text:
\begin{document}

	\maketitle

	\thanks{
		\begin{center}
			{ \it \FurukawaCommentThird{Dedicated to Professor Matthias Hieber on the occasion of his 60th birthday}}
		\end{center}
	}

	\begin{abstract}
		\FurukawaCommentSeventh{In this paper we justify the hydrostatic approximation of the primitive equations in the maximal $L^p$-$L^q$-setting in the three-dimensional layer domain $\Omega = \Torus^2 \times (-1, 1)$ under the no-slip (Dirichlet) boundary condition in any time interval $(0, T)$ for $T>0$.
		We show that the solution to the scaled Navier-Stokes equations with Besov initial data} \FurukawaCommentSecond{$u_0 \in B^{s}_{q,p}(\Omega)$ for $s > 2 - 2/p + 1/ q$} converges to the solution to the primitive equations with the same initial data in $\mathbb{E}_1 (T) = W^{1, p}(0, T ; L^q (\Omega)) \cap L^p(0, T ; W^{2, q} (\Omega)) $ with order $O(\epsilon)$ where $(p,q) \in (1,\infty)^2$ satisfies $ \frac{1}{p} \leq \min \bracket{ 1 - 1/q, 3/2 - 2/q }$.
		The global well-posedness of the scaled Navier-Stokes equations in $\mathbb{E}_1 (T)$ is also proved for sufficiently small $\epsilon>0$.	Note that $T = \infty$ is included.
	\end{abstract}

\section{Introduction}

	\FurukawaCommentThird{We consider the primitive equations of the form}
		\begin{align*}
			(PE) \left \{
				\begin{array}{rll}
					\dt u - \Delta u + u \cdot \nabla v + \nabla_H \pi \quad
					 =
					& 0
					& \In{ \SpaceTime }, \\
				 	\partial_z \pi \quad
					 =
				 	& 0
				 	& \In{\SpaceTime}, \\
					\Div u \quad
					 =
					&0
					& \In{\SpaceTime}, \\
					u \quad
					=
					& 0
					& \On \partial \Omega	\times (0 , \infty) \FurukawaCommentThird{,}
				\end{array}
			\right.
		\end{align*}
	\FurukawaCommentThird{where $u = (v, w) \in \euclid{2} \times \Real$ and $\pi$ are the unknown velocity field and pressure field,} respectively, $\nabla_H = \bracket{\partial_x, \partial_y}^T$, and $\Omega = \Torus^2 \times (-1, 1)$ \FurukawaCommentSixth{for $\Torus = \Real / 2 \pi \Integer$}.
	By divergence-free condition $w$ is given by the formula
	\begin{align*}
		w (x^\prime, x_3, t)
		= - \int_{-1}^{x_3}
			\mathrm{div}_H \, v (x^\prime, \FurukawaCommentSecond{\zeta}, t) d \zeta
		d \zeta
		= \int_{x_3}^{1}
			\mathrm{div}_H \, v (\FurukawaCommentSecond{x^\prime , \zeta}, t)
		\FurukawaCommentSecond{d \zeta} \FurukawaCommentThird{;}
	\end{align*}
	\FurukawaCommentThird{here we invoked physically reasonable condition $w (\cdot , \cdot , \pm 1 , \cdot) = 0$}.
	The primitive \FurukawaCommentSixth{equations are} fundamental model for geographic flow.
	Existence of \PGComment{the global} weak solution to the primitive equations \PGComment{on the sphere} with $L^2$-initial data was proved by Lions, Temam and Wang \cite{LionsTemamWangShou1992}.
	Local-in-time well-posedness was proved by \FurukawaComment{Guill\'{e}n-Gonz\'{a}lez, Masmoudi and Rodr\'{i}guez-Bellido} \cite{GuillenMasmoudiRodriguez2001}.
	Although global well-posedness of the 3-dimensional Navier-Stokes \FurukawaCommentFifth{equations are} the well-known open problem, for the primitive \FurukawaCommentFourth{equations,} this problem \FurukawaCommentFourth{has} been solved by Cao and Titi \cite{CaoTiti2007}.
	Hieber and Kashiwabara \cite{HieberKashiwabara2016} extended this result to prove global well-posedness for the \FurukawaCommentSixth{primitive equations} in $L^p$-settings.
	\FurukawaComment{
		In these papers boundary conditions are imposed no-slip (Dirichlet) on the bottom and slip (Neumann) on the top.}
	Recently, \FurukawaCommentSecond{the second and last authors together with} Gries, Hieber and Hussein \cite{GigaGriesHieberHusseinKashiwabara2017_analiticity} obtained global-in-time well-posedness in the maximal regularity spaces (mixed Lebesgue-Sobolev \FurukawaCommentFifth{spaces}) $ W^{1,p} \bracket{ 0, T ; L^q \bracket{\Omega} } \cap L^p \bracket{ 0, T ; W^{2,q} \bracket{\Omega} } $ for $T>0$ \FurukawaCommentFifth{and appropriate $1 < p,q < \infty$} under various boundary conditions.

	\FurukawaCommentSeventh{Our aim} in this paper is to give a rigorous justification of the derivation of the primitive equations under the Dirichlet boundary condition.
	We \PGComment{begin by explaining its derivation}.
	\FurukawaCommentThird{Let us consider the anisotropic viscous Navier-Stokes equations in a thin domain of the form}
		\begin{align*}
			(ANS) \left \{
				\begin{array}{rcll}
					\dt u - ( \Delta_H + \epsilon^2 \partial_z^2 ) u + u \cdot \nabla u + \nabla \pi
					& =
					& 0
					& \In{ \Omega_{\epsilon} \times (0, \infty) }, \\
					\Div u
					& =
					& 0
					& \In{ \Omega_{\epsilon} \times (0, \infty) },
				\end{array}
			\right.
		\end{align*}
	 where $\Omega_{\epsilon} = (- \epsilon, \epsilon) \times \mathbb{T}^2$.
	 If $\epsilon = 1$, (ANS) is the usual Navier-Stokes equations.
	\FurukawaCommentThird{
	The equations (ANS) \FurukawaCommentFifth{are} considered as a good model to describe motion of incompressible viscous fluid filled in a thin domain.
	}
	 Actually, if we put the Reynolds number $1$, since length and velocity are of $\epsilon$-order, apparent viscosity for vertical direction must be of $\epsilon^2$-order  \AmruComment{from the Reynolds number point of view}.
	 The primitive equations \FurukawaCommentThird{are} formally derived from \FurukawaCommentThird{(ANS)}.
	\FurukawaCommentThird{We introduce new unknowns of (ANS) by rescaling as}
		\begin{itemize}
			\item $ u_{\epsilon} := (v_{\epsilon}, w_{\epsilon}) $
			\item $ v_{\epsilon} (x, y, z, t) := v  (x, y, \FurukawaCommentFifth{\epsilon z}, t) $
			\item $ w_{\epsilon} (x, y, z, t) := w (x, y, \FurukawaCommentFifth{\epsilon z}, t) / \epsilon$
			\item $ \pi_{\epsilon} (x, y, z, t) := \pi (x, y, \FurukawaCommentFifth{\epsilon z}, t) $,
		\end{itemize}
 	\FurukawaCommentFifth{where $x, y \in \Torus$, $z \in (-1, 1)$ and $t> 0$.}
 	Then, $(u_{\epsilon}, \pi_{\epsilon})$ satisfy the scaled Navier-Stokes equations in a fixed domain
		\begin{align*}
			(SNS)
			\left \{
				\begin{array}{rcll}
					\dt v_{\epsilon} - \Delta v_{\epsilon} + u_{\epsilon} \cdot \nabla v_{\epsilon} + \nabla_H \pi_{\epsilon}
					& =
					& 0
					& \In{\SpaceTime}, \\
					\epsilon^2 \LongBracket{ \dt w_{\epsilon} - \Delta w_{\epsilon} + u_{\epsilon} \cdot \nabla w_{\epsilon} } + \partial_z \pi_{\epsilon}
					& =
					& 0
					& \In{\SpaceTime}, \\
					\FurukawaCommentFifth{\mathrm{div}}\, u
					& =
					& 0
					& \In{\SpaceTime}.
				\end{array}
			\right.
		\end{align*}
	Taking formally $\epsilon \rightarrow 0$ for the above equations, we get the primitive equations.

	\PGComment{
		The Navier-Stokes equations (SNS) \FurukawaCommentFifth{are} well-studied for $\epsilon = 1$ since the work of Leray \cite{Leray1934}, where a global weak solution is constructed in $\Omega = \Real^3$.
		For a general domain see Farwig, Kozono, Sohr \cite{FarwigKozonoSohr2005}.
		\FurukawaCommentThird{A local strong solution} is constructed by Fujita and Kato \cite{FujitaKato1964} \FurukawaCommentThird{when initial data is} in $H^{1/2}$.
		It is extended to various domains in various function spaces; see e.g. Ladyzenskaya \cite{Ladyzhenskaya1969}, Kato \cite{Kato1984}, Giga and Miyakawa \cite{GigaMiyakawa1989} for early development.
		\FurukawaCommentSixth{The reader refers} to a book of Lemari\'{e}-Rieusset \cite{Lemarie-Rieusset2016} and \FurukawaCommentFifth{review articles} by Farwig, Kozono and Sohr \cite{FarwigKozonoSohr2018} and Gallagher \cite{Gallagher2018} for resent development.
		Many results can be extended for general $\epsilon > 0$ but it is not often written explicitly \FurukawaCommentThird{except in} a book of Chemin, Desjardins, Gallagher and Grenier \cite{CheminDesjardinsGallagherGrenier2006}.
	}

	Rigorous justification of the primitive equations from the scaled Navier-Stokes equations was studied by Az\'{e}rad and Guill\'{e}n \cite{AzeradGuillen2001}.
	They obtained weak* convergence in the natural energy space $L^{\infty}( 0, T; L^2(\Omega)) \cap L^2( 0, T; H^1(\Omega)) $ for $\Omega = \Torus^2 \times (-1, 1)$ and $T>0$.
	Recently, Li and Titi \cite{LiTiti2017} improved their result to get strong convergence \PGComment{by energy method} with the aid of regularity of the solution to the primitive equations.
	\FurukawaCommentSecond{The authors together with }Hieber, Hussein and Wrona \cite{FurukawaGigaHieberHusseinKashiwabaraWrona2018} extended Li and Titi's result in maximal-regularity \FurukawaCommentFifth{spaces} $W^{1, p} (0, T ; L^q (\Torus^3)) \cap L^{p} (0, T ; W^{2, q} (\Torus^3))$ \FurukawaCommentSeventh{with initial trace in the Besov space $B^{2 - 2 / p}_{q,p}$} for $T > 0$ and $1 / p \leq \min (1 - 1 / q, 3 / 2 - 2 / q)$ \FurukawaCommentThird{by an operator} theoretic approach.
	The case of $p = q = 2$ is corresponding to Li and Titi's result.
	 \AmruComment{
	 	Note the case of the torus corresponding to Neumann boundary conditions on top and bottom part, moreover, the work of Az\'{e}rad and Guill\'{e}n treats mixed boundary conditions with Dirichlet boundary conditions on the bottom, while Li and Titi deal with Neumann boundary conditions only.
	 }
	As we already mentioned, the \FurukawaCommentFifth{primitive equations are} a model for geographic flow.
	Although it is more physically natural to consider the case of Dirichlet-Neumann and Dirichlet boundary conditions, there was no result of justification of derivation \FurukawaComment{in a strong topology} to the \FurukawaCommentFifth{primitive equations} from the Navier-Stokes equations.

	\AmruComment{
		Let
		\begin{align*}
			\mathbb{E}_1 (T)
			&= \{ u \in \MRspace \, ;  \, \mathrm{div} \, u = 0, \, u |_{x = \pm 1} = 0 \}, \\
			 \mathbb{E}_0 (T)
			 & = \{ u \in L^p (0, T ; L^q (\Omega)) \, ;  \, \mathrm{div} u = 0, \, u |_{x = \pm 1} = 0 \}, \\
			  \mathbb{E}_1^\pi (T)
			 & = \{ \pi \in L^p (0, T ; W^{1, q} (\Omega)) \, ;  \, \int_\Omega \pi \, dx = 0 \},
		\end{align*}
		and
		\begin{align*}
			 X_{\gamma}
			 & = \{ u \in B_{q, p}^{2 - 2 / p} \FurukawaCommentFourth{(\Omega)} \, ;  \, \mathrm{div} u = 0, \,  u |_{x = \pm 1} = 0 \}
		\end{align*}
	be the \FurukawaCommentFifth{initial} trace space of $\mathbb{E}_1 (T)$\FurukawaCommentSeventh{, where $B^s_{q,p}$ denotes the $L^q$-Besov space of order $s$}.
	In this paper, we frequently use $\Vert \cdot \Vert_{\mathbb{E}_0 (T)}$ as the norm of $L^p (0, T ; L^q (\Omega))$ \FurukawaCommentFifth{and $\Vert \cdot \Vert_{\mathbb{E}_1 (T)}$ as the norm of $\MRspace$} \FurukawaCommentThird{to simplify the notation}.
	}
	Let us seek the solution $U_\epsilon = (V_\epsilon, W_\epsilon)$ to

		\begin{equation}  \label{eq:A0502}
			\left \{
				\begin{array}{rcll}
					\dt \ep{V} - \Delta \ep{V} + \nabla_H \ep{P}
					& =
					& F_H
					&\In{ \Omega \times (0, T) }, \\
					\dt \bracket{ \epsilon \ep{W} } - \Delta \bracket{ \epsilon \ep{W} } + \frac{\partial_z}{\epsilon} \ep{P}
					& =
					& \epsilon F_z + \epsilon F
					&\In{ \Omega \times (0, T) }, \\
					\Div \ep{U}
					& =
					& 0
					&\In{ \Omega \times (0, T) }, \\
					\ep{U}
					& =
					& 0
					&\ON{\partial \Omega \times (0, T)},\\
					\ep{U}(0)
					& =
					& 0
					&\In{\Omega},
				\end{array}
			\right .
		\end{equation}
	where
		\begin{itemize}
			\item $F_H = - \LongBracket{ \Convection{ \ep{U} }{ \ep{V} } + \Convection{u}{ \ep{V} } + \Convection{ \ep{U} }{v} } $

			\item $ F_z = - \LongBracket{ \Convection{ \ep{U} }{ \ep{W} } + \Convection{u}{ \ep{W} } + \Convection{ \ep{U} }{w} }$

			\item $F = - \LongBracket{ \dt w - \Delta w + \Convection{u}{w} }$.
		\end{itemize}
	\PGComment{The system} (\ref{eq:A0502}) is the equation of the difference between the solution to the (PE) and (SNS).
		\begin{theorem} \label{thm:A0105}
			Let $T > 0$ \FurukawaCommentFourth{and $0< \epsilon \leq 1$}.
			Suppose $\FurukawaCommentSeventh{(p,q) \in (1, \infty)^2}$ satisfies $ \frac{1}{p} \leq \min \bracket{ 1 - 1/q, 3/2 - 2/q }$, \FurukawaComment{$u_0 = (v_0, w_0) \in X_{\gamma}$  and $v_0 \in B^{s}_{q, p} \FurukawaCommentFourth{(\Omega)}$ for $s > 2 - 2 / p + 1 / q$}.
			Let $u \in \mathbb{E}_1 (T)$ be a solution of (PE) with initial data $u_0 \in X_\gamma$.
			Then there exists constant $C = C (p, q, \Vert u \Vert_{\mathbb{E}_1 (T)})$ and a unique solution $U_\epsilon = (V_\epsilon, W_\epsilon)$ to (\ref{eq:A0502}) such that
			\begin{align}
				\Vert
					\left(
						V_\epsilon, \epsilon W_\epsilon
					\right)
				\Vert_{\mathbb{E}_1 (T)}
				\leq \epsilon C.
			\end{align}
			Moreover, $u_\epsilon = (v_\epsilon, w_\epsilon) := (v + V_\epsilon, w + W_\epsilon)$ is the unique solution to (SNS) in $\mathbb{E}_1 (T)$.
		\end{theorem}
	This theorem implies the justification of the hydrostatic approximation.
		\begin{corollary}  \label{cor:A0105}
			Let $T > 0$ \FurukawaCommentFourth{and $0 < \epsilon \leq 1$}.
			Suppose $\FurukawaCommentSeventh{(p,q) \in (1, \infty)^2}$ satisfies $ \frac{1}{p} \leq \min \bracket{ 1 - 1/q, 3/2 - 2/q }$, \FurukawaCommentFourth{$u_0 = (v_0, w_0) \in X_{\gamma}$  and $v_0 \in B^{s}_{q, p} (\Omega)$ for $s > 2 - 2 / p + 1 / q$}.
			Let $u$ and $ \ep{u} $ be a solution of (PE) and (SNS) in $\mathbb{E}_1\FurukawaComment{(T)}$ under the Dirichlet boundary condition with initial data $u_0$, respectively, such that
				\begin{align}
					\LongNorm{u}{\mathbb{E}_1 \bracket{T}}
					+ \LongNorm{ \bracket{ \ep{v}, \epsilon \ep{w} } }{\mathbb{E}_1 \bracket{T}}
					\leq C_{\FurukawaCommentFourth{0}}
				\end{align}
			for some $C_{\FurukawaCommentFourth{0}} = C_{\FurukawaCommentFourth{0}} \bracket{u_0, p, q}$.
			Then there exists a positive $ C = C(p, q, C_0 ) $ such that
				\begin{align*}
					\LongNorm{ \LongBracket{ \ep{v} - v, \epsilon \bracket{ \ep{w} - w } } }{ \mathbb{E}_1 \bracket{T} } \leq \epsilon C.
				\end{align*}
		\end{corollary}

	Our strategy to show Theorem \ref{thm:A0105} is based on the estimate for $(V_\epsilon, \epsilon W_\epsilon)$.
	\FurukawaCommentFifth{It consists of two key steps:} maximal regularity result of the anisotropic Stokes operator and improved regularity result for \FurukawaCommentFourth{the} vertical component of the solution to the \FurukawaCommentFifth{primitive equations}.
	\FurukawaCommentFifth{
	We consider the non-linear term in (SNS) as an external force term $f$ and set $u_\epsilon = (v_\epsilon, \FurukawaCommentSixth{\epsilon} w_\epsilon)$ to get
	\begin{equation} \label{eq_LE}
			\left \{
				\begin{array}{rcll}
					\dt u_{\FurukawaCommentFifth{\epsilon}} - \Delta u_{\FurukawaCommentFifth{\epsilon}} + \nabla_{\epsilon} \pi_{\FurukawaCommentFifth{\epsilon}}
					& =
					& f
					& \In{\Omega \times \bracket{0, T}}, \\
					\Dive u_{\FurukawaCommentFifth{\epsilon}}
					& =
					& 0
					&\In{\Omega \times \bracket{0, T}} ,\\
					u_{\FurukawaCommentFifth{\epsilon}}
					& =
					& 0
					&\ON{ \partial \Omega  \times (0, T) }, \\
					u_{\FurukawaCommentFifth{\epsilon}} (0)
					&=
					& u_0
					& \In{\Omega},
				\end{array}
			\right .
		\end{equation}
		\FurukawaCommentSixth{where $\nabla_\epsilon = (\partial_1, \partial_2 , \partial_3 / \epsilon)^T$ and $\mathrm{div}_\epsilon = \nabla_\epsilon \cdot$.}
		We define the function space $\mathbb{E}_{\epsilon, j} (T)$ for $j = 0, 1$ similarly as $\mathbb{E}_j (T)$ by replacing $\mathrm{div}$ by $\mathrm{div_\epsilon}$.
		Although the space $\mathbb{E}_{\epsilon, j} \FurukawaCommentSeventh{(T)}$ depends on $\epsilon$, the norm is just the norm in $\MRspace$, so we shall write the norm in $\mathbb{E}_{\epsilon , j} (T)$ simply by $\Vert \cdot \Vert_{\mathbb{E}_j (T)}$.
		The space $X_{\epsilon, \gamma}$ is the initial trace space of $\mathbb{E}_{\epsilon, j}(T)$ and it is almost the same as $X_\gamma$ by replacing $\mathrm{div}$ by $\mathrm{div}_\epsilon$.
		Since the norm of $X_\gamma$ is that of $B^{2 - 2 / p}_{q,p} (\Omega)$ and is independent of $\epsilon$, we shall write the norm in $X_{\epsilon, \gamma}$ simply by $\Vert \cdot \Vert_{X_{ \gamma}}$.}
	\FurukawaCommentFifth{We recall} some known results on maximal regularity of the Stokes operator, which is corresponding to the case $\epsilon = 1$.
	Solonnikov \cite{Solonnikov1964} first proved $\Lebesgue{q}$-$\Lebesgue{q}$ maximal regularity for the Stokes operator \PGComment{by a potential-theoretic approach}.
	\PGComment{
		\FurukawaCommentSecond{The second author} \cite{Giga1985} established a \FurukawaCommentSecond{bound} for the pure imaginary power of the Stokes operator in a bounded domain.
		This type of property will be \FurukawaCommentThird{simply called} a bounded imaginary power\FurukawaCommentFifth{, shortly} BIP.
		This BIP implies the maximal regularity $\Lebesgue{p}$-$\Lebesgue{q}$ regularity via Dore-Venni theory \cite{DoreVenni1990}.
		Indeed, \FurukawaCommentSecond{the second author} and Sohr \cite{GigaSohr1991} established a global-in-time maximal regularity in an exterior domain by estimating BIP.
	}
	Further studies on maximal regularity were done by many researchers, for instance, Dore and Venni \cite{DoreVenni1990} and Weis \cite{Weis2001}.
	See Denk, Hieber and Pr\"{u}ss \cite{DenkHieberPruss2003} for further comprehensive research.
	\PGComment{
		In our case, we have to clarify $\epsilon$-dependence in estimates for maximal regularity, which is a key point.
		Our key maximal regularity result is
	}
	\begin{lemma} \label{lem:A1001}
		Let $1 < p, q < \infty$, $0 < \epsilon \leq 1$ and $T > 0$.
		Let \FurukawaCommentFifth{$ f \in \mathbb{E}_{\epsilon, 0} (T)$ and $u_0 \in X_{\epsilon, \gamma}$}.
		Then \FurukawaCommentFourth{there exist} constants $C = C (p,q)>0$ and $C^\prime = C^\prime (p,q) > 0$, which are independent of $\epsilon$, and $(u, \pi)$ satisfying (\ref{eq_LE}) such that
		\begin{align} \label{eq:A1001}
			\LongNorm{ \dt u }{\mathbb{E}_0\FurukawaComment{(T)}}
			+ \LongNorm{ \nabla^2 u }{\mathbb{E}_0\FurukawaComment{(T)}}
			+ \LongNorm{ \ep{\nabla} \pi }{\mathbb{E}_0\FurukawaComment{(T)}}
			\leq C \LongNorm{f}{\mathbb{E}_0\FurukawaComment{(T)}}
			+ C^\prime \LongNorm{u_0}{ X_{\gamma} }.
		\end{align}
	\end{lemma}
	\PGComment{
	\FurukawaCommentFifth{Lemma \ref{lem:A1001} follows from a maximal regularity involving the Stokes operator, which follows from a bound for the pure imaginary power by Dore-Venni theory.}
	However, we need to clarify that $C$ and $C^\prime$ can be taken independent of $\epsilon$.
	For $\epsilon = 1$, \FurukawaCommentThird{a necessary BIP estimate} for the Stokes operator has been established \FurukawaCommentThird{by Abels \cite{Abels2002}, where a resolvent decomposition similar to \cite{Giga1985} is used.}
	Unfortunately, \FurukawaCommentFourth{the} $\epsilon$-dependent case is not discussed here.}
	However, the strategy in \cite{Abels2002} works \FurukawaComment{for} our problem.
	We construct the anisotropic Stokes operator by the method in \cite{Abels2002} and show the boundedness of imaginary power.
	Note that, in our previous paper \cite{FurukawaGigaHieberHusseinKashiwabaraWrona2018}, maximal regularity of the anisotropic Stokes operator is much easier since the corresponding Stokes operator is essentially the same as the Laplace operator on $\Torus^3$.
	In the case of the Dirichlet boundary condition, the corresponding Stokes operator becomes to be much more difficult by the effect of boundaries, \FurukawaCommentFifth{which is substantially different from the case} of the periodic boundary conditions.
	\FurukawaCommentThird{The maximal regularity was proved in a layer domain for the Stokes operator under various boundary conditions by Saito \cite{Saito2015} by proving $\mathcal{R}$-boundedness of the resolvent operator when $\epsilon =1$.
	Unfortunately, it seems very difficult to check the dependence of $\epsilon$, so we do not take this approach,}

	\FurukawaComment{The term} $F = \partial_t w - \Delta w + u \cdot \nabla w$ appears in \FurukawaCommentFourth{the right-hand side} of (\ref{eq:A0502}).
	Thus, we need to improve the regularity of $w$ and estimate this term in $L^p (0, T ; L^q (\Omega))$.
	\begin{lemma} \label{lem:A1002}
		Let $T > 0$ and $u_0 = (v_0, w_0) \in X_\gamma$ with $w_0 = - \TimeInt{-1}{x_3}{\mathrm{div}_H \, v_0 }{\zeta}$ and \FurukawaComment{$v_0 \in B^{s}_{q, p}(\Omega)$  for $s > 2 - 2 / p + 1 / q$} and $u = (v, w)$ be the solution to (PE).
		Assume $v \in \mathbb{E}_1 (T)$.
		Then there \FurukawaCommentFourth{exists} a \FurukawaComment{constant $C> 0$} such that
		\begin{align} \label{eq:B0800}
			\Vert
				w
			\Vert_{\mathbb{E}_1 (T)}
			\leq C.
		\end{align}
	\end{lemma}
	Since $v \in \mathbb{E}_1 (T)$, which is the \FurukawaCommentFifth{horizontal} component of the solution to the primitive equations, has already proved, it follows $w (\cdot, x_3) =  - \int_{-1}^{x_3} \mathrm{div}_H v (\cdot, \zeta) d\zeta \in \FurukawaComment{W}^{1, p}( 0, T ; \FurukawaComment{W}^{-1, q} (\Omega)) \cap L^p (0, T; \FurukawaComment{W}^{1, p} (\Omega))$ .
	This derivative loss is due to the absence of the equation of time-evolution of $w$ in the \FurukawaCommentFifth{primitive equations}.
	In our previous paper \cite{FurukawaGigaHieberHusseinKashiwabaraWrona2018}, \AmruComment{which treats the periodic boundary condition}, we recover the regularity of $w$ by deriving the equation which $w$ satisfies and applying maximal regularity of the Laplace operator to the equation.
	However, in the case of the Dirichlet boundary condition, this method is not applicable directly \FurukawaCommentThird{because of the presence of the second-order derivative term} at the boundary, which vanishes in the case of periodic boundary condition.
	\FurukawaComment{
	Thus, \FurukawaCommentThird{we are \FurukawaCommentFourth{forced} to impose} additional regularity for initial data to get regularity for $v$.
	If \FurukawaCommentSeventh{$v_0 \in B^{s}_{q,p}\FurukawaCommentSeventh{(\Omega)}$} for $s > 2 - 2 /p + 1 /q$, then we obtain $v \in L^p (0 , T; W^{s + 2 / p , q} \FurukawaCommentSeventh{(\Omega)})$ and the trace of \FurukawaCommentFourth{the} second derivative belongs to $\mathbb{E}_0 (T)$.
	}Let us \FurukawaCommentFifth{explain} our strategy to show Theorem \ref{thm:A0105}.
	\PGComment{
		By Lemma \ref{lem:A1002}, our main result Theorem \ref{thm:A0105}, can be proved \FurukawaCommentFourth{the} same way as  \cite{FurukawaGigaHieberHusseinKashiwabaraWrona2018}.
		The proof we give here is slightly different from that of \cite{FurukawaGigaHieberHusseinKashiwabaraWrona2018} in the sense of the constant $C$ \FurukawaCommentFourth{in} Theorem \ref{thm:A0105}
 is clarified.
 	}
	\FurukawaCommentThird{We first show the boundedness of non-linear terms $F_H$ and $F_z$ in (\ref{eq:A0502}) in the space $\mathbb{E}_0\FurukawaComment{(T)}$.}
	\FurukawaCommentThird{We know that $F$} is also bounded \FurukawaCommentThird{in} $\mathbb{E}_0 (T)$ by Lemma \ref{lem:A1001}.
	\FurukawaCommentThird{We next} apply Lemma \ref{lem:A1002} to (\ref{eq:A0502}) to get a quadratic inequality, which leads \FurukawaCommentSixth{to} $ \LongNorm{ \bracket{ \ep{V}, \epsilon \ep{W} } }{\mathbb{E}_1 \bracket{T^{\ast}}} \leq C \epsilon $ for some short time \FurukawaComment{$T^{\ast}> 0$} and $\epsilon$-independent constant $C>0$.
	Since $C$ depends only on $p$, $q$, $\Vert u_0 \Vert_{X_\gamma}$, $\Vert u \Vert_{\mathbb{E}_1 (T)}$ and $T$, if we take $\epsilon$ small, \FurukawaCommentThird{we are able to extend} the time to all finite time $T$ by finite step.

