text:
\begin{document}

\title
{{$L^p$-boundedness of wave operators for 2D
Schr\"odinger operators with point interactions}}

\author{Kenji Yajima \footnote{Department of Mathematics, Gakushuin University,
1-5-1 Mejiro Toshima-ku Tokyo 171-8588 (Japan), Supported by JSPS
grant in aid for scientific research No. 19K03589}}

\date{}

\allowdisplaybreaks
\maketitle

\vspace{-0.5cm}
\begin{center}
{\it Dedicated to Professor Arne Jensen on the occasion of
  his 70th birthday}
\end{center}

\vspace{0.2cm}

\begin{abstract}
For two dimensional Schr\"odinger operator $H$ with point interactions,
we prove that wave operators of scattering for the pair $(H,H_0)$,
$H_0$ being the free Schr\"odinger operator, are bounded in the Lebesgue space
$L^p(\R^2)$ for $1<p<\infty$ if and only if there are no generalized
eigenfunctions of $Hu(x)=0$ which satisfy $u(x)= C|x|^{-1}+ o(|x|^{-1})$
as $|x|\to \infty$, $C\not=0$. Otherwise they are bounded for $1<p\leq 2$
and unbounded for $2<p<\infty$.
\end{abstract}

\section{Introduction}

We consider Schr\"odinger operators in $\Hg= L^2(\R^2)$ with point interactions
(SOPI in short) at $Y=\{y_1, \dots, y_N\}\subset \R^2 $ with strength
$\a= (\a_1, \dots, \a_N)\in \R^N$, $1\leq N <\infty$ which are defined
symbolically by
\bqn \lbeq(1-1)
H_{\a,Y}= ``-\lap + \sum_{j=1}^N \a_j \d(x-Y_j)''
\eqn
and which will shortly be defined rigorously. Solutions of $H_{\a,Y}u=0$
which are bounded as $|x|\to 0$ are called (threshold) resonance. We  show
that a resonance satisfies $u(x) = a+ b\cdot{x}/|x|^2+ O(|x|^{-2})$
as $|x|\to \infty$ for a constant $a\in \C $ and a vector $b\in \C^2$;
we call it $s$-wave
resonance if $a\not=0$, $p$-wave resonance if $a=0$ and $b\not=0$; it is a
zero energy eigenfunction if $a$ and $b$ vanish but $u\not=0$. It wil be
shown which kind of resonances $H_{\a,Y}$ can possess is controled
by three $N \times N$ symmetric matrices defined in terms of $\a$ and $Y$.
We then prove that the wave operators of scattering for the pair
$(H_{\a,Y}, H_0)$, $H_0=-\lap$ being the free Schr\"odinger operator,
are bounded in the Lebesgue space $L^p(\R^2)$ for all $1<p<\infty$
if and only if $p$-wave resonances are absent from $H_{\a,Y}$ and,
otherwise they are bounded for $1<p \leq 2$ and unbounded for $2<p<\infty$.

For the roles played by SOPI in physics, in nuclear and solid state physics in
particular, and for the history of its mathematical studies,
we refer to the seminal monograph \cite{AGHH}, the introduction of
\cite{CMY} and references therein and we start with reviewing the rigorous
definition of $H_{\a,Y}$ and some of its basic properties (\cite{AGHH}).
The resolvent  $G_0(z)= (H_0-z^2)^{-1}$ of the free Schr\"odinger operator
with the momentum parameter
$z \in \C^{+}=\{z\in \C \colon \Im z >0\}$,
is the convolution operator with
\begin{equation}\lbeq(Gg)
\mathcal{G}_{z}(x)
\stackrel{\rm def}{=}
\frac1{(2\pi)^2}\int_{\R^2}\frac{e^{ix\xi}d\xi}{\xi^2-z^2}
= \frac14 H_0^{(1)}(z|x|),
\end{equation}
where $H_0^{(1)}(z)$ is the Hankel function of the first kind:
\begin{align}
& \frac14 H_0^{(1)}(z) = \left(-\frac1{2\pi}\log\Big(\frac{z}{2i}\Big)-
\frac{\gamma}{2\pi}\right) \sum_{k=0}^\infty \frac{(-1)^k}{(k!)^2}
\left(\frac{z^2}{4}\right)^k \notag \\
&\ \  -\frac1{2\pi}\left(
\frac{\frac{1}{4}z^2}{(1!)^2}
-\left(1+\frac12\right)\frac{\left(\frac14{z^2}\right)^2}{(2!)^2}
+\left(1+\frac12+\frac13\right)\frac{\left(\frac14{z^2}\right)^3}{(3!)^2}-
\cdots \right)  \lbeq(hankel-1) \\
& \quad = \frac{e^{iz}}{2^\frac32 \pi}\int_0^\infty e^{-t}t^{-\frac12}
\left(\frac{t}2 - iz\right)^{-\frac12}dt, \quad
z\in \Cb^{+}\setminus \{0\}, \lbeq(hankel-2)
\end{align}
where $\gamma$ is Euler's constant (\cite{Watson}).
We denote the prefactor in \refeq(hankel-1) by $g(z)$:
\bqn
g(z)= -\frac1{2\pi}\log\Big(\frac{z}{2}\Big)+ \frac{i}{4}-
\frac{\gamma}{2\pi}, \lbeq(g-def)
\eqn
where $\log(z/2)$ is real for $z>0$.
Notice that with two dimensional Newton potential $N_0(x)$
\bqn
g(z|x-y|)= g(z) + N_0(x-y), \quad N_0(x)= -(2\pi)^{-1}\log |x|
\eqn
is the leading term of the expansion of $\Gg_z(x-y)$ as $z \to 0$.
Define $N \times N$ matrix $\Ga_{\a,Y}(z)$ for
$z\in \Cb^{+}=\{z\in \C \colon \Im z\geq 0\}$ by
\bqn \lbeq(Ga-def)
\Ga_{\a,Y}(z) =
\left\{(\a_j -g(z))\d_{jk} - \Gg_z(y_j-y_k)\hat{\d}_{jk}\right\},
\eqn
where $\d_{jk}$ is the Kronecker delta and $\hat{\d}_{jk}= 1 -\d_{jk}$.
It is shown (cf. \cite{AGHH}) that $\Ga_{\a,Y}(z)$, $z\in \C^{+}$
is non-singular outside a finite subset $\Eg \subset i(0,\infty)$
and the operator valued
function $R(z^2)$ defined for $z \in \C^{+}\setminus \Eg$ by
\bqn  \lbeq(HaY-def)
R(z^2)=(H_0-z^2)^{-1} + \sum_{j,k=1}^N
[\Ga_{\a, Y}(z)^{-1}]_{jk} \Gg_z(\cdot - y_j) \otimes
\overline{\Gg_z(\cdot-y_k)}
\eqn
is the resolvent of the seladjoint operator $H_{\a,Y}$ in $\Hg$:
$R(z^2)=(H_{\a,Y}-z^2)^{-1}$; $H_{\a,Y}$ is the
selfadjoint extension of $-\lap\vert_{C_0^\infty(\R^2\setminus Y)}$
formally defined by \refeq(1-1);
it is a real local operator; domain $D(H_{\a,Y})$ is the set of $u$'s
of the form
\bqn \lbeq(DOMH)
u(x)= v(x)+ \sum_{j,k=1}^N
[\Ga_{\a, Y}(z)^{-1}]_{jk} v(y_k)\Gg_z(x-y_j), \ v \in H^2(\R^2);
\eqn
the function $u$ determines $v$ uniquely in \refeq(DOMH) and
$(H_{\a,Y}-z^2) u= (H_0 - z^2)v$,  $H^2(\R^2)$ being the Sobolev space
of secon order.

Spectrum of $H_{\a,Y}$ consists of the
absolutely continuous (AC for short) part $[0,\infty)$ and at most
$N$ number of non-positive eigenvalues.
The definition \refeq(HaY-def) shows that the rank of
$R(z^2)-(H_0-z^2)^{-1}$ is $N$ and, Kato-Rosenblum theorem (\cite{Kato, RS})
implies that the wave operators
defined by the strong limits in $L^2(\R^2)$:
\bqn \lbeq(limit)
W_{\a,Y}^{\pm}= \lim_{{ t\to\pm \infty}} e^{itH_{\a,Y}}e^{-itH_0}
\eqn
exist and are complete in the sense that
${\rm Range}\ W_\pm =L^2_{ac}(H_{\a,Y})$,
the AC subspace of $L^2(\R^2)$ for $H_{\a,Y}$.
In this paper we study if the wave operators $W_{\a,Y}^{\pm}$
are bounded in  $L^p(\R^2)$, $1\leq p\leq \infty$.

We introduce the three real symmetric matrices which will play important roles
in the rest of the paper:
\begin{gather}
\lbeq(deftD)
\tD= \Big(\d_{jk}\a_j + \frac{\hat{\d}_{jk}}{2\pi}\log|y_j-y_k|\Big),
\ \
\Gg_1(Y)= - \Big(\frac{\hat\d_{jk}}{4N}|y_j-y_k|^2\Big), \\
\Gg_2(Y)= - \left(\frac{\hat\d_{jk}}{8{\pi}N}|y_j-y_k|^2
\log\Big(\frac{e}{|y_j-y_k|}\Big) \right)\,
\end{gather}
and, which appear in the asymptotic expansion as $\lam \to 0$ of $\Ga(\lam)$:
\bqn \lbeq(expansion-1)
\Ga(\lam) = -Ng\Big(P - \frac{g^{-1}\tD}{N} + \lam^2 \Gg_1(Y) +
\lam^2 g(\lam)^{-1}
\Gg_2(Y) + O(\lam^4) \Big).
\eqn
Here $P$ and $S$ are projections in $\C^N$:
\begin{align}\lbeq(horia1)
\eb =
\frac{1}{\sqrt{N}}{{\bf 1}}, \quad
{\bf 1}= \begin{pmatrix} 1\\  \vdots \\ 1
\end{pmatrix}, \quad
P = \eb \otimes \eb, \quad S=1-P.
\end{align}
It is known (\cite{CMY}) that these matrices control also the
the asymptotic behavior as $z \to 0$ of $(H_{\a,Y}-z^2)^{-1}$ and
threshold resonances which $H_{\a,Y}$ can have.
We shall make the latter point clear by defining the resonanaces
as zero energy solutions $H_{\a,Y} \ph =0$ in an weighted $L^2$ spaces.

The following is the main theorem of this paper. It will be stated
by using the matrices defined above and it appears sightly differently
from what is stated at the beginning of the paper, however, it will
shortly become clear that they are actually equivalent.
\bgth \lbth(Main) \ben
\item[{\rm (1)}] Suppose that linear map $S{\tD} S$ in $S\C^N$ is
non-singular. Then  $W_{\a,Y}^{\pm}$ are bounded from $L^p(\R^2)$ to itself
for all $1<p<\infty$.
\item[{\rm (2)}] Suppose $S{\tD} S$ is singular in $S\C^N$ and let
$T$ be orthogonal projection in $S\C^n$ onto ${\rm Ker}_{S\C^N}\, S{\tD}S$.
Suppose $T\tD^2 T$ is non-singular in $T\C^N$.
Then  $W_{\a,Y}^{\pm}$ are bounded from $L^p(\R^2)$ to itself
for all $1<p<\infty$.
\item[{\rm (3)}] Suppose $S{\tD} S$ is singular in $S\C^N$ and that
$T\tD^2 T$ is also singular in $T\C^N$. Let $T_p$
be orthogonal projection in $T\C^n$ onto ${\rm Ker}_{T\C^N}\, T{\tD}^2 T$.
Suppose $T_p \Gg_1(Y)T_p$ is non-singular in $T_p\C^N$.
Then, $W_{\a,Y}^{\pm}$ are bounded from $L^p(\R^2)$ to itself
for $1<p\leq 2$ but are unbounded for $2<p<\infty$.
\item[{\rm (4)}] Suppose $S{\tD} S$ is singular in $S\C^N$,
$T\tD^2 T$ in $T\C^N$ and that $T_p \Gg_1(Y)T_p$ is also singular
in $T_p\C^N$. Let $T_e$ be the orthogonal projection in $T_p\C^N$ onto
$\Ker_{T\C^N} T_p \Gg_1(Y)T_p$. Suppose in addition
$T_e\not=T_p$, viz.
$T_p \Gg_1(Y)T_p\not=0$. Then,
$W_{\a,Y}^{\pm}$ are bounded from $L^p(\R^2)$ to itself
for $1<p\leq 2$ and unbounded for $2<p<\infty$.
\item[{\rm (5)}] Suppose $S{\tD} S$ is singular in $S\C^N$,
$T\tD^2 T$ in $T\C^N$ and $T_p \Gg_1(Y)T_p=0$.
Then, $W_{\a,Y}^{\pm}$ are bounded from
$L^p(\R^2)$ to itself for all $1<p<\infty$.
\een
\edth

\bgrm
\ben
\item[{\rm (1)}] Statements {\rm (3)} and  {\rm (4)} may of course be
unified simply by assuming
$T_p \Gg_1(Y)T_p\not=0$. We state \refth(Main) in this way
only for a later convenince.
\item[{\rm (2)}]
It is known {\rm (\cite{CMY})} that
${\rm rank}\, T\tD^2 T \leq 1$ under the condition of statement {\rm (2)}
and, in statements {\rm (4)} and {\rm (5)}, $T_e \Gg_2(Y)T_e$ is necessarily
non-singular in $T_e \C^N$.
\een
\edrm

For regular Schr\"odinger operators $H= -\lap + V(x)$ on $\R^d$,
$L^p$-boundedness of wave operators has long been studied
and many results are known under various assumptions on $V$.
Results depend on the dimensions $d$ and on the existence/absence of
eigenvalue and/or resonances at $z=0$. We list here some of the results.
In the following it is assumed that $|V(x)|\leq C \ax^{-\s}$ for
$\s>2$ or for a larger $\s$.
\ben
\item[{\rm (1)}] If $d=1$, $W^\pm$ are bounded in $L^p(\R)$ for $1<p<\infty$
but not for $p=1$ and $p=\infty$ (\cite{RW,GY, DF}).
\item[{\rm (2)}] If $H$ has no eigenvalue nor resonances at $z=0$,
$W^\pm$ are bounded in $L^p(\R^d)$ for $1\leq p\leq \infty$ if $d \geq 3$
and for $1<p<\infty$ if $d=2$ (\cite{Y-3dim, Y-2dim, JY-2}).
\item[{\rm (3)}]  If $H$ has an eigenvalue or resonances at $z=0$,
much is known if $d\geq 5$ or $d=3$ and the results depend on $d$ and
the types of sigularities of the resolvent at $z=0$
(\cite{KY,Y-3d-sing, FY, EG, GG, EGG}).
\item[{\rm (4)}] If $d=4$ and $H$ has an eigenvalue but no resonances at $0$,
$W^\pm$ are bounded in $L^p(\R^4)$ for $1\leq p \leq 4$ (\cite{JY-4, GG}).
\item[{\rm (5)}] If $d=2$ and $H$ has an $s$ wave resonances or only eigenvalue
at $0$, $W^\pm$ are bounded in $L^p(\R^2)$ for $1<p <\infty$ (\cite{EGG}).
\een
For SOPI, $W_{\a, Y}^\pm $ are bounded in
$L^p(\R)$ for $1<p <\infty$ for all $\a$ and $Y$ if $d=1$ (\cite{DMW});
if $H_{\a, Y}$ has no eigenvalue nor resonances
at zero, then $W_{\a, Y}^\pm $ are bounded in $L^p(\R^2)$ for $1<p <\infty$
if $d=2$ (\cite{CMY}) and in $L^p(\R^3)$ for $1<p<3$ if $d=3$ (\cite{DMSY}).
Thus, \refth(Main) gives a complete result for SOPI in two dimensions,
however, the problem for the end points $p=1$ and $p=\infty$ are still open.
We mention that for Schr\"odinger operators with regular potentials
in two dimensions, no results have been obtained when $H$ has $p$-wave resonances
which corresponds to the case of statements (3) and (4) of \refth(Main).

The three matrices $\tD, \Gg_1(Y)$ and $\Gg_2(Y)$ control
threshold resonances and the asymptotic behavior of the resolvent
$(H_{\a,Y}-z^2)^{-1}$ as $z \to 0$. We introduce some notation.
For $\s\in \R$, $L_{\s}^2$ and $H^2_{\s}$ are weighted spaces:
\begin{gather*}
L^2_{\s}(\R^2)\stackrel{\rm def}{=}
 \{\ax^{-\s}u(x) \colon u \in L^2(\R^2)\},
\quad \|u\|_{L^2_{\s}}\stackrel{\rm def}{=}  \|\ax^{\s}u\|_{L^2}, \\
H^2_{\s}(\R^2) \stackrel{\rm def}{=}
\{\ax^{-\s} u \colon u \in H^2(\R^2)\}, \quad
\|u\|_{H^2_\s}\stackrel{\rm def}{=}  \|\ax^{\s} u\|_{H^2}.
\end{gather*}
For $y\in \R^2$, $\t_{y} u(x) = u(x-y)$ is the translation by $y$
and we set
\[
\hat{v}_Y (x)=
\begin{pmatrix} \t_{y_1}v(x) \\ \vdots \\ \t_{y_N}v(x) \end{pmatrix}, \ \
\hat{\Gg}_{z,Y}(x)= \begin{pmatrix} \t_{y_1}\Gg_z(x) \\ \vdots
\\ \t_{y_N}\Gg_z(x) \end{pmatrix}, \ \
\hat{N}_{0,Y} (x)= \begin{pmatrix} \t_{y_1}N_0(x) \\ \vdots \\ \t_{y_N}N_0(x).
\end{pmatrix}
\]
In terms of these vectors, domain of $H_{\a,Y}$ is given by
\bqn \lbeq(domainof)
D(H_{\a,Y})=
\{u(x)= v(x)+ \la
\Ga_{\a, Y}(z)^{-1} v_Y, {\Gg}_{z,Y}(x)\ra  \colon v \in H^2(\R^2)\}.
\eqn
Here and hereafter $\la \ab, \bb \ra = a_1 b_1 + \cdots + a_N b_N$
without complex conjugation. We shall often write
\[
a \absleq b \ \ \mbox{for} \ \ |a|\leq |b|.
\]

In view of the proof of the corresponding statement for $H_{\a,Y}$
in \cite{AGHH} the following lemma should be obvious and the proof
will be omitted.
\bglm \lblm(1-1) Let $1<\s<2$ and $z\in \C^{+}\setminus \Eg$.
The operator $R(z^2)$ defined by {\rm \refeq(HaY-def)}
can be extended to a bounded operator
in $L^2_{-\s}(\R^2)$ by continuity, which we denote by $R_{-\s}(z^2)$.
Then, $R_{-\s}(z^2)$ is the resolvent of
a closed operator $H_{\a,Y}^{-\s}$
in $L^2_{-\s}(\R^2)$.  Domain of  $H_{\a,Y}^{-\s}$ is given
independently of $z\in \C^{+}\setminus \Eg$ by
\bqn \lbeq(dom-h)
{\rm Image}\, R_{-\s}(z^2)
=\{u= v+ \la
\Ga_{\a, Y}(z)^{-1} v_Y, \hat{\Gg}_{z,Y}\ra  \colon v \in H_{-\s}^2(\R^2)\},
\eqn
where $v\in H_{-\s}^2(\R^2)$ is uniquely determined by $u$  and
\bqn \lbeq(def-of-hweight)
(H_{\a,Y}^{-\s}-z^2) u = (H_0-z^2)v.
\eqn
\edlm

\bglm\lblm(1-2)  The null space of $H_{\a, Y}^{-\s}$ is given
independently of $1<\s<2$ by
\bqn \lbeq(phlog)
\Ker H_{\a, Y}^{-\s} =\Big\{\ph(x)
= \frac{\la\tD\ab, {{\bf 1}} \ra}{N}  -\frac1{2\pi}
\sum_{j=1}^N a_j \log |x-y_j| \colon \ab\in \Ker S\tD\Big\},
\eqn
where $a_1, \dots, a_N$ are components of $\ab \in \C^N$.
\edlm

We denote by $C_b(\R^2 \setminus Y)$ the set of continous functions in
$\R^2 \setminus Y$ which are bounded outside a bounded open set containing
$Y$ and define
$\Rg_{\a,Y}=\Ker H^{-\s}_{\a,Y} \cap C_b(\R^2 \setminus Y)$, which
is independent of $1<\s<2$ by virtue of \reflm(1-2).
We define $\hat{x}= x/|x|$ for $x\not=0$.

