Let us, first, prove some preliminary results.
Lemma 4.1
Let
\(u \in C^{\infty}_{0}(\overline{\Omega})\), with
\(\overline{\Omega}= [0, + \infty[\, \times\Omega' \times \mathbb{R}^{n-m}\), and let
\(\varphi\in C^{\infty}_{0}(\mathbb{R}^{n-m})\), with
\(\operatorname{supp} \varphi \subseteq\mathbb{R}^{n-m} \backslash\mathcal{U}_{x''}\). As a result
$$\Vert \varphi A_{s} u \Vert _{L^{2}(\overline{\Omega})} \leq \frac{c_{q,r,s}}{L^{q}} \Vert u \Vert _{H^{0,0,r}(\overline{\Omega})}, \quad \forall s \in \mathbb{R}, r \in\mathbb{Z}^{-}, q \geq s+r, $$
where
L
is the distance between suppφ
and
\(\mathcal{U}_{x''}\), supposed to be greater than 1.
Proof
Let us consider
$$\begin{aligned} \varphi A_{s} u =& \frac{1}{(2 \pi)^{n-m}} \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot\xi''} \varphi\bigl(x'' \bigr) \bigl(1 +\bigl|\xi''\bigr|^{2} \bigr)^{\frac{1}{2}} \widehat {u}\bigl(x_{0}, x', \xi''\bigr)\,d\xi'' \\ =& \frac{1}{(2 \pi)^{n-m}} \int_{\mathbb {R}^{2(n-m)}} e^{i (x''-y'') \cdot\xi''} \varphi\bigl(x'' \bigr) \bigl(1 +\bigl|\xi ''\bigr|^{2} \bigr)^{\frac{1}{2}} u\bigl(x_{0}, x', y''\bigr)\,dy''\,d\xi'' \\ =& \frac{i^{2p}}{(2 \pi)^{m-n}} \int_{\mathbb{R}^{2(n-m)}} e^{i (x''-y'') \cdot\xi''} \frac{\varphi(x'') u(x_{0}, x', y'')}{|x''-y''|^{2p}} \bigl( \triangle_{\xi''} \bigl(1 +\bigl|\xi''\bigr|^{2} \bigr)^{\frac {s}{2}}\bigr)^{[p]}\,dy''\,d\xi'' \\ =& \frac{i^{2p}}{(2 \pi)^{m-n}} \int_{\mathbb{R}^{n-m}} \bigl(\triangle_{\xi''} \bigl(1 +\bigl| \xi''\bigr|^{2}\bigr)^{\frac{s}{2}} \bigr)^{[p]}\,d\xi''\\ &{}\times \int_{\mathbb{R}^{n-m}} e^{i (x''-y'') \cdot\xi''} u\bigl(x_{0}, x', y''\bigr) \frac{\psi ( \frac{|x''-y''|}{L} ) \varphi(x'')}{|x''-y''|^{2p}}\,dy'', \end{aligned}$$
where \(m \in\mathbb{N}\) and \(\psi\in C^{\infty}(\mathbb{R})\) such that \(\psi(\tau)=1\) if \(|\tau|\geq1\), \(\psi(\tau)=0\) if \(|\tau| \leq\frac{1}{2}\).
This implies
$$\varphi A_{s} u= \frac{i^{2p} \varphi(x'')}{(2 \pi)^{n-m}} \int_{\mathbb{R}^{n-m}} \bigl(\triangle_{\xi''} \bigl(1 +\bigl| \xi''\bigr|^{2}\bigr)^{\frac{s}{2}} \bigr)^{[p]} u\bigl(x_{0}, x', x''\bigr) * \psi \biggl( \frac{|x''|}{L} \biggr) \frac{e^{i x'' \cdot\xi''}}{|x''|^{2p}}\,d\xi'', $$
where the convolution is done with respect to \(x''\), and also
$$\begin{aligned} \mathcal{F}_{x''}(\varphi A_{s} u)={}& \frac{i^{2p} \widehat{\varphi}(\eta'')}{2 \pi} \\ &{} * \int_{\mathbb{R}^{n-m}} \bigl(\triangle_{\xi''} \bigl(1 +\bigl| \xi''\bigr|^{2}\bigr)^{\frac{s}{2}} \bigr)^{[p]} \widehat{u}\bigl(x_{0}, x', \eta''\bigr) \mathcal{F}_{x''} \biggl(\psi \biggl( \frac{|x''|}{L} \biggr) \frac{e^{i x'' \cdot\xi''}}{|x''|^{2p}} \biggr)\,d\xi'', \end{aligned}$$
(22)
where
$$\mathcal{F}_{x''} \biggl(\psi \biggl( \frac{|x''|}{L} \biggr) \frac{e^{i x'' \cdot\xi''}}{|x''|^{2p}} \biggr)= \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot(\xi''- \eta'')} \psi \biggl( \frac{|x''|}{L} \biggr) \frac{1}{|x''|^{2p}}\,dx''. $$
It results
$$\begin{aligned} &\bigl(1+\bigl|\xi''-\eta''\bigr|^{2r'} \bigr) \mathcal{F}_{x''} \biggl(\psi \biggl( \frac {|x''|}{L} \biggr) \frac{e^{i x'' \cdot\xi''}}{|x''|^{2p}} \biggr) \\ &\quad = \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot(\xi''- \eta'')} \psi \biggl( \frac{|x''|}{L} \biggr) \frac{1}{|x''|^{2p}}\,dx'' \\ &\qquad{} + (-1)^{r} \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot(\xi''- \eta'')} \biggl( \triangle_{x''} \biggl( \psi \biggl( \frac{|x''|}{L} \biggr) \frac{1}{|x''|^{2p}} \biggr) \biggr)^{[r]}\,dx'' \end{aligned}$$
and then
$$ \biggl\vert \mathcal{F}_{x''} \biggl(\psi \biggl( \frac{|x''|}{L} \biggr) \frac{e^{i x'' \cdot\xi''}}{|x''|^{2p}} \biggr) \biggr\vert \leq \frac{c_{r,p}}{(1+|\xi''-\eta''|^{2r'})} \biggl( \frac{1}{L} \biggr)^{2p-n+m+1}. $$
