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Dirac operators, Weyl-Dirac operators, zero modes, zero resonances, the limiting absorption principle
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\documentclass[a4paper, 11pt,reqno]{amsart}  
\usepackage{amssymb}
\usepackage{amsmath}

\newtheorem{thm}{Theorem}[section]
\newtheorem{prop}{Proposition}[section]
\newtheorem{lem}{Lemma}[section]
\newtheorem{rem}{Remark}[section]
\newtheorem{cor}{Corollary}[section]

\makeatletter
   \renewcommand{\theequation}{\thesection.\arabic{equation}}
          \@addtoreset{equation}{section}
\makeatother

\begin{document}

%%%%%%%%\baselineskip=1.1\baselineskip

\title{ 
The zero modes and the zero resonances\\
\hspace{-13pt}of Dirac operators }

\author{Yoshimi Sait\={o}
and
 Tomio Umeda\\}


\address{
Department of Mathematics \\
University of Alabama at Birmingham, 
Birmingham, AL 35294, USA}
\email{saito@math.uab.edu}

\address{Department of Mathematics,
University of Hyogo, Himeji 671-2201, Japan
}
\email{umeda@sci.u-hyogo.ac.jp}
\date{}

\maketitle

\section{Introduction}



This paper is concerned with the Dirac operator
%
\begin{equation} \label{eqn:1-1}
H= \alpha \cdot D + Q(x), \quad D=\frac{1}{\, i \,} \nabla_x,
\,\,\, x \in {\mathbb R}^3,
\end{equation}
%
where $\alpha= (\alpha_1, \, \alpha_2, \, \alpha_3)$ is
the triple of  $4 \times 4$ Dirac matrices
%
\begin{equation*}\label{eqn:1-2}
\alpha_j = 
\begin{pmatrix}
 \mathbf 0 &\sigma_j \\ \sigma_j &   \mathbf 0 
\end{pmatrix}  \qquad  (j = 1, \, 2, \, 3)
\end{equation*}
with  the $2\times 2$ zero matrix $\mathbf 0$ and
the triple of  $2 \times 2$ Pauli matrices
%
\begin{equation*}\label{eqn:1-3}
\sigma_1 =
\begin{pmatrix}
0&1 \\ 1& 0
\end{pmatrix}, \,\,\,
\sigma_2 =
\begin{pmatrix}
0& -i  \\ i&0
\end{pmatrix}, \,\,\,
\sigma_3 =
\begin{pmatrix}
1&0 \\ 0&-1
\end{pmatrix},
\end{equation*}
%
and $Q(x)$ is a $4\times 4$ matrix-valued function
decaying at infinity.


We would like to emphasize that
one can regard the operator (\ref{eqn:1-1}) as 
a generalization of 
the operator
%
\begin{equation} \label{eqn:1-4}
\alpha\cdot \big(D - A(x) \big) + q(x) I_4 , 
\end{equation}
%
where $(q, A)$ is an electromagnetic potential and
$I_4$ is a $4\times 4$ identity matrix,
by  taking $Q(x)$ to be $- \alpha\cdot A(x) + q(x) I_4$.
In the case  where $q(x)\equiv 0$, the operator
(\ref{eqn:1-4}) becomes of the form
%
\begin{equation*}   \label{eqn:1-4-1}
\alpha\cdot \big(D - A(x) \big)
=
\begin{pmatrix}
 \mathbf 0 &\sigma \cdot (D -A(x)) \\ 
\sigma \cdot (D -A(x)) & \mathbf 0 
\end{pmatrix}.
\end{equation*}
%
The component $\sigma \cdot (D -A(x))$ is called 
the Weyl-Dirac operator. See Balinsky and Evans \cite{BalinEvan2}.

The zero modes of
the Dirac operator
$\alpha
\cdot (D -A(x))$,
the Weyl-Dirac operator $\sigma \cdot (D -A(x))$
and the Pauli operator $\{ \sigma \cdot (D -A(x)) \}^2 + q(x) I_2$
have attracted 
a considerable attention in recent
years, because
they have deep and fruitful
implications from the view point of
mathematics as well as physics. 
See Adam, Muratori and Nash \cite{AdamMuratoriNash1},
\cite{AdamMuratoriNash2}, \cite{AdamMuratoriNash3},
Balinsky and Evans \cite{BalinEvan1},
\cite{BalinEvan2},
\cite{BalinEvan3}
and
Elton \cite{Elton} for the three dimensional case,
and 
Rozenblum and Shirokov \cite{RozenblumShirokov}
for the two dimensional case. 
It should be   remarked that
the zero modes play significant roles
in the study of Coulomb systems with
magnetic fields and
in the study of the Lieb-Thirring
inequalities for the 
Pauli operators. See Fr\"ohlich, Lieb 
and Loss \cite{FrohlichLiebLoss},
 Loss and Yau \cite{LossYau},
and Erd\"os and Solovej \cite{ErdosSolovej1},
 \cite{ErdosSolovej2}, 
\cite{ErdosSolovej3}.


We also should like to note that 
the operator (\ref{eqn:1-1}) also generalizes
the Dirac operator of the form
%
\begin{equation} \label{eqn:1-5}
\alpha\cdot D  + m(x) \beta  + q(x) I_4, 
\end{equation}
%
where $m(x)$ is considered to be a variable mass, and
$\beta$ is the $4\times 4$ matrix defined by
%
\begin{equation*} \label{eqn:1-6}
\beta =
\begin{pmatrix}
 I_2 &  \mathbf 0  \\ \mathbf 0 & - I_2
\end{pmatrix}
\end{equation*}
%
Spectral properties of the operator (\ref{eqn:1-5})
have been extensively studied recent years.
See Kalf and Yamada \cite{KalfYamada},
Kalf, Okaji and Yamada \cite{KalfOkajiYamada},
Schmidt and Yamada \cite{SchmidtYamada},
Pladdy \cite{Pladdy} and
Yamada \cite{Yamada}.


Finally, we would like to emphasize 
the significant role of the zero modes 
and the zero resonances in the analysis of the asymptotic
behavior, around the origin of the
complex plane, of the  resolvent of the operator $H$ 
given by (\ref{eqn:1-1}).  
One can easily recognize the significance  as
the work \cite{JensenKato} by Jensen and Kato on
the Schr\"odinger operator suggests.




\vspace{10pt}




\noindent
\textbf{Notation.}

\noindent
The upper and lower half planes ${\mathbb C}_{\pm}$ are  
defined by
%
\begin{equation*}
{\mathbb C}_{+} 
  :=\{ \, z \in \mathbb C  \; | \;  \mbox{Im }z >0 \, \},
\qquad
{\mathbb C}_{-} 
  :=\{ \, z \in \mathbb C  \; | \;  \mbox{Im }z <0 \, \}
\end{equation*}
%
respectively.
By $S({\mathbb R}^3)$, we mean
 the Schwartz class of rapidly decreasing
functions on ${\mathbb R}^3$, and we set 
$\mathcal S = [S({\mathbb R}^3)]^4$.


By $L^2 =L^2({\mathbb R}^3)$, we mean the Hilbert space of
square-integrable functions on ${\mathbb R}^3$, and 
we introduce  a Hilbert space ${\mathcal L}^2$ by
     ${\mathcal L}^2 = [L^2({\mathbb R}^3)]^4$, where 
the inner product  
is given by
%
\begin{equation*}
    (f, g)_{{\mathcal L}^2}
 = \sum_{j=1}^4 (f_j, g_j)_{L^2}
\end{equation*}
%
for  $f = {}^t(f_1, f_2, f_3, f_4)$ 
and
$g = {}^t(g_1, g_2, g_3, g_4)$.
    
By $L^{2, s}({\mathbb R}^3)$, we mean the weighted $L^2$ space
defined by
%
\begin{equation*}
 L^{2, s}({\mathbb R}^3)
  :=\{ \, u \; | \; \langle x \rangle^s u 
 \in L^2({\mathbb R}^3) \, \}
\end{equation*}
%
with the inner product
%
\begin{equation*}
 (u, \, v)_{L^{2, s}}
:=
\int_{{\mathbb R}^3} 
   \langle x \rangle^{2s} u(x)  \, \overline{v(x)} \, dx, 
\end{equation*}
%
where
%
\begin{equation*}  \label{eqn:bracket-x}
\langle x \rangle = \sqrt{1 + |x|^2 \,}.
\end{equation*}
%
We introduce  the Hilbert space
     ${\mathcal L}^{2,s} = [L^{2,s}({\mathbb R}^3)]^4$ with
the inner product  
%
\begin{equation*}
    (f, g)_{{\mathcal L}^{2,s}}
 = \sum_{j=1}^4 (f_j, g_j)_{L^{2,s}}.
\end{equation*}
%
%
By $H^{\mu, s}({\mathbb R}^3)$, we mean the weighted 
Sobolev  space
defined by
%
\begin{equation*}
 H^{\mu, s}({\mathbb R}^3)
  :=\{ \, u \in {S^{\prime}({\mathbb R}^3)} \; | \; 
\langle x \rangle^s \langle D \rangle^{\mu}  \, u 
 \in L^2({\mathbb R}^3) \, \}
\end{equation*}
%
with the inner product
%
\begin{equation*}
 (u, \, v)_{H^{\mu, s}}
:=
\big(
   \langle x \rangle^{s} \langle D \rangle^{\mu} u ,
\langle x \rangle^{s}\langle D \rangle^{\mu} \, v
\big)_{L^2},
\end{equation*}
%
where 
%
\begin{equation}    \label{eqn:bracket-D}
\langle D \rangle = \sqrt{1 - \Delta \,}.
\end{equation}
%
In a similar fashion, 
we introduce  the Hilbert space
     ${\mathcal H}^{\mu,s} = [H^{\mu,s}({\mathbb R}^3)]^4$.
Note that $H^{\mu, 0}({\mathbb R}^3)$ coincides with
the Sobolev space of order $\mu$: $H^{\mu}({\mathbb R}^3)$,
and by  
${\mathcal H}^{\mu}$ we mean the
Hilbert space $[H^{\mu}({\mathbb R}^3)]^4$.
Also note
that
${\mathcal H}^{0,0} ={\mathcal L}^2$ and
${\mathcal H}^{0,s} ={\mathcal L}^{2,s}$.