\FurukawaComment{
	This paper is organized as follows.
	In section \ref{section_BIP}, \FurukawaCommentFourth{the} boundedness of pure imaginary power is proved.
	The resolvent operator of the anisotropic Stokes operator is decomposed into three parts, and for each part uniform bound on $\epsilon$ is proved.
	In section \ref{section_improved_regularity_w}, improved regularity for $w$ is proved.
	In section \ref{section_proof_of_main_thoerem}, we give a proof of our main theorem by iteration.
}

	In this paper, $|| \cdot||_{X \rightarrow Y}$ denotes the operator norm from a Banach space $X$ to a Banach space $Y$.
	\FurukawaCommentFifth{We denote by $C_0^\infty (\Omega)$ the set of compactly supported smooth functions in $\Omega$.
	We denote $L^q (\Omega)$ is the Lebesgue space for $1 \leq q \leq \infty$ \FurukawaCommentSixth{equipped} with the norm
	\begin{align*}
		\Vert
			f
		\Vert_{L^q (\Omega)}
		= \left(
			\int_\Omega
				\left \vert
					f (x)
				\right \vert^q
			d x
		\right)^{1/q}.
	\end{align*}
	We use the usual modification when $q = \infty$.
	For $m \in \Integer_{\geq 0}$ and $1 \leq q \leq \infty$ we denote by $W^{m, q} (\Omega)$ the $m$-th order Sobolev space \FurukawaCommentSixth{equipped} with the norm
	\begin{align*}
		\Vert
			 f
		\Vert_{W^{m, q} (\Omega)}
		= \Vert
			\nabla^m f
		\Vert_{L^q (\Omega)}.
	\end{align*}
	We define the fractional Sobolev spaces \FurukawaCommentSeventh{$W^{s, q} (\Omega) (= B^s_{qq} (\Omega))$ for $s \notin \Integer$} and $1 < q < \infty$ by the real interpolation $\left( W^{[s], q} (\Omega), W^{[s] + 1, q} (\Omega) \right)_{s - [s], q}$, where $[\cdot]$ denotes the Gauss symbol.
	}
	We define the Fourier transform by
	\begin{align*}
			\Fourier f \bracket{\xi} = \TimeInt{\Real^d}{}{ \Napier^{ - i x \cdot \xi } f \bracket{x} }{x}
	\end{align*}
	\FurukawaCommentFourth{and}
	the Fourier inverse transform by
	\begin{align*}
		\FourierInverse f \bracket{x} = \frac{1}{ \bracket{ 2 \pi }^d} \TimeInt{\Real^d}{}{ \Napier^{ i x \cdot \xi } f \bracket{\xi} }{\xi}.
	\end{align*}
	The Fourier transform on the \FurukawaComment{$d$-dimensional} torus $\Torus^d$ and its inverse transform are \FurukawaCommentSecond{defined by $[\Fourier_d f ] (n) = \int_{\Torus^d} e^{ - i x \cdot n} f (x) dx$ and $[\Fourier_d^{-1} g ]  (x) = \frac{1}{(2 \pi )^d} \sum_{n} g_n e^{i n \cdot x}$}, respectively.
	\FurukawaCommentFifth{We denote by $\Fourier_{\Prime{x}}$ the partial Fourier transform with respect to $\Prime{x} \in \Real^2$ and by $\FourierInverse_{\Prime{\xi}}$ the partial Fourier inverse transform with respect to $\Prime{\xi}$.}
	\FurukawaCommentFifth{We denote by $\Fourier_{d,\Prime{x}}$ the partial Fourier transform \FurukawaCommentSixth{with} respect to $\Prime{x} \in \Torus^2$ and by $\FourierInverse_{d, \Prime{n}}$ the partial Fourier inverse transform with respect to $\Prime{n} \in \Integer^2$.}
	Define $\Sigma_{\theta} := \LongSet{ \lambda \in \Complex}{ \abs{ \arg \lambda } < \pi - \theta }$.
	For a Fourier multiplier operator $\FourierInverse_{\xi} m \bracket{\xi} \Fourier_x$ in $\Real^3$, we denote by $[m]_{\mathcal{M}}$ the Mikhlin constant.
	$\FourierInverse_{\Prime{\xi}} m \bracket{\xi} \Fourier_{\FurukawaComment{x^\prime}}$ is a Fourier multiplier operator in $\Real^2$ with Mikhlin constant $[m]_{\Prime{\mathcal{M}}}$.
	For $0 < \epsilon \leq 1$, $\ep{\Delta} = \partial_1^2 + \partial_2^2 + \partial_3^2 / \epsilon^2$ denotes the anisotropic Laplace operator.
	\FurukawaCommentFifth{We denote by $E_0$ the zero-extension operator with respect to the vertical variable from $(-1 , 1)$ to $\Real$.
	We denote by $R_0$ the restriction operator with respect to the vertical variable from $\Real$ to $(-1, 1)$.}
	For \FurukawaCommentFourth{an} integrable function $f$ defined on $\Omega$, we write its vertical average by $\overline{f} = \frac{1}{2} \TimeInt{-1}{1}{ f (\cdot, \cdot , \zeta)}{\zeta}$.

	\section{\FurukawaCommentFifth{A uniform bound for pure imaginary power of the anisotropic Stokes operator and its maximal regularity}} \label{section_BIP}

	 \PGComment{In this section, we \FurukawaCommentFifth{first} establish a uniform bound  independent of $\epsilon$ for the pure imaginary power to the anisotropic Stokes operator along with \cite{Abels2002}.}
	\FurukawaCommentFifth{Then we shall give the proof of Lemma \ref{lem:A1001}.}
	\subsection{Boundedness of Fourier multipliers}

	Although the case of infinite \FurukawaCommentSeventh{the} layer $\Real^2 \times (-1. 1)$ is considered in \cite{Abels2002}, his method also works in \FurukawaCommentFifth{the} case of \FurukawaCommentFifth{the} periodic layer $\Omega = \Torus^2 \times (- 1, 1)$ thanks to Fourier multiplier theorem on the torus, e.g. Proposition 4.5 in \cite{HeckKimKozono2009} \FurukawaCommentFifth{and Section 4 of Grafakos's book \cite{Grafakos2008}.}
	\begin{proposition}[\cite{HeckKimKozono2009}] \label{prop_Fourier_Multiplier_Theorem_Discrete}
		Let $1 < p < \infty$ and $m \in C^{\FurukawaCommentSeventh{d}} (\Real^d \setminus \{ 0 \})$ satisfies the Mikhlin condition:
			\begin{align}
				[m]_{\PGComment{\mathcal{M}}}
				:= \sup_{\alpha \in \{ 0, 1 \}^d} \sup_{\xi \in \Real^d \setminus \{ 0 \} }	\left|
					\xi^\alpha
					\partial_{\xi}^\alpha m (\xi)
				\right|
				< \infty.
			\end{align}
		Let $a_k = m (k)$ for $k \in \mathbb{Z}^d \setminus \{ 0 \}$ and $a_0 \in \Complex$.
		\PGComment{
		For $f (x) = \sum_{n \in \mathbb{Z}^d} \widehat{f}_n e^{i n \cdot x} \in L^{\FurukawaCommentFifth{q}} (\Torus^d)$ \FurukawaCommentFifth{and a sequence $a = \{ a_n \}_{n \in \Integer^d}$}, we set  the Fourier multiplier operator of discrete type by
		\begin{align}
			[T f] (x)
			\FurukawaCommentFifth{
			:= \FourierInverse_{d} a \Fourier_d f
			}
			= \sum_{n \in \mathbb{Z}^d} a_n  \hat{f}_n e^{i n \cdot x}.
		\end{align}
		Then there exits a constant $C = C(p,d) > 0$ such that
		}
		\begin{align}
			\Vert
				T f
			\Vert_{L^{\FurukawaCommentFifth{q}} (\Torus^d)}
			\leq C \max ( [m]_{\PGComment{\mathcal{M}}}, a_0) \Vert
				f
			\Vert_{L^{\FurukawaCommentFifth{q}} (\Torus^d)}.
		\end{align}
	\end{proposition}

	Let us consider the resolvent problem to (\ref{eq_LE}) ;
		\begin{equation} \label{eq_RLE}
			\left \{
				\begin{array}{rrl}
					\lambda u - \Delta u + \ep{\nabla} \pi
					& = f
					&\In \Omega, \\
					\Dive u
					&= 0
					&\In \Omega , \\
					u
					&= 0
					&\ON \partial \Omega,
				\end{array}
			\right.
		\end{equation}
	for $\lambda \in \Sigma_\theta$ ($0 < \theta < \pi / 2$) and $f \in \Lebesgue{q} \bracket{\Omega}$.
	Let
	\begin{align*}
		H_\epsilon : L^{\FurukawaCommentFifth{q}}  (\Omega) \rightarrow L^{\FurukawaCommentFifth{q}}_{\sigma, \epsilon} (\Omega)
		= \Set{ u \in L^{\FurukawaCommentFifth{q}} (\Omega)}{ \mathrm{div}_\epsilon u = 0, u|_{x_3 = \pm 1 } = 0},
		\, (1 < p <\infty)
	\end{align*}
	 be the anisotropic Helmholtz projection on $\Omega$, its $L^{\FurukawaCommentFifth{q}}$-boundedness is proved later.
	Let $A_\epsilon = H_\epsilon (- \Delta)$ be the Stokes operator with the domain $D (A_\epsilon) = L^{\FurukawaCommentFifth{q}}_{\sigma, \epsilon} (\Omega) \cap W^{2, \FurukawaCommentFifth{q}} (\Omega)$.
	 \FurukawaCommentFourth{For $0 < a < 1 / 2$ and $ - a < \RealPart \, z < 0$, the fractional power of $A_{\epsilon}$ is defined via the Dunford calculus}
		\begin{align*}
			\ep{A}^z
			 = \frac{1}{ 2 \pi i } \TimeInt{\Gamma_{\theta}}{}{ \bracket{ - \lambda }^z \Inverse{ \lambda + \ep{A} } }{\lambda},
		\end{align*}
	where $ 0 < \theta < \pi /2 $ and $\Gamma_{\epsilon} =  \Real \Napier^{ i \bracket{ - \pi + \theta } } \cup \Real \Napier^{ i \bracket{ \pi - \theta } } $.
	Our aim in this section is to prove
		\begin{lemma} \label{lem:C02}
			Let $1 < q < \infty$, $\FurukawaCommentFifth{0 < \epsilon \leq 1}$, $ 0 < a < 1/2$, $z \in \Complex$ satisfying $ - a < \RealPart \, z < 0 $ and $ 0 < \theta < \pi /2 $.
			Then there  \FurukawaCommentFourth{exists} a constant $C = C \bracket{ q, a, \theta }$ such that
				\begin{align}
					\LongNorm{ \ep{A}^z }{ \Lebesgue{q} \bracket{\Omega} \rightarrow \Lebesgue{q} \bracket{\Omega} }
					\leq C \Napier^{ \theta \abs{ \ImaginaryPart z} }.
				\end{align}
		\end{lemma}
	\PGComment{Once} the above lemma is proved, then we obtain \FurukawaCommentFourth{the maximal regularity} of the anisotropic Stokes operator via the formula
		\begin{align}
			\left(
				\frac{
					d}{
					d t
					}
				+ A_\epsilon
			\right)^{-1}
			= \int_{c + i \infty}^{ c - i \infty}
				\frac{
					(d/dt)^z A_\epsilon^{1 - z}
				}{
					\sin \pi z
				}
			d z
		\end{align}
	for $0 < c < 1$ and the Dore-Venni theory \cite{DoreVenni1990}.

	To show Lemma \ref{lem:C02}, we decompose the solution $(u, \pi)$ \FurukawaCommentFifth{to} (\ref{eq_RLE}) into three parts;
		\begin{align} \label{def_v_j_pi_j}
			u
			& = R_0 v_1 - v_2 + \ep{\nabla} \pi_3, \\
			\ep{\nabla} \pi
			& = \ep{\nabla} \pi_1 + \ep{\nabla} \pi_2,
		\end{align}
	where $v_j$ and $\pi_j$ are solutions to
		\begin{align*}
			(I) \left \{
				\begin{array}{rcll}
					\lambda v_1 - \Delta v_1 + \ep{\nabla} \pi_1
					&=
					& E_0 f
					& \In \Torus^2 \times \Real,\\
					\Dive v_1
					&=
					& 0
					& \In \Torus^2 \times \Real,
				\end{array}
			\right.
		\end{align*}

		\begin{align*}
			(II)	\left \{
				\begin{array}{rcll}
					\lambda v_2 - \Delta v_2 + \ep{\nabla} \pi_2
					&=
					& 0
					& \In \Omega,\\
					\Dive v_2
					&=
					& 0
					& \In \Omega, \\
					v_2
					& =
					& \gamma v_1 - \bracket{ \gamma v_1 \cdot \nu } \nu
					& \On \partial \Omega,
				\end{array}
			\right.
		\end{align*}
	and
		\begin{align*}
			(III) \left \{
				\begin{array}{rcll}
					\ep{\Delta} \pi_3
					& =
					& 0
					& \In \Omega,\\
					\ep{\nabla} \pi_3 \cdot \nu
					& =
					& \bracket{ \gamma v_1 \cdot \nu } \nu
					& \On \partial \Omega,
				\end{array}
			\right.
		\end{align*}
	respectively, where $\gamma = \gamma_{\pm}$ is the trace operator to the upper and lower boundary, respectively, and $\nu$ is the unit outer normal. To show Lemma \ref{lem:C02}, we need to obtain
		\begin{align*}
			&\LongNorm{ \frac{1}{2 \pi i}
				\TimeInt{\Gamma}{}{
					\bracket{- \lambda}^z R_0 v_1
				}{\lambda}
			}{\Lebesgue{q} \bracket{\Omega}}
			+
			\LongNorm{  \frac{1}{2 \pi i}
				\TimeInt{\Gamma}{}{
					\bracket{- \lambda}^z v_2
				}{\lambda}
			}{\Lebesgue{q} \bracket{\Omega}} \\
			& +
			\LongNorm{ \frac{1}{2 \pi i}
				\TimeInt{\Gamma}{}{
					\bracket{- \lambda}^z \ep{\nabla} \pi_3
				}{\lambda}
			}{\Lebesgue{q} \bracket{\Omega}}
			 \leq C \Napier^{ \theta \abs{ \ImaginaryPart \, z } }
			\LongNorm{f}{\Lebesgue{q} \bracket{\Omega} }
		\end{align*}
	  \FurukawaCommentFourth{for some Constant $C >0$, which is independent of $\epsilon$.}

	  \FurukawaCommentFifth{
	  \begin{remark} \label{rem_average_free}
		For $f \in L^q (\Omega)$ and $\tilde{f} := \int_{\Torus^2} f d x^\prime$, we can solve the resolvent problem (\ref{eq_RLE}) with external force $\tilde{f}$ to get $\FurukawaCommentFifth{\Prime{u}} = \left( (\lambda - \partial_3^2)^{-1} \tilde{f}_H , 0 \right)^T$ and $\FurukawaCommentFifth{\Prime{\pi}} = \epsilon \int_{-1}^{x_3} \tilde{f}_3 d \zeta / \mathbb{R}$, where $\tilde{f}_H$ is \FurukawaCommentFourth{the} horizontal component of $\tilde{f}$ and $ / \mathbb{R}$ means average-free.
		Since $- \partial_3^2$ has BIP and the resolvent operator is linear, by taking the difference between the solution to (\ref{eq_RLE}) and \FurukawaCommentFifth{$(\Prime{u}, \Prime{\pi})$}, we can always assume without loss of generality that $f$ is \FurukawaCommentFifth{horizontal} average-free.
	\end{remark}
	}
	\FurukawaCommentFifth{We define the space of horizontally average-free $L^q$-vector fields by
	\begin{align*}
		L^q_{\FurukawaCommentSixth{\mathrm{af}}} (\Omega)
		: = \LongSet{
			f \in L^q (\Omega)
		}{
		\tilde{f} = 0}.
	\end{align*}
	Similarly we define
	\begin{align*}
		W^{s, q}_{\FurukawaCommentSixth{\mathrm{af}}} (\Omega)
		: = \LongSet{
			f \in W^{s, q} (\Omega)
		}{
		\tilde{f} = 0}.
	\end{align*}
	}
	Throughout this section we frequently use partial Fourier transform to construct solutions and estimate these partial Fourier multipliers.

	\FurukawaComment{
		\begin{proposition}[\cite{Abels2002}] \label{prop:C03}
			Let $1 < q < \infty$ and $a, \, b \in \{-1 , 1 \}$.
			Set a integral operator $M$ by
				\begin{align*}
					M f (x^\prime , x_3)
					=  \TimeInt{-1}{1}{
						\frac{
							f (x^\prime , \zeta)
						}{
							\abs{x_3 - a}
							+ \abs{\zeta - b}
						}
					}{\zeta}
				\end{align*}
			\FurukawaCommentSecond{for $f \in L^q (\Omega)$.}
			Then there exists a constant $C>0$ such that
				\begin{align*}
					\LongNorm{ M f }{\Lebesgue{q}}
					\leq C \LongNorm{f}{\Lebesgue{q}}.
				\end{align*}

		\end{proposition}
		\begin{proof}
			See Lemma 3.3 in \cite{Abels2002}.
		\end{proof}
	}

	Rescaled $L^{\FurukawaCommentFifth{q}}$-Fourier multipliers are also bounded $L^{\FurukawaCommentFifth{q}}$ multiplier by the direct consequence of the Mikhlin theorem.
	\begin{proposition} \label{prop_scaled_multiplier}
		\FurukawaCommentFifth{Let $1 < q  < \infty$ and $0 < \epsilon \leq 1$.}
		Let $m \in \FurukawaCommentSeventh{C^d} (\Real^d \setminus \{ 0 \})$ be a $L^{\FurukawaCommentFifth{q}}$-Fourier multiplier with the Mikhlin constant $[m]_\mathcal{M} \leq C$ for some $C>0$.
		Then rescaled one $m_\epsilon (\xi) := m (\epsilon \xi)$ is also bounded from $L^{\FurukawaCommentFifth{q}}$ into itself such that
		\begin{align*}
			[ m_\epsilon ]_{\PGComment{\mathcal{M}}} \leq C.
		\end{align*}
	\end{proposition}

	The above proposition is frequently used in this section to get $\epsilon$-independent estimate for scaled multipliers.
	We show boundedness of some Fourier multiplier operators in advance.
	We set
		\begin{align*}
			s_{\lambda}
			= \bracket{ \lambda + \abs{\xi^{\prime}}^2 }^{1/2}
		\end{align*}
	for $\Prime{\xi} \in \mathbb{R}^2$.
	\FurukawaCommentFourth{In this paper we use $s_\lambda$ to denote $(\lambda + \abs{n^\prime}^2)^{1/2}$ for $n^\prime \in \mathbb{Z}^2$ to simplify notation.}

		\begin{proposition} \label{prop:C05}
			\hspace{1mm}
			\begin{itemize}

				\item Let $0 < \theta < \pi /2$, $\lambda \in \Sigma_{\theta}$, $t > 0$ and $\alpha$ be a positive integer.
				Then there exist constants $c > 0$ and $C>0$ such that
 					\begin{align} \label{eq:C0501}
 						\sLongBracket{
						\abs{\xi^{\prime}}^{\alpha} \Napier^{- t s_{\lambda} }
					}_{
						\mathcal{M}^{\prime}
					}
					\leq C \frac{
						\Napier^{{ - c t \abs{ \lambda } }^{1/2} }
					}{
						t^{\alpha}
					}, \quad
						\sLongBracket{ \frac{ \Napier^{ - s_{\lambda} } }{ s_{\lambda} } }_{\mathcal{M}^{\prime}}
						\leq C \abs{ \lambda }^{ - 1/2 } \Napier^{ - c \abs{\lambda}^{1/2} }.
					\end{align}

				\item  Let $ -1 \leq x_3 \leq 1 $.
				Then there exists a constant $C>0$ which is independent of $\epsilon$, such that
					\begin{align} \label{eq:C0504}
						\sLongBracket{
							\frac{ \sinh \bracket{ \epsilon \abs{\xi^{\prime}} x_3 } }{ \sinh \bracket{ \epsilon \abs{\xi^{\prime}} } }
							\frac{ \epsilon \abs{\xi^{\prime}} }{ 1 + \epsilon  \abs{\xi^{\prime}} }
						}_{\mathcal{M}^{\prime}} \leq C, \quad
						\sLongBracket{
							\frac{ \cosh \bracket{ \epsilon \abs{\xi^{\prime}} x_3 } }{ \sinh \bracket{ \epsilon \abs{\xi^{\prime}} } }
							\frac{ \epsilon  \abs{\xi^{\prime}} }{ 1 + \epsilon  \abs{\xi^{\prime}} } }_{\mathcal{M}^{\prime}}
							\leq C
					\end{align}
				for all $0 < \epsilon \leq 1$.

				\item  Let $-1 \leq x_3 \leq 1$.
				Then there exists a constant $C>0$, which is independent of $\epsilon$, such that
					\begin{align} \label{eq:C0505}
						\sLongBracket{ \frac{ \sinh \bracket{ \epsilon \abs{\xi^{\prime}} x_3 } }{ \cosh \bracket{ \epsilon \abs{\xi^{\prime}} } } }_{\mathcal{M}^{\prime}} \leq C, \quad
						\sLongBracket{ \frac{ \sinh \bracket{ \epsilon \abs{\xi^{\prime}} x_3 } }{ \cosh \bracket{ \epsilon \abs{\xi^{\prime}} } } }_{\mathcal{M}^{\prime}}
						\leq C
					\end{align}
				for all $0 < \epsilon \leq 1$.
			\end{itemize}
		\end{proposition}

		\begin{proof}
			\FurukawaCommentFourth{The estimate} (\ref{eq:C0501}) is a direct consequence of Lemma 3.5 in \cite{Abels2002} and the Mikhlin theorem.

			By definition of $\sinh$ and $\cosh$, we find the formula
				\begin{align} \label{eq:C0526}
					\frac{
						\sinh \bracket{ \epsilon \abs{\Prime{\xi}} x_3 }
					}{
						\sinh \bracket{ \epsilon \abs{ \Prime{\xi} } }
					}
					= \frac{
						\Napier^{ \epsilon \abs{\Prime{\xi}} x_3 }
						- \Napier^{ - \epsilon \abs{\Prime{\xi}} x_3 }
					}{
						\Napier^{ \epsilon \abs{\Prime{\xi}} }
						- \Napier^{ - \epsilon \abs{\Prime{\xi}} }
					}
					= \frac{
						\Napier^{ - \epsilon \abs{\Prime{\xi}} \bracket{ x_3 - 1 } }
					}{
						1
						- \Napier^{ - 2 \epsilon \abs{\Prime{\xi}} }
					}
					- \frac{
						\Napier^{
							- \epsilon \abs{\Prime{\xi}} \bracket{
								x_3 + 1
							}
						}
					}{
						1
						- \Napier^{
							- 2 \epsilon \abs{\Prime{\xi}}
						}
					}
				\end{align}
			and
			\begin{align} \label{eq:C0526_2}
				\FurukawaComment{
				\frac{
						\cosh \bracket{ \epsilon \abs{\Prime{\xi}} x_3 }
					}{
						\sinh \bracket{ \epsilon \abs{ \Prime{\xi} } }
					}
				}
				= \frac{
						\Napier^{ - \epsilon \abs{\Prime{\xi}} \bracket{ x_3 - 1 } }
					}{
						1
						- \Napier^{ - 2 \epsilon \abs{\Prime{\xi}} }
					}
					+ \frac{
						\Napier^{
							- \epsilon \abs{\Prime{\xi}} \bracket{
								x_3 + 1
							}
						}
					}{
						1
						- \Napier^{
							- 2 \epsilon \abs{\Prime{\xi}}
						}
					} \FurukawaCommentFourth{.}
				\end{align}
			Thus, multiplying $ \frac{ \epsilon \abs{\Prime{\xi}} }{ 1 + \epsilon \abs{\Prime{\xi}} }$ by both sides of (\ref{eq:C0526}), \FurukawaCommentSeventh{we find from Proposition \ref{prop_scaled_multiplier} that}
				\begin{align}
					&\sLongBracket{
							\frac{
								\sinh \bracket{ \epsilon \abs{\Prime{\xi}} x_3 }
							}{
								\sinh \bracket{ \epsilon \abs{ \Prime{\xi} } }
							}
							\frac{
								\epsilon  \abs{\Prime{\xi}}
							}{
								1
								+ \epsilon \abs{\Prime{\xi}}
							}
						}_{\Prime{\mathcal{M}}} \notag \\ \label{eq:C0527}
					& \leq C
					\sLongBracket{
						\Napier^{ - \epsilon \abs{\Prime{\xi}} \bracket{ x_3 - 1 } }
						\frac{
							\epsilon  \abs{\Prime{\xi}}
						}{
							\bracket{
								1
								- \Napier^{
									- 2 \epsilon \abs{\Prime{\xi}} }
							}
						} \frac{
							1
						}{
							\bracket{
								1 + \epsilon  \abs{\Prime{\xi}}
							}
						}
					}_{\mathcal{M}^{\prime}} \notag \\
					& + \sLongBracket{
						\Napier^{ - \epsilon \abs{\Prime{\xi}} \bracket{ x_3 + 1 } }
						\frac{
							\epsilon \abs{\Prime{\xi}}
						}{
							\bracket{ 1 - \Napier^{ - 2 \epsilon \abs{\Prime{\xi}} } }
						} \frac{
							1
						}{
							\bracket{ 1 + \epsilon  \abs{\Prime{\xi}} }
						}
					}_{\mathcal{M}^{\prime}} \notag \\
					& \leq C.
				\end{align}
			\FurukawaCommentSeventh{The second} inequality of (\ref{eq:C0504}) is proved by the same as above using (\ref{eq:C0526_2}).
			\FurukawaCommentSeventh{Similarly, by definition of $\sinh$ and $\cosh$, the estimate (\ref{eq:C0505}) follows.}
		\end{proof}