(23)
Making use of (22) and (23), we obtain
$$\begin{aligned} & \bigl\Vert \mathcal{F}_{x''}(\varphi A_{s} u) \bigr\Vert \\ &\quad = \Vert \varphi A_{s} u \Vert \\ &\quad \leq\frac{1}{(2\pi)^{n-m}} \bigl\Vert \widehat{\varphi}\bigl( \eta''\bigr)\bigr\Vert _{L^{1}(\mathbb{R}^{n-m})} \\ &\qquad{}\cdot \biggl\Vert \int_{\mathbb{R}^{n-m}} \bigl(\triangle_{\xi''} \bigl(1 +\bigl| \xi''\bigr|^{2}\bigr)^{\frac{s}{2}} \bigr)^{[p]} \widehat{u}\bigl(x_{0}, x', \eta''\bigr) \mathcal{F}_{x''} \biggl(\psi \biggl( \frac{|x''|}{L} \biggr) \frac{e^{i x'' \cdot\xi''}}{|x''|^{2p}} \biggr)\,d\xi'' \biggr\Vert _{L^{2}(\Omega)} \\ &\quad \leq c \int_{\mathbb{R}^{n-m}} \biggl\Vert \bigl(\triangle_{\xi''} \bigl(1 +\bigl|\xi''\bigr|^{2}\bigr)^{\frac{s}{2}} \bigr)^{[p]} \widehat{u}\bigl(x_{0}, x', \eta''\bigr) \mathcal {F}_{x''} \biggl(\psi \biggl( \frac{|x''|}{L} \biggr) \frac{e^{i x'' \cdot\xi''}}{|x''|^{2p}} \biggr) \biggr\Vert _{L^{2}(\Omega)}\,d\xi'' \\ &\quad \leq\frac{c_{r,p}}{L^{2p-n+m+1}} \int_{\mathbb{R}^{n-m}} \biggl\Vert \frac{(\triangle_{\xi''} (1 +|\xi''|^{2})^{\frac{s}{2}})^{[p]} \widehat{u}(x_{0}, x', \eta'')}{(1+|\xi'' - \eta''|^{2r'})} \biggr\Vert _{L^{2}(\Omega)}\,d\xi''. \end{aligned}$$
Taking into account the previous inequality and the Peetre inequality (see [16], p. 17), it follows that
$$ \Vert \varphi A_{s} u \Vert _{L^{2}(\overline{\Omega})} \leq \frac{c_{r,p,s}}{L^{2p-n+m}} \int_{\mathbb{R}^{n-m}} \frac{\Vert (1 +|\xi''|^{2})^{\frac{s}{2} - \frac{2p+n-m}{2}} \widehat{u}(x_{0}, x', \eta'') \Vert _{L^{2}(\Omega)}}{(1+|\xi'' - \eta''|^{2r})}\,d\xi''. $$
(24)
If \(p \geq\frac{s+r+n-m+1}{2}\), setting \(q=2p-n+m+1\) in (24), \(r=2r'\), results in
$$\Vert \varphi A_{s} u \Vert _{L^{2}(\Omega)} \leq \frac{c_{q,r,s}}{L^{q}} \Vert u \Vert _{H^{0,0,r}(\Omega)}, $$
where the constant \(c_{q,r,s}\) is independent on L. □
Taking into account Lemma 4.1, it is easy to show the following.
Lemma 4.2
Let
\(\varphi\in C^{\infty}_{0}(\mathbb{R}^{n-m})\)
such that
\(\varphi(|\tau''|)=0\)
if
\(|\tau''| \leq1\). For every
\(\varepsilon >0\), \(r \in\mathbb{Z}^{-}\), \(s\in\mathbb{R}\)
and for every
\(u \in C^{\infty}_{0} (\overline{\Omega})\), with
\(\overline{\Omega}= [0, + \infty[\, \times\Omega' \times\mathbb{R}^{n-m}\), there exists
\(L>0\)
such that
$$\biggl\Vert \varphi \biggl( \frac{|x''|}{L} \biggr) A_{s} u \biggr\Vert _{L^{2}(\Omega)} \leq\varepsilon\|u \|_{H^{0,0,r}(\Omega)}. $$
Furthermore, we are able to prove the following.
Lemma 4.3
Let
\(\varphi\in C^{\infty}_{0}(\mathbb{R}^{n-m})\)
such that
\(\varphi(|\tau''|)=1\)
if
\(|\tau''| \leq 1\). For every
\(\varepsilon >0\), \(r \in \mathbb{Z}^{-}\), \(s\in\mathbb{R}\)
and for every
\(u \in C^{\infty}_{0} (\overline{\Omega})\), with
\(\overline{\Omega}= [0, + \infty[\, \times \Omega' \times\mathbb{R}^{n-m}\), there exists
\(L>0\)
such that
$$\biggl\Vert \biggl( 1- \varphi \biggl( \frac{|x''|}{L} \biggr) \biggr) A_{s} u \biggr\Vert _{L^{2}(\Omega)} \leq\varepsilon\|u \|_{H^{0,0,r}(\Omega)}. $$
Proof
In order to establish this result we can proceed as Lemma 4.1, but in (22) we need to consider the Fourier transform of the function \(\psi(|x''|)= 1- \varphi ( \frac{|x''|}{L} )\) instead of \(\widehat{\varphi}(\eta'')\) and keep in mind that
$$\widehat{\psi g} = \widehat{\psi} * \widehat{g} = ( 2\pi\delta- \widehat{ \varphi}) * \widehat{g} = 2 \pi\widehat{g} - \widehat{\varphi} * \widehat{g}, \quad \forall g \in S'(\mathbb{R}), $$
where \(S'(\mathbb{R})\) is the space of tempered distributions defined in \(\mathbb{R}\). □
Next, we prove a result concerning estimates near the boundary.
