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\define\bb{\bold{B}}
\define\pp{\Bbb{P}}
\define\hh{\Bbb{H}}
\centerline{\bf On Moments of Negative Eigenvalues for the Pauli Operator}
\bigskip

\centerline{Zhongwei Shen*\footnote{
Research  supported in part by the AMS Centennial
Research Fellowship, the MSRI at Berkeley, California, and
the NSF grant DMS-9596266.}}

\centerline{Department of Mathematics}
\centerline{University of Kentucky}
\centerline{Lexington, KY 40506}
\centerline{E-mail: shenz\@ms.uky.edu}
\centerline{Tel: (606) 257-3231}
\centerline{Fax: (606) 257-4078}
\bigskip
\bigskip
\centerline{\bf Dedicated to the Memory of Professor Ruilin Long}
\bigskip
\bigskip
\noindent{\bf Abstract.}\ \ 
This paper concerns
the three-dimensional Pauli operator 
$$
\Bbb{P}=\big( \bold{\sigma}
 \cdot (\bold{p}-\bold{A}(x))\big)^2 +V(x)
$$
with a non-homogeneous
magnetic field $\bold{B}=\text{curl\,}\bold{A}$.
The following Lieb-Thirring type inequality
for the moment of negative eigenvalues is established:
$$
\sum_{\lambda_j<0} |\lambda_j|
\le C_1
\int_{\Bbb{R}^3}|V(x)|_-^{5/2}dx
+
C_2\int_{\Bbb{R}^3} [b_p(x)]^{3/2} |V(x)|_-dx
$$
where $p>3/2$ and $b_p(x)$ is the $L^p$ average of $|\bb|$
over certain cube centered at $x$ with a side length scaling
like $|\bb|^{-1/2}$. We also show that, if $\bb$ has a
constant direction,
$$
\sum_{\lambda_j<0} |\lambda_j|^\gamma
\le 
C_{1,\gamma}
\int_{\Bbb{R}^3}|V(x)|_-^{\gamma+3/2}dx
+
C_{2,\gamma}
\int_{\Bbb{R}^3} b_p(x) |V(x)|_-^{\gamma +1/2}dx
$$
where $\gamma>1/2$ and $p>1$.
\vfill\eject

\centerline{\bf 1. Introduction}

Consider the Pauli operator
$$
\Bbb{P}=\Bbb{P}_0 +V(x)
=\big(\bold{\sigma}\cdot(\bold{p}-\bold{A}(x))\big)^2
+V(x), 
\tag 1.1
$$
acting on $L^2(\Bbb{R}^3,\Bbb{C}^2)$. Here
$\bold{\sigma}=(\sigma_1,\sigma_2,\sigma_3)$ denotes
the vector of Pauli matrices, $\bold{p}=-i\nabla$, and
$\bold{A}(x)=(A_1(x),A_2(x),A_3(x))\in L^2_{loc}(\Bbb{R}^3,\Bbb{R}^3)$ 
is a vector
potential.
We shall use $\bold{B}=\text{curl\,}\bold{A}$
to denote the magnetic field generated by the potential $\bold{A}$.
The goal of this paper is to establish the Lieb-Thirring type
estimates for the moments of negative eigenvalues of $\Bbb{P}$:
$$
\Cal{M}_\gamma=\sum_{\lambda_j<0}|\lambda_j|^\gamma,\ \ \ \ 
\ \ \gamma >0.
\tag 1.2
$$
We remark that in quantum mechanics $\Bbb{P}$ is used to describe
the motion of a charged spin-1/2 particle in an electromagnetic
field. Lieb-Thirring type estimates
are an important tool in the study of stability of matter
and in semi-classical analysis \cite{13,10,11,12,2,4,5}.

In the case of Schr\"odinger operator $-\Delta +V(x)$
in $\Bbb{R}^n$,
the classical Lieb-Thirring inequality \cite{13} states that
$$
\Cal{M}_\gamma
\le L_{\gamma,n}\int_{\Bbb{R}^n}
|V(x)|_-^{\gamma +n/2}dx
\tag 1.3
$$
where $\gamma >0$, $n\ge 2$, and $|V|_-=\max (-V,0)$ denotes
the negative part of $V$
(in the case $\gamma=0$ and $n\ge 3$, (1.3) is the well-known
Rosenblum-Lieb-Cwickel estimate \cite{15}).
For the Pauli operator $\Bbb{P}$ in $\Bbb{R}^3$
with a constant field, it was shown in \cite{11}
that
$$
\Cal{M}_\gamma
\le 
C_{1,\gamma}\int_{\Bbb{R}^3} |V(x)|_-^{\gamma+3/2}dx
+
C_{2,\gamma}\, 
|\bold{B}|\int_{\Bbb{R}^3} |V(x)|_-^{\gamma+1/2}dx
\tag 1.4
$$
for $\gamma >1/2$.
Subsequently,
L.~Erd\"os \cite{3} initiated 
the study of Lieb-Thirring estimates for Pauli operators
with non-homogeneous fields.
He observed that the direct extension of (1.4) to the case of
the non-constant fields:
$$
\Cal{M}_\gamma
\le 
C_{1,\gamma}\int_{\Bbb{R}^3} |V(x)|_-^{\gamma+3/2}dx
+
C_{2,\gamma}
\int_{\Bbb{R}^3}|\bold{B}(x)|\, |V(x)|_-^{\gamma+1/2}dx
\tag 1.5
$$
as well as its consequence (by H\"older's inequality)
$$
\Cal{M}_\gamma
\le 
\widetilde{C}_{1,\gamma}\int_{\Bbb{R}^3} |V(x)|_-^{\gamma+3/2}dx
+
\widetilde{C}_{2,\gamma}
\int_{\Bbb{R}^3}|\bold{B}(x)|^{3/2} |V(x)|_-^{\gamma}dx
\tag 1.6
$$
is false without substantial
regularity conditions on $\bb$. Moreover he conjectured
that it would be necessary
to replace $|\bold{B}|$ by some screened version of $|\bold{B}|$.

Motivated by Erd\"os' observation, 
A.~Sobolev \cite{19,20} obtained estimates in the forms of
(1.5) and (1.6), but with $|\bold{B}|$ 
replaced by a so-called effective (scalar)
magnetic field $b(x)$:
$$
\sum_{\lambda_j<0}
|\lambda_j|
\le C_1\int_{\Bbb{R}^3} |V(x)|_-^{5/2}dx
+C_2\int_{\Bbb{R}^3} b(x)^{3/2}
|V(x)|_-dx.\tag 1.7
$$
Roughly speaking,
 $b(x)$ is a 
slow varying function
which dominates $|\bold{B}(x)|$ pointwise.
We remark that the two-dimensional Pauli operator as well as the 
three-dimensional case with a constant direction field
was also investigated in \cite{3,19,20}.

Recently, L.~Bugliaro, C.~Fefferman, J.~Fr\"ohlich, G.~Graf,
and J.~Stubbe \cite{2} established (1.7)
with a $b(x)$ whose energy is comparable to that of $|\bb|$
: \ $\| b\|_{L^2}\approx \|\bb\|_{L^2}$.
This in particular implies the
following estimate of E.~Lieb,
M.~Loss, and J.~Solovej \cite{10}:
$$
\Cal{M}_1
\le 
C_1\int_{\Bbb{R}^3}|V(x)|_-^{5/2}dx
+
C_2\left(\int_{\Bbb{R}^3}|\bold{B}(x)|^2dx\right)^{3/4}
\left(\int_{\Bbb{R}^3}|V(x)|_-^4dx\right)^{1/4}.
\tag 1.8
$$

In this paper we further extend the results in \cite {3,19,20,2}.
The main novelty of our work is that
the effective field $b(x)$ we found is much simple and more
natural than the previous ones in \cite{20,2}.
Indeed, $b(x)=b_p(x)$ is defined to be the $L^p$ average
of $|\bb|$ over a suitable
cube centered at $x$ with a side length scaling
like $|\bold{B}|^{-1/2}$. 
We believe that this choice of the effective field is optimal.


