$k_{n}$ can lead to Landau-Zener transfers between topologically trivial and
nontrivial edge modes. The population of trivial edge modes explains the atoms
being left at the boundary. Hybridisation never changes the total number of
gapless edge modes at the boundary.

####

Fig. 2 shows experimental in-situ images of the atomic cloud as a function of
time $\tau$ at $U=0$. The data shows a quantised drift of $1.00(1)\times 2a/T$
up to about $60\,T$, which confirms the long coherence time of Bloch
oscillations which induce the transverse drift. At $\tau\simeq 75\;T$ the
atoms change their drift direction, which is a key observation of this work.
The expected topological boundary (red dashed line) represents the position at
which the local tilt from the external harmonic potential
$\Delta_{\rm{ext}}\left(j\right)\equiv\frac{1}{2}\left|V_{j}-V_{j-1}\right|=V_{0}\left|j-\frac{1}{2}\right|$
equals the maximum sublattice offset $\Delta_{0}$, thus,
$x_{\rm{edge}}/\left(2a\right)\simeq\frac{1}{2}\Delta_{0}/V_{0}=92(7)$. Beyond
this position the total sublattice offset ceases to change sign, rendering the
region outside $x_{\rm{edge}}$ topologically trivial. The boundary caused by
the harmonic confinement is not infinitely sharp, but smoothened over several
lattice sites. This leads to a small $T$–dependence of the reflection position
(Fig. S1), compared to its absolute value, and the calculation above should be
understood as the outermost point of the reflective region. The reflected
atoms exhibit a quantised drift of $-0.99(3)\times 2a/T$ in the opposite
direction, in agreement with a transfer to the first excited band with $C=-1$.
The linear relation between the position of topological boundary
$x_{\rm{edge}}$ and the maximum sublattice offset $\Delta_{0}$ is further
confirmed by measuring the reflection in different lattices (Fig. S1). The
reflection is observed under all parameter settings tested in this work,
highlighting that the existence of the topological boundary is robust.

####

In addition to the reflection, we also observe a cloud of atoms temporarily
remaining at the boundary before gradually dissolving. This process can be
understood via the presence of topologically trivial edge states, which
hybridise with the gapless edge modes. To simplify the picture, let us
consider a sharp domain wall between $C=1$ and $C=0$ (Fig. 2C). According to
the bulk-edge correspondence, the topologically nontrivial region contributes
exactly one gapless mode whereas the trivial region can contribute gapped edge
modes. Due to tunnel coupling at the interface, hybridisation takes place [34]
and gaps on the order of the pump frequency $2\pi/T$ emerge. Bloch
oscillations along $k_{n}$ can now lead to non-adiabatic ‘Landau-Zener’
transfers between topological and trivial edge modes, causing an incomplete
transfer to the higher band, and atoms remaining at the boundary. Subsequent
Bloch oscillations will transfer atoms back into the topological domain,
leading to the dissolution of the cloud at the boundary. While the harmonic
confinement leads to a more complex level structure [21], the underlying
process remains qualitatively the same.

Figure 3: Reflection of quantised Hall drifts from an interacting topological
edge. (A) An attractive Hubbard interaction of $U=-3.48(7)t$ leads to the same
reflection behaviour as observed for $U=0$ (measurement parameters otherwise
identical to Fig. 2A). (B) Repulsive Hubbard interactions of $U=3.48(7)t$ lead
to the emergence of a second reflection, closer to the trap centre, which we
attribute to an interacting topological boundary. A zoom-in (C) shows that the
early reflection happens after about twelve cycles. The white dashed lines are
guides to the eye, calculated as linear fits to the cloud position, extracted
as the sum of a skewed and a regular Gaussian. (D) Microscopic description of
the interaction-induced reflection for repulsive Hubbard $U$. When the maximum
energy offset between two neighbouring sites
$2\left(\Delta_{0}-\Delta_{\rm{ext}}\right)$ becomes smaller than the Hubbard
interaction, formation of double occupancies is prohibited and one atom is
left in the higher-energy site of a unit cell, which then drifts in the
opposite direction. (E) The critical point in the $\delta$–$\Delta$ plane is
split into two in the presence of repulsive Hubbard $U$ [35]. When the pump
trajectory encloses both critical points, a quantised drift is expected, as in
the non-interacting system. The local tilt given by the external potential
$\Delta_{\rm{ext}}$ shifts the trajectory along the $\Delta$–axis, eventually
enclosing just one critical point. Since a single split critical point
features half the topological charge of the orignal one, the material’s
topology changes and a boundary emerges.