By ${B}(\mu, s \, ; \,\nu, t )$, we mean the set of 
all bounded linear operators from
$H^{\mu, s}({\mathbb R}^3)$ into $H^{\nu, t}({\mathbb R}^3)$,
and by
$\mathcal{B}(\mu, s \, ; \,\nu, t)$,
the set of 
all bounded linear operators from
${\mathcal H}^{\mu, s}$ into 
${\mathcal H}^{\nu, t}$.
For an operator $W \in {B}(\mu, s \, ; \,\nu, t )$, 
we define a copy of $W \in \mathcal{B}(\mu, s \, ; \,\nu, t)$ by
%
\begin{gather}   \label{eqn:mapsto}
\begin{split}    
 {\mathcal H}^{\mu, s} \ni 
  f= \, &{}^t (f_1, \, f_2, \, f_3, \, f_4)    \\
{}&\mapsto
Wf ={}^t (Wf_1, \, Wf_2, \, Wf_3, \, Wf_4)
\in {\mathcal H}^{\nu, t}. 
\end{split}      
\end{gather}
%

\vspace{15pt}


\noindent
\textbf{Assumption (A).} 

\noindent
Each element $q_{jk}(x)$, 
$j, \, k =1, \, \cdots, \, 4$, of $Q(x)$ is 
a measurable function satisfying
%
\begin{equation} \label{eqn:2-1}
| q_{jk}(x) | \le C \langle x \rangle^{-\rho} 
\quad   ( \rho >1 ),
\end{equation}
%
where $C(=C_q)$ is a positive constant.
Moreover, $Q(x)$ is a Hermitian matrix for each 
$x \in {\mathbb R^3}$.

\vspace{15pt}

Note that under Assumption(A) the Dirac operator (\ref{eqn:1-1})
is a self-adjoint operator in $\mathcal L^2$ with  
$\mbox{Dom}(H) = {\mathcal H}^1$.
The self-adjoint realization will be denoted 
by $H$ again. 
With an abuse of notation, we shall write $Hf$  
\textit{in the distributional sense} for
$f \in {\mathcal S}^{\prime}$ 
whenenver it makes sense.

\vspace{15pt}

\noindent
D{\scriptsize EFINITION}. \ 
By a zero mode, we mean a function 
$f \in \mbox{Dom}(H)$ which satisfies
%
\begin{equation*}  \label{eqn:1-7}
Hf=0.
\end{equation*}
By a zero resonance, we mean a function 
$f \in {\mathcal L}^{2, -s}\setminus \mathcal L^2$, for
some $s$ with $1/2 < s <3/2$,
 which satisfies
%
\begin{equation*}  \label{eqn:1-8}
Hf=0 \;\;\text{ in the distributional sense.}
\end{equation*}
%

\vspace{15pt}


It is evident that a zero mode of $H$ is an eigenfunction of $H$
corresponding to the eigenvalue $0$, i.e.,
a zero mode is
 an element of $\mbox{Ker}(H)$,
 the kernel of the self-adjoint
operator $H$. 
We also remark that our definition of
the zero 
 resonance above is more general
than that of Jensen and Kato \cite{JensenKato} 
with respect to the smoothness,
 though we shall restrict ourselves to
$\rho > 5/2$ and 
$1/2 < s < \min \{ 3/2, \, \rho- 3/2 \}$ 
in dealing with zero 
resonances in section 2.



Balinsky and Evans \cite{BalinEvan2} is particulary interesting 
from our  view point in the sense that they dealt with 
the Weyl-Dirac  operator
$\sigma \cdot (D -A(x))$ and showed that the set of magnetic fields 
which give rise to zero modes
 is rather \lq\lq sparse.\rq\rq

In this paper, we investigate  the zero modes and
the zero resonances
of the operator (\ref{eqn:1-1})
under Assumption(A).
Our goal  
is to establish a pointwise
estimate of the zero modes
as well as the continuity of the zero modes, and
also to show that the zero resonasnces do not
exist.

\vspace{15pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Main results}



\begin{thm} \label{thm:th-2-2}
Suppose Assumption {\rm(A)} is verified. Let $f$ be  
a zero mode of the operator {\rm(\ref{eqn:1-1})}.
Then 

{\rm (i)} the  inequalities 
%
\begin{equation}   \label{eqn:2-2}
|f(x)| \le C
\begin{cases}  
\langle x \rangle^{-\rho +1}
                   & (1 < \rho <3 ) \\
{}    & \\
\langle x \rangle^{-2}\log ( 1 + \langle x \rangle)
& ( \rho = 3) \\
{}    & \\
\langle x \rangle^{-2}& ( \rho  >3).  
\end{cases}
\end{equation}
%
hold for all $x \in {\mathbb R}^3$, where the constant 
$C(=C_f)$ depends only on the zero mode $f$;

\vspace{4pt}

{\rm (ii)} the zero mode $f$ is a continuous function on ${\mathbb R}^3$.
\end{thm}
\vspace{15pt}


\begin{rem}
It is natural that zero modes
only decay polynomially at infinity. 
In Loss and Yau \cite{LossYau},
they considered the Weyl-Dirac operator
$\sigma \cdot (D-A(x))$,
and  constructed two examples of  pairs of
 a vector potential $A$ and a zero mode $\psi$. 
One of their examples 
shows that $A(x) = O(|x|^{-2})$ 
and $\psi(x) = O(|x|^{-2})$ at infinity.
Also, see examples in Adam, Muratori and
Nash \cite{AdamMuratoriNash1}.
It is not
clear for us if the decay rates
 in Theorem \ref{thm:th-2-2}
are optimal.
\end{rem}

\begin{rem}
In Bugliaro,Fefferman and Graf \cite{BugliaroFeffermanGraf} and
 Erd\"os and Solovej \cite{ErdosSolovej2}, they 
established estimates of the density of zero modes
of the Weyl-Dirac operator
$\sigma \cdot (D-A(x))$.
It is apparant that their estimates immediately imply
estimates of each zero mode.
These estimates of each zero
mode are, however, quite unclear in terms
of the decay rate at infinity.
\end{rem}

\vspace{10pt}
Theorem \ref{thm:th-2-1} below means that zero
resonances do not exist. In order to show this fact,
we need a larger $\rho$ than Theorem \ref{thm:th-2-2}.

\vspace{15pt}
\begin{thm} \label{thm:th-2-1}
Suppose Assumption {\rm(A)} is verified with
$\rho > 5/2$. If $f$ belongs to
${\mathcal L}^{2, \, -s}$ for some 
$s$ with $1/2 < s < \min\{3/2, \, \rho -3/2 \}$ and
satisfies $Hf=0$ in the 
distributional sense, then 
$f \in {\mathcal H}^1$.
\end{thm}


\vspace{15pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{A singular integral operator}  \label{sec:SIO}

One of the ingredients of the proofs of the main
theorems is a singular integral operator acting on 
four component vector functions.
The singular integral operator we deal with 
in this section is 
 defined by

%
\begin{equation} \label{eqn:sio-1}
A f(x) = 
\displaystyle{
\int_{{\mathbb R}^3}
  i \, \frac{\alpha \cdot (x - y)}{4\pi |x-y|^3} f(y) \, dy.
}
\end{equation}
% 
for 
%
\begin{equation*}   \label{eqn:sio-2}
f = {}^t(f_1, \, f_2, \, f_3, \, f_4) 
\in {\mathcal L}^2,
\end{equation*}
%
where  $\alpha \cdot (x - y)$  means 
 the sum of the matrix operation $\alpha_j$
for the four-vector $(x_j - y_j) f$:
%
\begin{equation*}   \label{eqn:sio-2+}
\alpha \cdot (x - y) f 
 = \sum_{j=1}^3 \alpha_j (x_j - y_j) f. 
\end{equation*}
%

We shall need a few estimates of $A$ on
${\mathcal L}^2$ and on its subspaces.


\begin{lem}   \label{lem:siolem-1}
For each $f \in {\mathcal L}^2$,
 $Af(x)$ is defined
for a.e. $x \in {\mathbb R}^3$.
Moreover,  $A$ is a bounded operator from
${\mathcal L}^2$ to ${\mathcal L}^6$, i.e., there 
exists a constant  $C$ such that
%
\begin{equation*} \label{eqn:sio-3}
\Vert A f \Vert_{{\mathcal L}^6} 
\le
 C \Vert  f \Vert_{{\mathcal L}^2}
\end{equation*}
% 
for all $f \in {\mathcal L}^2$.
\end{lem}

\begin{proof}
Since each $\alpha_j$, $j=1$, $2$, $3$,
is a unitary matrix, we have
%
\begin{gather*}   
\begin{split} %  \label{eqn:sio-4}
| \alpha \cdot (x - y) \, f(y)|
&\le
\sum_{j=1}^3  |x_j - y_j| \, |f(y)|   \\
{}& \le 3|x-y| \, |f(y)|.
\end{split}
\end{gather*}
%
(Note that $|x-y|$ and $|f(y)|$ 
are the Euclidean norms of ${\mathbb R}^3$ and  ${\mathbb R}^4$
respectively.)
%
Therefore we get
\begin{gather} 
\begin{split}     \label{eqn:sio-5}
|A f(x) | 
&\le 
\frac{3}{\,4\pi \,} 
\int_{{\mathbb R}^3}
\frac{1}{|x-y|^2}  |f(y)|\, dy  \\
{}
& \le \frac{\,3\pi \,}{2} I_1(\, |f| \, ) (x),
\end{split}
\end{gather}
% 
where $I_1$ is the Riesz potential; see
Stein \cite[p.117]{Stein}. We shall appeal two well-known
facts
(Stein \cite[p.119]{Stein}) 
that  $I_1(\,  u \, ) (x)$
is finite for a.e. $x \in {\mathbb R}^3$ 
if $u \in L^2({\mathbb R}^3)$, 
and that
 $I_1$ is
a bounded operator from $L^2({\mathbb R}^3)$
to $L^6({\mathbb R}^3)$. 
These facts, together with (\ref{eqn:sio-5}),
yield the conclusions of the lemma.
\end{proof}


\vspace{10pt}

\begin{lem} \label{lem:siolem-2}
Let $s > 3/2$. Then

\vspace{3pt}
{\rm(i)} $A\in{\mathcal B}( 0, s \, ; \, 0, 0)${\rm;}

\vspace{3pt}
{\rm(ii)} $A\in{\mathcal B}( 0, 0 \, ; \, 0, -s)$. 
\end{lem}
%

\begin{proof}
It is shown in Umeda \cite[Lemma 5.1]{Umeda} that 
the
Riesz potential
$I_1$ is a bounded operator from 
$L^{2, s}({\mathbb R}^3)$ to $L^{2}({\mathbb R}^3)$
and
that 
$I_1$ is a bounded operator from 
$L^{2}({\mathbb R}^3)$ to $L^{2, -s}({\mathbb R}^3)$.
 (Note that $I_1$
is written as $G_0$ in \cite{Umeda}.) The assertions (i) and
(ii) 
of the lemma respectively
follows from these  two facts  and
(\ref{eqn:sio-5}).
\end{proof}


\vspace{10pt}

We introduce a class of functions which
is necessary to establish an 
${\mathcal L}^{\infty}$ estimate 
of the operator $A$. For $q\ge 1$, 
we define
%
\begin{equation*}  \label{eqn:sio-6}
L^q_{ul}(\mathbb R^3)
=
\big\{\,
u \in L^q_{loc}(\mathbb R^3) \; \big| \;
 \Vert u \Vert_{L^q_{ul}} := 
\sup_{x \in {\mathbb R}^3}  
\Vert u \Vert_{L^q{( B(x; \, 1) )}} < \infty
\, \big\},
\end{equation*}
%
where $B(x;  1) 
= \{ \, y \in \mathbb R^3 \; | \;\, |x-y| \le 1 \, \}$, and
define
%
\begin{equation*}  \label{eqn:sio-7}
{\mathcal L}^q_{ul}
=
 \big[ L^q_{ul}(\mathbb R^3) \big]^4,
\quad 
\Vert  f \Vert_{{\mathcal L}^q_{ul}}
= \sum_{k=1}^4 
\Vert f_k \Vert_{L^q_{ul}}.
\end{equation*}
%