	\subsection{Estimate for $v_1$}

	Let us consider the equations (I).
	For \FurukawaCommentSeventh{$a \in \Real$} we denote by $\tau_a f = f (a \cdot)$ the rescaling operator by $a$.
	The anisotropic Helmholtz projection $\mathbb{\ep{P}}^{\Real^3}$ on $\Real^3$ with symbols
		\begin{align*}
			\mathcal{F} \mathbb{P}_{\epsilon}^{\Real^3}
			= I_3
			- \xi_{\epsilon} \otimes \xi_{\epsilon}, \quad
			\xi_{\epsilon}
			= \LongBracket{
				\xi_1, \xi_2, \frac{\xi_3}{\epsilon}
			} \in \Real^3,
		\end{align*}
	is bounded in $L^q(\Real^3)$ by boundedness of the Riesz operator and the formula
		\begin{align} \label{eq:0530}
			\FourierInverse_{\xi} m \bracket{ a \xi} \Fourier_x f
			= \tau_{a^{-1}}
				\sLongBracket{
					\FourierInverse_{\xi} m \bracket{ \xi } \Fourier_x \tau_a f
				}\FurukawaCommentFifth{.}
		\end{align}
		\FurukawaCommentFourth{Actually apply (\ref{eq:0530}) with respect to the third variable, then, \FurukawaCommentFifth{the symbol is no longer dependent on $\epsilon$}.
		Changing the variable with respect to and using boundedness of the Riesz operator, we find}
		\begin{align*}
			\Vert
				\mathbb{P}_\epsilon^{\Real^3} f
			\Vert_{L^{\FurukawaCommentFifth{q}} (\Real^3)}
			= \epsilon^{-1} \Vert
				\mathbb{P}_1 \sLongBracket{
					\tau_{1 / \epsilon}^3 f
				}
			\Vert_{L^{\FurukawaCommentFifth{q}} (\Real^3)}
			\leq C \Vert
				f
			\Vert_{L^{\FurukawaCommentFifth{q}} (\Real^3)},
		\end{align*}
		where $\tau^3_a$ is the rescaled operator with respect to the third variable for $a > 0$.
		We define the anisotropic Helmholtz projection \FurukawaComment{$\mathbb{P}_\epsilon^{\Torus^2 \times \Real}$ } on $\Torus^2 \times \Real$ with symbols \FurukawaCommentFourth{by}
		\begin{align*}
			\Fourier_{x_3} \Fourier_{d, \Prime{x} }\mathbb{P}_\epsilon^{\Torus^2 \times \Real}
			= I_3
				- \left(
					\begin{array}{c}
						n_1 \\
						n_2 \\
						\xi_3 / \epsilon
					\end{array}
				\right)
				\otimes \left(
					\begin{array}{c}
						n_1 \\
						n_2 \\
						\xi_3 / \epsilon
					\end{array}
				\right), \quad
				n_1, n_2 \in \mathbb{Z} , \, \, \xi_3 \in \Real.
		\end{align*}
		We find $\mathbb{P}_\epsilon^{\Torus^2 \times \Real}$ is bounded from $L^q (\Torus^2 \times \Real)$ into itself by boundedness of \FurukawaCommentThird{$\mathbb{P}^{\Real^3}_\epsilon$} and Proposition \ref{prop_Fourier_Multiplier_Theorem_Discrete} \FurukawaCommentSeventh{uniformly in $\epsilon \in (-1, 1)$}.

		\begin{proposition} \label{prop:C0510}
			Let $ 1 < q < \infty $, $0 < a < 1/2$, \FurukawaComment{$0<\epsilon \leq 1$}, $z \in \Complex$ satisfying $ - a < \RealPart z < 0$ and $0 < \theta < \pi / 2$. Then there exists a constant $C = C \bracket{ q, a, \theta }$ such that
				\begin{align}
					& \LongNorm{
						\frac{1}{ 2 \pi i } R_0 \TimeInt{
							\Gamma_\theta
						}{}{
							\bracket{- \lambda}^z \Inverse{ \lambda - \Delta_{\Torus^2 \times \Real} } \ep{ \mathbb{P} }^{\Torus^2 \times \Real}  E_0 f
						}{
							\lambda
						}
					}{\Lebesgue{q} \bracket{\Torus^2 \times \Real} } \notag \\ \label{eq:C0560}
					& \leq C \Napier^{\theta \abs{\ImaginaryPart \, z}} \LongNorm{
						f
					}{
						\Lebesgue{q}\bracket{\Omega}
					}
				\end{align}
			 for all $ f \in \Lebesgue{q} \bracket{ \Omega } $.
		\end{proposition}
		\begin{proof}
			It is known that the Laplace operator on a cylinder $\Torus^2 \times \Real$ has BIP.
			Combining this fact and $L^q$-boundedness of $\mathbb{P}_\epsilon^{\Torus^2 \times \Real}$, we have (\ref{eq:C0560}).
		\end{proof}

		\FurukawaCommentFifth{
		\begin{proposition} \label{prop_nabla2_resolvent_estimate}
			Let $1< q < \infty$, $0 < \epsilon \leq 1$, $0 < \theta < \pi /2$ and $\lambda \in \Sigma_\theta$.
			Then there exists a constant $C = C(q) > 0$, which is independent of $\epsilon$, \FurukawaCommentSeventh{such that}
			\begin{align*}
				\left \Vert
					\nabla^2 \left(
						\lambda - \Delta_{\Torus^2 \times \Real}
					\right)^{-1} \mathbb{P}_\epsilon^{\Torus^2 \times \Real} E_0 f
				\right \Vert_{L^q (\Omega)}
				\leq C \Vert
					f
				\Vert_{L^q (\Omega)}.
			\end{align*}
		\end{proposition}
		\begin{proof}
			\FurukawaCommentSeventh{This follows from Propositions \ref{prop_Fourier_Multiplier_Theorem_Discrete} and \ref{prop_scaled_multiplier} since $P_\epsilon^{\Torus^2 \times \Real}$ is uniformly bounded from $L^q (\Torus^2 \times \Real)$ into itself.}
		\end{proof}	}

	Let us calculate the partial Fourier transform for $v_1$ with respect to the horizontal variable.
	\FurukawaComment{This} is needed to obtain representation formula for $v_2$ later.

	Let $g \in L^q ( \FurukawaCommentFifth{\Torus^2 \times \Real})$.
	The solution $\FurukawaComment{\tilde{v}}$ to the equation
		\begin{align*}
			\left \{
				\begin{array}{rcll}
					\lambda \tilde{v} - \Delta \tilde{v} + \ep{\nabla} \tilde{\pi}
					&=
					& g
					& \In \FurukawaCommentFifth{\Torus^2 \times \Real},\\
					\Dive \tilde{v}
					&=
					& 0
					& \In \FurukawaCommentFifth{\Torus^2 \times \Real},
				\end{array}
			\right.
		\end{align*}
	is given by
		\begin{align*}
			\tilde{v}
			= \LongBracket{
				\lambda - \Delta_{\Real^3}
			}^{-1} \mathbb{P}_\epsilon^{\FurukawaCommentFifth{\Torus^2 \times \Real}} \FurukawaComment{g}.
		\end{align*}
	Moreover,
		\begin{align}
			K_{\lambda, \epsilon} g \notag
			& := \LongBracket{
				\lambda - \Delta_{\FurukawaCommentFifth{\Torus^2 \times \Real}}
				}^{-1} \mathbb{P}_\epsilon^{\FurukawaCommentFifth{\Torus^2 \times \Real}}  g \notag \\
			& = \mathcal{F}^{-1}
				\bracket{
					\lambda
					+ \FurukawaCommentFifth{ \abs{n^\prime}^2 + \xi_3^2}
				}^{- 1} \LongBracket{
					I_3 - \frac{\ep{\xi} \otimes \ep{\xi} }{ \abs{ \ep{\xi} }^2 }
				}
			\mathcal{F}  g \notag \\  \label{eq:C0550_discrete}
			& = \mathcal{F}^{- 1}_{\FurukawaCommentFifth{n^\prime}}  \TimeInt{\Real}{}{
					k_{\lambda, \epsilon} \bracket{
						\FurukawaCommentFifth{n^\prime}, x_3 - \zeta
					}
				\mathcal{F}_{\FurukawaCommentFifth{n^\prime}} g \bracket{\FurukawaCommentFifth{n^\prime}, \zeta }
			}{\zeta},
		\end{align}
	where \FurukawaCommentFifth{$\xi_\epsilon = (n^\prime, \xi_3 / \epsilon) \in \Integer^2 \times \Real$ and}
		\begin{align}
			& \Prime{k}_{\lambda, \epsilon} \bracket{\FurukawaCommentFifth{n^\prime}, x_3 } \notag \\
			& = \FourierInverse_{\xi_3} \sLongBracket{
				\bracket{
					\lambda
					+ \FurukawaCommentFifth{\abs{n^\prime}^2 + \xi_3^2}
				}^{- 1} \LongBracket{
						I_3
						- \frac{\ep{\xi} \otimes \ep{\xi} }{ \abs{ \ep{\xi} }^2 }
				}
			} \notag \\
			& =
			\frac{ \Napier^{ - s_{\lambda} } }{ 2 s_{\lambda} }
				\left(
					\begin{array}{cc}
						I & 0 \\
						0 & 0
					\end{array}
				\right) \notag \\
			&	-
				\left(
					\begin{array}{c}
						\FurukawaCommentFifth{n^\prime} \otimes \FurukawaCommentFifth{n^\prime} \frac{
							\epsilon^2}{
							\lambda
							+ \bracket{ 1 - \epsilon^2 } \abs{ \FurukawaCommentFifth{n^\prime} }^2
							}  \frac{
								- \epsilon \abs{\FurukawaCommentFifth{n^\prime}} \Napier^{ - \abs{x_3} s_{\lambda} }
								+ s_{\lambda} \Napier^{ - \abs{x_3} \epsilon \abs{ \FurukawaCommentFifth{n^\prime} } }
							}{
								2 s_{\lambda} \epsilon \abs{\FurukawaCommentFifth{n^\prime}}
							} \\
						- i { \FurukawaCommentFifth{n^\prime} }^T \frac{
							\epsilon^2
						 }{
						 	\lambda
						 	+ \bracket{ 1 - \epsilon^2 } \abs{ \FurukawaCommentFifth{n^\prime} }^2
						 } \frac{
						 	\Napier^{- \abs{x_3} s_{\lambda}} - \Napier^{ - \abs{x_3} \epsilon \abs{\FurukawaCommentFifth{n^\prime}} }
						 }{
						 	2
						 }
					\end{array}
				\right. \notag \\
			& \quad \quad \quad \quad \quad \quad \quad \left.
					\begin{array}{c}
						- i \FurukawaCommentFifth{n^\prime}  \frac{
							\epsilon^2
						}{
							\lambda
							+ \bracket{ 1 - \epsilon^2 } \abs{ \FurukawaCommentFifth{n^\prime} }^2
						} \frac{
							\Napier^{- \abs{x_3} s_{\lambda}}
							- \Napier^{
								- \abs{x_3} \epsilon \abs{\FurukawaCommentFifth{n^\prime}}
							}
						}{
							2
						} \notag \\
						- \abs{ \FurukawaCommentFifth{n^\prime} }^2 \frac{
						 	\epsilon^2
						 }{
						 	\lambda
						 	+ \bracket{ 1 - \epsilon^2 } \abs{ \FurukawaCommentFifth{n^\prime} }^2
						 } \frac{
						 	- \epsilon \abs{\FurukawaCommentFifth{n^\prime}} \Napier^{ - \abs{x_3} s_{\lambda} } + s_{\lambda} \Napier^{ - \abs{x_3} \epsilon \abs{ \FurukawaCommentFifth{n^\prime} } }
						 }{
							2 s_{\lambda} \epsilon \abs{\FurukawaCommentFifth{n^\prime}}
						 }
					\end{array}
				\right) \notag \\
			& = : \frac{
				\Napier^{ - s_{\lambda} }
			}{
				2 s_{\lambda}
			}
				\left(
					\begin{array}{cc}
						I_2 & 0 \\
						0 & 0
					\end{array}
				\right)
				-
				\left(
					\begin{array}{cc}
						\FurukawaCommentFifth{n^\prime} \otimes \FurukawaCommentFifth{n^\prime} \eta_{\lambda, \epsilon}^{\prime} \bracket{ \FurukawaCommentFifth{n^\prime}, x_3 }
						& - i \FurukawaCommentFifth{n^\prime} \partial_3 \eta_{\lambda, \epsilon}^{\prime} \bracket{ \FurukawaCommentFifth{n^\prime}, x_3 } \\ \label{eq:C0555}
						- i { \FurukawaCommentFifth{n^\prime} }^T \partial_3 \eta_{\lambda, \epsilon}^{\prime} \bracket{ \FurukawaCommentFifth{n^\prime}, x_3 }
						& - \abs{\xi^{\prime}}^2 \eta_{\lambda, \epsilon}^{\prime} \bracket{\FurukawaCommentFifth{n^\prime}, x_3 }
					\end{array}
				\right).
		\end{align}
	\FurukawaCommentFifth{The kernel function} $k_{\lambda, \epsilon} \bracket{\FurukawaCommentFifth{n^\prime}, x_3}$ is calculated by the residue theorem.
	Actually, since poles of $\Inverse{ \lambda + \FurukawaCommentFifth{\abs{n^\prime}^2 + \xi_3^2} }$ are $\xi_3 = \pm i s_{\lambda} $, the residue theorem implies the partial Fourier inverse transform of $ \Inverse{ \lambda + \FurukawaCommentFifth{\abs{n^\prime}^2 + \xi_3^2}  } $ with respect to $\xi_3$ is given by inserting $\xi_3 = i s_{\lambda} \, \mathrm{or} \, - i s_{\lambda}$ into $\Napier^{ i x_3 \xi_3}$ so that the real part become to be negative.
	Thus, we have
		\begin{align} \label{eq_definition_of_e_prime_lamda}
			\Prime{e}_\lambda (\FurukawaCommentSixth{n^\prime}, x_3)
			: = \FourierInverse_{\xi_3} \LongBracket{
				\lambda + \FurukawaCommentSixth{\abs{n^\prime}^2 + \abs{x_3}^2}
			}^{-1}
			= \frac{
				e^{- \abs{x_3} s_\lambda}}{
				s_\lambda
			}.
		\end{align}
	Moreover, this formula leads \FurukawaCommentSixth{to}
		\begin{align*}
			\FourierInverse_{\xi_3} \sLongBracket{
				\abs{\xi_\epsilon}^2
			}^{-1}
			= \FourierInverse_{\xi_3} \sLongBracket{
				\frac{
					\epsilon^2
				}{
					\epsilon^2 \abs{\FurukawaCommentFifth{n^\prime}}^2 + \xi_3^2
				}
			}
			= \frac{
				\epsilon e^{- \abs{x_3} \epsilon \abs{\FurukawaCommentFifth{n^\prime}} }
			}{
				\abs{\FurukawaCommentFifth{n^\prime}}
			}.
		\end{align*}
	Combining the above two calculations and the formula
		\begin{align*}
			I_3 - \frac{ \xi_\epsilon \otimes \xi_\epsilon }{ \abs{\xi_\epsilon}^2 }
			=	\left(
					\begin{array}{cc}
						I_2 & 0 \\
						0   & 0
					\end{array}
				\right)
			-
				\left(
					\begin{array}{cc}
						\frac{
							\FurukawaCommentFifth{n^\prime} \otimes \FurukawaCommentFifth{n^\prime}
						}{
							\abs{\FurukawaCommentFifth{\xi_\epsilon}}^2
						}
						& \frac{
							\xi_3 \FurukawaCommentFifth{n^\prime} / \epsilon
						}{
								\abs{\FurukawaCommentFifth{\xi_\epsilon}}^2
						} \\
						\frac{
							\xi_3 \FurukawaCommentFifth{n^\prime}^T / \epsilon
						}{
							\abs{\FurukawaCommentFifth{\xi_\epsilon}
						}^2
						}
						& - \frac{
							\abs{ \FurukawaCommentFifth{n^\prime}_\epsilon }^2
						}{
							\abs{\FurukawaCommentFifth{\xi_\epsilon}}^2
						}
					\end{array}
				\right)	,
		\end{align*}
	\FurukawaCommentFifth{we obtain (\ref{eq:C0555})}.
	\begin{comment}
	Thus, the solution $v_1$ to (I) with external force $g \in L^q (\Torus^2 \times \Real)$ is given by
		\begin{align} \label{eq:C0550_discrete}
			v_1
			= K_{\lambda, \epsilon} g
			:= \mathcal{F}^{- 1}_{ d, n^{\prime} }  \TimeInt{\Real}{}{
					k_{\lambda, \epsilon} \bracket{
						\FurukawaCommentFifth{n^\prime}, x_3 - \zeta
					}
				\mathcal{F}_{d, x^{\prime}} g \bracket{n^{\prime}, \zeta }
			}{\zeta} \FurukawaCommentFourth{.}
		\end{align}
		\end{comment}

	\subsection{Boundedness of the anisotropic Helmholtz projection}

	Next, we consider the equation (III) with boundary data $\phi = \bracket{\phi_+, \phi_-} \FurukawaCommentFifth{\in C_0^\infty (\Omega)}$.
	Applying the partial Fourier transform to (III), we have

		\begin{align} \label{eq:C0701}
			\left \{
				\begin{array}{rcl}
				\LongBracket{
						\frac{
							\partial_z^2
						}{
						\epsilon^2
						}
					- \abs{ n^\prime }^2 } \mathcal{F}_{d, x^{\prime} } \pi_3 (\Prime{n}, x_3)
					& =
					& 0 , \\
					\frac{\partial_z }{ \epsilon} \mathcal{F}_{d, x^{\prime} } \pi_3 \bracket{ n^\prime, \pm 1}
					& =
					& \Fourier_{d, \Prime{x}} {\phi}_{\pm} \bracket{\Prime{n}} \FurukawaCommentFourth{,}
				\end{array}
			\right .
		\end{align}

	for $n^\prime \in \mathbb{Z}^2 \FurukawaComment{ \setminus \{ 0 \}}$ and $x_3 \in (-1 , 1)$.
	The solution to (\ref{eq:C0701}) is of the form
		\begin{align*}
			\Fourier_{d, \Prime{x}} \pi_3  (n^\prime, x_3)
			= C_1 \Napier^{\epsilon x_3 \abs{\Prime{n}} }
			+ C_2 \Napier^{- \epsilon x_3 \abs{\Prime{n}} }
		\end{align*}
	for some constant $C_1$ and $C_2$.
	Take the constants so that (\ref{eq:C0701}) satisfied, namely
		\begin{align*}
			& C_1
			= \frac{
					\Fourier_{d, \Prime{x}} \phi_+
					+ \Fourier_{d, \Prime{x}} \phi_-
				}{
					4 \abs{\Prime{n}} \cosh \bracket{ \epsilon \abs{\Prime{n}} }
				}
			+ \frac{
					\Fourier_{d, \Prime{x}} \phi_+
					- \Fourier_{d, \Prime{x}} \phi_-
				}{
					4 \abs{\Prime{n}} \sinh \bracket{ \epsilon \abs{\Prime{n}} }
				}, \quad \\
			& C_2
			= -
				\frac{
					\Fourier_{d, \Prime{x}} \phi_+
					+ \Fourier_{d, \Prime{x}} \phi_-
				}{
					4 \abs{\Prime{\xi}} \cosh \bracket{ \epsilon \abs{\Prime{n}} }
				}
			+ \frac{
					\Fourier_{d, \Prime{x}} \phi_+ - \Fourier_{d, \Prime{x}} \phi_-
				}{ 4 \abs{\Prime{\xi}} \sinh \bracket{ \epsilon \abs{\Prime{n}} }
				},
		\end{align*}
	then the solution to (\ref{eq:C0701}) is given by
		\begin{align*}
			& \pi_3 \bracket{ x^{\prime}, x_3 } \\
			 & = \mathcal{F}^{ - 1 }_{d,  n^{\prime} } \LongBracket{
			 	\frac{
			 		\sinh \bracket{ \epsilon x_3 \abs{ \Prime{n}}}
			 	}{
			 		\abs{\Prime{n} } \cosh \bracket{\epsilon \abs{ \Prime{n}} }
			 	} \frac{
			 		\Fourier_{d, \Prime{x}} {\phi}_{+}
			 		+ \Fourier_{\Prime{x}} {\phi}_{-}
			 	}{
			 		2
			 	}
			 + \frac{
			 	\cosh \bracket{ \epsilon x_3 \abs{ \Prime{n}}}
			 }{
				\abs{\Prime{n} } \sinh \bracket{\epsilon \abs{ \Prime{n}} }
			 } \frac{
			 	\Fourier_{d, \Prime{x}} {\phi}_{+}
			 	- \Fourier_{d, \Prime{x}} {\phi}_{-}
			 }{
			 	2
			 } }.
		\end{align*}
	Moreover, its anisotropic gradient given by
		\begin{align}
			\nabla_{\epsilon} \pi_3
			& = \mathcal{F}^{ - 1 }_{d,  n^{\prime} }
				\left(
					\begin{array}{c}
						\frac{
							i n^{\prime} \sinh \bracket{ \epsilon x_3 \abs{ n^{\prime} } }
						}{
							\abs{ n^{\prime} } \cosh \bracket{ \epsilon \abs{ n^{\prime} } }
						}
						\frac{
							\Fourier_{d, \Prime{x}} {\phi}_{+}
							+ \Fourier_{d, \Prime{x}} {\phi}_{-}
						}{2}
					+ \frac{
						i n^{\prime} \cosh \bracket{ \epsilon x_3 \abs{ n^{\prime} } }
					}{
						\abs{ n^{\prime} } \sinh \bracket{ \epsilon \abs{ n^{\prime} } }
					} \frac{
						\Fourier_{d, \Prime{x}} {\phi}_{+}
						- \Fourier_{d, \Prime{x}} {\phi}_{-}
						}{2} \\
					\frac{
						\cosh \bracket{ \epsilon x_3 \abs{ n^{\prime} } }
					}{
						\cosh \bracket{ \epsilon \abs{ n^{\prime} } }
					} \frac{
						\Fourier_{d, \Prime{x}} {\phi}_{+}
						+ \Fourier_{d, \Prime{x}} {\phi}_{-}
					}{2}
					+ \frac{
						\sinh \bracket{ \epsilon x_3 \abs{ n^{\prime} } }
					}{
						\sinh \bracket{ \epsilon \abs{ n^{\prime} } }
					} \frac{
						\Fourier_{d, \Prime{x}}{\phi}_{+}
						- \Fourier_{d, \Prime{x}} {\phi}_{-} }{2}
					\end{array}
				\right) \notag \\ \label{eq:C0712}
			& =: \mathcal{F}^{ - 1}_{d,  n^{\prime} }
				\alpha_{ \epsilon, +} \bracket{ n^{\prime}, x_3 }
			\mathcal{F}_{d,  x^{\prime} } \phi_{+}
			+ \mathcal{F}^{ - 1}_{d,  n^{\prime} }
				\alpha_{ \epsilon, - } \bracket{ n^{\prime}, x_3}
			\mathcal{F}_{d,  x^{\prime} } \phi_{-}.
		\end{align}
	We \FurukawaComment{apply the trace} to (\ref{eq:C0712}) to get
		\begin{align*}
			\gamma_{ \pm } \ep{\nabla} \pi_3
			&= \mathcal{F}^{ - 1 }_{d,  n^{\prime} }
			\left(
				\begin{array}{c}
					\frac{ \pm i n^{\prime} \sinh \bracket{ \epsilon \abs{ n^{\prime} } } }{ \abs{ n^{\prime} } \cosh \bracket{ \epsilon \abs{ n^{\prime} } } }
					\frac{
						\Fourier_{d, \Prime{x}} {\phi}_{+}
						+ \Fourier_{d, \Prime{x}} {\phi}_{-}
					}{2}
					+ \frac{
						i n^{\prime} \cosh \bracket{ \epsilon \abs{ n^{\prime} } }
					}{
						\abs{ n^{\prime} } \sinh \bracket{ \epsilon \abs{ n^{\prime} } }
					} \frac{
						\Fourier_{d, \Prime{x}} {\phi}_{+} -
						\Fourier_{d, \Prime{x}} {\phi}_{-} }{2} \\
					\frac{
						\Fourier_{d, \Prime{x}} {\phi}_{+}
						+ \Fourier_{d, \Prime{x}} {\phi}_{-}
					}{2}
					\pm \frac{
						\Fourier_{d, \Prime{x}} {\phi}_{+}
						- \Fourier_{d, \Prime{x}} {\phi}_{-} }{2}
				\end{array}
			\right).
		\end{align*}
	We insert \FurukawaCommentFifth{$\phi_{\pm} = \gamma_{\pm} \ep{\mathbb{P}}^{\Torus^2 \times \Real} f$ to (\ref{eq:C0712}) for $f \in C_0^\infty (\Omega)$} \FurukawaCommentSeventh{satisfying $\tilde{f}=0$} and set
		\begin{align*}
			\Pi_{\epsilon} f
			& := \mathcal{F}^{-1}_{d, n^{\prime}}
				\sLongBracket{
					\alpha_{ \epsilon, + } \bracket{ n^{\prime}, x_3 } \gamma_+ \mathcal{F}_{d, x^{\prime}} \LongBracket{
						\Napier_3 \cdot \mathbb{P}_{\epsilon}^{\Torus^2 \times \Real} E_0 f
					}
				} \\
			 & +  \mathcal{F}^{-1}_{d, n^{\prime}}
				\sLongBracket{
					\alpha_{ \epsilon, - } \bracket{ n^{\prime}, x_3 } \gamma_-  \mathcal{F}_{d, x^{\prime}} \LongBracket{
						\Napier_3 \cdot \mathbb{P}_{\epsilon}^{\Torus^2 \times \Real} E_0 f
					}
				}.
		\end{align*}

		\begin{lemma} \label{lem:C0710}
			 \FurukawaCommentFifth{Let $1 < q < \infty$, $0 < \epsilon \leq 1$ and $s \geq 0$.}
			 Then there \FurukawaComment{exists} a constant $C = C \bracket{q}$, \FurukawaCommentFifth{which is independent of $\epsilon$, the operator  $\Pi_\epsilon$ can be extended to a bounded operator from $W^{s,q}_{\FurukawaCommentSeventh{\mathrm{af}}} (\Omega)$ into itself} such that
				\begin{align} \label{eq_bound_Pi}
					\LongNorm{
						\Pi_\epsilon f
					}{\FurukawaCommentFifth{W^{s, q}} \bracket{\Omega} }
					\leq C \LongNorm{f}{\FurukawaCommentFifth{W^{s, q}} \bracket{\Omega}}
				\end{align}
			for all \FurukawaCommentFifth{$f \in W^{s, q}_{\FurukawaCommentSixth{\mathrm{af}}} \bracket{\Omega}$}.
		\end{lemma}