####

We support the in-situ images with measurements of band population before,
during, and after the reflection, as shown in Fig. 2B [21]. Before the
reflection, we find a filled ground band, which is consistent with the
observation of a quantised Hall drift. At the reflection ($\tau\simeq 72\;T$),
we observe an inversion of the population to the first excited band. After the
reflection, the inverted population remains almost unchanged while the atoms
are travelling back, highlighting the absence of incoherent relaxation to the
ground band, even after more than a hundred Bloch oscillations.

####

We further explore the effect of attractive and repulsive interactions on the
topological boundary. For attractive Hubbard $U=-3.48(7)t=1.27(7)\Delta_{0}$
(Fig. 3A), the quantised Hall drift is reversed at the same position as in the
non-interacting situation. This can be explained in terms of the Rice-Mele
model in which fermions in the strongly attractive regime approach the limit
of hard-core bosons [22], and the condition for the emergence of a topological
boundary $\Delta_{\rm{ext}}\left(j\right)=\Delta_{0}$ remains unchanged.

####

For repulsive Hubbard $U=3.48(7)t$ (Fig. 3B), we observe a second reflection
in addition to the original one. Compared to the non-interacting case, this
reflection appears much closer to the trap centre. The zoomed-in image (Fig.
3C) shows that a proportion of the atoms start to move backwards after about
12 cycles. In contrast to the drift reversal in the non-interacting system, a
large fraction of the atom cloud still undergoes quantised drifting in the
original direction. In the following, we develop a microscopic picture of the
interaction-induced partial reflection in the limiting case of two isolated
spins $(\uparrow,\downarrow)$, which approximates our initial state in a unit
cell (Fig. 3D). As long as the maximum energy offset between two neighbouring
sites $2\left(\Delta_{0}-\Delta_{\rm{ext}}\right)$ is larger than the Hubbard
$U$, the formation of a double occupancy is energetically allowed and the
quantised drift persists [22], even in the presence of an external potential.
However, when $\Delta_{\rm{ext}}$ becomes larger, the energy offset between
two neighbouring sites remains always smaller than $U$, double occupancy
formation becomes prohibited. In the latter case, one atom of a singlet pair
is transferred to the energetically excited site of a unit cell, which will
subsequently drift in the opposite direction. The other atom, in the lower-
energy site, will move onwards because on-site interactions become irrelevant
if there is only one atom per unit cell. Since the underlying Hamiltonian (Eq.
‣ Reversal of quantised Hall drifts at non-interacting and interacting
topological boundaries) is $\text{SU}(2)$–symmetric, spin–$\uparrow$ and
spin–$\downarrow$ have equal probability of being reflected and they should
remain correlated after the splitting process.

####

The full many-body description of the interaction-induced reversal requires
the development of suitable topological invariants for smooth confinements and
strong interactions, which goes beyond the scope of this work. Nevertheless,
we obtain an intuition of the boundary’s topological origins using again the
idea of shifted pump trajectories in the $\delta$–$\Delta$ plane with a
staggered potential (c.f. Fig. 2A). Numerical simulations have shown that a
repulsive Hubbard $U$ can split the critical point at the origin into two
separate ones [35], each retaining half the original topological charge. The
distance between the new critical points is approximately $U$ up to a
correction on the order of the tunnelling $t$ [36]. When the trajectory
encloses both critical points, quantised drift of two spins
$(\uparrow,\downarrow)$ is expected, as in the non-interacting system. As the
position-dependent local tilt $\Delta_{\rm{ext}}(j)$ shifts the trajectory
upwards, it will enclose only one of the critical points beyond certain
position along $x$ (Fig. 3E). This indicates a transition of topological
properties and the emergence of an interacting topological edge in real space.
The estimation of the interacting boundary at
$\Delta_{\rm{ext}}(j)\simeq\Delta_{0}-U/2$ lies close to the centre and agrees
with the microscopic picture discussed above. Similar to the non-interacting
case, this boundary should be considered as the outermost position of the
reflective region.