\vspace{10pt}

\begin{lem} \label{lem:siolem-3}
Let $q >3$. Then there exists a constant 
$C_q$ such that
%
\begin{equation*}  \label{eqn:sio-8}
\Vert A f \Vert_{{\mathcal L}^{\infty}}
\le C_q
 \big( 
\Vert  f \Vert_{{\mathcal L}^2}
+\Vert  f \Vert_{{\mathcal L}^q_{ul}}
\big).
\end{equation*}
for all  $f \in {\mathcal L}^2 \cap {\mathcal L}^q_{ul}$.
\end{lem}
%
\begin{proof}
By virtue of (\ref{eqn:sio-5}), we only have to prove 
that there exists a constant $C_q^{\prime}$ such that
%
\begin{equation}   \label{eqn:sio-9}
\Vert I_1(u) \Vert_{{L}^{\infty}}
\le C_q^{\prime}
 \big( 
\Vert  u \Vert_{{L}^2}
+\Vert u \Vert_{{L}^q_{ul}}
\big)  
\mbox{ for }  u \in 
L^2(\mathbb R^3) \cap L^q_{ul}(\mathbb R^3).
\end{equation}
%
Since each $u \in 
L^2(\mathbb R^3) \cap L^q_{ul}(\mathbb R^3)$ can be
docomposed as
%
\begin{align*}  
u &=v_+ -v_- + i (w_+ -w_-)  \\%%%%%%\label{eqn:sio-10} \\
{}&
v_{\pm} \ge 0, \;\; w_{\pm}\ge 0  \\%%%%%\label{eqn:sio-11} \\
{}&
v_{\pm},\;\;  w_{\pm} 
\in  L^2(\mathbb R^3) \cap L^q_{ul}(\mathbb R^3) \\%%%\label{eqn:sio-12}
\end{align*}
%
we shall prove (\ref{eqn:sio-9}) for $u\ge0$.


Let $u \in 
L^2(\mathbb R^3) \cap L^q_{ul}(\mathbb R^3)$ be given,
and let satisfy $u\ge0$. 
Then one can find a sequence $\{ \, \varphi_n \, \} \subset
C_0^{\infty}(\mathbb R^3)$ such that
%
\begin{equation}   \label{eqn:sio-13}
0 \le \varphi_n \le u, \quad  \varphi_n  \to u  \;\; 
\mbox{ in } L^2(\mathbb R^3).
\end{equation}
%
(First cut $u$ as ${\chi}_{B(0;\, n)}(x) u$ by multipying
a characteristic function ${\chi}_{B(0;\, n)}$ 
of the ball with center at 
the origin and radius $n$, then use
the mollifier.) 
For each $n$, 
we decompose as
%
\begin{gather} 
\begin{split}    \label{eqn:sio-14}
I_1(\varphi_n)(x)
&= 
\int_{\mathbb R^3} \frac{1}{\, 2\pi^2 |x-y|^2 \,} \,
  \varphi_n (y) \, dy        \\
\noalign{\vskip 4pt}
&= h_0 * \varphi_n (x)  
  +  h_1 * \varphi_n (x),     
\end{split}
\end{gather}
%
where
%
\begin{gather*}
\begin{split}   %%%%%%%\label{eqn:sio-16}
h_0(x) &= \chi_{B(0;\, 1)}(x)
 \frac{1}{\,  2\pi^2 |x|^2 \,}      \\
h_1(x) &= (1 -\chi_{B(0;\, 1)}(x) )
 \frac{1}{\,  2\pi^2 |x|^2 \,}.
\end{split}
\end{gather*}
%
(One should note that the integral on the right hand side of
(\ref{eqn:sio-14})  converges because of the fact that
$\varphi_n \in C_0^{\infty}(\mathbb R^3)$.)
If we apply the H\"older inequality to $h_0 * \varphi_n$,
then we get
%
\begin{gather}   
\begin{split}\label{eqn:sio-17}
|h_0 * \varphi_n(x) | 
&\le
 \frac{1}{\,  2\pi^2 \,}  
\Big\{
 \int_{|x-y|\le 1} \frac{1}{\,  |x-y|^{2p} \,}  \, dy
\Big\}^{1/p} \,
\Vert \varphi_n \Vert_{L^q(B(x;1))}   \\
\noalign{\vskip 4pt}
{}&
\quad \le  
C_{q}^{\prime\prime}  \Vert u \Vert_{L^q(B(x;1))}  \\
\noalign{\vskip 4pt}
{}& \qquad  ( p^{-1}= 1 - q^{-1} )
\end{split}
\end{gather}
%
where we have used the fact that
 $2p < 3 \;\; (\because q >3$ by assumption) 
and  (\ref{eqn:sio-13}).
Similarly, if we apply the Schwarz inequality
to $h_1 * \varphi_n$, we obtain
%
%
\begin{gather}   
\begin{split}\label{eqn:sio-18}
|h_1 * \varphi_n(x) | 
&\le
 \frac{1}{\,  2\pi^2 \,}  
\Big\{
 \int_{|x-y|\ge 1} \frac{1}{\,  |x-y|^{4} \,}  \, dy
\Big\}^{1/2} \,
\Vert \varphi_n \Vert_{L^2({\mathbb R})}   \\
\noalign{\vskip 4pt}
{}&
\quad \le  
C  \Vert u \Vert_{L^2({\mathbb R}^3)} .
\end{split}
\end{gather}
%
If follows from (\ref{eqn:sio-14}), 
(\ref{eqn:sio-17})  and (\ref{eqn:sio-18})  that
%
\begin{gather}   
\begin{split}\label{eqn:sio-19}
|I_1(\varphi_n)(x)|
& \le  
C_{q}^{\prime} 
\big(  \Vert u \Vert_{L^q(B(x;1))}  
 + \Vert u \Vert_{L^2({\mathbb R}^3)}  \, \big) \\
& \le
C_q^{\prime}
\big(  \Vert u \Vert_{L^q_{ul}}  
 + \Vert u \Vert_{L^2}   \big)
\end{split}
\end{gather}
%
As was mentioned in the proof of
Lemma \ref{lem:siolem-2}, the Riesz potential
$I_1$ is a bounded operator 
from $L^2({\mathbb R}^3)$ to $L^{2, -s}({\mathbb R}^3)$
 with $s > 3/2$.
This fact, together with (\ref{eqn:sio-13}), implies that
there exists a subsequence $\{ \varphi_{n^{\prime}} \}$
such that
$I_1(\varphi_{n^{\prime}}) (x) \to I_1(u)(x)$ for a.e.  
$x \in {\mathbb R}^3$. 
Thus taking the limit of (\ref{eqn:sio-19}),
along with the subsequence,
gives (\ref{eqn:sio-9}).
\end{proof}




\vspace{15pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Estimates of the resolvents}  \label{sec:resolvent}

Another  ingredient of the proofs of the main theorems
is the limiting absorption principle (LAP) for
the free Dirac operator
%
\begin{equation} \label{eqn:reslvnDrc-00}
H_0= \alpha \cdot D.
\end{equation}
%
We note that $H_0$ with $\mbox{Dom}(H_0) ={\mathcal H}^1$ is 
a self-adjoint operator in ${\mathcal L}^2$. The self-adjoint realization will
be denoted by $H_0$ again. It is well-known
that the spectrum $\sigma(H_0)$ equals the whole 
real line $\mathbb R$.
With an abuse of notation again, we shall write
$H_0f$ for $f \in {\mathcal S}^{\prime}$.

 Our idea of proving the
LAP for the free Dirac operator $H_0$ 
is based on a decomposition  of the
resolvent 
%
\begin{equation}   \label{eqn:reslvnDrc-0}
R_0(z) = (H_0 +z ) \varGamma_0(z^2)
  \;\; \mbox{ on } \; {\mathcal L}^2, 
\quad \mbox{Im}\, z \not=0,
\end{equation}
%
where
%
\begin{equation} \label{eqn:reslvnDrc-1}
R_0(z) = (H_0 - z )^{-1},
\end{equation}
%
and  $\varGamma_0(z)$ in (\ref{eqn:reslvnDrc-0})
denotes the copy of 
the resolvent $\varGamma_0(z)$ of
the negative Laplacian
%
\begin{equation*}   \label{eqn:laplace-0}
-\Delta = -\big(
\frac{{\partial}^2}{\partial x_1^2} 
+
\frac{{\partial}^2}{\partial x_2^2} 
+
\frac{{\partial}^2}{\partial x_3^2} 
\big).
\end{equation*}
%
See  (\ref{eqn:mapsto}) for the definition
of the copy of an operator.
In other words,
we shall not distinguish between
$\varGamma_0 (z)$ in 
$L^2({\mathbb R}^3)$ 
and 
$\varGamma_0 (z)$ in ${\mathcal L}^2$.
We believe this will not cause any confusion.

A formal computation shows that
%
\begin{equation}   \label{eqn:diracsquare}
H_0^2 = -\Delta,
\end{equation}
%
from which one can deduce 
(\ref{eqn:reslvnDrc-0}).
The decomposition (\ref{eqn:reslvnDrc-0}) was
first exploited in 
Balslev and Hellfer \cite{BalslevHellfer}.
Similar decomposition was also adopted in Pladdy, Sait\={o} and
Umeda \cite{PSU1}, \cite{PSU2}.

