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 \begin{document}

\title[Subellipticity of complex vector fields]{Subellipticity of  some  complex vector fields related to Witten Laplacian}

\author[W.-X. Li]{Wei-Xi Li}

\address[W.-X. Li]{ School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China \& Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China
  }

 \email{wei-xi.li@whu.edu.cn}

 \author[C.-J. Xu]{Chao-Jiang Xu}

\address[C.-J. Xu]{Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China}

 \email{xuchaojiang@nuaa.edu.cn}

\begin{abstract}
  We consider  some system of complex vector fields   related to the semi-classical Witten Laplacian, and establish  the local subellipticity of  this system basing on   condition $(\Psi)$.
 \end{abstract}

\subjclass[2020]{35H20,35H10}
\keywords{subelliptic estimate, complex system, Condition $(\Psi)$}

 \maketitle

\section{Introduction and main results}

Let  $\Omega\subset\mathbb R^n$ be a neighborhood of $0$,  and denote by  $i$    the  square root   of $-1$. We consider in this paper the following system of complex vector fields :
 \begin{equation}\label{syst+++}
  \mathcal P_j=\partial_{x_j}-i \inner{\partial_{x_j}\varphi (x)}\partial_{t},\quad
  j=1,\cdots,n,\quad (x,t)\in\Omega\times\mathbb R,
\end{equation}
where  $\varphi(x)$ is  a
{\it real-valued} function defined in a neighborhood  $\Omega$ of $0$.   This system was first studied by Treves \cite{MR0426068},   and considered therein is more general case  for $t$ varying  in $\mathbb R^m$ rather than $\mathbb R.$
Denote by $(\xi,\lambda)$   the Fourier variables of $(x,t)$. Then
 the principle symbol $\sigma$ for the system $\set{\mathcal P_j}_{1\leq j\leq n}$ is
\begin{eqnarray*}
	\sigma (x,t; \xi,\lambda)=\inner{i \xi_1+\inner{\partial_{x_1}\varphi }\lambda,\cdots, i \xi_j+\inner{\partial_{x_j}\varphi }\lambda}\in \mathbb C^n
\end{eqnarray*}
with $ \inner{x,t; \xi,\lambda}\in T^*\inner{\Omega\times\mathbb R_t}\setminus\set{0},$
and thus the characteristic set is
\begin{eqnarray*}
	\big\{(x,t;\xi,\lambda)\in T^*\inner{\Omega\times\mathbb R_t}\setminus\set{0}~\big|~\xi=0, ~\lambda\neq 0, ~\nabla\varphi(x)=0\big\}.
\end{eqnarray*}
Since outside the characteristic set the system $\set{\mathcal P_j}_{1\leq j\leq n}$ is  elliptic,  we only need to study  the microlocal hypoellipticity  in the two components  $\set{\lambda>0}$ and  $\set{\lambda<0}$ under the assumption that
\begin{eqnarray*}\label{gra}
	\nabla \varphi(0)=0.
\end{eqnarray*}
Note we may assume $\varphi(0)=0$ if replacing $\varphi$ by $\varphi-\varphi(0).$     By maximal hypoellipticity (in
the sense of Helffer-Nourrigat \cite{MR897103}) it means the
existence of a neighborhood $\tilde\Omega\subset \Omega $ of $0$ and a
constant $C$ such that for any $u\in C_0^\infty(\tilde\Omega\times \mathbb R)$ we have
 \begin{multline*}\label{maxhyp}
 \sum_{j=1}^n\inner{ \norm{\partial_{x_j}u}_{L^2(\mathbb R^{n+1})}
    +\norm{\inner{\partial_{x_j}\varphi} \partial_{t}u}_{L^2(\mathbb R^{n+1})} }\\
    \leq
    C \Big(\sum_{j=1}^n\norm{\mathcal P_{j}u}_{L^2(\mathbb R^{n+1})}
    +   \norm{u}_{L^2(\mathbb R^{n+1})}\Big).
  \end{multline*}
  Note the  maximal hypoellipticity will yield the following subellptic estimate
  \begin{eqnarray*}
	\sum_{j=1}^n \norm{\partial_{x_j}u}_{L^2(\mathbb R^{n+1})}+\norm{\abs{D_t}^{\frac{1}{k+1}} u}_{L^2(\mathbb R^{n+1})}  \leq
    C  \sum_{j=1}^n\norm{\mathcal P_{j}u}_{L^2(\mathbb R^{n+1})}
    + C  \norm{u}_{L^2(\mathbb R^{n+1})},
\end{eqnarray*}
provided $\varphi$ is finite type, i.e., for some integer $k\geq 1$,
\begin{eqnarray*}
	\sum_{\abs{\alpha}\leq k+1}\abs{\partial_x^\alpha\varphi(0)}\neq 0.
\end{eqnarray*}
Thus the subellipticity is in some sense intermediate  between the maximal hypoellipticity and the local hypoellipticity.

We first recall the related works concerned with the subellipticity and  hypoellipticity as well as the links between these properties.
 For the system of real vector fields,  it is well-known (\cite{Hormander67, Kohn78, RothschildStein76}) that the H\"ormander's  bracket condition is sufficient to obtain the subellipticity  in terms of  the Lie algebra generated by the real vector fields.  Moreover for real vector fields with analytic coefficients,  Derridj's work \cite{MR0407433} implies that the  H\"ormander's condition and thus subellipticity  is also a necessary condition to obtain hypoellipticity.    But the situation is quite different  in the setting of  complex vector fields;   Kohn \cite{MR2183286} constructed a counterexample,  showing we may  lose the subellipticity even though  the H\"ormander's bracket condition is fulfilled.      The hypoellipticity for the system \eqref{syst+++} was  initiated  by Treves  \cite{MR0426068} , and then continued  by   Maire \cite{MR567778}.  The standard $L^2$  subellipticity was investigated in Derridj \cite{MR3603885, MR2298786} and  a series of works \cite{MR2868553, MR2885111,MR2471931}    of Derridj-Helffer,  where the crucial tool  is to find suitable escaping curves.   So far it remain unclear  for the links between hypoellipticity and subellipticity for   systems of complex vector fields.   In fact we may ask the same question as in the case of real vector fields,  wether hypoellipticity implies subellipticity for system \eqref{syst+++}  under the assumption that $\varphi$ is analytic in $\Omega$.   This is  true for $n=1,$ but may be false when $n>1$ due to the work  \cite{MR2276079} of Journ{\'e} and  Tr{\'e}preau,  where they showed for some specific polynomial potential $\varphi$ that the system is hypoelliptic without being subelliptic.  For specific  quasihomogeneous  analytic potential functions  Derridj-Helffer  \cite{MR2885111} proved  hypoellipticity system \eqref{syst+++} is   indeed subelliptic for the dimension $n=2$.     Finally we mention  the criterion of Helffer-Nourrigat   \cite{ MR897103},  which is based on   the nilpotent group techniques   to characterize  the maximal hypoellipticity  for   general systems of one order pseudodifferential operators; see also Nourrigat's work \cite{MR1044428, MR1098615} for further development and Helffer-Nier'   lecture \cite{HelfferNier05}  for the  application of  Helffer-Nourrigat's criterion to Witten Laplacian.

 Observe for the system $\set{\mathcal P_j}_{1\leq j\leq n} $,   we may perform partial Fourier transform with respect to $t$,  and  study  the  microhypoellipticity,  in the two directions $\lambda>0$  and $\lambda<0.$  Indeed we only need consider without loss of generality the   microhypoellipticity in the positive direction $\lambda>0,$ since the other direction $\lambda<0$ can be treated similarly by replacing $\varphi$ by $-\varphi.$   Consider the resulting system as follows after taking partial Fourier transform for $t\in\mathbb R$:
\begin{equation}\label{system2}
 \mathcal L_j=\partial_{x_j}+\lambda \inner{\partial_{x_j}\varphi},\quad
  j=1,\cdots,n.
\end{equation}
By maximal microhypoellipticity at $0$ in the positive direction in $\lambda>0$,  it means the existence of a positive number $\lambda_0>0,$   a
constant $C>0$  and  a neighborhood $\tilde\Omega\subset \Omega$ of $0,$  such that
\begin{multline}\label{hypo}
    \forall~ \lambda\geq \lambda_0,~\forall~u\in
  C_0^\infty(\tilde\Omega),\\
  \sum_{j=1}^n \inner{\norm{\partial_{x_j}u}_{L^2}
    +\norm{\lambda \inner{\partial_{x_j}\varphi} u}_{L^2}}\leq
    C\Big( \sum_{j=1}^n\norm{ \mathcal L_ju}_{L^2}
    +\norm{u}_{L^2}\Big),
  \end{multline}
where and throughout the paper we denote     $\norm{\cdot}_{L^2(\mathbb R^n)}$   by  $\norm{\cdot}_{L^2}$ for short  if no confusion occurs.  We remark  the  operators  defined in \eqref{system2}   is  closely related to the semi-classical Witten Laplacian
  \begin{eqnarray*}
  \triangle_{ \lambda \varphi}^{(0)} =-\triangle_x +\lambda^{2} \abs{\partial_x \varphi}^2-\lambda \triangle_x   \varphi
  \end{eqnarray*}
  with $\lambda^{-1}$ the semi-classical parameter,  by the relationship
  \begin{eqnarray*}
  	\sum_{j=1}^n \norm{ \mathcal L_j u}^2_{L^2}=\inner{\triangle_{ \lambda \varphi}^{(0)} u, \ u}_{L^2},
  \end{eqnarray*}
  where $\inner{\cdot,  \cdot}_{L^2}$ stands for the inner product in $L^2(\mathbb R^n).$   Helffer-Nier \cite{HelfferNier05}  conjectured $ \triangle_{ \lambda \varphi}^{(0)} $  is subelliptic near $0$ if  $\varphi$ is analytic and has no local minimum near $0,$ and  this still remains open so far.   Note  \eqref{hypo} is a local estimate concerning the sharp  regularity near $0\in \mathbb R^n$ for $\lambda>0$,  and we have also its global   counterpart,  which is of independent interest for  analyzing  the semi-classical  lower bound of  Witten Laplacian.    We refer to Helffer-Nier's work \cite{HelfferNier05}  for the  detailed presentation on the topic of global maximal hypoellipticity and its application to the spectral analysis on Witten Laplacian; see also  the first author's  work \cite{MR3775156} for the extension to the non-polynomial potentials.

In this work we aim to investigate the hypoellipticity by virtue of  condition $(\Psi)$. Recall condition   $(\Psi)$ was
initially  introduced to study  the solvability of a pseudo-differential
operator $p^w$ with complex-valued symbol $p_1+ip_2$; we refer to \cite{MR781537,MR2599384} for the comprehensive discussion on  condition  $(\Psi)$.   For a specific scalar  operator
\begin{eqnarray*}
	\partial_{s}+  \mu(s),
\end{eqnarray*}
with $\mu$ real-valued function defined in an interval $I\subset\mathbb R$,
condition $(\Psi)$ means the following change-of-sign condition:
\begin{eqnarray*}
	s\rightarrow \mu(s)\  \textrm{does not change sign from + to} - \textrm{as } s \ \textrm{increases in}\  I.
\end{eqnarray*}
Moreover we say $\partial_{s}+ \mu(s) $ satisfies condition $( \overline \Psi)$ if $\partial_{s}- \mu(s)$ satisfies condition $(\Psi)$, that is,
\begin{eqnarray*}
	s\rightarrow \mu(s)\ \textrm{does not change sign from}  -  \textrm{to + as } s \ \textrm{increases in}\  I.
\end{eqnarray*}
For the scalar operator $\partial_{s}-i \mu(s)\partial_{t}$,  the subellipticity is well explored by   condition $(\overline\Psi)$;  for instance if $\mu$ is finite type and
has no sign change from $-$ to $+$ as $s$ increases, then it is
subelliptic (c.f.
\cite{MR0290201} and  \cite[Chapter 27]{MR781536}).