####

In conclusion, we have experimentally observed a reversal of quantised Hall
drifts at a topological boundary in a harmonic potential. The reflection is a
direct manifestation of the gapless nature of topological edge modes between
Chern bands of opposite sign. We explore the effect of Hubbard interactions,
both attractive and repulsive, and find an asymmetric behavior with respect to
$U=0$. While on the attractive side the topological boundary is unaffected,
repulsive interactions lead to the emergence of a second interface, featuring
a splitting of quantised drifts. As a result, our experiments could enable the
realisation of circular current patterns for constructing novel many-body
phases [37]. More broadly, our work allows the exploration of the bulk-edge
correspondence in the presence of interactions [38], as well as the
investigation of edge reconstruction [39] in the quantum Hall effect and in
interacting topological insulators.

## Acknowledgments

We would like to thank Jason Ho, Gian-Michele Graf, Thomas Ihn, Fabian Grusdt,
Fabian Heidrich-Meisner, Armando Aligia, and Eric Bertok for valuable
discussions. We also thank Julian Léonard and Nur Ünal for comments on an
earlier version of the manuscript. We would like to thank Alexander Frank for
his contributions to the electronic part of the experimental setup. We
acknowledge funding by the Swiss National Science Foundation (Grant Nos.
182650, 212168, NCCR-QSIT, and TMAG-2 209376) and European Research Council
advanced grant TransQ (Grant No. 742579).

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##  I Supplemental materials

###  I.1 Dependence of the drift reversal on experimental parameters

The expected drift reversal happens when the maximum local site offset over
one pump-cycle $\Delta_{0}$ is equal to the local tilt given by the harmonic
trap. This position given by $x_{\rm{edge}}\simeq\Delta_{0}a/V_{0}$ with
$a=\lambda/2$ and $V_{0}=\frac{1}{2}m(2\pi\nu a)^{2}$. By measuring the
reflection point in lattices with different $\Delta_{0}$, we verify the
relevant scaling $x_{\rm{edge}}\propto\Delta_{0}$, as shown in Fig. S1. The
blue line in Fig. S1A marks the theoretically expected $x_{\rm{edge}}$ with
the uncertainty propagated from the uncertainty of the trap frequency $\nu$.
The disagreement between theory and experiment for larger values of
$\Delta_{0}$ can be explained by the finite waist of the lattice beams. In
order to explore the edge in our system, the atoms are pumped by almost 100
unit cells ($\sim$100\text{\,}\mu\mathrm{m}$$). Due to the Gaussian envelope
of the transverse beams, which are essential to realise the pump, the lattice
is effectively shallower far away from the centre. Thus, $\Delta_{0}$
decreases towards the edge and atoms are reflected sooner.

We also find a small dependence of the reflection point on the pump period,
compared to its absolute value, which spans roughly 10 unit cells when
changing $T$ from $2\text{\,}\mathrm{m}\mathrm{s}$ to
$10\text{\,}\mathrm{m}\mathrm{s}$ (Fig. S1B). This can be understood by
considering the energy spectrum of the Harper-Hofstadter-Hatsugai (HHH) model
in a harmonic potential, which will be discussed below.

Figure S1:  Experimental dependence of the reflection position. The reflection
point is expected to depend linearly on the maximal site-offset per pump cycle
$\Delta_{0}$ which is experimentally verified in (A). Deviations for large
values of $\Delta_{0}$ can be explained by the finite waist of the laser beams
forming the lattice. (B) shows the period dependence of the reflection
position. Changing the pump period $T$ over an order of magnitude only changes
the reflection point by about 10 unit cells, which is a result of the smooth
boundary of a harmonic potential.

###  I.2 Experimental sequence

We start by preparing a degenerate cloud of fermionic ${}^{40}\rm{K}$ in a
crossed dipole trap. We have a spin mixture of $m_{F}=\\{-9/2,-7/2\\}$ except
for the measurements in Fig. 2B and Fig. S1, where a spin-polarised cloud in
the magnetic state $F=9/2,\ m_{F}=-9/2$ is used. The spin-polarised cloud is
directly loaded into the pumping lattice, while the spin mixture is first
loaded into an intermediate chequerboard lattice with strongly attractive
interactions. The two-step loading precludes the presence of atoms in the
higher band and gives a larger fraction of atoms in doubly occupied unit cells
[22].