To state our main results, we first define
the basic length scale:
$$
\ell_p(x)
=\sup\left\{
\ell>0:\ \ell^2\left(\frac{1}{\ell^3}\int_{Q(x,\ell)}
|\bb(y)|^pdy\right)^{1/p}\le 1\right\}\tag 1.9
$$
where $Q(x,\ell)$ denotes the cube centered at $x$ with side
length $\ell$.

It is easy to see that if $|\bb|\in L^p_{loc}(\Bbb{R}^3)$ for some
$p>3/2$, then
$0<\ell_p(x)<\infty$ unless $|\bb|\equiv 0$. Our effective field is 
now given by
$$
b_p(x)\equiv\frac{1}{\{ \ell_p(x)\}^2}
=\left\{\frac{1}{\ell^3_p(x)}\int_{Q(x,\ell_p(x))}
|\bb(y)|^pdy\right\}^{1/p},
\tag 1.10
$$
i.e., $b_p(x)$ is the $L^p$ average of $|\bb|$ 
over the cube centered at
$x$ with side length $\ell_p(x)$. In particular, we have
$\|b_p\|_{L^q}\le C\|\bold{B}\|_{L^q}$
for any $q\ge p$.

\proclaim{\bf Proposition 1.1}
Let $p>3/2$. Then, for any $q\ge p$, there exists a constant
$C_{p,q}>0$ such that
$$
\int_{\Bbb{R}^3} |b_p(x)|^q dx
\le
C_{p,q}\int_{\Bbb{R}^3} |\bb(x)|^q dx.
\tag 1.11
$$
\endproclaim

The following are the main results
of the paper.

\proclaim{\bf Theorem 1.1}
Let $p>3/2$ and $\gamma\ge 1$. Then there exist constants
$C_1(\gamma,p) >0$ and $C_2 (\gamma, p)>0$ such that
$$
\sum_{\lambda_j<0} |\lambda_j|^\gamma
\le 
C_1(\gamma, p)\int_{\Bbb{R}^3}|V(x)|_-^{\gamma +3/2}dx
+
C_2(\gamma, p)\int_{\Bbb{R}^3} \left[
b_p(x)\right]^{3/2} |V(x)|_-^\gamma dx.
$$
\endproclaim

We have a stronger estimate for magnetic fields with constant directions.

\proclaim{\bf Theorem 1.2}
Let $p>1$ and $\gamma>1/2$. Suppose that $\bb$ has a constant
direction.
Then there exist constants
$C_3(\gamma,p) >0$ and $C_4 (\gamma, p)>0$ 
such that
$$
\sum_{\lambda_j<0} |\lambda_j|^\gamma
\le 
C_3(\gamma, p)\int_{\Bbb{R}^3}|V(x)|_-^{\gamma +3/2}dx
+
C_4(\gamma, p)\int_{\Bbb{R}^3} b_p(x) |V(x)|_-^{\gamma+1/2} dx.
$$
\endproclaim

A few remarks are in order.

\remark{\bf Remark 1.1}
The definition of $b_p(x)$ is motivated by
the auxiliary function $m(x,|\bb|)$, 
which is instrumental in the study of eigenvalue problems
for the magnetic Schr\"odinger operator 
$-(\bold{p}-\bold{A}(x))^2 +V(x)$
with certain degenerate potentials
\cite{16,17}.
We should point out that, although the definition (1.9)
is not rotation invariant, the basic length scales $\ell_p(x)$,
hence $b_p(x)$, are equivalent in different coordinates systems.
One can certainly use balls centered at $x$ instead of cubes in (1.9).
 We also remark that,
if the components of $\bold{B}$ are polynomials,
then
$$
b_p(x)\approx
\sum_{|\alpha|\le k}
|\nabla^\alpha \bold{B}(x)|^\frac{2}{|\alpha|+2}\tag 1.12
$$
where $k$ is the degree of $\bb$.
\endremark

\remark{\bf Remark 1.2}
Suppose that $\bold{B}=\bold{B}_1+\bold{B}_2$ where
$|\bold{B}_1|\in L^\infty(\Bbb{R}^3)$ and
$|\bold{B}_2|\in L^p (\Bbb{R}^3)$ for some $p>3/2$. 
Let 
$\ell =\ell_p(x)$, then
$$
\ell^2\left(\frac{1}{\ell^3}\int_{Q(x,\ell)}|\bold{B}(y)|^pdy\right)^{1/p}=1.
$$
It follows that
$$
\ell^2 \|\bold{B}_1\|_{L^\infty} +\ell^{2-3/p}\|\bold{B}_2\|_{L^p}\ge 1.
$$
A simple computation yields that
$$
b_p(x)
=\frac{1}{\ell^2}
\le C\big\{
\|\bold{B}_1\|_{L^\infty}
+\|\bold{B}_2\|_{L^p}^\frac{2p}{2p-3}\big\}.
\tag 1.13
$$
This, together with Theorem 1.1 ($p=2$), gives the estimate
in \cite{3, Theorem 2.4, p.635}. Same argument also shows that
Theorem 2.1 in \cite{3, p.632} follows from Theorem 1.2.
\endremark

\remark{\bf Remark 1.3}
To compare our results with that in \cite{2}, we note that
 the effective field in \cite {2} is defined by 
$\widetilde{b}(x)=1/[r(x)]^2$
where
$$
\frac{1}{r(x)}
=\int_{\Bbb{R}^3}
\varphi\left(\frac{y-x}{r(x)}\right) |\bb(y)|^2dy\tag 1.14
$$
and $\varphi(z)=(1+\frac{1}{2}z^2)^{-2}$.
Let $r=r(x)$. It is easy to see that
$$
\int_{Q(x,r)}|\bb(y)|^2dy\le \frac{C}{r}.
$$
Thus, if $\delta$ is small, we have
$$
(\delta r)^2\left\{\frac{1}{(\delta r)^3}
\int_{Q(x,\delta r)} |\bb(y)|^2dy\right\}^{1/2}
\le C\delta^{1/2}<1.
$$
By definition, $\ell_2(x) \ge \delta r$. Hence,
$$
b_2(x)=\frac{1}{\{ \ell_2(x)\}^2}
\le \frac{1}{(\delta r)^2}
=\frac{\widetilde{b}(x)}{\delta ^2}.
$$
Thus Theorem 1.1 extends the results in \cite {2},
with
an effective field much easier to compute.
One may deduce the estimates in \cite{20}
from Theorems 1.1 and 1.2 with $p=\infty$ by a similar argument.
\endremark

\remark{\bf Remark 1.4}
In Theorems 1.1 and 1.2, we have implicitly assumed that 
$\Bbb{P}_0+V$ is a self-adjoint operator. In fact,
if the right hand side of the estimate in 
Theorem 1.1 (or 1.2) is finite, then $\Bbb{P}_0+V$
has a unique self-adjoint realization on $L^2(\Bbb{R}^3,\Bbb{C}^2)$ associated
with its quadratic form.