Lemma 4.4
Let
\(\Omega= \,]0, + \infty[ \times\Omega_{0}\), where
\(\Omega_{0}\)
is an open subset of
\(\mathbb{R}^{n}\). For every
ε
and
δ
positive, there exists
\(k>0\)
such that if
$$I_{k,\delta} = \bigl\{ x \in\overline{\Omega}: x_{0} < k, \bigl|x_{0}+ \lambda- \alpha\bigl(x'\bigr) \bigr| > \delta \bigr\} , $$
as a result
$$\begin{aligned} &\Vert \partial_{x_{0}} u \Vert + \sum _{j=1}^{m} \Vert \partial_{x_{j}} u \Vert + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert + \Vert u \Vert \leq\varepsilon \Vert Pu \Vert , \\ &\quad \forall u \in C^{\infty}_{0}(\overline{\Omega}): \operatorname{supp} u \subseteq I_{k, \delta}. \end{aligned}$$
(25)
Proof
Integrating by parts and proceeding as in the first part of Theorem 3.1, we have
$$\begin{aligned} 2\bigl(e^{\tau x_{0}} \partial_{x_{0}} u(x), Pu\bigr) =& \tau\bigl\Vert e^{\frac{1}{2} \tau x_{0}} \partial_{x_{0}} u\bigr\Vert ^{2} + \int_{\Omega_{0}} \bigl(\partial_{x_{0}} u\bigl(0,x',x'' \bigr)\bigr)^{2}\,dx'\,dx'' \\ &{} + \int_{\Omega} e^{\tau x_{0}} A\bigl(x', x''\bigr) \nabla_{x'} u(x) \cdot\nabla _{x'} u(x)\,dx \\ &{} + \int_{\Omega_{0}} A\bigl(x', x'' \bigr) \nabla_{x'} u\bigl(0,x',x'' \bigr) \cdot \nabla_{x'} u\bigl(0,x',x'' \bigr)\,dx'\,dx'' \\ &{} + 2 \int_{\Omega} e^{\tau x_{0}} \bigl(x_{0}+\lambda- \alpha\bigl(x'\bigr)\bigr) B\bigl(x'' \bigr) \nabla_{x''} u(x) \cdot\nabla_{x''}u(x)\,dx \\ &{}+ \tau \int_{\Omega} e^{\tau x_{0}} \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr)\bigr)^{2} B\bigl(x'' \bigr) \nabla_{x''} u(x) \cdot\nabla_{x''}u(x)\,dx \\ &{} + \lambda \int_{\Omega_{0}} \bigl(\lambda- \alpha\bigl(x'\bigr) \bigr)^{2} B\bigl(x''\bigr) \nabla_{x''} u\bigl(0,x',x'' \bigr) \cdot\nabla_{x''} u\bigl(0,x',x'' \bigr)\,dx'\,dx'' \\ &{} - \int_{\Omega} \tau e^{\tau x_{0}} \gamma(x) u^{2}(x)\,dx - \int_{\Omega} e^{\tau x_{0}} u^{2}(x) \partial_{x_{0}} \gamma(x)\,dx \\ &{} - \int_{\Omega_{0}} \gamma\bigl(0,x',x'' \bigr) u^{2}\bigl(0,x',x'' \bigr)\,dx'\,dx''. \end{aligned}$$
(26)
This implies
$$\begin{aligned} & \tau \Biggl( \bigl\Vert e^{\frac{1}{2} \tau x_{0}} \partial_{x_{0}} u \bigr\Vert ^{2} + L_{1} \sum_{j=1}^{m} \bigl\Vert e^{\frac{1}{2} \tau x_{0}} \partial_{x_{j}} u \bigr\Vert ^{2} + L_{2} \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr) e^{\frac{1}{2} \tau x_{0}} \partial_{x_{j}} u \bigr\Vert ^{2} \Biggr) \\ &\quad \leq c \sum_{j=m+1}^{n} \biggl\Vert \frac{(x_{0}+\lambda-\alpha (x'))}{(x_{0}+\lambda-\alpha(x'))^{\frac{1}{2}}} e^{\frac{1}{2} \tau x_{0}} \partial_{x_{j}} u \biggr\Vert ^{2} + \tau\bigl\Vert e^{\frac{1}{2} \tau x_{0}} \bigl| \gamma (x)\bigr|^{\frac{1}{2}} u(x) \bigr\Vert ^{2} \\ &\qquad{} + \bigl\Vert e^{\frac{1}{2} \tau x_{0}} \bigl|\partial_{x_{0}} \gamma(x)\bigr|^{\frac{1}{2}} u(x) \bigr\Vert ^{2} + \int_{\Omega_{0}} \bigl|\gamma\bigl(0,x',x'' \bigr)\bigr| u^{2}\bigl(0,x',x'' \bigr)\,dx'\,dx'' \\ &\qquad{} + \bigl(ce^{\tau x_{0}} Pu, \partial_{x_{0}} u\bigr) \\ &\quad \leq\frac{c}{\delta} \sum_{j=m+1}^{n} \bigl\Vert e^{\frac{1}{2}\tau x_{0}} \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert ^{2} + c \biggl( \frac{1}{\tau^{2}} + \frac{1}{\tau} \biggr) \Vert \partial_{x_{0}} u \Vert ^{2} + \frac{1}{2} \bigl\Vert e^{\frac{1}{2} \tau x_{0}} Pu \bigr\Vert ^{2} \\ &\qquad{} + \frac{1}{2} \bigl\Vert e^{\frac{1}{2} \tau x_{0}} \partial_{x_{0}} u \bigr\Vert ^{2}. \end{aligned}$$
Choosing \(x_{0}< \frac{1}{\tau}\), it follows that
$$\begin{aligned} & \Vert \partial_{x_{0}} u \Vert ^{2} + \sum _{j=1}^{m} \Vert \partial_{x_{j}} u \Vert ^{2} + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} u \bigr\Vert ^{2} \\ &\quad \leq\frac{c}{\tau\delta} \sum_{j=m+1}^{n} \bigl\Vert e^{\frac{1}{2}\tau x_{0}} \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert ^{2} + c \frac{1}{\tau} \| Pu \|^{2} + \frac{c}{\tau} \| \partial_{x_{0}} u \|^{2}. \end{aligned}$$
(27)
For τ large enough, making use of (27) and (4) we obtain the claim. □
As a consequence, we establish the following result.