\vspace{10pt}
\subsection{The resolvent of the negative Laplacian}

It is well known that the resolvent of $-\Delta$ is 
represented as an integral operator:
%
\begin{equation}  \label{eqn:resolventLapc}
\varGamma_0 (z) u(x) =
\int_{{\mathbb R}^3} 
\frac{e^{i \sqrt{z} |x-y|} }{\, 4\pi |x-y| \,} u(y) \, dy,
\qquad
u \in L^2({\mathbb R}^3 )
\end{equation}
%
for $z \in \mathbb C \setminus [0, \,+\infty)$, 
where
$\mbox{Im} \sqrt{z} >0$.


\begin{lem}   \label{lem:hs}
Let  $s$, $s^{\prime} > 1/2$ and $s+ s^{\prime} >2$.
Then 
%
\begin{equation}  \label{eqn:hs-2}
\iint_{ \! {\mathbb R}^3 \times {\mathbb R}^3 } 
\,  \langle x \rangle^{-2s^{\prime}} 
     \frac{1}{ \, |x - y|^2 \,}
  \langle y \rangle^{-2s}  \, dxdy < +\infty.
\end{equation}
%
\end{lem}

\begin{proof}
We may assume, with no loss of generality, that $s < 3$. 
Then, by a well known
estimate (e.g. Umeda \cite[Lemma 11.1]{Umeda}), we have
%
\begin{equation}  \label{eqn:hs-3}
\int_{ {\mathbb R}^3} 
     \frac{1}{ \, |x - y|^2 \,}
  \langle y \rangle^{-2s}  \, dy 
\le 
C_s \langle x \rangle^{-2s +1}.
\end{equation}
%
Since $-2s^{\prime} -2s +1 < -3$ by assumption of 
the lemma, we see that (\ref{eqn:hs-3})
implies   (\ref{eqn:hs-2}).
\end{proof}

\vspace{10pt}

It follows from  (\ref{eqn:resolventLapc}) and
 Lemma \ref{lem:hs} that 
the operator
%
\begin{equation}   \label{eqn:khs-0}
K(z) :=\langle x \rangle^{-s^{\prime}} \!
 \varGamma_0(z)  
\langle x \rangle^{-s},
\end{equation}
%
which is represented as
%
\begin{equation*}   \label{eqn:khs-1}
K(z) u (x) =
\int_{{\mathbb R}^3} 
\langle x \rangle^{-s^{\prime}}
\frac{e^{i \sqrt{z} |x-y|} }{\, 4\pi |x-y| \,}
\langle y \rangle^{-s} u(y) \, dy,
\end{equation*}
%
belongs to the Hilbert-Schmidt class  on
${L}^2({\mathbb R}^3)$ for 
$z \in \mathbb C \setminus [0, \,+\infty)$:
%
\begin{equation*}   \label{eqn:khs-1+}
\Vert 
K(z)        
  \Vert^2_{\mbox{\tiny{HS}}} 
=
\iint_{ \! {\mathbb R}^3 \times {\mathbb R}^3 } 
\,  \langle x \rangle^{-2s^{\prime}} 
\Big| \,
\frac{e^{i \sqrt{z} |x-y|} }{\, 4\pi |x-y| \,} 
 \, \Big|^2
  \langle y \rangle^{-2s}  \, dxdy < \infty,
\end{equation*}
%
where $\Vert \cdot \Vert_{\mbox{\tiny{HS}}}$ denotes
the Hilbert-Schmidt norm.
Note that
%
\begin{gather}  \label{eqn:hs-4}
\begin{split}
\Vert 
K(z_1)  - K(z_2)        
  \Vert^2_{\mbox{\tiny{HS}}}   
       \qquad\qquad\qquad\quad \qquad\qquad
         \qquad\qquad\qquad\quad  \\
\noalign{\vskip 4pt}
=
\iint_{ \! {\mathbb R}^3 \times {\mathbb R}^3 } 
\,  \langle x \rangle^{-2s^{\prime}} 
\Big| \,
\frac{e^{i \sqrt{z_1} |x-y|} }{\, 4\pi |x-y| \,} 
-     
\frac{e^{i \sqrt{z_2} |x-y|} }{\, 4\pi |x-y| \,}
 \, \Big|^2
  \langle y \rangle^{-2s}  \, dxdy
\end{split}
\end{gather}
%
for all 
$z_1$, $z_2 \in \mathbb C \setminus [0, \,+\infty)$.
It follows from (\ref{eqn:hs-4}) that
$K(z)$
is continous,
 with respect to the Hilbert-Schmidt
norm topology, on 
$\mathbb C \setminus [0, \,+\infty)$.
Furthermore, we can deduce 
from  (\ref{eqn:hs-2}), 
(\ref{eqn:hs-4})  and the 
Lebesgue dominated convergence theorem 
that
$K(z)$ can be continuously extended, 
with respect to the Hilbert-Schmidt
norm topology, as follows:
%
\begin{equation}   \label{eqn:khs-2}
\widetilde K(z) 
=
\begin{cases}  
K(z) 
& \mbox{if } \; z \in \mathbb C \setminus [0, \, +\infty), 
\\ {}    & \\
K^+(\lambda) 
  & \mbox{if } \;  z= \lambda + i0, \; \lambda \ge 0,
\\ {}    & \\
K^-(\lambda) 
  & \mbox{if } \; z= \lambda - i0, \; \lambda \ge 0,  
\end{cases}
\end{equation}
%
where $K^+(\lambda)$ and $K^-(\lambda)$ for 
$\lambda >0$
are defined by
%
\begin{equation}  \label{eqn:khs-3}
K^{\pm} (\lambda) u(x) =
\int_{{\mathbb R}^3} 
\langle x \rangle^{-s^{\prime}} 
\frac{e^{\pm i \sqrt{\lambda} |x-y|} }{\, 4\pi |x-y| \,} 
\langle y \rangle^{-s} 
u(y) \, dy,
\end{equation}
%
and 
%
\begin{equation}  \label{eqn:khs-4}
K^+(0)u(x)= K^-(0)u(x)
=\int_{{\mathbb R}^3} 
 \langle x \rangle^{-s^{\prime}} 
\frac{1}{\, 4\pi |x-y| \,}  
 \langle y \rangle^{-s}  u(y) \, dy.
\end{equation}
%

For a later purpose, it is convenient to introduce 
a subset of the Riemann surface of $\sqrt{z \,}$ 
as follows:
%
\begin{gather}   \label{eqn:riemann-1}
\begin{split}
{}&\;\; \Pi_{(0, \, +\infty)}     \\
&:=
\big( \mathbb C \setminus (0, \, +\infty) \big)
\cup
\{ \,  z = \lambda + i0 \; | \; \lambda >  0  \, \}
\cup
\{ \, z = \lambda - i0 \; | \; \lambda >  0  \, \}.
\end{split}
\end{gather}
%
Thus, we can say that $\widetilde K(z)$ defined 
by (\ref{eqn:khs-2}) -- (\ref{eqn:khs-4}) 
is continuous on $\Pi_{(0, \, +\infty)}$
with respect to
the Hilbert-Schmidt norm topology.

In view of (\ref{eqn:khs-0}), 
we see that $\varGamma_0(z)$, 
$z \in \mathbb C \setminus [0, \,+\infty)$, 
is a Hilbert-Schmidt operator from
 $L^{2, s}({\mathbb R}^3)$ to 
$L^{2, -s^{\prime}}({\mathbb R}^3)$.
Hence, in particular,
$\varGamma_0(z) \in {B}(0, s \, ; \,0 , -s^{\prime})$, 
and
%
\begin{equation*}   \label{eqn:hs-5-0}
\Vert
\varGamma_0(z)\Vert_{{B}(0, s \, ; \,0 , -s^{\prime})}   
\le
\Vert
K(z) 
\Vert_{\mbox{\tiny{HS}}}  .
\end{equation*}
%
Since we have the inequality
%
\begin{gather}
\begin{split}   \label{eqn:hs-5}
\Vert
\varGamma_0(z_1)  - \varGamma_0(z_2)
\Vert_{{B}(0, s \, ; \,0 , -s^{\prime})}   
\le
\Vert
K(z_1)  - K(z_2) 
\Vert_{\mbox{\tiny{HS} } }  \\ 
\noalign{\vskip 4pt}
(z_1, \; z_2 \in   \mathbb C \setminus [0, \, &+\infty)) 
\end{split}
\end{gather}
%
we conclude from  (\ref{eqn:khs-2}) 
  and (\ref{eqn:hs-5}) that 
$\varGamma_0(z)\in 
{B}(0, s \, ; \,0 , - s^{\prime})$ 
can be continuously extended
 as
follows:
%
\begin{equation}   \label{eqn:laplace-1}
\widetilde\varGamma_0(z) 
=
\begin{cases}  
\varGamma_0(z) 
& \mbox{if } \; z \in \mathbb C \setminus [0, \, +\infty), 
\\ {}    & \\
\varGamma_0^+(\lambda) 
  & \mbox{if } \;  z= \lambda + i0, \; \lambda \ge 0,
\\ {}    & \\
\varGamma_0^-(\lambda) 
  & \mbox{if } \; z= \lambda - i0, \; \lambda \ge 0,  
\end{cases}
\end{equation}
%
where
%
\begin{equation*}   \label{eqn:laplace-2}
\varGamma_0^{\pm}(\lambda) :=
\langle x \rangle^{s^{\prime}} 
 K^{\pm}(\lambda)
\langle x \rangle^{s}
=
\lim_{\epsilon \downarrow 0}   
\varGamma_0(\lambda \pm i \epsilon)  
 \; \mbox{ in } \;   
{B}(0, s \, ; \,0 , - s^{\prime}).
 \end{equation*}
%
We must remark that 
%
\begin{equation}  \label{eqn:resolventLapc-1}
\varGamma_0^+(0)u(x)= \varGamma_0^-(0)u(x)
=\int_{{\mathbb R}^3} 
\frac{1}{\, 4\pi |x-y| \,} u(y) \, dy,
\end{equation}
%
and that
%
\begin{equation*}  \label{eqn:resolventLapc-2}
\varGamma_0^{\pm} (\lambda) u(x) =
\int_{{\mathbb R}^3} 
\frac{e^{\pm i \sqrt{\lambda} |x-y|} }{\, 4\pi |x-y| \,} u(y) \, dy.
\end{equation*}
%
Thus
%
\begin{equation*}  \label{eqn:resolventLapc-3}
\widetilde\varGamma_0(\lambda + i0)
 \not= \widetilde\varGamma_0(\lambda -i0), \quad \lambda >0.
\end{equation*}
%
Note that the equality (\ref{eqn:resolventLapc-1}) allows us 
to use the notation
%
\begin{equation}  \label{eqn:resolventLapc-3-1}
\widetilde\varGamma_0(0)
\big( =\varGamma_0^+(0)= \varGamma_0^-(0) \big).
\end{equation}
%
With the notation introduced in (\ref{eqn:riemann-1}),
we can  say that
$\widetilde\varGamma_0(z)$ is a\linebreak  
${B}(0, s \, ; \, 0, - s^{\prime})$-valued
continuous function on $\Pi_{(0, \, +\infty)}$.