Indeed, under the assumption $\bold{A}\in L^2_{loc}(\Bbb{R}^3, \Bbb{R}^3)$,
we may define $\Bbb{P}_0$ as the self-adjoint operator
associated with the closed quadratic form
$\| \Bbb{D}\psi\|_{L^2}^2$ where $\Bbb{D}=\sigma\cdot (\bold{p}-\bold{A})$,
i.e., $\Bbb{P}_0
=\Bbb{D}^*\Bbb{D}$.
Let $V_j=V\chi_{\{ |V|\le j\} }$. Since $V_j$ is bounded, the operator
$\Bbb{P}_0-\frac{1}{\varepsilon}|V_j|$ is 
 self-adjoint with a form core 
Domain$(\Bbb{D})$ for any $\varepsilon\in (0,1)$.
We now apply the variation principle and
Theorem 1.1 to $\Bbb{P}_0-\frac{1}{\varepsilon}|V_j|$. We obtain
$$
\aligned
\int_{\Bbb{R}^3}  |\sigma\cdot (\bold{p} & -\bold{A})\psi|^2dx
-\frac{1}{\varepsilon}\int_{\Bbb{R}^3} |V_j|\, |\psi|^2dx\\
&\ge \lambda_1(\varepsilon) \int_{\Bbb{R}^3}
|\psi|^2dx\ge -C(\delta, \varepsilon, V, \bold{B})
\int_{\Bbb{R}^3}|\psi|^2dx
\endaligned
$$
for any $\psi\in\text{Domain} (\Bbb{D})$. Thus,
$$
\int_{\Bbb{R}^3} |V_j|\, |\psi|^2dx
\le \varepsilon\int_{\Bbb{R}^3} |\sigma\cdot (\bold{p}  -\bold{A})\psi|^2dx
+\varepsilon\,
C(\delta, \varepsilon, V, \bold{B})
\int_{\Bbb{R}^3}|\psi|^2dx.
$$
Let $j\to \infty$. By Fatou's Lemma, we have
$$
\int_{\Bbb{R}^3} |V|\, |\psi|^2dx
\le \varepsilon\int_{\Bbb{R}^3} |\sigma\cdot (\bold{p} -\bold{A})\psi|^2dx
+\varepsilon\,
C(\delta, \varepsilon, V, \bold{B})
\int_{\Bbb{R}^3}|\psi|^2dx
$$
for $\psi\in \text{Domain} (\Bbb{D})$. It then follows from 
the KLMN Theorem \cite{14, p.167}
that $\Bbb{P}_0+V$ can be extended to the unique
self-adjoint operator associated with its quadratic form.
Furthermore, Domain$(\Bbb{D})$ is a form core for $\Bbb{P}_0+V$.
\endremark
To prove Theorem 1.1, we will compare the Pauli operator $\pp_0$
with the magnetic Schr\"odinger operator $\hh_0
=(\bold{p}-\bold{A})^2$ in an appropriate scale, as in \cite{2,8,19,20}.
This is possible since $\pp_0=\hh_0-\sigma\cdot\bb$.
To this end, our first step is to localize the operators
to cubes $Q$ over which the $L^p$ average of $|\bb|$ is small compare
to $\ell(Q)^{-2}$ (the localization error in the kinetic 
energy), by a Calder\'on--Zygmund decomposition.
More precisely, we
divide $\Bbb{R}^3$ into a grid of 
disjoint cubes $\{ Q_j\}$ where each $Q_j$ is
a maximal dyadic cube such that
$$
\left\{ \int_{12Q_j}|\bb(x)|^pdx\right\}^{1/p}
\le
\frac{\varepsilon}{[\ell (Q_j)]^{2-\frac{3}{p}}}.
\tag 1.15
$$
Here $\ell(Q_j)=\ell_j$ is the side length of $Q_j$,
and $12Q_j$ denotes the cube which has the same center as
$Q_j$ and side length $12 \ell(Q_j)$.
It can be proved that $\ell_j\approx \ell_k$ if
$4Q_j\cap 4Q_k \neq \emptyset$. 
Using this property, we construct a partition of unity for $\Bbb{R}^3$:
$\sum_j \phi_j^2(x)\equiv 1 $, with $\phi_j\in C_0^\infty
(2Q_j,\Bbb{R})$. 

Next we apply the Birman-Schwinger principle
which reduces the problem to the estimate of
singular values of $|V|^{1/2}(\pp_0+\lambda)^{-1/2}$ for $\lambda>0$.
We then use the 
resolvent identity to 
compare $\phi_j (\pp_0 +\lambda)^{-1}$
with $(\pp_0+\Phi +\frac{1}{\ell_j^2}
+\lambda)^{-1}\phi_j$
where $\Phi(x)=\sum_j \frac{1}{\ell_j^2}\, \phi_j^2(x)
\approx b_p(x) $.
We will show that, if $\varepsilon$ in (1.15) is sufficiently small, then
$$
\hh_0\le C\left\{ \pp_0+\Phi\right\}
\tag 1.16
$$
(Theorem 3.1). With (1.16) at our disposal, we are able to estimate
the contribution from $(\pp_0+\Phi+\frac{1}{\ell_j^2}
 +\lambda)^{-1}\phi_j$. This
leads to the first term in Theorem 1.1.
Finally, to deal with the error term
$
\phi_j (\pp_0 +\lambda)^{-1}-
(\pp_0+\Phi+\frac{1}{\ell_j^2}+
\lambda)^{-1}\phi_j
$ 
which can be written as 
$$
(\pp_0+\Phi+\frac{1}{\ell_j^2}+
\lambda)^{-1}
\big\{ \big(\Phi+\frac{1}{\ell_j^2}\big)
 \phi_j+[\pp_0,\phi_j]\big\} (\pp_0 +\lambda)^{-1},
$$
we use the resolvent identity 
again to compare $\psi_j (\pp_0+\Phi +\frac{1}{\ell_j^2})^{-1}$
with $(\hh_0+\frac{1}{\ell_j^2})^{-1}\psi_j$
where $\psi_j$ is a bump function such that
$\psi_j\phi_j\equiv \phi_j$.
The desired result follows from certain regularity estimates
for the operator $\psi_j(\pp_0+\Phi +\frac{1}{\ell_j^2})^{-1}$
(see Lemma 4.2).

A similar approach is used in the case of constant direction fields.
Assuming $\bb=(0,0,B(x_1,x_2))$ without the loss of generality,
we construct a partition of unity for $\Bbb{R}^2$
associated with $B$. The inequality
$$
\| (-\partial_{x_3}^2 +\lambda)^{1/2}
(\pp_0 +\lambda)^{-1/2}\|_{L^2\to L^2}
\le 1,\ \ \ \ \ \ \ \ \ \ \lambda>0
\tag 1.17
$$
is used to exploit the fact that $\pp_0$ 
commutes with $\partial_{x_3}$.

The paper is organized as follows. 
In Section 2 we collect some basic facts that will be used
concerning the norm in  
the Neumann-Schatten classes.
Section 2 also contains the proof of Proposition 1.1.
The partition of unity will be constructed in Section 3.
Section 4 is devoted to the proof of Theorem 1.1.
Finally we study the case of constant direction fields 
in Section 5 where Theorem 1.2 is proved.

Throughout this paper we will use $\|\psi\|_{L^p}$ to
denote the $L^p$ norm of the function $\psi$.
$\| T\|_{L^p\to L^q}$ will denote the operator
norm, while $\| T\|_p$ is reserved for the norm
in the Neumann-Schatten classes (see (2.1)).
Finally we will use $C$ to denote constants,
which are not necessarily the same at each occurrence,
which may depend on $p$.
\bigskip\medskip
\centerline{\bf 2. Some Preliminaries}

Most materials in this section on the Neumann-Schatten classes
can be found in
\cite {18}.
We include them here for the reader's convenience.
The proof of Proposition 1.1 will be given
at the end of the section.

