$\mathcal{P}_{\gamma}:H^{s+\frac{1}{2}}(\Omega;\mathbb{R})\to
H^{s+1}(\Omega;\mathbb{R})$ defined via

$\displaystyle\mathcal{U}_{\gamma}(\psi)=u,\;\mathcal{P}_{\gamma}(\psi)=p,$
(3.7)

where $(u,p)$ is the unique solution to the $-\gamma$-Stokes system

$\displaystyle\begin{cases}\operatorname{div}S(p,u)+\gamma\partial_{1}u=0,&\text{in}\;\Omega\\\
\operatorname{div}u=0,&\text{in}\;\Omega\\\ S(p,u)e_{n}=\psi
e_{n},&\text{on}\;\Sigma_{b}\\\ u=0,&\text{on}\;\Sigma_{0}.\end{cases}$ (3.8)

The existence and uniqueness of $(u,p)$ and the boundedness of
$\mathcal{U}_{\gamma},\mathcal{P}_{\gamma}$ are guaranteed by Theorem 3.3.

Now we may record properties of the pseudodifferential operator associated to
the normal stress to solution maps (3.7).

######  Theorem 3.5 (Symbols associated to the pseudodifferential operator and
their asymptotics).

Suppose $s\geqslant 0$. The linear maps
$\mathcal{U}_{\gamma},\mathcal{P}_{\gamma}$ defined in (3.7) are well-defined
and bounded. Moreover, there exists bounded and measurable functions
$V:\hat{\Sigma}\times[0,b]\times\mathbb{R}\to\mathbb{C}^{n},Q:\hat{\Sigma}\times[0,b]\times\mathbb{R}\to\mathbb{C}^{n}$,
and $m:\Sigma\times\mathbb{R}\to\mathbb{C}$ such that for all $\psi\in
H^{s+\frac{1}{2}}(\Sigma_{b};\mathbb{R})$, we have

$\displaystyle\widehat{\mathcal{U}_{\gamma}(\psi)}(\xi,x_{n})=V(\xi,x_{n},-\gamma)\widehat{\psi}(\xi),\quad\widehat{\mathcal{P}_{\gamma}(\psi)}(\xi,x_{n})=Q(\xi,x_{n},-\gamma)\widehat{\psi}(\xi),\quad
m(\xi,\gamma)=V_{n}(\xi,b,\gamma).$ (3.9)

Furthermore, the following hold.

  1. (1)

$V,Q,m$ are continuous, with $V(0,x_{n},\gamma)=0,Q(0,x_{n},\gamma)=1$, and
$m(0,\gamma)=0$.

  2. (2)

$\overline{V(\xi,x_{n},\gamma)}=V(-\xi,x_{n},\gamma),\overline{Q(\xi,x_{n},\gamma)}=Q(-\xi,x_{n},\gamma),\overline{m(\xi,\gamma)}=m(-\xi,\gamma)$
for all $\xi\in\hat{\Sigma}$.

  3. (3)

For each $\xi\in\hat{\Sigma}$, $V(\xi,\cdot,\gamma),Q(\xi,\cdot,\gamma)$ solve

$\displaystyle\begin{cases}\left(-\partial_{n}^{2}+4\pi^{2}|\xi|^{2}\right)V^{\prime}+2\pi
i\xi Q-2\pi i\xi_{1}\gamma V^{\prime}=0,&\text{in}\;(0,b)\\\
\left(-\partial_{n}^{2}+4\pi^{2}|\xi|^{2}\right)V_{n}+\partial_{n}Q-2\pi
i\xi_{1}\gamma V_{n}=0,&\text{in}\;(0,b)\\\ 2\pi i\xi\cdot
V^{\prime}+\partial_{n}V_{n}=0,&\text{in}\;(0,b)\\\
-\partial_{n}V^{\prime}-2\pi i\xi V_{n}=0,&\text{for}\;x_{n}=b\\\
Q-2\partial_{n}V_{n}=1,&\text{for}\;x_{n}=b\\\
V=0,&\text{for}\;x_{n}=0.\end{cases}$ (3.10)

  4. (4)

For $|\xi|\ll 1$, we have

$V^{\prime}(\xi,x_{n},\gamma)=-\pi\xi(2x_{n}b-x_{n}^{2})+O(|\xi|^{2}),\quad
V_{n}(\xi,x_{n},\gamma)=2\pi^{2}|\xi|^{2}x_{n}^{2}\left(\frac{x_{n}}{3}-b\right)+O(|\xi|^{3}),\\\
Q(\xi,x_{n},\gamma)=1+O(|\xi|^{2}),\quad
m(\xi,\gamma)=-\frac{4\pi^{2}|\xi|^{2}b^{3}}{3}+O(|\xi|^{3}),\quad\text{as}\;|\xi|\to
0$ (3.11)

where $F(\xi,x_{n})=O(|\xi|^{k})$ as $|\xi|\to 0$ means that

$\displaystyle\limsup_{|\xi|\to 0}\sup_{0\leqslant x_{n}\leqslant
b}\frac{|F(\xi,x_{n})|}{|\xi|^{k}}<\infty.$ (3.12)

  5. (5)

For each $\gamma\in\mathbb{R}$, there exists a constant $c=c(\gamma,b)>0$ and
$R=R(\gamma,b)>0$ such that for $x_{n}\in[0,b]$ and $|\xi|>R$, we have the
point-wise estimates

$\displaystyle|V^{\prime}(\xi,x_{n},\gamma)|$ $\displaystyle\leqslant
c\left(\frac{|\gamma|}{|\xi|^{2}}+(b-x_{n})\right)e^{-2\pi|\xi|(b-x_{n})}+ce^{-2\pi|\xi|b},$
(3.13) $\displaystyle|V_{n}(\xi,x_{n},\gamma)|$ $\displaystyle\leqslant
c\left(\frac{1}{|\xi|}+(b-x_{n})\right)e^{-2\pi|\xi|(b-x_{n})}+ce^{-2\pi|\xi|b},$
(3.14) $\displaystyle|m(\xi,\gamma)+\frac{1}{4\pi|\xi|}|$
$\displaystyle\leqslant c\frac{1}{|\xi|^{2}},$ (3.15)
$\displaystyle|Q(\xi,x_{n},\gamma)|$ $\displaystyle\leqslant
ce^{-2\pi|\xi|(b-x_{n})}+ce^{-2\pi|\xi|b}.$ (3.16)

  6. (6)

For each $\gamma\in\mathbb{R}$, there exists a constant $c=c(n,b,\gamma)$ such
that

$\displaystyle\sup_{|\xi|\leqslant 1,\xi\neq
0}\frac{1}{|\xi|^{2}}\left(\int_{0}^{b}|V(\xi,x_{n},\gamma)|^{2}+|Q(\xi,x_{n},\gamma)-1|^{2}\;dx_{n}+|V(\xi,b,\gamma)|^{2}\right)\leqslant
c$ (3.17)

and

$\displaystyle(1+|\xi|^{2})^{3/2}\int_{0}^{b}|V(\xi,x_{n},\gamma)|^{2}\;dx_{n}+(1+|\xi|^{2})^{1/2}\int_{0}^{b}|Q(\xi,x_{n},\gamma)|^{2}\;dx_{n}+(1+|\xi|^{2})|V(\xi,b,\gamma)|^{2}\leqslant
c,$ (3.18)

for all $\xi\in\hat{\Sigma}$.

###### Proof.

If $\hat{\Sigma}=\mathbb{R}^{n-1}$, all six items follow from Theorem 4.5,
Theorem 4.10, and Corollary 4.11 in [18]. In the other cases, we have
$\hat{\Sigma}\subseteq\widehat{\mathbb{R}^{n-1}}=\mathbb{R}^{n-1}$. Therefore,
the same conclusions will follow as we are sampling the frequencies on
subgroups of $\mathbb{R}^{n-1}$. ∎

We conclude this subsection by recording the properties of an auxiliary
function defined in terms of $m$.

######  Lemma 3.6.

Suppose $\gamma\in\mathbb{R}\setminus\\{0\\}$, and define

$\displaystyle\rho_{\gamma}(\xi)=2\pi
i\gamma\xi_{1}+(1+4\pi^{2}|\xi|^{2})\sigma\overline{m(\xi,-\gamma)}.$ (3.19)

Then the following hold.

  1. (1)

$\rho_{\gamma}(\xi)=0$ if and only if $\xi=0$, and
$\overline{\rho_{\gamma}(\xi)}=\rho_{\gamma}(-\xi)$ for all
$\xi\in\hat{\Sigma}$.

  2. (2)

For $\sigma>0$, there exists a constant $C=C(n,\gamma,\sigma,b)>0$ such that
for all $\xi\in\hat{\Sigma}$, we have

$\displaystyle
C^{-1}|\rho_{\gamma}(\xi)|^{2}\leqslant(\xi_{1}^{2}+|\xi|^{4})\mathbbm{1}_{B(0,1)}(\xi)+(1+|\xi|^{2})\mathbbm{1}_{B(0,1)^{c}}(\xi)\leqslant
C|\rho_{\gamma}(\xi)|^{2}.$ (3.20)

  3. (3)

For $\sigma=0$ and $n=2$, there exists a constant $C=C(\gamma,b)>0$ such that
for all $\xi\in\hat{\Sigma}$, we have

$\displaystyle
C^{-1}|\rho_{\gamma}(\xi)|^{2}\leqslant|\xi|^{2}\mathbbm{1}_{B(0,1)}(\xi)+(1+|\xi|^{2})\mathbbm{1}_{B(0,1)^{c}}(\xi)\leqslant
C|\rho_{\gamma}(\xi)|^{2}.$ (3.21)

###### Proof.

This follows from the proof for Lemma 6.1 in [18] and the asymptotics of
$m(\xi,-\gamma)$ on general $\hat{\Sigma}$ recorded as the third and fourth
items of Theorem 3.5. ∎

###  3.2. Compatibility conditions and the Hilbert isomorphism associated to
the overdetermined $\gamma$-Stokes system

In this subsection we record the Hilbert isomorphism that completely
characterize the solvability of the overdetermined system (1.37). First we
introduce a pair of function spaces for the data tuple $(f,g,h,k)$ that
encodes the compatibility conditions associated to (1.37).

######  Definition 3.7.

Let $s\geqslant 0$ and $\Sigma$ be defined as in (1.1). We say that a data
tuple $(f,g,h,k)\in H^{s}(\Omega;\mathbb{R}^{n})\times
H^{s+1}(\Omega;\mathbb{R})\times H^{s+3/2}(\Sigma;\mathbb{R})\times
H^{s+1/2}(\Sigma;\mathbb{R}^{n})$ satisfy the divergence-trace compatibility
condition if

$\displaystyle
h-\int_{0}^{b}g(\cdot,x_{n})\;dx_{n}\in\dot{H}^{-1}(\Sigma;\mathbb{R}).$
(3.22)

We define the Hilbert space $\mathcal{Y}^{s}$ to be

$\mathcal{Y}^{s}=\\{(f,g,h,k)\in H^{s}(\Omega;\mathbb{R}^{n})\times
H^{s+1}(\Omega;\mathbb{R})\times H^{s+3/2}(\Sigma;\mathbb{R})\times
H^{s+1/2}(\Sigma;\mathbb{R}^{n})\mid\\\ (f,g,h,k)\;\text{satisfy the
divergence-trace condition}\;\eqref{eq:divtrace}\\},$ (3.23)

with the associated norm defined via

$\displaystyle\left\|(f,g,h,k)\right\|_{\mathcal{Y}^{s}}^{2}=\left\|f\right\|_{H^{s}}^{2}+\left\|g\right\|_{H^{s+1}}^{2}+\left\|k\right\|_{H^{s+1/2}}^{2}+\left\|h\right\|_{H^{s+3/2}}^{2}+\left[h-\int_{0}^{b}g(\cdot,x_{n})\;dx_{n}\right]_{\dot{H}^{-1}}^{2}.$
(3.24)

######  Definition 3.8.

Let $s\geqslant 0$. We say that the data tuple $(f,g,h,k)\in
H^{s}(\Omega;\mathbb{R}^{n})\times H^{s+1}(\Omega;\mathbb{R})\times
H^{s+3/2}(\Sigma;\mathbb{R})\times H^{s+1/2}(\Sigma;\mathbb{R}^{n})$ satisfy
the adjoint compatibility condition if for every $\psi\in
H^{s+1/2}(\Sigma;\mathbb{R})$,

$\displaystyle\int_{0}^{b}\hat{f}(\xi,x_{n})\cdot\overline{V(\xi,x_{n},-\gamma)}-\hat{g}(\xi,x_{n})\overline{Q(\xi,x_{n},-\gamma)}\;dx_{n}-\hat{k}(\xi)\cdot\overline{V(\xi,b,-\gamma)}+\hat{h}(\xi)=0,$
(3.25)

where $V,Q$ are defined in terms of $\psi$ in (3.10). We define the closed
subspace $\mathcal{Z}^{s}$ of $\mathcal{Y}^{s}$ to be

$\displaystyle\mathcal{Z}^{s}=\\{(f,g,h,k)\in\mathcal{Y}^{s}\mid(f,g,h,k)\;\text{satisfy}\;\eqref{eq:
2nd comp fourier}\;\forall\psi\in H^{s+1/2}(\Sigma;\mathbb{R})\\}.$ (3.26)

Now we are ready to record the main result of this subsection.

######  Theorem 3.9.

Let $s\geqslant 0,\gamma\in\mathbb{R}$, and $\mathcal{Z}^{s}$ the Hilbert
space defined in (3.8). Then the bounded linear operator
$\Psi_{\gamma}:{}_{0}H^{s+2}(\Omega;\mathbb{R}^{n})\times
H^{s+1}(\Omega;\mathbb{R}^{n})\to\mathcal{Z}^{s}$ given by

$\displaystyle\Psi_{\gamma}(u,p)=(\operatorname{div}S(p,u)-\gamma\partial_{1}u,\operatorname{div}u,u_{n}\rvert_{\Sigma_{b}},S(p,u)e_{n}\rvert_{\Sigma_{b}})$
(3.27)

is an isomorphism.

###### Proof.

The case $\Sigma=\mathbb{R}^{n-1}$ follows from Theorem 3.4 of [18]. We note
that by Remark 3.2, the calculations performed in $\mathbb{R}^{d}$ in [18] are
also valid over the low frequency regime in $\hat{\Gamma}$, therefore the
arguments therein can be adapted with minimal modification to handle the cases
when $\Sigma\neq\mathbb{R}^{n-1}$. ∎

###  3.3. Linear analysis with $\eta$ and $\kappa$

In this section we would like to establish the solvability of the
$\gamma$-Stokes system with gravity capillary boundary conditions

$\displaystyle\begin{cases}\operatorname{div}S(p,u)-\gamma\partial_{1}u+(\nabla^{\prime}\eta,0)=f&\text{in}\;\Omega\\\
\operatorname{div}u=g,&\text{in}\;\Omega\\\
u_{n}+\gamma\partial_{1}\eta=h,&\text{on}\;\Sigma_{b}\\\
S(p,u)e_{n}+\sigma\Delta^{\prime}\eta e_{n}=k,&\text{on}\;\Sigma_{b}\\\
u=0,&\text{on}\;\Sigma_{0},\end{cases}$ (3.28)

for data tuples belonging to the space $\mathcal{Y}^{s}$. First, we introduce
the container space for the solution tuple $(u,p,\eta)$.

######  Definition 3.10.

For $s\geqslant 0$, we define the Hilbert space

$\displaystyle\mathcal{X}^{s}=\begin{cases}\\{(u,p,\eta)\in{}_{0}H^{s+2}(\Omega;\mathbb{R}^{n})\times
H^{s+1}(\Omega;\mathbb{R})\times
X^{s+5/2}(\Sigma;\mathbb{R})\\},&R_{\Sigma}\neq\varnothing\\\
\\{(u,p,\eta)\in{}_{0}H^{s+2}(\Omega;\mathbb{R}^{n})\times
H^{s+1}(\Omega;\mathbb{R})\times\mathring{X}^{s+5/2}(\Sigma;\mathbb{R})\\},&R_{\Sigma}=\varnothing,\end{cases}$
(3.29)

where

$\displaystyle\mathring{X}^{s+5/2}(\Sigma;\mathbb{R})=\\{\eta\in
X^{s+5/2}(\Sigma;\mathbb{R})\mid\hat{\eta}(0)=0\\}.$ (3.30)

We endow $\mathcal{X}^{s}$ with the natural product norm defined via

$\displaystyle\left\|(u,p,\eta)\right\|_{\mathcal{X}^{s}}^{2}=\left\|u\right\|_{{}_{0}H^{s+1}}^{2}+\left\|p\right\|_{H^{s+1}}^{2}+\left\|\eta\right\|_{X^{s+5/2}}^{2}.$
(3.31)

Next, we record an embedding result for $\mathcal{X}^{s}$.

######  Proposition 3.11.

Suppose $s\geqslant 0$ and $\mathcal{X}^{s}$ is the Banach space in Definition
3.10. If $s>n/2$, then we have the continuous inclusion

$\displaystyle\mathcal{X}^{s}\subseteq C_{b}^{\lfloor
s-n/2\rfloor}(\Omega;\mathbb{R}^{n})\times C_{b}^{1+\lfloor
s-n/2\rfloor}(\Omega;\mathbb{R})\times C_{0}^{3+\lfloor
s-n/2\rfloor}(\Sigma;\mathbb{R}).$ (3.32)

Moreover, if $(u,p,\eta)\in\mathcal{X}^{s}$, then

$\displaystyle\lim_{|x_{R}^{\prime}|\to\infty}\partial^{\alpha}u(x)$
$\displaystyle=0\;\text{for all}\;\alpha\in\mathbb{N}^{n}\;\text{such
that}\;|\alpha|\leqslant 2+\left\lfloor s-\frac{n}{2}\right\rfloor$ (3.33)
$\displaystyle\lim_{|x_{R}^{\prime}|\to\infty}\partial^{\alpha}p(x)$
$\displaystyle=0\;\text{for all}\;\alpha\in\mathbb{N}^{n}\;\text{such
that}\;|\alpha|\leqslant 1+\left\lfloor s-\frac{n}{2}\right\rfloor.$ (3.34)

###### Proof.

This follows from standard Sobolev embedding and the first item of Theorem
2.6. ∎

In the subsection to follow, we establish some preliminary results to be
utilized in the subsequent analysis.

###  3.4. Preliminary results

In this subsection we use the asymptotics recorded in Section 3.1 to show that
we can construct the free surface function $\eta$ from a given data tuple
$(f,g,h,k)$. First, we study an auxiliary function defined in terms of the
multipliers $V,Q$ defined in (3.9).

######  Lemma 3.12.

Let $s\geqslant 0$, $\gamma\in\mathbb{R}$ and $(f,g,h,k)\in\mathcal{Y}^{s}$,
where $\mathcal{Y}^{s}$ is the Hilbert space defined in (3.23). Consider
$V(\cdot,\cdot,-\gamma):\hat{\Sigma}\times[0,b]\to\mathbb{C}^{n}$,
$Q(\cdot,\cdot,-\gamma):\hat{\Sigma}\times[0,b]\to\mathbb{C}$ defined in
(3.9). We define the measurable function $\psi:\hat{\Sigma}\to\mathbb{C}$ via

$\displaystyle\psi(\xi)=\int_{0}^{b}\left(\hat{f}(\xi,x_{n})\cdot\overline{V(\xi,x_{n},-\gamma)}-\hat{g}(\xi,x_{n})\overline{Q(\xi,x_{n},-\gamma)}\right)\;dx_{n}-\hat{k}(\xi)\cdot\overline{V(\xi,b,-\gamma)}+\hat{h}(\xi).$
(3.35)

Then the following hold.

  1. (1)

If $R_{\Sigma}=\varnothing$, then $\psi(0)=0$.

  2. (2)

$\overline{\psi(\xi)}=\psi(-\xi)$ for every $\xi\in\hat{\Sigma}$.

  3. (3)

We have the estimate

$\displaystyle\int_{\hat{\Sigma}}\frac{1}{|\xi|^{2}}|\psi(\xi)|^{2}\mathbbm{1}_{B(0,r)}+(1+|\xi|^{2})^{s+3/2}|\psi(\xi)|^{2}\mathbbm{1}_{B(0,r)^{c}}\;d\xi\lesssim_{n,s,\gamma,b}\left\|(f,g,h,k)\right\|_{\mathcal{Y}^{s}}^{2}.$
(3.36)

###### Proof.

To prove the first item, we assume $R_{\Sigma}=\varnothing$ and rewrite

$\psi(\xi)=\int_{0}^{b}\left(\hat{f}(\xi,x_{n})\cdot\overline{V(\xi,x_{n},-\gamma)}-\hat{g}(\xi,x_{n})(\overline{Q(\xi,x_{n},-\gamma)}-1)\right)\;dx_{n}\\\
-\hat{k}(\xi)\cdot\overline{V(\xi,b,-\gamma)}+\left(\hat{h}(\xi)-\int_{0}^{b}\hat{g}(\xi,x_{n})\;dx_{n}\right).$
(3.37)

We note that since $h,g$ must satisfy (3.22), by Remark 3.2 we have
$\hat{h}(0)-\int_{0}^{b}\hat{g}(0,x_{n})\;dx_{n}=0$. We also note that by the
first item of Theorem 3.5, $\overline{V(0,x_{n},-\gamma)}=0$ and
$\overline{Q(\xi,x_{n},-\gamma)}=0$. The first item follows immediately from
these two observations.