After pumping the system for varying times, we either take a in-situ
absorption image to measure the density or detect the band population with
band-mapping. The latter is implemented with an exponential ramp to switch off
the lattice beam in $500\text{\,}\mu\mathrm{s}$, followed by a time-of-flight
expansion of $25\text{\,}\mathrm{m}\mathrm{s}$ before absorption imaging.

###  I.3 Realisation of a Thouless pump in the Rice-Mele model

The lattice setup is comprised of non-interfering standing waves in $x$, $y$,
and $z$ directions, together with interfering laser beams in the $x$–$z$
plane. All the lattice beams come from a single laser source at wavelength
$\lambda=$1064\text{\,}\mathrm{n}\mathrm{m}$$. These potential combine to form
a honeycomb lattice in the $x$–$z$ plane, which can be considered as isolated
tubes of one-dimensional superlattices along $x$ in the limit of deep
transverse lattices. In each 1D tube the potential can be modeled by a one-
dimensional superlattice with two sites per unit cell. With this setup, we
realise the Rice-Mele model [22]. In the tight-binding limit, the Rice-Mele
model can be described with three numbers: the offset energy $\Delta$ between
the two sites of a unit cell, the averaged nearest-neighbour tunneling $t$ and
the bond dimerisation $\delta$ which gives half the difference between the
inter- and intra-dimer tunnellings. By having a dynamical control of the
relative phase $\varphi$ between the laser beams generating the interfering
and the non-interfering lattice, we manage to shift the two with respect to
each other. This shift modulates $\Delta$ and $\delta$ periodically, which can
be depicted as a closed trajectory in the $\Delta$-$\delta$ coordinate (Fig.
S2). In the adiabatic limit, this realises a Thouless pump with its hallmark
quantised transport. In this case, the atomic displacement is given by the
number of revolutions around the origin of the $\Delta$-$\delta$ plane.

Figure S2: Realisation of a Thouless pump in the Rice-Mele model. In our
system the 1D lattice potential can be modelled by a superlattice with two
sites per unit cell, which is depicted in the bottom part of this figure as a
function of relative phase $\varphi$. The local site offset $\Delta$ as well
as the bond dimerisation $\delta$ is depicted in the $\Delta$-$\delta$ plots
corresponding to the respective potentials. The resulting pump displacement
corresponds to the number of revolutions around the origin in the
$\Delta$-$\delta$ plane.

###  I.4 Mapping a 1D Thouless pump to a 2D Hofstadter Model with quantum Hall
response

A 1D Thouless pump with a period of $T$, as realised in our experiment, can be
mapped to a 2D topological tight-binding (HHH) model with an applied electric
field $E=\frac{2\pi\hbar}{qT}$ where $q$ can be thought of as a fictitious
charge of netural atoms. Due to the topological bandstructure, this electric
field leads to a transverse current $I_{\rm{trans}}=\frac{q}{T}$ of one atom
per period, when considering a fully occupied band. The 2D model therefore has
a quantised transverse conductance
$\sigma_{\rm{trans}}=I_{\rm{trans}}/E=\frac{q^{2}}{2\pi\hbar}$ analogous to
the Hall conductance in the Quantum Hall Effect (QHE).

The time-periodicity of the Hamiltonian in Eq.  ‣ Reversal of quantised Hall
drifts at non-interacting and interacting topological boundaries with
${\hat{H}(\tau)=\hat{H}(\tau+T)}$ allows us to use Floquet’s theorem.
Solutions of the time-dependent Schrödinger equation

$\displaystyle i\hbar\partial_{\tau}\ket{\Psi(\tau)}=H(\tau)\ket{\Psi(\tau)}$
(S1)

can thus be written as

$\displaystyle\ket{\Psi(\tau)}=e^{-i\epsilon\tau/\hbar}\ket{u(\tau)}$ (S2)

with $\ket{u(\tau+T)}=\ket{u(\tau)}$ and $\epsilon\in\mathbb{R}$. Due to the
time-periodicity of $u(\tau)$ we expand it as a Fourier series,

$\displaystyle\ket{u(\tau)}=\sum_{n}e^{-i\omega n\tau}\ket{u_{n}}~{},$ (S3)

where $\omega=2\pi/T$ is the pump frequency. The change from the time-domain
into the Fourier-domain is the key ingredient to map the 1D Thouless pump to a
2D tight-binding model. The index $n$ is also called the photon number of the
mode $\ket{u_{n}}$.