Let $T$ be a compact operator on $L^2(\Bbb{R}^3,\Bbb{C}^2)$.
We will use
$n(s,T)$ to denote the number of singular values $\{ s_n(T)\}$
(counting multiplicity) of $T$ greater than $s$ where $s>0$.
For $p\ge 1$, let
$$
\| T\|_p^p\equiv\sum_n |s_n(T)|^p=p\int_0^\infty s^{p-1}n(s,T)ds.
\tag 2.1
$$
The functional $\|T\|_p$
defines a norm on the Neumann-Schatten class $\Cal{S}_p$
which consists of compact operators $T$ with $\|T\|_p<\infty$.

The following facts will be useful to us.
$$
\align
n(s^2, T^*T)&=n(s,T),
\tag 2.2\\
\|T\|_p^2&=\|T^* T\|_{p/2},\tag 2.3\\
n(s,T)&\le \frac{\| T\|_p^p}{s^p}.\tag 2.4
\endalign
$$ 
We will also need
$$
\align
&n(s_1+s_2,T_1+T_2)\le n(s_1,T_1)+n(s_2+T_2), \tag 2.5\\
&\|T_1 T_2\|_p\le \|T_1\|_p \|T_2\|_{L^2\to L^2},\tag 2.6\\
& 
\|T_1 T_2\|_p\le \| T_1\|_r \|T_2\|_s
\ \ \ \ \ \ \ \ \text {if } \ \ \ \frac1p =\frac1r +\frac 1s,\ \ 
r, \, s\ge 1\tag 2.7
\endalign
$$
where $\|T\|_{L^2\to L^2}$ denotes the operator norm of $T$.

The proof of the following lemma may be found in \cite{18}.

\proclaim{\bf Lemma  2.1}
If $f,\ g\in L^p(\Bbb{R}^3)$ with  $2\le p<\infty$.
Then
$$
\| f(x)g(-i\nabla)\|_p\le C_p \| f\|_{L^p} \| g\|_{L^p}.
$$
\endproclaim

It follows from Lemma 2.1 that, if $p\ge 2$,
$\alpha > \frac{3}{2p}$, and $\lambda >0$,
$$
\| V (-\Delta +\lambda )^{-\alpha}\|_p
\le C_{p, \alpha} \lambda^{\frac{3}{2p}-\alpha}\| V\|_{L^p}.
\tag 2.8
$$
Also, by Lemma 2.1, if $\alpha >1/2$ and $0 < \lambda_1\le \lambda_2$,
$$
\| V(-\partial _{x_3}^2+\lambda_1)^{-1/2}
(-\Delta+\lambda_2)^{-\alpha}\|_2
\le C_\alpha
\lambda_1^{-\frac14}\lambda_2^{-\alpha +\frac12}\|V\|_{L^2}.
\tag 2.9
$$
Thus, by the diamagnetic inequality \cite{9},
$$
|\big( \hh_0+\lambda\big)^{-\alpha}\psi|
\le (-\Delta +\lambda)^{-\alpha}|\psi|,
\tag 2.10
$$
where $\hh_0 =(\bold{p}-\bold{A}(x))^2$, and Theorem 2.13 in \cite{18, p.36},
we have
$$
\align
&\|V( \hh_0+\lambda)^{-\alpha}\|_2
\le C_\alpha \lambda^{\frac34-\alpha} \|V\|_{L^2}
\ \ \ \ \text{ for } \alpha >3/4,
\tag 2.11\\
&
\|V( \hh_0+\lambda)^{-\alpha}\|_4
\le C_\alpha \lambda^{\frac38-\alpha} \|V\|_{L^4}
\ \ \ \ \text{ for } \alpha >3/8,
\tag 2.12
\endalign
$$
and for $\alpha>1/2$ and $0<\lambda_1\le \lambda_2$,
$$
\| V(-\partial _{x_3}^2+\lambda_1)^{-1/2}
(\hh_0+\lambda_2)^{-\alpha}\|_2
\le C_\alpha
\lambda_1^{-\frac14}\lambda_2^{-\alpha +\frac12}\|V\|_{L^2}.
\tag 2.13
$$

The following two lemmas concern  the operator
norm of $V(-\Delta +\lambda)^{-\alpha}$ on $L^2(\Bbb{R}^3,
\Bbb{C}^2)$.

\proclaim{\bf Lemma 2.2} Let $\alpha >0$ and $\lambda >0$. Then
$$
\align
&
\| V (-\Delta +\lambda)^{-\alpha}\|_{L^2\to L^2}
\le C_\alpha \| V\|_{L^\frac{3}{2\alpha}}
\ \ \ \ \ \ \ \ \ \text{ if } \ \ \alpha <3/4,\tag 2.14\\
&
\| V (-\Delta +\lambda)^{-\alpha}\|_{L^2\to L^2}
\le C_\alpha \lambda^{\frac{3}{4}-\alpha}
\| V\|_{L^2}\ \ \ \ \text { if } \ \ \alpha >3/4.\tag 2.15
\endalign
$$
\endproclaim

\demo{\bf Proof}
Note that, if $\alpha <3/4$,
$$
\|V(-\Delta+\lambda)^{-\alpha}\psi\|_{L^2}
\le \|V\|_{L^{p_1}}\| (-\Delta +\lambda )^{-\alpha}\psi \|_{L^{p_2}}
\le C_\alpha
\|V\|_{L^{p_1}}\|\psi\|_{L^2}
$$
where $\frac{1}{p_1}+\frac{1}{p_2}=\frac12$,
$\frac{1}{p_2}=\frac12 -\frac{2\alpha}{3}$, and we have used
H\"older's inequality and
the well-known fractional integral estimate \cite{21}.
(2.14) is proved since $p_1=3/(2\alpha)$.

Now suppose $\alpha >3/4$. Using the Fourier transform, we have
$$
\aligned
\|(-\Delta& +\lambda)^{-\alpha}\psi\|_{L^\infty}
\le C\int_{\Bbb{R}^3}
(|\xi|^2 +\lambda)^{-\alpha} |\hat{\psi}(\xi)|d\xi\\
&\le C \|\psi\|_{L^2}
\left\{\int_{\Bbb{R}^3}
(|\xi|^2 +\lambda)^{-2\alpha} d\xi\right\}^{1/2}
\le C_\alpha \lambda^{\frac34 -\alpha}\|\psi\|_{L^2}.
\endaligned
\tag 2.16
$$
It follows that
$$
\|V(-\Delta+\lambda)^{-\alpha}\psi \|_{L^2}
\le
\|V\|_{L^2}
\| (-\Delta +\lambda)^{-\alpha}\psi \|_{L^\infty}
\le
C_\alpha \lambda^{\frac34 -\alpha}
\|V\|_{L^2}
\|\psi\|_{L^2}.
$$
\enddemo

\proclaim{\bf Lemma 2.3}
Let $0<\alpha <1/2$ and $\lambda >0$.
Suppose that $V$ depends only on the first two variables. Then
$$
\| V (-\Delta +\lambda )^{-\alpha}\|_{L^2\to L^2}
\le C_\alpha \| V\|_{L^{1/\alpha}(\Bbb{R}^2)}.
$$
\endproclaim