Note there is no condition $(\Psi)$ for systems, in particular for the system $\mathcal L_j, 1\leq j\leq n,$ defined by \eqref{system2}.   For clear presentation we  consider the case of $n=2$ and our argument may be applied to more general case of $n\geq 2.$
In what follows
we use $(x,y)$   instead of
$(x_1,x_2)$ as variables.  Then we can rewrite $\mathcal L_j, j=1,2,$ as
 \begin{equation}\label{syst}
 \left\{
 \begin{aligned}
& \mathcal L_1=\partial_{x}+\lambda \inner{\partial_{x}\varphi},\\
&\mathcal L_2=\partial_{y}+\lambda \inner{\partial_{y}\varphi}.
\end{aligned}
\right.
\end{equation}

To state our main result we first introduce the following assumption  which, roughly
speaking, says that the set of the points $(x,y)$ at which both of the functions
$s\rightarrow \partial_x\varphi (s,y)$ and   $s\rightarrow \partial_y\varphi (x,s)$ change
sign from $-$ to $+$ for increasing $s$, has measure $0$.

\begin{definition}
[{\bf Assumption} $\boldsymbol{H_1(\alpha)}$ and $\boldsymbol{H_2(\alpha)}$] \label{def}
 Let $\alpha>0$ and let $\varphi$ be defined in a neighborhood $\Omega$ of $0$. Write  $\Omega=\Omega_x\times\Omega_y$  with $\Omega_x,\Omega_y$ being  the projections of $\Omega$ onto $x$ and $y$ axes, respectively.          We say $\varphi$ satisfies  Assumption $\boldsymbol{H_1(\alpha)}$ in $\Omega$ if
    there exists  a family of disjoint open sets $\omega_{j} $  and   $C^1$-functions $r_j$,  $
    j=1,2,\cdots, N_1$ with $N_1$ an integer such that the following properties are fulfilled by $\omega_j$ and $r_j$.
    \begin{enumerate}[(i)]
    	\item Denote by $\bar\omega_j$   the closure of $\omega_j$.       Then
    	  \begin{eqnarray*}
    	 \bigcup_{1\leq j\leq N_1} \ \bar \omega_j =\bar\Omega_y.
    \end{eqnarray*}
    	\item Given $y\in\ \omega_j$ for $1\leq j\leq N_1$,   as $s$ increases in
  $\Omega_x$ the function
  $
    s\longmapsto\partial_x\varphi(s,y)
  $ either has no sign change from $-$ to $+$,
  or
  only changes sign from $-$ to $+$ at the point $r_{j}(y)$.  In the latter we suppose additionally that
  \begin{equation*}
s\longmapsto (\partial_y\varphi)(r_{j}(y), s)
  \end{equation*}
  doesn't change sign from $-$ to $+$ as $s$ increases.
  \item For any $1\leq j\leq N_1$ we have  $\big|\frac{d r_{j}}{dy}\big| \geq \alpha$
  in $\omega_{j}$.
    \end{enumerate}
Moreover Assumption $\boldsymbol{H_2(\alpha)}$ can be defined  symmetrically   by a family of pairs $(U_j,\tau_j)$, $1\leq j\leq N_2$, which satisfy similar properties as that of  $(\omega_j, r_j)$ such that for each $x\in U_j$,
 \begin{equation*}
s\longmapsto (\partial_x\varphi)(s,\tau_{j}(x))
  \end{equation*}
  does not change sign from $-$ to $+$ as $s$ increases if $s\longmapsto \partial_y\varphi(x,s)$ changes sign from $-$ to $+$ at point $\tau_j(x)$.
\end{definition}

\begin{theorem}\label{MainT2}
 Let $\alpha>0$ be given and  denote by
  $P_m$ the set of  polynomials with degree  $\leq m$. Then there exists
  a number $\delta$ depending only on  the degree $m$, and two
  constants $\lambda_0$ and $C$ depending only on $m$ and $\alpha$,  such that
  the following estimate
  \begin{equation*}
\forall \ u\in C_0^\infty(]-\delta,\delta[ \ \times\  ]-\delta,\delta[) ,\quad    \norm{\partial_{x}u}_{L^2}+\norm{\lambda  (\partial_x\varphi ) u}_{L^2}\leq
    C\sum_{1\leq j\leq 2}\norm{\mathcal L_{j}u}_{L^2}
  \end{equation*}
  holds for all $\lambda\geq \lambda_0$ and
  for all $\varphi\in P_m$  satisfying  Assumption
 $\boldsymbol{H_1(\alpha)}$ in  the  neighborhood $]-\delta,\delta[ \ \times\  ]-\delta,\delta[$ of $0$.  Symmetrically
 the   estimate
  \begin{equation*}
\forall \ u\in C_0^\infty(]-\delta,\delta[ \ \times\  ]-\delta,\delta[) , \quad    \norm{\partial_{y}u}_{L^2}+\norm{\lambda  (\partial_y\varphi ) u}_{L^2}\leq
    C\sum_{1\leq j\leq 2}\norm{\mathcal L_{j}u}_{L^2}
  \end{equation*}
  holds for all $\lambda>\lambda_0$ and
  for all $\varphi\in P_m$  satisfying  Assumption
 $\boldsymbol{H_2(\alpha)}$ in $]-\delta,\delta[ \ \times\  ]-\delta,\delta[$.
\end{theorem}

\begin{corollary}
If $\varphi\in  P_m$ satisfies  Assumption
 $\boldsymbol{H_1(\alpha)}$ in a neighborhood of $0$ and the condition that
 \begin{eqnarray*}
 \sum_{0\leq j\leq k}	\abs{\partial_x^{k+1} \varphi(0)}\neq 0
 \end{eqnarray*}
 for some $k$. Then system \eqref{syst} is (micro)subelliptic at positive direction, that is, there exists a constant $C>0,$ a neighborhood $\tilde\Omega$ of $0$ and a positive number $\lambda_0$  such that
\begin{eqnarray*}
    \forall~ \lambda\geq \lambda_0,~\forall~u\in
  C_0^\infty(\tilde\Omega),\quad
    \norm{\lambda^{1/(k+1)} u}_{L^2}\leq
    C \sum_{1\leq j\leq 2}\norm{\mathcal L_j u}_{L^2}.  \end{eqnarray*}
	If $\varphi\in  P_m$ satisfies  Assumptions
 $\boldsymbol{H_1(\alpha)}$  and $\boldsymbol{H_2(\alpha)}$ in a neighborhood of $0$   then we have the following maximal (micro)hypoellipticity:  for any $\lambda\geq \lambda_0$ and any $u\in
  C_0^\infty(\tilde\Omega),$
 \begin{multline*}
      \norm{\partial_{x}u}_{L^2}+  \norm{\partial_{y}u}_{L^2}
    +\norm{\lambda \inner{\partial_{x}\varphi} u}_{L^2}
    +\norm{\lambda \inner{\partial_{y}\varphi} u}_{L^2}\leq
    C \sum_{1\leq j\leq 2}\norm{\mathcal L_j u}_{L^2}.
    \end{multline*}
\end{corollary}

\begin{remark}
  This improves a result of Nourrigat (see Theorem 2.1 of \cite{MR1098615}) by allowing the sign change from $-$ to $+$
  for increasing variable.  The above results may hold for general $\varphi$ rather than polynomial case,  following the argument presented in \cite{MR3775156}.
\end{remark}

We would end up the introduction by an example of Maire (cf. \cite[Example 1.2]{MR567778}).
   These are the functions
  $\varphi$ defined by
  $$\varphi(x,y)=x^{2\ell+1}-xy^2,\quad \ell\in\mathbb Z_+\setminus \set{0}.$$
  Then the system \eqref{syst} has the form of
  \begin{equation*}
    \left\{
    \begin{array}{lll}
      \mathcal L_1=\partial_{x}+\lambda \inner{(2\ell+1)x^{2\ell}-y^2}\\
      \mathcal L_2=\partial_{y}-2\lambda xy.
    \end{array}
    \right.
  \end{equation*}
  Direct verification  shows  that  Assumption $\boldsymbol{H_1(\alpha)}$ in $\mathbb R^2$ is fulfilled if we
  we can take $\omega_{1}=]-\infty,0[$ and
  $r_{1}(y)=\inner{-\frac{y}{\sqrt{2\ell+1}}}^{1\over\ell}$, and $\omega_{2}=]0,\infty[$ and
  $r_{2}(y)=\inner{\frac{y}{\sqrt{ 2\ell+1}}}^{1\over\ell}$. Then it follows Theorem \ref{MainT2} that  there exists two constants $C>0,\lambda_0\geq 0$ such that
 for any $u\in C_0^\infty(\mathbb R^2)$  and for all $\lambda\geq\lambda_0$,
  \begin{equation*}
    \norm{\partial_{x}u}_{L^2}+\norm{\lambda (\partial_{x}\varphi)u}_{L^2}\leq
    C\norm{\mathcal L_{1}u}_{L^2}+C\norm{\mathcal L_{2}u}_{L^2}.
  \end{equation*}
  As a result
  \begin{equation*}
    \norm{ \lambda ^{\frac{1}{2\ell+1}}u}_{L^2}\leq
    C\norm{\mathcal L_{1}u}_{L^2}+C\norm{\mathcal L_{2}u}_{L^2}.
  \end{equation*}
  And hence the system $\mathcal L_j, 1\leq j\leq 2,$  is  subelliptic  with exponent $\frac{1}{2\ell+1}$.   We give here a new proof for the Maire's example basing on condition $(\Psi)$.    As proven by Helffer-Nier (see Chapter 11 of \cite{HelfferNier05}), the exponent
  $\frac{1}{2\ell+1}$ is optimal and can not be improved. This implies
  the system $(\mathcal L_1,\mathcal L_2)$ is not maximally
  hypoelliptic when $\ell>1$, otherwise a better exponent
  $\sigma=\frac{1}{3}$ would be deduced.

\section{Proof of Theorem \ref{MainT2}}

In this part we prove the main result.  Firstly we introduce some notations to be used frequently later. Let
$\varphi(x,y)$ be a polynomial of degree $m$. Then we can write
$\partial_x\varphi(x,y)$ and $\partial_y\varphi(x,y)$ as the following
forms
\begin{equation}\label{10101520}
  \partial_x\varphi(x,y)=\sum_{j=0}^{\ell} a_j(y)x^j,  ~
  \textrm{and} ~\partial_y\varphi(x,y)=\sum_{j=0}^{d}
  b_j(x)y^j,
\end{equation}
where $\ell,d\leq m$ and $a_j(y)$, $b_j(x)$ are polynomials of $y$ and $x$
respectively, such that the leading coefficients $a_{\ell}\not\equiv
0$ and $b_{d}\not\equiv0$. For each pair $(x,y)$ we set
 \begin{equation*}
 \left\{
 \begin{aligned}
    & M_{1}(x,y)=\sum_{j=0}^\ell
     \abs{\lambda \partial_x^{j+1} \varphi(x,y)}^{\frac{1}{j+1}},\\
     &M_{2}(x,y)=\sum_{j=0}^d\abs{\lambda \partial_y^{j+1}
 \varphi(x,y)}^{\frac{1}{j+1}},
     \end{aligned}
     \right.
  \end{equation*}
  and  set
  \begin{equation}\label{deg}
 	G(x,y)=\sum_{1\leq i+j\leq m}\abs{\lambda \partial_x^{i}
  \partial_y^j\varphi(x,y)}^{\frac{1}{ i+j}}.
  \end{equation}
Let $\mathcal N_1$, $\mathcal N_2$ be
  defined by
  \begin{equation}\label{n1n2}
  \mathcal N_1=\set{y\in\mathbb R ; ~M_1(x,y)\neq 0~\textrm{for
  all~} x}, \quad
   \mathcal N_2=\set{x\in\mathbb R ; ~M_2(x,y)\neq 0~\textrm{for all~} y} .
   \end{equation}
  We remark that  $\mathcal N_1$ and  $\mathcal N_2$ are
  dense in $\mathbb R$, since $\mathcal N_1$ and $\mathcal N_2$ include,respectively, the
  sets $\set{y ; ~a_{\ell}(y)\neq 0}$ and $\set{x ; ~b_{d}(x)\neq0}$ which are dense in $\mathbb R$.