Lemma 4.5
For every
ε
and
δ
positive, there exists
\(k>0\)
such that if
$$I_{k,\delta} = \bigl\{ x \in\overline{\Omega}: x_{0} < k, \bigl|x_{0}+ \lambda- \alpha\bigl(x'\bigr) \bigr| > \delta \bigr\} , $$
for every
\(s<0\), as a result
$$\begin{aligned} &\Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}(\Omega)} + \sum _{j=1}^{m} \Vert \partial_{x_{j}} u \Vert _{H^{0,0,s}(\Omega)} + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega)} + \Vert u \Vert _{H^{0,0,s}(\Omega)} \\ &\quad \leq\varepsilon \Vert Pu \Vert _{H^{0,0,s}(\Omega)},\quad \forall u \in C^{\infty}_{0}(\overline{\Omega}): \operatorname{supp} u \subseteq I_{k, \delta}. \end{aligned}$$
(28)
Proof
Let \(\varphi\in C^{\infty}_{0}(\mathbb{R})\), set \(v_{s}= \varphi(|x''|) A_{s} u\), as a result \(\operatorname{supp} v_{s} \subseteq I_{k, \delta}\) and \(\varphi(|x''|)=1\) in \(\mathcal{U}_{x''}\). Therefore we can rewrite (25) with \(v_{s}\), namely
$$ \Vert \partial_{x_{0}} v_{s} \Vert + \sum _{j=1}^{m} \Vert \partial_{x_{j}} v_{s} \Vert + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} v_{s} \bigr\Vert + \Vert v_{s} \Vert \leq\varepsilon \Vert Pv_{s} \Vert . $$
(29)
Let us compute
$$\begin{aligned} \Vert \partial_{x_{0}} v_{s} \Vert \geq& \Vert \partial_{x_{0}} A_{s} u \Vert - \bigl\Vert ( \varphi- 1) \partial_{x_{0}} A_{s} u \bigr\Vert \\ =& \Vert \partial_{x_{0}} A_{s} u \Vert - \Vert Ru \Vert , \end{aligned}$$
(30)
where \((\varphi- 1) \partial_{x_{0}} A_{s} =R\) is a regularizing operator. In the same way,
$$ \Vert \partial_{x_{j}} v_{s} \Vert \geq c \Vert A_{s} \partial_{x_{j}} u \Vert - \Vert Ru \Vert , \quad \forall j=1, \ldots, m. $$
(31)
Similarly, we obtain
$$\begin{aligned} \bigl\Vert \bigl(x_{0} +\lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{j}} v_{s} \bigr\Vert =& \bigl\Vert \bigl(x_{0}+\lambda -\alpha \bigl(x'\bigr)\bigr) \partial_{x_{j}} \varphi A_{s} u \bigr\Vert \\ =& \bigl\| A_{s} \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{j}} u + (\varphi-1) A_{s} \bigl(x_{0}+\lambda-\alpha\bigl(x' \bigr)\bigr) \partial_{x_{j}} u \\ &{} + \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr)\bigr) [ \partial_{x_{j}}, \varphi] (A_{s}u) \bigr\| \\ \geq& \bigl\Vert A_{s} \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr) \bigr) \partial_{x_{j}} u \bigr\Vert - c \Vert Ru\Vert , \end{aligned}$$
(32)
where we take into account that \((\varphi-1) A_{s} \partial_{x_{j}}\) and \([\partial_{x_{j}}, \varphi] A_{s}\) are regularizing operators. Finally, we have
$$\begin{aligned} \Vert v_{s} \Vert =& \bigl\Vert A_{s} u + (\varphi-1) A_{s} u \bigr\Vert \\ \geq& \Vert A_{s} u \Vert - \Vert Ru \Vert , \end{aligned}$$
(33)
\((\varphi-1) A_{s}\) being a regularizing operator.