To prove the second item, we first note that by using the second item of
Theorem 3.5 and by using the fact that $f,g,k,h$ are real-valued,

$\overline{\psi(\xi)}=\int_{0}^{b}\left(\overline{\hat{f}(\xi,x_{n})}\cdot
V(\xi,x_{n},-\gamma)-\overline{\hat{g}(\xi,x_{n})}Q(\xi,x_{n},-\gamma)\right)\;dx_{n}-\overline{\hat{k}(\xi)}\cdot
V(\xi,b,-\gamma)+\overline{\hat{h}(\xi)}\\\
=\int_{0}^{b}\left(\hat{f}(-\xi,x_{n})\cdot\overline{V(-\xi,x_{n},-\gamma)}-\hat{g}(-\xi,x_{n})\overline{Q(-\xi,x_{n},-\gamma)}\right)\;dx_{n}-\hat{k}(-\xi)\cdot\overline{V(-\xi,b,-\gamma)}+\hat{h}(-\xi)=\psi(-\xi).$
(3.38)

To prove the third, we first rewrite $\psi$ again as in (3.37) and apply the
Cauchy-Schwarz inequality to obtain

$|\psi(\xi)|^{2}\lesssim\left(\int_{0}^{b}|\hat{f}(\xi,x_{n})|^{2}\;d\xi\right)\left(\int_{0}^{b}|V(\xi,x_{n})|^{2}\;d\xi\right)+\left(\int_{0}^{b}|\hat{g}(\xi,x_{n})|^{2}\;d\xi\right)\left(\int_{0}^{b}|Q(\xi,x_{n},-\gamma)-1|^{2}\;d\xi\right)\\\
+|\hat{k}(\xi)|^{2}|V(\xi,b,-\gamma)|^{2}+|\hat{h}(\xi)-\int_{0}^{b}\hat{g}(\xi,x_{n})\;dx_{n}|^{2}.$
(3.39)

We note that by the definition of the $\dot{H}^{-1}$ seminorm, we have

$\displaystyle\int_{B(0,r)}\frac{1}{|\xi|^{2}}|\hat{h}(\xi)-\int_{0}^{b}\hat{g}(\xi,x_{n})\;dx_{n}|^{2}\;d\xi\leqslant\left[h-\int_{0}^{b}g(\cdot,x_{n})\;dx_{n}\right]_{\dot{H}^{-1}}^{2}.$
(3.40)

Combining (3.39), (3.40) with (3.17), Tonelli’s theorem, and Parseval’s
theorem then gives us

$\int_{B(0,r)}\frac{1}{|\xi|^{2}}|\psi(\xi)|^{2}\;d\xi\lesssim_{n,\gamma,b}\int_{B(0,r)}\int_{0}^{b}\left(|\hat{f}(\xi,x_{n})|^{2}+|\hat{g}(\xi,x_{n})|^{2}\right)dx_{n}d\xi\\\
+\int_{B(0,r)}|\hat{k}(\xi)|^{2}\;d\xi+\left[h-\int_{0}^{b}g(\cdot,x_{n})\;dx_{n}\right]_{\dot{H}^{-1}}^{2}\lesssim_{n,\gamma,b}\left\|(f,g,h,k)\right\|_{\mathcal{Y}^{s}}^{2}.$
(3.41)

If $|\xi|\geqslant 1$, then we apply the Cauchy-Schwarz inequality directly to
(3.35) and obtain

$|\psi(\xi)|^{2}\lesssim\left(\int_{0}^{b}|\hat{f}(\xi,x_{n})|^{2}\;d\xi\right)\left(\int_{0}^{b}|V(\xi,x_{n})|^{2}\;d\xi\right)+\left(\int_{0}^{b}|\hat{g}(\xi,x_{n})|^{2}\;d\xi\right)\left(\int_{0}^{b}|Q(\xi,x_{n},-\gamma)|^{2}\;d\xi\right)\\\
+|\hat{k}(\xi)|^{2}|V(\xi,b,-\gamma)|^{2}+|\hat{h}(\xi)|^{2}.$ (3.42)

Combining (3.42) with (3.18), Tonelli’s theorem, and Parseval’s theorem then
gives us

$\int_{B(0,1)^{c}}(1+|\xi|^{2})^{s+3/2}|\psi(\xi)|^{2}\;d\xi\lesssim_{n,\gamma,b}\int_{0}^{b}\int_{\hat{\Sigma}}(1+|\xi|^{2})^{s}|\hat{f}(\xi,x_{n})|^{2}\;d\xi
dx_{n}+\int_{0}^{b}\int_{\hat{\Sigma}}(1+|\xi|^{2})^{s}|\hat{g}(\xi,x_{n})|^{2}\;d\xi
dx_{n}\\\
+\int_{\hat{\Sigma}}(1+|\xi|^{2})^{s+1/2}|\hat{k}(\xi)|^{2}\;d\xi+\int_{\hat{\Sigma}}(1+|\xi|^{2})^{s+3/2}|\hat{h}(\xi)|^{2}\;d\xi\lesssim_{n,\gamma,b}\int_{0}^{b}\left\|f(\cdot,x_{n})\right\|^{2}_{H^{s}(\Sigma)}\;dx_{n}\\\
+\int_{0}^{b}\left\|g(\cdot,x_{n})\right\|^{2}_{H^{s}(\Sigma)}\;dx_{n}+\left\|k\right\|_{H^{s+1/2}}^{2}+\left\|h\right\|_{H^{s+3/2}}^{2}\lesssim_{n,s,\gamma,b}\left\|(f,g,h,k)\right\|_{\mathcal{Y}^{s}}^{2}.$
(3.43)

∎

Next we study the linear map
$\Upsilon_{\gamma,\sigma}:\mathcal{X}^{s}\to\mathcal{Y}^{s}$ defined via

$\displaystyle\Upsilon_{\gamma,\sigma}(u,p,\eta)=(\operatorname{div}S(p,u)-\gamma\partial_{1}u+(\nabla^{\prime}\eta,0),\operatorname{div}u,u_{n}\rvert_{\Sigma_{b}}+\gamma\partial_{1}\eta,S(p,u)e_{n}\rvert_{\Sigma_{b}}+\sigma\Delta^{\prime}\eta
e_{n})$ (3.44)

which is the solution operator corresponding to the system (3.28). The next
result shows that this linear map is well-defined, bounded, and also
injective.

######  Proposition 3.13.

Suppose $\gamma\in\mathbb{R}\setminus\\{0\\}$, $\sigma\geqslant 0$, and
$s\geqslant 0$. Then the linear map
$\Upsilon_{\gamma,\sigma}:\mathcal{X}^{s}\to\mathcal{Y}^{s}$ defined in (3.44)
is well-defined, continuous, and injective.

###### Proof.

We first check that the map is well-defined and continuous. By the fifth item
of Theorem 2.5, the first component
$f=\operatorname{div}S(p,u)-\gamma\partial_{1}u+(\nabla^{\prime}\eta,0)$
belongs to $H^{s}(\Omega;\mathbb{R}^{n})$ with the estimate

$\displaystyle\left\|f\right\|_{H^{s}}\lesssim\left\|u\right\|_{{}_{0}H^{s+2}}+\left\|p\right\|_{H^{s+1}}+\left\|\eta\right\|_{X^{s+5/2}}.$
(3.45)

The second component $g=\operatorname{div}u$ belongs to
$H^{s+1}(\Omega;\mathbb{R})$ with the estimate
$\left\|g\right\|_{H^{s+1}}\lesssim\left\|u\right\|_{{}_{0}H^{s+2}}$. By the
fifth item of Theorem 2.5 and standard trace theory, we deduce that the third
component $k=S(p,u)e_{n}\rvert_{\Sigma_{b}}+\sigma\Delta^{\prime}\eta e_{n}$
belongs to $H^{s+1/2}(\Sigma;\mathbb{R}^{n})$ with the estimate

$\left\|k\right\|_{H^{s+1/2}}\leqslant\left\|p\right\|_{H^{s+1/2}(\Sigma)}+\left\|\mathbb{D}u\right\|_{H^{s+1/2}(\Sigma)}+\left\|\sigma\Delta^{\prime}\eta\right\|_{H^{s+1/2}(\Sigma)}\lesssim\left\|p\right\|_{H^{s+1}(\Omega)}+\left\|u\right\|_{{}_{0}H^{s+2}}+\left\|\eta\right\|_{X^{s+5/2}}.$
(3.46)

By the fifth item of Theorem 2.5 and standard trace theory, the fourth
component $h=u_{n}\rvert_{\Sigma_{b}}+\gamma\partial_{1}\eta$ belongs to
$H^{s+3/2}(\Sigma_{b};\mathbb{R})$ with the estimate

$\displaystyle\left\|h\right\|_{H^{s+3/2}}\lesssim\left\|u\right\|_{{}_{0}H^{s+2}}+\left\|\eta\right\|_{X^{s+5/2}}.$
(3.47)

Next we note that since $g=\operatorname{div}u$,

$\displaystyle
u_{n}(x^{\prime},b)-\int_{0}^{b}g(x^{\prime},x_{n})\;dx_{n}=\int_{0}^{b}\partial_{n}u_{n}(x^{\prime},x_{n})\;dx_{n}-\int_{0}^{b}g(x^{\prime},x_{n})=-\operatorname{div}^{\prime}\int_{0}^{b}u^{\prime}(x^{\prime},x_{n})\;dx_{n}$
(3.48)

for a.e. $x^{\prime}\in\Sigma$. Writing $H^{s+3}(\Sigma;\mathbb{R})\ni
R(x^{\prime})=\int_{0}^{b}u^{\prime}(x^{\prime},x_{n})\;dx_{n}$, we first note
that if $R_{\Gamma}=\varnothing$, then $\widehat{\operatorname{div}R}(0)=0$
and $\widehat{\gamma\partial_{1}\eta}(0)=0$. Then we note that

$\displaystyle\left[u_{n}-\int_{0}^{b}g(\cdot,x_{n})\;dx_{n}\right]_{\dot{H}^{-1}}=\left[\operatorname{div}R\right]_{\dot{H}^{-1}}\lesssim\left\|u\right\|_{L^{2}},$
(3.49)

therefore

$\displaystyle\left[h-\int_{0}^{b}g(\cdot,x_{n})\;dx_{n}\right]_{\dot{H}^{-1}}\leqslant\left[u_{n}-\int_{0}^{b}g(\cdot,x_{n})\;dx_{n}\right]_{\dot{H}^{-1}}+[\gamma\partial_{1}\eta]_{\dot{H}^{-1}}\lesssim\left\|u\right\|_{{}_{0}H^{s+2}}+\left\|\eta\right\|_{X^{s+5/2}}.$
(3.50)

Combining the estimates above shows that $\Upsilon_{\gamma,\sigma}$ is well-
defined and continuous.

To show that $\Upsilon_{\gamma,\sigma}$ is injective, we suppose
$(u,p,\eta)\in\mathcal{X}^{s}$ and
$\Upsilon_{\gamma,\sigma}\left(u,p,\eta\right)=0$. We note that if
$\tilde{p}=p-\eta$, then $\nabla\tilde{p}=\nabla p-(\nabla^{\prime}\eta,0)$
and $\tilde{p}I=pI-\eta I$. Therefore $\Upsilon_{\gamma,\sigma}(u,p,\eta)=0$
if and only if $(u,\tilde{p},\eta)$ satisfies

$\displaystyle\begin{cases}\operatorname{div}S(\tilde{p},u)-\gamma\partial_{1}u=0,&\text{in}\;\Omega\\\
\operatorname{div}u=0,&\text{in}\;\Omega\\\
S(\tilde{p},u)e_{n}=(\eta-\sigma\Delta^{\prime}\eta)e_{n},&\text{on}\;\Omega_{b}\\\
u_{n}+\gamma\partial_{1}\eta=0,&\text{on}\;\Sigma_{b}\\\
u=0,&\text{on}\;\Sigma_{0}.\end{cases}$ (3.51)

We note that by Tonelli’s theorem, Parseval’s theorem, and the fifth item of
Theorem 2.5 we have $\hat{u}(\xi,\cdot)\in H^{s}((0,b);\mathbb{C}^{n})$ and
$\widehat{\tilde{p}}(\xi,\cdot)\in H^{1}((0,b);\mathbb{C})$, for a.e.
$\xi\in\hat{\Sigma}$. By the second item in Theorem 2.6, $\hat{\eta}\in
L^{1}(\Sigma;\mathbb{R})+L^{2}(\Sigma,(1+|\xi|^{2})^{(s+5/2)/2}d\xi;\mathbb{R})$.
Thus, we may apply the horizontal Fourier transform to (3.51) to deduce for
a.e. $\xi\in\hat{\Sigma}$,
$w=\hat{u}(\xi,\cdot),q=\widehat{\tilde{p}}(\xi,\cdot)$ satisfies

$\displaystyle\begin{cases}\left(-\partial_{n}^{2}+4\pi^{2}|\xi|^{2}\right)w^{\prime}+2\pi
i\xi q-2\pi i\xi_{1}\gamma w^{\prime}=0,&\text{in}\;(0,b)\\\
\left(-\partial_{n}^{2}+4\pi^{2}|\xi|^{2}\right)w_{n}+\partial_{n}q-2\pi
i\xi_{1}\gamma w_{n}=0,&\text{in}\;(0,b)\\\ 2\pi i\xi\cdot
w^{\prime}+\partial_{n}w_{n}=0,&\text{in}\;(0,b)\\\
-\partial_{n}w^{\prime}-2\pi i\xi w_{n}=0,&\text{for}\;x_{n}=b\\\
q-2\partial_{n}w_{n}=(1+4\pi^{2}|\xi|^{2}\sigma)\hat{\eta},&\text{for}\;x_{n}=b\\\
w_{n}+2\pi i\xi_{1}\gamma\hat{\eta}=0,&\text{for}\;x_{n}=b\\\
w=0,&\text{for}\;x_{n}=0.\end{cases}$ (3.52)

First consider the special case when $R_{\Sigma}=\varnothing$ and $\xi=0$. In
this case, the third and sixth equations imply that $w_{n}\equiv 0$. Since
$\eta\in\mathring{X}^{s+5/2}(\Sigma;\mathbb{R})$, we have $\hat{\eta}(0)=0$
and therefore by the second and fifth equations, $q\equiv 0$. The first,
fourth, and last equations then tell us that $w^{\prime}\equiv 0$. Therefore,
we can conclude that at $\xi=0$, $w,q\equiv 0$.

Next consider the general case for which $\xi\in\hat{\Sigma}\setminus\\{0\\}$
and (3.52) holds. Using (3.52) and performing integration by parts (for
further details, see Proposition 4.1 in [18].), we deduce that

$\int_{0}^{b}-\gamma 2\pi
i\xi_{1}|w|^{2}+2|\partial_{n}w_{n}|^{2}+|\partial_{n}w^{\prime}+2\pi i\xi
w_{n}|^{2}+\frac{1}{2}|2\pi i\xi\otimes w^{\prime}+w^{\prime}\otimes 2\pi
i\xi|^{2}\;dx_{n}=-2\pi
i\xi_{1}\gamma(1+4\pi^{2}|\xi|^{2}\sigma)|\hat{\eta}(\xi)|^{2}.$ (3.53)

By taking the real part of this expression, we see that we must have for a.e.
$\xi\in\hat{\Sigma}$, $\partial_{n}w_{n}\equiv 0$ and
$\partial_{n}w^{\prime}+2\pi i\xi w_{n}\equiv 0$ in $(0,b)$, but since
$w(0)=0$, we must have $w\equiv 0$ in $[0,b]$. Then by the first equation, we
must have $q\equiv 0$. By the second to last equation, we find that
$\eta\equiv 0$. From this we find that $(u,p,\eta)=(0,0,0)$, so we can
conclude that $\Upsilon_{\gamma,\sigma}$ is injective.

∎

Next we show that $\Upsilon_{\gamma,\sigma}$ is surjective. To do so we must
construct the free surface function $\eta$ from a given data tuple
$(f,g,h,k)\in\mathcal{Y}^{s}$. For the reader’s convenience, we record this
construction in the next subsection.

###  3.5. Construction of the free surface function and the isomorphism
associated to (3.28)

######  Lemma 3.14 (Construction of $\eta$ in the presence of surface
tension).

Suppose $\gamma\in\mathbb{R}\setminus\\{0\\}$, $\sigma>0$ and $n\geqslant 2$,
$s\geqslant 0$. Then for every $(f,g,h,k)\in\mathcal{Y}^{s}$, there exists an
$\eta\in X^{s+\frac{5}{2}}(\Sigma;\mathbb{R})$ for which the modified data
tuple

$\displaystyle(f-(\nabla^{\prime}\eta,0),g,h-\gamma\partial_{1}\eta,k-\sigma\Delta^{\prime}\eta
e_{n})\in H^{s}(\Omega;\mathbb{R}^{n})\times H^{s+1}(\Omega;\mathbb{R})\times
H^{s+\frac{3}{2}}(\Sigma_{b};\mathbb{R})\times
H^{s+\frac{1}{2}}(\Sigma_{b};\mathbb{R}^{n})$ (3.54)

belongs to the range of $\Upsilon_{\gamma,\sigma}$ defined in (3.44).
Moreover, we have the estimate

$\displaystyle\left\|\eta\right\|_{X^{s+\frac{5}{2}}}\lesssim\left\|(f,g,h,k)\right\|_{\mathcal{Y}^{s}}.$
(3.55)

###### Proof.

Given $(f,g,h,k)\in\mathcal{Y}^{s}$, we propose to define $\eta\in
X^{s+\frac{5}{2}}(\Sigma;\mathbb{R})$ through

$\displaystyle\hat{\eta}(\xi)=\begin{cases}\frac{\psi(\xi)}{\rho_{\gamma}(\xi)},&\xi\neq
0\\\ 0,&\xi=0,\end{cases}$ (3.56)

where $\rho_{\gamma}$ is defined in (3.19) and $\psi$ is defined in terms of
$(f,g,h,k)$ in (3.35). We note that the choice of $\hat{\eta}(0)=0$ is only
relevant in the case when $R_{\Gamma}=\varnothing$.

We first note that by the first item of Lemma 3.6 and the second item of Lemma
3.12, we have $\overline{\hat{\eta}(\xi)}=\hat{\eta}(-\xi)$. Furthermore, by
the second item of Lemma 3.6 and the third item of Lemma 3.12, we have the
estimate

$\int_{\hat{\Sigma}}\left(\frac{\xi_{1}^{2}+|\xi|^{4}}{|\xi|^{2}}\mathbbm{1}_{B(0,r)}+(1+|\xi|^{2})^{s+\frac{5}{2}}\mathbbm{1}_{B(0,r)^{c}}\right)|\hat{\eta}(\xi)|^{2}\;d\xi\\\
\lesssim\int_{\hat{\Sigma}}\left(\frac{1}{|\xi|^{2}}\mathbbm{1}_{B(0,r)}+(1+|\xi|^{2})^{s+\frac{3}{2}}\mathbbm{1}_{B(0,r)^{c}}\right)|\rho_{\gamma}(\xi)|^{2}|\hat{\eta}(\xi)|^{2}\;d\xi\\\
\lesssim\int_{\hat{\Sigma}}\left(\frac{1}{|\xi|^{2}}\mathbbm{1}_{B(0,r)}+(1+|\xi|^{2})^{s+\frac{3}{2}}\mathbbm{1}_{B(0,r)^{c}}\right)|\psi(\xi)|^{2}\;d\xi\lesssim\left\|(f,g,h,k)\right\|_{\mathcal{Y}^{s}}^{2}.$
(3.57)

Consequently, if we define $\eta=(\hat{\eta})^{\vee}$, then by (3.57), and
$\overline{\hat{\eta}(\xi)}=\hat{\eta}(\xi)$, we have $\eta\in
X^{s+5/2}(\Sigma;\mathbb{R})$ with the estimate (3.55).