To simplify the notation we will use the capital letter $C$
to denote some generic constant  that may vary from line to line and   depend only on the number $\alpha$  in Definition \ref{def} and the degree $m$ of the polynomial $\varphi$, but are independent of $\lambda$.

\begin{lemma}\label{legu}
  Let $\delta>0$ be a given number, and let $I$ be a subset of
  $\mathbb R$. Then there exists a constant $C$ such that for any $u\in C_0^\infty\inner{]-\delta,  \delta[\,\times ]-\delta,  \delta[}$ we have
  \begin{equation}\label{10101701}
    \norm{G  u}_{L^2(I\times ]-\delta,\delta[)}\leq C
    \sum_{1\leq j\leq 2}\norm{\mathcal L_{j} u}_{L^2}
    +C \inner{\norm{M_{1}  u}_{L^2}+
    \norm{M_{2}  u}_{L^2(I\times ]-\delta,\delta[)}},
  \end{equation}
  recalling $G$ is defined  by \eqref{deg}.
\end{lemma}

\begin{proof}
    If $I=]-\delta,\delta[$ the estimate \eqref{10101701} is a straightforward consequence
    of Baker-Campbell-Hausdorff formula (see e.g. \cite[Lemma 4.14]{MR2987644}).
    Next we treat the general
    case, following the argument presented in \cite{MR2987644}.  In this proof we always let $u\in
    C_0^\infty\inner{]-\delta,  \delta[\,\times ]-\delta,  \delta[}$.
    Direct verification  shows
    \begin{equation}\label{wfd}
    	\norm{\partial_x u}_{L^2}^2+\norm{\lambda(\partial_x\varphi) u}_{L^2}^2=
    \norm{\mathcal L_1 u}_{L^2}^2+
    \inner{ \lambda (\partial_x^2\varphi) u,\ u}_{L^2}^2.
    \end{equation}
    Then
     $$\norm{\partial_x u}_{L^2}^2\leq
    \norm{\mathcal L_1 u}_{L^2}^2+
    \norm{|\lambda \partial_x^2\varphi|^{1\over
    2}u}_{L^2}^2.$$ Furthermore since the function
  $y\longmapsto u(x,y)$ with $x\in I$ fixed  has compact support in the interval
  $]-\delta,\delta[$, then
  \begin{equation*}
    \norm{\partial_y u}_{L^2(I\times ]-\delta,\delta[)}^2\leq
    \norm{\mathcal L_2 u}_{L^2(I\times ]-\delta,\delta[)}^2+
    \norm{|\lambda \partial_y^2\varphi|^{1/2}u}_{L^2(I\times ]-\delta,\delta[)}^2.
  \end{equation*}
  It follows  that
  \begin{equation*}
    \norm{X u}_{L^2}^2+
    \norm{Y u}_{L^2}^2\leq
    \sum_{1\leq j\leq 2}\norm{\mathcal L_j u}_{L^2}^2+2
    \norm{M_{1}u}_{L^2}^2
    +\norm{M_{2}u}_{L^2(I\times ]-\delta,\delta[)}^2,
  \end{equation*}
  where the operators $X$ and $Y$ are defined by $X=\frac{\partial_x}{i}$  and
  $Y=\chi_I(x)\frac{\partial_y}{i}$ with $\chi_I(x)$ being the characteristic function of the set
  $I$.
  Thus the desired estimate will follow if there exists a constant $C$ such that for
  any integer $q\geq 1$  one has
  \begin{equation}\label{pq0}
    \sum_{p\geq 0}
    \norm{\abs{A_{p,q}}^{\frac{1}{p+q+1}} u}_{L^2}\leq C\inner{
    \norm{X u}_{L^2}+
    \norm{Y u}_{L^2}
    +\norm{ \lambda (\partial_x\varphi) u}_{L^2}},
  \end{equation}
  where $A_{p,q}$ are  purely imaginary-valued functions defined iteratively  by
  $$
  A_{0,q}= i^{q-1}  [\underbrace{X,~ [X,\cdots [X}_{q},~ \lambda \partial_x\varphi]~]\cdots],\quad
  A_{p,q}= i    [Y,~A_{p-1,q}] \textrm{~ for ~}p\geq 1.
  $$
  Here we denote by $[P,\ Q]$ the commutator of  two operators $P$ and $Q$,  which is defined by $
		[P,\ Q]=PQ-QP
$.
  Firstly note that for
   any $q\geq 1$ we have
  \begin{equation}\label{p0}
    \norm{\abs{A_{0,q}}^{\frac{1}{q+1}} u}_{L^2}\leq C\inner{
    \norm{\partial_x u}_{L^2}+
    \norm{ \lambda (\partial_x\varphi )u}_{L^2}+
    \norm{u}_{L^2}}.
  \end{equation}
  This can be deduced directly from Baker-Campbell-Hausdorff formula  (see e.g. \cite[Lemma 4.14]{MR2987644}). Next we will
  treat the case when $p\geq 1$.  Baker-Campbell-Hausdorff formula implies
  \begin{equation*}
  \begin{split}
    \exp\inner{tY}\exp\inner{t^{p+q}A_{p-1,q}}\exp\inner{-tY}\exp\inner{-t^{p+q}A_{p-1,q}}\\=
    \exp\biginner{t^{p+q+1}A_{p,q}+\sum_{j=1}^{m}c_j t^{p+q+j+1}A_{p+j, q}},
  \end{split}
  \end{equation*}
  where $c_j$ are a family of constants. As a result we use induction on $p$ to conclude
  that there exist two integers $n_1$ and $n_2$ depending only on $p$,
  and a sequence $(\eps_1,\eps_2,\cdots,\eps_{n_1})$ with $\eps_j=\pm
  1$, such that for each $p\geq 1$ one has
  \begin{equation*}
  \begin{split}
    \exp\inner{t^{p+q+1}A_{p,q}}
    &=\prod_{ j=1}^{n_2} Z_j
    \exp\biginner{\sum_{k=1}^m c_k t^{p+q+k+1}A_{p+k, q}}\\
    &=\prod_{ j=1}^{n_2} Z_j
    \prod_{k=1}^m\exp\inner{c_k t^{p+q+k+1}A_{p+k, q}}
  \end{split}
  \end{equation*}
   with $Z_j\in\set{\exp\inner{\eps_{1} tY}, \cdots,
  \exp\inner{\eps_{n_1} tY},
   \exp\inner{t^{q+1}A_{0,q}}}$. The last equality holds because $A_{p,q}$ are functions and hence
  commutative. Then $$\exp\inner{t^{p+q+1}A_{p,q}}-{\rm Id}$$   with ${\rm Id}$ the identity
  operator, can be rewritten as
  \begin{equation*}
  \begin{split}
    &\prod_{k=1}^{n_2} Z_k\sum_{i=1}^m\prod_{ j=1}^{i-1}
    \exp\biginner{c_j t^{q+j+1}A_{p+j, q}}
    \big(\exp\biginner{c_i t^{q+i+1}A_{p+i, q}}-{\rm
    Id}\big) \\
    &+\sum_{i=1}^{n_2}\prod_{ j=1}^{i-1} Z_j\inner{Z_i-{\rm Id}},
  \end{split}
  \end{equation*}
  which, together with fact that the norms  of operators $Z_j$ are smaller than $1$,
  implies
  \begin{multline} \label{10101705}
    \norm{\exp\inner{t^{p+q+1}A_{p,q}}u-u}_{L^2}^2\leq C
    \sum_{1\leq k\leq m}\norm{\exp\inner{c_k t^{k+q+1}A_{p+k,
    q}}u-u}_{L^2}^2\\
    +C\sum_{\eps_I\in\big\{\eps_1,\cdots,\eps_{n_1}\big\}} \norm{\exp\inner{\eps_{I} tY}u-u}_{L^2}^2
     +C\norm{\exp\inner{t^{q+1}A_{0,q}}u-u}_{L^2}^2.
  \end{multline}
  Since the functions $i  A_{p,q}$ are real-valued and hence
  self-adjoint as operators acting on $L^2$. So by the
  spectral theorem the norm
  $\norm{|i  A_{p,q} |^{\frac{1}{p+q+1}}u}_{L^2}$  is equivalent to
  \[
     (p+q+1)^{-1}\int_0^\infty
     \norm{\exp\inner{
     t^{p+q+1} A_{p,q}}u-u}_{L^2}^2  \frac{dt}{t^{3}}.
  \]
  As a result, integrating both sides of \eqref{10101705} with respect to $t$ with the measure $t^{-3}dt$
  gives that
  \begin{equation*}
    \norm{\abs{A_{p,q}}^{\frac{1}{p+q+1}} u}_{L^2}\leq C
    \sum_{k=1}^m\norm{\abs{A_{p+k,q}}^{\frac{1}{p+k+q+1}} u}_{L^2}\\
    +C\norm{Y u}_{L^2}
    +C\norm{\abs{A_{0,q}}^{\frac{1}{q+1}}u}_{L^2}.
  \end{equation*}
  Using reverse induction starting from $p=m$ and combing \eqref{p0},
  we get the desired estimate \eqref{pq0}, completing the proof.
\end{proof}

\begin{lemma}
  There exist two  constants $C_*\geq 1$ and $r_0\leq 1$ depending only on $m$,  such that for any $y\in \mathcal N_1$ and $x\in \mathcal N_2$  with $\mathcal N_j$ defined by \eqref{n1n2}, we have
  \begin{equation}\label{10101401}
    \forall~x_*,\tilde x\in \mathbb R, ~\abs{x_*-\tilde x}
    M_{1}(x_*,y)<r_0 \Longrightarrow C_*^{-1}  \leq \frac{ M_{1}(x_*,y)}{M_{1}(\tilde
    x,y)}\leq C_* ,
  \end{equation}
  \begin{equation}\label{10101401+}
    \forall~y_*,\tilde y\in \mathbb R, ~\abs{y_*-\tilde y} M_{2}(x,y_*)<r_0
    \Longrightarrow  C_*^{-1} \leq \frac{M_{2}(x,y_*)}{M_{2}( x,\tilde y)}\leq C_*.
  \end{equation}
 And for any $x\in\mathbb R,$
  \begin{equation}\label{10101401++}
    \forall~y_*,\tilde y\in \mathbb R, ~\abs{y_*-\tilde y} G(x,y_*)<r_0
    \Longrightarrow  C_*^{-1} \leq \frac{ G (x,y_*)}{G ( x,\tilde y)}\leq C_*.
  \end{equation}
\end{lemma}

\begin{proof}
  We  firstly show \eqref{10101401}, and claim that      for  any $y\in \mathcal N_1$
    we have, with each $  0\leq j\leq \ell$ with  $\ell$ the integer given in \eqref{10101520},
  \begin{equation}\label{10101402}
 \forall\ \abs{x_*-\tilde x} M_{1}(x_*,y)<1,\quad \abs{\lambda \partial_x^{j+1} \varphi(\tilde x,y)}  \leq
    (\ell+2) M_{1}(x_*,y)^{j+1}.
  \end{equation}
     In fact, by
 Taylor'expansion
  $$
    \partial_x^{j+1} \varphi(\tilde x,y)=
    \partial_x^{j+1}\varphi(x_*,y)+\sum_{i=0}^{\ell-j}
    \frac{\partial_x^{i+j+1}\varphi(x_*,y)}{i!}(\tilde
    x-x_*)^i,
  $$
 we have
  $$
    \abs{\lambda \partial_x^{j+1} \varphi(\tilde x,y)} \leq
    M_{1}(x_*,y)^{j+1}+ \sum_{i=0}^{\ell-j} M_{1}(x_*,y)^{i+j+1}\abs{\tilde
    x-x_*}^i,
  $$ which yields the assertion
  \eqref{10101402}. As a result it follows from \eqref{10101402} that
  \begin{equation}\label{ies}
  \forall\  x_*, \tilde x\in\mathbb R\  \textrm{with}\,\abs{x_*-\tilde x} M_{1}(x_*,y)<1,\quad	M_{1}(\tilde x,y)\leq (\ell+2)^2M_{1}(x_*,y).
  \end{equation}
  Now for any $x_*$ and $\tilde x$ with $ \abs{x_*-\tilde x} M_{1}(x_*,y)<r_0\leq 1$,  we use \eqref{ies} to get
  $$\abs{x_*-\tilde x} M_{1}(\tilde x,y)\leq (\ell+2)^2 \abs{x_*-\tilde x} M_{1}(x_*,y)<(\ell+2)^2r_0.$$
  As a result, if  we choose $r_0$ sufficiently small such that  $(\ell+2)^2r_0<1$ then we use \eqref{ies} to conclude
  \begin{eqnarray*}
  	M_{1}( x_*,y)\leq (\ell+2)^2M_{1}(\tilde x,y).
  \end{eqnarray*}
 We have proven \eqref{10101401}.
  The inequalities \eqref{10101401+} and \eqref{10101401++} can be deduced similarly. This completes the proof.
\end{proof}

\begin{lemma}\label{MH}
  Let $P_m$ be the set of all polynomials with degree
  $\leq m$, and let  $r\in C^1\inner{\omega; ~\mathbb
  R}$ be a given function defined in $\omega$.
  Then there
  exists a number $\delta>0$ and a constant  $C>0$, both depending only on $m$,
   such that the
  following estimate
  \begin{equation}\label{10101406}
    \norm{M_2u}_{L^2\inner{I\times ]-\delta,\delta[}}\leq
    C\norm{\mathcal L_2 u}_{L^2}
  \end{equation}
  holds for all $u\in C_0^\infty\inner{]-\delta,\delta[\,\times\,]-\delta,\delta[}$ and for all $\varphi\in P_m$ satisfying
  that for each $s\in\omega$ the function
  $y\longmapsto \partial_y\varphi\inner{r(s),  y}$
  has no sign change from $-$ to $+$  for increasing $y$, where  $I$ stands for the range of $r$, i.e.,  $I=r(\omega)$.
\end{lemma}

\begin{remark}
  If $r=Id$ then this is just a local version of the result of Nourrigat (see Theorem 2.1 of \cite{MR1098615}).
\end{remark}