\bgth \lbth(def-Rext)
The space $\Rg_{\a,Y}$ is equal to
\bqn \lbeq(res)
\Big\{\ph(x)
= \frac{\la\tD\ab, {\bf 1} \ra}{N}  -\frac1{2\pi}
\sum_{j=1}^N a_j \log |x-y_j| \colon \ab\in \Ker S\tD{S}\cap S\C^N\Big\}.
\eqn
The function $\ph(x)$ of \refeq(res) satisfies
\bqn \lbeq(reso-s)
\ph(x)= \frac{\la \tD\ab, {\bf 1}\ra}{N}  +
\frac1{2\pi}\sum_{j=1}^N \frac{\la \hat{x}, a_j y_j\ra}{|x|} + O(|x|^{-2})
\quad (|x|\to \infty).
\eqn
In particular, $\Rg_{\a, Y} = \{0\}$ if only if $S\tD{S}$ is non-singular
in $S\C^N$.
\edth

\bgdf A function $\ph \in \Rg_{\a,Y}$  is called (threshold) resonance
of $H_{\a,Y}$.
\ben
\item[{\rm (1)}] $H_{\a,Y}$ is said be regular at zero if
$\Rg_{\a,Y}=\{0\}$ otherwise singular at zero .
\item[{\rm (2)}] $\ph\in \Rg_{\a,Y}$ of \refeq(res) is an
$s$-wave resonance if $\la \tD\ab, {\bf 1}\ra\not=0$
and $p$-wave resonance if $\la \tD\ab, {\bf 1}\ra=0$ but
$\sum_{j=1}^N a_j y_j\not=0$.
\item[{\rm (3)}] $\ph\in \Rg_{\a,Y}\setminus\{0\}$
of in \refeq(reso-s) is an
eigenfunction of $H_{\a,Y}$ with eigenvalue $0$
if $\la \tD\ab, {\bf 1}\ra=0$ and
$\sum_{j=1}^N a_j y_j=0$.
\een
\eddf

In the following theorem we use the notation of \refth(Main).
\bgth \lbth(resonances)
Suppose $S\tD{S}$ is singular in $S\C^N$. Then:
\ben
\item[{\rm (1)}] $s$-wave resonances exist if and only if $P\tD T \not=0$.
\item[{\rm (2)}] All $\ph \in \Rg_{\a,Y}$ are $s$-wave resonances
if and only if $T{\tD}^2 T$ is non-singular in $T\C^N$.
\item[{\rm (3)}] Suppose $T{\tD}^2 T$ is singular in ${T\C^N}$.
Then, $\ph\in \Rg_{\a,Y}$ of \refeq(res) is
\ben
\item[{\rm (a)}]
an $s$-wave resonance if $\ab\in T\C^N \setminus T_p\C^N$.
\item[{\rm (b)}] a $p$-wave resonance if
$\ab \in T_p \C^N \setminus T_e \C^N$.
\item[{\rm (c)}] an eigenfunction with eigenvalue $0$ if
$\ab \in T_e \C^N$.
\een
\item[{\rm (4)}] The eigenspace of $H_{\a, Y}$ associated with eigenvalue zero
is the set of all $\ph(x)$ in \refeq(res) with $\ab\in T_e \C^N$.
\een
\edth

In virtue of \refth(Main), \refth(def-Rext) and \refth(resonances),
wave operators are bounded in $L^p(\R^2)$ for all
$1<p<\infty$ if $H_{\a,Y}$ has no $p$-wave resonances otherwise
they are bounded only for $1<p \leq 2$.

We briefly record here the result of \cite{CMY} on the threshold behavior
of $(H_{\a, Y}-z^2)^{-1}$ to show its relation to \refth(Main).
We refer to \cite{CMY} for more precise result. For Schr\"odinger
operators with regular potentials, the relation between resonances,
the threshold behavior of the resolvent and the large time
behavior of solutions of time dependent Schr\"odinger equation
is extensively studied (see e.g. \cite{JK, Mu, Schlag, JN, EG, EGG}).

Let $\s>1$ and let $\Bb_\s$ be the
Banach space of bounded operators from $L^2_{\s}(\R^2)$ to $L^2_{-\s}(\R^2)$.
Then, the well known limiting absorption principle for $(H_0 -z^2)$ and
the behavior of the Hankel function imply that
$(H_{Y,\a}-z^2)^{-1}$ regarded as a
$\Bb_\s$-valued function of $z \in \C^{+}\setminus\Eg $ can be
continuously extended to $\Cb^{+}\setminus (\Eg \cup \{0\})$.
Here and in what follows we use $\lam$ instead of $z$ when we emphasize
$z$ can also be real not only $z\in \C^{+}$.

\begin{theorem}[\cite{CMY}] \lbth(zerolimit)
{\rm (1)} Suppose that $H_{\a,Y}$ is regular at zero, then
$(H_{Y,\a}-\lam^2)^{-1}$
can be extended continuously to $0$. \\[5pt]
{\rm (2)} Suppose the condition of \refth(Main) {\rm (2)} is satisfied.
Then, $T= \fb \otimes \fb$ for a normalized $\fb\in T\R^N$ and
\[
(H_{Y,\a}-\lam^2)^{-1}  =
a^{-2} g(\lam)\ph \otimes \ph + O(1)  \quad (\lam \to 0),
\]
where $\ph(x)$ is an $s$-wave resonance defined in \refeq(res)
with $\fb$ in place of $\ab$.  \\[5pt]
{\rm (3)}
Suppose the condition of \refth(Main) {\rm (3)} is satisfied.
Then
\[
(H_{Y,\a}-\lam^2)^{-1}=  -(Ng\lam^{2})^{-1}
\sum_{j=1}^n  a_j \ph_j(x) \ph_j(y)  + O(\lam^{-2} ) \quad (\lam \to 0),
\]
where $n= {\rm rank}\, T_p$ and $\ph_j$, $j=1, \dots,, n$
are $p$-wave resonances. \\[5pt]
{\rm (4)} Suppose the condition of \refth(Main) {\rm (4)} is satisfied.
Then,
\begin{align*}
& (H_{Y,\a}-\lam^2)^{-1}(x,y)\notag \\
& = -  (N \lam^{2})^{-1} \la T_e \hat{N}_{0,Y}(x),
[T_e\Gg_2(Y) T_e]^{-1} T_e \hat{N}_{0,Y}(y)\ra +
O(\lam^{-2}g(\lam)^{-1} ).
\end{align*}
\end{theorem}

The rest of the paper is devoted to the proof of the lemmas and
theorems (but not of \refth(zerolimit)). In section 2, we prove results
on the resonances, \reflm(1-1), \reflm(1-2) and \refth(def-Rext).
In section 3, we collect results necessary for proving \refth(Main).
We first recall from \cite{CMY} the stationary
and the product decomposition formulas for the wave operators
and the result that the high energy part of the
$W_{\a,Y}^\pm \chi_{\geq \ep}(|D|)$ is bounded in $L^p(\R^2)$ for all
$1<p<\infty$. We then examine the result in \cite{CMY}
on the behavior of $\Ga(\lam)^{-1}$ as $\lam\to 0 $ and
give an estimate on the Fourier transform of a logarithmic function.
We prove in section 4 the statement of \refth(Main)
separately. In virtue of the the high energy results mentioned above
we prove them for the low energy part $W_{\a,Y}^\pm\chi_{\leq \ep}(|D|)$
only.
Statement (1) is a direct result of the product formula and Mikhlin's theorem
on Fourier multliplier. Proofs of statements (2) to (5) uses the cancellation
properties produced by the linear operators $S, T_p$ and $T_e$ of \refth(Main).

\section{Proof of results on resonances }
In this section we prove \reflm(1-2), \refth(def-Rext) and \refth(resonances).
The Fourier transform $\hat u(\xi) = {\Fg}u(\xi)$ is
defined by
\[
\Fg u(\xi)= \frac1{2\pi}\int_{\R^2} e^{-ix\xi}u(x) dx.
\]

\paragraph{Proof of \reflmb(1-2).}
Define for $\ab \in \C^N$ and $z\in \C^{+}$,
\bqn
h_z(x) \stackrel{\mathrm {def}}
{=} \la \ab, \hat{N}_{0,Y}(x)-\hat{\Gg}_{z,Y}(x) \ra .
\lbeq(h1)
\eqn
We have $h_z(x) \in H^2_{-\s}(\R^2)$ for $1<\s<2$
because \refeq(hankel-1) implies the $\log$-singularities at $x=y_j$ of
cancel and it grows only logarithmically as $|x|\to \infty$.
Moreover, ${N}_0(x-y_j)- {\Gg}_{z}(x-y_j)= g(z)+ O(g(z)^2|x-y_j|^2)$
as $x\to y_j$ and, the definitions of $\Ga_{\a,Y}(z)$ and $\tD$ imply
\begin{align}
h_z(y_j)& = \sum_{k=1 }a_k
\left(\big(-\frac1{2\pi}\log|y_k-y_j|- {\Gg}_z(y_j-y_k)\big)\hat{\d}_{jk}
- g(x)\d_{jk} \right) \lbeq(h2)  \\
& = [(\Ga_{\a,Y}(z)-\tD)\ab ]_j, \quad j=1, \dots, N  \lbeq(h3).
\end{align}
(a) We first show that $\ph(x)$ defined by \refeq(phlog) with
$\ab \in \Ker\, S\tD$ is a solution of
$H_{\a,Y}^{-\s} \ph =0$. Let $C = N^{-1}\la \tD\ab, {\bf 1}\ra$.
By virtue of \refeq(h1)
\bqn \lbeq(ph-ker)
\ph(x) = v_z(x) + \la \ab, \hat{\Gg}_{z,Y}(x) \ra \ \mbox{with}\
v_z(x)= h_z(x) + C \in H^2_{-\s}(\R^2)
\eqn
and, we have $\Ga_{\a,Y}(z)\ab=\hat{v}_{z,Y}$ because \refeq(h3) and
$S\tD\ab=0$ or $\tD\ab\in P\C^N$ imply
$\hat{v}_{z,Y}- \Ga_{\a,Y}(z)\ab= -\tD\ab + C{\bf 1}
=(-{N^{-1}}\la \tD\ab, {\bf 1}\ra -C){\bf 1} =0$.
It follows $\ph \in D(H_{\a,Y}^{-\s})$. Moreover, \refeq(def-of-hweight)
and \refeq(ph-ker) imply $H^{-\s}_{\a,Y}\ph(x)=0$ because
\[
(H^{-\s}_{\a,Y}-z^2)\ph(x) = (-\lap -z^2)v_z(x)
= -z^2 (C +\la \ab, \hat{N}_{0,Y}(x)\ra) = -z^2 \ph(x).
\]
(b) Assume conversely that $\ph(x)$ satisfies $H_{\a,Y}^{-\s}\ph=0$.
We show that $\ph(x)$ is necessarily of the form \refeq(phlog) with
$\ab$ such that $S\tD\ab=0$. Since $\ph \in D(H^{-\s}_{\a,Y})$,
for $z\in C^{+}\setminus \Eg$ there must exists $v_z \in H^2_{-\s}(\R^2)$
such that
\bqn \lbeq(phx)
\ph(x) = v_z (x) + \la \ab_z, \hat{\Gg}_{z,Y}(x)\ra \ \mbox{where} \
\ab_z= \Ga_{\a, Y}(z)^{-1}\hat{v}_{z,Y}
\eqn
and $(H_{\a,Y}^{-\s}-z^2) \ph = (-\lap -z^2)v_z$ or
$-z^2\ph(x) = (-\lap -z^2)v_z(x)$. hence,
\bqn \lbeq(2-6)
-\lap v_z(x) = z^2(v_z(x) -\ph(x))= -z^2 \la \ab, \hat{\Gg}_{z,Y}(x)\ra.
\eqn
We observe that
$(\ab_z)_j = -\lim_{x\to y_j} (2\pi)(\log |x-y_j|)^{-1}\ph(x)$
because $v_z\in H^{2}_{-\s}(\R^2)$ is continuous and $\ab_z$ is
independent of $z$. Thus, we write $\ab$
for $\ab_z$ and define $h_z(x)$ by \refeq(h1) with this $\ab$. We will
show $v_z(x) =  h_z(x) + C$ for a constant $C$ and,
hence $\ph(x)= \la \ab, \hat{N}_{0,Y}(z) \ra + C$, or it must be of the
form in \refeq(phlog). Indeed, it follows from \refeq(2-6) that
\[
-\lap \left(v_z+ \la \ab, \hat{\Gg}_{z,Y}\ra\right)
= \la \ab, (-\lap -z^2)\hat{\Gg}_{z,Y}\ra
= \la \ab, \d_{Y}\ra
= \la \ab, -\lap \hat{N}_{0,Y}\ra
\]
and $0= -\lap \left(v_z(x)+
\la \ab,(\hat{\Gg}_{z,Y}(x)-\hat{N}_{0,Y}(x)\ra \right)
=-\lap(v_z(x) -h_z (x))$.
Thus, $v_z(x) - h_z(x)$ must be a harmonic polynomial which belogns to
$H^2_{-\s}(\R^2)$ and, hence $v_z(x)-h_z(x) = C$. For determining $C$,
we recall \refeq(h3), which implies
\[
C{\bf 1}=
\hat{v}_{z,Y}- \hat{h}_{z,Y} = \Ga_{\a, Y}(z)\ab - (\Ga_{\a, Y}(z)-\tD)\ab
= \tD\ab.
\]
It follows that $\ab$ must be such that
$\tD\ab\in P\C^N$ and $C=N^{-1}\la \tD\ab, {\bf 1}\ra$. \qed

\paragraph{Proof of \refthb(def-Rext)}
The function $\ph(x)= C + \la \ab, \hat{N}_{0,Y}(x)\ra$, $S\tD\ab={\bf 0}$
of \refeq(phlog) is evidently continuous in $\R^2 \setminus Y$.
It is also evident that $\ph(x)$ is bounded near infinity if
and only if $\la {\bf 1}, \ab\ra=0$ or $\ab\in S\C^N$. This implies
that $\Rg_{\a,Y}$ is given by \refeq(res).
The relation \refeq(reso-s) is evident if
$\la {\bf 1}, \ab\ra=a_1+ \cdots+ a_N=0$. \qed

\paragraph{Proof of \refthb(resonances)}
(1) If $H_{\a,Y}$ has an $s$-wave resonance, then
there must exists non-zero $\ab \in T\C^N= S\C^N \cap \Ker\, S\tD$
such that $P\tD\ab\not=0$, hence $P\tD{T}\not=0$. Conversely, if
$P\tD{T}\not=0$, then there exists $\ab \in T\C^N$ such that
$P\tD\ab\not=0$ and $\ph(x)$
defined by \refeq(phlog) with this $\ab$ is an $s$-wave resonance.

(2) If $T{\tD}^2 T$ is non-singular,
$\Ker T{\tD}^2 T = \Ker \tD T =\{0\}$ since $\tD$ is real symmetric
and, for
$\ab\in T\C^N \setminus\{0\}$ we have $P\tD \ab = \tD\ab \not=0$.
Thus all resonances are $s$-wave resonances.
If $T{\tD}^2 T$ is singular on the other hand then, $\tD \ab =0$
for an $\ab \in T\C^N$ which trivially satisfies $P\Dg T\ab =0$
and, \refeq(phlog) with this $\ab$ produces a $p$-wave resonance.

(3a) If $\ab \in T\C^N$ is such that $T{\tD}^2 T\ab\not=0$,
then $\tD \ab \not=0$ and $P\tD \ab \not=0$ as $S\tD \ab=0$. Thus,
$\ph(x)$ produced by $\ab$ by \refeq(phlog) is an $s$-wave resonance.

(3b,c) For $\ab \in T_p \C^N\setminus \{0\}$, we have ${\tD}\ab={\tD}T\ab=0$
and $P\Dg\ab=0$ trivially. If $T_p \hat{\Gg}_1(Y) T_p \ab\not=0$,
$\la \hat{\Gg}_1(Y) \ab, \ab \ra = \la Y, \ab \ra^2 \not=0$
and $\ph(x)$ produced by $\ab$ by \refeq(phlog) is a $p$-wave resonance;
but if $T_p \hat{\Gg}_1(Y) T_p \ab=0$, $\ph(x) = O(|x|^{-2})$ as $|x|\to \infty$
and it is an eigenfunction.

(4)If $\ab \in T_e \C^N\subset T_p \C^N$, then we have seen in (3b) that
$\la \tD\ab, {\bf 1}\ra=0$
and $\la Y, \ab \ra={\bf 0}$. Hence $\ph(x)$ is an eigenfunction.
Conversely, eigenfunction is also of the form \refeq(phlog) with $\ab \in T\C^N$
by virtue of \refth(def-Rext) and $\ab$ must further satisfy
$\la \tD\ab, {\bf 1}\ra=0$ and $\la Y, \ab \ra={\bf 0}$.
The former implies $\tD \ab=0$, hence $T\tD^2 T\ab=0$ and
$\ab \in T_p\C^N$ and the latter $T_p \hat{\Gg}_1(Y)T_p \ab =0$,
which implies $\la \ab, T_p \hat{\Gg}_1(Y)T_p \ab\ra =0$
and $T_p \hat{\Gg}_1(Y)T_p\ab=0$ since $T_p\hat{\Gg}_1(Y)T_p$
is non-negative on $T_p\C^N$. Thus, we have $\ab \in T_e \C^N$.
\qed

\section{Preliminary for the proof of \refthb(Main)}
In this section, we collect several lemmas which we use for proving
\refth(Main), some of which are well known and are recorded
for readers convenince. The strength $\a$ and points of interaction $Y$
will be fixed hereafter and will be often omitted from various formulas,
e.g. $\hat\Gg_\lam(x)= \hat{\Gg}_{\lam,Y}$.
We prove \refth(Main) for $W^{-}$. We have $W^{+}= \Cg W^{-}\Cg^{-1}$
by the complex conjugation $\Cg u(x) = \overline{u(x)}$ and the result
for $W^{+}$ immediately follows from that for $W^{-}$.
We set
\bqn \lbeq(Dgast-def)
\Dg_\ast =
\{u \in \Sg(\R^2) \colon \hat{u}\in C_0^\infty(\R^2 \setminus\{0\}) \}.
\eqn
The space $\Dg_\ast$ is dense in $L^p(\R^2)$ for any $1<p<\infty$.
For Borel functions
$f$, $f(D)$ is the Fourier multiplier by $f(\xi)$:
$f(D)u(x)= \Fg^{-1}(f(\xi)\hat{u}(\xi))(x)$.
\[
\tGa(\lam) {=} {\Ga(\lam)}^{-1}
\]
and $\tGa(|D|)=(\tGa_{jk}(|D|))$ is the operator matrix of
Fourier multipliers
$\tGa_{jk}(|D|)$. For $y \in \R^2$, $\t_y u(x)=u(x-y)$ is translation by $y$.
We define the Riesz transform (\cite{Stein})
$R=\begin{pmatrix} R_1 \\ R_2 \end{pmatrix}$ by
$R_l= R_l(D)= \Fg^{-1}(\xi_l/|\xi|)\Fg$, $l=1,2$ and
for a vector $a\in \R^2$, $\la a, R \ra u= a_1 R_1 u + a_2 R_2 u$.
$\la a, R \ra$ is a bounded operator in $L^p(\R^2)$ for $1<p<\infty$.