Lemmas \ref{lem:lap-1} and \ref{lem:lap-1-1} below 
are variants of Lemma 2.1 of  
Jensen and Kato \cite{JensenKato},
although we shall give their proofs. 

\begin{lem} \label{lem:lap-1}
Let $s$, $s^{\prime}$ satisfy
the same assumption as in Lemma \ref{lem:hs},
 and let $\mu \in \mathbb R$.
 Then $\widetilde\varGamma_0(z)$ is a  
${B}(\mu, s \, ; \,\mu, - s^{\prime})$-valued
continuous function
on $\Pi_{(0, \, +\infty)}$.
\end{lem}

\begin{proof} 
As was mentioned before,
$\widetilde\varGamma_0(z)$ defined by (\ref{eqn:laplace-1})
is 
a ${B}(0, s \, ; \,0 ,-s^{\prime})$-valued
continuous function  on $\Pi_{(0, \, +\infty)}$.
 Then the lemma directly
follows from the inequalities
%
\begin{gather}   \label{eqn:laplace-3}
\begin{split}
\Vert
\varGamma_0(z)
\Vert_{{B}(\mu, s \, ; \,\mu , -s^{\prime})}
\le
\Vert
\varGamma_0(z)
\Vert_{{B}(0, s \, ; \,0 , -s^{\prime})},
\quad   
z \in   \mathbb C
 \setminus [0, \,+\infty)   
\end{split}
\end{gather}
%
and
%
\begin{gather}    \label{eqn:laplace-3+}
\begin{split}
\Vert
\varGamma_0(z_1) - \varGamma_0(z_2)
\Vert_{{B}(\mu, s \, ; \,\mu , -s^{\prime})}
\le
\Vert
\varGamma_0(z_1) - \varGamma_0(z_2)
\Vert_{{B}(0, s \, ; \,0 , -s^{\prime})}     \\
\noalign{\vskip 4pt}   
( z_1, \, z_2 \in 
\mathbb C \setminus [0, \,+\infty)).    \qquad\qquad\qquad
\end{split}
\end{gather}
%
In order to show  (\ref{eqn:laplace-3}),
we shall use the fact
that
%
\begin{equation}  \label{eqn:laplace-4}
\langle D \rangle^{\mu} \varGamma_0(z) u
=  \varGamma_0(z) \langle D \rangle^{\mu} u
\end{equation}
%
for $u\in  S({\mathbb R}^3)$ and 
$z\in \mathbb C \setminus [0, \,+\infty)$.
We then have
%
\begin{gather}  \label{eqn:laplace-5-0}
\begin{split}
\Vert \varGamma_0(z) u \Vert_{H^{\mu, -s{\prime}}}
=
\Vert
\langle D \rangle^{\mu} \varGamma_0(z) u 
\Vert_{L^{2, -s^{\prime}}}                  
=
\Vert
 \varGamma_0(z) \langle D \rangle^{\mu}u 
\Vert_{L^{2, -s^{\prime}}}          \qquad   \\
\noalign{\vskip 4pt}
\le
\Vert
 \varGamma_0(z) 
\Vert_{B(0, s \, ; \,0, -s^{\prime})} 
\,
\Vert 
\langle D \rangle^{\mu}u
\Vert_{L^{2, s}}           \qquad\qquad \qquad \qquad           \\
\noalign{\vskip 4pt}   
=
\Vert
 \varGamma_0(z) 
\Vert_{B(0, s \, ; \,0, -s^{\prime})} 
\,
\Vert u \Vert_{H^{\mu, s}},  \qquad\qquad \qquad \qquad   \quad
\end{split}
\end{gather}
%
which implies (\ref{eqn:laplace-3}). 
In a similar fashion, 
we can prove (\ref{eqn:laplace-3+}).
\end{proof}


\begin{rem}  \label{rem:independence-1}
We should remark that ${H}^{\mu, \, s}({\mathbb R}^3)$ 
 in Lemma \ref{lem:lap-1} is
 a subset of ${L}^2({\mathbb R}^3)$ for $\mu \ge 0$, but
not neccesarily for $\mu <0$. Thus $\widetilde\varGamma_0(z)$ 
may depend on $\mu$ and $s$. Nonetheless, we have
the unique representation of $\widetilde\varGamma_0(z)$ 
on  $S({\mathbb R}^3)$,
a dense subset of ${H}^{\mu, \, s}({\mathbb R}^3)${\rm :}
%
\begin{equation}   \label{eqn:kernelrep-1}
\widetilde\varGamma_0(z) u(x)
= 
\int_{{\mathbb R}^3} 
\frac{e^{i \sqrt{z} |x-y|} }{\, 4\pi |x-y| \,} u(y) \, dy,
\qquad
u \in S({\mathbb R}^3),
\end{equation}
%
for every $z \in \Pi_{(0, \, +\infty)}$,
where $\mbox{\rm Im}\sqrt{z} \ge 0$.
This representation, together with 
the fact that $S({\mathbb R}^3)$ is dense
in ${H}^{\mu, \, s}({\mathbb R}^3)$ for 
any pair of $\mu$ and $s$,
 ensures that  
the extension of 
$\widetilde\varGamma_0(z){\big|}_{S({\mathbb R}^3)}$ 
to
${H}^{\mu, \, s}({\mathbb R}^3)$ is independent of
 $\mu$ and $s$
in a certain sense.
Howver, we shall not discuss about the uniqueness
of the extension
any longer.
In the discussions below, we shall mostly
deal with the extension of 
$\widetilde\varGamma_0(z){\big|}_{S({\mathbb R}^3)}$ 
to
${H}^{-1, \, s}({\mathbb R}^3)$.
\end{rem}

\begin{lem} \label{lem:lap-1-1}
Let $s$, $s^{\prime}$ satisfy
the same assumption as in Lemma \ref{lem:hs},
 and let $\mu \in \mathbb R$.
 Then $\widetilde\varGamma_0(z)$ is a  
${B}(\mu -2 , s \, ; \,\mu, - s^{\prime})$-valued
continuous function
on $\Pi_{(0, \, +\infty)}$.
\end{lem}


\begin{proof} 
We first note that
%
\begin{equation}  \label{eqn:laplace-5}
\langle D \rangle^{2} \varGamma_0(z) u
= u + (z +1) \varGamma_0(z)  u
\end{equation}
%
for $u\in S({\mathbb R}^3)$ and 
$z\in \mathbb C \setminus [0, \,+\infty)$;
cf. Jensen and Kato \cite[Lemma 2.1]{JensenKato}.
(See (\ref{eqn:bracket-D}) for 
the definition of $\langle D \rangle$.) 
We then combine (\ref{eqn:laplace-5})
with Lemma \ref{lem:lap-1}, and obtain the conclusion
if we appeal to the fact that $S({\mathbb R}^3)$ is
dense in ${H}^{\mu -2, s}({\mathbb R}^3)$.
\end{proof}


\vspace{10pt}
In the rest of the paper, we shall need a variant of 
Lemma \ref{lem:lap-1-1}, namely a version for
four-component
vector-valued functions, with $\mu = 1$ in the
form described in Lemma \ref{lem:lap-1-2}. Thus
$\widetilde\varGamma_0(z)$ 
in Lemma \ref{lem:lap-1-2} below denotes
a copy of $\widetilde\varGamma_0(z)$; see (\ref{eqn:mapsto}).

\begin{lem} \label{lem:lap-1-2}
Let $s$, $s^{\prime}$ satisfy
the same assumption as in Lemma \ref{lem:hs}.
 Then $\widetilde\varGamma_0(z)$ is a  
${\mathcal B}(-1 , s \, ; \, 1, - s^{\prime})$-valued
continuous function
on $\Pi_{(0, \, +\infty)}$.
\end{lem}


\vspace{10pt}

\subsection{The resolvent of the free Dirac operator $H_0$}

In view of (\ref{eqn:reslvnDrc-0}),
it is convenient for us to introduce 
the following operator
valued-functions $\varOmega_0^{+}(z)$ 
defined on $\overline{{\mathbb C}}_{+}$
and $\varOmega_0^{-}(z)$
on $\overline{{\mathbb C}}_{-}$
as follows:
%
\begin{equation}       \label{eqn:laplace-6-0}
\varOmega_0^{\pm}(z) 
=\widetilde\varGamma_0(z^2), \quad z \in
 \overline{{\mathbb C}}_{\pm},
\end{equation}
%
in other words,
%
\begin{equation}   \label{eqn:laplace-6}
\varOmega_0^{\pm}(z) 
=
\begin{cases}  
\varGamma_0(z^2) 
& \mbox{if } \; z \in {\mathbb C}_{\pm}, 
\\ {}    & \\
\varGamma_0^{\pm}(\lambda^2) 
  & \mbox{if } \;  z= \lambda \ge 0,
\\ {}    & \\
\varGamma_0^{\mp}(\lambda^2) 
  & \mbox{if } \; z= \lambda \le 0.
\end{cases}
\end{equation}
%
It follows from Lemma \ref{lem:lap-1-2} 
 that $\varOmega_0^+(z)$ ( resp. $\varOmega_0^-(z)$) is
a $\mathcal{B}(-1 , s \, ; \,1, - s^{\prime})$-valued
continuous function on $\overline{\mathbb C}_+$
(resp. $\overline{\mathbb C}_-$). 
Also, it follows from (\ref{eqn:resolventLapc-3-1})
that
%
\begin{equation}   \label{eqn:addtional1}
\varOmega_0^+(0) = \varOmega_0^-(0)
=\widetilde\varGamma_0(0).
\end{equation}
%

In order to get expressions of the extended resolvents
of the free Dirac operator $H_0$ in terms of 
$\varOmega_0^{\pm}(z)$ inroduced in (\ref{eqn:laplace-6}),
we shall exploit the decomposition
(\ref{eqn:reslvnDrc-0})
and a boundedness estimate of $H_0$ in some 
weighted Sobolev spaces which
is given as follow. 

\begin{lem}  \label{lem:dirac-bdd}
Let $\mu$ and $s^{\prime}$ be in $\mathbb R$. Then
%
\begin{equation*}   \label{eqn:dirac-bdd-2}
H_0 \in  \mathcal B (\mu, -s^{\prime} ; \mu-1, -s^{\prime}).
\end{equation*}
% 
\end{lem}

\begin{proof}
To prove the lemma, it is sufficient to
show that   each $D_j$,
($j=1$, $2$, $3$) is a bounded operator
from ${\mathcal H}^{\mu, -s^{\prime}}$ to 
${\mathcal H}^{\mu-1, -s^{\prime}}$.
This fact is a direct consequence of 
Umeda \cite[Lemma 2.1]{Umeda0}.
\end{proof}




\vspace{5pt}
\begin{lem}   \label{lem:lap-2}
Let $s$, $s^{\prime} > 1/2$, and $s + s^{\prime}>2$.
Then
$R_0(z) \in 
{\mathcal B}(-1, s \, ; \, 0, - s^{\prime})$ is
continuous in $z \in {\mathbb C}_{\pm}$.
Moreover, 
as ${\mathcal B}(-1, s \, ; \, 0, - s^{\prime})$-valued
functions,
 they
can possess continuous extenstions 
$R_0^{\pm}(z)$ to
$\overline{\mathbb C}_{\pm}$ respectively,
and
%
\begin{equation} \label{eqn:dirac-1}
R_0^{\pm}(z) = (H_0 + z) \varOmega_0^{\pm}(z),  
 \quad z \in \overline{\mathbb C}_{\pm}. 
\end{equation}
\end{lem}
%

\begin{proof}
We shall give the proof only for 
$z \in \overline{\mathbb C}_{+}$.
The proof for $z \in \overline{\mathbb C}_{-}$ is
similar.