Using a multi-index $\alpha=(j,\sigma)$ we write the $T$-periodic 1D
Hamiltonian for $U=0$ in the Fourier-basis:

$\displaystyle\hat{H}(\tau)$
$\displaystyle=\sum_{\alpha,\beta}h_{\alpha\beta}(\tau)\ket{\alpha}\bra{\beta}$
(S4)
$\displaystyle=\sum_{\alpha,\beta,m}e^{-im\omega\tau}h_{\alpha\beta}^{m}\ket{\alpha}\bra{\beta}$

with
$h_{\alpha\beta}^{m}=\frac{1}{T}\int_{0}^{T}e^{im\omega\tau}h_{\alpha\beta}(\tau)d\tau$
and $\ket{\alpha}$ corresponding to an atom localised on site $j$ with spin
$\sigma$. Likewise, we use Fourier decomposition to express the solutions to
Eq. S1 as

$\displaystyle\ket{\Psi(\tau)}=e^{-i\epsilon\tau/\hbar}\sum_{n,\alpha}e^{-in\omega\tau}u_{n,\alpha}\ket{\alpha}~{}.$
(S5)

where $u_{n,\alpha}=\braket{\alpha}{u_{n}}$. As a result, we obtain an
eigenvalue equation for $u_{n,\alpha}$:

$\displaystyle\epsilon u_{n,\alpha}=-n\hbar\omega
u_{n,\alpha}+\sum_{\beta,m}h_{\alpha\beta}^{m}u_{n-m,\beta}~{}~{}\forall
n,\alpha$ (S6)

which can be understood as a time independent Schrödinger equation of a 2D
tight-binding model with a tilted potential energy along one axis. By
explicitly evaluating the $h_{\alpha\beta}^{m}$, we get

$\displaystyle
H_{\rm{2D}}=H_{\rm{real}}+H_{\rm{synth}}+H_{\rm{diag}}+H_{\rm{V}}+H_{\rm{tilt}},$
(S7)

with

$\displaystyle
H_{\rm{real}}=-t\sum_{j,n,\sigma}(\hat{c}^{\dagger}_{j,n,\sigma}\hat{c}_{j+1,n,\sigma}+h.c.),$
(S8) $\displaystyle
H_{\rm{diag}}=-\frac{\delta_{0}}{2}\sum_{j,n,\sigma}e^{-i\pi
j}(\hat{c}^{\dagger}_{j,n,\sigma}\hat{c}_{j+1,n+1,\sigma}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\hat{c}^{\dagger}_{j,n,\sigma}\hat{c}_{j+1,n-1,\sigma}+h.c.),$
$\displaystyle H_{\rm{synth}}=-\frac{\Delta_{0}}{2}\sum_{j,n,\sigma}e^{-i\pi
j}(i\hat{c}^{\dagger}_{j,n,\sigma}\hat{c}_{j,n+1,\sigma}+h.c.),$
$\displaystyle
H_{\rm{V}}=\sum_{j,n,\sigma}V(j)\hat{c}^{\dagger}_{j,n,\sigma}\hat{c}_{j,n,\sigma},$
$\displaystyle H_{\rm{tilt}}=-\sum_{j,n,\sigma}\hbar\omega
n\hat{c}^{\dagger}_{j,n,\sigma}\hat{c}_{j,n,\sigma}~{}.$

$H_{\rm{real}}$ and $H_{\rm{synth}}$ describe tunneling along the real ($x$)
and synthetic ($n$) dimension, respectively.

The diagonal tunnelling terms in $H_{\rm{diag}}$ are crucial because they open
a bandgap between the ground band and the first excited band, characterised by
the topological Chern number $C$ which is further related to the quantised
Hall conductance via $\sigma_{\rm{trans}}=\frac{q^{2}}{2\pi\hbar}C$. The terms
in $H_{\rm{V}}$ describe the external potential along the real-space
direction. $H_{\rm{tilt}}$ corresponds to a linear tilt in potential energy
along the synthetic dimension which can be thought of as originating from an
electric field $E=\frac{2\pi\hbar}{qT}$ pointing along $n$.