\demo{\bf Proof}
By the fractional integral estimates \cite{21},  if $\psi\in C^\infty
(\Bbb{R}^3,\Bbb{C}^2)$ and decays at $\infty$,
$$
\|\psi\|_{L^q(\Bbb{R}^2)}
\le C_\alpha
\| (-\Delta_2 +\lambda )^\alpha\psi\|_{L^2(\Bbb{R}^2)}
$$
where $\frac1q=\frac12 -\alpha$ and $\Delta_2
=\partial_{x_1}^2 +\partial_{x_2}^2$
denotes the Laplacian in $\Bbb{R}^2$.
It then follows from H\"older's inequality that
$$
\align
\|V\psi\|_{L^2(\Bbb{R}^2)}
&\le \|V\|_{L^p(\Bbb{R}^2)}
\|\psi\|_{L^q(\Bbb{R}^2)}\\
&\le
C_\alpha
 \|V\|_{L^p(\Bbb{R}^2)}
\| (-\Delta_2 +\lambda )^\alpha\psi\|_{L^2(\Bbb{R}^2)}
\endalign
$$
where $\frac1p +\frac1q =\frac12$.
Hence
$$
\align
\int_{\Bbb{R}^3} |V|^2 |\psi|^2dx
&\le C_\alpha^2
\|V\|_{L^p(\Bbb{R}^2)}^2
\int_{\Bbb{R}^3}
|(-\Delta_2 +\lambda )^\alpha\psi|^2dx\\
&\le C_\alpha^2
\|V\|_{L^p(\Bbb{R}^2)}^2
\int_{\Bbb{R}^3}
|(-\Delta +\lambda )^\alpha\psi|^2dx
\endalign
$$
where the second inequality can be verified through the
Fourier transform.
Thus
$$
\| V\psi \|_{L^2}
\le C_\alpha
\|V\|_{L^p(\Bbb{R}^2)}
\|(-\Delta +\lambda )^\alpha\psi\|_{L^2}.
$$
The lemma now follows easily
since $\frac1p =\frac12 -\frac1q=\alpha$.
\enddemo

We end this section with the
\demo{\bf Proof of Proposition 1.1}

It follows from (1.10) that
$b_p(x)\le \{ M(|\bb|^p)(x)\}^{1/p}$
where $M$ is the Hardy-Littlewood maximal operator.
This gives the desired estimate in the case $q>p$ since
$M$ is bounded on $L^s$ for $s>1$ \cite{21}.

For the case $q=p$, we appeal to an argument in \cite{2}.
Let $\widetilde{b}_p(x)=1/[r_p(x)]^2$ where $r_p(x)$ is
defined by
$$
\frac{1}{[r_p(x)]^{2p-3}}
=\int_{\Bbb{R}^3}
\varphi \left(\frac{x-y}{r_p(x)}\right)|\bb(y)|^pdy
\tag 2.17
$$
and $\varphi(z)=[1+\frac12 |z|^2]^{-2}$.
Since $b_p(x)\le C\, \widetilde{b}_p(x)$ (see Remark 1.4 for $p=2$),
it suffices to show that
$$
\int_{\Bbb{R}^3}
| \widetilde{b}_p(x)|^pdx
\le C\int_{\Bbb{R}^3}
|\bb(x)|^pdx.
\tag 2.18
$$
Using (2.17) and Fubini's Theorem, one may reduce (2.18) to
$$
\int_{\Bbb{R}^3}
\varphi \left(\frac{x-y}{r_p(x)}\right)
\frac{dx}{[r_p(x)]^3}
\le C<\infty.
\tag 2.19
$$

The proof of (2.19) relies on the following estimate
$$
r_p(y)g_-\left(\frac{|x-y|}{r_p(y)}\right)
\le r_p(x)
\le
r_p(y)g_+\left(\frac{|x-y|}{r_p(y)}\right)
\tag 2.20
$$
where $g_+(s)$ and $g_-(s)$ are solutions of the equation
$$
s^2=2(t^{p+\frac12}-1)(1-t^{\frac32-p})
\tag 2.21
$$
on $(1,\infty)$ and $(0,1)$ respectively.
Since $p>3/2$, it is easy to see that $g_+(s)$, $g_-(s)$
are well defined. We omit the proof of (2.20),
which is similar to that of Lemma 2 in
\cite{2, p.569}.

It is not hard to see that $g_+(s)\approx s^{\frac{4}{2p+1}}$
if $s\ge 1/2$, and $g_-(s)\ge c_0>0$ if $s<1/2$.
This, together with (2.20), implies that
$$
\alignedat2
&r_p(x)\le C\, [r_p(y)]^{1-\frac{4}{2p+1}} |x-y|^{\frac{4}{2p+1}}
\ \ & \text{ if }\ \ |x-y|\ge \frac12 r_p(y),\\
&r_p(x)\ge c \, r_p(y) & \text{ if }\ \ |x-y|<\frac12 r_p(y).
\endalignedat
$$
It then follows that
$$
\aligned
&\int_{\Bbb{R}^3}
\varphi \left(\frac{x-y}{r_p(x)}\right)
\frac{dx}{[r_p(x)]^3}
\le \int_{\Bbb{R}^3}
\frac{r_p(x)\, dx}
{\{ [r_p(x)]^2 +\frac12 |x-y|^2\}^2}\\
&\le 4
\int_{|x-y|\ge \frac12 r_p(y)}
\frac{r_p(x)\, dx}
{|x-y|^4}
+
\int_{|x-y|< \frac12 r_p(y)}
\frac{dx}{[r_p(x)]^3}\\
&\le C \, \big[r_p(y)\big]^{\frac{2p-3}{2p+1}}
\int_{|x-y|\ge \frac12 r_p(y)}
\frac{ dx}
{|x-y|^{3+\frac{2p-3}{2p+1}}}
+\frac{C}{[r_p(y)]^3}
\int_{|x-y|< \frac12 r_p(y)}
dx\\
&\le C.
\endaligned
$$
The proof is finished.
\enddemo
\bigskip

\centerline{\bf 3. A Partition of Unity}

Throughout this section we fix $p>3/2$.
We will assume that
$|\bb|\in L^p_{loc}(\Bbb{R}^3)$
and $|\bb|\not\equiv 0$.


Let $\varepsilon\in (0,1)$ be a small constant 
to be determined later.
Let $\Cal{A}$ be the set of all dyadic cubes in $\Bbb{R}^3$
such that
$$
\left(\int_{12 Q}|\bb(x)|^pdx\right)^{1/p}\le
\frac{\varepsilon}{[\ell(Q)]^{2-\frac{3}{p}}}
\tag 3.1
$$
where $\ell(Q)$ denotes the side length of $Q$. We say that $Q$ is
a maximal element of $\Cal{A}$ if $Q\in\Cal{A}$ and $Q$ is not properly
contained in any other cube in $\Cal{A}$.

Let $\Cal{B}$ denote the set of all maximal elements of $\Cal{A}$.
Clearly, by definition, the interiors of the cubes in $\Cal{B}$
are disjoint.

\proclaim{\bf Lemma 3.1}
Let $\Cal{B}=\left\{ Q_j,\ j=1,2,\dots\right\}$. Then,
$$
\Bbb{R}^3=\bigcup_j Q_j.
$$
\endproclaim

\demo{\bf Proof}
It suffices to show that any point in $\Bbb{R}^3$ belongs to
some cube $Q_j$ in $\Cal{B}$. To this end, fix $x\in \Bbb{R}^3$,
let 
$\{ Q_{\alpha_k}\}_{k=-\infty}^\infty$ be an increasing
sequence of dyadic cubes such that
$x\in Q_{\alpha_k}$ for all $k$, $\ell(Q_{\alpha_k})\to 0$
as $k\to -\infty$, and $\ell(Q_{\alpha_k})\to\infty$
as $k\to\infty$.