\subsection{Product decomposition of wave operators}

\paragraph{Stationary representation. }
The following representation of the wave operator $W^{+}$
via the resolvent kernel $\Gg_{\lam}(x)$ may be proved by following
the argument of the proof of Lemma 3.2 of \cite{DMSY}
for the three dimensional case.

\bglm \lblm(stationary)  Wave operator $W^{-}$ may be repesented in the form
\bqn
W^{-} u= u+
\frac1{\pi{i}}\int_0^\infty\int_{\R^2} \lam
\la \tGa(\lam)\hat{\Gg}_{\lam}(x),
\hat{\Gg}_{\lam}(y)-\hat{\Gg}_{-\lam}(y)\ra{u(y)}dy d\lam   \lbeq(Stat-2a)
\eqn
for $u\in \Dg_{\ast}$. Equivalently
$W^{-} u= u + \sum_{j,k=1}^N \tau_{y_j} \W_{jk}\tau_{y_k}^\ast $ where
\bqn \lbeq(second)
\W_{jk}u(x) = \frac1{\pi{i}}\int_0^\infty \lam \tGa_{jk}(\lam)\Gg_{\lam}(x)
\left(\int_{\R^2}(\Gg_{\lam}(y)-\Gg_{-\lam}(y))u(y)dy \right) d\lam .
\eqn
\edlm

\paragraph{Decomposition formula}
The following is a slight modification of the
result of Lemma 4.3 and 4.4 of \cite{CMY}.
Define the operator $K$ by
\bqn \lbeq(DEFK)
Ku(x) = \frac1{\pi^2}\int_0^\infty \Gg_{\lam}(x)\lam \left( \int_{{\mathbb S}^1}
\Fg{u}(\lam\w)d\w \right)d\lam.
\eqn

\bglm \lblm(prod) {\rm (1)} For $j,k=1, \dots, N$,
$\W_{jk}$ is the product of $\tGa_{jk}(|D|)$ and $K$:
\bqn \lbeq(prod)
(\W_{jk}u)(x)= (K \circ \tGa_{jk}(|D|))u(x), \quad u \in \Dg_{\ast}.
\eqn
{\rm (2)} $K$ is a singular integral operator:
\bqn \lbeq(Kxy)
Ku(x) = \lim_{\ep \downarrow 0}\frac{2}{\pi^2 i}
\int_{\R^2} \frac{u(y)dy}{x^2-y^2 + i\ep}.
\eqn
{\rm (3)}  For any $1<p<\infty$, there exists a constant $C_p>0$ such that
\bqn \lbeq(Kest)
\|Ku\|_{L^p(\R^2)}\leq C_p \|u\|_{L^p(\R^2)}, \quad u \in \Dg_{\ast}.
\eqn
\edlm

\paragraph{Mikhlin multiplier}
We recall the well-known result on the Fourier multiplier
which will be very often used in what follows.

\begin{lemma}[Mikhlin] Let $m \in C^2(\R^2 \setminus\{0\})$
satisfy $|\pa_{\xi}^\a m(\xi)|\leq C |\xi|^{-|\a|}$ for $|\a|\leq 2$.
Then, $m(D) \in \Bb(L^p(\R^2))$ for $1<p<\infty$. \lblm(Mikhlin)
\end{lemma}
We often say that $m(\lam)$ is a {\it good multiplier}  when it
satisfies the condition of \reflm(Mikhlin).

\subsection{High energy part $W^{-}\chi_{\geq \ep}(H_0)$.}

The $L^p$ property of the high energy part of $W^{-}$ does not
depend on the small $z$ behavior of $(H_{\a,Y}-z^2)^{-1}$
and the following is proved in \cite{CMY}. In what follows
$\chi$ will stand for the real function $\chi\in C_0^\infty(\R)$
which satisfies
\bqn \lbeq(chi-def)
\mbox{$\chi(\lam)=1$ for $|\lam|<1$ and
$\chi(\lam)=0$ for $|\lam|>2$}
\eqn
and, for $\ep>0$, we define
\bqn \lbeq(chiep-def)
\chi_{\leq \ep}(\lam)= \chi(\lam/\ep), \quad
\chi_{\geq \ep}(\lam)=1-\chi_\ep(\lam).
\eqn
When $\ep>0$ is fixed, then we often write
$\chi_{\leq}(\lam)= {\chi}_{\leq \ep}(\lam)$ and
$\chi{\geq}(\lam)={\chi}_{\geq \ep}(\lam)$ omitting $\ep>0$.

\begin{theorem}  \lbth(high) For any $\ep>0$,
$W_{\a,Y}^{-} (1-\chi_{\leq \ep}(|D|))$ is bounded
from $L^p(\R^2)$ to itself for any $1<p<\infty$.
\end{theorem}

We say for simplicity that an operator is a {\it good operator}
if it is bounded from $L^p(\R^2)$ to itself for any $1<p<\infty$.
By virtue of \refth(high), we need consider only the low energy part
$W_{\rm low}= W^{-}\chi_{\leq \ep}(|D|)$ in what follows.

\subsection{Expansion of $\Ga(\lam)^{-1}$}

For proving \refth(Main) we need precise information on
the behavior of $\tGa(\lam)= \Ga(\lam)^{-1}$ as $\lam \downarrow 0$.
We have already obtained some results in \cite{CMY} and \cite{CMYE},
however, because we shall need some more precise results and those
which are buried in proofs, we have decided to completely redo it.
The low energy behavior of $\Ga(\lam)^{-1}$ is different depending on the
conditions of statements of \refth(Main)
and we split the subsection into five paragraphs accordingly.
We remark that the conditions in each steps are mutually exclusive.

In what follows,  for two functions $f(\lam)$ and $h(\lam)$ on $(0,\infty)$,
$f(\lam)= O(h(\lam))$ means that for $j=0,1, \dots$ that
$f^{(j)}(\lam)\absleq C_j h^{(j)}(\lam)$ is satisfied for
$0<|\lam|<\ep_j$ for some $\ep_j>0$ and $C_j>0$.
Hereafter we shall indiscrimately denote by
$M(\lam)$ a good multiplier which may differ at each appearance

We shall repeatedly use the following lemma due to Jensen and Nenciu \cite{JN}.
The following trivial identities for matrices will be frequently used:
\begin{gather}
(1+X)^{-1}= 1-X(1+X)^{-1}= 1- (1+X)^{-1}X, \lbeq(X-1)\\
(1+X)^{-1}= 1- X + X(1+X)^{-1}X. \lbeq(X-2)
\end{gather}

\begin{lemma} \lblm(JN)
Let $A$ be a closed operator in a Hilbert space $\Hg$
and $S$ a projection. Suppose $A+S$ has a bounded inverse.
Then, $A$ has a bounded inverse if and only if
\[
B= S - S(A+S)^{-1}S
\]
has a bounded inverse in $S\Hg$ and, in this case,
\bqn \lbeq(JN-1)
A^{-1}= (A+S)^{-1}+ (A+S)^{-1}SB^{-1}S (A+S)^{-1}.
\eqn
\end{lemma}

We also use the well known Feshbach formula but only at the last step.
In what follows we use the notation of \refth(Main)
and omit the variable $\lam$ from various functions
when no confusion is feared. Identity matrices of various subspaces
are indiscrimately denoted by $1$ and orthogonal
projections $P$ in subspaces $V$ will be often regarded as the projection
$P \oplus 0$ in the full space $\C^N= V \oplus V^\perp$.

\paragraph{Step 1.}  Define $A(\lam)$ by
\[
\Ga(\lam) = -NgA(\lam)
\]
and
\bqn
F {=} - N^{-1}\tD, \quad
\Rg  = \Gg_1(Y) + g(\lam)^{-1}\Gg_2(Y) . \lbeq(A-a)
\eqn
Then, \refeq(expansion-1) implies
\bqn
A(\lam)+ S  = 1 + g^{-1} F + \Mg_0(\lam), \quad
\Mg_0(\lam){=} \lam^2 \Rg + O(\lam^4). \lbeq(A) \\
\eqn
Thus, $A+S$ is non-singular for small $\lam>0$ and by virtue of \refeq(X-2)
\begin{align} \lbeq(A+Sinv)
& (A+ S)^{-1}= (1+ g^{-1}F)^{-1}(1+ \Mg_0(1+ g^{-1}F)^{-1})^{-1} \\
& = (1+ g^{-1}F)^{-1} - (1+ g^{-1}F)^{-1}\Mg_0(1+O(\lam^2))(1+ g^{-1}F)^{-1}.
\lbeq(A+Sinv-a)
\end{align}
In particular $(A+S)^{-1}$ is a good multiplier and
\bqn \lbeq(A+S-Sinter)
(A+ S)^{-1}S  = S + O(g^{-1}), \quad
S (A+ S)^{-1}  = S + O(g^{-1}).
\eqn
In view of \reflm(JN), we define $B(\lam)$ and $A_1(\lam)$ by
\bqn
B {=} S- S(A+S)^{-1}S = - N^{-1}g(\lam)^{-1} A_1. \lbeq(B-0)
\eqn
We substitute \refeq(A+Sinv-a) for $(A+S)^{-1}$ in \refeq(B-0).
Since
\bqn \lbeq(L-def)
L \stackrel{\rm def}{=}-Ng(S-S(1+ g^{-1}F)^{-1}S)
= S{\tD}S + g^{-1}N^{-1} S\tD^2(1+ g^{-1}F)^{-1}S
\eqn
we have
\begin{align}
A_1 & =L - g SN(1+ g^{-1}F)^{-1}\Mg_0(1+O(\lam^2))(1+ g^{-1}F)^{-1}S
\lbeq(A1-exp) \\
& {=} S\tD{S}+ S\tD^2(gN)^{-1}(1+g^{-1}F)^{-1}S - g\lam^2 \Rg_1
\lbeq(G-3)
\end{align}
where we defined $\Rg_1=O(1)$ by
\bqn
\Rg_1 \stackrel{\rm def}{=}  N S (1+g^{-1}F)^{-1}
(\Rg+ O(\lam^2))(1+g^{-1}F)^{-1}S . \lbeq(tR-def)
\eqn

We have the following result.
\bglm \lblm(Step1)
If $S{\tD}S$ is non-singular in $S\C^N$, then
$\Ga(\lam)^{-1}$ is a good multiplier.
\edlm
\bgpf \refeq(G-3) implies $A_1= S\tD{S}+ O(g^{-1})$. Hence,
$A_1^{-1}$ exists in $S\C^N$ and it is a good multiplier.
It follows by virtue of \refeq(JN-1) of \reflm(JN) that
\bqn
\Ga(\lam)^{-1} = -N^{-1}g^{-1}(A+S)^{-1} +
(A+S)^{-1}S A_1^{-1}S (A+S)^{-1}. \lbeq(G-1)
\eqn
Since $(A+S)^{-1}$ is a good multilier, \refeq(G-1)
implies that $\Ga(\lam)^{-1}$ is also a good multiplier.
\edpf

\paragraph{Step 2.}
Suppose next $S{\tD}S$ is singular in $S\C^N$ and let $T$ be
the orthogonal projection to $\Ker_{S\C^N} S{\tD}S$.
>From \refeq(G-3) we have
\[
L+T =(S\tD{S}+T)+ N^{-1}g^{-1}S\tD(1+g^{-1}F)^{-1}\tD{S}
\]
and $S\tD{S}+T$ is clearly invertible in $S\C^N$. Hence
\bqn
L+T= (1+  g^{-1}\tilde{F})(S\tD{S}+T).
\eqn
where $\tilde{F}= O(1)$ is defined by
\bqn
\tilde{F} \stackrel{\rm def}{=}
N^{-1}S\tD(1+g^{-1}F)^{-1}\tD{S}(S\tD{S}+T)^{-1},
\lbeq(tFdef)
\eqn
It follows that $L+T$ is also invertible in $S\C^N$ for small $\lam>0$
and
\begin{multline}
(L+T)^{-1} = (S\tD{S}+T)^{-1} - g^{-1}(S\tD{S}+T)^{-1}\tilde{F} \\
+ g^{-2}(S\tD{S}+T)^{-1}\tilde{F}^2 (1+g^{-1}\tilde{F})^{-1}
\lbeq(L+T)
\end{multline}
by virtue of \refeq(X-2). Then, \refeq(A1-exp) implies that
$A_1+ T$ is also invertible in
$S\C^N$ and
\begin{align} \lbeq(A1+T)
& (A_1+ T)^{-1}= (L+T)^{-1}
(1- g\lam^2 \Rg_1(L+T)^{-1})^{-1} \\
& = (L+T)^{-1}+ g\lam^2(L+T)^{-1}
(\Rg_1+ O(\lam^2g))(L+T)^{-1} .
\lbeq(3-64)
\end{align}
\refeq(L+T) and \refeq(3-64) imply that
$(A_1+ T)^{-1}$ in $S\C^N$ is a good multiplier and that
\bqn
(A_1(\lam)+ T)^{-1} T = T + O(g^{-1}), \quad
T(A_1(\lam)+ T)^{-1} = T + O(g^{-1}).  \lbeq(A1-T-inter)
\eqn
For studying $A_1^{-1}$ by using \reflm(JN), define $B_1$ in $T\C^N$ by
\bqn \lbeq(B1-def)
B_1 {=} T - T(A_1+ T)^{-1}T.
\eqn
Inserting \refeq(3-64) into \refeq(B1-def), we have
\bqn
B_1= T- T(L+T)^{-1}T - g\lam^2 \Rg_2,  \lbeq(B1)
\eqn
where we defined $\Rg_2$ by
\bqn
\lbeq(R2-def)
\Rg_2 \stackrel{\rm def}{=} T(L+T)^{-1}(\Rg_1 + O(g\lam^2))(L+T)^{-1}T.
\eqn
Since $T(S\tD{S}+T)^{-1}= T$, \refeq(L+T) and \refeq(tFdef) imply
that
\begin{multline*}
T- T(L+T)^{-1}T \\
=
g^{-1}T\tilde{F}T - g^{-2}T \tilde{F}^2({{1}}
+g^{-1}\tilde{F})^{-1}T
{=}N^{-1}g^{-1} (T\tD^2{T}+ g^{-1}F_2) ,
\end{multline*}
where $F_2$ is defined by
\bqn \lbeq(F2-def)
F_2(\lam)= N^{-1}T\tD^2({{1}}+g^{-1}F)^{-1}\tD{T}- N
\tilde{F}^2({{1}}+g^{-1}\tilde{F})^{-1}T
\eqn
should be obvious.
With these definition $B_1$ becomes
\bqn \lbeq(A2-def)
B_1 = N^{-1}g^{-1} A_2, \ \ A_2 =
T\tD^2 T + g^{-1}F_2 - g^2 N \lam^2  \Rg_2 .
\eqn

\bglm \lblm(Ga-pwave)
Suppose $T{\tD}^2 T$ is non-singular in $T\C^N$. Then $\dim T\C^N=1$,
${\rm rank}\, T\tD^2 T=1$ and $H_{\a,Y}$ has an $s$-wave resonance only.
Moreover,
\bqn
\Ga(\lam)^{-1} =  N g T (T\tD^2 T)^{-1} T +  M(\lam)  .    \lbeq(S2-extra)
\eqn
where we wrote $T$ for $ST$ and $TS$ in \refeq(S2-extra).
\edlm
\bgpf The first part of the lemma is proved in \cite{CMYE}.
If $T\tD^2 T$ is non-singular in $T\C^N$, then \refeq(A2-def) implies
$A_2$ and, hence $B_1$ are non-singular in $T\C^N$,
\bqn \lbeq(B1-inv)
A_2^{-1} = (T\tD^2 T)^{-1} + O(g^{-1}), \quad
B_1^{-1} = Ng((T\tD^2 T)^{-1} + O(g^{-1}))
\eqn
and by virtue of \reflm(JN)
\bqn \lbeq(A1-inv)
A_1^{-1} = (A_1 + T)^{-1}+ (A_1 + T)^{-1} T B_1^{-1}T (A_1 + T)^{-1}.
\eqn
Combining \refeq(G-1) and \refeq(A1-inv), we have
\begin{align}
& \Ga(\lam)^{-1} = -N^{-1} g^{-1} (A+S)^{-1} +
(A+S)^{-1}S (A_1+ T)^{-1} S (A+S)^{-1} \lbeq(Step1-1)  \\
& \hspace{1cm}+ Ng (A+S)^{-1}S (A_1+ T)^{-1}T A_2^{-1}
T(A_1+T)^{-1} S (A+S)^{-1}. \lbeq(Step1-1a)
\end{align}
As $(A+S)^{-1}$ and $(A_1+ T)^{-1}$ are good multiplies as has been proved
previously, the two terms in \refeq(Step1-1) are good multipliers.
\refeq(B1-inv), \refeq(A1-T-inter) and \refeq(A+S-Sinter) implies
that \refeq(Step1-1a) is equal to the sum of
$N g ST (T\tD^2 T)^{-1} TS$ and a good multiplier. This proves \refeq(S2-extra).
\edpf

\paragraph{Step 3.}
We next assume that $S\tD{S}\vert_{S\C^N}$ and $T\tD^2 T\vert_{T\C^N}$  are
both singular. Let $T_p$ be the orthogonal projection onto
$\Ker\, T\tD^2 T$ in $T\C^N$.
Recall that we irrespectively write $M(\lam)$
for a good multiplier. Recalling \refeq(A2-def), we define
\bqn \lbeq(tildeA2-def)
\tilde A_2 \stackrel{\rm def}{=}  T\tD^2 T +  g^{-1}F_2 \ \
\mbox{\rm so\ that } \ A_2 = \tilde A_2 - g^2 \lam^2 N {\Rg}_2\, .
\eqn
It is evident that $T\tD^2 T + T_p$ is non-singular in $T\C^N$.
Hence both of $\tilde A_2+T_p$ and $A_2+ T_p$ are invertible in $T\C^N$
for small $\lam>0$ and
\begin{align}
(A_2+ T_p)^{-1} & = (\tilde A_2 + T_p)^{-1}
+ g^2 \lam^2 N (\tilde A_2 + T_p)^{-1}\Rg_2(1+ O(\lam^2 g^2))
(\tilde A_2 + T_p)^{-1}\notag  \\
& = (T\tD^2 T + T_p)^{-1} + O(g^{-1})
\lbeq(A2Tp)
\end{align}
In particular $(A_2+ T_p)^{-1}$ is a good multiplier.