		\begin{proof}
			\FurukawaCommentSixth{Let $f \in C_0^\infty (\Omega)$ satisfy $\tilde{f} = 0$.}
			We seek the multiplier of $\Pi_\epsilon$ by a direct calculation.
			Recall that the symbol of $\ep{\mathbb{P}}^{\Torus^2 \times \Real}$ is of the form
				\begin{align} \label{eq:C0715}
					\Fourier_{x_3} \Fourier_{d, x^\prime} \ep{\mathbb{P}}^{\Torus^2 \times \Real}
					=
						\left(
							\begin{array}{cc}
								I_2 & 0 \\
								0 & 0 \\
							\end{array}
						\right)
						-
							\left(
							\begin{array}{cc}
								\frac{ \Prime{n} \otimes \Prime{n} }{ \abs{ \ep{ \Prime{n}}}^2 }
								& \frac{ \Prime{n} \xi_3 / \epsilon }{ \abs{ \ep{ \Prime{n}}}^2 } \\
								\frac{{\Prime{n}}^T \xi_3 / \epsilon }{ \epsilon \abs{ \ep{ \Prime{n}}}^2 }
								& - \frac{ \abs{ \Prime{n}}^2 }{ \abs{ \ep{ \Prime{n}}}^2 } \\
							\end{array}
						\right).
				\end{align}
			Since the symbol of $\ep{\mathbb{P}}^{\Torus^2 \times \Real} $ have poles at $\xi_3 = \pm i \epsilon \abs{\Prime{n}}$, we apply $e_3 \cdot$ to (\ref{eq:C0715}) by the left hand side and use the residue theorem so that the power of $e$ is negative to get
				\begin{align} \label{eq:C0717}
					\Fourier_{d, x^{\prime}}
						\bracket{ \Napier_3 \cdot \mathbb{P}_{\epsilon}^{\Torus^2 \times \Real}  E_0 f }
					& =
					- \TimeInt{-1}{1}{
							\frac{
								i \Napier^{ - \abs{ x_3 - \zeta} \epsilon \abs{ n^{\prime} } }
							}{2} \epsilon n^{\prime} \cdot
							\Fourier_{d, x^{\prime}} f^{\prime} \bracket{ n^{\prime}, \zeta }
						}{\zeta}  \notag \\
					& +
					\TimeInt{-1}{1}{
						\frac{
							\Napier^{ - \abs{ x_3 - \zeta} \epsilon \abs{ \Prime{n} } }
						}{2}  \epsilon \abs{ \Prime{n} } \Fourier_{d, x^{\prime}} f_3 \bracket{ n^{\prime}, \zeta}
					}{\zeta}.
				\end{align}
			Note that the integration is due to the relationship between the Fourier transform and convolution.
			Applying trace operators $\gamma_{\pm}$ and $\alpha_{\epsilon, \pm} \bracket{\Prime{n}, x_3} $, respectively, and taking Fourier inverse transform with respect to $\Prime{n}$, we find
				\begin{align}
					& \Pi_{\epsilon} f \bracket{ {\Prime{x}, x_3} } \notag \\
					& =
					- \FourierInverse_{d, \Prime{n}}
							\TimeInt{-1}{1}{
								\alpha_{\epsilon, +} \bracket{ n^{\prime}, x_3 } \frac{ i \Napier^{ - \abs{ 1 - \zeta} \epsilon \abs{ n^{\prime} } } }{2} \epsilon n^{\prime} \cdot \Fourier_{\FurukawaCommentFifth{d,} x^{\prime}} f^{\prime} \bracket{ n^{\prime}, \zeta }
							}{\zeta} \notag \\
					& + \FourierInverse_{d, \Prime{n}}
							\TimeInt{-1}{1}{
								\alpha_{\epsilon, +} \bracket{ n^{\prime}, x_3 }
								\frac{ \Napier^{ - \abs{ 1 - \zeta} \epsilon \abs{ \Prime{n} } } }{2} \epsilon \abs{ \Prime{n} } \Fourier_{d, x^{\prime}} f_3 \bracket{ n^{\prime}, \zeta}
							}{\zeta}
					 \notag \\
					&
					- \FourierInverse_{d, \Prime{n}}
							\TimeInt{-1}{1}{
								\alpha_{\epsilon, - } \bracket{ n^{\prime}, x_3 }
								\frac{ i \Napier^{ - \abs{ -1 - \zeta} \epsilon \abs{ n^{\prime} } } }{2}
								\epsilon n^{\prime} \cdot \Fourier_{d, x^{\prime}} f^{\prime} \bracket{ n^{\prime}, \zeta }
							}{\zeta} \notag \\
					& + \FourierInverse_{d, \Prime{n}}
							\TimeInt{-1}{1}{
								\alpha_{\epsilon, -} \bracket{ n^{\prime}, x_3 }
								\frac{ \Napier^{ - \abs{ - 1 - \zeta} \epsilon \abs{ \Prime{n} } } }{2}
								\epsilon \abs{ \Prime{n} } \Fourier_{d, x^{\prime}} f_3 \bracket{ n^{\prime}, \zeta}
							}{\zeta} \notag \\ \label{eq:0720}
					& = : I_1 + I_2 + I_3 + I_4\FurukawaCommentFifth{.}
				\end{align}
			\FurukawaCommentFifth{By} the definition of $\alpha_{\pm, \epsilon}$,
				\begin{align*}
					I_1
					& = - \FourierInverse_{d, \Prime{n}}
						\int_{-1}^{1}
							\frac{1}{2}
							\left(
								\begin{array}{c}
									\frac{
										i n^{\prime} \sinh \bracket{ \FurukawaCommentSeventh{\epsilon} x_3 \abs{ n^{\prime} } }
									}{
										\abs{ n^{\prime} } \cosh \bracket{ \epsilon \abs{ n^{\prime} } }
									}
									+ \frac{
										i n^{\prime} \cosh \bracket{ \epsilon x_3 \abs{ n^{\prime} } }
									}{
										\abs{ n^{\prime} } \sinh \bracket{ \epsilon  \abs{ n^{\prime} } }
									} \\
									\frac{
										\cosh (\epsilon x_3 \abs{\Prime{n}})
									}{
										\cosh (\epsilon \abs{\Prime{n}})
									}
									+	\frac{
										\sinh (\epsilon x_3 \abs{\Prime{n}})
									}{
										\sinh (\epsilon \abs{\Prime{n}})
									}
								\end{array}
							\right) \\
							& \quad \quad \quad \quad \quad \quad \quad \quad
							\times \frac{
									i \Napier^{ - \abs{ 1 - \zeta} \epsilon \abs{ n^{\prime} } }
								}{2} \epsilon n^{\prime} \cdot
					\Fourier_{d, x^{\prime}} f^{\prime} \bracket{ n^{\prime}, \zeta }
							d\zeta.
				\end{align*}
			Symbols in the integral can be written by $A \bracket{\epsilon \Prime{n}} \bracket{ 1 + \epsilon \abs{\Prime{n}} } \Napier^{- \epsilon \abs{\Prime{n}} \bracket{ \abs{x_3 \pm 1} + \abs{\zeta \pm 1} }} $ for a symbol $A$ with \FurukawaCommentFourth{an} $\epsilon$-independent Mikhlin constant by \FurukawaCommentSeventh{Proposition \ref{prop:C05}}.
			The same argument is valid for $I_j$ ($j = 2, 3, 4$).
			Thus, we find from \FurukawaCommentSeventh{Propositions \ref{prop:C03} and \ref{prop_scaled_multiplier}} that
				\begin{align*}
					\LongNorm{
						\ep{\Pi} f
					}{L^q \bracket{\Omega} }
					& \leq C \LongNorm{
						f
					}{L^{\FurukawaCommentFourth{q}} (\Omega)}
					+ C\LongNorm{ \TimeInt{-1}{1}{
						\frac{ \LongNorm{f \bracket{ \cdot, \zeta }}{\Lebesgue{q} \bracket{\Torus^2}} }{ \abs{ x_3 \pm 1 } + \abs{ \zeta \pm 1  } } }{\zeta}
					}{\Lebesgue{q} \bracket{-1, 1}} \notag \\
					& \leq C \LongNorm{f}{\Lebesgue{q} \bracket{\Omega} }
				\end{align*}
			for all \FurukawaCommentSixth{$f \in C_0^\infty (\Omega)$ satisfying $\tilde{f} = 0$}, where the constant $C$ is independent of $\epsilon$.
			\FurukawaCommentFifth{Thus the estimate (\ref{eq_bound_Pi}) holds for $m = 0$.
			We find from the formula (\ref{eq:0720}) that $\partial_j$ commutes with $\Pi_\epsilon$ for $j = 1, 2$.
			Moreover, the equation (\ref{eq:C0701}) implies
			\begin{align*}
				\partial_3^2 \, \Pi_\epsilon f
				& = - \epsilon^2  (\partial_1^2 + \partial_2^2) \, \Pi_\epsilon f \\
				& = - \epsilon^2   \Pi_\epsilon \, (\partial_1^2 + \partial_2^2) f
			\end{align*}
			Thus we find (\ref{eq_bound_Pi}) holds for all positive even number $m$.
			We can obtain (\ref{eq_bound_Pi}) for all \FurukawaCommentSixth{$s>0$} by interpolation.}
		\end{proof}

	 \FurukawaCommentFifth{ We set the operator
	\begin{align*}
		P_{N, \epsilon}
		:= R_0 \left( \mathbb{P}_{\epsilon}^{\Torus^2 \times \Real} - \Pi_{\epsilon} \right) f
	\end{align*}
	for all $f \in C_0^\infty (\Omega)$ \FurukawaCommentSeventh{satisfying $\tilde{f}=0$}.}
	Then \FurukawaCommentThird{Lemma} \ref{lem:C0710} implies
	\begin{corollary} \label{cor_bound_P_N_epsilon}
		 Let $1 < q < \infty$, \FurukawaCommentFifth{$0 < \epsilon \leq 1$ and $s > 0$.}
		 Then there \FurukawaCommentFourth{exists} a constant $C > 0$, which is independent of $\epsilon$, \FurukawaCommentFifth{the operator  $P_{N, \epsilon}$ can be extended to a bounded operator from $W^{s,q}_{\FurukawaCommentSeventh{\mathrm{af}}} (\Omega)$ into itself} such that \FurukawaCommentFifth{
		 	\begin{align*}
		 		\LongNorm{
		 			P_{N, \epsilon} f
		 		}{
		 			W^{s, q} (\Omega)
		 		}
		 		\leq C \LongNorm{
		 			f
		 		}{
		 			W^{s, q} (\Omega)
		 		}
		 	\end{align*}
		} for all \FurukawaCommentFifth{$f \in W^{s, q}_{\FurukawaCommentSixth{\mathrm{af}}} (\Omega)$}.
	\end{corollary}

		\FurukawaComment{Note that} $P_{N, \epsilon}$ is not the anisotropic Helmholtz projection on $\Omega$.
		$P_{N, \epsilon}$ is the operator which maps from the \FurukawaCommentSeventh{$L^{\FurukawaCommentFifth{q}}$}-vector fields into \FurukawaCommentSeventh{$L^{\FurukawaComment{q}}$}-divergence-free vector fields  with tangential trace.
		However,  we find that the anisotropic Helmholtz projection is bounded from $L^{\FurukawaComment{q}} (\Omega)$ into itself by the same method of Lemma \ref{lem:C0710}.
		Let $u \in \FurukawaCommentSixth{C^\infty_0} (\Omega)$.
		Then, we obtain the solution $\pi_\epsilon$ to the Neumann problem
			\begin{align} \label{eq:C0725}
				\left \{
					\begin{array}{rll}
						\Delta_\epsilon \pi_{\epsilon}
						& = \mathrm{div}_\epsilon \, u
						& \mathrm{in} \quad \Omega, \\
						\gamma_\pm \frac{\partial_3 \pi_{\epsilon}}{\epsilon}
						& = u \cdot \nu_\pm
						& \mathrm{on} \quad \partial \Omega.
					\end{array}
				\right.
			\end{align}
		The anisotropic Helmholtz projection $H_\epsilon$ is defined by
			\begin{align*}
				H_\epsilon u = u - \nabla_\epsilon \pi_\epsilon.
			\end{align*}
		In the case of the Dirichlet boundary condition, i.e. $\gamma_\pm u = 0$, the right hand side of the second equality of (\ref{eq:C0725}) is zero.
		Let us consider the $L^{\FurukawaCommentFourth{q}}$-boundedness of $\nabla_\epsilon \pi_\epsilon$, which implies the boundedness of the anisotropic Helmholtz projection.
		\FurukawaCommentSixth{For the solution $\pi^0$ to the equation
		\begin{align*}
			\left \{
				\begin{array}{rcll}
					\partial_3^2 \pi^0 (x_3) / \epsilon^2
					& =
					& \partial_3 \tilde{u}_3 (x_3) / \epsilon,
					& x_3 \in (-1 , 1), \\
					\partial_3 \pi^0 (\pm 1) / \epsilon
					& =
					& 0,
				\end{array}
			 \right.
		\end{align*}
		where $\tilde{u}_3 = \int_{\Torus^2} u_3 \, dx^\prime$, we have
		\begin{align} \label{eq_C0730_0}
			\nabla_\epsilon \pi^0 = (0, 0, \tilde{u}_3 )^T.
		\end{align}
		}
		Let $\pi_\epsilon^1$ and $\pi_\epsilon^2$ be the solutions to
			\begin{align} \label{eq:C0730}

						\Delta_\epsilon \pi_{\epsilon}^1
						 = E_0 \mathrm{div}_\epsilon u \quad
						 \mathrm{in} \quad \Torus^2 \times \Real,

			\end{align}
		and
			\begin{align*}
				\left \{
					\begin{array}{rll}
						\Delta_\epsilon \pi_{\epsilon} ^2
						& = 0
						& \mathrm{in} \quad \Omega \FurukawaCommentFourth{,}\\
						\FurukawaCommentSixth{\gamma_\pm \partial_3 \pi_{\epsilon}^2 / \epsilon}
						& = - \gamma_\pm \nu_\pm \cdot \nabla_\epsilon \Delta_\epsilon^{-1} E_0 \mathrm{div}_\epsilon \, u
						& \mathrm{on} \quad \partial \Omega,
					\end{array}
				\right.
			\end{align*}
		respectively, \FurukawaCommentSixth{for $u \in C^\infty_0 (\Omega)$ satisfying $\tilde{u} = 0$.}

		Let us first consider (\ref{eq:C0730}).
		It follows from integration by parts
		\begin{align*}
			\Fourier_{x_3} \Fourier_{d, x^\prime} E_0 \mathrm{div}_\epsilon u
			& = \Fourier_{d, \Prime{x}} \TimeInt{-1}{1}{
				e^{- i x_3 \xi_3} \left(
					\mathrm{div}_H \Prime{u} (\cdot, x_3)
					+ \frac{\partial_3 u_3 (\cdot, x_3)}{\epsilon}
				\right)
			}{x_3} \\
			& = i \left(
				\begin{array}{cc}
					n^\prime
					& \xi_3 / \epsilon
				\end{array}
			\right)^T \cdot \Fourier_{d, x^\prime} ( E_0 u) \\
			& = \Fourier_{x_3} \Fourier_{d, x^\prime} \mathrm{div}_\epsilon (E_0 u).
		\end{align*}
		This formula, the Mikhlin theorem and Proposition \ref{prop_Fourier_Multiplier_Theorem_Discrete} imply
			\begin{align} \label{eq:CC0735}
				\LongNorm{
					\nabla_\epsilon \pi_\epsilon^1
				}{
					L^{\FurukawaCommentFourth{q}} (\Omega)
				}
				\leq C \LongNorm{
					u
				}{
					L^{\FurukawaCommentFourth{q}} (\Omega)
				},
			\end{align}
		where $C>0$ is independent of $\epsilon$.
		Moreover, since $e_3 \cdot \nabla_\epsilon \Delta_\epsilon^{-1} \mathrm{div}_\epsilon$ is given by the \FurukawaCommentFourth{left-hand} side of (\ref{eq:C0717}), we can use the same method as Lemma \ref{lem:C0710} to \FurukawaComment{get}
			\begin{align} \label{eq:CC0736}
				\LongNorm{
					\nabla_\epsilon \pi_\epsilon^2
				}{
					L^{\FurukawaCommentFourth{q}} (\Omega)
				}
				\leq C \LongNorm{
					u
				}{
					L^{\FurukawaCommentFourth{q}} (\Omega)
				},
			\end{align}
		where $C>0$ is also independent of $\epsilon$.
		\FurukawaCommentSixth{The formula (\ref{eq_C0730_0}) and} \FurukawaCommentSixth{estimates} (\ref{eq:CC0735}) and (\ref{eq:CC0736}) \FurukawaCommentFourth{imply} $L^p$-boundedness of the anisotropic Helmholtz projection on $\Omega$.

	\FurukawaCommentFifth{
	Summing up the above argument, we have
	\begin{lemma} \label{lem_bound_H_epsilon}
		 Let $1 < q < \infty$ and $0 < \epsilon \leq 1$. Then there \FurukawaCommentFifth{exists} a constant $C > 0$, which is independent of $\epsilon$, such that
		 	\begin{align*}
		 		\LongNorm{
		 			H_{ \epsilon} f
		 		}{
		 			L^q (\Omega)
		 		}
		 		\leq C \LongNorm{
		 			f
		 		}{
		 			L^q (\Omega)
		 		}
		 	\end{align*}
		 for all $f \in L^q (\Omega)$.
	\end{lemma}
	}

		\begin{proposition} \label{prop:C0715}
			Let $0 <\epsilon \leq 1$, $ 1 < q < \infty $, $0 < a < 1/2$, $z \in \Complex $ satisfying $- a < \RealPart z < 0$ and $ 0 < \theta < \pi / 2$.
			Then there exists a constant \FurukawaCommentFifth{$C = C \bracket{q, a, \theta}$, which is independent of $\epsilon$,} the solution $ \pi_3 $ to (III) with boundary data $\bracket{ \gamma K_{\lambda, \epsilon} f \cdot \nu } \nu$ satisfies
				\begin{align*}
					\LongNorm{
						\frac{1}{ 2 \pi i } \TimeInt{\Gamma_\theta}{}{
							\bracket{- \lambda }^{z} \ep{\nabla} \pi_3
						}{\lambda}
					}{\Lebesgue{q} \bracket{\Omega} }
					\leq C \Napier^{\theta \abs{\ImaginaryPart \, z}} \LongNorm{f}{\Lebesgue{q}\bracket{\Omega}}
				\end{align*}
			\FurukawaCommentFifth{for all $f \in \Lebesgue{q} \bracket{\Omega} $.}
		\end{proposition}

		\begin{proof}
			\FurukawaCommentFifth{In view of Remark \ref{rem_average_free}, we \FurukawaCommentSixth{may assume} $\tilde{f} = 0$ without loss of generality.}
			Since
				\begin{align} \label{eq_formula_nabla_epsilon_pi_3}
					\ep{\nabla} \pi_3 = \ep{\Pi} K_{\lambda, \epsilon} E_0 f
				\end{align}
			and the Cauchy integral commutes with $\ep{\Pi}$, \FurukawaCommentSixth{the conclusion is obtained from} Proposition \ref{prop:C0510} and Lemma \ref{lem:C0710}.
		\end{proof}

	\FurukawaCommentFifth{
	\begin{proposition} \label{prop_nabla2_resolvent_estimate_2}
			Let $0 <\epsilon \leq 1$, $ 1 < q < \infty $, $ 0 < \theta < \pi / 2$ and $\lambda \in \Sigma_\theta$.
			Then there exists a constant $C = C \bracket{q, \theta}$, which is independent of $\epsilon$, the solution $ \pi_3 $ to (III) with boundary data $\bracket{ \gamma K_{\lambda, \epsilon} f \cdot \nu } \nu$ satisfies
				\begin{align} \label{eq_nabla2_resolvent_estimate_2}
					\LongNorm{
						\nabla^2	 \ep{\nabla} \pi_3
					}{\Lebesgue{q} \bracket{\Omega} }
					\leq C  \LongNorm{f}{\Lebesgue{q}\bracket{\Omega}}
				\end{align}
			for all $f \in \Lebesgue{q} \bracket{\Omega} $.
		\end{proposition}
		\begin{proof}
			The estimate (\ref{eq_nabla2_resolvent_estimate_2}) is a direct consequence of (\ref{eq:C0550_discrete}), (\ref{eq_formula_nabla_epsilon_pi_3}),  \FurukawaCommentSeventh{Lemma \ref{lem:C0710} and Proposition \ref{prop_nabla2_resolvent_estimate}.}
		\end{proof}
		}
	\subsection{Estimate for $v_2$}

	Let us consider the equation (II) with tangential boundary data $g = \bracket{ g_{+}, g_{-}}$.
	Set
		\begin{align}
			y_{ \lambda, \epsilon}^{\prime} \bracket{n^{\prime}}
			& = 2 s_{\lambda} \LongBracket{ I_2 + \frac{ \epsilon \abs{n^{\prime}} }{s_{\lambda}} \frac{ n^{\prime} \otimes n^{\prime} }{\abs{ n^\prime }^2} }, \notag\\ \label{eq:y_lambda_epsilon}
			y_{\lambda, \epsilon} \bracket{\Prime{n}}
			& =
			\left(
				\begin{array}{cc}
					\Prime{y}_{ \lambda, \epsilon } \bracket{n^{\prime}} & 0 \\
					0 & 0
				\end{array}
			\right).
		\end{align}
	Then, $y_{ \lambda, \epsilon }$ satisfies
		\begin{align} \label{eq:C1505}
			\Prime{k}_{ \lambda, \epsilon } \bracket{ n^{\prime}, 0 } y_{\lambda, \epsilon} \bracket{\Prime{n}}
			= J_2
			:=
			\left(
				\begin{array}{cc}
					I_2 & 0 \\
					0 & 0
				\end{array}
			\right),
		\end{align}
	\FurukawaCommentFifth{where $k^\prime_{\lambda , \epsilon}$ is defined by (\ref{eq:C0555}).}
	Let us define a multiplier operator $L_{\lambda, \epsilon}$ as
		\begin{align}
			L_{\lambda, \epsilon} g (\Prime{n}, x_3)
			& = \ep{ \mathbb{P}}^{\Torus^2 \times \Real}  \FourierInverse_{d, n^{\prime}} \sLongBracket{
				e_{ \lambda }^{\prime} \bracket{ n^{\prime}, 1 - x_3 } y_{\lambda, \epsilon} \bracket{n^{\prime}} \Fourier_{d,  x^{\prime}} g_{+} \bracket{n^{\prime}}
			}  \notag \\ \label{eq:C1506}
			& + \ep{ \mathbb{P}}^{\Torus^2 \times \Real} \FourierInverse_{d, n^{\prime}} \sLongBracket{
				e_{ \lambda }^{\prime} \bracket{ n^{\prime}, - 1 - x_3 } y_{\lambda, \epsilon} \bracket{n^{\prime}} \Fourier_{d, x^{\prime}} g_{-} \bracket{n^{\prime}}
			},
		\end{align}
	where $e^\prime_\lambda$ is defined by (\ref{eq_definition_of_e_prime_lamda}).
	Let $p_{\FurukawaCommentFourth{\epsilon}}^{\prime} \bracket{\Prime{n}, x_3 }$ be a partial Fourier transform of the symbol of $\ep{\mathbb{P}}^{\Torus^2 \times \Real}$ with respect to $\xi_3$.
	Then
		\begin{align}
			L_{\lambda, \epsilon} g (\Prime{n}, \cdot)
			& = \FourierInverse_{d, \Prime{n}}
				\sLongBracket{
					\Prime{p}_{\epsilon} \bracket{ \Prime{n}, \cdot } \ast_{3} \Prime{e}_{\lambda} \bracket{ \Prime{n}, 1 - \cdot } \Prime{y}_{\lambda, \epsilon} \bracket{\Prime{n}} \Fourier_{d, \Prime{x}} g_{+} \bracket{\Prime{n}}
				} \notag \\ \label{eq:C1507}
			& + \FourierInverse_{d, \Prime{n}}
				\sLongBracket{
					\Prime{p}_{\epsilon} \bracket{ \Prime{n}, \cdot } \ast_{3} \Prime{e}_{\lambda} \bracket{ \Prime{n}, - 1 - \cdot } \Prime{y}_{\lambda, \epsilon} \bracket{\Prime{n}} \Fourier_{d, \Prime{x}} g_{-} \bracket{\Prime{n}}
				},
		\end{align}
	where $ \cdot \ast_3 \cdot$ is convolution with respect to $x_3$.
	We set
		\begin{align} \label{eq:C1509}
			W_{\lambda, \epsilon} = P_{N, \epsilon} L_{\lambda, \epsilon}.
	 	\end{align}
	Then, $W_{\lambda, \epsilon} g$ is a solution to (II) with boundary data $ \gamma W_{\lambda, \epsilon} g$.
	We first get \FurukawaComment{the} Fourier multiplier of $ \gamma W_{\lambda, \epsilon}$.
	Next, we show the map $S_{\lambda,\epsilon} : g \mapsto \gamma W_{\lambda,  \epsilon} g$ has \FurukawaComment{a} bounded inverse for large $\lambda$.
	Put
		\begin{align} \label{eq:C1510}
			V_{\lambda, \epsilon} g
			= W_{\lambda, \epsilon} S^{-1}_{\lambda, \epsilon} g,
		\end{align}
	then, $V_{\lambda, \epsilon} g$ gives the solution to (II) with boundary data $g$.

		\begin{proposition} \label{prop:C1500}
			\FurukawaCommentFifth{
			Let $r > 0$ be sufficiently large.
			Let $ 1 < q < \infty $, $0 < \epsilon \leq 1$, $s>0$, $0 < \theta < \pi/2 $ and $ \lambda \in \Sigma_{\theta} $.}
			Then, for $\abs{\lambda} > r $, there exists a bounded operator $R_{\lambda, \epsilon}$ from $\Lebesgue{q} (\Omega)$ into itself satisfying
				\begin{align}  \label{eq_existence_of_inverse}
					\LongNorm{R_{\lambda, \epsilon}}{
						\FurukawaCommentFifth{
						W^{s , q}_{\FurukawaCommentSixth{\mathrm{af}}} (\Torus^2)
						\rightarrow W^{s , q}_{\FurukawaCommentSixth{\mathrm{af}}} (\Torus^2)
						}
					}
					\leq \frac{C}{\abs{\lambda}^{1/2}},
				\end{align}
			and \FurukawaCommentFifth{
			\begin{align} \label{eq_existence_of_inverse_2}
				\LongNorm{
					R_{\lambda, \epsilon}}{
						\FurukawaCommentFifth{
						W^{s , q}_{\FurukawaCommentSixth{\mathrm{af}}} (\Torus^2)
						\rightarrow W^{s + 1 , q}_{\FurukawaCommentSixth{\mathrm{af}}} (\Torus^2)
					}
				}
				\leq C,
			\end{align}
			} where $C>0$ is independent of $\epsilon$, such that
				\begin{align} \label{eq:C1533}
					- S_{\lambda, \epsilon}^{-1} = I + R_{\lambda, \epsilon}.
				\end{align}
		\end{proposition}

		\begin{proof}
			 \FurukawaCommentFifth{Let $g \in C^\infty (\Torus^2)$ be horizontal average-free}.
			Since $e^\prime_{\lambda}$ is an even function with respect to $x_3$, we find from \FurukawaCommentFifth{the} change of variable that
				\begin{align*}
					\Prime{p}_{\epsilon} \bracket{\Prime{n}, \cdot} \ast_3 e^\prime_{\lambda} \bracket{\Prime{n}, 1 - \cdot} \notag
					& = \TimeInt{\Real}{}{
							\Prime{p}_{\epsilon} \bracket{\Prime{n}, \cdot - \zeta} e^\prime_{\lambda} \bracket{\Prime{n}, 1 - \zeta}
						}{\zeta} \notag \\
					& = - \TimeInt{\Real}{}{
							 		\Prime{p}_{\epsilon} \bracket{\Prime{n}, \eta} e^\prime_{\lambda} \bracket{\Prime{n}, - 1 + \cdot - \eta}
						 		}{\eta} \notag \\
					& = - \Prime{k}_{\lambda, \epsilon} \bracket{\Prime{n}, - 1 + \cdot },
				\end{align*}
			and similarly
				\begin{align*}
					\Prime{p}_{\epsilon} \bracket{\Prime{n}, \cdot} \ast_3 e^\prime_{\lambda} \bracket{\Prime{n}, - 1 - \cdot}
					= - \Prime{k}_{\lambda, \epsilon} \bracket{\Prime{n}, 1 + \cdot } \FurukawaCommentFourth{.}
				\end{align*}
			Thus, we find from (\ref{eq:C1507}) that
				\begin{align}
					L_{\lambda, \epsilon} g (\Prime{n}, \FurukawaCommentFifth{x_3})
					& = \FourierInverse_{d, {\FurukawaCommentFifth{n^\prime}}}
						\sLongBracket{
							- \Prime{k}_{\lambda, \epsilon} (\FurukawaCommentFifth{n^\prime}, -1 + \FurukawaCommentFifth{x_3})
							\Prime{y}_{\lambda, \epsilon} \bracket{\FurukawaCommentFifth{n^\prime}} \Fourier_{d, \Prime{x}} g_{+} \bracket{\FurukawaCommentFifth{n^\prime}}
						} \notag \\  \label{eq:C1520}
					& + \FourierInverse_{d, \FurukawaComment{\Prime{n}}}
						\sLongBracket{
							- \Prime{k}_{\lambda, \epsilon} (\FurukawaCommentFifth{n^\prime}, 1 + \FurukawaCommentFifth{x_3})
							\Prime{y}_{\lambda, \epsilon} \bracket{\FurukawaCommentFifth{n^\prime}} \Fourier_{d, \Prime{x}} g_{-} \bracket{\FurukawaCommentFifth{n^\prime}}
						}\FurukawaCommentFourth{.}
				\end{align}
			We apply $P_{N, \epsilon}$ to (\ref{eq:C1520}) to get

				\begin{align}
					S_{\lambda, \epsilon} g
					& = \gamma_{\pm} W_{\lambda, \epsilon} g \notag \\
					& = -
					\FourierInverse_{d, \Prime{n}}
						\sLongBracket{ \Prime{k}_{\lambda, \epsilon} \bracket{ \Prime{n}, - 1 \pm 1 } y_{\lambda, \epsilon} \bracket{\xi^{\prime}}
					\Fourier_{d, \Prime{x}} g_{+} \bracket{\Prime{n}} } \notag \\
					& -
					\FourierInverse_{d, n^{\prime}}
						\sLongBracket{
							\alpha_{+, \epsilon} \bracket{ \Prime{n }, \pm 1
							} e_3 \cdot
							k_{ \lambda, \epsilon }^{\prime}
							\bracket{
								n^{\prime}, 2
							} y_{\lambda, \epsilon} \bracket{n^{\prime}}
					\Fourier_{d, x^{\prime}} g_{-} \bracket{n^{\prime}} } \notag \\
					& -
					\FourierInverse_{d, \Prime{n}}
						\sLongBracket{
							\Prime{k}_{\lambda, \epsilon}
							\bracket{
								\Prime{n}, 1 \pm 1
							}
							y_{\lambda, \epsilon} \bracket{\xi^{\prime}}
					\Fourier_{d, \Prime{x}} g_{-} \bracket{\Prime{n}}
						} \notag \\
					& -
					\FourierInverse_{d, n^{\prime}}
						\sLongBracket{
							\alpha_{-, \epsilon} \bracket{\Prime{n}, \pm 1 } e_3 \cdot
							k_{ \lambda, \epsilon }^{\prime} \bracket{ n^{\prime}, - 2 } y_{\lambda, \epsilon} \bracket{n^{\prime}}
					\Fourier_{d, x^{\prime}} g_{+} \bracket{n^{\prime}}
						} \notag \\ \label{eq:C1525}
						& = I_1 + I_2 + I_3 + I_4.
				\end{align}
			Let us estimate $I_1$ and $I_3$.
			\FurukawaCommentFourth{The identity} (\ref{eq:C1505}) implies
				\begin{align} \label{eq:C1522}
					\FourierInverse_{d, \Prime{n}}
						\sLongBracket{
							\Prime{k}_{\lambda, \epsilon} \bracket{ \Prime{n}, 0 } y_{\lambda, \epsilon} \bracket{n^{\prime}}
					\Fourier_{d, \Prime{x}} g_{\pm} \bracket{\Prime{n}}
						}
					= g_{\pm}.
				\end{align}