Note that, as $\ell(Q)\to 0$, the l.h.s. of (3.1)
goes to $0$ while the r.h.s. goes to $\infty$.
This implies that $Q_{\alpha_k}\in\Cal{A}$
if $\ell(Q_{\alpha_k})$ is small. Also, as
$\ell(Q_{\alpha_k})\to\infty$, the l.h.s. of (3.1) goes to
$\|\bb\|_{L^p} >0$ and the r.h.s. goes to 0. Thus,
$Q_{\alpha_k}\notin\Cal{A}$ if $\ell(Q_{\alpha_k})$ is large.
Hence there must be a maximal element $Q_j$ in $\Cal{A}$
(which may not be in $\{ Q_{\alpha_k}\}$) such that $x\in Q_j$.
\enddemo

\proclaim{\bf Lemma 3.2}
Let $Q_j$, $Q_k \in \Cal{B}$. Suppose that $4Q_j\cap 4Q_k\neq\emptyset$.
Then 
$$
\frac{1}{2}\ell(Q_k)
\le \ell(Q_j)
\le 2\ell(Q_k).
$$
\endproclaim

\demo{\bf Proof}
We adapt an argument found in \cite {8}.

Suppose $\ell(Q_j)>2\ell (Q_k)$. Since $\ell(Q_j)$ and $\ell(Q_k)$
are powers of $2$, we have
$
\ell(Q_j)\ge 4\ell(Q_k).
$

Let $Q_k^+$ be the dyadic parent of $Q_k$, i.e., $Q_k$ can be obtained
by bisecting $Q_k^+$. Since $Q_k$ is a maximal element in
$\Cal{A}$, $Q_k^+\notin \Cal{A}$. It follows that
$$
\left(
\int_{12Q_k^+}|\bb|^pdx\right)^{1/p}
>\varepsilon
\left[\frac{1}{\ell(Q_k^+)}\right]^{2-\frac{3}{p}}
=\varepsilon
\left[\frac{1}{ 2\ell(Q_k)}\right]^{2-\frac{3}{p}}
\ge \varepsilon
\left[\frac{2}{\ell(Q_j)}\right]^{2-\frac{3}{p}}
\tag 3.2
$$
where we have used $p>3/2$ in the last inequality.

We claim that
$$
12Q_k^+\subset 12 Q_j.
\tag 3.3
$$
This, together with (3.2), would imply that
$$
\left(\int_{12Q_j}|\bb|^pdx\right)^{1/p}
\ge \left(\int_{12Q_k^+}|\bb|^pdx\right)^{1/p}
> \varepsilon
\left[\frac{2}{\ell(Q_j)}\right]^{2-\frac{3}{p}}
$$
which contradicts with the assumption that $Q_j\in \Cal{A}$.

To see (3.3), let $x_j$ and $x_k$ be the centers of $Q_j$ and $Q_k$
respectively. Since $4Q_j\cap 4Q_k\neq \emptyset$, there exists $z$
such that
$$
|z-x_j|_*\le 2\ell(Q_j)\ \ \ \text{ and } \ \ \ |z-x_k|_*\le 2\ell(Q_k)
$$
where $|\cdot|_*$ is the norm in $\Bbb{R}^3$ defined by
$$
|x|_*=|(a,b,c)|_*=\max \{ |a|,|b|,|c|\}.
$$
Thus,
$$
|x_j-x_k|_*\le 2\ell(Q_j)+2\ell(Q_k).
$$

Now suppose $y\in 12 Q_k^+$. Then $|y-x_k|_*
\le |y-x_k^+|_*+ |x_k^+-x_k|_* < 13 \ell(Q_k)$
where $x_k^+\in Q_k$ is the center of $Q_k^+$.
It follows that
$$
|y-x_j|_*\le |y-x_k|_*+|x_k-x_j|_*
< 15\ell(Q_k) +2\ell(Q_j)
< 6\ell(Q_j).
$$
Hence, $y\in 12 Q_j$. (3.3) is then proved.
\enddemo

\remark{\bf Remark 3.1}
It follows easily from Lemma 3.2 that
$\{ 4Q_j: Q_j\in \Cal{B}\}$ has the finite intersection
property: $\sum_j \chi_{4Q_j}\le C$ where $C$ is an absolute
constant.
\endremark
We are now in a position to construct the partition of unity
associated with the field strength $|\bb|$.

\proclaim{\bf Lemma 3.3}
There exists a sequence of functions $\{ \phi_j\}$ such that

(i)\ \  $\phi_j\in C_0^\infty(2Q_j)$ and $0\le \phi_j\le 1$,

(ii)\ \  $|\nabla^\alpha \phi_j(x)|\le c_\alpha/\ell_j^{|\alpha|}$
where $\ell_j=\ell(Q_j)$,

(iii)\ \  $\sum_j\phi_j^2\equiv 1$ in $\Bbb{R}^3$.
\endproclaim

\demo{\bf Proof}
Choose $\eta\in C_0^\infty(Q(0,2))$ such that $\eta\equiv1$ in $Q(0,1)$
and $0\le \eta \le 1$. Let $\Cal{B}=\{ Q_j\}_{j=1}^\infty$ and
$Q_j=Q(x_j,\ell_j)$. We define
$$
\eta_j(x)=\eta\left(\frac{x-x_j}{\ell_j}\right).
$$
and $\varphi (x)=\sum_j \eta_j^2(x)$. Clearly, by Lemma 3.1 and Remark 3.1,
$1\le \varphi \le C$ 
where $C$ is an absolute constant,
 and $\varphi\in C^\infty (\Bbb{R}^3)$.

Finally let $\phi_j(x)=\eta_j(x)/\sqrt{\varphi (x)}$. It is easy
to check that $\phi_j$ satisfies (i)-(iii). We omit the details.
\enddemo

Recall that $\ell_j=\ell(Q_j)$ and $Q_j$ is a maximal cube. Define
$$
\Phi=\sum_j\frac{1}{\ell_j^2}\, \phi_j^2.\tag 3.4
$$
We will show in Section 4 that $\Phi(x)\approx b_p(x)$ (see Lemma 4.5).

\proclaim{\bf Theorem 3.1} There exist constants $C>0$ and $\varepsilon_0>0$
such that, if $0<\varepsilon <\varepsilon_0$, we have
$$
\hh_0\le C\big\{ \pp_0+\Phi\big\}.
$$
\endproclaim

\demo{\bf Proof}
We first show that, if $\varepsilon$ in (3.1) is small, then
$$
\int_{\Bbb{R}^3} |\, |\bb|^{1/2}\phi_j\psi|^2dx
\le C\int_{\Bbb{R}^3}
|\sigma\cdot (\bold{p}-\bold{A})(\phi_j\psi)|^2dx\tag 3.5
$$
for any $\psi\in C^\infty_0(\Bbb{R}^3,\Bbb{C}^2)$.