\bglm \lblm(4-3)
{\rm (1)} The projection $T_p$ annihilates all of $\tD, \tilde{F}, L, F_2 $ and
$\tilde{A}_2$:
\bqn  \lbeq(zero)
\br{lll}
\tD{T}_p= T_p \tD =0, \quad & T_p \tilde{F}=\tilde{F}T_p=0 ,& T_p L = LT_p=0,
\\
T_p F_2= F_2 T_p=0, \quad & \tilde A_2 T_p = T_p \tilde A_2 =0. &
\er
\eqn
{\rm (2)} We have the following identities:
\begin{gather*}
(A+S)^{-1}S (A_1+ T)^{-1}T (A_2+ T_p)^{-1}T_p
= T_p + \lam^2 g^2 S M T_p + \lam^2 g M T_p. \\
 T_p (A_2+ T_p)^{-1} T (A_1+ T)^{-1}S (A+S)^{-1}
= T_p + \lam^2 g^2 T_p M S + \lam^2 g T_p M.
\end{gather*}
\edlm
\bgpf (1) Since $\tD$ is real symmetric, we have
$\Ker T\tD^2 T= \Ker \tD{T}$ and $\tD{T}_p= T_p \tD =0$.
Other identities of (1) follows from this immediately.
It follows from \refeq(tildeA2-def) that
$T_p(A_2+T_p)= T_p+ T_p O(\lam^2g^2)$,
$(A_2+T_p)T_p= T_p+ O(\lam^2g^2)T_p$ and hence
\bqn
T_p (A_2+T_p)^{-1}= T_p + T_p O(\lam^2g^2), \ \
(A_2+T_p)^{-1}T_p = T_p +O(\lam^2 g^2)T_p.   \lbeq(A2-Tpinter)
\eqn
Likewise we have from \refeq(G-3) and \refeq(A) that
\begin{gather}
(A_1+ T)^{-1} T_p = T_p + O(\lam^2 g)T_p, \ \
T_p (A_1+ T)^{-1}= T_p + T_p O(\lam^2 g).   \lbeq(A1-Tpinter) \\
(A+S)^{-1}T_p = T_p + O(\lam^2)T_p, \ \
T_p (A+S)^{-1} = T_p + T_p O(\lam^2). \lbeq(A+S-Tpinter)
\end{gather}
Then, we apply \refeqss(A2-Tpinter,A1-Tpinter,A+S-Tpinter) consecutively
in this order and then \refeq(A1-T-inter) and \refeq(A+S-Sinter) to obtain
\begin{align*}
&(A+S)^{-1}S (A_1+ T)^{-1}T (A_2+ T_p)^{-1}T_p  \\
& = (A+S)^{-1}S (A_1+ T)^{-1}T_p
+ (A+S)^{-1}S (A_1+ T)^{-1}T O(\lam^2 g^2)T_p \\
& =(A+S)^{-1}S (T_p + O(\lam^2 g)T_p) +
(A+S)^{-1}S (T + O(g^{-1})T) O(\lam^2 g^2)T_p  \\
& = T_p + O(\lam^{2})T_p+ SO(\lam^2 g)T_p+ SO(\lam^2 g^2)T_p+ O(\lam^2 g)T_p.
\end{align*}
This yields the first of (2). The second is the conjugate of the first.
\edpf

We study $A_2^{-1}$ via \reflm(JN) and we define
\bqn
B_2 = T_p - T_p (A_2+ T_p)^{-1}T_p . \lbeq(B2-1)
\eqn
By virtue of \refeq(zero) we have $T_p (\tilde A_2 + T_p)^{-1}
= (\tilde A_2 + T_p)^{-1}T_p = T_p$ and
\bqn

T_p(A_2+T_p)^{-1}T_p = T_p + N g^2 \lam^2  T_p
({\Rg}_2+O(\lam^2 g^2)) T_p. \lbeq(S5-3)
\eqn
Equations \refeq(zero) impy
$T_p (L+T)^{-1}= (L+T)^{-1}T_p = T_p $
and, recalling the definitions \refeq(R2-def) and \refeq(tR-def), we obtain
\bqn \lbeq(B-ob)
T_p \Rg_2 T_p = NT_p (\Rg + O(g\lam^2)) T_p=
NT_p(\Gg_1(Y) + g^{-1}\Gg_2(Y)+O(g\lam^2))T_p.
\eqn
It follows by applying \refeq(S5-3) and \refeq(B-ob) to \refeq(B2-1) that
\bqn \lbeq(A3def)
B_2
= -N^2 g^2 \lam^2 A_3, \ \
A_3=T_p {\Gg}_1(Y) T_p + g^{-1}T_p {\Gg}_2(Y) T_p + T_p O(\lam^2 g^2)T_p
\eqn

\bglm \lblm(p-wave)
Suppose $T{\tD}^2 T$ is singular in $T\C^N$ and let $T_p$ be
the projection onto $\Ker\, T{\tD}^2 T$ in $T\C^N$. Suppose
$T_p \hat{\Gg}_1(Y) T_p$ is non-singular in $T_p \C^N$. Then,
denoting $STT_p$ and $T_p T S$ simply by $T_p$,
\bqn \lbeq(Gam-S3)
\Ga(\lam)^{-1} =
-N^{-1}g^{-1}\lam^{-2}T_p
(T_p {\Gg}_1(Y) T_p + g^{-1}T_p {\Gg}_2(Y) T_p)^{-1}T_p
+ g SM S + M.
\eqn
\edlm
\bgpf If $T_p{\Gg}_1(Y) T_p$ is non-singular in $T_p \C^N$, then $B_2$ and
$A_3$ are invertible in $T_p \C^N$ by virtue of \refeq(A3def) and
\reflm(JN) implies
\bqn
A_2^{-1}= (A_2+T_p)^{-1} -N^{-2}g^{-2} \lam^{-2}
(A_2+T_p)^{-1}T_p A_3^{-1} T_p (A_2+T_p)^{-1}.
\lbeq(A2-1)
\eqn
We substitute  \refeq(A2-1) for $A_2^{-1}$ of \refeq(Step1-1a).
Then, modulo $M(\lam)$ which is produced by \refeq(Step1-1),
$\Ga(\lam)^{-1}$
is equal to
\begin{align}
& Ng (A+S)^{-1}S (A_1+ T)^{-1}T (A_2+T_p)^{-1} T(A_1 +T)^{-1} S (A+S)^{-1}
\lbeq(Step3-2)  \\
& - N^{-1}g^{-1}\lam^{-2} (A+S)^{-1}S (A_1+ T)^{-1}T(A_2+T_p)^{-1}  \notag \\
& \hspace{3cm} \times
T_p A_3^{-1} T_p (A_2+T_p)^{-1} T(A_1 +T)^{-1} S (A+S)^{-1}. \lbeq(Step3-3)
\end{align}
Substituting \refeq(A2Tp) that
$(A_2+ T_p)^{-1}=  (T\tD^2 T + T_p)^{-1} + O(g^{-1})$
for \refeq(Step3-2) and applying the relations
\refeq(A+S-Sinter) and \refeq(A1-T-inter), we obtain
\bqn \lbeq(L2)
\refeq(Step3-2)=Ng ST (T\tD^2 T + T_p)^{-1} T S +M(\lam).
\eqn
For studying \refeq(Step3-3) we introduce a short hand notation
\[
G_1 \stackrel{{\rm def}}{=} T_p {\Gg}_1(Y) T_p, \quad
G_2 \stackrel{{\rm def}}{=} T_p {\Gg}_2(Y) T_p, \quad
\tilde {\Rg}\stackrel{{\rm def}}{=} T_p\Rg T_p = G_1+ g^{-1} G_2 .
\]
The assumption of the lemma implies $\tilde {\Rg}$ is invertible in
$T_p \C^N$ and so is $A_3$ and
$A_3^{-1}= \tilde{\Rg}^{-1}+ O(\lam^2g^2)$, which we substitute for
for $A_3^{-1}$ in \refeq(Step3-3).  Then
\begin{multline}  \lbeq(Step3-3a)
\refeq(Step3-3) = -N^{-1}g^{-1}\lam^{-2} (A+S)^{-1}S(A_1+T)^{-1}T
(A_2+T_p)^{-1} \\
\times
T_p (\tilde{\Rg}^{-1}+ O(\lam^2g^2))T_p(A_2+T_p)^{-1}T(A_1+T)^{-1}S(A+S)^{-1}.
\end{multline}
Then, we replace functions on each side of
$(\tilde{\Rg}^{-1}+ O(\lam^2g^2))$
by the corresponding functions of \reflm(4-3) (2).
The term $O(\lam^2g^2)$ produces
$T_p O(g) T_p + O(\lam^2 g^3) = g S M(\lam)S + M(\lam)$  and
$\tilde{\Rg}^{-1}$ does
\[
-Ng^{-1}\lam^{-2}T_p \tilde{\Rg}^{-1} T_p +
g (SM(\lam)\tilde{\Rg}^{-1}T_p + T_p \tilde{\Rg}^{-1} M (\lam) S) + M(\lam)
\]
Combining these results with \refeq(L2), we obtain \refeq(Gam-S3).
\edpf

\newpage
\paragraph{Step 4.} Next we assume in addition to those of Step 3
that $G_1$ is singular in $T_p \C^N$ but
$G_1\not=0$. We let $T_e$ be the projection in
$T_p \C^N$ onto $\Ker_{T_p\C^N}\,G_1$.
We recall from \cite{CMY} that, then $T_e G_2 T_e $ is
necessarily non-singular in $T_e \C^N$.
In the following lemma we write $T_e^\perp$ for $T_p \ominus T_e$.
and we define the operator $\Bg$ on $T_p\C^N$ and $D_0$ on
$(T_p\ominus T_e) \C^N$ by
\begin{gather}
\Bg= T_e^\perp G_2 T_e (T_e G_2 T_e )^{-1} T_e, \\
D_0=(T_e^\perp G_1 T_e^\perp + g^{-1}
(T_e^\perp G_2 T_e^\perp - T_e^\perp G_2 T_e
(T_e G_2 T_e)^{-1}T_e G_2 T_e^\perp)^{-1}.
\end{gather}
By the remark above, $\Bg$ is well defined and, since
$T_e^\perp G_1 T_e^\perp$ is clearly invertible in $(T_p \ominus T_e)\C^N$,
$D_0$ is also well defined for small $\lam>0$.

\bglm \lblm(Step-4) Let $G_1$ and $T_e$ be as above.
Then, $\Ga(\lam)^{-1}$ has the following expression as $\lam \to 0$:
\begin{multline}  \lbeq(Step-4)
- N^{-1}\lam^{-2} T_e  (T_eG T_e)^{-1} T_e
-N^{-1}g^{-1}\lam^{-2}T_p (T_e^\perp - \Bg^\ast)T_e^\perp D_0
T_e^\perp (T_e^\perp - \Bg)T_p \\
+ g^3 T_p M T_p + g^2 S M S +
g S M + g M S + M.
\end{multline}
\edlm

For the proof we use the following well known formula from linear algebra.

\bglm \lblm(FS) Suppose $a_{11}$ and $a_{22}$ are closed
and $a_{12}$ and $a_{21}$ are bounded operators.
Suppose that $a_{22}^{-1}$ exists. Then $A^{-1}$ exists if and only if
$d= (a_{11}- a_{12}a_{22}^{-1}a_{21})^{-1}$ exists. In this case we have
\bqn \lbeq(FS-formula)
A^{-1} = \begin{pmatrix}  d & -d a_{12} a_{22}^{-1} \\
-a_{22}^{-1}a_{21} d & a_{22}^{-1}a_{21}d a_{12} a_{22}^{-1} + a_{22}^{-1}
\end{pmatrix}.
\eqn
\edlm

\bgpf
We may repeat the argument of Step 3 upto \refeq(A3def)
and notice that, as long as $A_3^{-1}$ exists in $T_p \C^N$, we have
$\Ga(\lam)^{-1}= \refeq(Step3-2) + \refeq(Step3-3)$
and that $\refeq(Step3-2)= g S M(\lam) S+ M(\lam)$ (see \refeq(L2)).
Thus, we have only to study \refeq(Step3-3).
We study $A_3^{-1}$ in $T_p\C^N$ by using \reflm(FS).
We write $A_3$ in the block matrix in the
direct decompotision $T_p\C^N= (T_p \ominus T_e)\C^N \oplus T_e \C_N $:
With obvious abusing notation
\bqn  \lbeq(mat-1)
A_3 =
\begin{pmatrix}  T_e^\perp G_1 T_e^\perp + g^{-1}
T_e^\perp G_2 T_e^\perp  &  g^{-1} T_e^\perp G_2 T_e  \\
g^{-1} T_e G_2 T_e^\perp   & g^{-1} T_e G_2 T_e  \\
\end{pmatrix} + O(g^2 \lam^2)
\eqn
Then, $a_{22}= g^{-1} T_e G_2 T_e$ is invertible in $T_e \C^N$
as mentioned above;
\begin{align*}
& a_{11}- a_{12}a_{22}^{-1}a_{21}
= T_e^\perp G_1 T_e^\perp + g^{-1}(T_e^\perp G_2 T_e^\perp
- T_e^\perp G_2 T_e
(T_e G_2 T_e)^{-1}T_e G_2 T_e^\perp)
\end{align*}
is also invertible for small $\lam>0$
in $T_p \C^N \ominus T_e\C^N$ because
$\Ker\,G_1 \cap T_e^\perp=\{0\}$ by the definition of $T_e$.
Then, \reflm(FS) yields that the matrix in the right of \refeq(mat-1)
is invertible in $T_p\C^N$ and is equal to
\bqn
\begin{pmatrix} D_0 & - D_0 \Bg \\ \Bg^\ast D_0 &
- \Bg^\ast D_0 \Bg + g (T_e G_2 T_e)^{-1} \end{pmatrix},
\eqn
whichis of order $O(g)$ as $\lam \to 0$. Then, the standard
perturbation theory implies
\begin{align}
A_3^{-1}& =
\begin{pmatrix} D_0 & - D_0 \Bg \\ \Bg^\ast D_0 &
\Bg^\ast D_0 \Bg+g (T_e G_2 T_e)^{-1} \end{pmatrix}
+ T_p O(g^4\lam^2)T_p  \notag \\
& = g (T_e G_2 T_e)^{-1} + (T_e^\perp-\Bg^\ast )D_0 (T_e^\perp + \Bg)
+ T_p O(g^4\lam^2)T_p .
\lbeq(A3inv)
\end{align}
We  substitute \refeq(A3inv) for $A_3^{-1}$ in \refeq(Step3-3)
and denote by $\Ga_1(\lam)$ and $\Ga_2(\lam)$ the functions
produced respectively by $g (T_e G_2 T_e)^{-1}$ and  by the other two terms.
Then, by virtue of \reflm(4-3) (2), $\Ga_1(\lam)$
is equal to
\begin{align}
& -N^{-1}\lam^{-2}
(1 + \lam^2 g^2 S M  + \lam^2 g M )T_e(T_e G_2 T_e)^{-1}
T_e(1 + \lam^2 g^2 M S + \lam^2 g M).
\notag \\
& = -N^{-1}\lam^{-2}T_e(T_e G_2 T_e)^{-1}T_e  \lbeq(principal) \\
& \hspace{1cm} + g^2(SMT_e+ T_e MS)+
g (MT_e + T_e M) + M.  \lbeq(principal-rest)
\end{align}
Likewise, denoting $ \Cg=(T_e^\perp-\Bg^\ast )D_0 (T_e^\perp-\Bg)$,
$\Ga_2(\lam)$ is equal to
\begin{align}
& -N^{-1}\lam^{-2}g^{-1}
(1 + \lam^2 g^2 S M  + \lam^2 g M )T_p
(\Cg + O(\lam^2 g^4))
T_p(1 + \lam^2 g^2 M S + \lam^2 g M)
\notag \\
& = -N^{-1}\lam^{-2}g^{-1}T_p \Cg T_p + g^3 T_p M T_p
+ g(SMT_p+ T_p MS) + M.  \lbeq(sub)
\end{align}
Combining \refeq(principal), \refeq(sub) with the remark stated
at the beginning, we obtain \refeq(Step-4)
and conclude the proof.
\edpf

\paragraph{Step 5.} We finally assume $G_1=0$ the opposite of
of \reflm(Step-4). Then, $T_p=T_e$ and $H_{\a,Y}$ has no $p$-wave resonances.

\bglm Suppose $G_1=0$. Then,  $T_e=T_p$ and
\begin{multline} \lbeq(5-1)
\Ga(\lam)^{-1}= -N^{-1}\lam^{-2}T_e G_2^{-1} T_e + g^3 T_p M(\lam) T_p  \\
+ g^2 (T_e M(\lam) S + S M(\lam)T_e)+
g (T_e M(\lam)+ M(\lam)T_e) + M(\lam) .
\end{multline}
\edlm

\bgpf In this case we still have the expression
$\Ga(\lam)^{-1}= \refeq(Step3-2) + \refeq(Step3-3)$
and \refeq(Step3-2) satisfies the estimate \refeq(L2).
As was remarked previously  $G_2$ is non-singular in $T_p \C^N$.
It follows from \refeq(A3def) that in $T_p \C^N$,
\bqn \lbeq(a3-1)
A_3^{-1} =
(g^{-1}G_2+ T_p O(\lam^2 g^2)T_p)^{-1}
= g G_2^{-1} + O(\lam^2 g^4).
\eqn
We substitute \refeq(a3-1) in \refeq(Step3-3) and apply \reflm(4-3) (2).
Then  $\refeq(Step3-3)$ becomes
\begin{align*}
& - N^{-1}g^{-1}\lam^{-2}(T_p + \lam^2 g^2 S M + \lam^2 g M)
T_p \\
& \hspace{2cm}
\times
(g G_2^{-1} + O(\lam^2 g^4))T_p (T_p + \lam^2 g^2 MS + \lam^2 g M) \\
& =  - N^{-1}\lam^{-2}(T_p + \lam^2 g^2 S M+ \lam^2 g M)G_2^{-1}
(T_p + \lam^2 g^2 MS+ \lam^2 g M) + g^3 T_p M T_p \\
& = - N^{-1}\lam^{-2}T_p G_2^{-1}T_p
-N^{-1}T_p G_2^{-1}(g^2 MS + g M) \\
& \hspace{2cm} - N^{-1}(g^2 SM + gM) G_2 T_p + M(\lam)
+ g^3 T_p M T_p.
\end{align*}
As $T_p= T_e$, this implies \refeq(5-1).
\edpf

\subsection{Fourier transform of a logarithmic function}

In the following section need a pointwise estimate on
the Fourier transform
\bqn \lbeq(pgF)
F(x){=}
\frac1{2\pi}\int_{\R^2} \frac{e^{ipx }\chi_{\leq \ep}(|p|){dp}}{|p| g(|p|)}.
\eqn
It is obvious from Hausdorff-Young's inequality
$F \in L^q(\R^2)$ for $2\leq q \leq \infty$.
The following estimate must be well
known and we give a proof for reader's convenience.