To conclude our proof it suffices to show that the modified data given in
(3.54) belongs to the range of $\Upsilon_{\gamma,\sigma}(u,p,\eta)$. By
Theorem 3.9, it suffices to show that the modified data tuple has the desired
regularity and satisfies the divergence-trace compatibility condition (3.22)
and the adjoint compatibility condition (3.25). We note that since $\eta\in
X^{s+5/2}(\Sigma;\mathbb{R})$, by the fifth item of Theorem 2.5 and the third
item of Theorem 2.6, we have
$(f,g,h-\gamma\partial_{1}\eta,k-\sigma\Delta^{\prime}\eta e_{n})\in
H^{s}(\Omega;\mathbb{R}^{n})\times H^{s+1}(\Omega;\mathbb{R})\times
H^{s+\frac{3}{2}}(\Sigma_{b};\mathbb{R})\times
H^{s+\frac{1}{2}}(\Sigma_{b};\mathbb{R}^{n})$ with
$(h-\gamma\partial_{1}\eta)-\int_{0}^{b}g(\cdot,x_{n})\in\dot{H}^{-1}(\Sigma;\mathbb{R})$.
To check (3.25), we write $0=\psi-\rho_{\gamma}\hat{\eta}$ for $\xi\neq 0$ and
use the definition of $\psi$ and $\rho_{\gamma}$ to obtain

$0=\int_{0}^{b}\left(\hat{f}(\xi,x_{n})\cdot\overline{V(\xi,x_{n},-\gamma)}-\hat{g}(\xi,x_{n})\overline{Q(\xi,x_{n},-\gamma)}\right)\;dx_{n}\\\
-(\hat{k}(\xi)+(1+4\pi^{2}|\xi|^{2}\sigma)\hat{\eta}(\xi)e_{n})\cdot\overline{V(\xi,b,-\gamma)}+\hat{h}(\xi)-2\pi
i\xi_{1}\gamma\hat{\eta}(\xi).$ (3.58)

Using the third equation in (3.10) we have $2\pi i\xi\cdot
V^{\prime}(\xi,x_{n},-\gamma)+\partial_{n}V_{n}(\xi,x_{n},-\gamma)=0$, and
since $V(\xi,0,-\gamma)=0$, we have

$\displaystyle\overline{V_{n}(\xi,b,-\gamma)}=\int_{0}^{b}\partial_{n}\overline{V_{n}(\xi,x_{n},-\gamma)}\;dx_{n}=\int_{0}^{b}2\pi
i\xi\cdot\overline{V^{\prime}(\xi,x_{n},-\gamma)}\;dx_{n}.$ (3.59)

Thus, we can rewrite (3.58) as

$0=\int_{0}^{b}\left(\left(\hat{f}(\xi,x_{n})-(\widehat{\nabla^{\prime}\eta}(\xi),0)\right)\cdot\overline{V(\xi,x_{n},-\gamma)}-\hat{g}(\xi,x_{n})\overline{Q(\xi,x_{n},-\gamma)}\right)\;dx_{n}\\\
-(\hat{k}(\xi)+\widehat{\sigma\Delta^{\prime}\eta}(\xi)e_{n})\cdot\overline{V(\xi,b,-\gamma)}+\hat{h}(\xi)-2\pi
i\xi_{1}\gamma\hat{\eta}(\xi),$ (3.60)

and the desired conclusion follows immediately. ∎

For the special case of $n=2$, we can also construct the free surface function
$\eta$ in the case without surface tension.

######  Lemma 3.15 (Construction of the free surface function without surface
tension).

Suppose $\gamma\in\mathbb{R}\setminus\\{0\\}$, $\sigma=0$ and $n=2$,
$s\geqslant 0$. Then for every $(f,g,h,k)\in\mathcal{Y}^{s}$, there exists an
$\eta\in H^{s+\frac{5}{2}}(\Sigma;\mathbb{R})$ for which the modified data
tuple

$\displaystyle(f-\partial_{1}\eta e_{1},g,h-\gamma\partial_{1}\eta,k)\in
H^{s}(\Omega;\mathbb{R}^{n})\times H^{s+1}(\Omega;\mathbb{R})\times
H^{s+\frac{3}{2}}(\Sigma_{b};\mathbb{R})\times
H^{s+\frac{1}{2}}(\Sigma_{b};\mathbb{R}^{n})$ (3.61)

belongs to the range of $\Upsilon_{\gamma,\sigma}$ defined in (3.44).
Moreover, we have the estimate

$\displaystyle\left\|\eta\right\|_{X^{s+\frac{5}{2}}}\lesssim\left\|(f,g,h,k)\right\|_{\mathcal{Y}^{s}}.$
(3.62)

###### Proof.

We follow the construction in the previous lemma and propose to define $\eta$
via (3.56). Then $\overline{\hat{\eta}(\xi)}=\hat{\eta}(-\xi)$,
$\hat{\eta}(0)=0$, and by the third item of Lemma 3.6 and the third item of
Lemma 3.12, we have

$\int_{\hat{\Sigma}}\left((1+|\xi|^{2})\mathbbm{1}_{B(0,1)}+(1+|\xi|^{2})^{s+\frac{5}{2}}\mathbbm{1}_{B(0,1)^{c}}\right)|\hat{\eta}(\xi)|^{2}\;d\xi\\\
\lesssim\int_{\hat{\Sigma}}\left(\frac{1}{|\xi|^{2}}\mathbbm{1}_{B(0,1)}+(1+|\xi|^{2})^{s+\frac{3}{2}}\mathbbm{1}_{B(0,1)^{c}}\right)|\rho_{\gamma}(\xi)|^{2}|\hat{\eta}(\xi)|^{2}\;d\xi\\\
\lesssim\int_{\hat{\Sigma}}\left(\frac{1}{|\xi|^{2}}\mathbbm{1}_{B(0,1)}+(1+|\xi|^{2})^{s+\frac{3}{2}}\mathbbm{1}_{B(0,1)^{c}}\right)|\psi(\xi)|^{2}\;d\xi\lesssim\left\|(f,g,h,k)\right\|_{\mathcal{Y}^{s}}^{2}.$
(3.63)

Consequently, we may define $\eta=(\hat{\eta})^{\vee}$. By (3.63) and Lemma
2.2, we have $\eta\in
H^{s+5/2}(\Sigma;\mathbb{R})=X^{s+5/2}(\Sigma;\mathbb{R})$ with the estimate
(3.62). By the third item of Theorem 2.6, we have $(f-\partial_{1}\eta
e_{1},g,h-\gamma\partial_{1}\eta,k)\in H^{s}(\Omega;\mathbb{R}^{n})\times
H^{s+1}(\Omega;\mathbb{R})\times
H^{s+\frac{3}{2}}(\Sigma_{b};\mathbb{R})\times
H^{s+\frac{1}{2}}(\Sigma_{b};\mathbb{R}^{n})$ and
$(h-\gamma\partial_{1}\eta)-\int_{0}^{b}g(\cdot,x_{2})\;dx_{2}\in\dot{H}^{-1}(\Sigma;\mathbb{R})$.
To conclude we follow the same line of calculations as in Lemma 3.14 to arrive
at

$0=\int_{0}^{b}\left((\hat{f}(\xi,x_{2})-2\pi
i\xi_{1}\hat{\eta}e_{1})\cdot\overline{V(\xi,x_{2},-\gamma)}-\hat{g}(\xi,x_{2})\overline{Q(\xi,x_{2},-\gamma)}\right)\;dx_{2}-\hat{k}(\xi)\cdot\overline{V(\xi,b,-\gamma)}\\\
+\hat{h}(\xi)-2\pi i\xi_{1}\gamma\hat{\eta}(\xi),$ (3.64)

which verifies the overdetermined compatibility condition (3.25). ∎

Now we are ready to prove that
$\Upsilon_{\gamma,\sigma}:\mathcal{X}^{s}\to\mathcal{Y}^{s}$ is an isomorphism
when $\sigma>0$ and $n\geqslant 2$, and when $\sigma=0$ and $n=2$.

######  Theorem 3.16 (Existence and uniqueness of solutions to (3.28)).

Suppose $\gamma\in\mathbb{R}\setminus\\{0\\},s\geqslant 0$.

  1. (1)

(Isomorphism in the case with surface tension) If $\sigma>0$ and $n\geqslant
2$, then the bounded linear map
$\Upsilon_{\gamma,\sigma}:\mathcal{X}^{s}\to\mathcal{Y}^{s}$ defined in (3.44)
is an isomorphism.

  2. (2)

(Isomorphism in the case without surface tension) If $\sigma=0$ and $n=2$,
then the bounded linear map
$\Upsilon_{\gamma,0}:\mathcal{X}^{s}\to\mathcal{Y}^{s}$ defined in (3.44) is
an isomorphism.

###### Proof.

To prove the first item, by Proposition 3.13, it suffices to show that
$\Upsilon_{\gamma,\sigma}$ is surjective. Suppose
$(f,g,h,k)\in\mathcal{Y}^{s}$, and define the free surface function $\eta\in
X^{s+5/2}(\Sigma;\mathbb{R})$ by the construction in Lemma 3.14. By Theorem
3.9, there exists $(u,p)\in{}_{0}H^{s+2}(\Omega;\mathbb{R}^{n})\times
H^{s+1}(\Omega;\mathbb{R})$ such that
$\Psi_{\gamma}(u,p)=(\operatorname{div}S(p,u)-\gamma\partial_{1}u,\operatorname{div}u,u_{n}\rvert_{\Sigma_{b}},S(p,u)e_{n}\rvert_{\Sigma_{b}})=(f-(\nabla^{\prime}\eta,0),g,h-\gamma\partial_{1}\eta,k-\sigma\Delta^{\prime}\eta
e_{n})$. Therefore, we find that
$\Upsilon_{\gamma,\sigma}(u,p,\eta)=(f,g,h,k)$. This shows that
$\Upsilon_{\gamma,\sigma}$ is surjective, and it follows that
$\Upsilon_{\gamma,\sigma}$ is an isomorphism.

To prove the second item we follow the same argument as above, using Lemma
3.15 in place of Lemma 3.14 and $\Upsilon_{\gamma,0}$ in place of
$\Upsilon_{\gamma,\sigma}$. ∎

###  3.6. Parameter regime for the linear isomorphism associated to (1.24)

In this subsection we consider the linearization of the flattened system
(1.24) around the trivial solution $u=0,p=0,\eta=0$, which is given by the map
$\mathcal{X}^{s}\ni(u,p,\eta)\mapsto L_{\kappa,\sigma}\in\mathcal{Y}^{s}$
defined in (1.42). Our goal is to show that there is a parameter regime in
terms of $\kappa$ for which $L_{\kappa,\sigma}$ remains an isomorphism between
$\mathcal{X}^{s}$ and $\mathcal{Y}^{s}$. To achieve this we first define
$L_{0,\sigma},M_{\kappa}:\mathcal{X}^{s}\to\mathcal{Y}^{s}$ via

$L_{0,\sigma}(u,p,\eta)=\Upsilon_{\gamma,\sigma}(u,p,\eta)=(\operatorname{div}S(p,u)-\gamma\partial_{1}u+(\nabla^{\prime}\eta,0),\operatorname{div}u,u_{n}\rvert_{\Sigma_{b}}+\gamma\partial_{1}\eta,S(p,u)e_{n}\rvert_{\Sigma_{b}}+\sigma\Delta^{\prime}\eta
e_{n}),$ (3.65)

and

$M_{\kappa}(u,p,\eta)=(-\gamma\kappa x_{n}\partial_{1}\eta e_{1}+\kappa
s_{0}(x_{n})\partial_{1}u+\kappa
s_{0}^{\prime}(x_{n})u_{n}e_{1}+\kappa^{2}x_{n}s_{0}(x_{n})\partial_{1}\eta
e_{1}-\kappa x_{n}\Delta^{\prime}\eta
e_{1}-\kappa(x_{n}\nabla^{\prime}\partial_{1}\eta,\partial_{1}\eta),\\\ \kappa
x_{n}\partial_{1}\eta,-\frac{\kappa b^{2}}{2}\partial_{1}\eta,0),$ (3.66)

so that
$L_{\kappa,\sigma}=L_{0,\sigma}+M_{\kappa}=L_{0,\sigma}(I+L_{0,\sigma}^{-1}M_{\kappa})$,
where $L_{\kappa,\sigma}$ is defined via (1.42) and $s_{0}$ is defined via
(1.6). Our goal in this subsection is to show that
$\left\|M_{\kappa}\right\|<\frac{1}{\left\|L_{0,\sigma}\right\|^{-1}}$ for
sufficiently small $\kappa$.

We first establish a preliminary lemma. While the following result is not at
all surprising and follows from routine applications of Tonelli’s theorem and
Hölder’s inequality, we record the proof below to show that the universal
constants in the estimates are purely combinatorial and do not depend on the
physical parameters.

######  Lemma 3.17.

Let $\mathbb{Z}\ni n\geqslant 2$ and suppose $f\in
C^{\infty}([0,b];\mathbb{R})$. The following hold.

  1. (1)

If $g\in H^{k}(\Omega;\mathbb{R}^{n})$ for some integer $k\geqslant 0$ and for
any $i\in\mathbb{N},j\in\mathbb{N}^{n}$ with $|j|\leqslant k$ we define
$(\partial^{i}f\partial^{j}g)(x)=\partial^{i}f(x_{n})\partial^{j}g(x)$, then
we have

$\displaystyle\left\|\partial^{i}f\partial^{j}g\right\|_{L^{2}}\leqslant\left\|\partial^{i}f\right\|_{L^{\infty}}\left\|g\right\|_{H^{j}}.$
(3.67)

Consequently, $fg\in H^{k}(\Omega;\mathbb{R}^{n})$ and there exists a
combinatorial constant $C=C(k)>0$ such that

$\displaystyle\left\|fg\right\|_{H^{k}}\leqslant C\max_{0\leqslant i\leqslant
k}\left\|\partial^{i}f\right\|_{L^{\infty}}\left\|g\right\|_{H^{k}}.$ (3.68)

  2. (2)

If $g\in H^{k}(\Sigma;\mathbb{R})$ for some integer $k\geqslant 0$ and for any
$i\in\mathbb{N},j\in\mathbb{N}^{n-1}$ with $|j|\leqslant k$ we define
$(\partial^{i}f\partial^{j}g)(x)=\partial^{i}f(x_{n})\partial^{j}g(x^{\prime})$,
then we have

$\displaystyle\left\|\partial^{i}f\partial^{j}g\right\|_{L^{2}}\leqslant\left\|\partial^{i}f\right\|_{L^{2}}\left\|g\right\|_{H^{j}}.$
(3.69)

Consequently, $fg\in H^{k}(\Omega;\mathbb{R})$ and there exists a
combinatorial constant $C=C(k)>0$ such that

$\displaystyle\left\|fg\right\|_{H^{k}}\leqslant C\max_{0\leqslant i\leqslant
k}\left\|\partial^{i}f\right\|_{L^{2}}\left\|g\right\|_{H^{k}}.$ (3.70)

###### Proof.

To prove the first item, we note that by Tonelli’s theorem and Hölder’s
inequality,

$\left\|\partial^{i}f\partial^{j}g\right\|_{L^{2}}=\left(\int_{\Sigma}\int_{0}^{b}|\partial^{i}f(x_{n})|^{2}|\partial^{j}g(x)|^{2}\;dx_{n}dx^{\prime}\right)^{\frac{1}{2}}\leqslant\left\|\partial^{i}f\right\|_{L^{\infty}(0,b)}\left(\int_{\Sigma}\int_{0}^{b}|\partial^{j}g(x)|^{2}\;dx_{n}dx^{\prime}\right)^{\frac{1}{2}}\\\
\leqslant\left\|\partial^{i}f\right\|_{L^{\infty}(0,b)}\left\|\partial^{j}g\right\|_{L^{2}\left(\Sigma\times(0,b)\right)}\leqslant\left\|\partial^{i}f\right\|_{L^{\infty}}\left\|g\right\|_{H^{j}}.$
(3.71)

Then by the Leibniz formula, we have

$\displaystyle\left\|fg\right\|_{H^{k}}\leqslant\sum_{\begin{subarray}{c}0\leqslant|j|\leqslant
k,\;m\leqslant
j\in\mathbb{N}^{n}\end{subarray}}\binom{j}{m}\left\|\partial^{j}f\right\|_{L^{\infty}}\left\|\partial^{j-m}g\right\|_{L^{2}}\leqslant
C\max_{0\leqslant i\leqslant
k\in\mathbb{N}}\left\|\partial^{i}f\right\|_{L^{\infty}}\left\|g\right\|_{H^{k}}$
(3.72)

where $C=C(k)>0$ is a combinatorial constant depending only on $k$. To prove
the second item, we note that by Tonelli’s theorem and Hölder’s inequality
again,

$\displaystyle\left\|\partial^{i}f\partial^{j}g\right\|_{L^{2}}=\left(\int_{0}^{b}|\partial^{i}f(x_{n})|^{2}\;dx_{n}\int_{\Sigma}|\partial^{j}g(x^{\prime})|^{2}\;dx^{\prime}\right)^{\frac{1}{2}}=\left\|\partial^{i}f\right\|_{L^{2}}\left\|\partial^{j}g\right\|_{L^{2}}\leqslant\left\|\partial^{i}f\right\|_{L^{2}}\left\|g\right\|_{H^{j}}.$
(3.73)

Then by the Leibniz formula,

$\displaystyle\left\|fg\right\|_{H^{k}}\leqslant\sum_{\begin{subarray}{c}0\leqslant|j|\leqslant
k,\;m\leqslant
j\in\mathbb{N}^{n-1}\end{subarray}}\binom{j}{m}\left\|\partial^{j}f\right\|_{L^{2}}\left\|\partial^{j-m}g\right\|_{L^{2}}\leqslant
C\max_{0\leqslant i\leqslant
k\in\mathbb{N}}\left\|\partial^{i}f\right\|_{L^{2}}\left\|g\right\|_{H^{k}}.$
(3.74)

∎

This gives us an immediate corollary that will prove useful in the next
section.

######  Corollary 3.18 (Multiplication with smooth functions in the vertical
variable).

Let $\Sigma$ be defined as in (1.1) and suppose $\mathbb{R}\ni s\geqslant 0$.
If $\varphi\in C^{\infty}([0,b];\mathbb{R})$, then the map
$T_{\varphi}:H^{s}(\Sigma;\mathbb{R})\to H^{s}(\Omega;\mathbb{R})$ defined via

$\displaystyle T_{\varphi}f(x)=\varphi(x_{n})f(x^{\prime})$ (3.75)

is well-defined and bounded.

###### Proof.

This follows directly from the second item of Lemma 3.17 when $s$ is an
integer. By interpolation, the second item also holds when $s$ is real-valued.
∎

Now we are ready to prove the main result of this subsection.

######  Theorem 3.19.

Let $\mathbb{R}\ni s\geqslant 0,\kappa\in\mathbb{R}$ and consider the linear
map $M_{\kappa}:\mathcal{X}^{s}\to\mathcal{Y}^{s}$ as defined in (3.66). Then
the following hold.

  1. (1)

There exists a constant $C>0$ depending on $s,n$, and in the periodic cases on
$L_{i}$, such that

$\displaystyle\left\|M_{\kappa}\right\|_{\mathcal{L}(\mathcal{X}^{s};\mathcal{Y}^{s})}\leqslant
C\left(\max\\{b^{\frac{1}{2}},b^{\frac{7}{2}}\\}|\kappa|^{2}+\left(|\gamma|\max\\{b^{1/2},b^{3/2}\\}+\max\\{1,b^{2}\\}\right)|\kappa|\right).$
(3.76)

  2. (2)

There exists $\kappa_{0}>0$ depending on $\gamma,s,b,n$, and in the toroidal
cases on $L_{i}$, such that for all $\kappa\in(-\kappa_{0},\kappa_{0})$, the
map $L_{\kappa,\sigma}:\mathcal{X}^{s}\to\mathcal{Y}^{s}$ as defined in (3.65)
is an isomorphism.