Making use of (30), (31), (32), and (33), it follows that
$$\begin{aligned} &\Vert \partial_{x_{0}} v_{s} \Vert + \sum _{j=1}^{m} \Vert \partial_{x_{j}} v_{s} \Vert + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} v_{s} \bigr\Vert + \Vert v_{s} \Vert \\ &\quad \geq \Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}(\Omega_{k})} + c \Biggl( \sum_{j=1}^{m} \Vert \partial_{x_{j}} u \Vert _{H^{0,0,s}(\Omega_{k})} + \sum _{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda-\alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega_{k})} \Biggr) \\ &\qquad{} + \Vert u \Vert _{H^{0,0,s}(\Omega_{k})} - c \Vert Ru \Vert . \end{aligned}$$
Since \(\Vert Ru \Vert \leq c \Vert u \Vert _{H^{0,0,s}(\Omega_{k})} \leq c k \Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}(\Omega_{k})}\), choosing k small enough, as a result
$$\begin{aligned} &\Vert \partial_{x_{0}} v_{s} \Vert + \sum _{j=1}^{m} \Vert \partial_{x_{j}} v_{s} \Vert + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} v_{s} \bigr\Vert + \Vert v_{s} \Vert \\ &\quad \geq c \Biggl( \Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}(\Omega_{k})} + \sum_{j=1}^{m} \Vert \partial_{x_{j}} u \Vert _{H^{0,0,s}(\Omega_{k})} + \sum _{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda-\alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega_{k})} \\ &\qquad{} + \Vert u \Vert _{H^{0,0,s}(\Omega_{k})} \Biggr). \end{aligned}$$
(34)
On the other hand, we have
$$Pv_{s} = \varphi\bigl(\bigl|x''\bigr|\bigr) A_{s} Pu+ \varphi\bigl(\bigl|x''\bigr|\bigr) [P,A_{s}](u) + \bigl[\varphi\bigl(\bigl|x''\bigr| \bigr),P\bigr](A_{s} u). $$
As a consequence, we obtain
$$ \Vert P v_{s} \Vert = \bigl\Vert \varphi \bigl(\bigl|x''\bigr|\bigr) A_{s} Pu+ \varphi \bigl(\bigl|x''\bigr|\bigr) B_{s+1}u + R u \bigr\Vert , $$
(35)
where we set \([P,A_{s}] = B_{s+1}\), this being a pseudodifferential operator endowed with the symbol with respect to the variable \(x''\) of order \(s+1\). Such a symbol has the following principal part:
$$c\bigl(x, \xi''\bigr) = - \frac{1}{i} \sum _{p=m+1}^{n} \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr)\bigr)^{2} \sum _{i,j=m+1}^{n} \partial_{x_{p}} b_{j} \bigl(x''\bigr) \xi_{i}'' \xi_{j}'' \partial_{\xi_{p}} \bigl(1+\bigl| \xi''\bigr|^{2}\bigr)^{\frac{s}{2}}. $$
Therefore the symbol \(b(x,\xi'')\) can be written as
$$b\bigl(x,\xi''\bigr) = c\bigl(x, \xi''\bigr) +d\bigl(x,\xi'' \bigr), $$
where \(d(x,\xi'')\) is a symbol of order s and we set
$$B_{s+1}= C_{s+1}+D_{s}. $$
Moreover, we set \(R=[\varphi(|x''|),P]A_{s}\), which is a regularizing operator.
At last, we remark that
$$C_{s+1} u = C_{s}' \bigl(x_{0}+ \lambda-\alpha\bigl(x'\bigr)\bigr) \sum _{i=m+1}^{n} \partial_{x_{j}} u $$
and the symbol \(c_{s}'\) of \(C_{s}'\) is given by
$$c_{s}'\bigl(x, \xi''\bigr) = - \frac{1}{i} \sum_{p=m+1}^{n} \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr)\bigr) \sum _{j=m+1}^{n} \partial_{x_{p}} b_{j}\bigl(x''\bigr) \xi_{j}'' \partial_{\xi_{p}} \bigl(1+ \bigl|\xi''\bigr|^{2} \bigr)^{\frac{s}{2}}. $$
By such insights and by (35), as a result
$$ \begin{aligned}[b] \Vert Pv_{s} \Vert &\leq c \Biggl( \Vert A_{s} Pu \Vert + \sum_{j=m+1}^{n} \bigl\Vert C_{s}'\bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert + \Vert D_{s} u \Vert + \Vert Ru \Vert \Biggr) \\ &\leq c \Biggl( \Vert Pu \Vert _{H^{0,0,s}(\Omega)} + \sum _{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda-\alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega)} + \Vert u \Vert _{H^{0,0,s}(\Omega)} \Biggr). \end{aligned} $$
(36)
By using (29), (34), (36), and for ε small enough, the claim follows. □
Now, we are able to prove the following theorem.
Theorem 4.1
Let
\(\Omega_{k} = [0,k[\, \times\Omega_{0}\), with
k
such that (21) holds, let
\(\Omega_{0}= \Omega' \times \mathbb{R}^{n-m}\), let
\(\Omega'\)
be an open set of
\(\mathbb{R}^{n-m}\)
and let
\(B(x'')\)
be a constant. Under assumptions (i) and (ii), for every
\(s \in\mathbb{Z}^{-}\)
there exists
\(c>0\)
such that
$$\begin{aligned} &\Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}(\Omega)} + \sum _{j=1}^{m} \Vert \partial_{x_{j}} u \Vert _{H^{0,0,s}(\Omega)} + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega)} + \Vert u \Vert _{H^{0,0,s}(\Omega)} \\ &\quad \leq c \Vert Pu \Vert _{H^{0,0,s}(\Omega_{k})},\quad \forall u \in C^{\infty}_{0}(\overline{\Omega}): \operatorname{supp} u \subseteq \Omega_{k}. \end{aligned}$$
(37)
Proof
Let \(\varphi\in C^{\infty}_{0}(\mathbb{R})\) and \(\varphi(\tau')=1\) if \(|\tau'| \leq1\) and \(\mathcal{U}_{x''} \subseteq[-L, L]^{n-m}\). Setting \(v_{s} = \varphi ( \frac{|x''|}{L} ) A_{s} u\) in (21), as a result
$$ \Vert \partial_{x_{0}} v_{s} \Vert + \sum _{j=1}^{m} \Vert \partial_{x_{j}} v_{s} \Vert + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} v_{s} \bigr\Vert + \Vert v_{s} \Vert \leq c \Vert P v_{s} \Vert . $$
(38)
Furthermore, we have
$$\begin{aligned} \Vert \partial_{x_{0}} v_{s} \Vert =& \bigl\Vert \partial_{x_{0}} A_{s} u + (\varphi-1) \partial_{x_{0}} A_{s} u \bigr\Vert \\ \geq& \Vert A_{s} \partial_{x_{0}} u \Vert - \bigl\Vert (\varphi-1) A_{s} \partial_{x_{0}} u \bigr\Vert . \end{aligned}$$
Taking into account Lemma 4.3, for L large enough, it follows that
$$ \Vert \partial_{x_{0}} v_{s} \Vert \geq c \Vert A_{s} \partial_{x_{0}} u \Vert . $$
(39)
Making use of the same technique, we have
$$ \Vert \partial_{x_{j}} v_{s} \Vert \geq c \Vert A_{s} \partial_{x_{j}} u \Vert ,\quad \forall j=1, \ldots, m, $$
(40)
and similarly
$$\begin{aligned} \bigl\Vert \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{j}} v_{s} \bigr\Vert =& \biggl\| A_{s}\bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{j}} u + (\varphi-1) A_{s}\bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} u \\ &{}+ \biggl[ \partial_{x_{j}}, \varphi \biggl( \frac{|x''|}{L} \biggr) \biggr] A_{s}\bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr) u \biggr\| \\ \geq& c \bigl\Vert A_{s}\bigl(x_{0}+ \lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert - \frac{c}{L} \bigl\Vert A_{s}\bigl(x_{0}+\lambda- \alpha\bigl(x'\bigr)\bigr) u \bigr\Vert , \end{aligned}$$
(41)
where L is large enough. Finally, we have
$$\begin{aligned} \Vert v_{s} \Vert =& \Vert A_{s} u \Vert + \biggl\Vert \biggl( 1- \varphi \biggl( \frac {|x''|}{L} \biggr) \biggr) A_{s} u \biggr\Vert \\ \geq& (1- \varepsilon) \Vert u \Vert _{H^{0,0,s}(\Omega)}, \end{aligned}$$
(42)
having used Lemma 4.2.