\bglm \lblm(plogp)
Let $\chi_{\leq \ep}$ be defined by \refeq(chiep-def). Then for
$0<\ep\leq 1$ there exists a constant $C_\ep >0$ such that
\bqn \lbeq(plogp)
|F(x)|\leq \frac{C_\ep}{\ax \log(2+|x|)}
\eqn
\edlm
\bgpf Since $F(x)$ is evidently a smooth function, it suffices to show
\refeq(plogp) when $|x|$ is sufficiently large and we assume
$|x|>(100e^{10\c}+4\pi/\ep)$, $\c= 0.577\dots$ being Euler's constant.  Since
$g(r)\geq (\log |x|)/200\pi +1/8$ for $0<r<2\pi/|x|$, we have
\[
\frac1{2\pi}\int_{\R^2} \frac{e^{i\xi x }
\chi_{\leq 2\pi/|x|}(|\xi |)\chi_{\leq \ep}(|\xi |){d\xi }}{|\xi | g(|\xi |)}
\absleq
\int_0^{2\pi/|x|} \frac{dr}{|g(r)|}\leq
\frac{C}{\ax \log(2+|x|)}
\]
for a constant $C>0$. Thus, it suffices to prove the lemma
after inserting $\chi_{\geq 2\pi/|x|}(|\xi |)$ in the integrand
of \refeq(plogp). We denote the function thus obtained by $\tilde{F}(x)$.
The Bessel function satisfies (see e.g. \cite{Stein}, page 338):
\[
J_0(r)= \frac1{\pi}\int_{0}^{\pi} e^{ir\cos\th}d\th
= (2/\pi)^{1/2}r^{-1/2}\cos(r-\pi/4)+ O(r^{-3/2}),  \quad r \to \infty.
\]
Let $\chi_{[\ep_1,\ep_2]}(r)
= \chi_{\geq \ep_1}(r)\chi_{\leq \ep_2}(r)$. Then, after a change variables
\begin{align}
\tilde{F}(x)&=
\frac1{\sqrt{2\pi}}\int_0^\infty \frac{\chi_{[2\pi/|x|,\ep]}(r)}{g(r)}
\Big(
\frac{\cos(|x|r-\pi/4)}{\sqrt{|x|r}}+ O((|x|r)^{-3/2})\Big)dr \notag \\
& = \frac1{\sqrt{2\pi}|x|}\int_{0}^\infty
\frac{\chi_{[2\pi,\ep|x|]}}{g(r/|x|)}\Big(\frac{\cos(r-\pi/4)}{\sqrt{r}}
+ O(r^{-3/2})\Big)dr . \lbeq(tilF)
\end{align}
Write $\tilde{\chi}$ for $\chi_{[2\pi,\ep|x|]}$.
Denote the integral produced by  $O(r^{-3/2})$ in \refeq(tilF) by $F_1(x)$.
Since $|g(r)|\geq 1/4$ for any $r>0$ and
$|g(r/|x|)|\geq C \log |x|$ for $2\pi \leq r<{\sqrt{x}}$  as
$|x|>100e^{10\c}$ then, we have
\begin{align*}
|F_1(x)| & \leq
\frac1{|x|}\left(
\int_0^{\sqrt{x}}+ \int_{\sqrt{x}}^{\ep |x|} \right)
\frac{\tilde{\chi}(r)} {|g(r/|x|)|}\la r \ra ^{-3/2}dr \\
&\leq
\frac{2}{|x|}\int_0^{\sqrt{x}}\frac1 {g(r/|x|)}\la r \ra ^{-3/2}dr
+ \frac{C}{|x|}\int_{\sqrt{x}}^{\ep |x|} \la r \ra ^{-3/2}dr
\leq \frac{C}{\ax \log(2+ |x|)}.
\end{align*}.
Since $\cos(s+\pi)= -\cos s$ and $2\pi<r<\ep|x|$, we have
\begin{align*}
&
\frac1{|x|}\int_0^\infty \frac{\tilde{\chi}(r)}{g (r/|x|)}
\frac{\cos(r-\tfrac{\pi}{4})}{r^{1/2}} dr
= - \frac1{|x|}\int_0^\infty
\frac{\tilde{\chi}(r+\pi)}{g ((r+\pi)/|x|)}
\frac{\cos(r-\tfrac{\pi}{4})}{(r+\pi)^{1/2}}dr \\
& = \frac1{2|x|}\int_0^\infty \cos(r-\tfrac{\pi}{4})
\left(\frac{\tilde{\chi}(r)}{g(r/|x|)} r^{-1/2}-
\frac{\tilde{\chi}(r+\pi)}{g((r+\pi)/|x|)} (r+\pi)^{-1/2}\right) dr
\end{align*}
The function inside $(\cdots )$ is $K_1+ K_2+ K_3$ where
\begin{align*}
K_1(r,x) & = \frac{\tilde{\chi}(r)-\tilde{\chi}(r+\pi)}{g (r/|x|)} r^{-1/2}. \\
K_2 (r,x) & = \tilde{\chi}(r+\pi)
\left(\frac1{g (r/|x|)} -\frac1{g((r+\pi)/|x|)}\right)
r^{-1/2} . \\
K_3 (r,x) & =
\frac{\tilde{\chi}(r+\pi)}{g((r+\pi)/|x|)}(r^{-1/2}-(r+\pi)^{-1/2}) .
\end{align*}
Since $\tilde{\chi}(r)-\tilde{\chi}(r+\pi)\not=0$ only on
$[\pi,2\pi]$ and on $[\ep|x|-\pi, \ep|x|]$ and,
$|g(r/|x|)|\geq c\log |x|$ on  $[\pi,2\pi]$ and
$|g(r/|x|)|\geq C$ on $[\ep|x|-\pi, \ep|x|]$, it follows that
\[
\frac1{2|x|}\int_0^\infty \cos(r-\tfrac{\pi}{4}) K_1(r,x)dr
\absleq \frac{C}{|x|}\left(
\int_{\pi}^{2\pi}\frac{dr}{\log |x|}dr +
\int_{\ep|x|-\pi}^{\ep|x|}\frac{dr}{r^{1/2}}dr\right)
\]
and this is bounded in modulus by $C\ax^{-1}(2+ \log |x|)^{-1}$ as
desired. Note that $K_2(r,x)\not=0$ for $\pi <r <\ep |x|-\pi$
and we estimate
\begin{align*}
&\frac1{g(r/|x|)} -\frac1{g((r+\pi)/|x|)}
\absleq \frac{\log(r+\pi)-\log r}
{2\pi g(r/|x|) \cdot g((r+\pi)/|x|)} \\
&= \frac{\log (1+\frac{\pi}{r})}{2\pi g(r/|x|)\cdot g((r+\pi)/|x|)}
\leq \left\{
\br{cl} \displaystyle
\frac{C_\ep }{(\log |x|)^2 r}, & \ 2\pi<r<\sqrt{|x|}, \\[10pt]
\displaystyle \frac{C_\ep}{r}, & \ \sqrt{|x|}\leq r \leq \ep |x| .
\er
\right.
\end{align*}
It follows for large $|x|$ that
\begin{align*}
& \frac1{2|x|}\int_0^\infty \cos(r-\tfrac{\pi}{4}) K_2(r,x)dr \\
& \absleq
\frac{C}{|x|} \int_{2\pi}^{\sqrt{x}}
\frac{1}{(\log x)^2 r^\frac32}dr +\frac{C}{|x|} \int_{\sqrt{x}}^{\ep |x|}
\frac{1}{r^\frac32}dr
\leq \frac{C_1}{|x|(\log |x|)^2}.
\end{align*}
We estimate for $\pi \leq r \leq \ep |x|-\pi$
\[
K_3 (r,x) \absleq \frac{\pi\tilde{\chi}(r+\pi)}
{r^{1/2}(r+\pi)|g((r+\pi)/|x|)|}
\leq \frac{C}{r^{3/2}}
\]
and
\[
\frac1{2|x|}\int_0^\infty \cos(r-\tfrac{\pi}{4}) K_3(r,x)dr
\absleq
\frac1{2|x|}\int_{\pi}^{\ep|x|-\pi}\frac{dr}{r^{3/2}}
\leq \frac{C}{\ax \log (2+|x|)}.
\]
Adding these up, we complete the proof. \edpf

\section{Proof of \refthb(Main)}

In this section we prove \refth(Main).
By virtue of \refth(high) it suffices to prove the statements for
the lower energy part $W_{\rm low}^{-} = W^{-}\chi_{\leq \ep}(|D|)$.
By virtue of \refeq(Stat-2a),
$W_{\rm low}^{-}= \chi_{\leq \ep}(|D|) u + \W_{\rm low}u$, where
\bqn
\W_{\rm low}u=
\frac1{\pi{i}}\int_0^\infty \chi_{\leq \ep}(\lam) \lam \big\la
\tGa(\lam) \hat{\Gg}_{\lam}(x),
\int_{\R^2}(\hat{\Gg}_{\lam}(y)-\hat{\Gg}_{-\lam}(y))u(y) dy \big\ra_{\C^N}
d\lam
\lbeq(low-W)
\eqn
and we study $\W_{\rm low}$. Recall $\tGa(\lam)  = \Ga(\lam)^{-1}$.
Here and hereafter we omit the index $Y$ and write $\hat{\Gg}_{\lam}(x)$ for
$\hat{\Gg}_{\lam,Y}(x)$  and etc. $\|u\|_p$ is the norm of $L^p(\R^2)$
and $\|u\|=\|u\|_p$. $\la u, v\ra$
is the coupling without complex conjugation. The inner product of
$L^2$ will be denoted by $(u,v)$. Recall the space $\Dg_\ast$ is defined
by \refeq(Dgast-def) and is dense in $L^p(\R^2)$ for any $1<p<\infty$.
In what follows  it is implicitly assumed that
$u \in \Dg_\ast$.

\subsection{Proof of statement (1)}
If $S\tD{S}$ is non-singular in $S\C^N$, then \reflm(Step1) implies that
$\tGa(\lam)\chi_{\leq \ep}(\lam)$ for a small $\ep>0$ is
a good multiplier. The product formula \refeq(prod) implies
\[
(\W_{jk}\chi_{\leq \ep}(|D|)u)(x)= (K \circ \tGa_{jk}(|D|)\chi_{\leq \ep}(|D|))u(x)
\]
and \reflm(prod) and \reflm(Mikhlin) imply statement (1).
This has been proved already in \cite{CMY}.

\subsection{Proof of statement (2)}

Under the condition of statement (2), \reflm(Ga-pwave) is satisfied and
we have \refeq(S2-extra) for $\lam $ in the support of the function
$\chi_{\leq \ep}$. We write $B$ for $(T\tD^2 T)^{-1}$
and substitute
\refeq(S2-extra) for $\tGa(\lam)$ in \refeq(low-W). The
term $M(\lam)$ produces a good operator as in the proof of statement (1)
and we are left with
\bqn \lbeq(low-Wa)
\frac{N}{i\pi}\int_0^\infty \chi_{\leq \ep}(\lam)
\lam g
\big\la TBT \hat{\Gg}_{\lam}(x) ,
\int_{\R^2}(\hat{\Gg}_{\lam}(y)-\hat{\Gg}_{-\lam}(y))u(y)dy \big\ra d\lam.
\eqn
For \refeq(low-Wa), we may still apply the product decomposition
of \reflm(prod), however, the multiplier
$\chi_{\leq\ep} (\lam)g(\lam)$ is not bounded
near $\lam=0$ and Mikhlin's theorem does not apply. To get around this
we use the cancellation produced by the projector $T$
among the components of
$\hat{\Gg}_{\lam}(y)-\hat{\Gg}_{-\lam}(y)$.
Define  the vector function $\hat{I}(\lam)$ by
\[
\hat{I}(\lam)= \frac{1}{i\pi}
\int_{\R^2}(\hat{\Gg}_{\lam}(y)-\hat{\Gg}_{-\lam}(y))u(y)dy
= \int_{{\mathbb S}^1} \begin{pmatrix}
e^{iy_1\w\lam} \\
\vdots \\
e^{iy_N\w\lam}
\end{pmatrix}\hat{u}(\lam\w)dw.
\]
Since $T=ST$, we may insert the orthogonal projection  $S$
in front of $I(\lam)$  without changing \refeq(low-Wa).
Since $NS=N-{\bf 1}\otimes {\bf 1}$, we then have
\bqn \lbeq(NSpi)
NS \hat{I}u(\lam)= \int_{{\mathbb S}^1}
\sum_{k=1}^N
\begin{pmatrix}
e^{iy_1\w\lam}-e^{iy_k\w\lam}  \\
\vdots \\
e^{iy_N\w\lam} -e^{iy_k\w\lam}
\end{pmatrix} \hat{u}(\lam\w)dw.
\eqn
Recall that $R=\begin{pmatrix} R_1 \\ R_2 \end{pmatrix}$
and $R_l= R_l(D)= \Fg^{-1}(\xi_l/|\xi|)\Fg$, $l=1,2$ are
Riesz transforms and $\la a, R u\ra= a_1 R_1 u + a_2 R_2 u$
for a vector $a\in \R^2$. Then, Taylor's formula implies that the $j$-th
component of \refeq(NSpi) is equal to
\begin{align}
& i \sum_{k=1}^N \int_{{\mathbb S}^1}\left(\int_0^1
e^{i(\th {y_j}+ (1-\th)y_k)\w\lam} (y_j-y_k) \w\lam\hat{u}(\lam\w)d\th\right)
d\w   \notag \\
&=i\lam \sum_{k=1}^N \int_0^1 \Fg \left(
\int_{{\mathbb S}^1}\la y_j-y_k, R \tau_{\th{y_j}+ (1-\th)y_k}u\ra )(\lam\w) d\w
\right) d\th, \lbeq(UseR)
\end{align}
where $\tau_y u(x)= u(x-y)$ is the translation by $y$.
Define
\bqn
m(\lam) = \lam g(\lam)\chi_{\leq \ep} (\lam).      \lbeq(m-def)
\eqn
Then, $m(|\xi|)$ is a good multiplier, $m(|D|)$ commutes
with translations and $R$ and we have
\bqn
m(\lam)
\Fg \big(\la a, R \tau_{y}u\ra \big)(\lam\w)
= \Fg \big(\la a, R \tau_{y}m(|D|)u\ra \big)(\lam\w).
\eqn
Thus, if we define
$(V_{jk}u)(x)
= \int_0^ 1 \la y_j-y_k, R \tau_{\th{y_j}+ (1-\th)y_k}m(|D|)u\ra d\th$
for $j,k=1, \dots, N$, then, $V_{jk}$ are evidently good operators
and \refeq(DEFK) implies
\begin{align}
\refeq(low-Wa)& = \int_0^\infty \chi_{\leq \ep}(\lam)\lam g(\lam)
\big\la TBT \hat{\Gg}_{\lam}(x) , NS  \hat{I}u(\lam) \big\ra
d\lam. \notag \\
& = \sum_{jkl}(TBT)_{lj}
\int_0^\infty \lam {\Gg}_{\lam}(x-y_l) \left(\int_{{\mathbb S}^1}
(\Fg V_{jk}u)(\lam\w)d\w \right)d\th \notag \\
& = \sum_{jkl} (TBT)_{jk} \tau_{y_l}(K\circ V_{jk}u)(x).  \lbeq(3-36a)
\end{align}
This  is a good operator and statement (2) is proved.
\qed

\subsection{Proof of statement (3)}

We next assume that $T_p \Gg_1(Y) T_p = G_1$ is non-singular in $T_p\C^N$
and $\Ga(\lam)^{-1}$ satisfies \reflm(p-wave),
We substitute \refeq(Gam-S3) for $\tGa(\lam)$ in
\refeq(low-W), which produces three operators. The proof of
statements (1) and (2) implies that $gS M(\lam) S$ and $M(\lam)$ produce
good operators. Ignoring unimportant constant, we write the integral produced by
$-N^{-1}g^{-1}\lam^{-2} T_p (G_1 + g^{-1}G_2)^{-1}T_p$ in the form
\bqn \lbeq(low-W-vec)
\W_{\rm low}u(x)=
\int_0^\infty \chi_{\leq \ep}(\lam)\lam^{-1} g(\lam)^{-1}
\big\la T_p B_\ast T_p \hat{\Gg}_{\lam}(x) ,
\frac{NS}{i\pi}\hat{I}u(\lam) \big\ra_{\C^N}
d\lam ,
\eqn
where we wrote $ST_p = T_p$ for simplicity and $B_\ast$ for
\[
B_\ast(\lam)=(G_1 + g^{-1}G_2)^{-1}.
\]
Remark that we have inserted $S$ in front of $\hat{I}u(\lam)$
which is allowed by the presence of $T_p$. Notice
the presence of the strong singularities $\lam^{-1} g(\lam)^{-1}$.

\subsubsection{Decomposition into good part and bad parts.}
By further expanding the exponential functions, we
decompose $NS\hat{I}u(\lam)$ into ``good'' and ``bad''
parts $\hat g(\lam)$ and $\hat{b}(\lam)$ as follows:
\begin{gather}
(NS\hat{I} u)(\lam) = \hat{g}(\lam) + \hat{b}(\lam)
= \begin{pmatrix} g_1(\lam) \\ \vdots \\ g_N(\lam)  \end{pmatrix}
+  \begin{pmatrix} b_1(\lam) \\ \vdots \\ b_N(\lam)   \end{pmatrix},
\lbeq(good-bad) \\
g_j(\lam)=\sum_{k=1}^N g_{jk}(\lam), \ \
b_j(\lam)= \sum_{k=1}^N b_{jk}(\lam) \ \ j=1, \dots, N  \\
g_{jk}(\lam)\stackrel{\mathrm{def}}{=}  i
\int_{{\mathbb S}^1}\left(
\int_0^1
(e^{i(\th {y_j}+ (1-\th)y_k)\w\lam} -1)
 (y_j-y_k) \w\lam\hat{u}(\lam\w)d\th\right)
d\w, \lbeq(good) \\
b_{jk}(\lam)\stackrel{\mathrm{def}}{=}  i
\int_{{\mathbb S}^1} (y_j-y_k) \w\lam\hat{u}(\lam\w)d\w \,. \lbeq(bad)
\end{gather}
Then, substituting \refeq(good-bad) for $(NS\hat{I} u)(\lam)$
in \refeq(low-W-vec), we obtain
\bqn \lbeq(gb-decomp)
\W_{\rm low}u = \W_{\rm low, g}u + \W_{\rm low,b}u
\eqn
where definition of $\W_{\rm low, g}u$ and $\W_{\rm low,b}u$ should be obvious.

\subsubsection{Good part produces a good operator}
\bglm \lblm(good)
For $1<p<\infty$, there exists a constant $C_p>0$ such that
\bqn \lbeq(good-part)
\|\W_{\rm low, g}u\|_p \leq C_p \|u\|_p, \quad u \in \Dg_\ast.
\eqn
\edlm
\bgpf Taylor's formula implies
\[
e^{i(\th {y_j}+ (1-\th)y_k)\w\lam} -1=
i\lam \la \th {y_j}+ (1-\th)y_k, \w \ra \cdot
\int_0^1
e^{i\mu(\th {y_j}+ (1-\th)y_k)\w\lam}d\mu,
\]
which produces an extra factor in \refeq(good) and
$g_{jk}(\lam)$ becomes the integral $d\th{d\mu}$ over
$(0,1) \times (0,1)$ of
\begin{multline*}
-\lam^2
\int_{\mathbb{S}^1}
e^{i\mu(\th {y_j}+ (1-\th)y_k)\w\lam}
(\la y_k, \w\ra \la y_j-y_k, \w\ra +\th\la y_j-y_k,\w\ra^2) \hat{u}(\lam\w)
d\w  \\
=-\lam^2
\int_{\mathbb{S}^1} \Fg(
\t_{\mu(\th {y_j}+ (1-\th)y_k)}
(\la y_k, R\ra \la y_j-y_k, R\ra +\th\la y_j-y_k,R\ra^2){u})(\lam\w) d\w.
\end{multline*}
Since $\{\tau_y : y \in \R^2\}$ is uniformly bounded in $\Bb(L^p)$, the operator
\[
W_{jk}u(x)\stackrel{\rm def}{=} \iint_{[0,1]^2}
\t_{\mu(\th {y_j}+ (1-\th)y_k)}
(\la y_k, R\ra \la y_j-y_k, R\ra +\th\la y_j-y_k,R\ra^2){u})(x)d\th{d\mu}.
\]
is a good operator. Define $m(\lam) = g^{-1}(\lam)\chi_{\leq \ep}(\lam)$.
$m(\lam)$ is a good multiplier and we can express $\W_{\rm low,g}u(x)$
is the form
\[
\sum_{j,k,l=1}^N
\int_0^\infty \lam  (T_p B_\ast(\lam) T_p)_{jk}\Gg_{\lam}(x-y_k)
\left(
\int_{{\mathbb S}^1}\Fg(W_{jl}m(|D|)u)(\lam\w) d\w
\right)
d\lam .
\]
It follows by virtue of the definition \refeq(DEFK) of $K$ that
\bqn \lbeq(Wgood)
\W_{\rm low,g}u(x) = \pi^2 \sum_{j,k,l=1}^N \tau_{y_k}
(K\circ W_{jl}m(|D|)(T_p B_\ast(|D|) T_p)_{jk} u)(x).
\eqn
Since $B_\ast(\lam)$ is a good multiplier under the assumption,
\reflm(prod) implies the lemma.
\edpf

\subsubsection{Decomposition of the bad part}
We decompose
$\W_{\rm low,b}u(x)$ into the low and high energy parts:
\begin{align}
\W_{\rm low,b}u(x)& = \int_0^\infty
\lam^{-1} \chi_{\leq \ep}(\lam) g(\lam)^{-1}
\big\la T_p B_\ast(\lam) T_p
\hat{\Gg}_{\lam}(x), \hat{b}(\lam)\big\ra d\lam
\lbeq(x-decomp0) \\
& = \chi_{\geq 2\ep}(|D|)\W_{\rm low,b}u(x) +
\chi_{\leq 2\ep}(|D|)\W_{\rm low,b}u(x)  \lbeq(x-decomp).
\end{align}
Note that supports of $\chi_{\geq 2\ep}$ and $\chi_{\leq \ep}$ do not
intersect.