As was mentioned before Lemma \ref{lem:dirac-bdd}, 
$\varOmega_0^{+}(z)$
is
a $\mathcal{B}(-1 , s \, ; \,1, - s^{\prime})$-valued
continuous function on $\overline{\mathbb C}_+$.
Combining this fact with (\ref{eqn:reslvnDrc-0}),
 (\ref{eqn:reslvnDrc-1}), 
the definition (\ref{eqn:laplace-6-0}) 
(or (\ref{eqn:laplace-6}))
of $\varOmega_0^{+}(z)$, 
 Lemma \ref{lem:lap-1-2} and 
Lemma \ref{lem:dirac-bdd} with
$\mu=1$, 
we see that
$R_0(z)= (H_0 + z) \varOmega_0^{+}(z) \in 
{\mathcal B}(-1, s \, ; \, 0, - s^{\prime})$ for
any $z \in {\mathbb C}_{+}$. 
Now it is evident that 
the second assertion of the lemma follows  from
Lemma \ref{lem:lap-1-2} and 
Lemma \ref{lem:dirac-bdd} with
$\mu=1$.
\end{proof}

Combining (\ref{eqn:dirac-1}) with (\ref{eqn:addtional1}),
we obtain a corollary to Lemma \ref{lem:lap-2}.

\begin{cor}  \label{cor:origin0}
Under the same assumption and the same notation as in
Lemma \ref{lem:lap-2},
%
\begin{equation*}   \label{eqn:additional2}
R_0^+(0)=R_0^-(0) = H_0\widetilde\varGamma(0).
\end{equation*}
%
\end{cor}

\vspace{4pt}


\begin{rem}
In \cite{IftMant}, Iftimovici and  M\u{a}ntoiu showed that
the limiting absorption principle
for the the free Dirac operator
$H_0= \alpha \cdot D + m \beta$, 
$m>0$, in ${\mathcal B}(0, 1 \, ; \, 0, - 1)$
holds  on the whole real line. With the result
exhibited in Lemma \ref{lem:lap-2},
together with the result in \cite{IftMant},
the limiting absorption principle
for the the free Dirac operator
$H_0= \alpha \cdot D + m \beta$
has been established 
for all $m \ge 0$.
\end{rem}


\begin{lem} \label{lem:ker3-2}
For $f \in { \mathcal S }$ and $z \in {\mathbb C}_{\pm}$
%
\begin{eqnarray} \label{eqn:3-2}
{}&R_0(z)f (x) \qquad \qquad \qquad \qquad
  \qquad \qquad \qquad \qquad \qquad \qquad \\
{}&= \displaystyle{
\int_{{\mathbb R}^3}  \Big(
i\, \frac{\alpha \cdot (x-y)}{|x-y|^2} \pm 
z \, \frac{\alpha \cdot (x-y)}{|x-y|}  + zI_4 \Big)
\frac{e^{\pm i z |x-y|}}{4\pi |x-y|}
} f(y) \, dy.   \nonumber
\end{eqnarray}
\end{lem}
%
\begin{proof}
We first recall (\ref{eqn:kernelrep-1}),
which we can write as
%
\begin{equation}   \label{eqn:kernelrep-1-1}
\varGamma_0(z^2) f(x)
= 
\int_{{\mathbb R}^3} 
\frac{e^{ \pm i z |y|} }{\, 4\pi |y| \,} f(x-y)  \, dy,
\qquad
 f \in {\mathcal S}, \;\; z \in {\mathbb C}_{\pm}.
\end{equation}
%
We then combine (\ref{eqn:laplace-6-0}) and 
(\ref{eqn:dirac-1}), and make differentiation
under the integral sign in (\ref{eqn:kernelrep-1-1}),
which gives
\begin{gather}   \label{eqn:3-2-0}
\begin{split}
{}&R_0(z)f (x) = 
\int_{{\mathbb R}^3}  
\frac{e^{\pm i z |y|}}{4\pi |y|}
             (\alpha \cdot D_x +z I_4)      f(x-y) \, dy ,
\quad z  \in {\mathbb
C}_{\pm}.  
\end{split} 
\end{gather}
%
Noting the fact that 
%
\begin{equation*}    \label{eqn:differentiation}
D_x f(x-y)= - D_y\big( f(x-y) \big),
\end{equation*}
%
and making integration by parts on the right hand
of (\ref{eqn:3-2-0}) 
implies that
%
\begin{gather}   \label{eqn:3-2-1}
\begin{split}
R_0(z)f (x)  &= 
\int_{{\mathbb R}^3}  \Big(
i\, \frac{\alpha \cdot y}{|y|^2} \pm 
z \, \frac{\alpha \cdot y}{|y|}  + zI_4 \Big)
\frac{e^{\pm i z |y|}}{4\pi |y|}
                   f(x-y) \, dy      \\
\noalign{\vskip 4pt}
{}& \qquad\quad (z  \in {\mathbb C}_{\pm}).  
\end{split} 
\end{gather}
%
A change of variables in (\ref{eqn:3-2-1})
yields (\ref{eqn:3-2}).
(See also Thaller \cite[p.39]{Thaller}.)
\end{proof}


\begin{lem} \label{lem:riesz3-3}
For $f \in { \mathcal S }$ 
%
\begin{equation*} \label{eqn:3-3}
R_0^{+}(0) f = R_0^{-}(0) f = A f,
\end{equation*}
where  $A$ is the singular integral operator 
defined by {\rm(\ref{eqn:sio-1})}.
\end{lem}
%
\begin{proof}
In view of Corollary \ref{cor:origin0},
we only need to give the proof for $R_0^+(0)$. 

Let $f \in { \mathcal S }$, and 
let $\{ z_n \} \subset {\mathbb C}_{+}$
be a sequence such that $z_n \to 0$ as $n \to \infty$.
It follows from Lemma \ref{lem:lap-2} that
$R_0(z_n)f \to R^+_0(0)f$ in 
${\mathcal L}^{2, -s^{\prime}}$ as $n \to
\infty$.
This fact implies that there exists a subsequece
$\{ z_{n^{\prime}} \} \subset\{ z_n \}$ 
suth that
%
\begin{equation}   \label{eqn:convergence-1}
R_0(z_{n^{\prime}})f(x) \to R^+_0(0)f(x)
\;\; \mbox{ a.e.} x \in {\mathbb R}^3 
\;\;\;  
\mbox{ as } n^{\prime} \to \infty.
\end{equation}
%

On the other hand, Lemma \ref{lem:ker3-2},
togher with the Lebesgue
dominated convergence theorem, implies
that 
%
\begin{equation}     \label{eqn:convergence-2}
R_0(z_{n})f(x) \to 
\int_{{\mathbb R}^3}  
i\, \frac{\alpha \cdot (x-y)}{4\pi |x-y|^3}
 f(y) \, dy = Af(x)
\;\;\;  
\mbox{ as } n  \to \infty
\end{equation}
%
for each $x \in {\mathbb R}^3$.
The conclusion of the lemma now 
follows from (\ref{eqn:convergence-1}) and 
(\ref{eqn:convergence-2}).
\end{proof}
 

\begin{lem} \label{lem:riesz3-4}
Let $s$, $s^{\prime} > 1/2$, and $s + s^{\prime}>2$.
 Then
$A$ can be continuously extended to
an operator in 
${\mathcal B}(-1, s \, ; \, 0, - s^{\prime})$. 
\end{lem}
%
%
\begin{proof}
Since $\mathcal S$ is dense in
${\mathcal H}^{-1, s}$, Lemmas \ref{lem:lap-2}
and  \ref{lem:riesz3-3} directly
imply the lemma.
\end{proof}

In the rest of the paper, we shall denote the extension
in Lemma \ref{lem:riesz3-4} 
by $A$ again. Thus we have
%
\begin{equation*}
R_0^+(0) =R_0^-(0) = A \; 
\mbox{ in } \; 
{\mathcal B}(-1, s \, ; \, 0, - s^{\prime}).
\end{equation*}
%


\begin{lem} \label{lem:riesz3-5-0}
Let $s > 1/2$. Then 
%
\begin{equation}   \label{eqn:adjoint-1}
H_0 A g = g
\end{equation}
%
for all $g \in {\mathcal L}^{2, s}$.
\end{lem}
%

\begin{proof}
Let $g \in {\mathcal L}^{2, s}$ be given.
We  then start with the fact that
%
\begin{equation}   \label{eqn:adjoint-1+}
(H_0 - z) R_0(z)g =g  \quad ( \forall z \in {\mathbb C}_+ ).
\end{equation}
%
Choose $s^{\prime} > 1/2$ so that
$s + s^{\prime} >2$. 
We see from Lemmas \ref{lem:lap-2},
\ref{lem:riesz3-3} and \ref{lem:riesz3-4}
that
%
\begin{equation}    \label{eqn:pair-4}
R_0(i/n) g \to R_0^+(0)g = Ag  \;\; \mbox{ \rm in }  \;\; 
{\mathcal L}^{2, -s^{\prime}}
\quad
{\mbox \rm as } \;\; n   \to \infty .
\end{equation}
%
Lemma \ref{lem:dirac-bdd}, with $\mu=0$,
and (\ref{eqn:pair-4})
imply that
%
\begin{equation}    \label{eqn:pair-5}
(H_0 -\frac{i}{\, n \,}) 
 R_0(\frac{i}{\, n \,}) g \to  H_0 Ag  \;\; \mbox{ \rm
in }  \;\;  {\mathcal H}^{-1, -s^{\prime}}
\quad
{\mbox \rm as } \;\; n \to \infty.
\end{equation}
%
Since, by (\ref{eqn:adjoint-1+}),
%
\begin{equation*}    \label{eqn:pair-5+}
(H_0 - \frac{i}{\, n \,} ) R_0(\frac{i}{\, n \,})g 
=g  \quad  \mbox{ for }\forall n,
\end{equation*}
%
we find that 
(\ref{eqn:pair-5}) yields (\ref{eqn:adjoint-1}).
\end{proof}


\vspace{5pt}

We shall need  
Lemma 2.4 of Jensen and Kato \cite{JensenKato},
which we shall rewrite in a suitable form
to our setting (cf. Lemma \ref{lem:riez3-5-7} below),
where the operators
$-\Delta$
and 
$\widetilde\varGamma_0(0)$
act on four-component vector functions. 
The reader should note
that $\widetilde\varGamma_0(0)$
is the same as $G_0$ in the 
Jensen-Kato paper. 
See (\ref{eqn:resolventLapc-1}) and
(\ref{eqn:resolventLapc-3-1}).



\begin{lem}[\textbf{Jensen-Kato}]   \label{lem:riez3-5-7}
Let $s >1/2$. Then

\vspace{3pt}

{\rm(i)} $(-\Delta) \widetilde\varGamma_0(0) g = g$ \ 
 for all \  $g \in {\mathcal H}^{-1, s}.$

\vspace{2pt}

{\rm(ii)} $ \widetilde\varGamma_0(0) (-\Delta)f = f \;$ 
if $\; f \in  {\mathcal L}^{2, -3/2} \ and \ 
(-\Delta)f \in {\mathcal H}^{-1, s}$.
\end{lem}
%

\vspace{4pt}

\begin{lem} \label{lem:riez3-5-8}
Let $s >1/2$. Then \ 
 $\widetilde\varGamma_0(0) H_0 \, g = Ag$  \ 
 for all $g \in {\mathcal L}^{2, s}.$
\end{lem}
%
%
\begin{proof}
Let $g \in {\mathcal L}^{2, s}$ be given.
Noting that  $H_0^2 = -\Delta$ (cf. (\ref{eqn:diracsquare})),
 we have
%
\begin{equation}   \label{eqn:identity-10}
(-\Delta) A g 
=H_0 (H_0 Ag)            
=H_0 \, g  
\end{equation}
%
where we have used Lemma \ref{lem:riesz3-5-0}
in the second equality.
Since $H_0 \, g \in {\mathcal H}^{-1, s}$ by Lemma \ref{lem:dirac-bdd},
it follows from (\ref{eqn:identity-10}) that
$(-\Delta) A g  \in {\mathcal H}^{-1, s}$.