###### Proof.

To prove the first item, we first suppose that $(u,p,\eta)\in\mathcal{X}^{s}$
for some integer $s\geqslant 0$ and consider $(f,g,h,k)=M_{\kappa}(u,p,\eta)$,
where $M_{\kappa}$ is defined in (3.66). Then by the fifth item of Theorem
2.5, the third item of Theorem 2.6, and Lemma 3.17 we have
$(f,g,h,k)=M_{\kappa}(u,p,\eta)\in\mathcal{Y}^{s}$. Recall that
$s_{0}(x_{n})=bx_{n}-\frac{x_{n}^{2}}{2}$ and note that $s_{0}^{(i)}(x_{n})=0$
for $i\geqslant 3$ and $(x_{n}s_{0}(x_{n}))^{(i)}=0$ for $i\geqslant 4$.
Therefore, by Lemma 3.17 and the fifth item of Theorem 2.5 we have

$\left\|f\right\|_{H^{s}}\lesssim_{s}|\gamma||\kappa|\max_{0\leqslant
i\leqslant
1}\left\|x_{n}^{(i)}\right\|_{L^{2}}\left\|\partial_{1}\eta\right\|_{H^{s}}+|\kappa|\max_{0\leqslant
i\leqslant
2}\left\|s_{0}^{(i)}\right\|_{L^{\infty}}\left\|u\right\|_{H^{s+1}}+|\kappa|\max_{1\leqslant
i\leqslant
2}\left\|s_{0}^{(i)}\right\|_{L^{\infty}}\left\|u\right\|_{H^{s}}\\\
+|\kappa|^{2}\max_{0\leqslant i\leqslant
3}\left\|(x_{n}s_{0})^{(i)}\right\|_{L^{2}}\left\|\partial_{1}\eta\right\|_{H^{s}}+|\kappa|\max_{0\leqslant
i\leqslant
1}\left\|x_{n}^{(i)}\right\|_{L^{2}}\left\|\Delta^{\prime}\eta\right\|_{H^{s}}+|\kappa|\max_{0\leqslant
i\leqslant
1}\left\|x_{n}^{(i)}\right\|_{L^{2}}\left\|\partial_{1}\eta\right\|_{H^{s+1}}\\\
+|\kappa|\left\|\partial_{1}\eta\right\|_{H^{s}}\lesssim_{s,n}\left[\max\\{b^{\frac{1}{2}},b^{\frac{7}{2}}\\}|\kappa|^{2}+\left(|\gamma|\max\\{b^{1/2},b^{3/2}\\}+\max\\{1,b^{2}\\}\right)|\kappa|\right]\left(\left\|u\right\|_{H^{s+1}}+\left\|\eta\right\|_{X^{s+2}}\right)\\\
\lesssim_{s,n}\left(\max\\{b^{\frac{1}{2}},b^{\frac{7}{2}}\\}|\kappa|^{2}+\left(|\gamma|\max\\{b^{1/2},b^{3/2}\\}+\max\\{1,b^{2}\\}\right)|\kappa|\right)\left\|(u,p,\eta)\right\|_{\mathcal{X}^{s}}.$
(3.77)

We also have

$\displaystyle\left\|g\right\|_{H^{s+1}}\lesssim_{s}|\kappa|\max\\{b^{\frac{1}{2}},b^{\frac{3}{2}}\\}\left\|\partial_{1}\eta\right\|_{H^{s+1}}\lesssim_{s,n}|\kappa|\max\\{b^{\frac{1}{2}},b^{\frac{3}{2}}\\}\left\|\eta\right\|_{X^{s+2}}$
(3.78)

and
$\left\|h\right\|_{H^{s+\frac{3}{2}}}\lesssim_{s,n}|\kappa|b^{2}\left\|\eta\right\|_{X^{s+\frac{5}{2}}}$.
Therefore,

$\left\|M_{\kappa}(u,p,\eta)\right\|_{\mathcal{Y}^{s}}\lesssim_{s,n}\left(\max\\{b^{\frac{1}{2}},b^{\frac{7}{2}}\\}|\kappa|^{2}+\left(|\gamma|\max\\{b^{1/2},b^{3/2}\\}+\max\\{1,b^{2}\\}\right)|\kappa|\right)\left\|(u,p,\eta)\right\|_{\mathcal{X}^{s}}\\\
=:B(\gamma,b,\kappa)\left\|(u,p,\eta)\right\|_{\mathcal{X}^{s}}$ (3.79)

which implies that
$\left\|M_{\kappa}\right\|_{\mathcal{L}(\mathcal{X}^{s};\mathcal{Y}^{s})}\lesssim_{s,n}B(\gamma,b,\kappa)$.
We note that by the fifth item of Theorem 2.5, in the toroidal cases the
universal constants in the preceding equations also depend on $L_{i}$.

For any real valued $s\geqslant 0$, we may find an integer $m\geqslant 0$ and
some $\theta\in[0,1]$ such that $s=\theta m+(1-\theta)(m+1)$. Then by
interpolation, there exists some constant $C>0$ depending on $s,n$, and in the
periodic cases on $L_{i}$ such that

$\displaystyle\left\|M_{\kappa}\right\|_{\mathcal{L}(\mathcal{X}^{s};\mathcal{Y}^{s})}\leqslant\left\|M_{\kappa}\right\|_{\mathcal{L}(\mathcal{X}^{m};\mathcal{Y}^{m})}^{\theta}\left\|M_{\kappa}\right\|_{\mathcal{L}(\mathcal{X}^{m+1};\mathcal{Y}^{m+1})}^{1-\theta}\leqslant
CB(\gamma,b,\kappa).$ (3.80)

This proves the first item. To prove the second item, we note that

$\displaystyle
B(\gamma,b,\kappa)=\max\\{b^{\frac{1}{2}},b^{\frac{7}{2}}\\}|\kappa|^{2}+\left(|\gamma|\max\\{b^{1/2},b^{3/2}\\}+\max\\{1,b^{2}\\}\right)|\kappa|=\begin{cases}b^{7/2}|\kappa|^{2}+(|\gamma|b^{3/2}+b^{2})|\kappa|,&b>1\\\
b^{1/2}|\kappa|^{2}+(|\gamma|b^{1/2}+1)|\kappa|,&0<b\leqslant 1\end{cases}.$
(3.81)

From (3.81), we may infer that for
$c=C^{-1}\left\|L_{0,\sigma}\right\|^{-1}>0$, where $C>0$ is the constant
appearing in (3.80), there exists $\kappa_{0}$ depending on $\gamma,s,b,n$ and
in the toroidal cases on $L_{i}$ for which $B(\gamma,b,\kappa)<c$ if
$\kappa\in(-\kappa_{0},\kappa_{0})$.

Then by (3.80), if $\kappa\in(-\kappa_{0},\kappa_{0})$ then
$\left\|M_{\kappa}\right\|_{\mathcal{L}(\mathcal{X}^{s},\mathcal{Y}^{s})}<\left\|L_{0,\sigma}\right\|_{\mathcal{L}\left(\mathcal{Y}^{s};\mathcal{X}^{s}\right)}^{-1}$,
thus the linear map
$L_{0,\sigma}^{-1}M_{\kappa}:\mathcal{X}^{s}\to\mathcal{X}^{s}$ as defined in
(3.65) satisfies
$\left\|L_{0,\sigma}^{-1}M_{\kappa}\right\|\leqslant\left\|L_{0,\sigma}^{-1}\right\|\left\|M_{\kappa}\right\|<1$.
Hence, $I+L_{0,\sigma}^{-1}M_{\kappa}:\mathcal{X}^{s}\to\mathcal{X}^{s}$ is an
isomorphism, and we conclude that
$L_{\kappa,\sigma}=L_{0,\sigma}(I+L_{0,\sigma}^{-1}M_{\kappa})$ is an
isomorphism from $\mathcal{X}^{s}$ to $\mathcal{Y}^{s}$. ∎

##  4\. Nonlinear analysis

###  4.1. Preliminaries

We first record a set of results on various maps defined in terms of $\eta$,
including the flattening map $\mathfrak{F}_{\eta}$, that we will use in the
subsequent analysis.

######  Theorem 4.1.

Let $\Sigma$ be defined as in (1.1), $\mathbb{N}\ni n\geqslant 2$,
$\mathbb{N}\ni k>\frac{n}{2}$, and $V$ be a real finite dimensional inner
product space.

  1. (1)

Let $\upzeta\in C_{b}^{0,1}(\Sigma;\mathbb{R})$ such that $\inf\upzeta>0$.
Then there exists $r_{1}>0$ depending on $n,b,s,\upzeta$, and in the toroidal
cases on $L_{i}$, such that the maps
$\Gamma_{1},\Gamma_{2}:B_{X^{s}}(0,r_{1})\times
H^{s}(\Omega_{\upzeta};\mathbb{R})\to H^{s}(\Omega_{\upzeta};\mathbb{R})$
given by

$\displaystyle\Gamma_{1}(f,g)=\frac{g}{b+f},\quad\Gamma_{2}(f,g)=\frac{gf}{b+f}$
(4.1)

are well-defined and smooth. There also exists a constant $r_{2}>0$ depending
on $n,s$, and in the toroidal cases on $L_{i}$, such that the map
$\Gamma:B_{X^{s}(0,r_{2})}\to H^{s}(\Omega;\mathbb{R}^{n})$ given by

$\displaystyle\Gamma(f)=\frac{f}{\sqrt{1+|f|^{2}}}$ (4.2)

is well-defined and smooth.

  2. (2)

(The flattening map $\mathfrak{F}_{\eta}$ and its inverse) Let $\eta\in
X^{k+\frac{5}{2}}(\Sigma;\mathbb{R})$ be such that
$\left\|\eta\right\|_{C^{0}_{b}}\leqslant\frac{b}{2}$. Define
$\mathfrak{G}_{\eta}:\Omega_{b+\eta}\to\Omega_{b}$ via

$\displaystyle\mathfrak{G}_{\eta}(x)=\left(x^{\prime},\frac{x_{n}b}{b+\eta(x^{\prime})}\right).$
(4.3)

Then the following hold.

    1. (a)

$\mathfrak{G}_{\eta}\in C^{r}(\Omega_{b+\eta};\Omega_{b})$ is a diffeomorphism
for $r=3+\left\lfloor k-\frac{n}{2}\right\rfloor$, with its inverse being
$\mathfrak{F}_{\eta}\in C^{r}(\Omega;\Omega_{b+\eta})$.

    2. (b)

If $0\leqslant s\leqslant k+2$ and $F\in H^{s}(\Omega_{b};V)$, then
$F\circ\mathfrak{G}_{\eta}\in H^{s}(\Omega_{b+\eta};V)$. Moreover, there
exists a constant $c>0$ depending on $n,s,k,\left\|\eta\right\|_{X^{k+2}}$,
and in the toroidal cases on $L_{i}$, such that

$\displaystyle\left\|F\circ\mathfrak{G}_{\eta}\right\|_{H^{s}(\Omega_{b+\eta};V)}\leqslant
c\left\|F\right\|_{H^{s}(\Omega_{b};V)},$ (4.4)

and the map $r\mapsto c(n,s,k,r)$ is non-decreasing.

    3. (c)

If $0\leqslant s\leqslant k+2$ and $F\in H^{s}(\Omega_{b+\eta};V)$, then
$F\circ\mathfrak{F}_{\eta}\in H^{s}(\Omega_{b};V)$. Moreover, there exists a
constant $c>0$ depending on $n,s,k,\left\|\eta\right\|_{X^{k+2}}$, and in the
toroidal cases on $L_{i}$, such that

$\displaystyle\left\|F\circ\mathfrak{F}_{\eta}\right\|_{H^{s}(\Omega_{b};V)}\leqslant
c\left\|F\right\|_{H^{s}(\Omega_{b+\eta};V)},$ (4.5)

and the map $r\mapsto c(n,s,k,r)$ is non-decreasing.

  3. (3)

($\omega$-lemma for compositions) For $\eta\in
X^{k+\frac{1}{2}}(\Sigma_{b};\mathbb{R})$ we define the flattening map
$\mathfrak{F}_{\eta}$ as in (1.17). Then there exists some $0<\delta_{*}<1$
such that the following hold:

    1. (a)

The map $\Lambda_{\Omega}:H^{k+1}(\Sigma\times\mathbb{R};\mathbb{R}^{n})\times
B_{X^{k+\frac{1}{2}}}(0,\delta_{*})\to H^{k}(\Omega_{b};\mathbb{R}^{d})$ given
by

$\displaystyle\Lambda_{\Omega}(f,\eta)=f\circ\mathfrak{F}_{\eta}$ (4.6)

is well-defined and $C^{1}$, with
$D\Lambda_{\Omega}(f,\eta)(g,\zeta)=\frac{\varphi}{b}\partial_{n}f\circ\mathfrak{F}_{\eta}\zeta+g\circ\mathfrak{F}_{\eta}$.

    2. (b)

The map $\mathfrak{S}_{b}:H^{k+2}(\Sigma\times\mathbb{R};\mathbb{R}^{n})\times
B_{X^{s+\frac{3}{2}}}(0,\delta_{*})\to
H^{s+\frac{1}{2}}(\Sigma_{b};\mathbb{R}^{d})$ defined via

$\displaystyle\mathfrak{S}_{b}(f,\eta)=f\circ\mathfrak{F}_{\eta}\rvert_{\Sigma_{b}}$
(4.7)

is well-defined and $C^{1}$, with
$D\mathfrak{S}_{b}(f,\eta)(g,\zeta)\left(\frac{\varphi}{b}\partial_{n}f\circ\mathfrak{F}_{\eta}\zeta+g\circ\mathfrak{F}_{\eta}\right)\rvert_{\Sigma_{b}}$.

    3. (c)

The map $\mathfrak{T}_{b}:H^{k+2}(\Sigma\times\mathbb{R};\mathbb{R}^{n\times
n}_{\operatorname{sym}})\times B_{X^{k+5/2}}(0,\delta_{*})\to
H^{k+1/2}(\Sigma_{b};\mathbb{R}^{d})$ defined via

$\displaystyle\mathfrak{T}_{b}(\mathcal{T},\eta)=(\mathcal{T}\circ\mathfrak{F}_{\eta}\rvert_{\Sigma_{b}})\mathcal{N},$
(4.8)

where $\mathcal{N}$ is defined via (1.13), is well-defined and $C^{1}$.

###### Proof.

The three items in the case $\Sigma=\mathbb{R}^{d}$ for $d\geqslant 1$ follow
from Theorems 5.16, 5.17, A.12, Corollary 5.21, and Proposition 7.4 in [18];
the proofs therein can be adopted with minimal modification to handle to cases
when $\Sigma\neq\mathbb{R}^{d}$. ∎

Now we can synthesize the aforementioned results to show that all the
nonlinear maps appearing in (1.24) are well-defined and smooth.

######  Theorem 4.2.

Let $\mathbb{R}\ni s>\frac{n}{2}$ and $\sigma\geqslant 0$. For $\delta>0$,
define the open set of $\mathcal{X}^{s}$

$\displaystyle
U^{s}_{\delta}=\\{(u,p,\eta)\in\mathcal{X}^{s}\mid\left\|\eta\right\|_{X^{s+\frac{5}{2}}}<\delta\\}.$
(4.9)

For $\gamma\in\mathbb{R},T\in H^{s+\frac{1}{2}}(\Sigma;\mathbb{R}^{n\times
n}_{\operatorname{sym}})$, $(u,p,\eta)\in U^{s}_{\delta}$,
$U^{1}=W_{1}\circ\mathfrak{F}_{\eta},U_{2}\circ=W_{2}\circ\mathfrak{F}_{\eta}$
with $W_{1},W_{2}$ as defined in (1.7), (1.14), $\mathfrak{F}_{\eta}$ as
defined in (1.17), and $\mathcal{J},\mathcal{A},\mathcal{N},\mathcal{H}$ as
defined in (1.20), (1.21), (1.13) and (1.2), we define
$f=\operatorname{div}_{\mathcal{A}}S_{\mathcal{A}}(p,u)-\gamma
e_{1}\cdot\nabla_{\mathcal{A}}u-\gamma\kappa(x_{n}+\eta\frac{x_{n}}{b})\partial_{1}\eta
e_{1}+\left(u+U^{1}+U^{2}\right)\cdot\nabla_{\mathcal{A}}\left(u+U^{1}+U^{2}\right)+(\nabla^{\prime}\eta,0)-\kappa\left(x_{n}+\eta\frac{x_{n}}{b}\right)\Delta^{\prime}\eta
e_{1}-\kappa((x_{n}+\eta\frac{x_{n}}{b})\nabla^{\prime}\partial_{1}\eta,\partial_{1}\eta)$,
$g=\mathcal{J}[\operatorname{div}_{\mathcal{A}}u+\kappa\left(x_{n}+\eta(x^{\prime})\frac{x_{n}}{b}\right)\partial_{1}\eta(x^{\prime})]$,
$h=u\cdot\mathcal{N}-(-\gamma+s(\eta+b)+\kappa(\eta+b)\eta)\partial_{1}\eta$,
and
$k=S_{\mathcal{A}}(p,u)\mathcal{N}-[\sigma\mathcal{H}(\eta)I+T+\kappa(\eta+b)(e_{1}\otimes(\nabla^{\prime}\eta,0)+(\nabla^{\prime}\eta,0)\otimes
e_{1})]\mathcal{N}$. Then $(f,g,h,k)\in\mathcal{Y}^{s}$, and the map

$\displaystyle\mathbb{R}\times\mathbb{R}\times
H^{s+\frac{1}{2}}(\Sigma;\mathbb{R}^{n\times n}_{\operatorname{sym}})\times
U^{s}_{\delta}\ni(\gamma,T,u,p,\eta)\mapsto(f,g,h,k)\in\mathcal{Y}^{s}$ (4.10)

is smooth.

###### Proof.

By the first item in Theorem 2.6 and standard Sobolev embedding, there exists
a constant $\delta_{0}>0$ depending on $n,s,b$, and in the toroidal cases on
$L_{i}$, such that if $s>n/2$ and $\eta\in X^{s+5/2}(\Sigma;\mathbb{R})$ with
$\left\|\eta\right\|_{X^{s+5/2}}<\delta_{0}$, then
$\left\|\eta\right\|_{C^{0}_{b}}\leqslant b/2$. We define
$\delta=\min\\{\delta_{0},r_{1},r_{2}/c_{1}\\}$, where $r_{1},r_{2}$ are the
radii from the first item of Theorem 4.1, and $c_{1}$ is the embedding
constant from (2.24). By Theorem 7.3 in [18] and standard results from the
theory of Sobolev spaces, the map

$\mathbb{R}\times\mathbb{R}\times H^{s+\frac{1}{2}}(\Sigma;\mathbb{R}^{n\times
n}_{\operatorname{sym}})\times U^{s}_{\delta}\ni(\gamma,T,u,p,\eta)\mapsto\\\
(\operatorname{div}_{\mathcal{A}}S_{\mathcal{A}}(p,u)+(u-\gamma
e_{1})\cdot\nabla_{\mathcal{A}}u+u\cdot\nabla_{\mathcal{A}}u,\mathcal{J}\operatorname{div}_{\mathcal{A}}u,u\cdot\mathcal{N}+\gamma\partial_{1}\eta,S_{\mathcal{A}}(p,u)\mathcal{N}-(\sigma\mathcal{H}(\eta)I+T)\mathcal{N})\in\mathcal{Y}^{s}$
(4.11)

is well-defined and smooth, so it suffices to show that the map

$\mathbb{R}\times H^{s+\frac{1}{2}}(\Sigma;\mathbb{R}^{n\times
n}_{\operatorname{sym}})\times U^{s}_{\delta}\ni(\gamma,T,u,p,\eta)\mapsto\\\
(-\gamma\kappa\mathfrak{F}_{\eta}(x^{\prime},x_{n})e_{n}\partial_{1}\eta
e_{1}+u\cdot\nabla_{\mathcal{A}}(U^{1}+U^{2})+(U^{1}+U^{2})\cdot\nabla_{\mathcal{A}}(u+U^{1}+U^{2})\\\
+(\nabla^{\prime}\eta,0)-\kappa\mathfrak{F}_{\eta}(x^{\prime},x_{n})e_{n}\Delta^{\prime}\eta(x^{\prime})e_{1}-\kappa(\mathfrak{F}_{\eta}(x^{\prime},x_{n})e_{n}\partial_{1}\nabla^{\prime}\eta,\partial_{1}),\mathcal{J}\kappa(x_{n}+\eta(x^{\prime})\frac{x_{n}}{b})\partial_{1}\eta(x^{\prime}),\\\
-s(\eta+b)\partial_{1}\eta-\kappa(\eta+b)\eta\partial_{1}\eta,\;\kappa(\eta+b)\left[e_{1}\otimes(\nabla^{\prime}\eta,0)+(\nabla^{\prime}\eta,0)\otimes
e_{1}\right]\mathcal{N})\in\mathcal{Y}^{s}$ (4.12)

is well-defined and smooth, where
$\mathfrak{F}_{\eta}(x^{\prime},x_{n})e_{n}=x_{n}+\eta(x^{\prime})\frac{x_{n}}{b}$.
By the fifth item of Theorem 2.5, the second item of Theorem 2.6, and
Corollary 3.18 the map