To prove the above lemma we need the following
\begin{lemma}
	[Lemmas 3.1.1-3.1.3 of  \cite{MR2599384}] \label{lemsig} Let $[a, b]\subset\mathbb R$ be a given interval, and let  $\phi(s)\in C^0\inner{[a,b];\,\mathbb R}$ such that  $\phi$ does not change sign from $-$ to $+$ as $s$ increases in $[a,b]$. Then for any $v\in C^1\inner{[a,b]}$ with $v(a)=0$ we have
	\begin{eqnarray*}
		\max_{s\in[a,b]}\abs{v(s)}^2+2\int_a^b\abs{\lambda\phi(s)}\abs{v(s)}^2ds\leq \abs{v(b)}^2+2\int_a^b\Big|\big(\frac{d}{ds}+\lambda\phi\big)v(s)\Big|\abs{v(s)}ds,
	\end{eqnarray*}
	where $\lambda>0$ is a given number.  In particular, for any for any $v\in C_0^1\inner{[a,b]}$  we have
	\begin{eqnarray*}
		\max_{s\in[a,b]}\abs{v(s)}^2+2\int_a^b\abs{\lambda\phi(s)}\abs{v(s)}^2ds\leq  2\int_a^b\Big|\big(\frac{d}{ds}+\lambda\phi\big)v(s)\Big|\abs{v(s)}ds.
	\end{eqnarray*}
Moreover if
	\begin{eqnarray*}
		\min_{s\in[a,b]}\Big|\frac{d^k\phi}{ds^k}(s)\Big|\geq c_*
	\end{eqnarray*}
	for some   $k\in\mathbb Z_+$ and some constant $c_*>0$, then for any $v\in C_0^1\inner{[a,b]}$, we have
	\begin{eqnarray*}
\lambda^{\frac{1}{k+1}}\int_a^b  \abs{v(s)}^2ds \leq  C\Big(\max_{s\in[a,b]}\abs{v(s)}^2+\int_a^b\abs{\lambda\phi(s)}\abs{v(s)}^2ds\Big),
	\end{eqnarray*}
	with $C$ a constant depending only on $k$ and $c_*$ above.
\end{lemma}
\begin{proof}
	The just follows from the argument for proving     \cite [Lemma 3.1.1]{MR2599384} and the assertions in   \cite [Lemmas 3.1.2-3.1.3]{MR2599384}.
	\end{proof}

\begin{proof}[Proof of Lemma \ref{MH}] Recall  $\mathcal N_2$ is defined by \eqref{n1n2}.
  Since $r(\omega)\setminus\mathcal N_2$ has measure $0$ then we may assume without loss of generality that  $r(\omega)\subset\mathcal
  N_2$.
  For $x_0=r(s_0)\in r(\omega)\subset \mathcal
  N_2$,  $y_0\in \mathbb R$ and $\varphi\in P_m$
  we define a new function
  $\zeta(x_0,\cdot)$ by setting
  \[
    \zeta(x_0,y)=\frac{ (\partial_y \varphi)\inner{x_0,
    y_0+\frac{y}{M_{2}(x_0,y_0)}}}{M_{2}(x_0,y_0)}.
  \]
  Then we use Taylor's expansion to write
     \begin{eqnarray*}
  	    \zeta(x_0,y)=\sum_{0\leq j\leq d}\frac{  \partial_y^{j+1} \varphi\inner{x_0,
    y_0}}{M_{2}(x_0,y_0)^{j+1}}\frac{y^j}{j!},
  \end{eqnarray*}
 where the integer $d$ is given in \eqref{10101520}.  By direct verification,
  \begin{equation*}
  \forall\ 0\leq j \leq d,\quad\frac{ \abs{  \partial_y^{j+1} \varphi\inner{x_0,
    y_0}}}{M_{2}(x_0,y_0)^{j+1}}\leq 1	,
  \end{equation*}
  which yields
  \begin{equation}\label{ues}
  \forall\ 0\leq j \leq d,\quad\abs{ \partial_y^{j} \zeta\inner{x_0,
    0}}\leq 1.
  \end{equation}
  On the other hand, we can find an integer $k$ with $0\leq k\leq d\leq m$ such that
  \begin{eqnarray*}
  	\abs{ \partial_y^{k+1} \varphi\inner{x_0,
    y_0}}^{1/(k+1)} =\max_{0\leq j\leq d}\abs{ \partial_y^{j+1} \varphi\inner{x_0,
    y_0}}^{1/(j+1)} .
  \end{eqnarray*}
  Thus
  \begin{equation*}
    \abs{\inner{\partial_y^k\zeta}
    \inner{x_0,0}}=\frac{ \partial_y^{k+1} \varphi\inner{x_0,
    y_0}}{M_{2}(x_0,y_0)^{k+1}}\geq c_0
  \end{equation*}
   for some constant $c_0>0$ depending only on $m$.
This, with \eqref{ues}, implies
    $$\abs{\inner{\partial_y^k\zeta}
    \inner{x_0,y}}\geq  c_0 -\sum_{1\leq j\leq d-k} \abs{y}^{j}. $$
  As a result one can find a number $\delta>0$  depending only on
  $m$,  such that
  \begin{equation}\label{10101405}
    \inf_{y\in]-\delta,\delta[} \abs{\inner{\partial_y^k\zeta}
    \inner{x_0,y}}\geq c_0/2.
  \end{equation}
  Moreover by assumption $\partial_y\varphi(x_0,y)$ has no sign change from $-$ to $+$ for increasing $y$.
  Then the function
  $\zeta(x_0,y)$ doesn't change sign from $-$ to $+$ as $y$
  increases in $]-\delta,\delta[$. As a result we apply Lemma \ref{lemsig} with $\phi=\zeta$ to conclude    for any $v\in C_0^\infty(]-\delta,\delta[)$ we have
  \begin{equation*}\label{10101501}
    \lambda^{\frac{1}{k+1}}\int_{-\delta}^\delta \abs{v(y)}^2dy \leq
     C \int_{-\delta}^\delta
    \abs{(\partial_y+\lambda \zeta(x_0,y))v(y)} ^2dy+C\int_{-\delta}^\delta\abs{v(y)}^2
     d y.
  \end{equation*}
   Inspired by \cite{MR781536},  for any function $v\in C^\infty\inner{[-\delta,\delta]}$  having no compact
  support in $]-\delta,\delta[$, we can introduce a cut-off function
  $\chi(y)$ supported in the interval $]-\delta,\delta[$  and equal to $1$ in
  $]-\delta/2,\delta/2[$.
  Then applying the above inequality to the function $\chi(y)  v(y)$ gives
  that, for any $v\in C^\infty([-\delta,\delta])$,
    \begin{equation}\label{10101502}
    \lambda^{\frac{1}{k+1}}\int_{-{\delta\over2}}^{\delta\over 2}
     \abs{v(y)}^2 d y
    \leq
     C \int_{-\delta}^\delta
    \abs{\inner{\partial_y + \lambda \zeta\inner{x_0,y}}v(y)}^2
   dy+C \int_{-\delta}^\delta \abs{v(y)}^2 d y.
  \end{equation}
 Now for any given  $u\in C_0^\infty\inner{]-\delta,\delta[\,\times\,]-\delta,\delta[}$ we define
 $v\in C^\infty([-\delta,\delta])$ by setting
  \[
    v(y)=u\Big(x_0, y_0+\frac{y}{M_{2}(x_0,y_0)}\Big).
  \]
  For such a function $v$ we can verify that
  \[
    \big(\partial_y+ \lambda \zeta(x_0,y_0)\big)v (y)= \frac{(\mathcal L_2 u)\inner{x_0, y_0+y/M_{2}(x_0,y_0)}}{M_{2}(x_0,y_0)}.
  \]
  As a result, applying \eqref{10101502} to the function $v$ above   and then using the change of variables, we get
  \begin{multline*}
     \lambda^{\frac{1}{k+1}}\int_{I_{y_0}}
  M_{2}(x_0,y_0)^3 \abs{u(x_0,y)}^2d y\\     \leq
    C\int_{J_{y_0}} M_{2}(x_0,y_0)
    \abs{(\mathcal L_2 u)(x_0,y)}^2
     d y  + C \int_{J_{y_0}}  M_{2}(x_0,y_0)^3 \abs{  u(x_0,y)}^2
     d y,
  \end{multline*}
  where
  \begin{eqnarray*}
  I_{y_0}=\big\{y ; ~ \abs{y-y_0}M_{2}(x_0,y_0)<\delta/2\big\},\quad J_{y_0}=\big\{y ; ~ \abs{y-y_0}M_2(
  x_0,y_0)<\delta\big\}.
 \end{eqnarray*}
  By \eqref{10101401+} we see
  $C_*^{-1}\leq  M_2(x_0,y_0)/M_{2}(x_0,y)\leq C_*$
  when $y\in J_{y_0}$,  shrinking $\delta$ if necessary. Then we may replay $M_{2}(x_0,y_0)$ by
  $M_{2}(x_0,y)$ in the above integrands, that is,
\begin{multline*}\label{m2es}
     \lambda^{\frac{1}{k+1}}\int_{I_{y_0}}
  M_{2}(x_0,y)^3 \abs{u(x_0,y)}^2d y\\     \leq
    C\int_{J_{y_0}} M_{2}(x_0,y)
    \abs{(\mathcal L_2 u)(x_0,y)}^2
     d y  + C \int_{J_{y_0}}  M_{2}(x_0,y)^3 \abs{  u(x_0,y)}^2
     d y.
  \end{multline*}
  Thus taking integration with respect to $y_0\in\mathbb R$ on the both sides of above inequality and  observing  \begin{eqnarray*}
  	\set{y ; ~
  \abs{y-y_0}M_2(x_0,y)<\delta/(2C_*)}\subset I_{y_0}, \quad    J_{y_0}\subset \set{y ; ~
  \abs{y-y_0}M_2(x_0,y)<C_*\delta },
  \end{eqnarray*}
   we obtain, applying Fubini's theorem,
  \begin{multline*}
    \lambda^{\frac{1}{k+1}}\int_{-\delta}^{\delta}
    \abs{M_{2}(x_0,  y)u(x_0,y)}^2d y \leq
    C\int_{-\delta}^{\delta}
    \abs{(\mathcal L_2 u)(x_0,y)}^2dy \\
     + C \int_{-\delta}^{\delta}  \abs{M_{2}(x_0, y)u(x_0,y)}^2
     d y.
  \end{multline*}
  Note that the constant $C$ is independent of $\lambda$, and hence we
  can choose $\lambda$ large enough such that $\lambda^{\frac{1}{k+1}}\geq
  2C$. This implies
  \begin{equation*}
    \lambda^{\frac{1}{k+1}}\int_{-\delta}^{\delta}
    \abs{M_{2}(x_0,  y)u(x_0,y)}^2d y \leq
    C\int_{-\delta}^{\delta}
    \abs{(\mathcal L_2 u)(x_0,y)}^2dy
  \end{equation*}
  Now integrating  both sides with respect to $x_0\in r(\omega)$ gives the
  desired estimate \eqref{10101406}.
  The proof is hence completed.
\end{proof}

\begin{lemma}
 Let $\delta, \alpha>0$   be two given numbers with $\delta$ small sufficiently and let  $r(y)\in C^1\inner{\omega; ~\mathbb
  R}$ such that  $\big|\frac{dr}{dy}(y)\big|\geq \alpha$ for any
  $y\in \omega$. Suppose $\varphi\in P_m$ satisfies
  that for each $y_0\in\omega$ the function
  $y\longmapsto (\partial_y\varphi)\inner{r(y_0),  y}$
  has no sign change from $-$ to $+$ for increasing $y$.
  Then  the following esitmate
  \begin{equation}\label{10101505}
    \int_{\omega} G(r(y),y)\abs{u(r(y),y)}^2d y
    \leq C  \sum_{1\leq j\leq 2}\norm{\mathcal L_j u}_{L^2}^2+C
    \norm{M_{1}u}_{L^2}^2
  \end{equation}
  holds for all $u\in C_0^\infty\inner{]-\delta,\delta[\,\times\,]-\delta,\delta[}$.  Recall $G=\sum_{i+j\geq 1}\abs{  \lambda\partial_x^{i}
 \partial_y^{j}\varphi}^{\frac{1}{i+j}}$.
\end{lemma}