On the other hand, we find, by Lemma \ref{lem:riesz3-4},
that $Ag \in {\mathcal L}^{2, -3/2}$,
because we can choose $s^{\prime}$ so 
that $1/2 < s^{\prime} \le 3/2$ \ and
$s + s^{\prime} >2$. (Choose 
$s^{\prime}$ so 
that $\max (s, \, 2-s) < s^{\prime} \le 3/2$.)

Now we can apply Lemma \ref{lem:riez3-5-7}(ii) with $f$ 
replaced by $Ag$, and obtain
%
\begin{equation}   \label{eqn:identity-11}
\widetilde\varGamma_0(0)(-\Delta) A g 
=Ag. 
\end{equation}
%
It follows from (\ref{eqn:identity-10}) that
the left hand side of (\ref{eqn:identity-11})
 equals $\widetilde\varGamma_0(0)
H_0g$. This proves the conclusion of the lemma.
\end{proof}
%
%
\vspace{2pt}
%

\begin{lem} \label{lem:riesz3-5}
If $f \in {\mathcal L}^{2, -3/2}$ 
  and  $H_0 f \in {\mathcal L}^{2, s}$ 
for some $\, s > 1/2$,
 then $AH_0 f = f$.
\end{lem}
%
%
\begin{proof}
Put
%
\begin{equation*}  \label{eqn:identity-12}
g= H_0 f.
\end{equation*}
%
By assumption of the lemma, we see that $g \in {\mathcal L}^{2, s}$ 
for some $\, s > 1/2$.
It follows from Lemma \ref{lem:riez3-5-8}
that $Ag = \widetilde\varGamma_0(0) H_0 \, g$, {\it i.e.},
%
\begin{equation*}    \label{eqn:identity-13}
AH_0f = \widetilde\varGamma_0(0) H_0 H_0f
  = \widetilde\varGamma_0(0) (-\Delta)f.
\end{equation*}
%
Since $(-\Delta)f=H_0g \in {\mathcal H}^{-1, s}$  by
Lemma \ref{lem:dirac-bdd}, 
it follows from assertion (ii) of 
Lemma \ref{lem:riez3-5-7} that
$\widetilde\varGamma_0(0) (-\Delta)f=f$. Thus $AH_0f =f$.
\end{proof}







\vspace{15pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Pointwise estimates for zero modes and their continuity}

\begin{proof}[{\bf Proof of Theorem \ref{thm:th-2-2}}]
We first prove assertion (i).
Let $f$ be a zero mode of the operator
(\ref{eqn:1-1}). Then we have
%
\begin{equation}   \label{eqn:proof-1}
Hf = \big( \alpha \cdot D + Q(x) \big) f =0,
\quad
f \in \mbox{Dom}(H) = {\mathcal H}^1 .
\end{equation}
%
It follows from (\ref{eqn:proof-1}) and Assumption (A)
that
%
\begin{equation}   \label{eqn:proof-2}
H_0f =(\alpha \cdot D) f = - Q(x) f
 \in 
{\mathcal L}^{2, \rho}.
\end{equation}
%
(Recall  (\ref{eqn:reslvnDrc-00}) for the definiton of $H_0$.)
Since $\rho >1 >1/2$ by assumption of the
theorem, we can apply Lemma \ref{lem:riesz3-5} 
to (\ref{eqn:proof-2})
and get
%
\begin{equation}   \label{eqn:proof-3}
f = AH_0 f = - A  Q f.
\end{equation}
%%%%%where we have used Lemmas \ref{lem:riesz3-3} and 
%%%%%%\ref{lem:riesz3-4} in the third equality of 
%%%%%%%(\ref{eqn:proof-3}). 
It follows from (\ref{eqn:proof-3}) and
 Lemma \ref{lem:siolem-1} 
that
$f \in {\mathcal L^2} \cap {\mathcal L}^6$.
It follows from (\ref{eqn:proof-3}) 
again and Lemma \ref{lem:siolem-3} 
that $f \in {\mathcal L}^{\infty}$.
This fact, together with  (\ref{eqn:proof-3})
and Assumption(A),
implies that
%
\begin{equation}   \label{eqn:proof-4}
| f(x) |
 \le 
C \Vert f \Vert_{{\mathcal L}^{\infty}} 
\int_{{\mathbb R}^3}
 \frac{1}{\, |x-y|^2 \, \langle y \rangle^{\rho} \,} 
\, dy.
\end{equation}
%
Noting that $\rho >1$ by assumption, 
and  applying Lemma 11.1 in Appendix
of Umeda \cite{Umeda} to the integral 
in (\ref{eqn:proof-4}),
we get the desired inequalities.

We next prove assertion (ii) by
utilizing  (\ref{eqn:proof-3}):
%
\begin{equation*}   \label{eqn:proof-4+1}
f(x) = 
-\displaystyle{
\int_{{\mathbb R}^3}
  i \, \frac{\alpha \cdot (x - y)}{4\pi |x-y|^3} Q(y) f(y) \, dy.
}
\end{equation*}
%
Let $x_0$ be any point in ${\mathbb R}^3$, and
let
$\varepsilon >0$ be given.
We choose  $r>0$ so that
%
\begin{equation}   \label{eqn:proof-4+2}
\displaystyle{
\int_{|y| \le 2r}
  \frac{1}{\, |y|^2 \,} \, dy < \varepsilon}.
\end{equation}
%
We then decompose $f(x)$ into two
parts:
%
\begin{gather}   \label{eqn:proof-4+3}
\begin{split}
f(x)
&= - \Big( 
\int_{B(x, \,2r)} 
+ \int_{E(x,\,2r)}  
\Big)
 \, i \, 
   \frac{\alpha \cdot (x - y)}{4\pi |x-y|^3} Q(y) f(y) \, dy   \\
\noalign{\vskip 4pt}
&=: f_b(x) + f_e(x),     
\end{split}
\end{gather}
%
where
%
\begin{equation*}   \label{eqn:proof-4+4}
B(x, \, 2r) = \{ \, y \; \big| 
\,|x - y| \le 2r \, \}, 
\quad 
E(x, \,2r) =  \{ \, y \; \big| 
\,|x - y| > 2r \, \}.
\end{equation*}
%
Since each $\alpha_j$ is a unitary matrix,
it follows from (\ref{eqn:proof-4+2})
and (\ref{eqn:proof-4+3}) that
%
\begin{equation}    \label{eqn:proof-4+5}
|f_b(x)| < \frac{3}{\, 4\pi \,} C_q C_f \,\varepsilon  
\quad \mbox{ for }\; \forall x
\in {\mathbb R}^3 ,
\end{equation}
%
where $C_q$ is a constant determined 
by  (\ref{eqn:2-1}) in Assumption (A)
and $C_f$ is a constant described in 
the inequality (\ref{eqn:2-2}),
which we have just proved in
the first half of the proof.
It follows from the 
definition of $f_e(x)$ that
%
\begin{gather}   \label{eqn:proof-4+6}
\begin{split}
{}&f_e(x) - f_e(x_0)    \\
\noalign{\vskip 4pt}
&= 
\int_{{\mathbb R}^3} \!
\Big\{
1_{E(x_0, \, 2r)}(y) 
\frac{\alpha \cdot (x_0 - y)}{4\pi |x_0-y|^3}  
-
1_{E(x, \, 2r)}(y) 
\frac{\alpha \cdot (x - y)}{4\pi |x-y|^3} 
\Big\}   \\
\noalign{\vskip 4pt}
&\qquad\qquad \qquad\qquad \qquad\qquad \qquad\qquad \qquad\qquad 
   \times Q(y) f(y) \, dy .
\end{split}
\end{gather}
%
To apply the Lebesgue dominated convergence theorem
to the integral in (\ref{eqn:proof-4+6}),
we need the following two facts that
%
\begin{equation}  \label{eqn:proof-4+7}
|x_0 - y | \ge  r
\quad \mbox{ if } \; 
 |x_0 - x| <r, \; |x-y| > 2r.
\end{equation}
%
and that
%
\begin{equation}  \label{eqn:proof-4+8}
|x - y | \ge  \frac{2}{\, 3 \,} |x_0 -y|
\quad \mbox{ if } \; 
 |x_0 - x| <r, \; |x-y| > 2r
\end{equation}
%
(use the inequality $|x-y| \ge |x_0 -y| - |x_0 -x|$).
We can deduce from
(\ref{eqn:proof-4+7}) and (\ref{eqn:proof-4+8}) that
%
\begin{gather}    \label{eqn:proof-4+9}
\begin{split}
{}&\Big| 
1_{E(x, \, 2r)}(y) 
\frac{\alpha \cdot (x - y)}{4\pi |x-y|^3} Q(y) f(y)  \Big| \\  
&\quad \le
1_{E(x_0, \, r)}(y) 
\frac{3}{\, 4\pi \,} 
\Big(
\frac{2}{\, 3 \,} |x_0 -y|
\Big)^{-2}   \,
 \big|  Q(y) f(y)  \big|
\end{split}
\end{gather}
%
whenever $|x_0 - x| < r$.
It is
straightforward that
the estimate (\ref{eqn:proof-4+9}) 
implies 
%
\begin{gather}   \label{eqn:proof-4+10}
\begin{split}
{}&\big|
\mbox{the integrand in (\ref{eqn:proof-4+6})}
\big|  \\
\noalign{\vskip 4pt}
&\le 
1_{E(x_0, \, r)}(y) 
\frac{3}{\, 4\pi \,} 
\Big(
1+ (\frac{3}{\, 2 \,})^2 
\Big)
|x_0 -y|^{-2}   \,
 \big|  Q(y) f(y)  \big| 
\end{split}
\end{gather}
%
whenever $|x_0 - x| < r$.
In view of
(\ref{eqn:2-1}) in Assumption (A)
and the inequality (\ref{eqn:2-2}),
the function on the right hand side
of (\ref{eqn:proof-4+10})
is integrable on ${\mathbb R}^3$.
Thus, we can apply  
the Lebesgue dominated convergence theorem
to the integral in (\ref{eqn:proof-4+6}),
and conclude that
%
\begin{equation}    \label{eqn:proof-4+11}
\lim_{x \to x_0} \big( f_e(x) - f_e(x_0) \big)=0.
\end{equation}
%
Combining (\ref{eqn:proof-4+11})
with 
both (\ref{eqn:proof-4+3}) and (\ref{eqn:proof-4+5}) yields
%
\begin{equation*}   \label{eqn:proof-4+12}
 \limsup_{x \to x_0}
\big| f(x) - f(x_0) \big| 
\le 
2\times
\frac{3}{\, 4\pi \,} C_q C_f \,\varepsilon .     
\end{equation*}
%
Since $\varepsilon$ was arbitrary,
this completes the proof of assertion (ii).
\end{proof}