\bglm \lblm(Bad-B)
For any $\ep>0$, $\chi_{\geq 2\ep}(|D|)\W_{\rm low,b}$
is bounded from $L^p(\R^2)$ to itself for $1<p\leq 2$.
\edlm
\bgpf Denote $\tilde{B}_{jk}=(T_p B_\ast T_p)_{jk}$ and
express $\chi_{\geq 2\ep}(|D|)\W_{\rm low,b}u(x)$ as the sum over
$j,k=1, \dots, N$ of
\bqn \lbeq(Xjk)
X_{jk}u(x)\stackrel{\rm def}{=}\int_0^\infty
\lam^{-1} \chi_{\leq \ep}(\lam) g(\lam)^{-1} \tilde{B}_{jk}(\lam)
\t_{y_j}\chi_{\geq 2\ep}(|D|){\Gg}_{\lam}(x)b_k(\lam) d\lam .
\eqn
Define $\m(\xi)= \chi_{\geq 2\ep}(|\xi|)|\xi|^{-2}$. $\m$ is a good multiplier.
By applying the inverse Fourier tansform to
$\chi_{\geq 2\ep}(\xi)(\xi^2-\lam^2)^{-1}=
\mu(\xi) + \lam^2 \mu(\xi) (\xi^2-\lam^2)^{-1}$ we have
\bqn
\chi_{\geq 2\ep}(|D|){\Gg}_{\lam}(x)= \hat{\m}(x) +
\m(|D|) \lam^2 {\Gg}_{\lam}(x). \lbeq(chigeq)
\eqn
We substitute \refeq(chigeq) for the
$\chi_{\geq 2\ep}(|D|){\Gg}_{\lam}(x)$ in \refeq(Xjk). The second summand
$\m(|D|) \lam^2 {\Gg}_{\lam}(x)$ cancels the singularity $\lam^{-1}$
and produces
\bqn \lbeq(mDint)
\sum_{l=1}^N \m(D)
\int_0^\infty \lam
{\Gg}_{\lam}(x-y_j) \left(\int_{{\mathbb S}^1}
\la y_k-y_l, \w \ra \r_{jk}(\lam)
\hat{u}(\lam\w)d\w \right) d\lam
\eqn
where $\r_{jk}(\lam)= \lam\chi_{\leq \ep}(\lam)g(\lam)^{-1}\tilde{B}_{jk}(\lam)$
is obviously a good multiplier.
By using $K$ of \refeq(DEFK), we may express \refeq(mDint) in the form
\[
\sum_{l=1}^N\m(D)\t_{y_j} K \circ (\la y_k -y_l, R \ra \r_{jk}(|D|)) u
\]
and \reflm(prod) implies that this is a good operator.

The first summand $\hat{\m}(x)$ produces
\bqn \lbeq(Int-1)
\sum_{l=1}^N  \int_0^\infty \chi_{\leq \ep}(\lam) g(\lam)^{-1}
\hat{\m}(x-y_j) \tilde{B}_{jk}(\lam)\big(\int_{{\mathbb S}^1}
i \la y_k-y_l, \w\ra \hat{u}(\lam\w)d\w \big) d\lam .
\eqn
Here $\hat{\m}(x)$ is $\lam$-independent and in the polar coordinates
$\xi = \lam \w$, $\lam>0$ and $\w \in {\mathbb S}^1$,
$d\xi=\lam d\lam d\w $. Thus, by using the Parseval formula, we may express
\bqn \lbeq(N-2)
\refeq(Int-1) = \sum_{l=1}^N  i \hat{\m}(x-y_j)
\la \Fg \tilde \r_{jk}, \la y_k-y_l,R \ra u\ra_{L^2},
\eqn
where $\tilde\rho_{jk}(\xi)= |\xi|^{-1}\chi_{\leq \ep} (|\xi|)
g(|\xi|)^{-1}\tilde{B}_{jk}(|\xi|)$. It is evident that
$\tilde\rho_{jk}\in L^p(\R^2)$ for $1<p \leq 2$ and,
we have $\hat{\mu} \in L^p(\R^2)$ for $1\leq p<\infty$ because
$\hat\m(x) = N_0(x) - (N_0 \ast \widehat{\chi_{\leq 2\ep}})(x)$
implies
\bqn \lbeq(hatm)
|\hat{\mu}(x)| \leq C_N |\log |x|| \ax^{-N}, \quad N=0,1, \dots.
\eqn
Thus H\"older's  and Hausdorff-Young's inequalities imply
for $1<p\leq 2$ and its dual exponent $q=(p-1)/p$ that
 \begin{multline*}
\|\hat{\m}(x-y_j)
\la \Fg \tilde \r_{jk}, \la y_k-y_l,R \ra u\ra\|_p \\
\leq \|\hat{\mu}\|_p \|\Fg \tilde \r_{jk}\|_{L^q}
\|\la y_k-y_l,R \ra u\ra\|_p
\leq C \|\hat{\mu}\|_p \|\tilde \r_{jk}\|_{p}\la u\ra\|_p
\end{multline*}
and \refeq(Int-1) is bounded in $L^p(\R^2)$ for $1\leq p\leq 2$.
\end{proof}

We next show that
$\chi_{\geq 2\ep}(|D|)\W_{\rm low,b}$
is unbounded in $L^p(\R^2)$ for $2<p<\infty$.

\bglm \lblm(Bad-C) Suppose $\ep>0$ is sufficiently small.
Then, $\chi_{\geq 2\ep}(|D|)\W_{\rm low,b}$
is unbounded from $L^p(\R^2)$ to itself for any $2<p<\infty$.
\edlm
\bgpf By virtue of the proof of previous \reflm(Bad-B), it suffices to
show that the sum over $j,k=1, \dots, N$ of \refeq(Int-1)  or \refeq(N-2)
is unbounded in $L^p$ for $p>2$. Introduce the notation:
$\r(\lam) = \chi_{\leq \ep}(\lam) g(\lam)^{-1}|{\lam}|^{-1}$ and
\bqn \lbeq(notat-hat)
\hat{\m}_Y(x) = \begin{pmatrix}\hat{\m}(x-y_1)
\\ \vdots \\ \hat{\m}(x-y_N) \end{pmatrix}, \quad
R_Y(\w) =
\begin{pmatrix}\la y_1, \w \ra \\ \vdots \\ \la y_N, \w \ra \end{pmatrix},
\eqn
Because of the presence of $T_p$ in $\tilde{B}= T_p {B_\ast}(\lam) T_p$
which annihilates $P$, we have again with vector notation and
$\w=\xi/|\xi|$ that
\begin{align}
& \sum_{jkl}\refeq(Int-1) =
\int_0^\infty \r(\lam)
\la (T_p {B_\ast}(\lam)T_p) \hat{\m}_Y(x), \hat{b}(\lam)\ra_{\C^N} d\lam \notag \\
& = \Big\la T_p \hat{\m}_Y(x),
\int_0^\infty \r(\lam)\lam^{-1}
\left(\int_{{\mathbb S}^1} {B_\ast(\lam)} T_p R_Y(\w) \hat{u}(\lam\w)d\w \right)
d\lam \Big\ra
\notag \\
& = \Big\la T_p \hat{\m}_Y(x),
\int_{\R^2} {B_\ast(|\xi|)}
T_p R_Y(\w) \r(|\xi|)\hat{u}(\xi)d\xi \Big\ra , \notag  \\
& = \Big\la T_p \hat{\m}_Y(x),
\int_{\R^2} \Fg(B_\ast(|\xi|)T_p R_Y(\w) \r(|\xi|))(y)u(y) dy \Big\ra
\lbeq(Bad-Ia)
\end{align}
where we used Parseval identity in the last step.
Take an orthonormal basis $\{\eb_1, \dots, \eb_n \}$ of $T_p \C^N$
such that $T_p \Gg_1(Y) T_p \eb_j
= \k_j \eb_j$, $j=1, \dots, n$, $n ={\rm rank\ }T_p$ (we shall see
$\k_j>0$ shortly). Then, \refeq(Bad-Ia) is equal to
\bqn
\sum_{j=1}^n \la \eb_j, \hat{\m}_Y(x) \ra \int_{\R^2}
\Fg (\la \eb_j, B_\ast R_Y(\w) \ra \r(|\xi|))(y)u(y) dy
\lbeq(Bad-I).
\eqn
We note that,  if $y_j\not=y_k$ for $j\not=k$,
any non-trivial linear combination  of
$\hat{\m}(x-y_1), \dots, \hat{\m}(x-y_N)$ does not vanish because that
\[
\Fg(c_1 \hat{\m}(x-y_1)+  \cdots + c_N \hat{\m}(x-y_N))(\xi)
= |\xi|^{-2}\chi_{\geq 2\ep}(\xi) \sum_{j=1}^N c_j e^{iy_j \xi}= 0
\]
implies
$\sum_{j=1}^N c_j e^{iy_j \xi}=0$ for $|\xi|\geq 2\ep$ and hence
$c_1= \dots = c_N =0$. It follows that
$\sum_{j=1}^n c_j \la \eb_j, \hat{\m}_Y(x)\ra =0$
implies $\sum_{j=1}^n c_j \eb_j=0$,
hence $c_1= \dots= c_n=0$, viz.
$\{\la \eb_j, \hat{\m}_Y(x) \ra \colon j=1, \dots, n\}$ is linearly independent.
Then, Hahn-Banach theorem implies for any $j=1, \dots, n$, there exists
$ z_j \in L^q$, $q=p/p-1$ such that
\[
\int_{\R^2}z_j(x) \la \eb_k, \hat{\m}_Y(x) \ra dx=\d_{jk}, \quad k=1, \dots, n.
\]
It follows that if $\sum_{jkl}\refeq(Int-1)$ is bounded in $L^p(\R^2)$ then
\bqn \lbeq(by2)
u \mapsto \int_{\R^2}
\Fg (\la \eb_j, B_\ast R_Y(\w) \ra \r(|\xi|))(y)u(y) dy, \quad j=1, \dots, n
\eqn
must be bounded linear functionals on $L^p(\R^2)$ or, by virtue of the Riesz
representation theorem,
$\Fg (\la \eb_j, B_\ast R_Y(\w) \ra \r(|\xi|)) \in L^q(\R^2)$,
$j=1, \dots, n$,
$q$ being the dual exponent of $p$, hence $1<q<2$. Then,
by virtue of Hausdorff-Young's inequality we must have
$B_\ast(|\xi|) R_Y(\w) \r(|\xi|) \in L^p(\R^2)$,
which is equivalent to
$T_p R_Y(\w)\r(|\xi|)\in L^p(\R^2)$, or
$\la \eb_j, R_Y(\w) \ra \r(|\xi|)\in L^p(\R^2)$, $j=1, \dots,n $
because $\la \eb_j, B_\ast(|\xi|) \eb_k\ra = \k_j \d_{jk} + O(g^{-1})$,
$\k_j>0$, if $\ep>0$ is sufficiently small.
However, if $e^{(j)}_l$ is the $l$-th component of $\eb_j$, we have
\bqn \lbeq(by1)
\k_j = \la \eb_j, T_p \Gg_1(Y) T_p \eb_j\ra = -\frac12 \sum_{k,l=1}^N
|y_k-y_l|^2 e^{(j)}_k e^{(j)}_l = \big(\sum_{l=1}^N e^{(j)}_l y_l \big)^2
\eqn
and for every $j=1, \dots, n$,
\bqn \lbeq(by)
\la \eb_j ,\yb \ra
= \sum_{l=1}^N e^{(j)}_l y_l \not=0, \quad \yb=
\begin{pmatrix}y_1 \\ \vdots \\ y_N \end{pmatrix} .
\eqn
Thus, $\la \eb_j, R_Y(\w) \ra \r(|\xi|)\not\in L^p(\R^2)$, $2<p<\infty$
for any $1\leq j \leq n$ because
\[
\int_{|\xi|<\ep} \frac{|\la \ab, \w \ra|^p}{|\xi|^p|g(|\xi|)|^p}d\xi = \infty
\]
for any $\ep>0$ and any $\ab \in \R^2 \setminus\{0\}$.
This complete the proof.
\edpf

The proof of statement (3) will be finished if we have proven the
following lemma.

\bglm \lblm(2-final) Operator $\chi_{\leq 2\ep}(|D|)\W_{\rm low,b}$
is bounded in $L^p(\R^2)$ if $1<p< 2$.
\edlm
\bgpf The proof uses the fact that
$\tilde B(\lam) = T_p B_\ast(\lam) T_p$ has the factor $T_p$ also on the
right. $\chi_{\leq 2\ep}(|D|)\W_{\rm low,b}(x)$
may be expressed in the form
\begin{align} \lbeq(6-16)
& i \int_0^\infty \int_{{\mathbb S}^1} \chi_{\leq \ep}
(\lam) g(\lam)^{-1}
\big\la \tilde{B} \chi_{\leq 2\ep}(|D|)\hat{\Gg}_{\lam} (x),
\la\yb, \w\ra \big\ra
\hat{u}(\lam\w)
d\lam d\w  \\
& = \sum_{jk}\int_{\R^4}
\frac{\chi_{\leq {2\ep}}(\xi)\chi_{\leq\ep}(|\eta|)\tilde{B}_{jk}(|\eta|)}
{(\xi^2-\eta^2-i0)|\eta|g(|\eta|)}
e^{i(x-y_j)\xi}\la y_k, \hat{\eta}\ra \hat{u}(\eta) d\eta d\xi . \lbeq(6-16a)
\end{align}
Recall that we are assuming $u \in \Dg_\ast$
and $(\xi^2-\eta^2-i0)^{-1}$ in \refeq(6-16a) is well defined as
limit $\k \downarrow  0$ of $\lim (\xi^2-\eta^2-i\k)^{-1}$.
Hereafter in the proof we write
$\r_{jk}(\lam)=\chi_{\leq\ep}(\lam)\tilde{B}_{jk}(\lam)g(\lam)^{-1}$.
$\r_{jk}$ is a good multiplier.

Since $\tilde{B}= T_p B_\ast T_p$ and $T_p$ annihilates
${\bf 1}$  we may replace
$\hat{\Gg}_{\lam}$ in \refeq(6-16)
by $\hat{\Gg}_{\lam}(x)-
{N^{-1}} \big(\sum_{l=1}^N {\Gg}_{\lam}(x-y_l)\big)
{\bf 1}$
without changing the result. However, this changes $e^{-iy_j \xi}$ in
\refeq(6-16a) to
\bqn \lbeq(17-1)
\frac1{N}\sum_{l=1}^N (e^{-iy_j \xi}-e^{-iy_l \xi})
=
\frac{-i}{N}\sum_{l=1}^N
\int_0^1 e^{-i\xi(\th{y_j}+ (1-\th){y_l})}d\th \cdot (y_j-y_l)\xi.
\eqn
Then, \refeq(6-16a) becomes the sum over $j,k,l=1, \dots, N $ of
\bqn
U_{jk} \int_{\R^6}
\chi_{\leq {2\ep}}(\xi)\r_{jk}(|\eta|)
\frac{(y_j-y_l)\cdot \xi}{(\xi^2-\eta^2-i0)|\eta|}
e^{ix \xi}\la y_k, \hat{\eta}\ra \hat{u}(\eta)d\eta d\xi  \lbeq(16-6b)
\eqn
where ${\displaystyle U_{jk}=-iN^{-1} \int_0^1 \t_{\th{y_j}+ (1-\th){y_l}} d\th}$
is obviously a good operator. Since
$(y_j-y_l)\xi=
(y_j-y_l)\hat{\xi}(|\eta|+ (|\xi|-|\eta|))$, $\hat{\xi}= \xi/|\xi|$,
we have
\bqn
\frac{(y_j-y_l)\xi}{\xi^2-\eta^2-i0}
= (y_j-y_l)\,\hat{\xi}\cdot
\frac{|\eta|}{\xi^2-\eta^2-i0}+ (y_j-y_l)\hat{\xi} \cdot
\frac{1}{|\xi|+ |\eta|},  \lbeq(16-6c)
\eqn
which we use in \refeq(16-6b). Then the first term yields
\bqn  \lbeq(16-7)
\la y_j-y_l, R_x \ra
\iint_{\R^4} \frac{e^{ix \xi}\chi_{\leq {2\ep}}(\xi)}{(\xi^2-\eta^2-i0)}
\rho_{jk}(|\eta|)\la y_k, \hat{\eta}\ra \hat{u}(\eta) d\eta d\xi
\eqn
If we integrating with respect $\xi$ first and use the polar coordinate
$\eta= \lam \w$, (16-7) becomes
\begin{multline}
\la y_j-y_l, R_x \ra \chi_{\leq {2\ep}}(|D|)
\int_{0}^\infty \lam \Gg_{\lam}(x) \left(\int_{\mathbb S^1}
\Fg(\rho_{jk}(|D|)\la y_k, R\ra u )(\lam\w)d\w \right) d\lam \\
= 2\pi \la y_j-y_l, R_x \ra \chi_{\leq {2\ep}}(|D|)
K \circ (\rho_{jk}(|D|)\la y_k, R_y \ra) u(x).
\end{multline}
This is a good operator since $\r_{jk}$ is a good multiplier.
The operator produced by the second term may be expressed as follows:
\[
\la y_j-y_l, R_x \ra \int_{\R^2}
\left(\iint_{\R^4}
e^{ix\xi-iy\eta}\frac
{\chi_{\leq {2\ep}}(\xi)\chi_{\leq\ep}(|\eta|)}{(|\xi|+ |\eta|)|\eta|g(|\eta|)}d\eta d\xi\right)
(\tilde{\rho}_{jk}(D)u)(y) dy
\]
where $\tilde{\rho}_{jk}(|D|)= \tilde{B}_{jk}(|D|)\la y_k, R\ra$ is
a good operator.
Thus, the proof of \reflm(2-final) will be completed if we have
proven the following lemma.