\begin{proof}
This proof is quite similar as  Lemma \ref{MH}.  Let $y_0\in\omega $ be given and
we define a new function
  $y\longmapsto \eta(r(y_0),y)$ by setting
  \[
    \eta(r(y_0),y)=\frac{  (\partial_y \varphi )\inner{r(y_0),
    y_0+y/G(r(y_0), y_0)}}{G(r(y_0),
    y_0)}.
  \]
  Then by the assumption, the function $y\longmapsto \eta\inner{r(y_0), y}$ does not change sign from $-$ to $+$ as   $y$ increases in $]-\delta,\delta[$.  Then we apply Lemma \ref{lemsig} to conclude, for any $v\in C_0^\infty\inner{]-\delta,\delta[}$ we have
  \begin{eqnarray*}
 \max_{y\in ]-\delta,\delta[}    \abs{v(y)}^2  \leq
   2 \int_{-\delta}^\delta
    \abs{(\partial_y+\lambda \eta(r(y_0),y))v(y)} \abs{v(y)} dy.
     \end{eqnarray*}
     Thus applying  the above inequality to $\chi(y)  v(y)$, with $\chi\in C_0^{\infty}\inner{]-\delta,\delta[}$ and $\chi\equiv 1$ on $[-\delta/2,\delta/2]$, we obtain, for any $v\in C^\infty([-\delta,\delta])$
  \begin{eqnarray*}
 \abs{v(0)}^2\leq \max_{y\in ]-\frac{\delta}{2},\frac{\delta}{2}[}    \abs{v(y)}^2  \leq
   C \int_{-\delta}^\delta
    \abs{(\partial_y+\lambda \eta(r(y_0),y))v(y)}^2dy+C\int_{-\delta}^\delta \abs{v(y)}^2 dy.
     \end{eqnarray*}
  As a result for any $u\in C_0^\infty\inner{]-\delta,\delta[\,\times\,]-\delta,\delta[}$, we apply the above estimate to
  \[
    v(y):=u\Big(r(y_0), y_0+\frac{y}{G(r(y_0),y_0)}\Big);
  \]
  this yields, by virtue of the change of variables,
\begin{multline*}
    G(r(y_0),y_0)^2\abs{u(r(y_0),y_0)}^2\leq
    C  \int_{K_{y_0}}
    \abs{(\mathcal L_2 u)(r(y_0),y)}^2dy\\
    +C  \int_{K_{y_0}} G(r(y_0),y_0)^2\abs{u(r(y_0),y)}^2
     d y,
  \end{multline*}
   where $K_{y_0}=\set{y ; ~ \abs{y-y_0}G(
  r(y_0),y_0)<\delta}$.
  It follows from \eqref{10101401++}  that
  \begin{eqnarray*}
  	\forall\ y\in K_{y_0},\quad C_*^{-1}G(r(y_0),y)\leq  G(r(y_0),y_0)\leq C_*G(r(y_0),y).
  \end{eqnarray*}
   Then we may replay $G(r(y_0),y_0)$ by
  $G(r(y_0),y)$ in the above integrands.
  This gives
\begin{multline*}
    G(r(y_0), y_0)^2\abs{u(r(y_0),y_0)}^2\\ \leq
    C  \int_{-\delta}^\delta
    \abs{(\mathcal L_2 u)(r(y_0),y)}^2dy+C\int_{-\delta}^\delta  G(r(y_0), y) ^2\abs{ u(r(y_0),y)}^2
     d y.
  \end{multline*}
  Integrating both sides
  with respect to $y_0\in\omega$ gives
  \begin{equation*}
  \begin{aligned}
   & \int_{\omega} G(r(y_0),y_0)^2\abs{u(r(y_0),y_0)}^2d y_0\\
   &\leq C  \int_{\omega} \int_{-\delta}^\delta \inner{
    \abs{(\mathcal L_2 u)(r(y_0),y)}^2 +   G(r(y_0), y)^2\abs{ u(r(y_0),y)}^2}
     d ydy_0\\
   &\leq C
    \norm{ \mathcal L_2 u}_{L^2}^2 +  C\norm{G     u }_{L^2(I\times]-\delta,\delta[)}^2    \end{aligned}
  \end{equation*}
  with $I=r\inner{\omega}$,
  the last inequality using the change of variables $r(y_0)\rightarrow x$ and the fact that $\abs{r'(y)}\geq \alpha$ for all $y\in\omega$.    Moreover  we apply Lemma \ref{legu} and Lemma \ref{MH} to conclude
  \begin{eqnarray*}
  	\begin{aligned}
  		\norm{G     u }_{L^2(I\times]-\delta,\delta[)} & \leq  C
    \sum_{1\leq j\leq 2}\norm{\mathcal L_{j} u}_{L^2}
    +C \inner{\norm{M_{1}  u}_{L^2}+
    \norm{M_{2}  u}_{L^2(I\times ]-\delta.\delta[)}}\\& \leq  C
    \sum_{1\leq j\leq 2}\norm{\mathcal L_{j} u}_{L^2}
    +C  \norm{M_{1}  u}_{L^2}.
  	\end{aligned}
  \end{eqnarray*}
   Combining the above inequalities  gives the desired estimate \eqref{10101505}.
  The proof is completed.
  \end{proof}

\begin{proof}[Proof of Theorem \ref{MainT2}]
  Let $\varphi\in P_m$ satisfy Assumption $\boldsymbol{H_1(\alpha)}$ and let $\omega_j, r_j, 1\leq j\leq N_1,$ be given therein.  For any $y_0\in \omega_j \cap \mathcal N_1$ with $1\leq j\leq N_1$ and
  $x_0\in\mathbb R$ we define a new
  function $x\longmapsto \xi(x,y_0)$ by setting
  \[
     \xi(x, y_0)=\frac{  (\partial_x \varphi)\inner{x_0+\frac{x}{M_{1}(x_0,y_0)},
     y_0}}{M_{1}(x_0,y_0)}.
  \]
  Similar to the proof of \eqref{10101405}, we may find an integer $k$ such that
  \begin{equation}\label{10101508}
    \inf_{x\in]-\delta,\delta[} \abs{\inner{\partial_x^k\xi}
    \inner{x,y_0}}\geq  c_0  /2
  \end{equation}
  with $c_0,\delta$ depending only on $m$.
  Denote
  $$\mathcal A(y_0)=\set{x\in\mathbb R;  ~\abs{x-r(y_0)}M_{1}(x,y_0)<\delta}.$$
  We have two cases that either $x_0\in\mathcal A(y_0)$ or $x_0\not\in\mathcal
  A(y_0)$.

\noindent {\it Case (a)}.  We  firstly consider the case when $x_0\in\mathcal A(y_0)$, that is,
\begin{equation}\label{eqdel}
	\abs{r(y_0)-x_0}M_{1}(x_0,
  y_0)\leq \delta.
\end{equation}
Since   $\varphi\in P_m$ satisfies Assumption $\boldsymbol{H_1(\alpha)}$ then  the function $ x\longmapsto
  \partial_x\varphi\inner{x,y_0}$ only changes sign from $-$ to $+$
  at $r(y_0)$  for increasing $x$  and the function
  $y\longmapsto \partial_y\varphi\inner{r(y_0),  y} $
  has no sign change from $-$ to $+$ for increasing $y$.
    This with \eqref{eqdel}   implies that  $x\longmapsto \xi(x,y_0)$  only changes sign
  from $-$ to $+$ at the point $\inner{r(y_0)-x_0}M_{1}(x_0,
  y_0)$ for $x$ increases in $]-\delta,\delta[$.
  Then we apply the  first assertion in Lemma \ref{lemsig} with $[a, b]=[-\delta, \inner{r(y_0)-x_0}M_{1}(x_0,
  y_0) ]$ or $[a, b]=[\inner{r(y_0)-x_0}M_{1}(x_0,
  y_0), \delta]$, to conclude
  for any $v\in C_0^\infty(-]\delta,\delta[)$,
  \begin{equation*}\label{10101512}
  \begin{aligned}
   &  \sup_{x\in]-\delta,\delta[}\abs{v(x)}^2+2\int_{-\delta}^\delta
     \abs{\lambda\xi(x,y_0)}\abs{v(x)}^2 d x\\
     &\leq
    2\abs{v \big(\inner{r(y_0)-x_0}M_{1}(x_0,y_0)\big)}^2
     +4\int_{-\delta}^\delta
    \abs{(\partial_x+\lambda\xi(x,y_0))v(x)} \abs{v(x)}
     d x.
  \end{aligned}
  \end{equation*}
  which, with \eqref{10101508}
   and the last assertion in Lemma \ref{lemsig}, yields, for any $v\in C_0^\infty(]-\delta,\delta[)$,
   \begin{equation*}\label{10101512}
  \begin{aligned}
   &  \lambda^{\frac{1}{k+1}}\int_{-\delta}^\delta
       \abs{v(x)}^2 d x\\
     &\leq
    C\abs{v \big(\inner{r(y_0)-x_0}M_{1}(x_0,y_0)\big)}^2
     +C\int_{-\delta}^\delta
    \abs{(\partial_x+\lambda\xi(x,y_0))v(x)} \abs{v(x)}
     d x.
  \end{aligned}
  \end{equation*}
  Then we follow the argument for proving  Lemma \ref{MH} to conclude,  for any $v\in C^\infty([-\delta,\delta])$
  \begin{equation*}\label{10101512}
  \begin{aligned}
   &  \lambda^{\frac{1}{k+1}}\int_{-\delta/2}^{\delta/2}
       \abs{v(x)}^2 d x\\
     &\leq
    C\abs{v \big(\inner{r(y_0)-x_0}M_{1}(x_0,y_0)\big)}^2
     +C\int_{-\delta}^\delta
    \abs{(\partial_x+\lambda\xi(x,y_0))v(x)} \abs{v(x)}
     d x.
  \end{aligned}
  \end{equation*}
  Now for any $u\in C_0^\infty(]-\delta,\delta[\,\times\,]-\delta,\delta[) $ we apply the above estimate to
  \begin{eqnarray*}
  	v(x):=u\Big(x_0+\frac{x}{M_1(x_0,y_0)}, y_0\Big);
  \end{eqnarray*}
  this gives,
  for any  $u\in C_0^\infty(]-\delta,\delta[\,\times\,]-\delta,\delta[) $,
  \begin{equation}\label{mla}
  \begin{split}
    &\lambda^{\frac{1}{k+1}}\int_{\tilde I_{x_0}}
   M_{1}(x_0,y_0)^3\abs{u(x,y_0)}^2d x \leq C
   M_{1}(x_0,y_0)^3\abs{u\inner{r(y_0),y_0}}^2\\
    &+C\int_{\tilde J_{x_0}}M_{1}(x_0,y_0)
    \abs{\mathcal L_1 u(x,y_0)} ^2
     d x+C \int_{\tilde J_{x_0}} M_{1}(x_0 ,y_0)^3\abs{u(x,y_0)}^2
     d x,
  \end{split}
  \end{equation}
  where
  \begin{eqnarray*}
  	\tilde I_{x_0}=\big\{x ; ~ \abs{x-x_0}M_{1}(x_0,y_0)<\frac{\delta}{2}\big\},\ \tilde  J_{x_0}=\big\{x ; \  \abs{x-x_0}M_{1}(
  x_0,y_0)<\delta\big\}.
  \end{eqnarray*}
It follows from \eqref{10101401} that
  $C_*^{-1}\leq  M_1(x_0,y_0)/M_{1}(x,y)\leq C_*$
  when   $x\in \tilde J_{x_0}$.
  Moreover in view of \eqref{eqdel},
  \begin{equation*}\label{10101509}
   C_*^{-1} M_{1}(r(y_0), y_0) \leq M_{1}(x_0, y_0)\leq C_* M_{1}(r(y_0), y_0).
  \end{equation*}
  Consequently  we   replay $M_{1}(x_0,y_0)$ by
  $M_{1}(x,y_0)$ in the above integrands and and  by
  $M_{1}(r(y_0),y_0)$ in the first term on the right side, and then  integrate both side with respect to $x_0\in \mathcal
  A(y_0)$; this gives
  \begin{equation}\label{10101510}
  \begin{aligned}
   & \lambda^{\frac{1}{k+1}}\int_{\mathcal A(y_0)}\int_{\tilde I_{x_0}}
   M_{1}(x,y_0)^3\abs{u(x,y_0)}^2d x dx_0  \leq C
   M_{1}(r(y_0),y_0)^2\abs{u\inner{r(y_0),y_0}}^2\\
    &\quad +C \int_{\mathcal A(y_0)}\int_{\tilde J_{x_0}}
 \Big(M_{1}(x,y_0)   \abs{\mathcal L_1 u(x,y_0)}^2
  +
    M_{1}(x
    ,y_0)^3\abs{u(x,y_0)}^2 \Big)
     d x dx_0.
     \end{aligned}
  \end{equation}
  Here we have used the fact that $\mathcal A(y_0)\subset\set{x ; ~\abs{x-r(y_0)} M_{1}(r(y_0),y_0)<C_*\delta}$
  and hence the measure of $\mathcal A(y_0)$ is less than
  $2C_*\delta/M_{1}(r(y_0),y_0)$.