$\mathbb{R}\times\mathbb{R}\times
X^{s+5/2}(\Sigma;\mathbb{R})\ni(\gamma,\kappa,\eta)\mapsto-\gamma\kappa\mathfrak{F}_{\eta}(x^{\prime},x_{n})e_{n}\partial_{1}\eta
e_{1}+(\nabla^{\prime}\eta,0)\\\
-\kappa\mathfrak{F}_{\eta}(x^{\prime},x_{n})e_{n}\Delta^{\prime}\eta(x^{\prime})e_{1}-\kappa(\mathfrak{F}_{\eta}(x^{\prime},x_{n})e_{n}\partial_{1}\nabla^{\prime}\eta,\partial_{1}\eta)\in
H^{s+1/2}(\Omega;\mathbb{R}^{n})$ (4.13)

is well-defined and smooth. By (1.7), (1.14), (1.17), and (1.21), we have

$u\cdot\nabla_{\mathcal{A}}(U^{1}+U^{2})+(U^{1}+U^{2})\cdot\nabla_{\mathcal{A}}(u+U^{1}+U^{2})=\sum_{j,k=1}^{n}u_{j}\mathcal{A}_{jk}\partial_{k}(U^{1}+U^{2})+\sum_{k=1}^{n}(U_{1}^{1}+U_{1}^{2})\mathcal{A}_{1k}\partial_{k}(u+U^{1}+U^{2})\\\
=\sum_{j=1}^{n-1}u_{j}\partial_{j}(U^{1}+U^{2})+\sum_{k=1}^{n-1}u_{n}\mathcal{A}_{nk}\partial_{k}(U^{1}+U^{2})+u_{n}\mathcal{K}\partial_{n}(U^{1}+U^{2})+(U_{1}^{1}+U_{2}^{2})\partial_{1}(u+U^{1}+U^{2})\\\
-(U_{1}^{1}+U_{1}^{1})x_{n}\mathcal{K}\frac{\partial_{1}\eta(x^{\prime})}{b}\partial_{n}(U^{1}+U^{2}),$
(4.14)

where $\mathcal{K}=1/\mathcal{J}$. By (1.7), (1.14), and (1.17),
$(U^{1}+U^{2})(x,\eta)=\kappa(b\mathfrak{F}_{\eta}(x)e_{n}-(\mathfrak{F}_{\eta}(x)e_{n})^{2}/2+\mathfrak{F}_{\eta}(x)e_{n}\eta(x^{\prime}))e_{1}$
where
$\mathfrak{F}_{\eta}(x)e_{n}=x_{n}(1+\eta(x^{\prime})/b)=x_{n}\mathcal{J}$, so
we find that

$\displaystyle\partial_{k}(U^{1}+U^{2})(x^{\prime},x_{n})=\begin{cases}\kappa\mathcal{J}\frac{x_{n}(2b-x_{n})}{b}\partial_{k}\eta(x^{\prime})e_{1},&k\neq
n\\\ \kappa(b+x_{n}\mathcal{J}+\eta(x^{\prime}))\mathcal{J},&k=n.\end{cases}.$
(4.15)

Thus, by the second item of Theorem 2.6, Corollary 3.18, and the algebra
properties of standard Sobolev spaces for $s>n/2$ the map
$X^{s+5/2}(\Sigma;\mathbb{R})\times
H^{s}(\Omega;\mathbb{R})\ni(\eta,\psi)\mapsto\psi\partial_{k}(U^{1}+U^{2})\in
H^{s}(\Omega;\mathbb{R}^{n})$ is well-defined and smooth. By the first item of
Theorem 4.1, the map $B_{X^{s}}(0,\delta)\times
H^{s}(\Omega;\mathbb{R})\ni(\eta,\psi)\mapsto\psi\mathcal{K}\in
H^{s}(\Omega;\mathbb{R})$ is well-defined and smooth for any $\psi\in
H^{s}(\Omega;\mathbb{R}^{n})$. We also note that every non-trivial term in the
components of (4.14) is either a product of functions in
$X^{s+5/2}(\Sigma;\mathbb{R})$ and functions in $H^{s}(\Omega;\mathbb{R})$, or
functions in $X^{s+5/2}(\Sigma;\mathbb{R})$ and derivatives of functions in
$X^{s+5/2}(\Sigma;\mathbb{R})$. To summarize, by the observations made above,
(4.15), the second item of Theorem 2.6, Corollary 3.18, and the first item of
Theorem 4.1, the map

$\displaystyle\mathbb{R}\times U^{s}_{\delta}\ni(\kappa,u,p,\eta)\mapsto
u\cdot\nabla_{\mathcal{A}}(U^{1}+U^{2})+(U^{1}+U^{2})\cdot\nabla_{\mathcal{A}}(u+U^{1}+U^{2})\in
H^{s}(\Omega;\mathbb{R}^{n})$ (4.16)

is well-defined and smooth. Similarly, the maps

$\displaystyle\mathbb{R}\times
X^{s+5/2}(\Sigma;\mathbb{R})\ni(\kappa,\eta)\mapsto\mathcal{J}\kappa(x_{n}+\eta(x^{\prime})\frac{x_{n}}{b})\partial_{1}\eta(x^{\prime})\in
H^{s+3/2}(\Sigma;\mathbb{R})$ (4.17)

and

$\mathbb{R}\times X^{s+5/2}(\Sigma;\mathbb{R})\ni(\kappa,\eta)\mapsto\\\
(s(\eta+b)\partial_{1}\eta+\kappa(\eta+b)\eta\partial_{1}\eta,\kappa(\eta+b)(e_{1}\otimes(\nabla^{\prime}\eta,0)+(\nabla^{\prime}\eta,0)\otimes
e_{1})\mathcal{N})\in H^{3/2}(\Sigma;\mathbb{R})\times
H^{3/2}(\Sigma;\mathbb{R}^{n})$ (4.18)

are also well-defined and smooth. Finally, it remains to show that

$\displaystyle-s(\eta+b)\partial_{1}\eta-\kappa(\eta+b)\eta\partial_{1}\eta-\int_{0}^{b}\mathcal{J}(\cdot,x_{n})\kappa\left(x_{n}+\eta(\cdot)\frac{x_{n}}{b}\right)\partial_{1}\eta(\cdot)\;dx_{n}\in\dot{H}^{-1}(\Sigma;\mathbb{R}).$
(4.19)

By (1.20), we have

$\displaystyle-s(\eta+b)\partial_{1}\eta-\kappa(\eta+b)\eta\partial_{1}\eta-\int_{0}^{b}\mathcal{J}(\cdot,x_{n})\kappa\left(x_{n}+\eta(\cdot)\frac{x_{n}}{b}\right)\partial_{1}\eta(\cdot)\;dx_{n}=-\kappa(b^{2}\partial_{1}\eta+b\partial_{1}\eta^{2}+\frac{1}{3}\partial_{1}\eta^{3}).$
(4.20)

Thus, by Theorem 1.1 and the third item of Theorem 2.6 we have
$b^{2}\partial_{1}\eta+b\partial_{1}\eta^{2}+\frac{1}{3}\partial_{1}\eta^{3}\in\dot{H}^{-1}(\Sigma;\mathbb{R})$.
Therefore, we have shown that $(f,g,h,k)\in\mathcal{Y}^{s}$, and the map
defined by (4.10) is smooth. ∎

###  4.2. Solvability of the flattened system (1.24)

Now we are ready to construct solutions to (1.24) by using the implicit
function theorem.

###### Proof of Theorem 1.2.

We first consider the case with surface tension, $\sigma>0$ and $n\geqslant
2$. Let $\delta$ be the minimum of the $\delta_{1}>0$ from Theorem 4.2 and
$\delta_{*}>0$ from the third item of Theorem 4.1. Consider the open subset
$U^{s}_{\delta}$ of $\mathcal{X}^{s}$ defined via

$\displaystyle
U^{s}_{\delta}=\\{(u,p,\eta)\in\mathcal{X}^{s}\mid\left\|\eta\right\|_{X^{s+\frac{5}{2}}}<\delta\\}.$
(4.21)

Using Proposition 3.11 and standard Sobolev embedding, any open subset of
$U^{s}_{\delta}$ containing $(0,0,0)$ satisfies the first assertion of the
theorem. This proves the first item.

To prove the remaining items, we consider the Hilbert space

$\displaystyle\mathcal{E}^{s}=\mathbb{R}\times\mathbb{R}\times
H^{s+2}(\Sigma\times\mathbb{R};\mathbb{R}^{n\times
n}_{\operatorname{sym}})\times H^{s+\frac{1}{2}}(\Sigma;\mathbb{R}^{n\times
n}_{\operatorname{sym}})\times
H^{s+1}(\Sigma\times\mathbb{R};\mathbb{R}^{n})\times
H^{s}(\Sigma;\mathbb{R}^{n})$ (4.22)

and the solution map $\Xi:\mathcal{E}^{s}\times
U^{s}_{\delta}\to\mathcal{Y}^{s}$ associated to (1.24) defined via

$\Xi(\gamma,\kappa,\mathcal{T},T,\mathfrak{f},f,u,p,\eta)=(\operatorname{div}_{\mathcal{A}}S_{\mathcal{A}}(p,u)-\gamma
e_{1}\cdot\nabla_{\mathcal{A}}u-\gamma\kappa x_{n}\partial_{1}\eta
e_{1}+\left(u+U^{1}+U^{2}\right)\cdot\nabla_{\mathcal{A}}\left(u+U^{1}+U^{2}\right)\\\
+(\nabla^{\prime}\eta,0)-\kappa\mathfrak{F}_{\eta}(x)e_{n}\Delta^{\prime}\eta
e_{1}-\kappa(\mathfrak{F}_{\eta}(x)e_{n}\nabla^{\prime}\partial_{1}\eta,\partial_{1}\eta)-\mathfrak{f}\circ\mathfrak{F}_{\eta}-L_{\Omega_{b}}f,\\\
\mathcal{J}[\operatorname{div}_{\mathcal{A}}u+\kappa\mathfrak{F}_{\eta}(x)e_{n}\partial_{1}\eta],u\cdot\mathcal{N}-(-\gamma+s(\eta+b)+\kappa(\eta+b)\eta)\partial_{1}\eta,\\\
S_{\mathcal{A}}(p,u)\mathcal{N}-[\sigma\mathcal{H}(\eta)I+\mathcal{T}\circ\mathfrak{F}_{\eta}+S_{b}T\rvert_{\Sigma_{b}}+\kappa(\eta+b)(e_{1}\otimes(\nabla^{\prime}\eta,0)+(\nabla^{\prime}\eta,0)\otimes
e_{1})]\mathcal{N}),$ (4.23)

where $L_{\Omega_{b}}f(x)=f(x^{\prime}),S_{b}T(x^{\prime},b)=T(x^{\prime})$.
By Theorem 4.1, Theorem 4.2, and Lemmas A.10 and A.11 in [18] the map $\Xi$ is
well-defined and $C^{1}$.

By the product structure of $\mathcal{E}^{s}\times U^{s}_{\delta}$, we can
define $D_{1}\Xi:\mathcal{E}^{s}\times
U^{s}_{\delta}\to\mathcal{L}(\mathcal{E}^{s};\mathcal{Y}^{s})$ and
$D_{2}\Xi:\mathcal{E}^{s}\times
U^{s}_{\delta}\to\mathcal{L}(\mathcal{X}^{s};\mathcal{Y}^{s})$ to be the
derivatives of $\Xi$ with respect to $\mathcal{E}^{s}$ and $U^{s}_{\delta}$,
respectively. Note that by the second item of Theorem 4.1, we have
$D_{2}\mathfrak{S}_{b}(0,0)=0$ and $D_{2}\Lambda_{\Omega}(0,0)=0$. Therefore,
for any $\gamma\in\mathbb{R}$, $\Xi(\gamma,0,0,0,0,0,0,0,0)=(0,0,0,0)$ since
we also have $U^{2}=0$,
$U^{1}\cdot\nabla_{\mathcal{A}}U^{1}=(U_{1}^{1})\partial_{1}U^{1}=0$ for
$\eta=0$, and $D_{2}\Xi(\gamma,0,0,0,0,0,0,0,0)=L_{\kappa,\sigma}$ where
$L_{\kappa,\sigma}$ is defined in (3.65). By Theorem 3.19, for every
$\gamma_{*}\neq 0$ there exists some $\kappa_{0}>0$ for which
$D_{2}\Xi(\gamma,0,0,0,0,0,0,0,0)$ is a linear isomorphism for every
$\kappa\in(-\kappa_{0},\kappa_{0})$. Thus, by the implicit function theorem,
there exists open sets $\mathcal{U}(\gamma_{*})\subseteq\mathcal{E}^{s}$ and
$\mathcal{O}(\gamma_{*})\subseteq U^{s}_{\delta}$ such that
$(\gamma_{*},0,0,0,0,0)\in\mathcal{U}(\gamma_{*})$ and
$(0,0,0)\in\mathcal{O}(\gamma_{*})$, and a $C^{1}$ and Lipschitz map
$\varpi_{\gamma_{*}}:\mathcal{U}(\gamma_{*})\to\mathcal{O}(\gamma_{*})\subseteq
U^{s}_{\delta}$ such that
$\Xi(\gamma,\kappa,\mathcal{T},T,\mathfrak{f},f,\varpi_{\gamma_{*}}(\gamma,\kappa,\mathcal{T},T,\mathfrak{f},f))=(0,0,0,0)$
for all
$(\gamma,\kappa,\mathcal{T},T,\mathfrak{f},f)\in\mathcal{U}(\gamma_{*})$.
Moreover,
$(u,p,\eta)=\varpi_{\gamma_{*}}(\gamma,\kappa,\mathcal{T},T,\mathfrak{f},f)$
is the unique solution to
$\Xi(\gamma,\kappa,\mathcal{T},T,\mathfrak{f},f,u,p,\eta)=(0,0,0,0)$ in
$\mathcal{O}(\gamma_{*})$.

Next, we define the open sets

$\displaystyle\mathcal{U}^{s}=\bigcup_{\gamma_{*}\in\mathbb{R}\setminus\\{0\\}}\mathcal{U}(\gamma_{*})\subseteq\mathcal{E}^{s}\;\text{and}\;\mathcal{O}^{s}=\bigcup_{\gamma_{*}\in\mathbb{R}\setminus\\{0\\}}\mathcal{O}(\gamma_{*})\subseteq
U^{s}_{\delta}.$ (4.24)

We note that by construction,
$(\mathbb{R}\setminus\\{0\\})\times\\{0\\}\times\\{0\\}\times\\{0\\}\times\\{0\\}\times\\{0\\}\subset\mathcal{U}^{s}$.
Furthermore, for every
$(\gamma,\kappa,\mathcal{T},T,\mathfrak{f},f)\in\mathcal{U}^{s}$, there exists
a $\gamma_{*}\in\mathbb{R}\setminus\\{0\\}$ for which
$(\gamma,\kappa,\mathcal{T},T,\mathfrak{f},f)\in\mathcal{U}(\gamma_{*})$ and
$(u,p,\eta)=\varpi_{\gamma_{*}}(\gamma,\kappa,\mathcal{T},T,\mathfrak{f},f)\in\mathcal{O}(\gamma_{*})$.
By the observation above and the implicit function theorem, the map
$\overline{\varpi}:\mathcal{U}^{s}\to\mathcal{O}^{s}$ defined via
$\overline{\varpi}(\gamma,\kappa,\mathcal{T},T,\mathfrak{f},f)=\varpi_{\gamma_{*}}(\gamma,\kappa,\mathcal{T},T,\mathfrak{f},f)$,
where $\gamma_{*}\in\mathbb{R}\setminus\\{0\\}$ is such that
$(\gamma,\kappa,\mathcal{T},T,\mathfrak{f},f)\in\mathcal{U}(\gamma_{*})$, is
well-defined, $C^{1}$, and locally Lipschitz. This proves the remaining items
for $\sigma>0$ and $n\geqslant 3$.

To prove the remaining items in the case without surface tension and $n=2$, we
argue along the same lines but use the second item of Theorem 3.16 instead of
the first and use the isomorphism $L_{\kappa,0}$. ∎

###  4.3. Solvability of the unflattened system (1.16)

We now examine the solvability of the system (1.16) in the original Eulerian
coordinates.

###### Proof of Theorem 1.3.

Consider the map $\varpi:\mathcal{U}^{s}\to\mathcal{O}^{s}$ constructed in
Theorem 1.2. By Theorem 1.2, for every
$(\gamma,\kappa,\mathcal{T},T,\mathfrak{f},f)\in\mathcal{U}^{s}$ the tuple
$\mathcal{O}^{s}\ni(u,p,\eta)=\varpi(\gamma,\kappa,\mathcal{T},T,\mathfrak{f},f)$
solves (1.24) classically. Since $(u,p,\eta)\in\mathcal{O}^{s}$, we have
$\left\|\eta\right\|_{C_{b}^{0}}\leqslant b/2$, therefore by the second item
in Theorem 4.1, the flattening map $\mathfrak{F}_{\eta}$ and its inverse
$\mathfrak{F}_{\eta}^{-1}$ are both $C^{\lfloor s-n/2\rfloor+3}$
diffeomorphisms. Now we fix
$(\gamma,\kappa,\mathcal{T},T,\mathfrak{f},f)\in\mathcal{U}^{s}$ and set
$(u,p,\eta)=\varpi(\gamma,\kappa,\mathcal{T},T,\mathfrak{f},f),v=u\circ\mathfrak{F}_{\eta}^{-1}$,
and $q=p\circ\mathfrak{F}_{\eta}^{-1}$. Then by the second item of Theorem 4.1
and Sobolev embedding, we have
$v\in{}_{0}H^{s+2}(\Omega_{b+\eta};\mathbb{R}^{n})\cap C^{\lfloor
s-n/2\rfloor+2}(\Omega_{b+\eta};\mathbb{R}^{n})$ and $q\in
H^{s+1}(\Omega_{b+\eta};\mathbb{R})\cap C_{b}^{\lfloor
s-n/2\rfloor+1}(\Omega_{b+\eta};\mathbb{R})$. Since
$(\mathfrak{F}_{\eta}^{-1}(x))^{\prime}=x^{\prime}$, we have
$(\mathfrak{f}\circ\mathfrak{F}_{\eta}+L_{\Omega_{b}}f)\circ\mathfrak{F}_{\eta}^{-1}(x)=\mathfrak{f}(x)+L_{\Omega_{b+\eta}}f(x)$
and
$(\mathcal{T}\circ\mathfrak{F}_{\eta}\rvert_{\Sigma_{b}}+S_{b}T)\circ\mathfrak{F}^{-1}_{\eta}(x)=\mathcal{T}\rvert_{\Sigma_{b+\eta}}(x)+S_{b+\eta}T(x)$
for all $x\in\Omega_{b+\eta}$. Thus, if $(u,p,\eta)$ is a solution tuple to
(1.24) then $(v,q,\eta)$ is a solution tuple to (1.16). The last item follows
from the fact that $\varpi$ is locally Lipschitz. ∎

##  Appendix A Analysis tools

###  A.1. Permutations of Cartesian products

Here we collect a number of simple tools and bits of notation related to
Cartesian products, especially products of groups.

######  Definition A.1.

Suppose that $d\geqslant 1$ and that for $1\leqslant i\leqslant d$ we have a
commutative monoid $X_{i}$ with identity element $0_{i}\in X_{i}$. Further
suppose that $E\subseteq\\{1,\dotsc,d\\}$ and $x\in X$. We define $x_{E}\in X$
via

$(x_{E})_{j}=\begin{cases}x_{j}&\text{if }j\in E\\\ 0_{j}&\text{if }j\notin
E.\end{cases}$ (A.1)

The utility of this notation is seen in the formula $x=x_{E}+x_{E^{c}}$, valid
for any set $E\subseteq\\{1,\dotsc,d\\}$.

We will often use this notation when $X=\Gamma$ or $X=\hat{\Gamma}$, where
$\Gamma$ is of the form (1.43). In this context, we introduce further
notation.

######  Definition A.2.

Let $\Gamma$ be as in (1.43).

  1. (1)

We define the sets
$R_{\Gamma},R_{\hat{\Gamma}},T_{\Gamma},T_{\hat{\Gamma}}\subseteq\\{1,\dotsc,d\\}$
via

$\displaystyle R_{\Gamma}$
$\displaystyle=R_{\hat{\Gamma}}=\\{i\in\\{1,\dots,d\\}:\Gamma_{i}=\mathbb{R}\\}=:\\{\mathfrak{r}_{1},\dots,\mathfrak{r}_{|R_{\Gamma}|}\\}$
(A.2) $\displaystyle T_{\Gamma}$
$\displaystyle=T_{\hat{\Gamma}}=\\{i\in\\{1,\dots,d\\}:\Gamma_{i}=L_{i}\mathbb{T}\\}=\\{1,\dotsc,d\\}\backslash
R_{\Gamma}=\\{1,\dotsc,d\\}\backslash
R_{\hat{\Gamma}}=:\\{\mathfrak{t}_{1},\dots,\mathfrak{t}_{|T_{\Gamma}|}\\},$
(A.3)

where we order these such that
$\mathfrak{r}_{1}<\dots<\mathfrak{r}_{|R_{\Gamma}|}$ and
$\mathfrak{t}_{1}<\dots<\mathfrak{t}_{|T_{\Gamma}|}$. This allows us to write
$\Gamma\ni x=x_{R_{\Gamma}}+x_{T_{\Gamma}}$ and
$\hat{\Gamma}\ni\xi=\xi_{R_{\hat{\Gamma}}}+\xi_{T_{\hat{\Gamma}}}$.