\bglm \lblm(L-integral kernel)
Let $L(x,y)$ be defined by
\bqn \lbeq(L=def)
L(x,y)= \iint_{\R^4}
e^{ix\xi-iy\eta}\frac
{\chi_{\leq {2\ep}}(\xi)\chi_{\leq\ep}(|\eta|)}{(|\xi|+ |\eta|)|\eta|g(|\eta|)}d\eta d\xi.
\eqn
Then, the integral operator
\bqn \lbeq(L-intop)
Lu(x)  = \int_{\R^2}L(x, y) u(y) dy
\eqn
is bounded in $L^p(\R^2)$ for $1<p<2$.
\edlm
\bgpf The Fourier transform of
$\chi_{\leq {2\ep}}(\xi)\chi_{\leq\ep}(|\eta|)(|\xi|+ |\eta|)^{-1}$
is smooth and in virtue of Lemma B of appendix
bounded by $C\ax^{-1}\ay^{-1}(\ax+\ay)^{-1}$.
It follows by virtue of \reflm(plogp) that
\[
L(x,y) \absleq C \int_{\R^2}\frac{C}
{\ax\la {y-y'}\ra (\ax+ \la {y-y'}\ra)\la y'\ra \log (1+ \la y'\ra)}dy'
\]
Then, by using Minkowski's inequality twice we have for $1<p<2$ that
\begin{align}
& \Big\| \int_{\R^2} L(x,y) |f(y)|dy \Big\|_p
\leq \int_{\R^2}
\left(\int_{\R^2}|L(x,y)|^p dx \right)^{1/p} |f(y)|dy \notag \\
& \leq C\iint_{\R^4}
\left(\int_{\R^2}
\frac{dx}{\ax^{p}(\ax+ \la {y-y'}\ra)^{p}}\right)^{1/p}
\frac{|f(y)|dy'dy}{\la y'\ra \log (1+ \la y'\ra)\la y-y'\ra} \notag \\
& \leq C \int_{\R^2}
\left(\int_{\R^2}
\frac{dy'}
{\la y-y'\ra^{3-2/p}
\la y'\ra \log (1+ \la y'\ra)} \right)
|f(y)|dy   \lbeq(MF)
\end{align}
We show for the dual exponent $q=p/(p-1)$ of $1< p< 2$,
\bqn \lbeq(MLQ)
Q(y) \stackrel{\rm def}{=}
\int_{\R^2}
\frac{dy'}
{\la y-y'\ra^{3-2/p}
\la y'\ra \log (1+ \la y'\ra)} \in L^q(\R^2),
\eqn
which will prove that the right of \refeq(MF) is bounded by
$\|Q\|_q \|f\|_p$ by H\"older's inequality and which will complete the proof.
Since $p>1$, $3-2/p>1$, Schwarz's inequality implies
\[
|Q(y)|\leq \|\la y\ra^{3-2/p}\|_2
\|(\la y'\ra \log (1+ \la y'\ra))^{-1}\|_2 < \infty
\]
and $Q(y)\in L^\infty(\R^2)$. Thus, we need estimate $Q(y)$
for large $|y|>100$ only, which we assume in what follows.
We split $\R^2$ into three regions
$D_1=\{y' \colon |y'-y|\leq |y|/2\}$,
$D_2=\{y' \colon  |y-y'|>|y|/2 \ \mbox{and} \  |y'|\leq 2|y|\}$ and
$D_3=\{y'\colon |y-y'|>|y|/2 \ \mbox{and} \ |y'|>2|y|\}$
so that
\[
Q(y)=
\left(\int_{D_1} +
\int_{D_2} +
\int_{D_3} \right)
\frac{\la y-y'\ra^{2/p-3} dy'} {\la y'\ra \log (1+ \la y'\ra)}
\stackrel{\rm def}{=} A(y) + B(y) + C(y).
\]
On $D_1$ we have $|y|/2<|y'|<2|y|$ and $-1<2/p-2<0$ if
$1<p< 2$. Hence
\begin{align*}
A(y) & \leq \frac{C}{\la y\ra \log (1+ \la y\ra)}\int_0^{|y|/2}
\la r \ra^{2/p-3} r dr
\leq \frac{C\la y\ra^{2/p-2}}{\log (1+ \la y \ra)}\in L^q(\R^2)
\end{align*}
since $q(2/p-2)=-2$ and $q>2$. For $B(y)$ we have
\begin{align*}
B(y)\leq \frac{C}{\la y \ra^{3-\frac2{p}}} \int_0^{2|y|}\frac{rdr}
{\la r\ra \log (2+ r)}
\leq \frac{C}{\la y\ra^{3-\frac2{p}}}
\left(C + \int_{e^2}^{2|y|} \frac{dr}{\log r}\right)
\end{align*}
Here integration by parts shows
\[
\int_{e^2}^{2|y|} \frac{dr}{\log r}
= \left. \frac{r}{\log r}\right]^{2|y|}_{e^{2}}
+ \int_{e^2}^{2|y|} \frac{dr}{(\log r)^2}
\leq \frac{2|y|}{\log 2|y|} + \frac12
\int_{e^2}^{2|y|} \frac{dr}{\log r}
\]
and the integral is bounded by $4|y|(\log 2|y|)^{-1}$. It follows once
more that
\[
B(y) \leq \frac{C}{\la y\ra^{2-\frac2{p}}\log (1+ \la y \ra)} \in L^q(\R^2)
\]
Finally as $|y'|>2|y|$ implies $|y-y'|>|y'|/2$ and $3-\frac2{p}>1$
\[
C(y) \leq \int_{2|y|}^\infty
\frac{rdr} {
\la r\ra^{4-\frac{2}{p}}
\log (1+ \la r \ra)}
\leq \frac{C}{\la y \ra^{2-\frac{2}{p}}\log (1+ \la y\ra)}
\in L^q(\R^2).
\]
Thus, $Q \in L^q(\R^2)$ as desired and the lemma is proved.
\edpf

\subsection{Proof of statement (4)}
We use the notation of \reflm(Step-4).
By virtue of \reflm(Step-4) $\Ga(\lam)^{-1}$
under the assumption of statement (4) satisfies
\begin{multline}  \lbeq(E-1)
- N^{-1}\lam^{-2} T_e  (T_eG T_e)^{-1} T_e
-N^{-1}g^{-1}\lam^{-2}T_p (T_e^\perp - \Bg^\ast)T_e^\perp D_0
T_e^\perp (T_e^\perp - \Bg)T_p \\
+ g^3 T_p M T_p + g^2 S M S +
g S M + g M S + M.
\end{multline}
We substitute \refeq(E-1) for $\tGa(\lam)$ of \refeq(low-W),
which produces seven operators. The one produced by $M(\lam)$
is a good operator by the proof of statement (1); those
produced by $g S M$, $g^2 S M S$ and $g^3 T_p M T_p$ which have
the factor $S$ {\it on the left} are also good operators.
This can be seen by repeating the proof statement (2)
by observing that (i) out of two T's in $TBT$ of \refeq(low-Wa)
the one {\it on the left} is used for introducing $S$
in front of $\hat{I}(\lam)$, which produces \refeq(UseR) with the
extra factor $\lam$ and that (ii)
$\lam g(\lam)^j \chi_{\leq \ep}(\lam) M(\lam)$
is a good multiplier for any $j\in {\mathbb N}$ and it can play the role
played by $m(\lam)$ of \refeq(m-def).

For the operator produced by $ g M(\lam) S$ we have the following lemma.
\bglm \lblm(rs)
The operator $\W_{rs}$ defined by \refeq(low-Wa) with
$ g(\lam)M(\lam) S $ in place of $\tGa(\lam)$ is a good operator.
\edlm
\bgpf Denote by $M(\lam)^\ast$ the conjugate of $M(\lam)$. It suffices
to prove the lemma when $M(\lam)= 1$ since
\begin{align}
& \W_{rs}u(x)=
\frac1{\pi{i}}\int_0^\infty \chi_{\leq\ep} g \lam \big\la
M(\lam)S\hat{\Gg}_{\lam}(x),
\int_{\R^2}(\hat{\Gg}_{\lam}(y)-\hat{\Gg}_{-\lam}(y))u(y) dy \big\ra_{\C^N}
d\lam  \notag  \\
& =
\frac1{\pi{i}}\int_0^\infty \chi_{\leq\ep} g \lam \big\la
S\hat{\Gg}_{\lam}(x),
\int_{\R^2}(\hat{\Gg}_{\lam}(y)-\hat{\Gg}_{-\lam}(y))M^\ast(|D|)u(y)
dy \big\ra_{\C^N}
d\lam
\lbeq(rs)
\end{align}
and $M^\ast(|D|)$ is a good operator. We split $\W_{rs}u$ as follows:
\[
\W_{rs}u = \chi_{\geq 2\ep}(D_x)\W_{rs}u+ \chi_{\leq 2\ep}(D_x)\W_{rs}u.
\]
(1) We first prove that $\chi_{\geq 2\ep}(D_x)\W_{rs}$ is a good operator.
For this, it suffices to show the same for the operator $Zu(x)$ defined by
\bqn \lbeq(Wrs)
Zu(x)= \int_0^\infty \chi_{\leq \ep} \lam g
\chi_{\geq 2\ep}(|D|){\Gg}_{\lam}(x)
\left(\int_{\mathbb S}\hat{u}(\lam\w)d\w \right) d\lam
\eqn
is a good operato. We repeat the argument of the proof of \reflm(Bad-B).
Substitute \refeq(chigeq) for $\chi_{\geq 2\ep}(|D|){\Gg}_{\lam}(x)$ in
\refeq(Wrs),
which produces two integrals.
The one produced by $\hat{\m}(x)$ is equal to
\bqn \lbeq(Wrs-1)
\hat{\mu}(x)
\int_0^\infty \chi_{\leq \ep}(\lam) \lam g(\lam)
\left(\int_{\mathbb S}(\Fg u)(\lam\w)d\w \right) d\lam
= \hat{\mu}(x) \la \Fg(\chi_{\ep} {g}), {u}\ra.
\eqn
As previously $\hat{\mu} \in L^p(\R^2)$ for all $1\leq p <\infty$ and
\bqn \lbeq(Fchig)
|\Fg(\chi_{\ep}(|\xi|)g(|\xi|))(x)|
= |(2\pi)^{-1}(\hat{g}\ast \hat{\chi})(x)| \leq C \ax^{-2}.
\eqn
Hence, \refeq(Wrs-1) is a good operator.
Define $\tilde{\m}(\lam)= \lam^2 \chi_{\leq \ep}(\lam)$. Then
$\tilde{\m}$ is a good multiplier and
the operator produced by $\m(|D|) \lam^2 {\Gg}_{\lam}(x)$ of
\refeq(chigeq) is equal to
\[
{\m(|D|)}\int_0^\infty \chi_{\leq\ep}(\lam) \lam {\Gg}_{\lam}(x)
\left(
\int_{\mathbb S}\Fg(\tilde{\m}(|D|)u)(\lam\w) d\w \right) d\lam
=\m(|D|) K \tilde{\m}(|D|)u,
\]
which is a good operator.
Thus, $\chi_{\geq 2\ep}(D_x)\W_{rs}$ is a good operator. \\[5pt]
(2) We next show $\chi_{\leq 2\ep}(D_x)\W_{rs}$ is also a good operator.
The proof below resembles the one of \reflm(2-final).
Recall $\t_Y u(x)$ of \refeq(notat-hat). We have
\begin{align}\lbeq(Wrs-2)
& \chi_{\leq 2\ep}(D_x)\W_{rs}u(x) \\
& =\int_0^\infty \chi_{\leq \ep}(\lam) \lam g(\lam)
\Big\la \chi_{\leq 2\ep}(|D|)S \hat{\Gg}_{\lam}(x)
\int_{\mathbb S}\Fg(M(|D|)\tau_Y u)(\lam\w)d\w \Big\ra_{\C^N}
d\lam.  \notag
\end{align}
Here $S \chi_{\leq 2\ep}(|D|)\hat{\Gg}_{\lam}(x)$ is a vector
whose $j$-th component is given by
\bqn \lbeq(Kj-def)
K_j(x)= \frac1{N}
\sum_{k=1}^N \frac1{2\pi}\int_{\R^2}
\frac{\chi_{\leq 2\ep}(\xi)e^{ix\xi}(e^{-iy_j\xi}-e^{-iy_k \xi})}
{\xi^2 -\lam^2 -i0} d\xi .
\eqn
Using Taylor's formula, write
$e^{-iy_j \xi}-e^{-iy_k\xi}$ in the form
\[
 -i(y_j-y_k)\xi + |\xi|^2
\iint_{[0,1]^2} e^{-i\m(\th y_j + (1-\th)y_k)}
(y_j-y_k, \hat{\xi})(\th{y_j}+ (1-\th)y_k, \hat{\xi})
d\th{d\m}
\]
and define two functions $L_{\lam}^{(1)}(x)$ and $L_{\lam}^{(2)}(x)$ by
\[
L_{\lam}^{(1)}(x) = \frac1{2\pi}\int_{\R^2}
\frac{\chi_{\leq 2\ep}(\xi)e^{ix\xi}|\xi|}{\xi^2 -\lam^2 -i0} d\xi, \quad
L_{\lam}^{(2)}(x) = \frac1{2\pi}\int_{\R^2}
\frac{\chi_{\leq 2\ep}(\xi)e^{ix\xi}|\xi|^2}{\xi^2 -\lam^2 -i0} d\xi .
\]
Then, $K_j(x)$ may be expressed as a sum
\bqn
K_j(x) = \frac{1}{N}
\sum_{k=1}^N ( -i \la y_j-y_k, R_x\ra L_{\lam}^{(1)}(x)+
U_{jk} L_{\lam}^{(2)}(x)).
\lbeq(Kj1)
\eqn
Here $R_x= (R_{x_1}, R_{x_2})$ is the Riesz transform
and for $j,k=1, \dots, N$
\bqn
\tilde{U}_{kj}=  \int_0^1 \int_0^1 \tau_{\m(\th y_k + (1-\th)y_j)}
(y_k-y_j, R_x)(\th{y_k}+ (1-\th)y_j, R_x) d\m {d\th}.
\eqn
Note that $\la y_j-y_k, R_x\ra$, $U_{kj}$, $M(|D|)$ and
$\tau_Y$ are all good operators. We shall prove that $Q_1$ and $Q_2$ defined by
\bqn
Q_j u(x) = \int_0^\infty \chi_{\leq\ep}(\lam) \lam g(\lam)
L_\lam^{(j)} (x) \left(\int_{\mathbb S}\hat{u}(\lam\w)d\w \right) d\lam,
\quad j=1,2
\eqn
are good operators, which will finish the proof of the lemma.

As in \refeq(16-6c) we have the identity:
\[
L_{\lam}^{(1)}(x) = \lam \chi_{\leq 2\ep}(D)\Gg_{\lam}(x)
+ \frac1{2\pi}\int_{\R^2}
\frac{\chi_{\leq 2\ep}(\xi)e^{ix\xi}}{|\xi|+\lam} d\xi.
\]
Define $\n(\lam) = \lam \chi_{\leq \ep}(\lam)g(\lam)$. Then
$\n(\lam)$ is a good multiplier and  we have
\[
Q_1u(x) = \chi_{\leq 2\ep}(D) (K\circ \n(|D|))u(x) + \tilde{L}u(x),
\]
where $\tilde{L}$ is the integral operator with the integral kernel
\[
\tilde{L}(x,y) =
\iint_{\R^4}
e^{ix\xi-iy\eta}\frac
{\chi_{\leq {2\ep}}(\xi)\chi_{\leq\ep}(|\eta|)g(|\eta|)}{4\pi^2 (|\xi|+ |\eta|)}d\eta d\xi.
\]
It is evident that $\chi_{\leq 2\ep}(D) K\circ (\n(|D|)u(x)$ is a good operator
and the proof of \reflm(L-integral kernel) implies $\tilde{L}$ is a also
good operator. Indeed, by using Lemma B in appendix and \refeq(Fchig) we obtain
\[
|\tilde{L}(x,y)|\leq C \int_{\R^2}
\frac{dy'}{\ax \la y-y'\ra (\ax+ \la y-y'\ra) \la y'\ra^{2}}
\]
and, the argument which led to \refeq(MLQ) implies
\[
\|\tilde{L}f\|_p \leq
C \int_{\R^2}
\left(\int_{\R^2}
\frac{dy'}
{\la y-y'\ra^{3-2/p}\la y'\ra^2} \right)
|f(y)|dy.
\]
Then, Young's inequality implies that the function
in the parentheses is in $L^q(\R^2)$ for all $1<q<\infty$.
Thus, H\"older's inequality implies $\tilde{L}$ and, hence
that $Q_1$ is a good operator.

We have
\[
L_\lam^{(2)} (x) = \widehat{\chi_{\leq 2\ep}}(x)+ \lam^2 \Gg_{\lam}(x).
\]
Define
$\m(\xi)= \chi_{\leq 2\ep}(|\xi|)g(|\xi|)$ and
$\tilde\m(\xi)= |\xi|^2 \m(\xi)$. Then $Q_2 u(x)$ is expressed in the form
\[
Q_2 u(x) = \widehat{\chi_{\leq 2\ep}}(x)
\left(\int_{\R^2}\la \m(|\xi|)\hat{u}(\xi) d\xi\right)
+ K\circ \tilde{\mu}(|D|)u,
\]
$\hat{\mu}\in L^p(\R^2)$ for any $1<p<\infty$
and $\tilde{\mu}$ is a good multiplier.
Thus, $Q_2$ is also a good operator and the proof of \reflm(rs)
is completed.
\edpf

\bglm \lblm(We)
Let $\W_{e}$ be defined by \refeq(low-W) with
$- N^{-1}\lam^{-2} T_e(T_eG_2 T_e)^{-1} T_e $
in place of $\tGa(\lam)$. Then, $\W_e$ is a good operator.
\edlm
\bgpf Write
$H(\lam) = (T_eG_2 T_e)^{-1}$ for shortening formulas .
Let $\{\eb_1, \dots, \eb_n\}$ be an orthonormal basis of $T_e \C^N$,
$n = {\rm rank}\, T_e$ and $H_{lm}= \la \eb_l, H \eb_m\ra$,
$l,m =1, \dots, n$. Then, $W_{e} u(x)$ is equal to
\begin{align}
& \frac{-1}{N}\int_0^\infty \chi_{\leq\ep}(\lam) \lam^{-1}\big\la
H(\lam)T_e \hat{\Gg}_{\lam}(x), \int_{{\mathbb S}^1} T_e \widehat
{\t_Y u}(\lam\w) d\w \big\ra_{\C^N} d\lam  \notag \\
& =
\sum_{lm}\frac{-1}{N}
\int_0^\infty \chi_{\leq\ep} \lam^{-1}H_{lm}
\la \eb_m, \hat{\Gg}_{\lam}(x)\ra
\left(\int_{{\mathbb S}^1}
\la \eb_l, \widehat
{\t_Y u}(\lam\w) \ra_{\C^N}d\w\right) d\lam
\lbeq(We)
\end{align}
Since $S\eb_l= \eb_l$, we may replace in \refeq(We)
$\widehat{\t_Y u}(\lam\w)$ by
$S\widehat{\t_Y u}(\lam\w)$ whose $j$-th component
is given by ${N}^{-1}\sum_{k=1}^N (e^{iy_j {\lam\w}}-e^{iy_k {\lam\w}}) \hat{u}({\lam\w})$.
As previously we have
\begin{gather*}
e^{iy_j {\lam\w}}-e^{iy_k {\lam\w}}=
-i(y_j-y_k){\lam\w} + \lam^2 v_{jk}({{\lam\w}}), \\
v_{jk}({{\lam\w}})= \iint_{[0,1]^2} e^{-i\m(\th y_j + (1-\th)y_k){\lam\w}}
(y_j-y_k, \w)(\th{y_j}+ (1-\th)y_k, \w)
d\th{d\m}.
\end{gather*}
Since $\eb_l\in T_e \C^N$, we have $\sum_{j=1}^N \eb_l^{(j)}=0$ and
\[
\la \Gg_1(Y)\eb_l, \eb_l \ra= -\frac{1}{4N}\sum_{j,k=1}^N
|y_j-y_k|^2 \eb_l^{(j)}\eb_j^{(k)}
= \frac{1}{2N}\left(\sum_{j=1}^N \eb_l^{(j)}y_j \right)^2=0.
\]
It follows that $\sum_{k,j=1}^N \eb_l^{(j)}(y_j-y_k){\lam\w} =0$
and
\bqn
\la \eb_l, \widehat{\t_Y u}(\lam\w) \ra=
\lam^2 \sum_{k,j=1}^N \eb_l^{(j)} v_{jk}(\lam\w) \hat{u}(\lam\w)
= \lam^2 \Fg(\Vg_l u)(\lam\w)
\eqn
where $\Vg_l $ is a good operator defined by
\[
\Vg_l u (x) = \sum_{j,k=1}^N \eb_l^{(j)}
\iint_{[0,1]^2} \t_{\m(\th y_j + (1-\th)y_k)}
(y_j-y_k, R)(\th{y_j}+ (1-\th)y_k, R) u(x).
\]
Thus, $W_{e} u(x)$ is equal to
\begin{align*}
& \frac{-1}{N}\sum_{lm n}
\int_0^\infty \chi_{\leq\ep} \lam  H_{lm}(\lam) \eb_m^{(n)}\Gg_{\lam}(x-y_n)
\left(\int_{{\mathbb S}^1} \Fg(\Vg_l u)(\lam\w)
d\w \right) d\lam \\
&\hspace{1cm} = \frac{-1}{N} \sum_{lm n} \eb_m^{(n)} \t_{y_n}
K \circ (\chi_{\leq\ep}(|D|)H_{lm}(|D|) \Vg_l u)(x) ,
\end{align*}
which is a good operator.
\edpf