 \noindent {\it Case (b)}.   Next we consider the case that $x_0\not\in\mathcal
  A(y_0)$. Then  $$\abs{r(y_0)-x_0}M_{1}(x_0,
  y_0)\geq \delta.$$  This implies $\xi(x,y_0)$  has no sign change
  from $-$ to $+$ as $x$ increases in $]-\delta,\delta[$. Then the
  estimate \eqref{mla} still holds with the absence of the first term on the
  right side. Then repeating the above arguments, we can
  show that, with $x_0$ varying in $\mathcal A^c(y_0)\stackrel{\rm def}{=}\mathbb R\setminus \mathcal
  A( y_0 )$,
  \begin{equation}\label{10101515}
  \begin{aligned}
   & \lambda^{\frac{1}{k+1}}\int_{\mathcal A^c(y_0)}\int_{\tilde I_{x_0}}
   M_{1}(x,y_0)^3\abs{u(x,y_0)}^2d x dx_0  \\
    &\leq C \int_{\mathcal A^c(y_0)}\int_{\tilde J_{x_0}}
 \Big(M_{1}(x,y_0)   \abs{\mathcal L_1 u(x,y_0)}^2
  +
    M_{1}(x
    ,y_0)^3\abs{u(x,y_0)}^2 \Big)
     d x dx_0.
     \end{aligned}
  \end{equation}

\noindent {\it Completeness of the proof of Theorem \ref{MainT2}}.  In conclusion we  take  sum of \eqref{10101510} and \eqref{10101515}, to get
  \begin{equation*}
  \begin{split}
    &\lambda^{\frac{1}{k+1}}\int_{\mathbb R}\int_{\tilde I_{x_0}}
   M_{1}(x,y_0)^3\abs{u(x,y_0)}^2d x dx_0 \leq C
   M_{1}(r(y_0),y_0)^2\abs{u\inner{r(y_0),y_0}}^2\\
    &+C \int_{\mathbb R}\int_{\tilde J_{x_0}}
    \inner{M_{1}(x,y_0)\abs{\mathcal L_1 u(x,y_0)}^2  +M_{1}(x
    ,y_0)^3\abs{u(x,y_0)}^2}
     d x dx_0,
  \end{split}
  \end{equation*}
  which, with Fubini'theorem as well as the facts  that \begin{eqnarray*}
  \Big\{x ; ~ \abs{x-x_0}M_{1}(x,y_0)<\frac{\delta}{2C_*}\Big\}\subset	\tilde I_{x_0},\ \tilde  J_{x_0}\subset\big\{x ; \  \abs{x-x_0}M_{1}(
  x,y_0)<C_*\delta\big\},
  \end{eqnarray*}
   implies
    \begin{equation*}
  \begin{split}
    &\lambda^{\frac{1}{k+1}}\int_{-\delta}^{\delta}
   M_{1}(x,y_0)^2\abs{u(x,y_0)}^2d x \leq C
   M_{1}(r(y_0),y_0)^2\abs{u\inner{r(y_0),y_0}}^2\\
    &+C \int_{-\delta}^{\delta}
    \inner{\abs{\mathcal L_1 u(x,y_0)}^2  +M_{1}(x
    ,y_0)^2\abs{u(x,y_0)}^2}
     d x.
  \end{split}
  \end{equation*}
  Observe the above inequality holds for all $y_0\in \omega_j \cap \mathcal N_1$ with $\mathcal N_1$ dense in $\omega_j$. Then after integration with respect to
  $y_0$ we obtain
  \begin{eqnarray*}
  \begin{aligned}
    &\lambda^{\frac{1}{k+1}}\norm{
    M_{1}  u}_{L^2\inner{]-\delta,\delta[\,\times \omega_j}}\\
   & \leq C
    \int_{\omega_j}M_{1}(r(y),y)\abs{u\inner{r(y),y}}^2  d y+C
    \norm{\mathcal L_1 u}_{L^2}^2+C\norm{
   M_{1}  u}_{L^2}^2\\
    &\leq C \sum_{1\leq j\leq 2}
    \norm{\mathcal L_j u}_{L^2}^2+C\norm{
   M_{1}  u}_{L^2}^2,
   \end{aligned}
  \end{eqnarray*}
  the last inequality following from \eqref{10101505} since $M_1\leq G$.  Observe $]-\delta,\delta[ \subset \bigcup_{1\leq j\leq N_1} \bar \omega_j$ with $\omega_j $ disjoint, and thus
  \begin{eqnarray*}
  \begin{aligned}
     \lambda^{\frac{1}{k+1}}\norm{
    M_{1}  u}_{L^2} \leq C \sum_{1\leq j\leq 2}
    \norm{\mathcal L_j u}_{L^2}^2+C\norm{
   M_{1}  u}_{L^2}^2.
   \end{aligned}
  \end{eqnarray*}
   Now we choose $\lambda_0$  such that $\lambda_0^{\frac{1}{k+1}}=4C$; this gives
  \begin{eqnarray*}
 \forall\ u\in C_0^\infty(]-\delta,\delta[\,\times\,]-\delta,\delta[),\  \forall\ \lambda\geq \lambda_0, \quad \norm{
    M_{1}  u}_{L^2} \leq C \sum_{1\leq j\leq 2}
    \norm{\mathcal L_j u}_{L^2}^2.
  \end{eqnarray*}
 In view of \eqref{wfd} we have for any $u\in C_0^\infty(]-\delta,\delta[\,\times\,]-\delta,\delta[)$,
 \begin{eqnarray*}
 	\norm{\partial_xu}_{L^2}^2+\norm{\lambda(\partial_x\varphi)u}_{L^2}^2\leq  \norm{\mathcal L_1 u}_{L^2}^2+\norm{M_1 u}_{L^2}^2,
 \end{eqnarray*}
  and thus the desired estimate follows.  The proof of Theorem \ref{MainT2} is  completed.
\end{proof}

{\bf Acknowledgments}
 The research of the first author was supported by NSFC (Nos. 11871054,11961160716, 11771342) and the Fundamental Research Funds for the Central Universities(No.2042020kf0210).

redpajama_set_name: RedPajamaArXiv


Have you ever thought that a guest at your church might, in fact, be a spy?
It's official. I no longer like e-mail.
I make no claim to an expert at time management. What I am is a seminary dean, education consultant, church consultant, and local church pastor who has been forced to learn how to budget time.
I have read that as many as one billion tweets are sent every 2-3 days. That's amazing, especially for an application that began less than a decade ago.
For more than thirty years, I have been a full-time minister. Most of that time, I have been a senior pastor, a seminary professor, or an interim pastor. For a brief time, I served as a ministry staff member (in this case, an associate pastor) in a local church. That time was important to me because it gave me some insight into what staff members experience.
Evangelicals know that theology matters, and we're quick to remind others of this fact. What we're not so quick to acknowledge is that we do a poor job of teaching the very theology we claim is so important.


In the beginning Elohim created the heavens and the earth.
The first sentence of the Scriptures is about spacetime. The Book of Genesis is ancient but its truth has only been confirmed by science recently. The phrase "In the beginning" refers to time. Because nothing comes from nothing the phrase "Elohim created" reveals that everything came into existence by the Creator God. The phrase "The heavens and the earth" refers to space and matter. For matter to exist it needs space to be. The Creation of the universe was also the beginning of time. This explains that God is eternal, or timeless, or independent of time (Psalms 90:1-4, Psalms 102:24-27, 2 Peter 3:8). Because He created it. See also big bang. Genesis 1:1 is a summary of the Creation of the fine-tuned universe we live in. The rest of Genesis 1 and the first verses of Genesis 2 are a general overview of God's Creation step by step.
So the verb which is used in Genesis 1:1 reveals that it talks about one God. Apparently the word 'elohim' is not only used in the Scriptures to point to several gods, but it is also used pointing to one God. Wikipedia and Britannica confirm this.
When used of the true God, "Elohim" denotes what is called by linguists a plural of majesty, honor, or fullness. That is, he is GOD in the fullest sense of the word. He is "GOD of gods" or literally, "ELOHIM of elohim" (Deut 10:17; Ps 136:2).
And the messenger whom I saw standing on the sea and on the land lifted up his right hand to the heaven, and swore by Him who lives forever and ever, who created the heaven and what is in it, the earth and what is in it, and the sea and what is in it, that there shall be no further delay, but in the days of the sounding of the seventh messenger, when he is about to sound, the secret of Elohim shall also be ended, as He declared to His servants the prophets.
So clearly it talks of One God the Creator. This is also confirmed by the Greek interlinear which speaks of 'the [One]' in verse 6 and of 'God' (singular) in verse 7. It is important to look at the context. The Scriptures do not contradict themselves.
And the earth came to be formless and empty, and darkness was on the face of the deep. And the Spirit of Elohim was moving on the face of the waters.
Peter said that "the heavens were of old, and the earth standing out of water and in the water, by the Word of Elohim" (2 Peter 3:5). Water covers 71% of the Earth's surface. It must have been there in some form close to the creation of the earth. Planet earth is the only planet known which contains such an amount of water. The origin of water on the earth is a mystery to science. The first element of the periodic table is Hydrogen. It is the most basic and common of all elements in the universe. Scientists are now slowly coming up with evidence that points to the idea that only light elements, like hydrogen and helium, were created at the start of the universe. How it happened is a matter of speculation based on assumptions.
And Elohim said, "Let light come to be," and light came to be. And Elohim saw the light, that it was good. And Elohim separated the light from the darkness. And Elohim called the light 'day' and the darkness He called 'night.' And there came to be evening and there came to be morning, one day.
Throughout the Scriptures there is a contrast between light and darkness, between good and evil (Isaiah 5:20). Light is used to symbolize God or the Son of God (Psalm 27:1, John 8:12, 1 John 1:5, Colossians 1:15).
And Elohim said, "Let an expanse come to be in the midst of the waters, and let it separate the waters from the waters." And Elohim made the expanse, and separated the waters which were under the expanse from the waters which were above the expanse. And it came to be so. And Elohim called the expanse 'heavens.' And there came to be evening and there came to be morning, the second day.
Scientists have theories about the early conditions on earth. Of course much of it is based on assumptions derived from their preconceived worldview. But it's clear that those early conditions involve water and atmosphere. The Scriptures clearly mention changing conditions.
And Elohim said, "Let the waters under the heavens be gathered together into one place, and let the dry land appear." And it came to be so. And Elohim called the dry land 'earth,' and the collection of the waters He called 'seas.' And Elohim saw that it was good.
Slowly science is catching up with ancient truth. Scientists now say that the early earth was covered with water. But of course most modern-day scientits are brainwashed with the evolutionary worldview, taught in schools and universities, which is completely based on the unprovable philosophy naturalism which assumes without evidence that there is nothing beyond the natural world and are therefore forced to believe that everything came from nothing.
And Elohim said, "Let the earth bring forth grass, the plant that yields seed, and the fruit tree that yields fruit according to its kind, whose seed is in itself, on the earth." And it came to be so. And the earth brought forth grass, the plant that yields seed according to its kind, and the tree that yields fruit, whose seed is in itself according to its kind. And Elohim saw that it was good. And there came to be evening and there came to be morning, the third day.
And Elohim said, "Let lights come to be in the expanse of the heavens to separate the day from the night, and let them be for signs and appointed times, and for days and years, and let them be for lights in the expanse of the heavens to give light on the earth." And it came to be so. And Elohim made two great lights: the greater light to rule the day, and the lesser light to rule the night, and the stars. And Elohim set them in the expanse of the heavens to give light on the earth, and to rule over the day and over the night, and to separate the light from the darkness. And Elohim saw that it was good. And there came to be evening and there came to be morning, the fourth day.
If the plants needed sunlight then this is a strong indication for the idea that the Creation days were literal days. Also note that water was created before the sun.
And Elohim said, "Let the waters teem with shoals of living beings, and let birds fly above the earth on the face of the expanse of the heavens." And Elohim created great sea creatures and every living being that moves, with which the waters teemed, according to their kind, and every winged bird according to its kind. And Elohim saw that it was good. And Elohim blessed them, saying, "Be fruitful and increase, and fill the waters in the seas, and let the birds increase on the earth." And there came to be evening and there came to be morning, the fifth day.
It is generally believed that life originated from sea. According to the Scriptures God created sea life first.
And Elohim said, "Let the earth bring forth the living being according to its kind: livestock and creeping creatures and beasts of the earth, according to its kind." And it came to be so. And Elohim made the beast of the earth according to its kind, livestock according to its kind, and all that creep on the earth according to its kind. And Elohim saw that it was good.
After the sea creatures and the birds God created land animals.
And Elohim said, "Let Us make man in Our image, according to Our likeness, and let them rule over the fish of the sea, and over the birds of the heavens, and over the livestock, and over all the earth and over all the creeping creatures that creep on the ground." And Elohim created the man in His image, in the image of Elohim He created him – male and female He created them.
With whom was God when He created man? The Son of God is the first-born of all creation and all have been created through Him and for Him (Colossians 1:15-16). The Son of God was in the beginning with Elohim and all came to be through Him, and without Him not even one came to be that came to be (John 1:1-3). The Son of God was the first living being created by God. What does it mean to be created in the image of God and His Son? God is spirit (John 4:24). Some defining characteristics of God are love (Romans 5:8), justice (Isaiah 30:18) and righteousnous (Psalms 119:137). These are characteristics that are also found in humans. Unlike animal life, God designed mankind as spiritual beings in physical bodies. Human intelligence is far supreme to that of animals. Human beings have the ability to learn and understand a complex language and understand symbolism. This makes communication possible between God and mankind. He gave His Word in order to communicate to all human beings who care to know Him.
And Elohim blessed them, and Elohim said to them, "Be fruitful and increase, and fill the earth and subdue it, and rule over the fish of the sea, and over the birds of the heavens, and over all creeping creatures on the earth." And Elohim said, "See, I have given you every plant that yields seed which is on the face of all the earth, and every tree whose fruit yields seed, to you it is for food.
The task which God gave to mankind was to take care of the planet and everything in it and to multiply. The first human beings were vegetarians.
And to every beast of the earth, and to every bird of the heavens, and to every creeping creature on the earth, in which there is a living being, every green plant is for food." And it came to be so. And Elohim saw all that He had made, and see, it was very good. And there came to be evening and there came to be morning, the sixth day.
The animals were herbivores and the Creation was very good. There was no corruption.
Thus the heavens and the earth were completed, and all their array. And in the seventh day Elohim completed His work which He had done, and He rested on the seventh day from all His work which He had made. And Elohim blessed the seventh day and set it apart, because on it He rested from all His work which Elohim in creating had made. These are the births of the heavens and the earth when they were created, in the day that YHWH Elohim made earth and heavens.
Genesis 2 starts with a summary of the Creation described in Genesis 1. It adds the seventh day to it which completes all of God's Creation. This is confirmed in Exodus 20:11 and Exodus 31:17.