  2. (2)

We define the sets $\Gamma_{R},\Gamma_{T},\hat{\Gamma}_{R},\hat{\Gamma}_{T}$
via

$\displaystyle\Gamma_{R}=\\{x\in\Gamma\mid
x_{T_{\Gamma}}=0\\},\;\Gamma_{T}=\\{x\in\Gamma\mid
x_{R_{\Gamma}}=0\\},\;\hat{\Gamma}_{R}=\\{\xi\in\hat{\Gamma}\mid\xi_{T_{\hat{\Gamma}}}=0\\},\;\hat{\Gamma}_{T}=\\{\xi\in\hat{\Gamma}\mid\xi_{R_{\hat{\Gamma}}}=0\\}.$
(A.4)

It follows that we have the direct sum decompositions

$\displaystyle\Gamma=\Gamma_{R}\oplus\Gamma_{T},\;\hat{\Gamma}=\hat{\Gamma}_{R}\oplus\hat{\Gamma}_{T}.$
(A.5)

At numerous points in the paper it is convenient to reorder the factors
appearing in a Cartesian product. We introduce notation for this and related
ideas now.

######  Definition A.3.

Suppose that $X_{1},\dotsc,X_{d}$ are sets and consider the Cartesian product
$X=\prod_{i=1}^{d}X_{i}$. Let $S_{d}$ denote the symmetric group of
permutations of the indices $\\{1,\dotsc,d\\}$.

  1. (1)

For $g\in S_{d}$ we write $gX$ for the reordered Cartesian product
$gX=\prod_{i=1}^{d}X_{g(i)}$.

  2. (2)

Let $g\in S_{d}$. We define the induced reordering map $\mathcal{P}_{g}:X\to
gX$ via

$\mathcal{P}_{g}(x)=\mathcal{P}_{g}(x_{1},\dotsc,x_{d})=(x_{g(1)},\dotsc,x_{g(d)}).$
(A.6)

This map is clearly a bijection. Sometimes it will be convenient to include
the domain in the notation, in which case we write $\mathcal{P}_{g,X}$ in
place of $\mathcal{P}_{g}$.

  3. (3)

Given sets $E$ and $Y$, write $\mathcal{F}(E;Y)=\\{f:E\to Y\\}$. Let $g\in
S_{d}$ and $Y$ be a set. We define the map
$\sigma_{g}:\mathcal{F}(X;Y)\to\mathcal{F}(gX;Y)$ via
$\sigma_{g}f=f\circ\mathcal{P}_{g}^{-1}$. Since $\mathcal{P}_{g}$ is
invertible, the map $\sigma_{g}$ is as well, and
$\sigma_{g}^{-1}:\mathcal{F}(gX;Y)\to\mathcal{F}(X;Y)$ is given by
$\sigma_{g}^{-1}f=f\circ\mathcal{P}_{g}$. As above, we sometimes indicate the
domain by writing $\sigma_{g,X}$ in place of $\sigma_{g}$.

It will be convenient to record the following obvious result as a lemma.

######  Lemma A.4.

Suppose that $X_{1},\dotsc,X_{d}$ are sets and consider the Cartesian product
$X=\prod_{i=1}^{d}X_{i}$. Let $g\in S_{d}$ and let $\mathcal{P}_{g}$ be the
induced reordering map defined in the second item in Definition A.3. Then the
following hold.

  1. (1)

If each $X_{i}$ is a group, then $\mathcal{P}_{g}$ is a group isomorphism.

  2. (2)

If each $X_{i}$ is a topological space and $X$ is endowed with the product
topology, then $\mathcal{P}_{g}$ is a homeomorphism.

  3. (3)

If each $X_{i}$ is a metric space and $X$ is endowed with a product metric,
then $\mathcal{P}_{g}$ is an isometry.

  4. (4)

If each $X_{i}$ is a $\sigma-$finite measure space and $X$ is endowed with the
product measure, then $\mathcal{P}_{g}$ is a measure preserving map.

  5. (5)

If each $X_{i}$ is a smooth manifold and $X$ is endowed with the smooth
product manifold structure, then $\mathcal{P}_{g}$ is a smooth diffeomorphism.

Next we introduce the notion of the canonical reordering of $\Gamma$ and
$\hat{\Gamma}$, where $\Gamma$ is as in (1.43).

######  Definition A.5.

Let $\Gamma$ be as in (1.43), and let
$R_{\Gamma},R_{\hat{\Gamma}},T_{\Gamma},T_{\hat{\Gamma}}\subseteq\\{1,\dotsc,d\\}$
be as in Definition A.2.

  1. (1)

We define the canonical permutation associated to $\Gamma$ to be $g_{\ast}\in
S_{d}$ defined by

$\displaystyle g_{\ast}(i)=\begin{cases}\mathfrak{r}_{i},&\text{if }1\leqslant
i\leqslant|R_{\Gamma}|\\\
\mathfrak{t}_{i-|R_{\Gamma}|},&\text{otherwise}.\end{cases}$ (A.7)

We use this to define the canonical reorderings of $\Gamma$ and $\hat{\Gamma}$
via

$\displaystyle\Gamma_{\ast}=g_{*}\Gamma=\begin{cases}\mathbb{R}^{|R_{\Gamma}|}\times\prod_{i\in
T_{\Gamma}}L_{g(i)}\mathbb{T}&|R_{\Gamma}|\geqslant 1,\\\ \prod_{i\in
T_{\Gamma}}L_{g(i)}\mathbb{T},&R_{\Gamma}=\varnothing\end{cases}\;\text{and}\;\hat{\Gamma}_{\ast}=g_{*}\hat{\Gamma}=\begin{cases}\mathbb{R}^{|R_{\Gamma}|}\times\prod_{i\in
T_{\Gamma}}L_{g(i)}^{-1}\mathbb{Z}&|R_{\Gamma}|\geqslant 1,\\\ \prod_{i\in
T_{\Gamma}}L_{g(i)}^{-1}\mathbb{Z},&R_{\Gamma}=\varnothing.\end{cases}$ (A.8)

Following the convention in Definition A.2, we may write

$\displaystyle\Gamma_{*}=(\Gamma_{*})_{R}\oplus(\Gamma_{*})_{T}=:\Gamma_{*,R}\oplus\Gamma_{*,T},\;\hat{\Gamma}_{*}=(\hat{\Gamma}_{*})_{R}\oplus(\hat{\Gamma}_{*})_{T}:=\hat{\Gamma}_{*,R}\oplus\hat{\Gamma}_{*,T}.$
(A.9)

The essential point here is that all toroidal or integer factors are moved to
the end of the canonical reordered product with the real factors preceding. In
turn, we define the canonical reordering maps

$\mathcal{P}_{\Gamma}:=\mathcal{P}_{g_{\ast},\Gamma}:\Gamma\to\Gamma_{\ast}\text{
and
}\mathcal{P}_{\hat{\Gamma}}:=\mathcal{P}_{g_{\ast},\hat{\Gamma}}:\hat{\Gamma}\to\hat{\Gamma}_{\ast}$
(A.10)

as well as the induced maps

$\sigma_{\Gamma}:=\sigma_{g_{\ast},\Gamma}:\mathcal{F}(\Gamma;Y)\to\mathcal{F}(\Gamma_{\ast};Y)\text{
and
}\sigma_{\hat{\Gamma}}:=\sigma_{g_{\ast},\hat{\Gamma}}:\mathcal{F}(\hat{\Gamma};Y)\to\mathcal{F}(\hat{\Gamma}_{\ast};Y)$
(A.11)

for any set $Y$, where $\mathcal{F}(X;Y)$ is defined as in the third item of
Definition A.3.

  2. (2)

In the case when $1\in R_{\Gamma}$, we define the surjective map
$\uppi_{\hat{\Gamma}_{*}}:\hat{\Gamma}_{*}\to\mathbb{R}^{|R_{\hat{\Gamma}}|}$
via

$\displaystyle\uppi_{\hat{\Gamma}_{*}}(\xi)$
$\displaystyle=\uppi_{\hat{\Gamma}_{*}}(\xi_{\mathfrak{r}_{1}},\dots,\xi_{\mathfrak{r}_{|R_{\Gamma}|}},\xi_{\mathfrak{t}_{1}},\dots,\xi_{\mathfrak{t}_{|T_{\Gamma}|}})=(\xi_{\mathfrak{r}_{1}},\dots,\xi_{\mathfrak{r}_{|R_{\Gamma}|}})$
(A.12)

where $\mathfrak{r}_{i},\mathfrak{t}_{i}$ are defined via (A.2) and (A.3). We
also define the injective map
$l_{\hat{\Gamma}_{*}}:\mathbb{R}^{|R_{\hat{\Gamma}}|}\to\hat{\Gamma}_{*}$ via
$l_{\hat{\Gamma}_{*}}(\omega)=(\omega,0)$.

In Section 2, we generalize many of results in [18] concerning the low-
frequency behavior of functions belonging to the anisotropic Sobolev space
$X^{s}(\Gamma;\mathbb{F})$. To introduce the analogue of the low-frequency
ball over $\hat{\Gamma}$, we introduce the low frequency cutoff $0<r\leqslant
1$ via

$\displaystyle r=\begin{cases}1,&\Gamma=\mathbb{R}^{d}\\\
\min\\{1,\min_{i\in\\{1,\dots,|T_{\Gamma}|\\}}L_{i}^{-1}\\},&\text{otherwise},\end{cases}$
(A.13)

and we note that $\xi\in B_{\hat{\Gamma}}(0,r)\implies\xi_{T_{\Gamma}}=0$.

To simplify some key computations when $1\in R_{\Gamma}$, we reduce the
computation over the low-frequency ball $B_{\hat{\Gamma}}(0,r)$ in
$\hat{\Gamma}$ to the low frequency ball
$B_{\mathbb{R}^{|R_{\hat{\Gamma}}|}}(0,r)$ in
$\mathbb{R}^{|R_{\hat{\Gamma}}|}$. We introduce the map to make this reduction
in the following lemma.

######  Lemma A.6.

Let $\Gamma$ be defined as in (1.43) and assume $1\in R_{\Gamma}$. We define
the surjective map
$\uppi_{\hat{\Gamma}}:\hat{\Gamma}\to\mathbb{R}^{|R_{\hat{\Gamma}}|}$ via

$\displaystyle\uppi_{\hat{\Gamma}}=\uppi_{\hat{\Gamma}_{\ast}}\circ\mathcal{P}_{\hat{\Gamma}}.$
(A.14)

The following hold.

  1. (1)

The restriction of the map $\uppi_{\hat{\Gamma}_{*}}$ on
$\hat{\Gamma}_{\ast,R}$ is an isometric measure-preserving group isomorphism
between $\hat{\Gamma}_{\ast,R}$ and $\mathbb{R}^{|R_{\hat{\Gamma}}|}$, where
$\hat{\Gamma}_{\ast,R}$ is the first summand in the direct sum decomposition
of $\hat{\Gamma}_{*}$ in (A.9).

  2. (2)

As a consequence of the first item and Lemma A.4, the restriction of the map
$\uppi_{\hat{\Gamma}}$ on $\hat{\Gamma}_{R}$ is an isometric measure-
preserving group isomorphism between $\hat{\Gamma}_{R}$ and
$\mathbb{R}^{|R_{\hat{\Gamma}}|}$, with the map
$\mathcal{P}_{\hat{\Gamma}}^{-1}\circ l_{\hat{\Gamma}_{\ast}}$ as its inverse.

  3. (3)

The restriction of $\uppi_{\hat{\Gamma}}$ on $B(0,r)\subset\hat{\Gamma}_{R}$
is also an isometric measure-preserving group isomorphism between
$B_{\hat{\Gamma}}(0,r)$ to $B_{\mathbb{R}^{|R_{\hat{\Gamma}}|}}(0,r)$, with
the restriction of the map $\mathcal{P}_{\hat{\Gamma}}^{-1}\circ
l_{\hat{\Gamma}_{\ast}}$ on $B_{\mathbb{R}^{|R_{\hat{\Gamma}}|}}(0,r)$ as its
inverse.

In other words, we have $\xi\in
B_{\hat{\Gamma}}(0,r)\implies\uppi_{\hat{\Gamma}}\xi\in
B_{\mathbb{R}^{|R_{\hat{\Gamma}}|}}(0,r)$ and $\omega\in
B_{\mathbb{R}^{|R_{\hat{\Gamma}}|}}(0,r)\implies\mathcal{P}_{\hat{\Gamma}}^{-1}l_{\hat{\Gamma}_{\ast}}\omega\in
B_{\hat{\Gamma}}(0,r)$, and the maps
$\uppi_{\hat{\Gamma}},\mathcal{P}_{\hat{\Gamma}}^{-1}l_{\hat{\Gamma}_{\ast}}$
are isometric measure-preserving group isomorphisms.

###  A.2. Tempered distributions and the Fourier transform

In this subsection we carefully define the class of tempered distributions on
$\Gamma$ and its Pontryagin dual $\hat{\Gamma}$. First, we note that for any
$\alpha\in\mathbb{N}^{d}$, we may follow the convention in (A.1) and write
$\alpha=\alpha_{R_{\Gamma}}+\alpha_{T_{\Gamma}}$. We then define

$\displaystyle\mathbb{N}^{d}_{R_{\Gamma}}=\\{\alpha\in\mathbb{N}^{d}\mid\alpha_{T_{\Gamma}}=0\\},\;\mathbb{N}^{d}_{T_{\Gamma}}=\\{\alpha\in\mathbb{N}^{d}\mid\alpha_{R_{\Gamma}}=0\\}.$
(A.15)

######  Definition A.7.

Let $\Gamma$ be defined as in (1.43).

  1. (1)

We define the Schwartz class on $\Gamma$ to be

$\displaystyle\mathscr{S}(\Gamma;\mathbb{C})=\\{f\in
C^{\infty}(\Gamma;\mathbb{C})\mid[f]_{\Gamma,\alpha,\beta}=\sup_{x\in\Gamma}|x^{\alpha}\partial^{\beta}f(x)|<\infty\;\text{for
all }\alpha\in\mathbb{N}^{d}_{R_{\Gamma}},\beta\in\mathbb{N}^{d}\\},$ (A.16)

and on $\hat{\Gamma}$ to be

$\mathscr{S}(\hat{\Gamma};\mathbb{C})=\\{f:\hat{\Gamma}\to\mathbb{C}\mid
f(\mathcal{P}_{\hat{\Gamma}}^{-1}(\cdot,0)+\xi_{T})\in
C^{\infty}(\mathbb{R}^{|R_{\Gamma}|};\mathbb{C})\;\text{for
all}\;\xi_{T}\in\hat{\Gamma}_{T_{\hat{\Gamma}}}\;\text{and}\;\\\
[f]_{\hat{\Gamma},\alpha,\beta}=\sup_{\xi\in\hat{\Gamma}}|\xi^{\alpha}\partial^{\beta}f(\xi)|<\infty\;\text{for
all}\;\alpha\in\mathbb{N}^{d},\beta\in\mathbb{N}_{R_{\hat{\Gamma}}}^{d}\\}.$
(A.17)

We equip $\mathscr{S}(\Gamma;\mathbb{C})$ and
$\mathscr{S}(\hat{\Gamma};\mathbb{C})$ with the Fréchet topology induced by
the countable family of seminorms
$\\{[\cdot]_{\Gamma,\alpha,\beta}\\}_{\alpha\in\mathbb{N}^{d}_{R_{\Gamma}},\beta\in\mathbb{N}^{d}}$
and
$\\{[\cdot]_{\hat{\Gamma},\alpha,\beta}\\}_{\alpha\in\mathbb{N}^{d},\beta\in\mathbb{N}_{R_{\hat{\Gamma}}}^{d}}$,
by endowing $\mathscr{S}(\Gamma;\mathbb{C})$ and
$\mathscr{S}(\hat{\Gamma};\mathbb{C})$ with the metrics

$\displaystyle
d_{\mathscr{S}(\Gamma)}(f,g)=\sum_{\alpha\in\mathbb{N}^{d}_{R_{\Gamma}}}\sum_{\beta\in\mathbb{N}^{d}}\frac{1}{2^{|\alpha|+|\beta|}}\frac{[f-g]_{\Gamma,\alpha,\beta}}{1+[f-g]_{\Gamma,\alpha,\beta}},\;d_{\mathscr{S}(\hat{\Gamma})}(f,g)=\sum_{\alpha\in\mathbb{N}^{d}}\sum_{\beta\in\mathbb{N}_{R_{\hat{\Gamma}}}^{d}}\frac{1}{2^{|\alpha|+|\beta|}}\frac{[f-g]_{\hat{\Gamma},\alpha,\beta}}{1+[f-g]_{\hat{\Gamma},\alpha,\beta}}.$
(A.18)

  2. (2)

We define the unitary Fourier and inverse Fourier transforms
$\mathscr{F}^{\pm}_{\Gamma}:\mathscr{S}(\Gamma;\mathbb{C})\to\mathscr{S}(\hat{\Gamma};\mathbb{C})$,
$\mathscr{F}^{\pm}_{\hat{\Gamma}}:\mathscr{S}(\hat{\Gamma};\mathbb{C})\to\mathscr{S}(\Gamma;\mathbb{C})$
on $\mathscr{S}(\Gamma;\mathbb{C})$ and $\mathscr{S}(\hat{\Gamma};\mathbb{C})$
via

$\mathscr{F}^{\pm}_{\Gamma}\\{f\\}(\xi)=\frac{1}{\sqrt{\prod_{i\in
T_{\Gamma}}L_{i}}}\int_{\Gamma}f(x)e^{\mp 2\pi i\xi\cdot
x}\;dx=\mathscr{F}_{\Gamma}^{\mp}\\{f\\}(-\xi),\\\
\mathscr{F}^{\pm}_{\hat{\Gamma}}\\{f\\}(x)=\frac{1}{\sqrt{\prod_{i\in
T_{\Gamma}}L_{i}}}\int_{\hat{\Gamma}}f(\xi)e^{\mp 2\pi i\xi\cdot
x}\;d\xi=\mathscr{F}_{\hat{\Gamma}}^{\mp}\\{f\\}(-x).$ (A.19)

  3. (3)

We define the class of tempered distributions
$\mathscr{S}^{\prime}(\Gamma;\mathbb{C})$ and
$\mathscr{S}^{\prime}(\hat{\Gamma};\mathbb{C})$ to be the set of continuous
linear functionals on $\mathscr{S}(\Gamma;\mathbb{C})$ and
$\mathscr{S}(\hat{\Gamma};\mathbb{C})$, respectively.

The next lemma shows that the restriction of $\sigma_{g,\Gamma}$ on
$\mathscr{S}(\Gamma;\mathbb{F})$ is an isometric isomorphism between
$\mathscr{S}(\Gamma;\mathbb{F})$ and $\mathscr{S}(g\Gamma;\mathbb{F})$, where
$\sigma_{g,\Gamma}$ is defined in the third item of Definition A.3.

######  Lemma A.8.

Let $\Gamma$ be defined as in (1.43), $g\in S_{d}$ be a permutation, and let
$g\Gamma$ be as defined in the first item of Definition A.3. Then the map
$\sigma_{g,\Gamma}:\mathscr{S}(\Gamma;\mathbb{F})\to\mathscr{S}(g\Gamma;\mathbb{F})$
defined in the third item of Definition A.3 is an isometric isomorphism, with
its inverse given by the map
$\sigma_{g,\Gamma}^{-1}:\mathscr{S}(g\Gamma;\mathbb{F})\to\mathscr{S}(\Gamma;\mathbb{F})$,
also defined in the third item of Definition A.3.