To this end, we note that, by  H\"older's inequality
$$
\left\{\int_{2Q_j}|\bb|^{3/2}dx\right\}^{2/3}
\le |2Q_j|^{\frac23-\frac1p}
\left\{\int_{2Q_j}|\bb|^p
dx\right\}^{1/p}
\le 4\varepsilon.\tag 3.6
$$
It then follows from
the Sobolev embedding and the inequality
$|\nabla |\psi|\,|\le |(\bold{p}-\bold{A})\psi|$ that
$$
\aligned
\bigg\{ 
\int_{\Bbb{R}^3} |\, &|\bb|^{1/2}\phi_j\psi|^2dx\bigg\}^{1/2}
\le \left\{\int_{2Q_j}|\bb|^{3/2}dx\right\}^{1/3}
\left\{\int_{\Bbb{R}^3}|\phi_j\psi|^6dx\right\}^{1/6}\\
&\le
C\varepsilon^{1/2}
\left\{\int_{\Bbb{R}^3}|\nabla|\phi_j\psi|\,|^2dx\right\}^{1/2}\\
&\le
C\varepsilon^{1/2}
\left\{\int_{\Bbb{R}^3}|(\bold{p}-\bold{A})(\phi_j\psi)|^2
dx\right\}^{1/2}\\
&\le
C\varepsilon^{1/2}
\left\{\int_{\Bbb{R}^3}|\sigma\cdot
(\bold{p}-\bold{A})(\phi_j\psi)|^2
dx\right\}^{1/2}
+C\varepsilon^{1/2}
\left\{ 
\int_{\Bbb{R}^3} |\, |\bb|^{1/2}\phi_j\psi|^2dx\right\}^{1/2}
\endaligned
$$
where we have used the fact that $\pp_0 =\hh_0 -\sigma\cdot \bb$.
This gives (3.5) if $C\varepsilon^{1/2}<1/2$.

Next we use Lemma 3.3 and (3.5) to obtain
$$
\aligned
&\int_{\Bbb{R}^3} |(\bold{p}-\bold{A})\psi|^2dx
=\sum_j\int_{\Bbb{R}^3} |\phi_j(\bold{p}-\bold{A})\psi|^2dx\\
&
\le 2\sum_j
\int_{\Bbb{R}^3} |(\bold{p}-\bold{A})(\phi_j\psi)|^2dx
+2\sum_j
\int_{\Bbb{R}^3}|\nabla\phi_j|^2|\psi|^2dx\\
&\le 
2\sum_j
\int_{\Bbb{R}^3} |\sigma\cdot
(\bold{p}-\bold{A})(\phi_j\psi)|^2dx
+2\sum_j\int_{\Bbb{R}^3}
|\, |\bb|^{1/2}\phi_j\psi|^2dx
+
2\sum_j
\int_{\Bbb{R}^3}|\nabla\phi_j|^2|\psi|^2dx\\
&\le
C\sum_j
\int_{\Bbb{R}^3} |\sigma\cdot
(\bold{p}-\bold{A})(\phi_j\psi)|^2dx
+2\sum_j \int_{\Bbb{R}^3}|\nabla\phi_j|^2|\psi|^2dx\\
&\le 
C\int_{\Bbb{R}^3} |\sigma\cdot
(\bold{p}-\bold{A})\psi|^2dx
+C\sum_j \frac{1}{\ell_j^2}
\int_{2Q_j}|\psi|^2dx.
\endaligned
$$
Clearly, the theorem follows if we have
$$
\sum_j\frac{1}{\ell_j^2}\, \chi_{2Q_j}\le C\, \Phi.
\tag 3.7
$$
We claim that 
(3.7) is an easy consequence of Lemma 3.2. Indeed, if $x\in Q_k$,
the l.h.s. of (3.7) is bounded by
$$
\sum_{2Q_j\cap Q_k\neq\emptyset}
\frac{1}{\ell_j^2}
\le \frac{C}{\ell_k^2}\cdot
\# \{ Q_j: \ 2Q_j\cap Q_k\neq\emptyset\}
\le\frac{C}{\ell_k^2}\le C\, \Phi(x).
$$
The proof is now complete.
\enddemo

It follows easily from Theorem 3.1 that, if $0<\lambda_1\le\lambda_2$,
$$
\| (\hh_0+\lambda_1)^{1/2} (\pp_0+\Phi +\lambda_2)^{-1/2}\|_{L^2\to L^2}
\le C.
\tag 3.8
$$

The following lemma will be used in the next section.

\proclaim{\bf Lemma 3.4}
There exists a constant $C>0$ such that, for $\lambda>0$,
$$
\| (\pp_0+\Phi+\frac{1}{\ell_j^2}+\lambda)^{-1/2}\phi_j
(\pp_0+\lambda)^{1/2}\|_{L^2\to L^2}\le C.
$$
\endproclaim

\demo{\bf Proof}
By duality, it suffices to show that 
$$
\|(\pp_0+\lambda)^{1/2}\phi_j
(\pp_0+\Phi+\frac{1}{\ell_j^2}+\lambda)^{-1/2}\|_{L^2\to
L^2}
\le C.
\tag 3.9
$$

To this end, we note that, for any 
$\psi\in C^\infty_0(\Bbb{R}^3,\Bbb{C}^2)$,
$$
\aligned
&\int_{\Bbb{R}^3}
|(\pp_0+\lambda)^{1/2}\phi_j
(\pp_0+\Phi+\frac{1}{\ell_j^2}+\lambda)^{-1/2}\psi|^2dx\\
&
=
\int_{\Bbb{R}^3}|\sigma\cdot (\bold{p}-\bold{A})\phi_j
(\pp_0+\Phi+\frac{1}{\ell_j^2}+\lambda)^{-1/2}\psi|^2dx
+\lambda 
\int_{\Bbb{R}^3}
|\phi_j(\pp_0+\Phi+\frac{1}{\ell_j^2}+\lambda)^{-1/2}\psi|^2dx\\
&
\le 4\int_{\Bbb{R}^3}|\psi|^2
+2\int_{\Bbb{R}^3}
|\nabla \phi_j|^2
|(\pp_0+\Phi+\frac{1}{\ell_j^2}+\lambda)^{-1/2}\psi|^2dx\\
&
\le C\int_{\Bbb{R}^3} |\psi|^2dx
\endaligned
$$
where we have used $|\nabla \phi_j|\le C/\ell_j$ and 
$$
\align
&\| \sigma \cdot (\bold{p}-\bold{A})(\pp_0+\Phi+\lambda)^{-1/2}\|_{L^2
\to L^2}\le 1,\tag 3.10\\
&\|(\pp_0+\Phi+\lambda)^{-1/2}\|_{L^2
\to L^2}\le \frac{1}{\sqrt{\lambda }}.
\tag 3.11
\endalign
$$
The proof is finished.
\enddemo
\bigskip

\centerline{\bf 4. The Proof of Theorem 1.1}

For $\lambda >0$, we denote by $N(\lambda, \Bbb{P}_0+V)$
the number of eigenvalues
(counting multiplicity) of $\Bbb{P}_0+V$ smaller than $-\lambda$.
Since $V\ge -\lambda -|V+\lambda|_-$,
we have
$$
N(2\lambda, \Bbb{P}_0+V)\le N(\lambda, \Bbb{P}_0-|V+\lambda|_-).
$$
It then follows from the Birman-Schwinger principle and (2.2) that
$$
N(2\lambda, \Bbb{P}_0+V)
\le n(1,Y(\Bbb{P}_0+\lambda)^{-1}Y )
= n(1,Y(\pp_0+\lambda)^{-1/2})
\tag 4.1
$$
where $Y=|V+\lambda|^{1/2}_-$.