On the other hand we have the result
$$ Pv_{s} = \varphi \biggl( \frac{|x''|}{L} \biggr) A_{s} Pu + \biggl[ \varphi \biggl( \frac{|x''|}{L} \biggr), P \biggr] (A_{s}u). $$
(43)
Let us observe that
$$ \Vert A_{s}Pu\Vert =\Vert Pu\Vert _{H^{0,0,s}(\Omega)}, $$
(44)
from the continuity property of the pseudodifferential operators (see [16], Theorem 2.1) and making use of Lemma 4.2
$$ \biggl\Vert \biggl[ \varphi \biggl( \frac{|x''|}{L} \biggr) , P \biggr] (A_{s} u ) \biggr\Vert \leq\varepsilon \Vert u\Vert _{H^{0,0,s}(\Omega)}. $$
(45)
By using (43), (44), and (45), we have
$$\begin{aligned} \Vert Pv_{s} \Vert \leq& \biggl\Vert \varphi \biggl( \frac{|x''|}{L} \biggr) A_{s} Pu \biggr\Vert + \biggl\Vert \varphi \biggl( \frac{|x''|}{L} \biggr) B_{s} u \biggr\Vert + \biggl\Vert \biggl[ \varphi \biggl( \frac{|x''|}{L} \biggr), P \biggr] (A_{s} u) \biggr\Vert \\ \leq& c \|Pu \|_{H^{0,0,s}(\Omega)} + \varepsilon\| u \|_{H^{0,0,s}(\Omega)}. \end{aligned}$$
(46)
Making use of (38), (39), (40), (41), and (42), choosing ε small enough and taking into account Lemma 3.1, as a result
$$\begin{aligned} & \Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}(\Omega)} + \sum _{j=1}^{m} \Vert \partial _{x_{j}} u \Vert _{H^{0,0,s}(\Omega)} + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega)} + \Vert u \Vert _{H^{0,0,s}(\Omega)} \\ &\quad \leq c \Vert Pu\Vert _{H^{0,0,s}(\Omega)} + \varepsilon \Vert u \Vert _{H^{0,0,s}(\Omega)} \\ &\quad \leq c \Vert Pu\Vert _{H^{0,0,s}(\Omega)} + \varepsilon \Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}(\Omega_{k})}, \end{aligned}$$
(47)
and for ε small enough the claim is established. □
Now, we prove an estimate in Sobolev spaces with \(s<0\).
Theorem 4.2
Under assumptions (i), (ii), (iii), and (iv), for every
\(s \in \mathbb{R}_{0}^{-}\)
there exists
\(c>0\)
such that
$$\begin{aligned} & \Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}(\Omega)} + \sum_{j=1}^{m} \Vert \partial_{x_{j}} u \Vert _{H^{0,0,s}(\Omega)} + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega)} + \Vert u \Vert _{H^{0,0,s}(\Omega)} \\ &\quad \leq c \Vert Pu \Vert _{H^{0,0,s}(\Omega_{k})},\quad \forall u \in C^{\infty}_{0}(\overline{\Omega}): \operatorname{supp} u \subseteq \Omega_{k}= [0,k[ \times\Omega_{0}. \end{aligned}$$
(48)
Proof
Let \(\varphi\in C^{\infty}_{0}(\mathbb{R})\) such that \(\varphi(|x''|)=1\) on \(\mathcal{U}_{x''}\) and let k such that (21) holds.
Then, for every \(u \in C^{\infty}_{0}(\overline{\Omega})\) such that \(\operatorname{supp} u \subseteq\Omega_{k}\), as a result
$$\Vert \partial_{x_{0}} v_{s} \Vert + \sum _{j=1}^{m} \Vert \partial_{x_{j}} v_{s} \Vert + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} v_{s} \bigr\Vert + \Vert v_{s} \Vert \leq c \Vert P v_{s} \Vert , $$
where \(v_{s}= \varphi(|x''|) A_{s} u\).