###### Proof.

For $g\in S_{d}$, let $\mathcal{P}_{g,\Gamma}:\Gamma\to
g\Gamma,\mathcal{P}_{g,\mathbb{N}^{d}}:\mathbb{N}^{d}\to g\mathbb{N}^{d}$ be
the induced reordering maps defined via (A.6). We note that for any
$F\in\mathscr{S}(\Gamma;\mathbb{F}),\alpha\in\mathbb{N}_{R_{g\Gamma}}^{d}$,
and $\beta\in\mathbb{N}^{d}$ we have $F(\mathcal{P}_{g}^{-1}\cdot)\in
C^{\infty}(g\Gamma;\mathbb{F})$ with

$\displaystyle[F(\mathcal{P}_{g}^{-1}\cdot)]_{g\Gamma,\alpha,\beta}=\sup_{x\in
g\Gamma}|x^{\alpha}\partial_{x}^{\beta}F(\mathcal{P}_{g,\Gamma}^{-1}x)|=\sup_{y\in\Gamma}|y^{\mathcal{P}_{g,\mathbb{N}^{d}}^{-1}\alpha}\partial_{y}^{\mathcal{P}_{g,\mathbb{N}^{d}}^{-1}\beta}F(y)|=[F]_{\Gamma,\mathcal{P}_{g,\mathbb{N}^{d}}^{-1}\alpha,\mathcal{P}_{g,\mathbb{N}^{d}}^{-1}\beta}<\infty,$
(A.20)

and for any
$G\in\mathscr{S}(g\Gamma;\mathbb{F}),\alpha\in\mathbb{N}^{d}_{R_{\Gamma}}$,
and $\beta\in\mathbb{N}^{d}$ we have $G(\mathcal{P}_{g,\Gamma}^{-1}\cdot)\in
C^{\infty}(\Gamma;\mathbb{F})$ with

$\displaystyle[G(\mathcal{P}_{g,\Gamma}^{-1}\cdot)]_{\Gamma,\alpha,\beta}=\sup_{x\in\Gamma}|x^{\alpha}\partial_{x}^{\beta}G(\mathcal{P}_{g,\Gamma}^{-1}x)|=\sup_{y\in
g\Gamma}|y^{\mathcal{P}_{g,\mathbb{N}^{d}}\alpha}\partial_{y}^{\mathcal{P}_{g,\mathbb{N}^{d}}\beta}G(y)|=[G]_{g\Gamma,\mathcal{P}_{g,\mathbb{N}^{d}}\alpha,\mathcal{P}_{g,\mathbb{N}^{d}}\beta}<\infty.$
(A.21)

This shows that $\sigma_{g,\Gamma}$ and $\sigma_{g,\Gamma}^{-1}$ are well-
defined. Since $\sigma_{g,\Gamma}$ is a bijection, and
$\mathcal{P}_{g,\mathbb{N}^{d}}$ is a bijection between
$\mathbb{N}^{d}_{R_{\Gamma}}$ and $\mathbb{N}^{d}_{R_{g\Gamma}}$, and between
$\mathbb{N}^{d}$ and $g\mathbb{N}^{d}=\mathbb{N}^{d}$, with (A.20) and (A.21)
we can also conclude that

$\displaystyle
d_{\mathscr{S}(\Gamma)}(F,G)=d_{\mathscr{S}(g\Gamma)}(F(\mathcal{P}_{g}^{-1}\cdot),G(\mathcal{P}_{g}^{-1}\cdot)).$
(A.22)

This shows that $\sigma_{g,\Gamma}$ is an isometric isomorphism. ∎

Following the exact same argument as above we can conclude that the
restriction of $\sigma_{g,\hat{\Gamma}}$ on
$\mathscr{S}(\hat{\Gamma};\mathbb{F})$ is an isometric isomorphism between
$\mathscr{S}(\hat{\Gamma};\mathbb{F})$ and
$\mathscr{S}(\hat{\Gamma}_{\ast};\mathbb{F})$.

Next, we record some important properties of the Schwartz classes
$\mathscr{S}(\Gamma;\mathbb{F})$ and $\mathscr{S}(\hat{\Gamma};\mathbb{F})$.

######  Proposition A.9.

Let $\Gamma$ be defined as in (1.43). Then the following hold.

  1. (1)

We have the continuous inclusions
$\mathscr{S}(\Gamma;\mathbb{C})\hookrightarrow L^{p}(\Gamma;\mathbb{C})$ and
$\mathscr{S}(\hat{\Gamma};\mathbb{C})\hookrightarrow
L^{p}(\hat{\Gamma};\mathbb{C})$ for any $1\leqslant p\leqslant\infty$.

  2. (2)

We have the continuous inclusion
$L^{p}(\Gamma;\mathbb{C})\hookrightarrow\mathscr{S}^{\prime}(\Gamma;\mathbb{C})$
and
$L^{p}(\hat{\Gamma};\mathbb{C})\hookrightarrow\mathscr{S}^{\prime}(\hat{\Gamma};\mathbb{C})$
for all $1\leqslant p\leqslant\infty$.

  3. (3)

For any $f\in\mathscr{S}(\Gamma;\mathbb{C})$,
$\alpha\in\mathbb{N}^{d}_{R_{\Gamma}}$, and $\beta\in\mathbb{N}^{d}$ the
functions $g_{\alpha,\beta},h_{\alpha,\beta}:\Gamma\to\mathbb{C}$ defined via
$g_{\alpha,\beta}(x)=x^{\alpha}\partial^{\beta}f(x)$ and
$h_{\alpha,\beta}=\partial^{\alpha}(x^{\beta}f(x))$ belong to
$\mathscr{S}(\Gamma;\mathbb{C})$. Furthermore, the maps $f\mapsto
g_{\alpha,\beta}$ and $f\mapsto h_{\alpha,\beta}$ are continuous. Similarly,
for any $f\in\mathscr{S}(\hat{\Gamma};\mathbb{C})$, $\alpha\in\mathbb{N}^{d}$,
and $\beta\in\mathbb{N}_{R_{\hat{\Gamma}}}^{d}$ the functions
$g_{\alpha,\beta},h_{\alpha,\beta}:\Gamma\to\mathbb{C}$ defined via
$g_{\alpha,\gamma}(\xi)=\xi^{\alpha}\partial^{\beta}f(\xi)$ and
$h_{\alpha,\beta}=\partial^{\alpha}(\xi^{\beta}f(\xi))$ belong to
$\mathscr{S}(\hat{\Gamma};\mathbb{C})$. Furthermore, the maps $f\mapsto
g_{\alpha,\beta}$ and $f\mapsto h_{\alpha,\beta}$ are continuous.

  4. (4)

If $f\in\mathscr{S}(\Gamma;\mathbb{C})$, then
$\mathscr{F}^{\pm}_{\Gamma}\\{f\\}\in\mathscr{S}(\hat{\Gamma};\mathbb{C})$ and
the map $f\mapsto\mathscr{F}_{\Gamma}^{\pm}\\{f\\}$ is continuous. Likewise,
if $f\in\mathscr{S}(\hat{\Gamma};\mathbb{C})$, then
$\mathscr{F}^{\pm}_{\hat{\Gamma}}\\{f\\}\in\mathscr{S}(\Gamma;\mathbb{C})$ and
the map $f\mapsto\mathscr{F}_{\hat{\Gamma}}^{\pm}\\{f\\}$ is continuous.

  5. (5)

If $f\in\mathscr{S}(\Gamma;\mathbb{C})$ and
$g\in\mathscr{S}(\hat{\Gamma};\mathbb{C})$, then

$\displaystyle\int_{\hat{\Gamma}}\mathscr{F}^{\pm}_{\Gamma}\\{f\\}(\xi,k)g(\xi,k)\;d\xi
dk=\int_{\Gamma}f(x,y)\mathscr{F}^{\pm}_{\hat{\Gamma}}\\{g\\}(x,y)\;dxdy.$
(A.23)

  6. (6)

If $f\in\mathscr{S}(\Gamma;\mathbb{C})$, then
$f=\mathscr{F}^{\mp}_{\hat{\Gamma}}\circ\mathscr{F}^{\pm}_{\Gamma}\\{f\\}$.
Similarly, if $f\in\mathscr{S}(\hat{\Gamma};\mathbb{C})$, then
$f=\mathscr{F}^{\mp}_{\Gamma}\circ\mathscr{F}^{\pm}_{\hat{\Gamma}}\\{f\\}$.

  7. (7)

The Fourier and inverse Fourier transforms
$\mathscr{F}^{\pm}_{\Gamma}:\mathscr{S}(\Gamma;\mathbb{C})\to\mathscr{S}(\hat{\Gamma};\mathbb{C})$
on $\mathscr{S}(\Gamma;\mathbb{C})$ are isomorphisms from
$\mathscr{S}(\Gamma;\mathbb{C})$ to $\mathscr{S}(\hat{\Gamma};\mathbb{C})$.
Likewise, the Fourier and inverse Fourier transforms
$\mathscr{F}^{\pm}_{\hat{\Gamma}}:\mathscr{S}(\hat{\Gamma};\mathbb{C})\to\mathscr{S}(\Gamma;\mathbb{C})$
on $\mathscr{S}(\hat{\Gamma};\mathbb{C})$ are isomorphisms from
$\mathscr{S}(\hat{\Gamma};\mathbb{C})$ to $\mathscr{S}(\Gamma;\mathbb{C})$

###### Proof.

Define

$\displaystyle\tau=\begin{cases}1,&\Gamma=\mathbb{R}^{d}\\\
\max\\{1,2\max_{i\in\\{1,\dots,d_{2}\\}}L_{i}\\},&\text{otherwise}.\end{cases}$
(A.24)

To prove the first item, we note that for any $1\leqslant p<\infty$,
$f\in\mathscr{S}(\Gamma;\mathbb{C})$, and
$\alpha,\beta\in\mathbb{N}^{d}_{R_{\Gamma}}$ with
$|\alpha|=d+1,|\beta|=\lfloor(d+1)/p\rfloor+1$ we have

$\left\|f\right\|_{L^{p}(\Gamma)}\lesssim\left(\int_{B(0,\tau)}|f(x)|^{p}\;dx+\int_{B(0,\tau)^{c}}d_{\Gamma}(x,0)^{\alpha}|f(x)|^{p}d_{\Gamma}(x,0)^{-\alpha}\;dx\right)^{1/p}\\\
\lesssim\left(\left\|f\right\|_{L^{\infty}}^{p}+\left(\sup_{x\in
B(0,\tau)^{c}}d_{\Gamma}(x,0)^{\alpha/p}|f(x)|\right)^{p}\int_{B(0,\tau)^{c}}d_{\Gamma}(x,0)^{-\alpha}\;dx\right)^{1/p}\\\
\lesssim\left\|f\right\|_{L^{\infty}}+\sup_{x\in
B(0,\tau)^{c}}d_{\Gamma}(x,0)^{\beta}|f(x)|\lesssim[f]_{\Gamma,0,0}+\sum_{\beta\in\mathbb{N}^{d}_{R_{\Gamma}},|\beta|=\lfloor(d+1)/p\rfloor+1}[f]_{\Gamma,\beta,0},$
(A.25)

and when $p=\infty$ we have
$\left\|f\right\|_{L^{\infty}(\Gamma)}=[f]_{\Gamma,0,0}$. It follows then that
$\mathscr{S}(\Gamma;\mathbb{C})\hookrightarrow L^{p}(\Gamma;\mathbb{C})$ for
all $1\leqslant p\leqslant\infty$. A similar set of computations can be
performed to prove $\mathscr{S}(\hat{\Gamma};\mathbb{C})\hookrightarrow
L^{p}(\hat{\Gamma};\mathbb{C})$ for $1\leqslant p\leqslant\infty$. For the
second item, suppose that we have a sequence
$\\{f_{k}\\}_{k=1}^{\infty}\subset L^{p}(\Gamma;\mathbb{C})$ for $1\leqslant
p\leqslant\infty$ such that $f_{k}\to f$ in $L^{p}(\Gamma;\mathbb{C})$. We
note that the canonical distribution
$T_{f_{k}}:\mathscr{S}(\Gamma;\mathbb{C})\to\mathbb{C}$ associated to each
$f_{k}$ defined via

$\displaystyle\langle
T_{f_{k}},\varphi\rangle_{\mathscr{S^{\prime}},\mathscr{S}}=\int_{\Gamma}f_{k}\varphi\;dx\;\text{for
all}\;\varphi\in\mathscr{S}(\Gamma;\mathbb{C})$ (A.26)

is clearly linear. By the first item and Hölder’s inequality, if
$\\{\varphi_{m}\\}_{m=1}^{\infty}\subset\mathscr{S}(\Gamma;\mathbb{C})$ and
$\varphi_{m}\to\varphi$ as $m\to\infty$ in $\mathscr{S}(\Gamma;\mathbb{C})$,
then for each $k$ we have

$\displaystyle|\langle
T_{f_{k}},\varphi_{m}-\varphi\rangle_{\mathscr{S^{\prime}},\mathscr{S}}|\leqslant\left\|f_{k}\right\|_{L^{p}}\left\|\varphi_{m}\to\varphi\right\|_{L^{q}}\to
0.$ (A.27)

This implies that
$\mathcal{T}_{f_{k}}\in\mathscr{S}^{\prime}(\Gamma;\mathbb{C})$. Furthermore,
since $f_{k}\to f$ in $L^{p}(\Gamma;\mathbb{C})$, by the first item and
Hölder’s inequality again, we have

$\displaystyle|\langle
T_{f_{k}}-T_{f},\varphi\rangle_{\mathscr{S^{\prime}},\mathscr{S}}|\leqslant\left\|f_{k}-f\right\|_{L^{p}}\left\|\varphi\right\|_{L^{q}}\to
0\;\text{for all}\;\varphi\in\mathscr{S}(\Gamma;\mathbb{C}).$ (A.28)

Since the map $L^{p}(\Gamma;\mathbb{C})\ni f\mapsto
T_{f}\in\mathscr{S}^{\prime}(\Gamma;\mathbb{C})$ is also clearly injective,
the inclusion map
$i:L^{p}(\Gamma;\mathbb{C})\to\mathscr{S}^{\prime}(\Gamma;\mathbb{C})$ as is
well-defined and continuous. This proves the second item for
$\mathscr{S}(\Gamma;\mathbb{C})$, and the same argument can be applied to
prove the second item for $\mathscr{S}(\hat{\Gamma};\mathbb{C})$. The third
item follows directly from the Leibniz rule.

For the remaining items, we note that by using the isometric isomorphisms
$\sigma_{\Gamma}:\mathscr{S}(\Gamma;\mathbb{F})\to\mathscr{S}(\Gamma_{\ast};\mathbb{F})$
and
$\sigma_{\hat{\Gamma}}:\mathscr{S}(\hat{\Gamma};\mathbb{F})\to\mathscr{S}(\hat{\Gamma}_{\ast};\mathbb{F})$
induced by the canonical permutation $g_{*}$, it suffices to prove the
remaining items for $\mathscr{S}(\Gamma_{*};\mathbb{F})$ and
$\mathscr{S}(\hat{\Gamma}_{\ast};\mathbb{F})$. Indeed, if the remaining items
are true for $\mathscr{S}(\Gamma_{*};\mathbb{F})$ and
$\mathscr{S}(\hat{\Gamma}_{\ast};\mathbb{F})$, then it suffices to check that

$\displaystyle\mathscr{F}_{\Gamma}^{\pm}=\sigma_{\hat{\Gamma}}^{-1}\circ\mathscr{F}_{\Gamma_{\ast}}^{\pm}\circ\sigma_{\Gamma},\;\mathscr{F}_{\hat{\Gamma}}^{\pm}=\sigma_{\Gamma}^{-1}\circ\mathscr{F}_{\hat{\Gamma}_{\ast}}^{\pm}\circ\sigma_{\hat{\Gamma}},$
(A.29)

and the remaining items also follow for $\mathscr{S}(\Gamma;\mathbb{F})$ and
$\mathscr{S}(\hat{\Gamma};\mathbb{F})$ as the following diagram commutes:

${\mathscr{S}(\Gamma;\mathbb{C})}$${\mathscr{S}(\hat{\Gamma};\mathbb{C})}$${\mathscr{S}(\Gamma;\mathbb{C})}$${\mathscr{S}(\Gamma_{\ast};\mathbb{C})}$${\mathscr{S}(\hat{\Gamma}_{\ast};\mathbb{C})}$${\mathscr{S}(\Gamma_{\ast};\mathbb{C})}$$\scriptstyle{\mathscr{F}^{\pm}_{\Gamma}}$$\scriptstyle{\sigma_{\Gamma}}$$\scriptstyle{\sigma_{\hat{\Gamma}}}$$\scriptstyle{\mathscr{F}^{\mp}_{\hat{\Gamma}}}$$\scriptstyle{\sigma_{\Gamma}}$$\scriptstyle{\mathscr{F}^{\pm}_{\Gamma_{\ast}}}$$\scriptstyle{\mathscr{F}^{\mp}_{\hat{\Gamma}_{\ast}}}$
(A.30)

To verify (A.29), we note that for any $f\in\mathscr{S}(\Gamma;\mathbb{C})$,
we have

$(\sigma_{\hat{\Gamma}}^{-1}\mathscr{F}_{\Gamma_{\ast}}^{\pm}\sigma_{\Gamma}f)(\xi)=\sigma_{\hat{\Gamma}}^{-1}\mathscr{F}_{\Gamma_{\ast}}^{\pm}[f(\mathcal{P}_{\Gamma}^{-1}\cdot)](\xi)=\sigma_{\hat{\Gamma}}^{-1}\frac{1}{\sqrt{\prod_{i\in
T_{\Gamma}}L_{i}}}\int_{\Gamma_{\ast}}f(\mathcal{P}_{\Gamma}^{-1}x)e^{\mp 2\pi
i\xi\cdot x}\;dx\\\ =\sigma_{\hat{\Gamma}}^{-1}\frac{1}{\sqrt{\prod_{i\in
T_{\Gamma}}L_{i}}}\int_{\Gamma}f(x)e^{\mp 2\pi
i\mathcal{P}_{\hat{\Gamma}}^{-1}\xi\cdot
x}\;dx=\sigma_{\hat{\Gamma}}^{-1}\mathscr{F}_{\Gamma}^{\pm}[f(\cdot)](\mathcal{P}_{\hat{\Gamma}}^{-1}\xi)=\mathscr{F}_{\Gamma}^{\pm}[f(\cdot)](\xi).$
(A.31)

A similar set of calculations can be performed to verify that
$\sigma_{\Gamma}^{-1}\circ\mathscr{F}_{\hat{\Gamma}_{\ast}}^{\pm}\circ\sigma_{\hat{\Gamma}}$.
Consequently, for the rest of the proof we may assume without loss of
generality that
$\Gamma=\mathbb{R}^{d_{1}}\times\prod_{i=1}^{d_{2}}L_{i}\mathbb{T}$, with
$d=\dim\Gamma=d_{1}+d_{2}$.