\vspace{15pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Nonexistence of zero resonances}


\begin{lem} \label{lem:sobolev-1}
If $f \in {\mathcal L}^{2}$ and  
 $(\alpha \cdot D) f \in {\mathcal L}^{2}$,
 then $f \in{\mathcal H}^{1}$.
\end{lem}
%
%
\begin{proof}
We take the Fourier transform of $(\alpha \cdot D)f$,
and we have
%
\begin{equation}    \label{eqn:proof-5}
{\mathcal F}[ (\alpha \cdot D)f]
= (\alpha \cdot \xi) \hat f,
\end{equation}
%
where
%
\begin{equation*}    \label{eqn:proof-5+}
{\mathcal F}[f](\xi) = \hat f(\xi)
= (2\pi)^{-2/3} \int_{{\mathbb R}^3}
e^{-ix\cdot \xi}  \, f(x) \, dx.
\end{equation*}
%
Then by using assumption of the lemma 
and (\ref{eqn:proof-5}), we see
that
%
\begin{gather}   \label{eqn:proof-6}
\begin{split}
\infty > \Vert (\alpha \cdot D)f \Vert_{{\mathcal L}^2}^2
&=
\int_{{\mathbb R}^3}  
| (\alpha \cdot \xi) \hat f  (\xi) | ^2 \, d\xi   \\
&=
\int_{{\mathbb R}^3}
 < \!\! (\alpha \cdot \xi) \hat f  (\xi) ,
 (\alpha \cdot \xi) \hat f  (\xi) \!\! >_{\!{}_{\mathbb C}}  \, d\xi \\
&=
\int_{{\mathbb R}^3}
 |\xi|^2 |\hat f  (\xi)|^2  \, d\xi,
\end{split}
\end{gather}
%
where $< \cdot, \, \cdot>_{\!{}_{\mathbb C}}$ 
denotes the inner product of ${\mathbb C}^4$.
In the third equality of (\ref{eqn:proof-6}), we have used the
anti-commutation  relation of the Dirac matrices: 
 $\alpha_j \alpha_k + \alpha_k \alpha_j = 2\delta_{jk}$,
$\delta_{jk}$ being $=1$ if $j=k$,
 and $=0$ if $j\not=k$.
Since, by assumption of 
the lemma, $f \in {\mathcal L}^{2}$,
 the coclusion of the lemma
follows  from (\ref{eqn:proof-6}).
\end{proof}


\begin{proof}[{\bf Proof of Theorem \ref{thm:th-2-1}}]
Let $f$ satisfy the assumption of the theorem:
 $f\in {\mathcal L}^{2,-s}$
for  some $s$ with $1/2 < s < \min\{3/2, \, \rho -3/2\}$. 
In the same manner as in (\ref{eqn:proof-2}) and 
(\ref{eqn:proof-3}), we can show that
%
\begin{equation}  \label{eqn:proof-7}
H_0f=(\alpha \cdot D) f =- Qf \in  
{\mathcal L}^{2,\rho-s},
\end{equation}
%
and that
%
\begin{equation}  \label{eqn:proof-8}
f= -A Qf.
\end{equation}
%
Note that $\rho -s >3/2>1/2$, which we have used 
to apply Lemma \ref{lem:riesz3-5}
 in showing 
 (\ref{eqn:proof-8}).
It follows from Lemma \ref{lem:siolem-2}(i)
that $f \in {\mathcal L}^2$.
This fact, together with (\ref{eqn:proof-7})
and Lemma \ref{lem:sobolev-1}, gives 
the conclusion of the theorem.
\end{proof}



\vspace{20pt}

{\bf Acknowledgement:}  T.U. would like to express his gratitude to 
Michael Loss
for the hospitality during his visit to 
Georgia Institute of Technology, USA, in
April, 2002. Discussions with Michael were one of the 
motivations of the present
paper.
Also, he would like to express his thanks to the Department
of Mathematics, the University of 
Alabama at Birmingham, USA, 
for their hospitality.
Part of the present paper was done during
his stay there in March and September, 2006.


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\begin{thebibliography}{99}

\bibitem{AdamMuratoriNash1} C. Adam, B. Muratori and C. Nash,
    {\em  Zero modes of the Dirac operator in three dimensions},
    Phys. Rev. D \textbf{60} (1999), 125001-1 -- 125001-8.
%
\bibitem{AdamMuratoriNash2} C. Adam, B. Muratori and C. Nash,
    {\em  Degeneracy of zero modes of the Dirac operator 
   in three dimensions},
    Phys. Lett. B \textbf{485} (2000), 314--318
%
\bibitem{AdamMuratoriNash3} C. Adam, B. Muratori and C. Nash,
    {\em  Multiple zero modes of the Dirac operator in three dimensions},
    Phys. Rev. D \textbf{62} (2000), 085026-1 -- 085026-9.
%
\bibitem{BalinEvan1} A.A. Balinsky and W.D. Evans, 
   {\em On the zero modes of Pauli operators}, J. Funct. Analysis, 
    \textbf{179}
   (2001), 120--135.
%
\bibitem{BalinEvan2} A.A. Balinsky and W.D. Evans, 
   {\em On the zero modes of Weyl-Dirac operators and 
   their multiplicity}, Bull. London Math. Soc.,  
    \textbf{34} (2002), 236--242.
%
\bibitem{BalinEvan3} A.A. Balinsky and W.D. Evans, 
    {\em Zero modes of Pauli and Weyl-Dirac operators},
   Advances in differential equations and  mathematical
  physics (Birmingham, AL, 2002), 1--9, Contemp. Math.,
   \textbf{327}, Amer. Math. Soc., Providence, Rhode Island,
   2003.
%
\bibitem{BalslevHellfer} E. Balslev and B. Helffer,  
{\em Limiting
absorption principle and 
          resonances for the Dirac operator},
 Adv. Appl. Math. \textbf{13} (1992), 186--215.
%
\bibitem{BugliaroFeffermanGraf} L. Bugliaro, C. Fefferman
and G.M. Graf,
{\em A Lieb-Thirring bound for a magnetic Pauli Hamiltonian, II},
Rev. Mat. Iberoamericana \textbf{15} (1999), 593--619.
%
\bibitem{Elton}  D.M. Elton,
   {\em   The local structure of zero mode producing
   magnetic potentials},
   Commun. Math. Phys. \textbf{229} (2002), 121--139.
%
\bibitem{ErdosSolovej1} L. Erd\"os and J.P. Solovej,
   {\em  The kernel of Dirac operators on ${\mathbb S}^2$
     and ${\mathbb R}^2$},
   Rev. Math. Phys. \textbf{13}  (2001),  1247--1280.
%
\bibitem{ErdosSolovej2} L. Erd\"os and J.P. Solovej,
   {\em  Uniform Lieb-Thirring inequality for the 
   three-dimensional Pauli operator with a  strong
  non-homogeneous magnetic field},
   Ann. Henri Poincar\'e \textbf{5}  (2004),  671--741.
%
\bibitem{ErdosSolovej3} L. Erd\"os and J.P. Solovej,
   {\em  Magnetic Lieb-Thirring inequalities with optimal
   dependence on the field strength},
   J. Statist. Phys. \textbf{116}  (2004),  475--506.
%
\bibitem{IftMant} A. Iftimovici and M. M\u{a}ntoiu,
  {\em Limiting absorption principle at critical values for
   the Dirac operator}, Lett. Math. Phys., \textbf{49} (1999),
  235--243.
%
\bibitem{JensenKato} A. Jensen and T. Kato,
   {\em Spectral properties of Schr\"odinger operators
    and time-decay of the wave functions}, Duke Math. J.,
       \textbf{46} (1979), 583--611.
%
\bibitem{KalfYamada} H. Kalf and O. Yamada, 
  {\em Essential self-adjointness of $n$-dimensional
   Dirac operators with a variable mass term},  
  J. Math. Phys. \textbf{42} (2001), 2667--2676.
%
\bibitem{KalfOkajiYamada} H. Kalf, T. Okaji and O. Yamada, 
  {\em Absence of eigenvalues of Dirac operators with 
   potentials diverging at infinity},  Math. Nachr. 
   \textbf{259} (2003), 19--41.
%
\bibitem{FrohlichLiebLoss} J. Fr\"ohlich, E.H. Lieb and M. Loss,
  {\em Stability of Coulomb systems with magnetic fields. I.
    The one-electron Atom},
    Commun. Math. Phys. \textbf{104} (1986),251--270.
%
\bibitem{LossYau} M. Loss and H.T. Yau,
   {\em Stability of Coulomb systems with magnetic fields. III.  
      Zero energy bound states of the Pauli operators}, Commun. 
       Math. Phys. \textbf{104} (1986), 283--290.
\bibitem{Pladdy} C. Pladdy, 
  {\em Asymptotics of the resolvent of the Dirac operator with
  a scalor short-range potential},  Analysis \textbf{21} (2001),
 79-97.
%
\bibitem{PSU1} C. Pladdy, Y. Sait\={o} and T. Umeda,  
{\em Resolvent
estimates of the Dirac
           operators},  Analysis \textbf{15} (1995),
 123--149.
\bibitem{PSU2}  C. Pladdy, Y. Sait\={o} and T. Umeda,
  {\em Radiation
condition for Dirac operators}, J. Math. Kyoto Univ.
\textbf{37} (1997), 567--584.
%
\bibitem{RozenblumShirokov} G. Rozenblum and N. Shirokov,
   {\em Infiniteness of zero modes for the Pauli operator
   with singular magnetic field},
   J. Funct. Analysis \textbf{233} (2006), 135--172.
%
\bibitem{SchmidtYamada} K. M. Schmidt and O. Yamada, 
   {\em Spherially symmetric Dirac operators with
  variable mass and potentials infinity at infinity},
   Publ. Res. Inst. Math. Sci., Kyoto Univ. \textbf{34} (1998),
211--227.
%
\bibitem{Stein} E.M. Stein, 
    {\em Singular integrals and differential properties
   of functions}, Princeton University Press, Princeton, 
  New Jersey, 1970.
%
\bibitem{Thaller} B. Thaller,
     {\em The Dirac equation}, Springer-Verlag, Berlin Heidelberg,
      1992.
%
\bibitem{Umeda0}  T. Umeda,
  {\em The action of $\sqrt{-\Delta \,}$ on weighted Sobolev
  spaces},  Lett. Math. Phys. \textbf{54} (2000), 301--313.
%
\bibitem{Umeda}  T. Umeda,
  {\em Generalized eigenfunctions of relativistic Schr\"odinger
operators I}, Electron. J. Diff. Eqns., \textbf{2006} (2006), 
No. 127,
pp.1--46. 

%
\bibitem{Yamada} O. Yamada,   {\em  On the spectrum of
  Dirac operators with unbounded potential at infinity},
   Hokkaido Math. J. \textbf{26} (1997), 439--449.

\end{thebibliography}
\end{document}
 
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