Figure S3: Energy spectra for different confining potentials. States localised
to the left edge are omitted for clarity. Energy spectra of $H_{\rm{2D}}$ in
the limit $\omega\rightarrow 0$ for $\kappa=24$ (A), $\kappa=10$ (B), and
$\kappa=2$ (C). Topological edge modes which connect the two bands with Chern
number $1$ and $-1$ in (A) and (B) are marked in red. The upper inset in (C)
marks the topological boundary where the reflection is observed as described
in the main text. The inset in the center of (C) shows the tiny avoided
crossings which can lead to a slight period dependence of the observed
reflection point (Fig. S1). (D) Energy spectrum for a linearly increasing
staggered potential. The gapless, topological edge mode is marked in red.

###  I.5 Edge modes and their reflection properties

To illustrate the topological edge modes in the presence of an external
potential, we evaluate the spectrum of $H_{\rm{2D}}$ in the adiabatic limit
($\omega\rightarrow 0$) for different potentials

$\displaystyle V(j)=\frac{1}{2}m(2\pi\nu a)^{2}j^{\kappa}$ (S9)

with $m$ being the mass of ${}^{40}\rm{K}$, trap frequency
$\nu=$134\text{\,}\mathrm{H}\mathrm{z}$$, lattice spacing
$a=$532\text{\,}\mathrm{n}\mathrm{m}$$, and lattice site $j$. The parameter
$\kappa$, an even integer, characterises steepness of the trap; the limit
$\kappa\rightarrow\infty$ corresponds to the textbook case of infinitely sharp
walls [28]. Fig. S3A-C shows the numerically calculated energy spectra,
omitting states on localised to the left edge for clarity.

Fig. S3A shows the spectrum for a box-like potential with $\kappa=24$. In this
case there is a family of topological edge states, marked in red, which
connect the lower and the upper band (black), separated from topologically
trivial states above $5\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ (also in
black). All red states are localised along the right edge in $x$-direction.
The lower and upper band have Chern number $1$ and $-1$, respectively.
Considering the dynamics in this model, an applied electric field along $n$ as
defined in $H_{\rm{tilt}}$ leads to Bloch oscillations with a period $T$ along
$k_{n}$. At the same time, the center-of-mass of the atoms moves by one unit
cell per Bloch oscillation period along $x$, which corresponds to the
quantised bulk Hall drift [14, 15]. This drift can be evaluated in the
numerics by following the eigenstates in Fig. S3A in real space. Since the red
edge states are gapped from the next higher-lying trivial (black) states, the
atoms ‘Bloch-oscillate’ from the ground to the excited band via the red-marked
edge modes over several periods. Once they are in the excited band they are
transported backwards along $x$.

Fig. S3B shows the situation for $\kappa=10$. It behaves similarly to Fig.
S3A, except that there are more localised states marked in red, compared to
$\kappa=2$. Likewise, these states are transported along $x$ as they undergo
Bloch oscillations. As before, this family of edge states is gapped from
trivial states and connects right-moving to left-moving states, which leads to
the reflection phenomenon.

The experimentally relevant case is a harmonic trap with $\kappa=2$ (see also
refs. [29, 30, 23, 31]). Fig. S3C shows that for $\kappa=2$ the number of
localised sates outside of the bands is even larger than for $\kappa=10$. As
before, we adiabatically follow these localised states along $k_{n}$ and
evaluate their centre-of-mass along $x$. By numerically observing these drifts
we confirm that the states describe quantised drifting in a large region,
which manifests their nontrivial topological nature. Thus, the $\kappa=2$ case
is ideal to observe the reflection after long-distance quantised Hall drifts.
However, the gaps between topological (right-moving and left-moving) and
trivial (stationary) states become smaller, compared to the $\kappa=24$ and
$\kappa=10$ cases, as shown in the insets of Fig. S3C ($\kappa=2$). As a
result, the reflection point for $\kappa=2$ is spread out over several unit
cells but the reflection itself remains intact. Faster pumping leads to non-
adiabatic crossings of the energy gaps between right-moving and left-moving
states, causing the reflection to happen later in time and further up in
energy. We confirm this dependence experimentally in Fig. S1B, which shows a
later reversal for smaller pump periods.