To prove the fourth item, we first note that if
$f\in\mathscr{S}(\Gamma;\mathbb{C})$, by the first and third items we have
$\partial^{\alpha}((-2\pi
ix)^{\gamma}f)\in\mathscr{S}(\Gamma;\mathbb{C})\subset
L^{1}(\Gamma;\mathbb{C})$ for any $\alpha,\gamma\in\mathbb{N}^{d_{1}}$. This
shows that $\mathscr{F}^{\pm}_{\Gamma}{f}(\cdot,k)\in
C^{\infty}(\mathbb{R}^{d_{1}};\mathbb{C})$ for any
$k\in\prod_{i=1}^{d_{2}}L_{i}^{-1}\mathbb{Z}$. Then for any
$\alpha,\gamma\in\mathbb{N}^{d_{1}},\beta\in\mathbb{N}^{d_{2}}$, we have

$(-2\pi i\xi)^{\alpha}(-2\pi
ik)^{\beta}\partial_{\xi}^{\gamma}\mathscr{F}^{\pm}\\{f\\}(\xi,k)=(-2\pi
i\xi)^{\alpha}(-2\pi
ik)^{\beta}\partial_{\xi}^{\gamma}\int_{\Gamma}f(x,y)e^{\mp 2\pi i\xi\cdot
x}e^{\mp 2\pi ik\cdot y}\;dxdy\\\ =\int_{\Gamma}f(x,y)(\mp 2\pi
ix)^{\gamma}\partial_{x}^{\alpha}e^{\mp 2\pi i\xi\cdot
x}\partial_{y}^{\beta}e^{\mp 2\pi ik\cdot
y}\;dxdy=(-1)^{|\alpha|+|\beta|}\int_{\Gamma}\partial_{x}^{\alpha}[(\mp 2\pi
ix)^{\gamma}\partial_{y}^{\beta}f(x,y)]e^{\mp 2\pi i\xi\cdot x}e^{\mp 2\pi
ik\cdot y}\;dxdy.$ (A.32)

By the first and third items, we find that the preceding inequality implies
$\mathscr{F}^{\pm}\\{f\\}\in\mathscr{S}(\hat{\Gamma};\mathbb{C})$ and if
$\\{f_{k}\\}_{k=1}^{\infty}\subset\mathscr{S}(\Gamma;\mathbb{C})$ and
$f_{k}\to f$ in $\mathscr{S}(\Gamma;\mathbb{C})$, then
$\mathscr{F}_{\Gamma}^{\pm}\\{f_{k}\\}\to\mathscr{F}_{\Gamma}^{\pm}\\{f\\}$ in
$\mathscr{S}(\hat{\Gamma};\mathbb{C})$. Therefore, we can conclude that if
$f\in\mathscr{S}(\Gamma;\mathbb{C})$ then
$\mathscr{F}_{\Gamma}^{\pm}\\{f\\}\in\mathscr{S}(\hat{\Gamma};\mathbb{C})$,
and the map $f\mapsto\mathscr{F}_{\Gamma}^{\pm}\\{f\\}$ is continuous.
Likewise, by a similar calculation we find that if
$f\in\mathscr{S}(\hat{\Gamma};\mathbb{C})$ then
$\mathscr{F}_{\hat{\Gamma}}^{\pm}\\{f\\}\in\mathscr{S}(\Gamma;\mathbb{C})$,
and the map $f\mapsto\mathscr{F}_{\hat{\Gamma}}^{\pm}\\{f\\}$ is continuous.
To prove the fifth item, we note that by the first item and Fubini’s theorem,

$\int_{\hat{\Gamma}}\mathscr{F}^{\pm}_{\Gamma}\\{f\\}(\xi,k)g(\xi,k)\;d\xi
dk=\int_{\hat{\Gamma}}\left(\int_{\Gamma}f(x,y)e^{\mp 2\pi i\xi\cdot x}e^{\mp
2\pi ik\cdot y}\;dxdy\right)g(\xi,k)\;d\xi dk\\\
=\int_{\Gamma}f(x,y)\left(\int_{\hat{\Gamma}}g(\xi,k)e^{\mp 2\pi i\xi\cdot
x}e^{\mp 2\pi ik\cdot y}\;d\xi
dk\right)=\int_{\Gamma}f(x,y)\mathscr{F}^{\pm}_{\hat{\Gamma}}\\{g\\}(x,y)\;dxdy.$
(A.33)

To prove the sixth item, suppose $f\in\mathscr{S}(\Gamma;\mathbb{C})$ and for
any fixed $t>0$ and $(x,y)\in\Gamma$ we consider the function

$\displaystyle\phi^{\pm}(\xi,k)=e^{-\pi t^{2}(|\xi|^{2}+|k|^{2})}e^{\pm 2\pi
i\xi\cdot x}\frac{e^{\pm 2\pi ik\cdot y}}{\sqrt{\prod_{i=1}^{d_{2}}L_{i}}}.$
(A.34)

We note that

$\displaystyle\mathscr{F}^{\pm}_{\hat{\Gamma}}\\{\phi^{\pm}\\}(r,s)=\mathscr{F}^{\pm}_{\hat{\Gamma}}\\{g_{t}\\}(r-x,s-y),$
(A.35)

where $g_{t}(\xi,k)=(\prod_{i=1}^{d_{2}}L_{i})^{-1/2}e^{-\pi
t^{2}(|\xi|^{2}+|k|^{2})}$. Furthermore, we note that the dual lattice of
$\Delta=\prod_{i=1}^{d_{2}}L_{i}^{-1}\mathbb{Z}$ is
$\Delta^{\prime}=\prod_{i=1}^{d_{2}}L_{i}\mathbb{Z}$ and the volume of the
fundamental domain of $\Delta^{\prime}$ is $\prod_{i=1}^{d_{2}}L_{i}$.
Therefore,

$\mathscr{F}^{\pm}_{\hat{\Gamma}}\\{g_{t}\\}(x,y)=\mathscr{F}^{\pm}_{\mathbb{R}^{d_{1}}}\\{e^{-\pi
t^{2}|\cdot|^{2}}\\}(x)\cdot\mathscr{F}^{\pm}_{\prod_{i=1}^{d_{2}}L_{i}^{-1}\mathbb{Z}}\\{e^{-\pi
t^{2}|\cdot|^{2}}\\}(y)\\\
=t^{-d_{1}}e^{-\pi|x|^{2}/t^{2}}\left(\prod_{i=1}^{d_{2}}L_{i}\right)^{-1}\sum_{k\in\prod_{i=1}^{d_{2}}L_{i}^{-1}\mathbb{Z}}e^{-\pi
t^{2}|k|^{2}}e^{2\pi ik\cdot
y}=t^{-d_{1}}e^{-\pi|x|^{2}/t^{2}}t^{-d_{2}}\sum_{k\in\prod_{i=1}^{d_{2}}L_{i}\mathbb{Z}}e^{-\pi|y+k|^{2}/t^{2}}\\\
=t^{-(d_{1}+d_{2})}e^{-\pi|x|^{2}/t^{2}}\sum_{k\in\prod_{i=1}^{d_{2}}L_{i}\mathbb{Z}}e^{-\pi|y+k|^{2}/t^{2}},$
(A.36)

where we used the Poisson summation formula for $h(y)=e^{-\pi t^{2}|y|^{2}}$
on lattices to justify the second to last equality. By the fifth item, (A.35)
and (A.36), we have

$\int_{\hat{\Gamma}}e^{-\pi t^{2}(|\xi|^{2}+|k|^{2})}e^{\pm 2\pi i\xi\cdot
x}\frac{e^{\pm 2\pi ik\cdot
y}}{\sqrt{\prod_{i=1}^{d_{2}}L_{i}}}\mathscr{F}^{\pm}_{\Gamma}\\{f\\}(\xi,k)\;d\xi
dk=\int_{\Gamma}f(r,s)\mathscr{F}^{\pm}_{\hat{\Gamma}}\\{g_{t}\\}(r-x,s-y)\;drds\\\
=\int_{\Gamma}f(r,s)\mathscr{F}^{\pm}_{\hat{\Gamma}}\\{g_{t}\\}(x-r,y-s)\;drds=f*\mathscr{F}_{\hat{\Gamma}}^{\pm}\\{g_{t}\\}(x,y).$
(A.37)

We note that by the dominated convergence theorem, we have

$\displaystyle\lim_{t\to 0^{+}}\int_{\hat{\Gamma}}e^{-\pi
t^{2}(|\xi|^{2}+|k|^{2})}e^{\pm 2\pi i\xi\cdot x}\frac{e^{\pm 2\pi ik\cdot
y}}{\sqrt{\prod_{i=1}^{d_{2}}L_{i}}}\mathscr{F}^{\pm}_{\Gamma}\\{f\\}(\xi,k)\;d\xi
dk=\mathscr{F}^{\mp}_{\hat{\Gamma}}\circ\mathscr{F}^{\pm}_{\Gamma}\\{f\\}(x,y)$
(A.38)

for all $(x,y)\in\Gamma$. On the other hand, we note that
$\\{\mathscr{F}^{\pm}_{\hat{\Gamma}}\\{g_{t}\\}\\}_{t>0}$ form an approximate
identity on $\Gamma$ since for all $t>0$,
$\mathscr{F}^{\pm}\\{g_{t}\\}(r,s)\geqslant 0$ on $\Gamma$ and by (A.36),

$\int_{\Gamma}\mathscr{F}^{\pm}_{\hat{\Gamma}}\\{g_{t}\\}(r,s)\;drds=t^{-(d_{1}+d_{2})}\int_{\Gamma}e^{-\pi|r|^{2}/t^{2}}\sum_{k\in\prod_{i=1}^{d_{2}}L_{i}\mathbb{Z}}e^{-\pi|s+k|^{2}/t^{2}}\;drds\\\
=t^{-(d_{1}+d_{2})}\int_{\mathbb{R}^{d_{1}}}e^{-\pi|r|^{2}/t^{2}}\;dr\int_{\mathbb{R}^{d_{2}}}e^{-\pi|s|^{2}/t^{2}}\;ds=1,$
(A.39)

and by (A.39) we also have $\lim_{t\to 0}\int_{\Gamma\setminus
B(0,\delta)}\mathscr{F}_{\hat{\Gamma}}^{\pm}\\{g_{t}\\}(r,s)\;drds\to 0$ for
all $\delta>0$. Therefore, for all $(x,y)\in\Gamma$ we have

$\displaystyle\mathscr{F}^{\mp}_{\hat{\Gamma}}\circ\mathscr{F}^{\pm}_{\Gamma}\\{f\\}(x,y)=\lim_{t\to
0^{+}}f*\mathscr{F}\\{g_{t}\\}(x,y)=f(x,y).$ (A.40)

To prove the analogous inversion formula on
$\mathscr{S}(\hat{\Gamma};\mathbb{C})$, for any given $t>0$ and
$(\xi,k)\in\hat{\Gamma}$ we consider the function

$\displaystyle\varphi^{\pm}(x,y)=\left(\prod_{i=1}^{d_{2}}L_{i}\right)^{1/2}e^{-\pi
t^{2}|x|^{2}}e^{\pm 2\pi i\xi\cdot x}e^{\pm 2\pi ik\cdot y}.$ (A.41)

We note that

$\displaystyle\mathscr{F}_{\Gamma}^{\pm}\\{\varphi^{\pm}\\}(\eta,m)=\mathscr{F}_{\Gamma}^{\pm}\\{h_{t}\\}(\eta-\xi,m-k)$
(A.42)

where $h_{t}(x,y)=\left(\prod_{i=1}^{d_{2}}L_{i}\right)^{1/2}e^{-\pi
t^{2}|x|^{2}}$, and by a change of variables we have

$\displaystyle\mathscr{F}_{\Gamma}^{\pm}\\{h_{t}\\}(\xi,k)=\left(\prod_{i=1}^{d_{2}}L_{i}\right)^{1/2}\mathscr{F}_{\mathbb{R}^{d_{1}}}\\{e^{-\pi
t^{2}|\cdot|^{2}}\\}(\xi)\cdot\mathscr{F}_{\prod_{i=1}^{d_{2}}L_{i}\mathbb{T}}\\{1\\}(k)=t^{-d_{1}}e^{-\pi|\xi|^{2}/t^{2}}\delta_{0,k},$
(A.43)

where $\delta_{0,k}$ is the Kronecker delta. Thus, by the fifth item, we have

$\int_{\Gamma}\left(\prod_{i=1}^{d_{2}}L_{i}\right)^{1/2}e^{-\pi
t^{2}|x|^{2}}e^{\pm 2\pi i\xi\cdot x}e^{\pm 2\pi ik\cdot
y}\mathscr{F}^{\pm}_{\hat{\Gamma}}\\{f\\}(x,y)\;dxdy=\int_{\hat{\Gamma}}f(\eta,m)\mathscr{F}^{\pm}_{\Gamma}\\{h_{t}\\}(\eta-\xi,m-k)\;d\eta
dm\\\
=\int_{\hat{\Gamma}}f(\eta,m)\mathscr{F}^{\pm}_{\Gamma}\\{g_{t}\\}(\xi-\eta,k-m)\;d\eta
dm=f*\mathscr{F}_{\Gamma}^{\pm}\\{h_{t}\\}(\xi,k).$ (A.44)

We note that by (A.42), $\\{\mathscr{F}_{\Gamma}^{\pm}\\{h_{t}\\}\\}_{t>0}$
form an approximate identity on $\hat{\Gamma}$, therefore by repeating the
same argument on $\mathscr{S}(\Gamma;\mathbb{C})$ we find that
$f(\xi,k)=\mathscr{F}^{\mp}_{\Gamma}\circ\mathscr{F}^{\pm}_{\hat{\Gamma}}\\{f\\}(\xi,k)$
for all $(\xi,k)\in\hat{\Gamma}$. This proves the sixth item.

The last item follows immediately from the fourth item and the seventh item. ∎

We then extend the definition of the Fourier transform to the class of
tempered distributions.

######  Definition A.10.

Suppose $T_{1}\in\mathscr{S}^{\prime}(\Gamma;\mathbb{C})$ and
$T_{2}\in\mathscr{S}^{\prime}(\hat{\Gamma};\mathbb{C})$, where $\Sigma$ is
defined via (1.43).

  1. (1)

We define the Fourier and inverse Fourier transform of $T_{1}$ to be
$\mathscr{F}_{\Gamma}^{\pm}\\{T_{1}\\}\in\mathscr{S}^{\prime}(\hat{\Gamma};\mathbb{C})$
given by

$\displaystyle\langle\mathscr{F}_{\Gamma}^{\pm}\\{T_{1}\\},\varphi\rangle=\langle
T_{1},\mathscr{F}_{\Gamma}^{\pm}\\{\varphi\\}\rangle\;\text{for
all}\;\varphi\in\mathscr{S}(\hat{\Gamma};\mathbb{C})$ (A.45)

  2. (2)

We define the Fourier and inverse Fourier transform of $T_{2}$ to be
$\mathscr{F}_{\hat{\Gamma}}^{\pm}\\{T_{2}\\}\in\mathscr{S}^{\prime}(\Gamma;\mathbb{C})$
given by

$\displaystyle\langle\mathscr{F}_{\hat{\Gamma}}^{\pm}\\{T_{2}\\},\varphi\rangle=\langle
T_{2},\mathscr{F}_{\hat{\Gamma}}^{\pm}\\{\varphi\\}\rangle\;\text{for
all}\;\varphi\in\mathscr{S}(\Gamma;\mathbb{C})$ (A.46)

We note that by Proposition A.9, the maps
$\mathscr{F}_{\Gamma}^{\pm}:\mathscr{S}^{\prime}(\Gamma;\mathbb{C})\to\mathscr{S}^{\prime}(\hat{\Gamma};\mathbb{C})$
and
$\mathscr{F}_{\hat{\Gamma}}^{\pm}:\mathscr{S}^{\prime}(\hat{\Gamma};\mathbb{C})\to\mathscr{S}^{\prime}(\Gamma;\mathbb{C})$
are also isomorphisms.

Throughout the paper we will follow standard notation by denoting the Fourier
and inverse Fourier transforms for a Schwartz function
$f\in\mathscr{S}(\Gamma;\mathbb{C})$ or a tempered distribution
$f\in\mathscr{S}^{\prime}(\Gamma;\mathbb{C})$ by
$\hat{f}=\mathscr{F}_{\Gamma}^{+}\\{f\\},\;\check{f}=\mathscr{F}_{\Gamma}^{-}\\{f\\}$.
Sometimes we will also write
$\mathscr{F}_{\Gamma}[f]=\mathscr{F}_{\Gamma}^{+}\\{f\\},\;\mathscr{F}_{\Gamma}^{-1}[f]=\mathscr{F}_{\Gamma}^{-}\\{f\\}$.

###  A.3. Spaces defined via Fourier multipliers

We note that since $\hat{\Gamma}$ and $\hat{\Gamma}_{\ast}$ are always
subgroups of $\mathbb{R}^{d}$, we can extend the reordering map
$\mathcal{P}_{g,\hat{\Gamma}}$ trivially to $\mathbb{R}^{d}$. In the next
lemma, we show that the restriction of the map $\sigma_{g,\Gamma}$ on a space
$\mathscr{X}_{\omega}(\Gamma;\mathbb{F})$, where $\sigma_{g,\Gamma}$ is
defined in the third item of Definition A.3, is an isometric isomorphism
between $\mathscr{X}_{\omega}(\Gamma;\mathbb{F})$ and
$\mathscr{X}_{\omega}(g\Gamma;\mathbb{F})$, where
$\mathscr{X}_{\omega}(\Gamma;\mathbb{F})$ is a function space defined in terms
of a Fourier multiplier $\omega:\mathbb{R}^{d}\to[0,\infty)$ that is invariant
under $\mathcal{P}_{g,\hat{\Gamma}}$.

######  Lemma A.11.

Let $\Gamma$ be defined as in (1.43). Let $g\in S_{d}$, $g\Gamma$ be defined
as in the first item of Definition A.3, and
$\mathcal{P}_{g,\Gamma},\mathcal{P}_{g,\hat{\Gamma}}$ be the reordering maps
defined as in (A.6). Let $\omega:\mathbb{R}^{d}\to[0,\infty)$ be a Fourier
multiplier that is positive $\mathcal{L}^{d}$-almost everywhere such that
$\omega(\cdot)=\omega(\mathcal{P}_{g,\hat{\Gamma}}\cdot)$. We define the
function space $\mathscr{X}_{\omega}(\Gamma;\mathbb{F})$ via

$\displaystyle\mathscr{X}_{\omega}(\Gamma;\mathbb{F})=\begin{cases}\\{{f\in\mathscr{S}^{\prime}(\Gamma;\mathbb{C})\mid\hat{f}\in
L^{1}_{\mathrm{loc}}(\hat{\Gamma};\mathbb{C}),\;f=\overline{f},\;\lVert\sqrt{\omega}\hat{f}\rVert_{L^{2}}<\infty}\\},&\mathbb{F}=\mathbb{R}\\\
\\{{f\in\mathscr{S}^{\prime}(\Gamma;\mathbb{C})\mid\hat{f}\in
L^{1}_{\mathrm{loc}}(\hat{\Gamma};\mathbb{C}),\;\lVert\sqrt{\omega}\hat{f}\rVert_{L^{2}}<\infty}\\},&\mathbb{F}=\mathbb{C},\end{cases}$
(A.47)

Then the map
$\sigma_{g,\Gamma}:\mathscr{X}_{\omega}(\Gamma;\mathbb{F})\to\mathscr{X}_{\omega}(g\Gamma;\mathbb{F})$
defined via $\sigma_{g,\Gamma}f=f\circ\mathcal{P}_{g,\Gamma}^{-1}$ is an
isometric isomorphism, with its inverse given by the map
$\sigma_{g,\Gamma}^{-1}:\mathscr{X}_{\omega}(g\Gamma;\mathbb{F})\to\mathscr{X}_{\omega}(\Gamma;\mathbb{F})$
defined via $\sigma_{g,\hat{\Gamma}}^{-1}f=f\circ\mathcal{P}_{g,\Gamma}$.

###### Proof.

We first note that by Lemma A.8, the map
$\sigma_{g,\Gamma}:\mathscr{S}^{\prime}(\Gamma;\mathbb{F})\to\mathscr{S}^{\prime}(g\Gamma;\mathbb{F})$,
with its inverse given by
$\sigma_{g,\Gamma}^{-1}:\mathscr{S}^{\prime}(g\Gamma;\mathbb{F})\to\mathscr{S}^{\prime}(\Gamma;\mathbb{F})$.
Then for any $f\in\mathscr{X}_{\omega}(\Gamma;\mathbb{F})$, we have
$\sigma_{g,\Gamma}f\in\mathscr{S}^{\prime}(g\Gamma;\mathbb{C}),\mathscr{F}_{g\Gamma}^{+}[\sigma_{g,\Gamma}f]\in
L^{1}_{\mathrm{loc}}(\hat{\Gamma}_{\ast};\mathbb{C})$, and
$\sigma_{g,\Gamma}f=\overline{\sigma_{g,\Gamma}f}$. Furthermore, by (A.31) we
have
$\mathscr{F}_{g\Gamma}^{+}[\sigma_{g,\Gamma}f]=\sigma_{g,\hat{\Gamma}}\mathscr{F}_{\Gamma}^{+}[f]$.
Then

$\left\|\sigma_{g,\Gamma}f\right\|_{\mathscr{X}_{\omega}(g\Gamma)}^{2}=\int_{g\hat{\Gamma}}\omega(\xi)|\mathscr{F}^{+}_{\Gamma}[f](\mathcal{P}_{g,\hat{\Gamma}}^{-1}\xi)|^{2}\;d\xi=\int_{\hat{\Gamma}}\omega(\mathcal{P}_{g,\hat{\Gamma}}\xi)|\mathscr{F}^{+}_{\Gamma}[f](\xi)|^{2}\;d\xi\\\
=\int_{\hat{\Gamma}}\omega(\xi)|\mathscr{F}^{+}_{\Gamma}[f](\xi)|^{2}\;d\xi=\left\|f\right\|_{\mathscr{X}_{\omega}(\Gamma)}^{2}.$
(A.48)

This shows that
$\sigma_{g,\Gamma}f\in\mathscr{X}_{\omega}(g\Gamma;\mathbb{F})$ with
$\left\|\sigma_{g,\Gamma}f\right\|_{\mathscr{X}_{\omega}(g\Gamma)}=\left\|f\right\|_{\mathscr{X}_{\omega}(\Gamma)}$.
Following the exact same calculations as above we also find that for any
$f\in\mathscr{X}_{\omega}(g\Gamma;\mathbb{F})$, we have
$\sigma_{g,\Gamma}^{-1}f=f\circ\mathcal{P}_{g,\Gamma}\in\mathscr{X}_{\omega}(\Gamma;\mathbb{F})$
with
$\|\sigma_{g,\Gamma}^{-1}f\|_{\mathscr{X}_{\omega}(\Gamma)}=\left\|f\right\|_{\mathscr{X}_{\omega}(g\Gamma)}$.
This shows that
$\sigma_{g,\Gamma}:\mathscr{X}_{\omega}(\Gamma;\mathbb{F})\to\mathscr{X}_{\omega}(g\Gamma;\mathbb{F})$
is an isometric isomorphism with $\sigma_{g,\Gamma}^{-1}$ as its inverse. ∎

### Acknowledgement

The authors would like to thank Benoît Pausader for pointing out the proof for
Theorem 2.17.

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