			We need to show the other terms are $O(1 / \abs{\lambda}^{1/2})$.
			By (\ref{eq:C0555}) and (\ref{eq:y_lambda_epsilon}), we have
				\begin{align*}
					 & \Prime{k}_{\lambda, \epsilon} \bracket{ \Prime{\xi}, \pm 2 }
					 y_{\lambda, \epsilon} \bracket{ \Prime{n} } \notag \\
				 	 & =
					 \Napier^{- s_{\lambda}}
					 	\LongBracket{
						 	J_2 + \frac{ \epsilon \abs{\Prime{n}} }{ s_{\lambda} } \frac{ J_2 n \otimes J_2 n }{ \abs{\Prime{n}}^2 }
						} \notag \\

						& \, -
					 J_2 n \otimes J_2 n
					 \frac{ \epsilon^2 }{ \lambda + \bracket{ 1 - \epsilon^2 } \abs{\Prime{n}}^2 }

					 	\Napier^{  - 2 s_{\lambda} }

					  \LongBracket{
					  	 J_2 +
					  	\frac{ \epsilon \abs{\Prime{n}} }{ s_{\lambda} } \frac{  J_2 n \otimes  J_2 n }{ \abs{\Prime{n}}^2 }
					  } \\
					  & \, -
					 J_2 n \otimes J_2 n
					 \frac{ \epsilon^2 }{ \lambda + \bracket{ 1 - \epsilon^2 } \abs{\Prime{n}}^2 }
					 \frac{

							\Napier^{ - 2 \epsilon \abs{\Prime{n}}  }
						}{
							\epsilon \abs{\Prime{n}}
						} s_{\lambda}
					  \LongBracket{
					  	 J_2 +
					  	\frac{ \epsilon \abs{\Prime{n}} }{ s_{\lambda} } \frac{  J_2 n \otimes  J_2 n }{ \abs{\Prime{n}}^2 }
					  } \notag \\
						& =: II_1 + II_2 + II_3. \notag
				\end{align*}
			We find from (\ref{eq:C0501}) and (\ref{eq:C0504}) in Proposition \ref{prop:C05} and
			\begin{align} \label{eq_C1530_multiplier}
				\sLongBracket{
					\frac{
						\abs{\xi^\prime}
					}{
						s_\lambda
					}
				}_{\mathcal{M}^\prime}
				+ \sLongBracket{
					\frac{
						J_2 \xi \otimes J_2 \xi
					}{
						\abs{\xi^\prime}^2
					}
				}_{\mathcal{M}^\prime}
				\leq C
				\FurukawaCommentFifth{, \quad\xi = (\xi^\prime , \xi_3) \in \Real^3,}
			\end{align}
		 	that
				\begin{align} \label{eq:boundedness_Mikhlin_constant_A_1}
					\sLongBracket{
						II_1
					}_{\mathcal{M}^\prime}
					\leq C e^{ - c \abs{\lambda}^\frac{1}{2}},
					\FurukawaCommentFifth{
					\quad \sLongBracket{
						\, \abs {\xi^\prime} \, II_1
					}_{\mathcal{M}^\prime}
					\leq C e^{ - c \abs{\lambda}^\frac{1}{2}},
					}
				\end{align}
			\FurukawaCommentSixth{where we interpret that the multiplier $II_1$ is extended from $\Integer^2$ to $\Real^2$ in the canonical way.}
			\FurukawaComment{
			Since
				\begin{align} \label{eq_C1530_multiplier_2}
					\sLongBracket{
					\frac{
						1
					}{
						\lambda + (1 - \epsilon^2) \abs{\xi^\prime}^2
					}
				}_{\mathcal{M}^\prime}
				\leq \frac{C}{\abs{\lambda}},
				\end{align}
			by the same way as above we find
			}
				\begin{align} \label{eq:boundedness_Mikhlin_constant_A_2}
					\sLongBracket{
						II_2
					}_{\mathcal{M}^\prime}
					\leq \frac{C e^{- c \abs{\lambda}^{1/2}}}{\abs{\lambda}}, \quad
					\FurukawaCommentFifth{
					\sLongBracket{
						\, \abs {\xi^\prime} \, II_2
					}_{\mathcal{M}^\prime}
					\leq C e^{ - c \abs{\lambda}^\frac{1}{2}},}
				\end{align}
			for \FurukawaComment{$\lambda \in \Sigma_\theta$}, where constants $c, C > 0$ are independent of $\epsilon$.
			Note that \FurukawaCommentFifth{$II_3$} \FurukawaComment{has a} little bit problem near $\epsilon = 0$ since, at this point, we can not use \FurukawaCommentFourth{the} decay of $\Napier^{- 2 \epsilon \abs{\Prime{\xi}}}$ to obtain uniform boundedness of the Mikhlin constant.
			However, we can use \FurukawaCommentFourth{the} decay of $1/ \bracket{ \lambda + (1 - \epsilon^2) \abs{\Prime{\xi}}^2}$ around $\epsilon = 0$.
			On the other hand, when $\epsilon$ is away from $0$, we have no problem \FurukawaComment{to use decay of $e^{- 2 \epsilon \abs{\xi^\prime}}$}.
			Thus, combining \FurukawaComment{this} observation with \FurukawaCommentSeventh{Proposition \ref{prop_scaled_multiplier}, (\ref{eq_C1530_multiplier}) and (\ref{eq_C1530_multiplier_2})}, we conclude that
				\begin{align} \label{eq:boundedness_Mikhlin_constant_A_3}
					\sLongBracket{
						II_3
					}_{\mathcal{M}^\prime}
					\leq \frac{C}{\abs{\lambda}^\frac{1}{2}}, \quad
					\FurukawaCommentFifth{
					\sLongBracket{
						\, \abs{\xi^\prime} \, II_3
					}_{\mathcal{M}^\prime}
					\leq C,}
				\end{align}
			where $C>0$ is independent of $\epsilon$.
			Thus we find from (\ref{eq:boundedness_Mikhlin_constant_A_1}), (\ref{eq:boundedness_Mikhlin_constant_A_2}) and \FurukawaComment{(\ref{eq:boundedness_Mikhlin_constant_A_3}) }that \FurukawaCommentFifth{
				\begin{align}
					&
					\LongNorm{
						I_1 + I_3 - g_+ - g_-
					}{L^q (\Torus^2)} \notag \\
					& \leq  \LongNorm{
						\FourierInverse_{d, \Prime{n}}
							\sLongBracket{
								\Prime{k}_{\lambda, \epsilon} \bracket{ \Prime{n}, 2 } y_{\lambda, \epsilon} \bracket{n^{\prime}}
						\Fourier_{d, \Prime{x}} g \bracket{\Prime{n}}
							}
					}{\FurukawaCommentFifth{L^q (\Omega)}} \notag \\
					& + \LongNorm{
						\FourierInverse_{d, \Prime{n}}
							\sLongBracket{
								\Prime{k}_{\lambda, \epsilon} \bracket{ \Prime{n}, - 2 } y_{\lambda, \epsilon} \bracket{n^{\prime}}
						\Fourier_{d, \Prime{x}} g \bracket{\Prime{n}}
							}
					}{\FurukawaCommentFifth{L^q (\Omega)}} \notag \\
					& \leq \frac{C}{\abs{\lambda}^\frac{1}{2}} \LongNorm{
						g
					}{
						L^{\FurukawaCommentFifth{q}} (\FurukawaCommentFifth{\Torus^2})
					}.\label{eq:C1531}
				\end{align}
				}
			Next, we estimate $I_2$ and $I_4$.
			It follows from (\ref{eq:C0555}) that
 				\begin{align}
					& e_3 \cdot
							k_{ \lambda, \epsilon }^{\prime} \bracket{ n^{\prime}, \pm 2 } y_{\lambda, \epsilon} \bracket{n^{\prime}}
					\Fourier_{d, x^{\prime}} g_{\mp} \bracket{n^{\prime}} \notag \\
					& = e_3 \cdot \sLongBracket{
						\frac{e^{- s_\lambda}}{2 s_\lambda} y_{\lambda, \epsilon} (\Prime{n}) \Fourier_{d, \Prime{x}} g_\mp
						- \left(
						\begin{array}{cc}
							\eta_{\lambda, \epsilon}^{\prime} \bracket{ n^{\prime}, \pm 2 }
							n^{\prime} \otimes n^{\prime}
							\Prime{y}_{\lambda, \epsilon} (\Prime{n})
							& 0 \\
							- \partial_3 \eta_{\lambda, \epsilon}^{\prime} \bracket{ n^{\prime}, \pm 2 }
							i { n^{\prime} }^T
							\Prime{y}_{\lambda, \epsilon} (\Prime{n})
							& 0
						\end{array}
						\right)
						\Fourier_{d, \Prime{x}} g_\mp
					} \notag \\ \label{eq_e_k_y_F_g}
					& = - \left(
						\begin{array}{ccc}
							- \partial_3 \eta_{\lambda, \epsilon}^{\prime} \bracket{ n^{\prime}, \pm 2}
							i { n^{\prime} }^T
							\Prime{y}_{\lambda, \epsilon} (\Prime{n})
							& 0
						\end{array}
						\right)
						\Fourier_{d, \Prime{x}} g_\mp.
				\end{align}
			Recall
				$ \partial_3 \eta_{\lambda, \epsilon} (\Prime{n}, \pm 2) = \frac{
							\epsilon^2
						 }{
						 	\lambda
						 	+ \bracket{ 1 - \epsilon^2 } \abs{ n^{\prime} }^2
						 } \frac{
						 	\Napier^{- 2 s_{\lambda}} - \Napier^{ - 2 \epsilon \abs{n^{\prime}} }
						 }{
						 	2
						 }
				$.
			Then, we find from \FurukawaCommentSeventh{the first inequality of (\ref{eq:C0501})} and \FurukawaComment{(\ref{eq_C1530_multiplier_2})} that
				\begin{align*}
					& \sLongBracket{
						\LongBracket{
							1 + \epsilon \abs{\Prime{\xi}}
						}
						\partial_3 \eta_{\lambda, \epsilon} (\Prime{\xi}, \pm 2)
						\Prime{y}_{\lambda} (\Prime{\xi})
					}_{\mathcal{M}^\prime} \notag \\
					& = \sLongBracket{
						\frac{
							\epsilon^2
						 }{
						 	\lambda
						 	+ \bracket{ 1 - \epsilon^2 } \abs{ \Prime{\xi} }^2
						 }
						\LongBracket{
							1 + \epsilon \abs{\Prime{\xi}}
						}
						s_{\lambda}
						 \LongBracket{
						 	\Napier^{- 2 s_{\lambda}}
						 	- \Napier^{ - 2 \epsilon \abs{\xi^{\prime}} }
						 	}
						 	\LongBracket{
						 		I_2 + \frac{ \epsilon \abs{\Prime{\xi}} }{s_{\lambda}} \frac{ \Prime{\xi} \otimes \Prime{\xi} }{\abs{ \Prime{\xi} }^2}
						 	}
					}_{\mathcal{M}^\prime} \notag \\
					& \leq \frac{C}{\abs{\lambda}^\frac{1}{2}},
				\end{align*}
			\FurukawaCommentFifth{
			and
			\begin{align} \label{eq_regularity_R_epsilon_lamda}
				\sLongBracket{
					\abs{\xi^\prime}
						\LongBracket{
							1 + \epsilon \abs{\Prime{\xi}}
						}
						\partial_3 \eta_{\lambda, \epsilon} (\Prime{\xi}, \pm 2)
						\Prime{y}_{\lambda} (\Prime{\xi})
					}_{\mathcal{M}^\prime}
				\leq C,
			\end{align}}
			\hspace{-1.5mm}where $C>0$ is independent of $\epsilon$.
			\FurukawaCommentSeventh{The formula (\ref{eq:C0712}), estimates (\ref{eq:C0504}) and (\ref{eq:C0505})} lead \FurukawaCommentSixth{to}
				\begin{align*}
						\sLongBracket{
							\frac{
								\alpha_{\pm, \epsilon} \bracket{\Prime{\xi}, \pm \FurukawaCommentFifth{1}} \epsilon \abs{\Prime{\xi}}
							}{
								1 + \epsilon \abs{\Prime{\xi}}
							}
						}_{\mathcal{M}^\prime}
						< \infty,
				\end{align*}
		uniformly on $\epsilon$.
		\FurukawaCommentSixth{We find from Proposition \ref{prop_Fourier_Multiplier_Theorem_Discrete} that}
				\begin{align}
						& \FurukawaCommentFifth{\LongNorm{
							I_2 + I_4
						}{L^q (\Torus^2)}} \notag \\
						& \FurukawaCommentFifth{\leq} \LongNorm{
							\FourierInverse_{d, \Prime{n}}
								\alpha_{+, \epsilon} \bracket{\Prime{n}, \pm 1 } e_3 \cdot
								k_{ \lambda, \epsilon }^{\prime} \bracket{ \Prime{n},  2 }
								y_{\lambda, \epsilon} \bracket{\Prime{n}}
							\Fourier_{d, x^{\prime}} g_{-}
						}{\Lebesgue{q} (\Torus^2)} \notag \\
						& + \LongNorm{
							\FourierInverse_{d, \Prime{n}}
								\alpha_{-, \epsilon} \bracket{\Prime{n}, \pm 1 } e_3 \cdot
								k_{ \lambda, \epsilon }^{\prime} \bracket{ \Prime{n}, - 2 }
								y_{\lambda, \epsilon} \bracket{\Prime{n}}
							\Fourier_{d, x^{\prime}} g_{+}
						}{\Lebesgue{q} (\Torus^2)} \notag \\ \label{eq:C1532}
						& \leq \frac{
							C
						}{
							\abs{\lambda}^\frac{1}{2}
						} \LongNorm{
							g
						}{
							L^q (\Torus^2)
						},
				\end{align}
			where $C$ is independent of $\epsilon$.
			Thus, taking $\abs{\lambda}$ sufficiently large, clearly \FurukawaCommentFourth{the} choice of $\lambda$ is also independent of $\epsilon$, we can conclude by \FurukawaComment{(\ref{eq:C1525}), (\ref{eq:C1522}\FurukawaCommentFourth{)}, (\ref{eq:C1531}) and (\ref{eq:C1532}) }that
				\begin{align*}
					- S_{\lambda,\epsilon}
					= I + O \bracket{ \abs{\lambda}^{ - 1/2} }.
				\end{align*}
			\FurukawaCommentFifth{By the Neumann series argument we obtain (\ref{eq_existence_of_inverse}) for $s = 0$.
			Moreover, we find from (\ref{eq:boundedness_Mikhlin_constant_A_1}), (\ref{eq:boundedness_Mikhlin_constant_A_2}), (\ref{eq:boundedness_Mikhlin_constant_A_3}) and (\ref{eq_regularity_R_epsilon_lamda}) that (\ref{eq_existence_of_inverse_2}) holds for $s = 0$.
			Since $\partial_j$ ($j = 1,2$) commutes Fourier multiplier operators, we obtain (\ref{eq_existence_of_inverse}) and (\ref{eq_existence_of_inverse_2}) for $s > 0$.
			}
		\end{proof}

	\begin{proposition} \label{prop:C1507}
		Let \FurukawaCommentFifth{$1 < q < \infty$, $0 < \epsilon \leq 1$,} $0 < \theta < \pi / 2$ and $\lambda \in \Sigma_\theta$.
		Then there exist $r > 0$ and a constant $C > 0$, which is independent of $\epsilon$ and $\lambda$, if $\abs{\lambda} \geq r$, $V_{\lambda , \epsilon}$ defined by (\ref{eq:C1510}) satisfies
		\begin{align} \label{eq:C1534}
			\LongNorm{V_{\lambda, \epsilon} g}{\Lebesgue{q} \bracket{\Omega}}
			\leq C \abs{\lambda}^{- 1 / 2q} \LongNorm{g}{\Lebesgue{q}  \bracket{\partial \Omega}}
		\end{align}
		for all \FurukawaCommentSixth{$g \in \Lebesgue{q} \bracket{\partial \Omega}$ satisfying $\tilde{g} = 0$.}
	\end{proposition}
	\begin{proof}
		We take $r>0$ so that $R_{\lambda, \epsilon}$ exists.
		Then $S_{\lambda, \epsilon}^{-1}$ is bounded on $L^q (\Omega)$.
		\FurukawaCommentSeventh{
		 We find from the resolvent estimate for the Dirichlet Laplacian on $\Omega$, see Lemma 5.3 in \cite{Abels2002}, and Proposition \ref{prop:C1500} that
		 \begin{align*}
				\LongNorm{
					L_{\lambda, \epsilon} S_{\lambda, \epsilon}^{-1} g
				}{L^q (\Omega)}
				\leq C \abs{\lambda}^{- 1 / 2 q} \LongNorm{
					g
				}{L^q (\Omega)},
			\end{align*}
			where $C>0$ is independent of $\epsilon$.
		By Corollary \ref{cor_bound_P_N_epsilon}, we obtain (\ref{eq:C1534}).
		}

	\end{proof}

	\begin{proposition} \label{prop:C1510}
		Let $1 < q < \infty$ and $0 < \epsilon \leq 1$.
		Then there exists a constant $C > 0$, which is independent of $\epsilon$, such that
			\begin{align} \label{eq:C1535}
				\LongNorm{
					\FourierInverse_{d, \Prime{n}}
						\frac{ 1 + \epsilon \abs{\Prime{n}} }{ \epsilon \abs{\Prime{n}} }
						\LongBracket{ e_3 \cdot
					\Fourier_{\FurukawaCommentFifth{d,\Prime{x}}}
					\LongBracket{ \ep{\mathbb{P}}^{\Torus^2 \times \Real} E_0 f }}
				}{\Lebesgue{q} (\Omega)}
				\leq C
				\LongNorm{f}{\Lebesgue{q} (\Omega)}
			\end{align}
		for all \FurukawaCommentFifth{$f \in \Lebesgue{q}_{\FurukawaCommentSixth{\mathrm{af}}} (\Omega )$}.
	\end{proposition}

	\begin{proof}

		Since the symbol have poles at $\xi_3 = \pm i \epsilon \abs{\Prime{n}}$, we obtain its partial Fourier transform with  respect to $\xi_3$ by the residue theorem.
		Thus, we have
			\begin{align*}
				& \FourierInverse_{d, \Prime{n}}
					\frac{ 1 }{ \epsilon \abs{\Prime{n}} }
					\LongBracket{ e_3 \cdot
				\Fourier_{\FurukawaCommentFifth{d,\Prime{x}}}
				\LongBracket{ \ep{\mathbb{P}}^{\Torus^2 \times \Real}  E_0 f }} \notag \\
				& = \frac{1}{2} \FourierInverse_{d, \Prime{n}}
				\int_{-1}^{1}
					\frac{ 1 }{ \epsilon \abs{\Prime{n}} }
					\left[
						\Napier^{- \abs{x_3 - \zeta} \epsilon \abs{\Prime{n}} }
						i \epsilon \Prime{n} \cdot
						\Fourier_{d, \Prime{x}} \Prime{f} \bracket{\Prime{n}, \zeta}
					\right. \\
					& \quad \quad \quad \quad \quad \quad \left.
					+ \Napier^{- \abs{x_3 - \zeta} \epsilon \abs{\Prime{n}} }
						\epsilon \abs{\Prime{n}}
						\Fourier_{d, \Prime{x}} f_3 \bracket{\Prime{n}, \zeta}
					\right]
				d \zeta.
			\end{align*}
		\FurukawaCommentSeventh{This formula and Proposition \ref{prop_Fourier_Multiplier_Theorem_Discrete} imply}
			\begin{align*}
				\LongNorm{
					\FourierInverse_{d, \Prime{n}}
					\frac{ 1 }{ \epsilon \abs{\Prime{n}} }
					\LongBracket{ e_3 \cdot
				\Fourier_{d, \Prime{x}}
				\LongBracket{ \ep{\mathbb{P}}^{\Torus^2 \times \Real} E_0 f }}
				}{
					L^q (\Torus^2)
				}
				\leq C \LongNorm{
					f
				}{
					L^q (\Real^2)
				},
			\end{align*}
		where $C$ is independent of $\epsilon$.
		\FurukawaCommentFifth{
		Combining \FurukawaCommentSixth{this estimate with} the boundedness of $\mathbb{P}_\epsilon^{\Torus^2 \times \Real}$, we obtain (\ref{eq:C1535}).
		}
	\end{proof}

	Let us show BIP for the solution operator for \FurukawaComment{the equation (II)}.

		\begin{proposition} \label{prop:C1515}
			Let \FurukawaCommentFifth{$1 < q < \infty$, $0 < \epsilon \leq 1$, $0 < \theta < \pi /2$,} $\lambda \in \Sigma_\theta$,  $0 < a < 1/2$, \FurukawaCommentFifth{ and $z$ satisfying $ - a < \RealPart z <0 $.}
			Then there exists a constant $C = C \bracket{ q, a, \theta }$, it holds that
				\begin{align} \label{eq:C1550}
					\LongNorm{ \frac{
						1
					}{
						2 \pi i
					} \TimeInt{
							\Gamma_\theta
						}{}{
							\bracket{
								- \lambda
							}^z V_{\lambda, \epsilon} \sLongBracket{
								\gamma v_1 - \bracket{ \gamma v_1 \cdot \nu } \nu
							}
						}{\lambda}
					}{\Lebesgue{q}(\Omega)}
					\leq C \Napier^{ \abs{ \ImaginaryPart z } \theta } \LongNorm{
						f
					}{
						\Lebesgue{q} (\Omega)
					}
				\end{align}
			\FurukawaCommentFourth{for all $f \in \Lebesgue{q} (\Omega)$, where $ v_1 = K_{\lambda, \epsilon} f$.}
		\end{proposition}

		\begin{proof}

			\FurukawaCommentFifth{In view of Remark \ref{rem_average_free}, we \FurukawaCommentSixth{may} assume $\tilde{f} = 0$ without loss of generality.}
			It holds \FurukawaComment{by (\ref{eq:C0550_discrete}) that}
				\begin{align*}
					\gamma v_1 - \bracket{ \gamma v_1 \cdot \nu } \nu

					= \gamma K_{\lambda, \epsilon} E_0 f
					- \gamma \Pi_{\epsilon} K_{\lambda, \epsilon} E_0 f.
				\end{align*}
			We find from this formula, \FurukawaCommentSeventh{(\ref{eq:C0712}), (\ref{eq:C1506}), (\ref{eq:C1509}), (\ref{eq:C1510}) and (\ref{eq:C1533})} that the integrand of the left hand side of (\ref{eq:C1550}) can be essentially written as
				\begin{align}
					& P_{N, \epsilon} \ep{\mathbb{P}}^{\Torus^2 \times \Real}
					\FourierInverse_{d, \Prime{n}}
						\int_{-1}^{1}
							\Prime{e}_{\lambda} \bracket{\Prime{\xi}, \pm 1 - x_3}
							y_{\lambda, \epsilon} \bracket{\Prime{n}}
							\Prime{e}_{\lambda} \bracket{\Prime{\xi}, \pm 1 - \zeta} \notag \\
				& \quad \quad \quad \quad 	\quad \quad \quad \quad
					\times \Fourier_{d, \Prime{x}} \LongBracket{ \ep{\mathbb{P}}^{\Torus^2 \times \Real} E_0 f } \bracket{\Prime{n}, \zeta}
						d \zeta \notag \\
				+ & P_{N, \epsilon} \ep{\mathbb{P}}^{\Torus^2 \times \Real}
					\FourierInverse_{d, \Prime{n}}
						\int_{-1}^{1}
							\Prime{e}_{\lambda} \bracket{\Prime{n}, \pm 1 - x_3}
							y_{\lambda, \epsilon} \bracket{\Prime{n}}
							\alpha_{\epsilon, \pm} \bracket{\Prime{n}, \pm 1}
							\Prime{e}_{\lambda} \bracket{\Prime{n}, \pm 1 -\zeta} \notag \\
				&	 \quad\quad\quad\quad\quad\quad\quad\quad
							\times \frac{\epsilon \abs{\Prime{n}}}{ 1 + \epsilon \abs{\Prime{n}} }
							\frac{1 + \epsilon \abs{\Prime{n}} }{ \epsilon \abs{\Prime{n}}}
					\Fourier_{d, \Prime{x}} \LongBracket{  \ep{\mathbb{P}}^{\Torus^2 \times \Real} E_0 f } \bracket{\Prime{n}, \zeta}
						d \zeta \notag \\
				+ & W_{\lambda, \epsilon} R_{\lambda, \epsilon}
					\sLongBracket{
							\gamma K_{\lambda, \epsilon} E_0 f
							- \gamma \Pi_{\epsilon} K_{\lambda, \epsilon} E_0 f
					}
					 \notag \\
				=: & I_1 + I_2 + I_3, \label{eq_formula_V_lambda_epsilon_tangential_term}
				\end{align}
			where $\pm$ should be take properly.