Proceeding as in the proof of (34) (see from (29) to (34)), we obtain
$$\begin{aligned} &\Vert \partial_{x_{0}} v_{s} \Vert + \sum _{j=1}^{m} \Vert \partial_{x_{j}} v_{s} \Vert + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr) \partial_{x_{j}} v_{s} \bigr\Vert + \Vert v_{s} \Vert \\ &\quad \geq c \Biggl( \Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}(\Omega_{k})} + \sum_{j=1}^{m} \Vert \partial_{x_{j}} u \Vert _{H^{0,0,s}(\Omega_{k})} + \sum _{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda-\alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega_{k})} \\ &\qquad{} + \Vert u \Vert _{H^{0,0,s}(\Omega_{k})} \Biggr). \end{aligned}$$
On the other hand, we have
$$\begin{aligned} P v_{s} =& \varphi\bigl(\bigl|x''\bigr|\bigr) A_{s} Pu + \bigl[\varphi\bigl(\bigl|x'\bigr|\bigr), P \bigr](A_{s} u) + \varphi \bigl(\bigl|x'\bigr|\bigr) [P, A_{s}]u = \varphi\bigl(\bigl|x'\bigr|\bigr) A_{s} Pu + Ru + B_{s+1} u, \end{aligned}$$
where \(R = [\varphi(|x'|), P] A_{s}\) is a regularizing operator and \(B_{s+1} = [P, A_{s}]\) is a pseudodifferential operator with respect to the variables \(x''\) of order \(s+1\) endowed with symbol \(b(x, \xi'')\) with principal part equal to
$$c\bigl(x, \xi''\bigr) = - \frac{1}{i} \sum _{p=m+1}^{n} \bigl(x_{0}+ \lambda- \alpha\bigl(x'\bigr)\bigr)^{2} \sum _{i,j=m+1}^{n} \partial_{x_{p}} b_{j} \bigl(x''\bigr) \xi_{i}'' \xi_{j}'' \partial_{\xi_{p}} \bigl(1+\bigl| \xi''\bigr|^{2}\bigr)^{\frac{s}{2}}. $$
Hence,
$$b\bigl(x, \xi''\bigr) = c\bigl(x, \xi''\bigr) +d\bigl(x, \xi'' \bigr), $$
where \(c(x, \xi'')\) is the symbol of order s. Therefore, we have
$$B_{s+1} =C_{s+1} +D_{s}. $$
Then as a result
$$ [P, A_{s}](u) = C_{s+1}u +D_{s} u. $$
(49)
Taking into account (41), (42), (43), and (49), we obtain
$$ \begin{aligned}[b] \Vert Pv_{s} \Vert &\leq \biggl\Vert \varphi \biggl( \frac{|x''|}{L} \biggr) A_{s} Pu \biggr\Vert + \biggl\Vert \varphi \biggl( \frac{|x''|}{L} \biggr) B_{s+1}u \biggr\Vert + \biggl\Vert \biggl[ \varphi \biggl( \frac{|x''|}{L} \biggr), P \biggr] (A_{s} u) \biggr\Vert \\ &\leq c \| Pu \|_{H^{0,0,s}(\Omega)} + \biggl\Vert \varphi \biggl( \frac{|x''|}{L} \biggr) C_{s+1}u \biggr\Vert + \biggl\Vert \varphi \biggl( \frac{|x''|}{L} \biggr) D_{s}u \biggr\Vert + \varepsilon \| u \|_{H^{0,0,s}(\Omega)} \\ &\leq c \bigl( \| Pu \|_{H^{0,0,s}(\Omega)} + \| u \|_{H^{0,0,s}(\Omega)} \bigr) + \varepsilon\| u \|_{H^{0,0,s}(\Omega)} + \biggl\Vert \varphi \biggl( \frac{|x''|}{L} \biggr) C_{s+1}u \biggr\Vert . \end{aligned} $$
(50)
We remember that \(C_{s+1}\) is a pseudodifferential operator endowed with the symbol
$$c\bigl(x,\xi'\bigr) = - \frac{1}{i} \sum _{p=m+1}^{n} \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr)^{2} \sum _{i,j=m+1}^{n} \partial_{x_{p}} b_{j} \bigl(x''\bigr) \xi_{i}'' \xi_{j}'' \partial_{\xi_{p}} \bigl(1+\bigl|\xi''\bigr|^{2}\bigr)^{\frac{1}{2}}. $$
Therefore, we have
$$\begin{aligned} C_{s+1}u =& \frac{1}{(2\pi)^{n-m}} \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot\xi''} c\bigl(x,\xi'' \bigr) \widehat{u}\bigl(x_{0},x',\xi'' \bigr)\,d\xi'' \\ =& \frac{1}{(2\pi)^{n-m}} \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot\xi ''} \chi\bigl(\bigl|\xi''\bigr| \bigr) c\bigl(x,\xi''\bigr) \widehat{u} \bigl(x_{0},x',\xi''\bigr)\,d\xi'' \\ &{} + \frac{1}{(2\pi)^{n-m}} \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot \xi''} \bigl( 1- \chi\bigl(\bigl| \xi''\bigr|\bigr) \bigr) c\bigl(x,\xi'' \bigr) \widehat{u}\bigl(x_{0},x',\xi'' \bigr)\,d\xi'', \end{aligned}$$
where \(\chi\in C^{\infty}_{0}(\mathbb{R})\) such that \(\chi(t)=1\) for \(|t|<1\). Therefore, we have
$$\begin{aligned} C_{s+1}u =& Ru \\ &{} + \frac{1}{(2\pi)^{n-m}} \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot \xi''} \bigl( 1- \chi\bigl(\bigl| \xi''\bigr|\bigr) \bigr) c\bigl(x,\xi'' \bigr) \widehat{u}\bigl(x_{0},x',\xi'' \bigr)\,d\xi'', \end{aligned}$$
where R is a regularizing operator. On the other hand, we have