Next lemma completes the proof of statement (4) of \refth(Main).
For shortening the formula we write
\[
B_{\ast\ast}= -\frac{1}{iN\pi}
(T_e^\perp - \Bg^\ast)T_e^\perp D_0 T_e^\perp (T_e^\perp - \Bg).
\]
\bglm \lblm(Wu)
Let $\W_{u}$ be defined by \refeq(low-W) with
$g^{-1}\lam^{-2}T_p B_{\ast\ast}T_p$
in place of $\tGa(\lam)$. Then, $\W_{u}$ is bounded
in $L^p(\R^2)$ for $1<p \leq 2$ but is unbounded
for $2<p<\infty$.
\edlm
\bgpf By the definition $\W_{u}u(x)$ is equal to
\bqn
\int_0^\infty \chi_{\leq \ep}(\lam) g^{-1}\lam^{-1} \big\la
T_p B_{**}T_p \hat{\Gg}_{\lam}(x),
\int_{\R^2}(\hat{\Gg}_{\lam}(y)-\hat{\Gg}_{-\lam}(y))u(y) dy \big\ra_{\C^N}
d\lam
\lbeq(eqWu)
\eqn
Then, we repeat the argument of Section 4.3 for $\W_{\rm  low}$ of
\refeq(low-W-vec) replacing $B_{*}$ by $-i{\pi}N B_{**}$ everywhere.
Then, if we decompose $W_{u}$ into the sum $W_{u,g}+ W_{u,b}$ as in
\refeq(gb-decomp) by using the identity \refeq(good-bad), then: \\[5pt]
(i) Proofs of \reflm(good) and \reflm(Bad-B) respectively imply
without any more changes that $W_{u,g}$ is a good operator and that
$\chi_{\geq 2\ep}(|D|)W_{u, b}$ is bounded in
$L^p(\R^2)$ for $1<p<2$. \\[5pt]
(ii) The proof of \reflm(2-final) implies that
$\chi_{\leq 2\ep}(|D|)W_{u, b}$ is bounded in $L^p(\R^2)$ for $1<p<2$. \\[5pt]
(i) and (ii) should be obvious because the only property
of $B_\ast$ used in the proof of these lemmas is that
$T_p B_\ast(\lam)T_p$ is a good multiplier which is shared by
$T_p B_{**}(\lam)T_p$. \\[5pt]
(iii) We prove that $\chi_{\geq 2\ep}(|D|)W_{u,b}$ is unbounded
$L^p(\R^2)$ for $2<p<\infty$ by modifying the argument of
the proof of \reflm(Bad-C) slightly as follows:

Let $l={\rm rank}\, D_0 $ and take an orthonormal basis
$\{\eb_1, \dots, \eb_n\}$ of $T_p\C^N$
such that $\eb_1, \dots, \eb_l$ are eigenvectors of $T_e^\perp G_1 T_e^\perp$
with positive eigenvalues
and $\eb_{l+1}, \dots, \eb_n \in T_e\C^N $.
Then, equation \refeq(Bad-I) is satisfied
and the argument after \refeq(Bad-I) implies that,
for $\chi_{\leq 2\ep}(|D|)W_{u,b}$ to be bounded $L^p(\R^2)$ for $2<p<\infty$,
it must be satisfied that $B_{**}(|\xi|)R_Y(\w)\r(|\xi|)\in L^p(\R^2)$.
Since $B_{**}(|\xi|)$ has a bounded inverse
in $T_e^\perp \C^N$ for small $|\xi|$ such that $\rho(|\xi|)\not=0$,
it must be then that
$(T_p \ominus T_e) R_Y(\w)\r(|\xi|)\in L^p(\R^2, T_p\C^N)$ or
$\la \eb_j, R_Y(\w)\r(|\xi|)\ra \in L^p(\R^2)$, $j=1, \dots, l$.
However, we have shown in the last part of the proof of \reflm(Bad-C)
that this is impossible for $j=1, \dots, l$.
\edpf \edpf

\subsection{Proof of statement (5) of \refthb(Main)}
Under the condition of statement (5), $T_p=T_e$ and $\Ga(\lam)^{-1}$
satisfies \refeq(5-1), which we substitute in \refeq(low-W).
Since $T_p S = ST_p= T_p$, the argument of section 4.4 implies that
the operator produced by
$g^2 (T_p M(\lam) S + S M(\lam)T_p)+
g (T_p M(\lam)+ g M(\lam)T_p) + M(\lam)$ is a good operator.
An easy modification of the argument in section 4.2 implies
that $g^3 T_p M(\lam) T_p$ produces a good operator.
The argument of the proof of \reflm(We) appliesto show that
$-N^{-1}\lam^{-2}T_e G_2^{-1} T_e$
also produces a good operator. We skip the repetitive details.
This proves statement (5) and completes the proof of \refth(Main).

\subsection*{Appendix }

In this appendix we show the following lemma: \\[7pt]
{\bf Lemma A.} {\it For any $\ep>0$ there exits a constant $C_\ep>0$ such that }
\bqn
\lbeq(Appen)
\int_{\R^4}e^{ix\xi-ip y}
\frac{\chi_{\leq {2\ep}}(\xi)\chi_{\leq\ep}(|p|)}{|\xi|+ |p|} d\xi{dp}
\leq \frac{C_\ep}{(\ax+ \ay)^3}\log\left(\frac{(\ax+ \ay)^2}{\ax\ay}\right).
\eqn
\bgpf If we use the identity
\[
\frac1{|\xi|+|p|} = \int_0^\infty e^{-t(|\xi|+ |p|)} dt
\]
and Fubini's theorem, then the left side of \refeq(Appen) becomes
\bqn \lbeq(Appen-2)
\int_0^\infty
\left(\frac1{2\pi}
\int_{\R^2}e^{ix\xi-t|\xi|} \chi_{\leq {2\ep}}(\xi)d\xi\right)
\left(\frac1{2\pi}
\int_{\R^2}e^{-ip y-t|p|} \chi_{\leq\ep}(|p|){dp} \right)dt.
\eqn
The functions inside parentheses
are convolutions of the Poisson kernel with bump functions
$\Fg \chi_{\leq 2{\ep}}(x)$ and $\Fg \chi_{\leq {\ep}}(y)$
respectively
(see \cite{Stein}, p. 61). They are bounded by
$C_1 t(\ax^2+ t^2)^{-3/2}$ and $C_2 t(\ay^2+ t^2)^{-3/2}$ respectively.
It follows by changing variable $t$ to $\ax^{1/2}\ay^{1/2} t $ that
\begin{align} \lbeq(A-3)
\refeq(Appen-2) & \leq
C \int_0^\infty \frac{t^2 dt}{(\ax^2+ t^2)^{3/2}(\ay^2+ t^2)^{3/2}} \notag \\
& = \frac{C}{\ax^{3/2}\ay^{3/2}}
\int_0^\infty \frac{t^2dt}{(t^4 + s^2 t^2 +1)^{3/2}}, \quad
s^2= {\frac{\ax^2+\ay^2}{\ax\ay}}.
\end{align}
We estimate the integral in
the right hand side of  \refeq(A-3)
by slitting $(0,\infty)$ into intervals into
$(0,1/s)$, $(1/s,s)$ and $(s,\infty)$ where the denominator is bounded
from below by $1$, $s^3 t^3$ and $t^6$ respectively. Then
\[
\int_0^\infty \frac{t^2dt}{(t^4 + s^2 t^2 +1)^{3/2} }
\leq \int_0^{1/s}{t^2dt}+
\int_{1/s}^s \frac{dt}{s^3 t} +
\int_{s}^\infty \frac{dt}{t^4}
= \frac{2}{3s^3}(1+3 \log{s}) .
\]
Since $s\geq \sqrt{2}$, the right side may be further estimated
by $C s^{-3}\log{s^2}$ and $s^2\leq (\ax+\ay)^2/\ax \ay$.
Combining this with \refeq(A-3), we obtain the lemma.
\edpf

For applications in the text we need only the following weaker version
which trivially follows from Lemma A. \\[7pt]
{\bf Lemma B.} {\it For any $\ep>0$ there exits a constant $C_\ep>0$ such that }
\bqn \lbeq(Appen-1)
\frac1{(2\pi)^2} \int_{\R^4}e^{ix\xi-ip y}
\frac{\chi_{\leq {2\ep}}(\xi)\chi_{\leq\ep}(|p|)}{|\xi|+ |p|} d\xi{dp}
\leq \frac{C_\ep}{\ax\ay(\ax+ \ay)}.
\eqn


text:
import control.traversable.instances
import control.traversable.lemmas

subset_name: curated


TITLE: Find all rational numbers $r$ for which $\log_{2}(r)$ is rational as well.
QUESTION [1 upvotes]: Find all rational numbers $r$ for which $\log_{2}(r)$ is rational as well.
I found this problem in one of the Serbian competition books marked with a star. I looked at the solution, but it's confusing. Could someone help me by providing a detailed solution and how to approach problems similar to this one?

REPLY [3 votes]: This is the same thing as asking: for which rational numbers $r$ is it true that $2^r$ is also rational? The answer is: when $r\in\mathbb Z$. In fact, if $r\in\mathbb{Q}\setminus\mathbb Z$, $r=\frac ab$ with $b>1$ and $\gcd(a,b)=1$. The number $2^{\frac ab}$ is a root of the polynomial $x^b-2^a$. By the rational root theorem, all rational roots of this polynomial are integer numbers, but $2^{\frac ab}$ is not an integer.

REPLY [1 votes]: Assume $\log_2(r)=\frac{p}{q}$, with $q\in\mathbb{Z}^+,p\in\mathbb{Z}$ and $\gcd(p,q)=1$. We have
$$ 2^\frac{p}{q} = r \in\mathbb{Q} \tag{A}$$
$$ 2^{k+\frac{p}{q}} = \frac{a}{b},\qquad a,b\in2\mathbb{Z}+1,\gcd(a,b)=1\tag{B}$$
$$ b^q\cdot 2^{qk+p} = a^q \tag{C}$$
hence $qk+p=0$, since otherwise the LHS and RHS of $(C)$ would have a different value of $\nu_2$, where $\nu_2(m)=\max\{n\in\mathbb{N}: 2^n\mid m\}$. This implies
$$ r\in\mathbb{Q},\;\log_2(r)\in\mathbb{Q}\quad\Longrightarrow\quad \log_2(r)\in\mathbb{Z},\;r\in 2^{\mathbb{Z}}.\tag{D}$$


text:
\section{Stable DSC processes}\label{S:sec3}

In the following, DSC schemes are represented as paired $\,\alpha$-passive
processes such as dealt with in Theorem ~\ref{T.2.1}, and which hence are
stable.

\begin{lproposition}\label{P.3.1}
For every $\,r,s\in J\,$ and $\,T_{q}:f(t)\,\mapsto\,f(t+q)\,$ denoting the
\emph{shift operator} on $\,\mathcal{L}^{I\;\cup\,J}\,$,
\vspace{-8pt}
\begin{equation}\centering\notag
\text{if}\quad r+s\;\leqslant\;0\quad\text{and}\quad
\begin{cases}
F_{_{R}}\\[-2pt]
F_{_{C}}
\end{cases}
\!\!\!\!\text{is $\,\alpha$-passive on}\quad
\begin{cases}
\mathcal{L}^{I}\\[-2pt]
\mathcal{L}^{J}
\end{cases}
\!,
\end{equation}
\vspace{-5pt}
\begin{equation}\centering\notag
\text{then}\quad
\begin{cases}
\,T_{r}\circ F_{_{R}}\circ T_{s}\\[-2pt]
\,T_{r}\circ F_{_{C}}\circ T_{s}
\end{cases}
\!\!\!\!\text{is $\,\alpha$-passive on}\quad
\begin{cases}
\mathcal{L}^{J}\\[-2pt]
\mathcal{L}^{I}
\end{cases}
\!.
\end{equation}
\end{lproposition}

\hspace{-.55cm}
The statement is a direct consequence of Definition ~\ref{D.2.3}
\hfill $\Box$

\begin{ldefinition}\label{D.3.1}
\begin{equation}\centering\notag
\text{The}\quad
\begin{cases}
\text{reflection map $\mathcal{R}$}\\[-2pt]
\text{connection map $\mathcal{C}$}
\end{cases}
\text{is called $\,\alpha$-passive,\;iff}\quad
\begin{cases}
F_{_{R}}\\[-2pt]
F_{_{C}}
\end{cases}
\text{is $\,\alpha$-passive}
\end{equation}
in the sense of Definition ~\ref{D.2.3}.
\end{ldefinition}

\begin{ltheorem}\label{T.3.1}
With every time limited excitation, the DSC process generated by
$\,\alpha$-passive reflection and connection maps is uniformly bounded,
hence stable.
\end{ltheorem}

\begin{proof}
It is sufficient to show that every finitely excited DSC process
which is generated by $\,\alpha$-passive $\mathcal{R}\,$ and $\mathcal{C}\,$
can be written as a pair of processes, either of which satisfies
Theorem ~\ref{T.2.1}.

With $\,H\,:\,=\,I\cup J\,=\,{\{k \tau/2\,|\,k\in\mathbb{Z}\}}\,$
and the measure on $\,H\,$ inherited from $\,I\,$ and $\,J\,$ jointly,
the space of all DSC processes is
\begin{equation}\centering\notag
\begin{aligned}
\mathcal{E}\;&:\,=\;\{\;z=(\,z_{in},\,z_{out}\,)
\in (\mathcal{L}^{2})^{\,H}\quad |
\quad z_{in}((2k+1)\,\tau/2\,)\,=\,z_{in}(\,k\tau\,)\\
\quad &\text{and}\quad
z_{out}(\,k\tau\,)\,=\,z_{out}((2k-1)\,\tau/2\,)\;,\quad
\text{for every $\,k\in\mathbb{Z}$}\;\}\;,
\end{aligned}
\end{equation}
i.e. $\mathcal{E}$ consists of all pairs of functions
${\,z\,=\,(\,z_{in}\,,\,z_{out}\,)\,
\in(\mathcal{L}^2)^{H}\,}$,
such that $\,z_{in}:H\to\mathcal{L}\,$ and
$\,z_{out}:H\to\mathcal{L}\,$, respectively, 'switch' at even and odd
integer multiples of half the time step.
So, there is a natural bijection
\vspace{-.05cm}
\begin{equation}\centering\notag
\begin{aligned}
\mathcal{E}\;
&\to\;\mathcal{L}^{I}\times\mathcal{L}^{J}\\[-3pt]
z\,=\,(\,z_{in}\,,\,z_{out}\,)\;
&\mapsto\;(\,z_{in}\downharpoonright I\,,\,z_{out}\downharpoonright J\,)\quad,
\end{aligned}
\end{equation}
in virtue of which the first and second components in
$\,\mathcal{E}\,$ can be naturally identified with $\,\mathcal{L}^{I}\,$
and $\,\mathcal{L}^{J}\,$, respectively.

From section ~\ref{S:sec1} (Definition\,\ref{D.1.1}) we recall that for
any incident function
$\,e\,\in\,\mathcal{L}^{I}\times\{\,0\,\}\,\subset\mathcal{E}\,$
supported on a finite interval $\,[\,0\,,\,N\tau\,)\subset I\,$,
the DSC process excited with $\,e\,$ and generated by
$\,\mathcal{R}\,$ and $\,\mathcal{C}\,$ is the well-defined function
$\,z\in\,(\,\mathcal{L}^{2}\,)^{H}\,$
such that $\,z\,(\,t\,)\,=\,0\,$ for $\,t\,\leqslant\,0\,$ and recursively
for $\,0\,<\,t\,\in\,H\,$
\vspace{-.2cm}
\begin{equation}\centering\label{3.11}
\begin{aligned}
z\,(\,t\,+\,\frac{\tau}{2}\,)\,=\,
\begin{cases}
(\,z_{in}\,(\,t\,)\,,\;T_{_{-\frac{\tau}{2}}}\,F_{_{R}}\,T_{_{-\frac{\tau}{2}}}
\,[\,e\,+\,z_{in}\,]\,(\,t\,)\,)
&\,\text{if $\,t\,\in\,I\,$}\\
(\,T_{_{-\frac{\tau}{2}}}\,F_{_{C}}\,T_{_{-\frac{\tau}{2}}}\,
[\,z_{out}\,]\,(\,t\,)\,,\;z_{out}(\,t\,)\,)
&\,\text{if $\,t\,\in\,J\,$}\,.
\end{cases}
\end{aligned}
\end{equation}
In fact, $\,z\,$ is uniquely defined by these relations which provide
separate recurrence relations for the two processes
$\,z_{in}\,,\,z_{out}\,$;
for instance for $\,z_{in}\,$ and $\,t\,\in I\,$
\vspace{-4pt}
\begin{equation}\centering\label{3.12}
\begin{aligned}
z_{in}\,(\,t\,+\,\tau\,)\;
&=\; z_{in}\,(\,t^{\sptilde}\!+\,\frac{\tau}{2}\,)
\quad\text{with}\quad t^{\sptilde}:\,=\,t\,+\,\frac{\tau}{2}\,\in\,J &&\,\\
&=\; T_{-\frac{\tau}{2}}F_{_{C}}\,T_{\frac{-\tau}{2}} \hspace{-4.9cm}
\underbrace{[\,z_{out}\,]_{_{\leq t^{\sptilde}}}}_{\hspace{5.8cm}
=\,T_{\frac{\tau}{2}}\;T_{-\frac{\tau}{2}}F_{_{R}}\,T_{-\frac{\tau}{2}}
[e + z_{in}]_{_{\leq t^{\sptilde}\!\!-\frac{\tau}{2}\,=\,t}}\;
\text{by \eqref{3.11}}}
\hspace{-4.9cm}
(\;t^{\sptilde}\,) &&\text{by \eqref{3.11}}\\
&=\; F_{_{C}}\,T_{-\frac{t\tau}{2}}\,F_{_{R}}\,T_{-\frac{\tau}{2}}
[\,e\,+\,z_{in}\,]\,(\,t\,)\quad .\,&&
\end{aligned}
\end{equation}
By causality $\,z_{in}\,$ enters the last expression only up to argument
$\,t\,$, hence \eqref{3.12} is a well defined recursion formula for
$\,z_{in}(\,t\,)\,$, $\,t\in I$.
Moreover, since products of $\,\alpha$-passive operators are
$\alpha$-passive,
${F_{_{C}}\,T_{-\tau/2}\,F_{_{R}}\,T_{-\tau/2}}$ is $\,\alpha$-passive
in virtue of Proposition ~\ref{P.3.1}. Hence Theorem~\ref{T.2.1}
applies to $\,z_{in}\,$ which thus is stable, just as then also is
$\,z_{out}\,=\,F_{_{R}}\,T_{-\tau/2}\,[\,z_{in}\,+\,e\,]\,$.
\end{proof}

config: arxiv