			It follows from \FurukawaCommentSeventh{(\ref{eq:C0501}) and (\ref{eq_C1530_multiplier})} that
				\begin{align} \label{eq:C1560}
					& \sLongBracket{
						\Prime{e}_{\lambda} \bracket{\FurukawaCommentFifth{\FurukawaCommentSixth{\xi}^\prime}, \pm 1 - x_3 }
						y_{\lambda, \epsilon} \bracket{\FurukawaCommentFifth{\FurukawaCommentSixth{\xi}^\prime}}
						\Prime{e}_{\lambda} \bracket{\FurukawaCommentFifth{\FurukawaCommentSixth{\xi}^\prime}, \pm 1 - \zeta }
					}_{\Prime{\mathcal{M}}} \notag \\
					& = 2 \sLongBracket{

							e^{ \FurukawaCommentFifth{-} \abs{\pm 1 - x_3} s_\lambda}

						\LongBracket{
							I_2
							+ \frac{
								\epsilon \abs{\FurukawaCommentFifth{\FurukawaCommentSixth{\xi}^\prime}}
							}{
								s_{\lambda}
							} \frac{
								\FurukawaCommentFifth{\FurukawaCommentSixth{\xi}^\prime} \otimes \FurukawaCommentFifth{\FurukawaCommentSixth{\xi}^\prime}
							}{
								\abs{\FurukawaCommentFifth{\FurukawaCommentSixth{\xi}^\prime}}^2
							}
						}
						\frac{
							e^{ \FurukawaCommentFifth{-} \abs{\pm 1 - \zeta} s_\lambda}
						}{
							s_\lambda
						}
					}_{\mathcal{M}^\prime} \notag \\
					& \leq C
					\frac{
						\Napier^{- c \abs{\lambda}^{1/2} \bracket{ \abs{ \pm 1 - x_3 } + \abs{ \pm 1 - \zeta } }}
					}{\abs{\lambda}^{1/2}}.
				\end{align}
			\FurukawaCommentFifth{Let $R>0$ be large enough so that $S_{\lambda, \epsilon}^{-1}$ in Proposition \ref{prop:C1500} exists.
			Then we find from the change of integral curve  around the origin to ensure $\abs{\lambda} > R$ and Proposition \ref{prop_Fourier_Multiplier_Theorem_Discrete} that}
				\begin{align*}
					& \LongNorm{ \frac{1}{ 2 \pi i}
						 \TimeInt{\Gamma}{}{
							 \bracket{- \lambda}^z I_1
							 }{\lambda}
					 }{\Lebesgue{q} (\FurukawaCommentFifth{\Omega)}} \notag \\
					& \leq C \LongNorm{
							\TimeInt{-1}{1}{
								\TimeInt{\Gamma}{}{	\abs{ \lambda^z}
										 \frac{ \Napier^{ - c \abs{\lambda}^{1/2} \bracket{ \abs{ x_3 - a}  + \abs{ \zeta - b } } } }{ \abs{\lambda}^{1/2} } \LongNorm{
										\mathbb{P}_\epsilon^{\FurukawaCommentFifth{\Torus^2 \times \Real}} E_0 f \bracket{\cdot, \zeta} }{ \Lebesgue{q} \bracket{\Torus^2} }
								}{\lambda}
							}{\zeta}
							}{\Lebesgue{q} \bracket{-1, 1} } \notag \\
					& \leq C \FurukawaComment{e^{\theta \abs{\ImaginaryPart z}}}  \LongNorm{f}{\Lebesgue{q} \bracket{\Omega}} \notag \\
					& + C \left \Vert
						\int_{-1}^{1}
								\int_{R}^{\infty}
									\Napier^{ \theta \abs{ \ImaginaryPart z } }
									r^{ \RealPart z - 1/2 } \Napier^{ - c r^{1/2} \bracket{ \abs{ x_3 - a}  + \abs{ \zeta - b } } } \LongNorm{f \bracket{\cdot, \zeta}}{\Lebesgue{q} \bracket{\Torus^2}}
							dr
						d \zeta
					\right \Vert_{\Lebesgue{q} \bracket{-1, 1} } \\
					& \leq C \FurukawaComment{e^{\theta \abs{\ImaginaryPart z}}} \LongNorm{
						f
					}{L^q (\Omega)}
					+ C_R e^{\theta \abs{\ImaginaryPart z}} \LongNorm{
						\int_{-1}^{1}
							\frac{\LongNorm{f (\cdot , \zeta)}{L^q (\Torus^2)}}{\abs{ x_3 - a}  + \abs{ \zeta - b }}
						d r
					}{L^q (\Omega)}
				\end{align*}
			\FurukawaComment{for some $a , b \in \{ -1 , 1 \}$,} where \FurukawaCommentSeventh{$C$ and $C_R$ are} independent of $\epsilon$.
			Applying \FurukawaCommentSeventh{Proposition} \ref{prop:C03}, we obtain
				\begin{align} \label{eq:C1570}
					\LongNorm{
						 \TimeInt{\Gamma}{}{
							 \bracket{- \lambda}^z I_1
							 }{\lambda}
					 }{\Lebesgue{q} \bracket{\Omega} }
					\leq C \Napier^{ \theta \abs{\ImaginaryPart z} } \LongNorm{f}{\Lebesgue{q} \bracket{\Omega}} \FurukawaCommentFourth{.}
				\end{align}
			\FurukawaCommentSeventh{It follows from (\ref{eq_definition_of_e_prime_lamda}), (\ref{eq:C0712}), (\ref{eq:y_lambda_epsilon}) and Proposition \ref{prop:C05} that}
				\begin{align}
					&\sLongBracket{
						\Prime{e}_{\lambda} \bracket{\FurukawaCommentFifth{\Prime{\FurukawaCommentSixth{\xi}}}, \pm 1 - x_3}
							y_{\lambda, \epsilon} \bracket{\FurukawaCommentFifth{\Prime{\FurukawaCommentSixth{\xi}}}}
							\alpha_{\epsilon, \pm} \bracket{\FurukawaCommentFifth{\Prime{\FurukawaCommentSixth{\xi}}}, \pm 1}
							\Prime{e}_{\lambda} \bracket{\FurukawaCommentFifth{\Prime{\FurukawaCommentSixth{\xi}}}, \pm 1 -\zeta}
							\frac{\epsilon \abs{\FurukawaCommentFifth{\Prime{\FurukawaCommentSixth{\xi}}}}}{ 1 + \epsilon \abs{\FurukawaCommentFifth{\Prime{\FurukawaCommentSixth{\xi}}}} }
					}_{\Prime{\mathcal{M}}} \notag \\  \label{eq_Resolvent_estimate_2}
					& \leq C
					\frac{
						\Napier^{- c \abs{\lambda}^{1/2} \bracket{ \abs{ \pm 1 - x_3 } + \abs{ \pm 1 - \zeta } }}
					}{\abs{\lambda}^{1/2}}.
				\end{align}
			Thus we find from Proposition \ref{prop:C1510} \FurukawaCommentFifth{that}
				\begin{align*}
					& \LongNorm{ \frac{1}{2 \pi i }
						\TimeInt{\Gamma}{}{
							 \bracket{ - \lambda }^z I_2
						}{\lambda}
					}{
						\Lebesgue{q} \bracket{\Omega}
					} \\
					& \leq C \Napier^{ \theta \abs{\ImaginaryPart z} }
					\LongNorm{
						\FourierInverse_{\FurukawaCommentFifth{d,} \Prime{n}}
						\frac{
							1 + \epsilon \abs{\FurukawaComment{\Prime{n}}}
						}{
							\epsilon \abs{\Prime{n}}
						}
						\Fourier_{\FurukawaCommentFifth{d,} \Prime{x}} \LongBracket{
							 \mathbb{P}_\epsilon^{\FurukawaCommentFifth{\Torus^2 \times \Real}} E_0 f
						} \bracket{\Prime{n}, \zeta}
					}{
						\Lebesgue{q}\bracket{\Omega}
					} \notag \\
					& \leq C \Napier^{ \theta \abs{\ImaginaryPart z} }
					\LongNorm{
						f
					}{\Lebesgue{q} \bracket{\Omega} }.
				\end{align*}
			\FurukawaCommentSeventh{By Proposition \ref{prop:C1500}, the trace theorem, Lemma \ref{lem:C0710} and the resolvent estimate for the Laplace operator on $\Torus^2 \times \Real$, we have
			\begin{align*}
					& \LongNorm{ R_{\lambda ,\epsilon}
					 \gamma	\sLongBracket{
							K_{\lambda, \epsilon} E_0 f
							- \gamma \Pi_{\epsilon} K_{\lambda, \epsilon} E_0 f
					}
					}{\Lebesgue{q} \bracket{ \partial \Omega}}  \\
					& \leq C \abs{\lambda}^{ -3/2 + 1/ 2 q + \delta}
					\LongNorm{
						f
					}{\Lebesgue{q} \bracket{ \Omega}}
				\end{align*}
			for some small $\delta>0$.
			 The resolvent estimate for the Dirichlet Laplacian on $\Omega$, see Lemma 5.3 in \cite{Abels2002}, \FurukawaCommentSeventh{and Lemma \ref{lem:C0710}} imply
			 \begin{align*}
					\LongNorm{
						P_{N, \epsilon} L_{\lambda, \epsilon}
					}{\Lebesgue{q} \bracket{\partial  \Omega} \rightarrow \Lebesgue{q} \bracket{\Omega}}
					 \leq C \abs{\lambda}^{- 1 /2q }
				\end{align*}
			for some small $\delta>0$.
			We find from the above \FurukawaCommentSeventh{two inequalities}
			\begin{align} \label{eq:C1575}
			\LongNorm{I_3}{\Lebesgue{q} \bracket{\Omega}}
					& \leq C \abs{\lambda}^{- 3/2 + \delta}
					\LongNorm{
						f
					}{\Lebesgue{q} \bracket{ \Omega}}.
				\end{align}
			}

			Thus \FurukawaCommentThird{we find from the change of integral line around the origin that}
				\begin{align*}
					\LongNorm{
						\frac{1}{2 \pi i}
						\TimeInt{\Gamma}{}{
							\bracket{ - \lambda}^z I_3
						}{\lambda}
					}{\Lebesgue{q} \bracket{\Omega}}
					\leq C \LongNorm{f}{\Lebesgue{q} \bracket{\Omega}},
				\end{align*}
		\end{proof}
		\FurukawaCommentSeventh{where $C>0$ is independent of $\epsilon$.}
		\begin{proof}[Proof of Lemma \ref{lem:C02}]
			Lemma \ref{lem:C02} is a direct consequence of \AmruComment{Propositions \ref{prop:C0510}, \ref{prop:C0715} and \ref{prop:C1515}}.
		\end{proof}
			\FurukawaCommentSixth{We next prove Lemma \ref{lem:A1001} from Lemma \ref{lem:C02}.
		For this purpose we need further uniform estimate for the resolvent to compare $\Vert \nabla^2 u \Vert_{L^q (\Omega)}$ and $\Vert A_\epsilon  u \Vert_{L^q (\Omega)}$.
		For resolvent estimates we begin with
		}
		\FurukawaCommentFifth{
		\begin{proposition} \label{prop_nabla_2_V_lambda_epsilon}
			Let $1 < q < \infty$, $0 < \epsilon \leq 1$, $0 < \theta < \pi /2$.
			Let $\lambda \in \Sigma_\theta$ be sufficiently large so that $S_{\lambda, \epsilon}^{-1}$ exists in Proposition \ref{prop:C1500}.
			Then there exists a constant $C = C \bracket{ q, \theta }$, it holds that
				\begin{align*}
					\LongNorm{
						 \nabla^2 V_{\lambda, \epsilon} \sLongBracket{
								\gamma v_1 - \bracket{ \gamma v_1 \cdot \nu } \nu
						}
					}{\Lebesgue{q}(\Omega)}
					\leq C \LongNorm{
						f
					}{
						\Lebesgue{q} (\Omega)
					}
				\end{align*}
			for all $f \in \Lebesgue{q} (\Omega)$, where $ v_1 = K_{\lambda, \epsilon} f$.
		\end{proposition}
		}\FurukawaCommentFifth{
		\begin{proof}
			\FurukawaCommentFifth{In view of Remark \ref{rem_average_free}, we \FurukawaCommentSixth{may} assume $\tilde{f} = 0$ without loss of generality.}
			It is enough to estimate the second derivative of the left-hand side of (\ref{eq_formula_V_lambda_epsilon_tangential_term}) in $L^q (\Omega)$.
			We find from (\ref{eq:C0501}) and (\ref{eq:C1560}) that
			\begin{align*}
					& \sLongBracket{
						\abs{n^\prime}^2 \, \Prime{e}_{\lambda} \bracket{\FurukawaCommentFifth{\FurukawaCommentSixth{\xi}^\prime}, \pm 1 - x_3 }
						y_{\lambda, \epsilon} \bracket{\FurukawaCommentFifth{\FurukawaCommentSixth{\xi}^\prime}}
						\Prime{e}_{\lambda} \bracket{\FurukawaCommentFifth{\FurukawaCommentSixth{\xi}^\prime}, \pm 1 - \zeta }
					}_{\Prime{\mathcal{M}}} \notag \\
					& = 2 \sLongBracket{
							\abs{\FurukawaCommentSixth{\xi}^\prime}^2 \, e^{ \FurukawaCommentFifth{-} \abs{\pm 1 - x_3} s_\lambda}
						\LongBracket{
							I_2
							+ \frac{
								\epsilon \abs{\FurukawaCommentFifth{\FurukawaCommentSixth{\xi}^\prime}}
							}{
								s_{\lambda}
							} \frac{
								\FurukawaCommentFifth{\FurukawaCommentSixth{\xi}^\prime} \otimes \FurukawaCommentFifth{\FurukawaCommentSixth{\xi}^\prime}
							}{
								\abs{\FurukawaCommentFifth{\FurukawaCommentSixth{\xi}^\prime}}^2
							}
						}
						\frac{
							e^{ \FurukawaCommentFifth{-} \abs{\pm 1 - \zeta} s_\lambda}
						}{
							s_\lambda
						}
					}_{\mathcal{M}^\prime} \notag \\
					& \leq \frac{C}{
						 	\abs{ \pm 1 - x_3 }
						 	+ \abs{ \pm 1 - \zeta }
					},
				\end{align*}
			where $C> 0$ is independent of $\epsilon$.
			 Similarly, it follows from  (\ref{eq:y_lambda_epsilon}), (\ref{eq:C0712}) and Proposition \ref{prop:C05} that
				\begin{align*}
					&\sLongBracket{
						\abs{\FurukawaCommentSixth{\xi}^\prime}^2 \, \Prime{e}_{\lambda} \bracket{\FurukawaCommentFifth{\Prime{\FurukawaCommentSixth{\xi}}}, \pm 1 - x_3}
							y_{\lambda, \epsilon} \bracket{\FurukawaCommentFifth{\Prime{\FurukawaCommentSixth{\xi}}}}
							\alpha_{\epsilon, \pm} \bracket{\FurukawaCommentFifth{\Prime{\FurukawaCommentSixth{\xi}}}, \pm 1}
							\Prime{e}_{\lambda} \bracket{\FurukawaCommentFifth{\Prime{\FurukawaCommentSixth{\xi}}}, \pm 1 -\zeta}
							\frac{\epsilon \abs{\FurukawaCommentFifth{\Prime{\FurukawaCommentSixth{\xi}}}}}{ 1 + \epsilon \abs{\FurukawaCommentFifth{\Prime{\FurukawaCommentSixth{\xi}}}} }
					}_{\Prime{\mathcal{M}}} \notag \\
					&  \leq \frac{C}{
						 	\abs{ \pm 1 - x_3 }
						 	+ \abs{ \pm 1 - \zeta }
					}.
				\end{align*}
				Thus we find from Corollary \ref{cor_bound_P_N_epsilon} and Proposition \ref{prop:C03} that
				\begin{align*}
					\LongNorm{
						\nabla_H \otimes \nabla_H I_j
					}{L^q(\Omega)}
					\leq C \Vert
						f
					\Vert_{L^q (\Omega)}, \quad j = 1,2,
				\end{align*}
				where $\nabla_H = (\partial_1, \partial_2)^T$, $I_j$ is defined in (\ref{eq_formula_V_lambda_epsilon_tangential_term}) and $C>0$ is independent of $\epsilon$.
				Since
				\begin{align*}
					\partial_3 e_{\lambda}^\prime (n^\prime, \pm  1- x_3)
					& = \frac{\pm e^{- (1 \mp x_3)s_\lambda}}{2}, \\
					\partial_3^2 e_{\lambda}^\prime (n^\prime, \pm 1- x_3)
					& = \frac{ s_\lambda e^{- (1 \mp x_3)s_\lambda}}{2},
				\end{align*}
				\FurukawaCommentSeventh{we} can use the same way as above to get
				\begin{align*}
					\LongNorm{
						\nabla_H \partial_3 I_j
					}{L^q(\Omega)}
					+ \LongNorm{
						\partial_3^2 I_j
					}{L^q(\Omega)}
					\leq C \Vert
						f
					\Vert_{L^q (\Omega)}, \quad j = 1,2,
				\end{align*}
				where $I_j$ is defined in (\ref{eq_formula_V_lambda_epsilon_tangential_term}) and $C>0$ is independent of $\epsilon$.
				Propositions \ref{prop_nabla2_resolvent_estimate} \FurukawaCommentSeventh{and} \ref{prop:C1500}, Lemma \ref{lem:C0710} and the trace theorem imply
				\begin{align*}
					R_{\lambda, \epsilon}
					\sLongBracket{
							\gamma K_{\lambda, \epsilon} E_0 f
							- \gamma \Pi_{\epsilon} K_{\lambda, \epsilon} E_0 f
					} \in W^{3 - 1 / q, q} (\Torus^2)
				\end{align*}
				and its norm is bounded uniformly on $\epsilon$.
				\FurukawaCommentSeventh{By} the definition of the operator $L_{\lambda, \epsilon}$, see (\ref{eq:C1506}), \FurukawaCommentSeventh{we have}
				\begin{align*}
					L_{\lambda, \epsilon} R_{\lambda, \epsilon}
					\sLongBracket{
							\gamma K_{\lambda, \epsilon} E_0 f
							- \gamma \Pi_{\epsilon} K_{\lambda, \epsilon} E_0 f
					}
				\end{align*}
				solves the elliptic equations $\lambda u - \Delta u =0$.
				Moreover, the boundary data belongs to $W^{3 - 1 / q, q} (\Torus^2)$ by (\ref{eq_definition_of_e_prime_lamda}), (\ref{eq:y_lambda_epsilon}) and Proposition \ref{prop_Fourier_Multiplier_Theorem_Discrete}.
				Thus we find from (\ref{eq:C1510}), Corollary \ref{cor_bound_P_N_epsilon} and smoothing effect of the solution operator to the elliptic equation that
				\begin{align*}
					& \LongNorm{W_{\lambda, \epsilon} R_{\lambda, \epsilon}
					\sLongBracket{
							\gamma K_{\lambda, \epsilon} E_0 f
							- \gamma \Pi_{\epsilon} K_{\lambda, \epsilon} E_0 f
					}}{W^{2, q} (\Omega)} \\
					& \leq C \LongNorm{L_{\lambda, \epsilon} R_{\lambda, \epsilon}
					\sLongBracket{
							\gamma K_{\lambda, \epsilon} E_0 f
							- \gamma \Pi_{\epsilon} K_{\lambda, \epsilon} E_0 f
					}}{W^{2, q} (\Omega)} \\
					& \leq C \LongNorm{R_{\lambda, \epsilon}
					\sLongBracket{
							\gamma K_{\lambda, \epsilon} E_0 f
							- \gamma \Pi_{\epsilon} K_{\lambda, \epsilon} E_0 f
					}}{W^{2 - 1/ q + \delta, q} (\Torus^2)} \\
					& \leq C \LongNorm{
						f
					}{L^q (\Omega)},
				\end{align*}
			where $\delta>0$ is small and $C$ is independent of $\epsilon$.
		\end{proof}}
		\FurukawaCommentFifth{
		\begin{lemma} \label{lem_nabla2_resolvent_f}
			Let $1 < q < \infty$, $0 < \epsilon \leq 1$, \FurukawaCommentSeventh{$0< \theta < \pi / 2$} and $\lambda \in \Sigma_\theta$ satisfying $\abs{\lambda} > R$ for sufficiently large $R>0$.
			Then there exists a constant $C = C \bracket{ q, \theta }$ \FurukawaCommentSeventh{such that}
				\begin{align*}
					\LongNorm{
						\nabla^2 \left(
							\lambda + A_\epsilon
						\right)^{-1} f
					}{\Lebesgue{q}(\Omega)}
					\leq C \LongNorm{
						f
					}{
						\Lebesgue{q} (\Omega)
					}
				\end{align*}
			for all $f \in \Lebesgue{q} (\Omega)$.
		\end{lemma}
		\begin{proof}
			This is a direct consequence of Propositions \ref{prop_nabla2_resolvent_estimate}, \ref{prop_nabla2_resolvent_estimate_2} and \ref{prop_nabla_2_V_lambda_epsilon}.
		\end{proof}
		}

	\FurukawaCommentFifth{
	\begin{lemma} \label{lem_apriori_estimate_nabla2}
		Let $1 < q < \infty$, $0 < \epsilon \leq 1$.
		Then there exists a constant \FurukawaCommentSixth{$C = C \bracket{ q}$} \FurukawaCommentSeventh{such that}
		\begin{align*}
			\Vert
				\nabla^2 u
			\Vert_{\FurukawaCommentSixth{L^q (\Omega)}}
			\leq C \Vert
				A_\epsilon u
			\Vert_{L^q (\Omega)}
		\end{align*}
		for all $u \in D(A_\epsilon)$.
	\end{lemma}}
	\FurukawaCommentFifth{
	\begin{proof}[Proof of Lemma \ref{lem:A1001}]
		Let $u$ be a solution of (\ref{eq_LE}).
		Our uniform BIP yields
		\begin{align*}
			\Vert
				\partial_t u
			\Vert_{\mathbb{E}_0 (T)}
			+ \Vert
				A_\epsilon u
			\Vert_{\mathbb{E}_0 (T)}
			\leq C \left(
				\Vert
					f
				\Vert_{\mathbb{E}_0 (T)}
				+ \Vert
					u_0
				\Vert_{X_\gamma}
			\right)
		\end{align*}
		by the Dore-Venni theory, where $C>0$ is independent of $\epsilon$ and $T$.
		Applying an a priori estimate Lemma \ref{lem_apriori_estimate_nabla2}, we can replace $\Vert A_\epsilon u \Vert_{\mathbb{E}_0 (T)}$ by $\Vert \nabla^2 u \Vert_{\mathbb{E}_0 (T)}$.
		Since $(u, \pi)$ solves (\ref{eq_LE}) and $\partial_t u$ and $\nabla^2 u$ are controlled, we are able to estimate $\Vert \nabla_\epsilon \pi \Vert_{\mathbb{E}_0 (T)}$.
		This completes the proof of Lemma \ref{lem:A1001}.
	\end{proof}
	}
	\FurukawaCommentFifth{
	It remains to prove Lemma \ref{lem_apriori_estimate_nabla2}.
	We first observe an a priori estimate slightly weaker than Lemma \ref{lem_apriori_estimate_nabla2}, which can be proved by using the resolvent estimate Lemma \ref{lem_nabla2_resolvent_f}.
	}
	\FurukawaCommentFifth{
	\begin{proposition} \label{prop_anisotropic_Stokes_lambda_equal_0}
		Let $1 < q < \infty$ and $0 < \epsilon \leq 1$.
		There exists a unique solution $(u, \pi) \in D (A_\epsilon) \times L^q (\Omega) / \Real$ to
		\begin{align} \label{eq_anisotropic_Stokes_lambda_equal_0}
			\begin{array}{rcll}
				 - \Delta u + \nabla_\epsilon \pi
				&=
				& f
				&  \quad \mathrm{in} \quad \Omega , \\
				\mathrm{div}_\epsilon \, u
				&=
				& 0
				& \quad \mathrm{in} \quad \Omega, \\
				u
				&=
				& 0
				& \quad  \mathrm{on} \quad \partial \Omega,
			\end{array}
		\end{align}
		\FurukawaCommentSixth{for $f \in L^q (\Omega)$,} such that
		\begin{align*}
			\Vert
				\nabla^2 u
			\Vert_{L^q (\Omega)}
			+ \Vert
				\nabla_\epsilon \pi
			\Vert_{L^q (\Omega)}
			\leq C \Vert
				f
			\Vert_{L^q (\Omega)}
			+ C \Vert
				u
			\Vert_{L^q (\Omega)},
		\end{align*}
		where $C > 0$ is independent of $\epsilon$ \FurukawaCommentSixth{and $f$}.
	\end{proposition}}
	\FurukawaCommentFifth{
	\begin{proof}
		The equations are equivalent to
		\begin{align*}
			\begin{array}{rcll}
				\lambda_0 u - \Delta u + \nabla_\epsilon \pi
				&=
				& f + \lambda_0 u
				&\quad \mathrm{in} \quad \Omega,  \\
				\mathrm{div}_\epsilon \, u
				&=
				& 0
				&\quad \mathrm{in} \quad \Omega, \\
				u
				&=
				& 0
				& \quad \mathrm{on} \quad \partial \Omega,
			\end{array}
		\end{align*}
		\FurukawaCommentSixth{for sufficiently large $\lambda_0 > 0$.}
		We find from Lemma \ref{lem_nabla2_resolvent_f} that
		\begin{align*}
			\Vert
				\nabla^2 u
			\Vert_{L^q (\Omega)}
			& \leq C \Vert
				\nabla^2 (\lambda_0 + A_\epsilon)^{-1}
			\Vert_{L^q (\Omega) \rightarrow L^q (\Omega)}
			\Vert
				f + \lambda_0 u
			\Vert_{L^q (\Omega)} \\
			& \leq C \left(
			 \Vert
					f
				\Vert_{L^q (\Omega)}
				+ \lambda_0 \Vert
					u
				\Vert_{L^q (\Omega)}
			\right)
		\end{align*}
		for some constant $C> 0$, which is independent of $\epsilon$.
		The first equation in (\ref{eq_anisotropic_Stokes_lambda_equal_0}) implies
		\begin{align*}
			\Vert
				\nabla_\epsilon \pi
			\Vert_{L^q (\Omega)}
			& \leq\Vert
				\nabla^2 u
			\Vert_{L^q (\Omega)}
			+ \Vert
				f
			\Vert_{L^q (\Omega)} \\
			& \leq C \left(
			 \Vert
					f
				\Vert_{L^q (\Omega)}
				+ \lambda_0 \Vert
					u
				\Vert_{L^q (\Omega)}
			\right).
		\end{align*}
		\FurukawaCommentSixth{
	For uniqueness we multiply $u$ with the first equation and integrating by parts yields $\nabla u = 0$.
	By the Poincar\'{e} inequality it implies $u = 0$.
	This argument works for $q \geq 2$ since $\Omega$ is bounded.
	Since $(\lambda_0 + A_\epsilon)^{-1}$ is compact in $L^q (\Omega)$, the Riesz-Schauder theorem implies that $0$ is in resolvent since $\mathrm{ker} A_\epsilon = \{ 0 \}$.
	In particular, (\ref{eq_anisotropic_Stokes_lambda_equal_0}) is uniquely solvable for any $f \in L^q (\Omega)$ for $q \geq 2$.
	By duality argument the solvability of $q \geq 2$ implies the uniqueness of (\ref{eq_anisotropic_Stokes_lambda_equal_0}) for $1 < q < 2$.
	Again by compactness of $(\lambda_0 + A_\epsilon)^{-1}$ the solvability for (\ref{eq_anisotropic_Stokes_lambda_equal_0}) follows.
	}
	\end{proof}	}
	\FurukawaCommentFifth{
	\begin{proof}[Proof of Lemma \ref{lem_apriori_estimate_nabla2}]
		Assume that the statement were false then there would exist a sequence $\{ \epsilon_k \}_{k\in \Integer_{\geq 1}}$, ($0 < \epsilon_k \leq 1$) and $u_k \in D(A_\epsilon)$ such that
		\begin{align*}
			\Vert
				\nabla^2 u_k
			\Vert_{L^q (\Omega)}
			> k \Vert
				f_k
			\Vert_{L^q (\Omega)}, \quad
			f_k
			= A_{\epsilon_k} u_k.
		\end{align*}
		Since the problem is linear we may assume that
		\begin{align*}
			\Vert
				\nabla^2 u_k
			\Vert_{L^q (\Omega)}
			\equiv 1, \quad
			\Vert
				f_k
			\Vert_{L^q (\Omega)}
			\leq \frac{1}{k}
			\rightarrow 0,  \,
			(k \rightarrow 0).
		\end{align*}
		By $A_{\epsilon_k} u_k = f_k$ and Proposition \ref{prop_anisotropic_Stokes_lambda_equal_0}, we have
		\begin{align*}
			1
			\leq \alpha \left(
				\Vert
					f_k
				\Vert_{L^q (\Omega)}
				+ \Vert
					u_k
				\Vert_{L^q (\Omega)}
			\right)
		\end{align*}
		for some constant $\alpha>0$, which is independent of $\epsilon_k$.
		Letting $k \rightarrow \infty$ implies
		\begin{align} \label{eq_lower_bound_1_u_k}
			\frac{1}{\alpha}
			\leq \liminf_{k \rightarrow \infty} \Vert
				u_k
			\Vert_{L^q (\Omega)}.
		\end{align}
		By the Poincar\'{e} inequality for $u_k$ our bound $\Vert \nabla u_k \Vert_{L^q (\Omega)}$ implies that $u_k$ and $\nabla u_k$ are bounded in $L^q (\Omega)$.
		By Rellich's compactness theorem, we observe that $u_k \rightarrow u$ for some $u \in L^q (\Omega)$ strongly in $L^q (\Omega)$ by taking a subsequence.
		The estimate (\ref{eq_lower_bound_1_u_k}) implies that
		\begin{align*}
			\Vert
				u
			\Vert_{L^q (\Omega)}
			\geq \frac{1}{\alpha}.
		\end{align*}
		We may assume $\epsilon_k \rightarrow \epsilon_\ast \in [0, 1]$ and $u_k \rightarrow u$ as $k \rightarrow \infty$ by taking a subsequence.
		The situation is divided into two cases, i.e. $\epsilon_\ast = 0$ or $\epsilon_\ast > 0$.
		By definition,
		\begin{align*}
			\begin{array}{rcll}
				 - \Delta u_k + \nabla_{\epsilon_k} \pi_k
				&=
				& f_k
				& \quad  \mathrm{in} \quad \Omega , \\
				\mathrm{div}_\epsilon \, u_k
				&=
				& 0
				& \quad  \mathrm{in} \quad \Omega, \\
				u_k
				&=
				& 0
				&  \quad \mathrm{on} \quad \partial \Omega,
			\end{array}
		\end{align*}
		with some function $\pi_k$ satisfying $\int_{\Omega} \pi_k dx = 0$.
		Since $\Vert \nabla^2 u_k \Vert_{L^q (\Omega)} \leq 1$, we see that
		\begin{align*}
			\Vert
				\nabla_{\epsilon_k} \pi_k
			\Vert_{L^q (\Omega)}
			\leq \Vert
				f
			\Vert_{L^q (\Omega)}
			+ 1.
		\end{align*}
		By the Poincar\'{e} inequality $\{ \pi_k \}$ is bounded in $L^q (\Omega)$.
		By Rellich's compactness theorem we may assume $\pi_k \rightarrow \pi$ in $L^q (\Omega)$ for some \FurukawaCommentSixth{$\pi \in L^q (\Omega)$} strongly by taking a subsequence.
		If $\epsilon_\ast = 0$, this implies $\pi$ is independent of \FurukawaCommentSixth{$z$}.
		Since $\mathrm{div}_{\epsilon_k} \, u_k = 0$ and the vertical component $w_k = 0$ on $x_3 = \pm 1$, integration vertically on $(-1 , 1)$ yields that the horizontal limit $v$ satisfies
		\begin{align*}
			\mathrm{div}_H \overline{v} = 0,
		\end{align*}
		where $\mathrm{div}_H = \nabla_H \cdot$.
		Thus the horizontal component $v$ satisfies the hydrostatic Stokes equations
		\begin{align*}
			\begin{array}{rcll}
				 - \Delta u + \nabla_H \pi
				&=
				& 0
				& \quad  \mathrm{in} \quad \Omega , \\
				\mathrm{div}_H \, \overline{v}
				&=
				& 0
				&  \quad \mathrm{in} \quad \Omega, \\
				v
				&=
				& 0
				&  \quad \mathrm{on} \quad \partial \Omega.
			\end{array}
		\end{align*}
		Since we know the only possible $W^{2 , q}$-solution is zero, so we conclude that $v = 0$.
		Since $\Vert \nabla v_k \Vert_{L^q (\Omega)}$ is bounded, $\mathrm{div}_{\epsilon_k} $-free condition implies that the horizontal limit $w$ is independent of the vertical variable.
		By the boundary condition $w = 0$ at $x_3 = \pm 1$, this implies $w$ must be zero.
		We thus observe that $u_k \rightarrow 0$ strongly in $L^q (\Omega)$, this contradicts $\Vert u \Vert_{L^q (\Omega)} \geq 1 / \alpha > 0$.
		The case $\epsilon_\ast$ is easier since the limit satisfies the anisotropic Stokes equations
		\begin{align*}
			\begin{array}{rcll}
				 - \Delta u + \nabla_{\epsilon_\ast} \pi
				&=
				& 0
				&  \quad \mathrm{in} \quad \Omega , \\
				\mathrm{div}_{\epsilon_\ast} \, u
				&=
				& 0
				&  \quad \mathrm{in} \quad \Omega, \\
				u
				&=
				& 0
				& \quad  \mathrm{on} \quad \partial \Omega.
			\end{array}
		\end{align*}
		By the uniqueness $u \equiv 0$ in $\Omega$.
		This again contradicts $\Vert u \Vert_{L^q (\Omega)} \geq 1 / \alpha > 0$.
		The proof of Lemma \ref{lem_apriori_estimate_nabla2} is now complete.
	\end{proof}	}
	\FurukawaCommentSixth{As a direct application of Lemma \ref{lem:A1001} we obtain}
		\begin{corollary} \label{cor_mr_estimate}
		Let $p,q\in (1,\infty )$, $T>0$, $F = (f_H, f_z) \in \mathbb{E} _0(T)$, $U_0\in X_\gamma $ and \FurukawaCommentFourth{$0<\epsilon \leq 1$}.
		Then there is a unique solution $( U_\epsilon,
		P_\epsilon) \in \mathbb{E}_1 (T) \times \mathbb{E}_0(T)$ to the \FurukawaCommentFourth{equations}
			\begin{equation} \label{eq_diff-eq_abstract_2}
				\left \{
					\begin{array}{rclll}
						\partial _t V-\Delta V + \nabla_{H} P
						& =
						& f_H
						&\mathrm{ in }
						& \Omega \times  (0,T) ,\\
						\partial _t (\epsilon W) -\Delta (\epsilon W) + \frac{\partial_3 P}{\epsilon}
						& =
						& f_z
						&\mathrm{in}
						& \Omega \times (0, T) ,\\
						\mathrm{div}_H \, V + \frac{\partial_3}{\epsilon
						} ({\FurukawaCommentFourth{\epsilon}} W)
						&=
						& 0
						&\mathrm{ in }
						& \Omega \times (0, T),\\
						U
						& =
						& 0
						&  \mathrm{on}
						& \partial \Omega \times (0, T) \FurukawaCommentFourth{,} \\
						U (0)
						& =
						& U_0
						&\mathrm{ in }
						& \Omega ,
					\end{array}\right .
			\end{equation}
		where $P$ is unique up to a constant.
		Moreover, there exist constants $C>0$ and $C_T>0$, which is independent of $\epsilon$,  such that
		\begin{equation} \label{maximal_regularity_of_anisotropic_Stokes_2}
			\Vert \left(
				V, {\FurukawaCommentFourth{\epsilon}} W
			\right)
			\Vert _{\mathbb{E}_1(T)}
			+ \Vert
				\nabla_{{\FurukawaCommentFourth{\epsilon}}} P \Vert_{\mathbb{E}_0(T)}
			\le  C\Vert F \Vert _{\mathbb{E}_0(T)}
			+ C_T \Vert
			\left(
				V_0, \epsilon W_0
			\right)
			\Vert _{X_\gamma }.
		\end{equation}
	\end{corollary}