$$\begin{aligned} & \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot\xi''} \bigl( 1- \chi\bigl(\bigl| \xi''\bigr|\bigr) \bigr) c\bigl(x,\xi'' \bigr) \widehat{u}\bigl(x_{0},x',\xi'' \bigr)\,d\xi'' \\ &\quad = \sum_{j=m+1}^{n} \int_{\mathbb{R}^{n-m}} e^{i x'' \cdot \xi''} c'\bigl(x, \xi''\bigr) \widehat{\bigl(x_{0}+\lambda- \alpha\bigl(x'\bigr)\bigr)^{2} \partial_{x_{j}}u \bigl(x_{0},x',\xi''\bigr)}\,d\xi'', \end{aligned}$$
where \(c'(x,\xi'')= \dfrac{ ( 1- \chi(|\xi''|) ) c(x,\xi'')}{(x_{0}+\lambda-\alpha(x'))^{2} |\xi''|^{2}} (\xi_{1}+ \cdots+\xi_{n})\) is a symbol of order s. As a consequence, it follows that
$$\begin{aligned} C_{s+1}u =& Ru + \sum_{j=1}^{n} C'_{s} \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr)^{2} \partial _{x_{j}} u \\ =& Ru + \sum_{j=m+1}^{n} C'_{s} \bigl(x_{0}+\lambda-\alpha \bigl(x'\bigr)\bigr)^{2} \partial_{x_{j}} \chi \biggl( \frac{|x_{0}+\lambda-\alpha(x')|}{\delta} \biggr) u \\ &{} + \sum_{j=m+1}^{n} C'_{s} \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr)^{2} \partial_{x_{j}} \biggl[ 1- \chi \biggl( \frac{|x_{0}+\lambda -\alpha(x')|}{\delta} \biggr) \biggr] u. \end{aligned}$$
Then, taking into account Lemma 4.5,
$$\begin{aligned} \Vert C_{s+1}u \Vert \leq& \Vert Ru \Vert + c \sum_{j=m+1}^{n} \biggl\Vert \frac{(x_{0}+\lambda-\alpha(x'))^{2}}{\delta} \delta\chi \biggl( \frac{|x_{0}+\lambda-\alpha(x')|}{\delta} \biggr) \partial_{x_{j}} u \biggr\Vert _{H^{0,0,s(\Omega)}} \\ &{} +c \sum_{j=m+1}^{n} \biggl\Vert \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr) \bigr)^{2} \partial_{x_{j}} \biggl[ 1- \chi \biggl( \frac{|x_{0}+\lambda-\alpha(x')|}{\delta} \biggr) \biggr] u \biggr\Vert _{H^{0,0,s}(\Omega)} \\ \leq& \Vert u \Vert _{H^{0,0,s}(\Omega)} + c \delta\sum _{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda-\alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega)} \\ &{} + c \varepsilon\biggl\Vert P \biggl( 1-\chi \biggl( \frac{|x_{0}+\lambda -\delta(x')|}{\delta} \biggr) \biggr) u \biggr\Vert \\ \leq& \|u \|_{H^{0,0,s}(\Omega)} + c \delta\sum_{j=m+1}^{n} \|\bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \|_{H^{0,0,s}(\Omega)} + c \varepsilon\| Pu \|_{H^{0,0,s}(\Omega)} \\ &{}+ c \varepsilon\biggl\Vert \biggl[ P, \biggl( 1- \chi \biggl( \frac{|x_{0}+\lambda-\alpha(x')|}{\delta} \biggr) \biggr) \biggr] (u) \biggr\Vert _{H^{0,0,s}(\Omega)} \\ \leq& \Vert u \Vert _{H^{0,0,s}(\Omega)} + c \delta\sum _{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+ \lambda-\alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega)} \\ &{} + c \varepsilon \Biggl( \| Pu \|_{H^{0,0,s}(\Omega)} + \Vert \partial_{x_{0}} u \Vert _{H^{0,0,s}(\Omega)} + \sum _{j=1}^{m} \| \partial_{x_{j}} u \|_{H^{0,0,s}(\Omega)} \Biggr). \end{aligned}$$
(51)
Making use of (50), (51), and Lemma 3.1, we have
$$\begin{aligned} \| Pv_{s} \| \leq& c \bigl( \| Pu \|_{H^{0,0,s}(\Omega_{k})} + k \| \partial_{x_{0}} u \|_{H^{0,0,s}(\Omega_{k})} \bigr) \\ &{} + c \varepsilon \Biggl( \| Pu \|_{H^{0,0,s}(\Omega_{k})} + \| \partial_{x_{0}} u \|_{H^{0,0,s}(\Omega_{k})} + \sum_{j=1}^{m} \| \partial_{x_{j}} u \|_{H^{0,0,s}(\Omega_{k})} \Biggr) \\ &{} + c \delta\sum_{j=m+1}^{n} \bigl\| \bigl(x_{0}+\lambda-\alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\| _{H^{0,0,s}(\Omega_{k})}. \end{aligned}$$
(52)
Finally, by (52), (38), (39), (40), and (41), for δ and ε small enough and, hence, k small enough, as a result
$$\begin{aligned} & \Vert \partial_{x_{0}} u \Vert _{H^{0,0,s(\Omega_{k})}} + \sum _{j=1}^{m} \Vert \partial_{x_{j}} u \Vert _{H^{0,0,s}(\Omega_{k})} + \sum_{j=m+1}^{n} \bigl\Vert \bigl(x_{0}+\lambda- \alpha\bigl(x'\bigr)\bigr) \partial_{x_{j}} u \bigr\Vert _{H^{0,0,s}(\Omega_{k})} \\ &\quad{} + \Vert u \Vert _{H^{0,0,s}(\Omega_{k})}\leq c \Vert Pu \Vert _{H^{0,0,s}(\Omega_{k})}, \quad\forall u \in C^{\infty}_{0} (\overline{\Omega}): \operatorname{supp} u \subseteq\Omega_{k}. \end{aligned}$$
□