Fig. S3D shows the spectrum for the linearly increasing staggered potential,
described in the following sections. This potential allows a straightforward
identification of the gapless edge mode (red line). The states corresponding
to this gapless edge mode are localised around the topological boundary.

Figure S4: Linearly increasing staggered potential. To elucidate the topology
in our system, a linearly increasing staggered potential is considered (blue):
$V(j)=jV_{0}(-1)^{j}$ with $V_{0}=1/2\times m(2\pi\nu a)^{2}$, as before. It
is chosen such that the local tilt always equals the tilt from the harmonic
potential (orange), but alternates in sign. The staggered potential allows a
simple pictorial representation of the emergence of the topological boundary.
In a local density approximation the trap linearly shifts the pump trajectory
upwards in the $\Delta$-$\delta$ plane, as depicted in the upper part of the
figure. As soon as the trajectory ceases to enclose the critical point, a
topological–trivial boundary develops.

###  I.6 Staggered potential

Another possibility to identify the topological boundary in our system makes
use of a staggered potential. First, we consider a potential with uniform
staggering, given by $V_{c}(j)=V(-1)^{j}$, where $j$ indexes the lattice-site
and $2V$ corresponds to the energy difference between adjacent sites. Adding
such a potential to the Rice-Mele Hamiltonian (Eq.  ‣ Reversal of quantised
Hall drifts at non-interacting and interacting topological boundaries) changes
its trajectory in the $\Delta$-$\delta$ plane. The onsite energy in such a
system is given by $(\Delta(\tau)+V)(-1)^{j}$, which ranges from $-\Delta+V$
to $\Delta+V$. The tunnellings are unchanged. Therefore, the trajectory
remains circular and it is simply shifted upwards by an amount $V$.

A topological boundary emerges for a linearly increasing staggered potential,
given by $V(j)=jV_{0}(-1)^{j}$, with $V_{0}=\frac{1}{2}m(2\pi\nu a)^{2}$ as
before. $V(j)$ is chosen such that it has the same local tilt as the harmonic
trap in the experiment. Within the local density approximation we assign a
$\Delta$-$\delta$ trajectory locally to each unit cell. The trajectories are
thus linearly shifted upwards as function of $j$ (Fig. S4), describing a
change of topology in real space. We expect the local density approximation to
be valid since the atomic eigenstates in the experiment are strongly
localised. Similar models with linearly increasing staggered potential have
been studied in refs. [29, 32, 38].

###  I.7 Local Chern marker

The mathematical formulation of the Chern number as a topological invariant
requires translational invariance, which does not apply to realistic
experiments. Instead, we use a local quantity, known as Chern marker [33, 8].
The local Chern marker depends on the real-space position and it is defined
by:

$\displaystyle
c(r_{\gamma})=-\frac{4\pi}{A_{c}}Im\sum_{s=A,B}\bra{r_{\gamma_{s}}}\hat{P}\hat{x}\hat{Q}\hat{y}\hat{P}\ket{r_{\gamma_{s}}},$
(S10)

where $r_{\gamma}$ is the position of the unit cell $\gamma$ with sub-lattice-
sites at positions $r_{\gamma_{A}}$ and $r_{\gamma_{B}}$,
$\ket{r_{\gamma_{s}}}=c^{\dagger}_{\gamma_{s}}\ket{0}$ is the state localised
on the corresponding lattice site , $A_{c}$ is the area of a real-space unit
cell, $\hat{Q}=1-\hat{P}$ and $\hat{P}$ is the projector onto the ground band.
Defining $\hat{P}$ is not unambiguously possible in our system (Eq. S7)
because of the energy shift from the harmonic confinement. Instead, we use a
linearly increasing staggered potential, as described in the previous
paragraph. This model leaves the bands intact and a ground band can be
unambiguously defined. Experimentally, a local probe of the band topology is
the velocity of the Hall drift, plotted in Fig. 2A. Theory and experiment
agree approximately with one another. The local velocity is extracted from the
atomic positions by fitting linear functions to groups of three adjacent
datapoints in ten pump cycles. The resulting velocities are plotted against
position and smoothed through a running average of width three (ten cycles).