text:
Harriet Angela Victoria Duckworth1
F, #91091, b. 2 May 1951
Consanguinity Index=0.0%
Harriet Angela Victoria Duckworth was born on 2 May 1951.1 She is the daughter of Henry George Austen de l'Etaing Herbert Duckworth and Hon. Mary Katharine Medina Chatfield.2 She married Andreas Jean-Paul Graf von Einsiedel, son of Wittigo Graf von Einsiedel and Walburga Gräfin von Oberndorff, in 1979 at West Hoathly, Sussex, EnglandG.1
From 1979, her married name became von Einsiedel. She lived in 2003 at 63 Wood Vale, London, EnglandG.1
Children of Harriet Angela Victoria Duckworth and Andreas Jean-Paul Graf von Einsiedel
Orlando Ernie Benedict von Einsiedel2 b. 1980
Evelyn henry Alexander Graf von Einsiedel2 b. 26 Aug 1982, d. 2 Sep 2004
Gwendolen von Einsiedel2 b. 1985
Robin von Einsiedel2 b. 1988
[S37] BP2003 volume 1, page 755. See link for full details for this source. Hereinafter cited as. [S37]
Fabian Graf von Einsiedel
M, #91092, b. 1959
Fabian Graf von Einsiedel was born in 1959. He is the son of Wittigo Graf von Einsiedel and Walburga Gräfin von Oberndorff. He married Daisy Felicitas von Strasser in 1987 at New York, U.S.A.G.
Daisy Felicitas von Strasser
Daisy Felicitas von Strasser married Fabian Graf von Einsiedel, son of Wittigo Graf von Einsiedel and Walburga Gräfin von Oberndorff, in 1987 at New York, U.S.A.G.
Her married name became von Einsiedel.
Georgia Gräfin von Einsiedel
F, #91094, b. 1954
Georgia Gräfin von Einsiedel was born in 1954. She is the daughter of Wittigo Graf von Einsiedel and Walburga Gräfin von Oberndorff. She married Michael von Schmidt-Pauli in 1978.
Her married name became von Schmidt-Pauli.
Michael von Schmidt-Pauli
Michael von Schmidt-Pauli was born in 1954. He married Georgia Gräfin von Einsiedel, daughter of Wittigo Graf von Einsiedel and Walburga Gräfin von Oberndorff, in 1978.
Hedwig Gräfin von Oberndorff
F, #91096, b. 1890, d. 1933
Hedwig Gräfin von Oberndorff was born in 1890. She was the daughter of Franz Albert Alfred Maria Fortunatus Graf von Oberndorff and Marie Therese Gräfin Wolff Metternich zur Gracht. She died in 1933.
Irene Gräfin von Oberndorff
Irene Gräfin von Oberndorff was born in 1895. She was the daughter of Franz Albert Alfred Maria Fortunatus Graf von Oberndorff and Marie Therese Gräfin Wolff Metternich zur Gracht. She died in 1920.
Maximilian Graf von Oberndorff
M, #91098, b. 1867, d. 1900
Maximilian Graf von Oberndorff was born in 1867. He was the son of Friedrich Carl Joseph Lopold Maria Fortunatus Graf von Oberndorff and Irene Maria Elisabetha Gräfin von und zu Arco-Zinneberg. He died in 1900 at TogoG.
Wolfgang Graf von Oberndorff
Wolfgang Graf von Oberndorff was born in 1871. He was the son of Friedrich Carl Joseph Lopold Maria Fortunatus Graf von Oberndorff and Irene Maria Elisabetha Gräfin von und zu Arco-Zinneberg. He married Maria Gräfin von Schönborn-Wiesentheid in 1906 at WissentheidG. He died in 1906.
Maria Gräfin von Schönborn-Wiesentheid
Maria Gräfin von Schönborn-Wiesentheid was born in 1872 at WissentheidG. She married Wolfgang Graf von Oberndorff, son of Friedrich Carl Joseph Lopold Maria Fortunatus Graf von Oberndorff and Irene Maria Elisabetha Gräfin von und zu Arco-Zinneberg, in 1906 at WissentheidG. She died in 1920 at WissentheidG.
Her married name became von Oberndorff.

redpajama_set_name: RedPajamaCommonCrawl


text:
Genius Detective
Chapter 547: Suicide By Jumping Off A Building
Volume 36: Why Did Ah Zhen Die?
Ah[1] Zhen: Are you there?
Ah Zhen: I can't sleep. I'm so miserable. Can you talk with me?
Ah Zhen:?
Mieba: Fuck. Do you have to be like this? You just got jilted. Why haven't you gotten over it yet? You're too narrow-minded, aren't you?
Ah Zhen: I want to die!
Mieba: Oh save that crap. Go wash up and go to sleep!
Mieba: When you're sad, try not to listen to sad songs and watch those weepy Korean dramas. You should read more unconstrained poems. A person needs to look further ahead and just be heartless like me. See how well I live!
Ah Zhen: Thank you!
Mieba: It's okay, don't think too much. Just go to sleep. That man is a scum. There's no need to shed tears for him. Who hasn't experienced such a thing as a breakup? Be more mature!
Ah Zhen: Alright.
Ah Zhen: Sorry for bothering you.
Ah Zhen: Hello, this is Zhang Jianjun, a policeman from the Long'an Public Security Bureau. The owner of this phone died after falling off a building half an hour ago. I need to verify her identity with you.
Mieba: *A message was withdrawn*
Mieba: Stop teasing me. How could she commit suicide.
Ah Zhen: Please tell me your mobile phone number!!!
Mieba: Fuck, really?!
In the middle of the night on September 25th, there were police sirens flashing in the Taohuayuan housing district. The woman who had fallen from the building was surrounded by the police cordon. The police were taking photos and fixing the area. Some residents ran down to watch in their pajamas and were stopped by the police outside the cordon.
Old Zhang had a mobile phone with a broken screen in his hands. On WeChat, the friend who had chatted with her last finally sent over information on her identity. The deceased's name was Ai Lizhen. She graduated from XX University and had been working for three years. She worked as a telemarketer in an investment company.
Old Zhang copied the information down and looked at the female corpse on the ground with emotion, "Haii, so young."
A policeman came down. "No outsiders have been in the house. The door was locked from the inside and the window was open. This was found on the table!" It was a suicide note, and some of the writing had already been blurred. Old Zhang sniffed it to confirm that it was wet from tears.
"It seems to be a suicide. Take the body away quickly! Hey, don't take pictures. There's nothing to see here. Go back to sleep!" The last sentence was directed towards a grinning young man who was taking photos just outside the cordon.
A car drove into the housing district and two youngsters came out, rushing into the cordon regardless of the police's obstruction. Old Zhang said, "What are you doing? Hurry and get out. The police are handling the case!"
A guy in a red plaid shirt who reeked of alcohol cried and pointed at himself. "I was the one on WeChat just now, Mieba. I killed her! I should have figured it out earlier, why didn't I figure it out... Ah Zhen, Ah Zhen, wake up!"
The guy knelt beside the body and cried. The guy in the leather jacket stood next to him, looking sad. He shook his head and sighed for a while, then took out a cigarette and got ready to light it up.
Old Zhang said, "No smoking here!"
"I'm not trying to smoke. I want to mourn Ah Zhen."
"Stop shedding crocodile tears!" The guy in the red plaid shirt stood up and pushed him. "Officer, arrest him. It was this scumbag who played with her emotions and caused Ah Zhen to commit suicide! It's all your fault!"
"What do you mean? The last person that she contacted was you. If it weren't for your words, would she die?" The guy in the leather jacket countered.
The two of them looked like they were about to get into a fight. The police quickly persuaded them to stay away from each other. Old Zhang smelled alcohol on them and asked, "Who drove the car?"
The guy in the leather jacket pointed to the guy in the red plaid. "He did!"
"Drinking and driving? Take him away!"
The guy in the red plaid wailed and shook them off, trying to kick the guy in the leather jacket in order to vent his anger. When he was taken into the police car, he was still shouting, "Ah Qiang, karma will get you! It definitely will!"
Hearing the word "karma", the guy in the leather jacket felt a little lingering fear. Old Zhang came over and asked, "What is your name? What is your relationship with the deceased?"
"Gu Daqiang. I just broke up with her."
"Because of what?"
"Because I had found someone else. You can ask any one of our friends and they'd tell you that, but it's not true! I know it's bad to speak ill of the dead, but Ah Zhen was too insecure and was suspicious of everything. As long as I was one second late replying to a text message, she'd suspect that I was with another woman. Do you know what the reason the last time that we quarreled was? Wasn't a province in the south flooded? Everyone in our company donated as a group. I donated a thousand dollars. After she found out about it, she started overthinking. She said that my ex-girlfriend was from that province and asked if it was because of this reason that I donated so much. I said that she was mad and refused to answer her unreasonable questions. She cried and scolded me. I couldn't stand it anymore, so I suggested that we break up. About a week after the breakup, I found another girlfriend. This incident reached her ears, and she started to overthink it, thinking that I had met someone else long ago, which caused us to break up. She would vent her grievances to people everywhere, just like Xianglin's widow[2]. Our friends all came to criticize me. Don't you think I'm being wrongly accused? Officer, she committed suicide, right? It's got nothing to do with me!"
"We're still investigating whether she committed suicide. What is the name of the guy from just now?"
"Liu Qin. We were all classmates."
"You two were together?"
"We were just drinking, and we were at the Bluebird Bar on Erma Road from 10:00 in the evening until we received the text message."
Old Zhang recorded the information down while listening. He finally said, "Okay, thank you for your cooperation. Leave down your contact information and we will inform you if there's progress."
Gu Daqiang shook his head. "No need, she must have committed suicide. I should have known. I want some peace and quiet for the next two days."
Gu Daqiang took his leave.
After the site investigation was completed, the police loaded the body into the van and returned to the bureau. Old Zhang was on duty that night. In fact, he had been exposed to too many such suicide cases ever since he became an officer. Compared with those earth-shattering murders, it didn't seem like a big event.
Back at the bureau, a police officer brought a detention document over and asked Old Zhang to sign it. When he read "Liu Qin" on it, Old Zhang said, "He's a drunk driver. Why was he detained here?"
"You didn't tell us earlier!"
"Forget it. Leave him here first. Tomorrow, I'll contact the traffic department and let them deal with it."
Just before he got off work, Old Zhang went to take a look at Liu Qin. He sat very dejectedly in a solo detention room. Old Zhang asked, "Are you sober yet?"
Liu Qin nodded.
"Although there aren't many cars at night, it was still too dangerous for you to do so. I know that things have happened suddenly, but there are no exceptions before the law."
"Did she leave a suicide note or something?"
"She did, but I can't show it to you right now. We have to wait until we've completely ruled out homicide."
"I really shouldn't have said those words back then, really... I'm practically a pig, why didn't I notice it?!" Liu Qin covered his face with great pain. "You can punish me. Shoot or jail me, I'd take it. I was the one who killed her!"
Old Zhang shook his head. "Don't blame yourself. She's already gone. Get some rest!"
1. Used in front of a shortened form of peoples' given names to express familiarity
2. A very pitiful woman who complained to everyone about the misfortunes in her life. To be fair, she did meet with a lot of misfortune.
New advance chapter tier!
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The ACRE Blue-Green Summit will host politicians, academics, activists, businesses and others to explore how conservatives truly make the best conservationists.
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Within the automotive recycling industry there is growing concern about future economic developments, according to a recent survey of participants at the International Automobile Recycling Congress IARC 2019. The survey shows that around 38 per cent expect economic conditions for the automotive recycling industry to deteriorate over the next two years. A similarly high percentage, however, think business will remain stable. Only just over one quarter of those surveyed expect economic conditions to improve.
Europe's eyes on Earth is the vision for Copernicus, the European Union's Earth observation programme. Building on this vision, the Horizon 2020 GROW Observatory is training CITIZENS' eyes on Earth, by mobilising thousands of citizens to collect, share and make use of data on the Earth system.