	\begin{proof}
		Lemma \ref{lem:A1001} implies there exists a solution $(\tilde{U}, \tilde{P})$ to (\ref{eq_LE}) with initial data $U_0$ such that
		\begin{equation*}
			\Vert
				\widetilde{U}

			\Vert _{\mathbb{E}_1(T)}
			+ \Vert
				\nabla_{{\FurukawaCommentFourth{\epsilon}}} \widetilde{P} \Vert_{\mathbb{E}_0(T)}
			\le  C\Vert F \Vert _{\mathbb{E}_0(T)}
			+ C_T \Vert

				U_0

			\Vert _{X_\gamma }.
		\end{equation*}
		Set
		\begin{align*}
			V = \tilde{V}, \quad
			W = \epsilon \tilde{W}, \quad
			P = \tilde{P}.
		\end{align*}
		Then $(U, P)$ is the desired solution satisfying (\ref{maximal_regularity_of_anisotropic_Stokes_2}).
		\FurukawaComment{Note that $\lim_{T \rightarrow \infty} C_T < \infty$.}
	\end{proof}

	\section{Non-linear Estimates and Regularity of w} \label{section_improved_regularity_w}

	In this section\FurukawaCommentFourth{,} we introduce some Propositions on non-linear estimates to estimate $F_H$, $F_z$ and $F$ and on the regularity of $w$, which is \FurukawaCommentFourth{the} vertical component of the solution to the primitive equations.
	Although the following Propositions have already proved in \cite{FurukawaGigaHieberHusseinKashiwabaraWrona2018}, we introduce them to explain our restriction for $p$ and $q$ and for \FurukawaCommentFourth{the reader's convenience}.

	\begin{proposition}[\cite{FurukawaGigaHieberHusseinKashiwabaraWrona2018}] \label{prop:0201}
		Let $T > 0$, $p, q \in \bracket{ 1, \infty } $ such that $2 / 3p + 1 / q \leq 1$.
		Then there exist a constant $C = C (p, q) > 0$ such that
			\begin{align*}
				\LongNorm{ v_1 \partial_x v_2}{\mathbb{E}_0(T)} \leq C \LongNorm{v_1}{\mathbb{E}_1 (T)} \LongNorm{v_2}{\mathbb{E}_1 (T)}
			\end{align*}
		\FurukawaCommentFourth{for all $v_1, v_2 \in \mathbb{E}_1(T)$.}
	\end{proposition}

	\begin{proposition}[\cite{FurukawaGigaHieberHusseinKashiwabaraWrona2018}] \label{prop:0202}
		Let $T > 0$ and $z \in \bracket{ - 1, 1}$. Let $p, q \in \bracket{ 1, \infty }$ such that $ 1 / p + 1 / q \leq 1 $. Then there exist a constant $C = C (p, q) > 0$ such that
			\begin{align*}
				\LongNorm{ w_1 \partial_3 v_2}{\mathbb{E}_0 (T)} \leq C \LongNorm{ v_1 }{\mathbb{E}_1 (T)} \LongNorm{ v_2 }{\mathbb{E}_1 (T)}
			\end{align*}
		\FurukawaCommentFourth{for all $v_1, v_2 \in \mathbb{E}_2 (T)$ and $ w_1 : = - \TimeInt{z}{-1}{ \mathrm{div}_H v_1 }{\zeta} $.}
 	\end{proposition}

		The \FurukawaCommentFourth{restriction} for $p$ and $q$ in our theorem is due to \FurukawaCommentFifth{Propositions \ref{prop:0201} and \ref{prop:0202}}.

	Let us show $w \in \mathbb{E}_1 \bracket{T}$.
	In our previous paper \cite{FurukawaGigaHieberHusseinKashiwabaraWrona2018}, we first derive the equation which $w$ satisfies by applying $\TimeInt{-1}{x_3}{\mathrm{div}_H \, \cdot}{\zeta}$ to the equations $v$ satisfies.
	Then, estimating the corresponding non-linear terms and applying the maximal regularity principle, we obtain $w \in \mathbb{E}_1 \bracket{T}$.
	\FurukawaCommentThird{
		Note that, in the present paper, we invoke additional regularity for $v$ to deal with the trace of the second derivative.
	}
	\FurukawaCommentSecond{
		Although, in \cite{GigaGriesHieberHusseinKashiwabara2017_analiticity}, the authors treat higher order regularity of the solution to the primitive equations, \FurukawaCommentThird{they do not explicitly write the maximal regularity in fractional Sobolev spaces.}
		\FurukawaCommentThird{However, it is easy to modify their proof to get the maximal regularity in the fractional Sobolev spaces.}
		In \cite{GigaGriesHieberHusseinKashiwabara2017_H_infty}, the argument to get $H^\infty$-calculus of hydrostatic Stokes operator is based on $H^\infty$-calculus for the Laplace operator and perturbations arguments.
		Since the Laplace operator admits $H^\infty$-calculus in fractional Sobolev spaces, \FurukawaCommentThird{it is not difficult to establish $H^{\infty}$-calculus of the hydrostatic Stokes operator in fractional Sobolev spaces.}
		We also find local well-posedness of the primitive equations in \FurukawaCommentThird{fractional maximal regularity space $W^{1, p} (0, T ;W^{s \FurukawaCommentFifth{, q}} (\Omega) ) \cap L^{p} (0, T ;W^{2 + s, q} (\Omega) )$ for $s > 1 / q$ \FurukawaCommentThird{in the same way} \cite{GigaGriesHieberHusseinKashiwabara2017_analiticity} to get local well-posedness, namely, use Lemma 6.1, Corollary 6.2 and Theorem 5.1 in \cite{GigaGriesHieberHusseinKashiwabara2017_analiticity}.}}

	\begin{remark}
		It \FurukawaCommentFifth{is} already known that $v \in \mathbb{E}_1 \FurukawaCommentFifth{(T)}$ with initial data $v_0 \in X_{\gamma}$ by Giga, et al. \cite{GigaGriesHieberHusseinKashiwabara2017_H_infty}.
	\end{remark}

	\begin{proof}[Proof of Lemma \ref{lem:A1002}]

 		Integrating (PE) both sides over $\bracket{-1, 1}$, we find $\bracket{\overline{v}, \overline{\pi}}$ \FurukawaCommentFifth{satisfy}
			\begin{align} \label{eq:B0801}
				\begin{array}{rcll}
					\dt \overline{v} - \Delta \overline{v} + \nabla_H \overline{\pi}
					& =
					& - \TimeInt{-1}{1}{ v \cdot \nabla_H v + w \partial_3 v
						}{\zeta}
					&  \\
					\,
					& \,
					&  + \bracket{\partial_3 v} |_{x_3 = -1}^{x_3 = 1}
					& \In{\Omega \times \bracket{0, T}}\FurukawaCommentFourth{,} \\
					\mathrm{div}_H \overline{v}
					& =
					& 0
					&\In{\Omega \times \bracket{0, T}}\FurukawaCommentFourth{,} \\
					\overline{v}
					& =
					& 0
					&\On{ \partial \Omega \times \bracket{0, T}}\FurukawaCommentFourth{,} \\
					\overline{v} (0)
					& =
					& \overline{v_0}
					& \quad \mathrm{in} \quad \Omega.
				\end{array}
			\end{align}
 		 Put $\tilde{v} =  v - \overline{v}$.
 		 Then, $\tilde{u} = (\tilde{v}, w)$ satisfies
 		\begin{align} \label{eq;B1001}
 			\begin{array}{rcll}
	 			\dt \tilde{v} - \Delta \tilde{v}
 				& =
 				& - \tilde{v} \cdot \nabla_H\tilde{v}
 				- w \partial_z \tilde{v}
 				- \overline{v} \cdot \nabla_H \tilde{v}
 				- \tilde{v} \cdot \nabla_H \overline{v}
 				& \, \\
 				\,
 				& \,
 				& - \frac{1}{2}
 				\TimeInt{-1}{1}{
 					\tilde{v} \cdot \nabla_H \tilde{v}
 					- \bracket{ \mathrm{div}_H \, \tilde{v} }
 				}{\zeta}
 				& \, \\
 				\,
 				& \,
 				& +  \frac{1}{2} \bracket{\partial_3 v} |_{x_3 = -1}^{x_3 = 1}
 				& \In{\Omega \times \bracket{0, T}}, \\
 				\mathrm{div}_H \, \tilde{v} + \partial_z w
 				& =
 				& 0
 				& \In{\Omega \times \bracket{0, T}} \FurukawaCommentFourth{,} \\
 				\tilde{v}
				& =
				& 0
				&\On{ \partial \Omega \times \bracket{0, T}} \FurukawaCommentFourth{,} \\
 				\tilde{v} (0)
 				& =
 				& v (0) - \overline{v_0}
 				& \In{\Omega} \FurukawaCommentFourth{.}
 			\end{array}
 		\end{align}
 		Note that the pressure term no longer appears in the above equations and
 		\begin{align*}
 			\mathrm{div}_H \, \tilde{v} + \partial_3 w
 			= 0.
		\end{align*}
 		Applying $ - \mathrm{div}_H $ to (\ref{eq;B1001}) and integrating over $(-1, x_3)$ with respect to vertical variable, we find
 		\begin{align*}
 			& \dt w - \Delta w \notag \\
 			& = \FurukawaComment{
 				\partial_z \mathrm{div}_H \tilde{v} |_{x_3 = - 1}
 				- \int_{-1}^{x_3}
 					\frac{1}{2} \mathrm{div}_H  \left[
 						\bracket{\partial_3 v} |_{x_3 = -1}^{x_3 = 1}
 					\right]
 					d \zeta \notag
 				}\\
 			& + \int_{-1}^{x_3}
 				\FurukawaCommentFourth{\mathrm{div}_H}
 					\bracket{
 						- \tilde{v} \cdot \nabla_H\tilde{v}
 						- w \partial_{\zeta} \tilde{v}
 						- \overline{v} \cdot \nabla_H \tilde{v}
 						- \tilde{v} \cdot \nabla_H \overline{v} }
 			d \zeta \notag \\
 			& - \frac{1}{2}
			\TimeInt{-1}{x_3}{
	 			\mathrm{div}_H
 				\TimeInt{-1}{1}{
 					\tilde{v} \cdot \nabla_H \tilde{v} - \bracket{ \mathrm{div}_H \, \tilde{v} } \tilde{v}
 				}{\zeta}
 			}{\eta}  \notag \\
 			& = : I_1 + I_2 + I_3,
		\end{align*}
		with initial data $w_0$.
		\FurukawaCommentSecond{
			Since $v_0 \in X_\gamma \cap B^{s}_{q,p}(\Omega)$ for $s > 2 - 2/ p + 1/q$, we have $v \in \mathbb{E}_1 (T) \cap L^p (0, T ; W^{\FurukawaCommentFourth{2 + 1/q +\delta}, q}(\Omega))$ \FurukawaCommentFourth{for some $\delta>0$ by \cite{GigaGriesHieberHusseinKashiwabara2017_analiticity} and \cite{GigaGriesHieberHusseinKashiwabara2017_H_infty} }, and thus $\nrm{I_1}{\mathbb{E}_0 \bracket{T}} \leq C$ for some $C>0$.}
		We use integration by parts to get
		\begin{align*}
			I_2
			& = \tilde{v} \cdot \nabla_H w
			- w \mathrm{div}_H v
			+ \overline{v} \cdot \nabla_H w \\
			& + \int_{-1}^{x_3}

					\partial_j \tilde{v} \cdot \nabla_H \tilde{v}_{\FurukawaCommentFourth{j}}
					-
					\bracket{
						\partial_{\zeta} \tilde{v}

					\cdot \nabla_H w
				}
				+ \nabla_H w \cdot \partial_{\zeta} \tilde{v}
				-

						\partial_{\zeta} w \,
						\mathrm{div}_H \tilde{v}

			\, d \zeta \\
			& + \TimeInt{-1}{x_3}{
				\partial_j \overline{v} \cdot \nabla_H \tilde{v} _{\FurukawaCommentFourth{j}}
				+ \partial_j \tilde{v} \cdot \nabla_H \overline{v}_j
			}{\zeta}.
		\end{align*}
	\FurukawaCommentFourth{for $j = 1, 2$.}
	We can apply \FurukawaCommentFourth{Propositions} \ref{prop:0201} and \ref{prop:0202} to $I_2$ to get
		\begin{align} \label{eq_estimate_I_2_improved_reg}
			\LongNorm{
				I_2
			}{\mathbb{E}_0 \bracket{T}}
			\leq C
			\LongBracket{
				\nrm{w}{\mathbb{E}_1 \bracket{T}} \LongNorm{\tilde{v}}{\mathbb{E}_1 \bracket{T}}
				+ \LongNorm{\tilde{v}}{\mathbb{E}_1 \bracket{T}}^2
			}.
		\end{align}
	Similarly, we have
		\begin{align} \label{eq_estimate_I_3_improved_reg}
			\LongNorm{
				I_3
			}{\mathbb{E}_0 \bracket{T}}
			\leq C
			\LongBracket{
				\nrm{w}{\mathbb{E}_1 \bracket{T}} \LongNorm{\tilde{v}}{\mathbb{E}_1 \bracket{T}}
				+ \LongNorm{\tilde{v}}{\mathbb{E}_1 \bracket{T}}^2
			}.
		\end{align}
	Note that constants in (\ref{eq_estimate_I_2_improved_reg}) and (\ref{eq_estimate_I_3_improved_reg}) are independent of $T$ since constants in \FurukawaCommentFourth{Propositions} \ref{prop:0201} and \ref{prop:0202} are independent of \FurukawaComment{$T$}.
	Thus we find from the maximal regularity of the heat equation, implicit function theorem and Neumann series argument, which is the same way as in Proposition 4.8 in \cite{FurukawaGigaHieberHusseinKashiwabaraWrona2018}, that
		\begin{align*}
			\Vert
				w
			\Vert_{\mathbb{E}_1 (T)}
			\leq C
		\end{align*}
	for some \FurukawaComment{$C > 0$}.
 	\end{proof}

 	\section{Justification of the Hydrostatic approximation and Global-well-posedness of the anisotropic Navier-Stokes Equations} \label{section_proof_of_main_thoerem}

	Let us prove our main theorem.

	\begin{proof}[Proof of Theorem \ref{thm:A0105}]
		Let $\mathcal{T} > 0$.
		Let $C_1$ be the maximum of constants $C$ in \FurukawaCommentFourth{Propositions} \ref{prop:0201} and \ref{prop:0202},  (\ref{maximal_regularity_of_anisotropic_Stokes_2}) and the constant in the trace theorem.
		Let us construct a solution $(V_{\FurukawaCommentFourth{\epsilon}}, {\FurukawaCommentFourth{\epsilon}} W_{\FurukawaCommentFourth{\epsilon}})$ to (\ref{eq:A0502})
		with zero initial data on $[0, \mathcal{T}]$.
		Set $(u_{\FurukawaCommentFourth{\epsilon}}, p_{\FurukawaCommentFourth{\epsilon}}) := (v + V_{\FurukawaCommentFourth{\epsilon}}, w + W_{\FurukawaCommentFourth{\epsilon}}, p + P_{\FurukawaCommentFourth{\epsilon}})$, then this is the desired solution to (SNS).
		We denote by $\Vert \cdot \Vert_{\mathbb{E}_1 (mT, (m + 1)T)}$ and $\Vert \cdot \Vert_{\mathbb{E}_0 (mT, (m + 1)T)}$ the $\mathbb{E}_1$-norm and $\mathbb{E}_0$-norm on the time interval $[m T, (m + 1)T]$, respectively.
		Let us take $0 < T \leq 1$ \FurukawaCommentFifth{satisfying $\mathcal{T} = N T$ for sufficiently large integer $N$ and}
 		\begin{align} \label{Choice_of_T}
 			\Vert
 				u
 			\Vert_{\mathbb{E}_1 (m T, (m + 1) T)}
 			\leq \frac{1}{\FurukawaCommentFifth{10 C_1}},
 		\end{align}
 		for all integer \FurukawaCommentFifth{$m \in (1, N)$}.
 		This choice of $T$ is clearly independent of ${\FurukawaCommentFourth{\epsilon}}$.
 		We divide the time interval $[0, \mathcal{T}]$ into $\cup_{m = 0}^N [m T, (m + 1) T ]$.
		Put $F = F ( V_{\FurukawaCommentFourth{\epsilon}}, W_{\FurukawaCommentFourth{\epsilon}}, u) := \left( F_H ( V_{\FurukawaCommentFourth{\epsilon}}, W_{\FurukawaCommentFourth{\epsilon}}, u)), F_z ( V_{\FurukawaCommentFourth{\epsilon}}, W_{\FurukawaCommentFourth{\epsilon}}, u) \right)$ be the left hand side of (\ref{eq:A0502}).
		We denote by $\mathcal{R} (F, U_0) = (\mathcal{R}^u (F, U_0), \mathcal{R}^p (F, U_0 ) \FurukawaComment{)}= ( U, P)$ the solution to (\ref{eq_diff-eq_abstract_2}) with initial data $U_0$.
		Set inductively
		\begin{align*}
			& U_{{\FurukawaCommentFourth{\epsilon}}, 1}
			= \mathcal{R}^u (F (0, u), 0) , \quad
			P_{{\FurukawaCommentFourth{\epsilon}}, 1}
			=  \mathcal{R}^p (F (0, u), 0),\\
			& U_{{\FurukawaCommentFourth{\epsilon}}, j + 1}
			= \mathcal{R}^u (F ( U_j, u), 0), \quad
			P_{{\FurukawaCommentFourth{\epsilon}}, j + 1}
			= \mathcal{R}^p (F (U_j, u), 0) \FurukawaCommentFourth{.}
		\end{align*}
		\FurukawaCommentFourth{Propositions} \ref{prop:0201}, \ref{prop:0202} and \FurukawaCommentSeventh{Corollary} \ref{cor_mr_estimate} lead \FurukawaCommentFifth{to}
		\begin{align}
			&\Vert
				\left(
				V_{{\FurukawaCommentFourth{\epsilon}}, j + 1}
				, {\FurukawaCommentFourth{\epsilon}} W_{ {\FurukawaCommentFourth{\epsilon}}, j + 1}
				\right)
			\Vert_{\mathbb{E}_1 (T)}
			+ \Vert
				\nabla_{\FurukawaCommentFourth{\epsilon}} P_{{\FurukawaCommentFourth{\epsilon}}, j + 1}
			\Vert_{\mathbb{E}_0 (T)} \notag \\
			& \leq C_1 T^\eta \left(
				\Vert
					u
				\Vert_{\mathbb{E}_1 (T)}
				\Vert
					\left(
						V_{{\FurukawaCommentFourth{\epsilon}}, j}
						, {\FurukawaCommentFourth{\epsilon}} W_{ {\FurukawaCommentFourth{\epsilon}}, j }
					\right)
				\Vert_{\mathbb{E}_1 (T)}
				+ \Vert
					\left(
						V_{{\FurukawaCommentFourth{\epsilon}}, j}
						, {\FurukawaCommentFourth{\epsilon}} W_{ {\FurukawaCommentFourth{\epsilon}}, j }
					\right)
				\Vert_{\mathbb{E}_1 (T)}^2
			\right) \notag \\ \label{ineq_quadratic_estimate}
 			& + {\FurukawaCommentFourth{\epsilon}} C_1 T^\eta \left(
				\Vert
					u
				\Vert_{\mathbb{E}_1 (T)}
				+ \Vert
					u
				\Vert_{\mathbb{E}_1 (T)}^2
 			\right) \FurukawaCommentFourth{.}
 		\end{align}
 		This quadratic inequality and (\ref{Choice_of_T}) \FurukawaCommentFifth{imply}
 		\begin{align} \label{estimate_max_norm_on_0_T}
			\Vert
				\left(
						V_{{\FurukawaCommentFourth{\epsilon}}, j}
						, {\FurukawaCommentFourth{\epsilon}} W_{ {\FurukawaCommentFourth{\epsilon}}, j }
					\right)
			\Vert_{\mathbb{E}_1 (T)}
			+ \Vert
				\nabla_{\FurukawaCommentFourth{\epsilon}} P_{{\FurukawaCommentFourth{\epsilon}}, j}
			\Vert_{\mathbb{E}_0 (T)}
			\leq 2 {\FurukawaCommentFourth{\epsilon}} C^\ast
 		\end{align}
 		for $C^\ast = \left( 1/4 C_1 + 1/ 16 C_1^2 \right)$ and small ${\FurukawaCommentFourth{\epsilon}} > 0$.
 		Put
 		\begin{align*}
 			& \widetilde{U}_{{\FurukawaCommentFourth{\epsilon}}, j}
 			= U_{{\FurukawaCommentFourth{\epsilon}}, j+1} - U_{{\FurukawaCommentFourth{\epsilon}}, j} \, ( j \geq 1), \quad
 			\widetilde{U}_{{\FurukawaCommentFourth{\epsilon}}, 0}
 			= U_{{\FurukawaCommentFourth{\epsilon}}, 1}\FurukawaCommentFourth{,} \\
 			& \widetilde{P}_{{\FurukawaCommentFourth{\epsilon}}, j}
 			= P_{{\FurukawaCommentFourth{\epsilon}}, j + 1} - P_{{\FurukawaCommentFourth{\epsilon}}, j } \, (j \geq 1 ), \quad
 			\widetilde{P}_{{\FurukawaCommentFourth{\epsilon}}, 0}
 			= P_{{\FurukawaCommentFourth{\epsilon}}, 1}.
 		\end{align*}
 		Then seeking the equation which $(\widetilde{U}_{{\FurukawaCommentFourth{\epsilon}}, j}, \widetilde{P}_{{\FurukawaCommentFourth{\epsilon}}, j})$ satisfies and applying Propositions \ref{prop:0201}, \ref{prop:0202} and \FurukawaCommentFifth{Corollary} \ref{cor_mr_estimate}, we have
 		\begin{align*} \label{ineq_quadratic_estimate_for_diff_eq}
 			& \Vert
 				(
						\widetilde{V}_{{\FurukawaCommentFourth{\epsilon}}, j + 1}
						, {\FurukawaCommentFourth{\epsilon}} \widetilde{W}_{ {\FurukawaCommentFourth{\epsilon}}, j + 1 }
				)
 			\Vert_{\mathbb{E}_1 (T)}
 			+ \Vert
 				\nabla_{\FurukawaCommentFourth{\epsilon}} \tilde{P}_{j + 1}
 			\Vert_{\mathbb{E}_0 (T)} \\
 			& \leq C_1 T^\eta \left(
 				\Vert
 					\left(
						V_{{\FurukawaCommentFourth{\epsilon}}, j}
						, {\FurukawaCommentFourth{\epsilon}} W_{ {\FurukawaCommentFourth{\epsilon}}, j }
					\right)
 				\Vert_{\mathbb{E}_1 (T)}
 				+ \Vert
 					\left(
						V_{{\FurukawaCommentFourth{\epsilon}}, j+ 1}
						, {\FurukawaCommentFourth{\epsilon}} W_{ {\FurukawaCommentFourth{\epsilon}}, j + 1 }
					\right)
 				\Vert_{\mathbb{E}_1 (T)}
 			\right. \\
 			& \left.
 				\quad \quad \quad \quad
 				+ 2 \Vert
 					u
 				\Vert_{\mathbb{E}_1 (T)}
 			\right) \Vert
 				(
						\widetilde{V}_{{\FurukawaCommentFourth{\epsilon}}, j}
						, {\FurukawaCommentFourth{\epsilon}} \widetilde{W}_{ {\FurukawaCommentFourth{\epsilon}}, j }
				)
 			\Vert_{\mathbb{E}_1 (T)} \\
 			& \leq \frac{3}{4}
 			\left(
 				\Vert
 				(
						\widetilde{V}_{{\FurukawaCommentFourth{\epsilon}}, j}
						, {\FurukawaCommentFourth{\epsilon}} \widetilde{W}_{ {\FurukawaCommentFourth{\epsilon}}, j }
				)
 			\Vert_{\mathbb{E}_1 (T)}
 			+ \Vert
 				\nabla_{\FurukawaCommentFourth{\epsilon}} \tilde{P}_{j}
 			\Vert_{\mathbb{E}_0 (T)}
 			\right).
 		\end{align*}
 		Thus $(U_{\FurukawaCommentFourth{\epsilon}}, P_{\FurukawaCommentFourth{\epsilon}}) : = ( \lim_{j \rightarrow \infty} U_j, \lim_{j \rightarrow \infty} P_j ) = (\sum_{j = 0} \tilde{U}_{{\FurukawaCommentFourth{\epsilon}}, j}, \sum_{j = 0} \tilde{P}_{{\FurukawaCommentFourth{\epsilon}}, j})$ exists on $[0, T]$ and satisfies
 		\begin{align}
 			\Vert
 				\left(
						V_{{\FurukawaCommentFourth{\epsilon}}}
						, {\FurukawaCommentFourth{\epsilon}} W_{ {\FurukawaCommentFourth{\epsilon}} }
					\right)
 			\Vert_{\mathbb{E}_1 (T)}
 			+ \Vert
 				\nabla_{\FurukawaCommentFourth{\epsilon}} P_{\FurukawaCommentFourth{\epsilon}}
 			\Vert_{\mathbb{E}_0 (T)}
 			\leq 2 {\FurukawaCommentFourth{\epsilon}} C^\ast.
 		\end{align}
 		By construction $(U_{\FurukawaCommentFourth{\epsilon}}, P_{\FurukawaCommentFourth{\epsilon}})$ satisfies (\ref{eq:A0502}) on $[0, T]$.
 		Moreover, by trace theorem there exists a constant $C_{tr} > 0$ such that
 		\begin{align} \label{ineq_trace_at_T}
 			\Vert
 				\left(
						V_{{\FurukawaCommentFourth{\epsilon}}} (T)
						, {\FurukawaCommentFourth{\epsilon}} W_{ {\FurukawaCommentFourth{\epsilon}} } (T)
					\right)
 			\Vert_{X_\gamma}
 			\leq C_{tr} \Vert
 				\left(
						V_{{\FurukawaCommentFourth{\epsilon}}}
						, {\FurukawaCommentFourth{\epsilon}} W_{ {\FurukawaCommentFourth{\epsilon}} }
					\right)
 			\Vert_{E_1 (0, T)}
 			\leq 2 {\FurukawaCommentFourth{\epsilon}} C^\ast C_{tr}.
 		\end{align}
 		Next let us construct the solution to (\ref{eq:A0502}) on $[ T, 2T]$ with initial data $U_{\FurukawaCommentFourth{\epsilon}} (T)$.
 		By \FurukawaCommentFourth{(\ref{ineq_trace_at_T}), we have $\Vert U_{\FurukawaCommentFourth{\epsilon}} (T) \Vert_{X_\gamma} \leq 2 {\FurukawaCommentFourth{\epsilon}}  C^\ast C_{tr} $}.
		Put $a_{{\FurukawaCommentFourth{\epsilon}}, 1} = (b_{{\FurukawaCommentFourth{\epsilon}}, 1}, c_{{\FurukawaCommentFourth{\epsilon}}, 1}) = \mathcal{R}^u (0, U_{\FurukawaCommentFourth{\epsilon}} (T))$ and $\pi_{{\FurukawaCommentFourth{\epsilon}}, 1} = \mathcal{R}^p (0, U_{{\FurukawaCommentFourth{\epsilon}}} (T))$.
		Corollary \ref{cor_mr_estimate} implies
		\begin{align} \label{ineq_a_1_pi_1_MR}
			\Vert
				\left(
					b_{{\FurukawaCommentFourth{\epsilon}}, 1},
					{\FurukawaCommentFourth{\epsilon}} c_{{\FurukawaCommentFourth{\epsilon}}, 1}
				\right)
			\Vert_{\mathbb{E}_1 (T, 2 T)}
			+ \Vert
				\nabla_{\FurukawaCommentFourth{\epsilon}} \pi_{{\FurukawaCommentFourth{\epsilon}}, 1}
			\Vert_{\mathbb{E}_0 (T, 2 T)}
			\leq 2 {\FurukawaCommentFourth{\epsilon}} C^\ast C_{tr} C_T.
		\end{align}
 		Let the vector field $a_{\FurukawaCommentFourth{\epsilon}} = (b_{\FurukawaCommentFourth{\epsilon}}, c_{\FurukawaCommentFourth{\epsilon}})$ be the solution to
		\begin{equation} \label{eq_DE_modified_2}
			\left \{
			\begin{array}{rl}
				\partial_t b_{\FurukawaCommentFourth{\epsilon}} - \Delta b_{\FurukawaCommentFourth{\epsilon}} + \nabla_H \pi_{\FurukawaCommentFourth{\epsilon}}
				& = F_H ( b_{1, {\FurukawaCommentFourth{\epsilon}}} + b_{\FurukawaCommentFourth{\epsilon}}, c_{{\FurukawaCommentFourth{\epsilon}}, 1} +c_{\FurukawaCommentFourth{\epsilon}}, u) \FurukawaCommentFourth{,} \\
				\partial_t ( {\FurukawaCommentFourth{\epsilon}} c_{\FurukawaCommentFourth{\epsilon}} ) - \Delta ( {\FurukawaCommentFourth{\epsilon}} c_{\FurukawaCommentFourth{\epsilon}} ) + \frac{\partial_3}{{\FurukawaCommentFourth{\epsilon}}} \pi_{\FurukawaCommentFourth{\epsilon}}
				& = {\FurukawaCommentFourth{\epsilon}} F_z ( b_{1, {\FurukawaCommentFourth{\epsilon}}} + b_{\FurukawaCommentFourth{\epsilon}}, c_{{\FurukawaCommentFourth{\epsilon}}, 1} +c_{\FurukawaCommentFourth{\epsilon}}, u) \FurukawaCommentFourth{,} \\
				\mathrm{div} \, a_{\FurukawaCommentFourth{\epsilon}}
				& = 0 \FurukawaCommentFourth{,} \\
				a_{\FurukawaCommentFourth{\epsilon}} (T)
				& =  0 \FurukawaCommentFourth{.}
 			\end{array}
			\right.
		\end{equation}
		Then $U_{{\FurukawaCommentFourth{\epsilon}}} = a_{{\FurukawaCommentFourth{\epsilon}}, 1}  + a_{{\FurukawaCommentFourth{\epsilon}}}$ and $P_{\FurukawaCommentFourth{\epsilon}} = \pi_{{\FurukawaCommentFourth{\epsilon}}, 1} + \pi_{{\FurukawaCommentFourth{\epsilon}}}$ is a solution to ((\ref{eq:A0502}) with initial data $U_{\FurukawaCommentFourth{\epsilon}} (T)$.
		Let us construct the solution to (\ref{eq_diff-eq_abstract_2}).