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text:
September 22, 2022 — Alliston, ON.
Honda of Canada Mfg. is lead global plant for all-new CR-V; production to follow in the coming days at the Indiana Auto Plant and the East Liberty Auto Plant in Ohio
All-new CR-V hybrid now available in Canada
CR-V gets bolder styling, increased performance, and greater functionality
CR-V hybrid production will begin in Alliston, Ontario next month, with the Indiana and Ohio plants also building the hybrid model
Associates at Honda of Canada Mfg. (HCM) celebrated the start of mass production of the all-new, 6th generation, 2023 Honda CR-V. HCM is the global lead plant for CR-V production, which has been built in Canada since 2012. In the coming days, production¹ of the all-new CR-V will follow at the Indiana Auto Plant (IAP) and the East Liberty Auto Plant (ELP) in Ohio.
Next month, production of the Honda CR-V hybrid model will start in Canada, followed by IAP and ELP. Honda plans to increase production of the hybrid model to 50 percent of CR-V sales in North America, which is a key step in Honda's overall electrification strategy.
"Becoming the global lead plant for the 2023 Honda CR-V was a big challenge, and I'm proud of our experienced team at Honda of Canada Manufacturing and the level of teamwork we had in North America to create this incredible new product for our customers," said Clement D'Souza, senior vice president of HCM. "Next up is the CR-V hybrid that is more important than ever to the CR-V lineup and a model we are very excited to deliver to our Canadian customers for the first time."
The all-new 2023 Honda CR-V debuts with a rugged yet sophisticated design, including three turbocharged trim levels, LX, Sport, EX-L and the Touring Hybrid trim.
Perfect for daily life or weekend escapes, the new CR-V is longer and wider with premium proportions and a significantly broader stance, further advancing the new Honda design direction with a low horizontal beltline and a long, powerful front end. Honda's most popular SUV backs up its strong new looks with a more fun-to-drive personality, more confident and refined dynamics, and increased torque and capability. For product details, please click here.
Manufacturing the 2023 Honda CR-V
A year ago, HCM Line 1 served as the lead plant in North America for the production launch of the all-new 2022 Civic. Now, Line 2 at HCM is the global lead plant for CR-V. CR-V production will launch at the Indiana Auto Plant and the East Liberty Auto Plant in Ohio in the coming weeks. All three plants worked closely together to implement new technologies and processes for CR-V to deliver an innovative and consistently high-quality product for Honda's North American customers.
HCM employed a "perfect body" fixture for the first time to more quickly achieve optimal fit and finish using a data-driven process. An approach first employed at ELP for the 2017 CR-V new model, this full-size mock-up of the CR-V body is milled from a single piece of steel with the highest dimensional accuracy to help mature the fit of interior parts and components, as well as the tailgate. Time is critical during the trial production stage and, in the past, the team would need to tear apart a trial unit to diagnose fit and finish issues. The perfect fixture enables the team to quickly apply countermeasures on the main assembly line.
The new CR-V has the largest hood in product history, a key element of the bold styling that fulfils Honda's rugged direction for the brand's light truck lineup. To deliver on this hood design in a mass production environment, Honda production engineers worked with the development team to increase rigidity by increasing the thickness of the steel sheet and adjusting the hood frame design so that it could be handled through the stamping and weld departments. The result is a hood that delivers on the bold and sophisticated design direction.
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With the R&D team in Japan, the initial prototype build for CR-V would typically take place there, but with development taking place during the early stages of the COVID-19 pandemic, travel restrictions were in place. Since the North American production teams could not travel to Japan, the prototype build was moved to HCM where the use of digital tools were applied to collaborate between sites. This activity led to design and process changes that now make it easier for Honda production associates to build CR-V to a high level of quality.
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2023 CR-V Touring Hybrid - Canyon River Blue Metallic.


Born in Lincoln. Died in Birmingham.
First listed as President in the Society's exhibition catalogues in Spring 1932. Last listed in Autumn 1934. Listed as Honorary President from Spring 1950.
Elected as an Associate member in Autumn 1884.
Elected as a full member in Spring 1891. Last listed as Member in Spring 1950. From Spring 1927 listed as R.B.S.A.


text:
Roy Lichtenstein painting hidden in private collection for 25 years to be unveiled
Frightened Girl is part of a London exhibition devoted to the Ben-Day dot technique commonly used in pulp fiction comics
23rd February 2018 10:56 GMT
Roy Lichtenstein's Frightened Girl (1964) Estate of Roy Lichtenstein/DACS 2018
Roy Lichtenstein's Ben Day dot painting of a fear-stricken woman is going on show for the first time in 25 years in London next month. Frightened Girl (1964) has been tucked away in a private collection in Europe since its last outing in 1993 when it was shown in the US pop artist's retrospective at the Guggenheim Museum in New York.
The work is the centrepiece of an exhibition at the newly expanded Lévy Gorvy gallery devoted to three artists who used the Ben-Day dot: Lichtenstein in the US, Sigmar Polke in Germany and Gerald Laing in the UK. All three adopted the printing technique in the early 1960s, although it is hard to pinpoint exactly who took it up first.
"I don't think there was a first," the gallery's co-owner, Dominique Lévy, says. "One of the most striking things about the exhibition is just how international the art world was back then. Polke was much closer to Lichtenstein than we ever thought. There were far less frontiers."
The Ben-Day dot was first invented in the late 19th century by the US illustrator and publisher Benjamin Day. The cost-effective printing process, whereby dots are rendered in different densities to produce an image, went mainstream in the 1950s and 1960s when it was used to print newspapers, advertisements and pulp fiction comics.
At that time artists were devouring the media for inspiration, both in terms of technique and subject matter. "Lichtenstein, Polke and Laing all came to the decision to use the Ben-Day dot as a vehicle, not because they were looking at each other, but because they were looking at the mass media around them. Particularly Laing, he was seeing posters on the underground and reconfiguring them in his works," says the gallery's senior director, Lock Kresler, who organised the first major British Pop art show in London at Christie's in 2013.
In the Lévy Gorvy exhibition, Source and Stimulus: Polke, Lichtenstein, Laing (5 March-21 April, named after Laing's 1964 exhibition at the Slade School of Fine Art), two of Laing's works are to be reunited for the first time since they were created in 1965. Shout and Rain Check, two paintings of bikini-clad women, were sold the same year by the New York dealer Richard Feigen to two different collectors in the US. Neither work has been seen publicly since then.
Images of sexually liberated, coiffed and made-up women were popular among all three artists. Lévy sees this as a celebration, rather than an objectification, of women. "It took a long time to heal after the war and suddenly there's a new energy," she says. "With that, there's a return to figuration after years of Abstract Expressionism. Beauty and desire were back." The death of John F Kennedy and the Space Race were among the other newsworthy events that permeated the works of all three artists.
Other highlights of the show, of which around a fifth is for sale, include Polke's Untitled (Couple) (1965-66), which was purchased directly from the artist and has never been seen or published before, and Girlfriends (1964-65), which featured prominently in Polke's retrospectives in 2014 at New York's Museum of Modern Art and London's Tate Modern.
More NewsTopicsExhibitionsCommercial galleriesPaintingRoy LichtensteinLévy GorvySigmar PolkeGerald Laing


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Grand Opening of the Mariott TownePlace Suites - A new long-stay hotel operated by Marriott opened in downtown Windsor in May 2018 following a $10-million dollar renovation of the former downtown Quality Suites hotel located on Dougall Ave., Windsor. The new hotel features 128 suites and is owned and managed by Sunray Group of Hotels, based in Toronto.
Devonshire Mall – The city's largest enclosed mall announced $70 million in renovations, including a new food court and retail space expected to be completed in phases through 2018.
YMCA – Moving to its new 40,000 square foot facility at the Central Park Athletics complex in east Windsor meant that the YMCA needed to hire 48 new employees.
U-Haul – An excellent example of repurposing an older industrial building, U-Haul purchased a 35,117-sq. m. (378,000-sq. ft.) (building on Walker Road which had been part of the General Motors Transmission Plant and was locally known as the GM "H" building. Currently the site is offering truck and trailer rentals, towing equipment, moving supplies and boxes. This was U-Haul's second purchase of a large former industrial site in the city.
Gordie Howe International Bridge – Request for Proposals has been issued for this project with a new estimated cost of $4 billion, plaza site preparation on the Canadian side is well underway.
University of Windsor is continuing the renovation of the former Armouries building to accommodate University of Windsor's School of Creative Arts, scheduled to open in September 2017, providing 5,574 sq. m. (60,000 sq. ft._ of space.
Windsor City Hall – construction is underway on this $44 million project, due to be completed in 2018.


text:
Not Getting It Right Versus Getting It Wrong
March 8, 2016 / JD Byrne
A little while back someone in one of the online writers' forums I haunt asked about what, within our particular area of real life expertise, makes us throw up our hands when we see it in fiction. This was particularly in a sci-fi and fantasy context, so my own area of expertise – criminal law – doesn't really come into play. But reading the discussion and thinking about the question made me realize there is a fine line in fiction between not getting something right and actually getting it wrong.
Exhibit A, since it's right in my wheelhouse – Law & Order. I'm talking the mothership here, the one that effectively split its time between the courtroom and the police investigation. To put it bluntly, Law & Order very rarely actually got things right, but it didn't very often actually get things wrong. The former is forgivable, understandable, and even to be encouraged in the interest of drama. The latter just makes me want to punch the TV.
An episode that was on the other day while I was knocking around the house provides great examples of each. "Charm City" is the first part of the first crossover between Law & Order and my favorite cop show of all time, Homicide: Life On the Street. A gas bomb attack in New York has ties to a similar attack years earlier in Baltimore, bringing Baltimore detectives to town to crack the case.
The eventual trial provides an example of the first situation – not getting it right, but in the understandable service of the narrative. Curtis testifies at trial and, among other things, tells the jury that a partial thumbprint was found at the crime scene and it matched the defendant. In no way would a detective testify about this in a real trial. That evidence would come in via an expert witness, probably from the state or city's crime lab. But we've never seen a crime lab expert in this episode, so it's a waste of narrative resources to introduce an entirely new character to pass on this single (uncontested) fact. Curtis is a main character, by contrast, so give it to him to carry. It's not right, but it's not all that wrong, either.
By contrast, earlier on is the kind of thing that Law & Order routinely does that makes me howl. The two New York detectives question the suspect, who gives up nothing. Their boss lets the Baltimore guys take a run at him. Homicide viewers knew that Pembleton's great skill was extracting confessions in "the box," and, he does, but only after the suspect says he "can't talk about it." Pembleton responds, "you mean you're unable to talk about it, or you just don't want to talk about it?"
The confession winds up getting suppressed because, in the Law & Order universe, "I can't talk about it" is an invocation of the right to remain silent and the right to counsel. There's even a good back and forth about how it doesn't matter that such things will "fly" down in Baltimore. In truth, such things "fly" everywhere that Supreme Court precedent controls. Invocation of counsel, or the right to remain silent, has to be explicit before it keeps the cops from barreling on with their inquiries.
The ruling was, simply, TV-land bullshit, an attempt to throw an obstacle in the path of our heroes. This was getting it wrong, seriously wrong, more than just not getting it right. And, in the end, it didn't really make a difference (the guy was convicted anyway, without much drama). The overly-defense friendly law is a staple on Law & Order, but it's usually at least closer to getting it right than this.
In the end, does it really matter? No, because the vast majority of people watching Law & Order aren't lawyers, much less criminal defense lawyers who might zero in on that kind of thing. For most viewers it's just an obstacle for our heroes to overcome. That, after all, is why we're watching. Which is why it's important for "experts" to back off a little bit and give fiction some room to breathe. Everybody sing along:
film, reading, TV
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