		Let $F ( b_{1, {\FurukawaCommentFourth{\epsilon}}} + b_{\FurukawaCommentFourth{\epsilon}}, c_{{\FurukawaCommentFourth{\epsilon}}, 1} +c_{\FurukawaCommentFourth{\epsilon}}, u) = \left( F_H ( b_{1, {\FurukawaCommentFourth{\epsilon}}} + b_{\FurukawaCommentFourth{\epsilon}}, c_{{\FurukawaCommentFourth{\epsilon}}, 1} +c_{\FurukawaCommentFourth{\epsilon}}, u), {\FurukawaCommentFourth{\epsilon}} F_z ( b_{1, {\FurukawaCommentFourth{\epsilon}}} + b_{\FurukawaCommentFourth{\epsilon}}, c_{{\FurukawaCommentFourth{\epsilon}}, 1} +c_{\FurukawaCommentFourth{\epsilon}}, u) \right)$
		Set inductively
 		\begin{align*}
 			& a_{{\FurukawaCommentFourth{\epsilon}}, j + 1}
 			= a_{{\FurukawaCommentFourth{\epsilon}}, 1}
 			+ \mathcal{R}^u (F ( b_{1, {\FurukawaCommentFourth{\epsilon}}} + b_{{\FurukawaCommentFourth{\epsilon}} + j}, c_{{\FurukawaCommentFourth{\epsilon}}, 1} +c_{{\FurukawaCommentFourth{\epsilon}} + j}, u), 0), \\
 			& \pi_{{\FurukawaCommentFourth{\epsilon}}, j + 1}
 			= \mathcal{R}^p (F ( b_{1, {\FurukawaCommentFourth{\epsilon}}} + b_{{\FurukawaCommentFourth{\epsilon}} + j}, c_{{\FurukawaCommentFourth{\epsilon}}, 1} +c_{{\FurukawaCommentFourth{\epsilon}} + j}, u), 0) \FurukawaCommentFourth{,}
		\end{align*}
		for $j \geq 1$.
		Applying \FurukawaCommentFourth{Propositions} \ref{prop:0201}, \ref{prop:0202} and \FurukawaCommentSeventh{Corollary} \ref{cor_mr_estimate} to (\ref{eq_DE_modified_2}), we find
		\begin{align*}
			& \Vert
				\left(
					b_{{\FurukawaCommentFourth{\epsilon}} ,j + 1},
					{\FurukawaCommentFourth{\epsilon}} c_{{\FurukawaCommentFourth{\epsilon}} ,j + 1}
				\right)
			\Vert_{\mathbb{E}_1 (T, 2T)}
			+ \Vert
				\nabla_{\FurukawaCommentFourth{\epsilon}} \pi_{{\FurukawaCommentFourth{\epsilon}}, j + 1}
			\Vert_{\mathbb{E}_0 (T, 2T)} \notag \\
			& \leq C_1 T^\eta
				\Vert
					u
				\Vert_{\mathbb{E}_1 (T, 2T)}
				\Vert
					\left(
						b_{{\FurukawaCommentFourth{\epsilon}}, 1} +  b_{{\FurukawaCommentFourth{\epsilon}} ,j},
						{\FurukawaCommentFourth{\epsilon}} ( c_{{\FurukawaCommentFourth{\epsilon}}, 1} + c_{{\FurukawaCommentFourth{\epsilon}} ,j} )
					\right)
				\Vert_{\mathbb{E}_1 (T, 2T)} \\
			&	+ C_1 T^\eta \Vert
					\left(
						b_{{\FurukawaCommentFourth{\epsilon}}, 1} +  b_{{\FurukawaCommentFourth{\epsilon}} ,j},
						{\FurukawaCommentFourth{\epsilon}} ( c_{{\FurukawaCommentFourth{\epsilon}}, 1} + c_{{\FurukawaCommentFourth{\epsilon}} ,j} )
					\right)
				\Vert_{\mathbb{E}_1 (T, 2T)}^2 \\
			& + {\FurukawaCommentFourth{\epsilon}} C_1 T^\eta \left[
				\Vert
					u
				\Vert_{\mathbb{E}_1 (T, 2T)}
				+ \Vert
					u
				\Vert_{\mathbb{E}_1 (T, 2T)}^2
			\right] \notag \\
			& \leq C_1 T^\eta
				\Vert
					\left(
						b_{{\FurukawaCommentFourth{\epsilon}} ,j},
						{\FurukawaCommentFourth{\epsilon}} c_{{\FurukawaCommentFourth{\epsilon}} ,j}
					\right)
				\Vert_{\mathbb{E}_1 (T, 2T)}^2 \\
				& + C_1 T^\eta \left(
					\Vert
						u
					\Vert_{\mathbb{E}_1 (T, 2T)}
					+ 2	\Vert
						\left(
							b_{{\FurukawaCommentFourth{\epsilon}} ,1},
							{\FurukawaCommentFourth{\epsilon}} c_{{\FurukawaCommentFourth{\epsilon}} ,1}
						\right)
					\Vert_{\mathbb{E}_1 (T, 2T)}
				\right) \Vert
					\left(
						b_{{\FurukawaCommentFourth{\epsilon}} ,j},
						{\FurukawaCommentFourth{\epsilon}} c_{ {\FurukawaCommentFourth{\epsilon}} ,j}
					\right)
				\Vert_{\mathbb{E}_1 (T, 2T)} \\
				& + {\FurukawaCommentFourth{\epsilon}} C_1 T^\eta \left[ \Vert
					u
				\Vert_{\mathbb{E}_1 (T, 2T)}
				\Vert
					\left(
						b_{{\FurukawaCommentFourth{\epsilon}} ,1},
						{\FurukawaCommentFourth{\epsilon}} c_{{\FurukawaCommentFourth{\epsilon}} ,1}
					\right)
				\Vert_{\mathbb{E}_1 (T, 2T)}
				+ \Vert
					\left(
						b_{{\FurukawaCommentFourth{\epsilon}} ,1},
						{\FurukawaCommentFourth{\epsilon}} c_{{\FurukawaCommentFourth{\epsilon}} ,1}
					\right)
				\Vert_{\mathbb{E}_1 (T, 2T)}^2
			\right] \\
			& + {\FurukawaCommentFourth{\epsilon}} C_1 T^\eta \left[
				\Vert
					u
				\Vert_{\mathbb{E}_1 (T, 2T)}
				+ \Vert
					u
				\Vert_{\mathbb{E}_1 (T, 2T)}^2
			\right] \notag \FurukawaCommentFourth{.}
		\end{align*}
		If we take ${\FurukawaCommentFourth{\epsilon}}$ so small that
		\begin{align*}
			\Vert
				a_{{\FurukawaCommentFourth{\epsilon}}, 1}
			\Vert_{\mathbb{E}_1 (T, 2T)}
			\leq2 {\FurukawaCommentFourth{\epsilon}} C^\ast C_{tr} C_T
			\leq \frac{1}{8 C_1} \FurukawaCommentFourth{,}
		\end{align*}
		we have
		\begin{align*}
			& \Vert
				\left(
					b_{{\FurukawaCommentFourth{\epsilon}} ,j + 1},
					{\FurukawaCommentFourth{\epsilon}} c_{{\FurukawaCommentFourth{\epsilon}} ,j + 1}
				\right)
			\Vert_{\mathbb{E}_1 (T, 2T)}
			+ \Vert
				\nabla_{\FurukawaCommentFourth{\epsilon}} \pi_{{\FurukawaCommentFourth{\epsilon}}, j + 1}
			\Vert_{\mathbb{E}_0 (T, 2T)}  \\
			& \leq C_1 \Vert
				\left(
						b_{{\FurukawaCommentFourth{\epsilon}} ,j},
						{\FurukawaCommentFourth{\epsilon}} c_{{\FurukawaCommentFourth{\epsilon}} ,j}
					\right)
			\Vert_{\mathbb{E}_1 (T, 2T)}^2
			+ \frac{1}{2} \Vert
				\left(
						b_{{\FurukawaCommentFourth{\epsilon}} ,j},
						{\FurukawaCommentFourth{\epsilon}} c_{{\FurukawaCommentFourth{\epsilon}} ,j}
					\right)
			\Vert_{\mathbb{E}_1 (T, 2T)}
			+ {\FurukawaCommentFourth{\epsilon}} C^\ast C_T C_{tr}
			+ {\FurukawaCommentFourth{\epsilon}} C^\ast \notag \\
			& \leq C_1 \Vert
				\left(
						b_{{\FurukawaCommentFourth{\epsilon}} ,j},
						{\FurukawaCommentFourth{\epsilon}} c_{{\FurukawaCommentFourth{\epsilon}} ,j}
					\right)
			\Vert_{\mathbb{E}_1 (T, 2T)}^2
			+ \frac{1}{2} \Vert
				\left(
						b_{{\FurukawaCommentFourth{\epsilon}} ,j},
						{\FurukawaCommentFourth{\epsilon}} c_{{\FurukawaCommentFourth{\epsilon}} ,j}
					\right)
			\Vert_{\mathbb{E}_1 (T, 2T)}
			+ {\FurukawaCommentFourth{\epsilon}} C^\ast (1 + C_T C_{tr}). \notag
		\end{align*}
		Thus, we have by induction
		\begin{align*}
			\Vert
				\left(
						b_{{\FurukawaCommentFourth{\epsilon}} ,j},
						{\FurukawaCommentFourth{\epsilon}} c_{{\FurukawaCommentFourth{\epsilon}} ,j}
					\right)
			\Vert_{\mathbb{E}_1 (T, 2T)}
			+ \Vert
				\nabla_{\FurukawaCommentFourth{\epsilon}} \pi_{{\FurukawaCommentFourth{\epsilon}}, j}
			\Vert_{\mathbb{E}_0 (T, 2T)}
			\leq 2 {\FurukawaCommentFourth{\epsilon}} C^\ast (1 + C_T C_{tr})
		\end{align*}
		for all $j \geq 1$.
		Set
		\begin{align*}
			& \tilde{a}_{{\FurukawaCommentFourth{\epsilon}}, j}
			= a_{{\FurukawaCommentFourth{\epsilon}}, j + 1}
			- a_{{\FurukawaCommentFourth{\epsilon}}, j} \quad (j \geq 1), \quad
			\tilde{a}_{{\FurukawaCommentFourth{\epsilon}}, 0}
			= a_{{\FurukawaCommentFourth{\epsilon}}, 0}  \FurukawaCommentFourth{,} \\
			& \tilde{\pi}_{{\FurukawaCommentFourth{\epsilon}}, j}
			= \pi_{{\FurukawaCommentFourth{\epsilon}}, j + 1}
			- \pi_{{\FurukawaCommentFourth{\epsilon}}, j} \quad (j \geq 1), \quad
			\tilde{\pi}_{{\FurukawaCommentFourth{\epsilon}}, 0}
			= \pi_{{\FurukawaCommentFourth{\epsilon}}, 0} \FurukawaCommentFourth{.}
		\end{align*}
		Applying \FurukawaCommentFourth{Propositions} \ref{prop:0201}, \ref{prop:0202} and \FurukawaCommentFifth{Corollary} \ref{cor_mr_estimate} to the equations that
		\begin{align*}
			(\tilde{a}_{{\FurukawaCommentFourth{\epsilon}}, j + 1}, \tilde{\pi}_{{\FurukawaCommentFourth{\epsilon}}, j + 1})
		\end{align*}
		satisfies, we find
		\begin{align*}
			&\Vert
				(
					\tilde{b}_{{\FurukawaCommentFourth{\epsilon}}, j + 1},
					{\FurukawaCommentFourth{\epsilon}} \tilde{c}_{{\FurukawaCommentFourth{\epsilon}}, j + 1}
				)
			\Vert_{\mathbb{E}_1 (T, 2 T)}
			+ \Vert
 				\nabla_{\FurukawaCommentFourth{\epsilon}} \tilde{\pi}_{{\FurukawaCommentFourth{\epsilon}}, j + 1}
			\Vert_{\mathbb{E}_0 (T, 2 T)} \\
			& \leq C_1 T^\eta \left(
				\Vert
					\left(
						b_{{\FurukawaCommentFourth{\epsilon}} ,j},
						{\FurukawaCommentFourth{\epsilon}} c_{{\FurukawaCommentFourth{\epsilon}} ,j}
					\right)
				\Vert_{\mathbb{E}_1 (T, 2 T)}
				+ \Vert
					\left(
						b_{{\FurukawaCommentFourth{\epsilon}} ,j + 1},
						{\FurukawaCommentFourth{\epsilon}} c_{{\FurukawaCommentFourth{\epsilon}} ,j + 1}
					\right)
				\Vert_{\mathbb{E}_1 (T, 2 T)}
				+ 2 \Vert
					u
				\Vert_{\mathbb{E}_1 (T, 2 T)}
			\right) \\
			& \quad \quad \times \Vert
				(
					\tilde{b}_{{\FurukawaCommentFourth{\epsilon}}, j },
					{\FurukawaCommentFourth{\epsilon}} \tilde{c}_{{\FurukawaCommentFourth{\epsilon}}, j },
				)
			\Vert_{\mathbb{E}_1 (T, 2 T)} \\
			& \leq \left[
				C_1 \left(
					\Vert
						\left(
						b_{{\FurukawaCommentFourth{\epsilon}} ,j},
						{\FurukawaCommentFourth{\epsilon}} c_{{\FurukawaCommentFourth{\epsilon}} ,j}
					\right)
					\Vert_{\mathbb{E}_1 (T, 2 T)}
					+ \Vert
						\left(
						b_{{\FurukawaCommentFourth{\epsilon}} ,j + 1},
						{\FurukawaCommentFourth{\epsilon}} c_{{\FurukawaCommentFourth{\epsilon}} ,j + 1}
					\right)
					\Vert_{\mathbb{E}_1 (T, 2 T)}
				\right)
				+ \frac{1}{2}
			\right] \\
			& \quad \quad  \times \Vert
				(
					\tilde{b}_{{\FurukawaCommentFourth{\epsilon}}, j},
					{\FurukawaCommentFourth{\epsilon}} \tilde{c}_{{\FurukawaCommentFourth{\epsilon}}, j},
				)
			\Vert_{\mathbb{E}_1 (T, 2 T)} \\
			& \leq \frac{3}{4} \Vert
				(
					\tilde{b}_{{\FurukawaCommentFourth{\epsilon}}, j},
					{\FurukawaCommentFourth{\epsilon}} \tilde{c}_{{\FurukawaCommentFourth{\epsilon}}, j },
				)
			\Vert_{\mathbb{E}_1 (T, 2 T)}.
		\end{align*}
		The last inequality holds if ${\FurukawaCommentFourth{\epsilon}}$ is sufficiently small.
		Thus,
		\begin{align*}
			( a_{{\FurukawaCommentFourth{\epsilon}}}, \pi_{\FurukawaCommentFourth{\epsilon}})
			: = (
				\lim_{ j \rightarrow \infty} a_{{\FurukawaCommentFourth{\epsilon}}, j} ,
				\lim_{ j \rightarrow \infty} \pi_{{\FurukawaCommentFourth{\epsilon}}, j}
			)
			= (
				\sum_{j = 0}\tilde{a}_{{\FurukawaCommentFourth{\epsilon}}, j},
				\sum_{j = 0} \tilde{\pi}_{{\FurukawaCommentFourth{\epsilon}}, j}
			)
		\end{align*}
		exists and satisfies (\ref{eq_DE_modified_2}) such that
		\begin{align*}
			\Vert
				\left(
						b_{{\FurukawaCommentFourth{\epsilon}}},
						{\FurukawaCommentFourth{\epsilon}} c_{{\FurukawaCommentFourth{\epsilon}}}
					\right)
			\Vert_{\mathbb{E}_1 (T, 2T)}
			+ \Vert
				\nabla_{\FurukawaCommentFourth{\epsilon}} \pi_{{\FurukawaCommentFourth{\epsilon}}}
			\Vert_{\mathbb{E}_0 (T, 2T)}
			\leq 2 {\FurukawaCommentFourth{\epsilon}} C^\ast (1 + C_T C_{tr}).
		\end{align*}
		\FurukawaCommentFourth{The functions} $(U_{\FurukawaCommentFourth{\epsilon}}, P_{\FurukawaCommentFourth{\epsilon}})$ solves (\ref{eq:A0502}) on the time interval $[T, 2 T]$ with initial data $U_{\FurukawaCommentFourth{\epsilon}} (T)$ such that
		\begin{align*}
			& \Vert
 				\left(
						V_{{\FurukawaCommentFourth{\epsilon}}}
						, {\FurukawaCommentFourth{\epsilon}} W_{ {\FurukawaCommentFourth{\epsilon}} }
					\right)
 			\Vert_{\mathbb{E}_1 (T)}
 			+ \Vert
 				\nabla_{\FurukawaCommentFourth{\epsilon}} P_{\FurukawaCommentFourth{\epsilon}}
 			\Vert_{\mathbb{E}_0 (T)} \\
			& \leq \Vert
				\left(
					b_{{\FurukawaCommentFourth{\epsilon}}, 1}
					, {\FurukawaCommentFourth{\epsilon}} c_{{\FurukawaCommentFourth{\epsilon}}, 1}
				\right)
			\Vert_{\mathbb{E}_1 (T, 2 T)}
			+ \Vert
				\left(
					b_{{\FurukawaCommentFourth{\epsilon}}}
					, {\FurukawaCommentFourth{\epsilon}} c_{{\FurukawaCommentFourth{\epsilon}}}
				\right)
			\Vert_{\mathbb{E}_1 (T, 2 T)} \\
			& + \Vert
				\nabla_{\FurukawaCommentFourth{\epsilon}} \pi_{{\FurukawaCommentFourth{\epsilon}}, 1}
			\Vert_{\mathbb{E}_0 (T, 2 T)}
			+ \Vert
				\nabla_{\FurukawaCommentFourth{\epsilon}} \pi_{\FurukawaCommentFourth{\epsilon}}
			\Vert_{\mathbb{E}_0 (T, 2 T)} \\
			& \leq C_T \Vert
				U_{\FurukawaCommentFourth{\epsilon}} (T)
			\Vert_{X_\gamma}
			+ 2 {\FurukawaCommentFourth{\epsilon}} C^\ast (1 +  C_{tr} C_T)
			\leq 2 {\FurukawaCommentFourth{\epsilon}} C^\ast ( 1 + 2 C_{tr} CT)  \FurukawaCommentFourth{.}
		\end{align*}
		By induction, the solution $(U_{\FurukawaCommentFourth{\epsilon}}, P_{\FurukawaCommentFourth{\epsilon}})$ constructed by the same way on the time interval $[m T , (m + 1) T]$ satisfies
		\begin{align*}
			& \Vert
 				\left(
						V_{{\FurukawaCommentFourth{\epsilon}}}
						, {\FurukawaCommentFourth{\epsilon}} W_{ {\FurukawaCommentFourth{\epsilon}} }
					\right)
 			\Vert_{\mathbb{E}_1 (m T, (m + 1) T)}
 			+ \Vert
 				\nabla_{\FurukawaCommentFourth{\epsilon}} P_{\FurukawaCommentFourth{\epsilon}}
 			\Vert_{\mathbb{E}_0 (m T, (m + 1) T)} \\
			& \leq 2 {\FurukawaCommentFourth{\epsilon}} C^\ast \left[
				1 + 3 C_T C_{tr} \left(
					1 + 3 C_T C_{tr} ( \cdots )

				\right)
			\right]
			=: 2 {\FurukawaCommentFourth{\epsilon}} \alpha_j \FurukawaCommentFourth{.}
		\end{align*}
		Since $\mathcal{T}$ is finite, this induction ends in finite steps.
		Thus we conclude
		\begin{align*}
		 \Vert
		 	\left(
						V_{{\FurukawaCommentFourth{\epsilon}}}
						, {\FurukawaCommentFourth{\epsilon}} W_{ {\FurukawaCommentFourth{\epsilon}} }
					\right)
		 \Vert_{\mathbb{E}_1 (\mathcal{T})}
		 + \Vert
		 	\nabla_{\FurukawaCommentFourth{\epsilon}} P_{\FurukawaCommentFourth{\epsilon}}
		 \Vert_{\mathbb{E}_1  (\mathcal{T})}
		 \leq 2 {\FurukawaCommentFourth{\epsilon}} \sum_{1 \leq j \leq \FurukawaCommentFifth{N}} \alpha_j.
		\end{align*}
	\end{proof}
\FurukawaCommentThird{
	\section*{Acknowledgements}

	The authors are grateful to Professor Matthias Hieber and Professor Amru Hussein for helpful discussions and comments.
}

config: arxiv