As in \cite {19,20}, our approach will be based on the
following resolvent identity:
$$
\varphi H_1^{-1} =H_2^{-1}\varphi
+
H_2^{-1}
\big\{
(H_2-H_1)\varphi +[H_1,\varphi]\big\} H_1^{-1}
\tag 4.2
$$
where $\varphi\in C^\infty_0(\Bbb{R}^3,\Bbb{R})$,
$H_1,\,  H_2$ are two operators, and $[H_1,\varphi]=H_1\varphi-\varphi H_1$
denotes the commutator between $H_1$ and $\varphi$.

Using (4.2), we may write
$$
\aligned
Y(\pp_0+\lambda)^{-1/2}&=\sum_j
Y\phi_j(\pp_0+\Phi +\frac{1}{\ell_j^2}+\lambda)^{-1}
\phi_j(\pp_0+\lambda)^{1/2}\\
&\ \ \ \ \ \ 
+\sum_j
Y\phi_j(\pp_0+\Phi +\frac{1}{\ell_j^2}+\lambda)^{-1}
(\Phi +\frac{1}{\ell_j^2})\phi_j (\pp_0+\lambda)^{-1/2}\\
&\ \ \ \ \ \
+\sum_jY\phi_j(\pp_0+\Phi +\frac{1}{\ell_j^2}+\lambda)^{-1}
[\pp_0,\phi_j](\pp_0+\lambda)^{-1/2}\\
&=I_1 +I_2 +I_3
\endaligned
$$
where $\Phi$ is the function defined by (3.4).
Thus, by (2.5),
$$
n(1, Y(\pp_0+\lambda)^{-1/2})
\le n(1/3, I_1)+n(1/3,I_2)+n(1/3, I_3).
\tag 4.3
$$

We begin with the estimate of $n(1/3, I_1)$.

\proclaim{\bf Lemma 4.1}
There exists a constant $C>0$ such that, for any $\lambda>0$,
$$
n(1/3, I_1)\le
\frac{C}{\sqrt{\lambda}}\int_{\Bbb{R}^3}|Y|^4 dx.
$$
\endproclaim

\demo{\bf Proof}
First observe that, by (2.4),
$$
n(1/3, I_1)\le C\|\sum_j Y\phi_j(\pp_0+\Phi +\frac{1}{\ell_j^2}+\lambda)^{-1}
\phi_j(\pp_0+\lambda)^{1/2}\|_4^4.
\tag 4.4
$$
Let
$$
I_{1j}=Y\phi_j(\pp_0+\Phi +\frac{1}{\ell_j^2}+\lambda)^{-1}
\phi_j(\pp_0+\lambda)^{1/2}.
\tag 4.5
$$
It follows from (2.6) and Lemma 3.4 that
$$
\aligned
\|I_{1j}\|_4
&\le \|Y\phi_j(\pp_0+\Phi +\frac{1}{\ell_j^2}+\lambda)^{-1/2}
\|_4
\|(\pp_0+\Phi +\frac{1}{\ell_j^2}+\lambda)^{-1/2}
\phi_j(\pp_0+\lambda)^{1/2}\|_{L^2\to L^2}\\
&\le C\|Y\phi_j(\pp_0+\Phi +\frac{1}{\ell_j^2}+\lambda)^{-1/2}\|_4\\
&\le 
C\|Y\phi_j (\hh_0+\lambda)^{-1/2}\|_4
\| (\hh_0+\lambda)^{1/2}
(\pp_0+\Phi +\frac{1}{\ell_j^2}+\lambda)^{-1/2}\|_{L^2\to L^2}\\
&\le C\|Y\phi_j (\hh_0+\lambda)^{-1/2}\|_4
\endaligned
$$
where we also used (3.8) in the last inequality.
We now apply (2.12) with $\alpha =1/2$ to obtain
$$
\|I_{1j}\|_4
\le 
C\lambda^{-1/8}\|Y\phi_j\|_{L^4}.
\tag 4.6
$$

Next we note that, by (2.3),
$$
\|\sum_jI_{1j}\|_4^4
=\|\sum_j\sum_k\sum_m\sum_n I_{1j} I_{1k}^* I_{1m}I_{1n}^*\|_1
\le 
\sum_j\sum_k\sum_m\sum_n \|I_{1j} I_{1k}^* I_{1m}I_{1n}^*\|_1.
$$
Since $\phi_j$ is supported in $2Q_j$ and $\phi_j
(\pp_0+\lambda)^{1/2} (\pp_0+\lambda)^{1/2}\phi_k
= \phi_j
(\pp_0+\lambda)\phi_k=0$ unless $2Q_j\cap 2Q_k
\neq \emptyset$, it is not very hard to see that
$$
I_{1j} I_{1k}^* I_{1m}I_{1n}^*=0
$$
unless
$$
2Q_j\cap2Q_k\neq\emptyset,\ \  2Q_k\cap 2Q_m\neq\emptyset,\ \ 
\text {and } \ \ 
2Q_m\cap 2Q_n\neq \emptyset .
\tag 4.7
$$
On the other hand, by Lemma 3.2, if we fix any one of
indices $j,k,m,n$, the number of remaining indices
which satisfy (4.7) is bounded by an absolute constant.
Thus, by (2.7),
$$
\aligned
n(1/3,I_1)
&\le C\|\sum_jI_{1j}\|_4^4
\le C\sum \|I_{1j}\|_4 \|I_{1k}\|_4\|I_{1m}\|_4\|I_{1n}\|_4\\
&\le C\sum \big\{
\|I_{1j}\|_4^4+ \|I_{1k}\|_4^4
+\|I_{1m}\|_4^4
+\|I_{1n}\|_4^4\big\}\\
&\le C\sum_j\|I_{1j}\|^4_4
\endaligned
$$
where the second and third
 sums are over all $(j,k,m,n)$ satisfying (4.7).
The lemma now follows from (4.6).
\enddemo

Using $|Y|\le |V|^{1/2}\chi_{\{ x\in\Bbb{R}^3: V(x) <-\lambda\} }$, 
we may deduce easily from Lemma 4.1 that
$$
\int_0^\infty
n(1/3, I_1)\, d\lambda
\le C\int_{\Bbb{R}^3}
|V(x)|^{5/2}dx.
\tag 4.8
$$

Let $\psi_j\in C^\infty_0(3Q_j, \Bbb{R})$
such that $0\le \psi_j\le 1$, $\phi_j\psi_j\equiv \phi_j$ and $
|\nabla^\alpha \psi_j |\le C_\alpha/\ell_j^{|\alpha|}$.

We will need the following lemma in the estimates of
$n(1/3, I_2)$ and $n(1/3,I_3)$. 

\proclaim{\bf Lemma 4.2}
There exist constants $C>0$ and $\varepsilon_0>0$ such that,
if $0<\varepsilon<\varepsilon_0$,
$$
\align
&\| \, [\pp_0,\psi_j] (\pp_0+\Phi +\frac{1}{\ell_j^2})^{-1}
\|_{L^2\to L^2} \le C,
\tag 4.9
\\
&\| \psi_j
(\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}\|_{L^2\to L^\infty}
\le C \ell_j^{\frac12},
\tag 4.10
\\
&
\| (\hh_0 +\frac{1}{\ell_j^2})^{-\frac14 +\delta}
(\sigma\cdot \bb)\psi_j
(\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}
\|_{L^2\to L^2}
\le C_\delta \, \ell_j^{\frac12 -2\delta}
\tag 4.11
\endalign
$$
where $0<\delta <\min\big(1/4, (p-(3/2))/2\big)$.
\endproclaim

\demo{\bf Proof}
To show (4.9), we note that
$$
[\pp_0,\psi_j]=
\big[\Bbb{D}^*,[\Bbb{D},\psi_j]\big]
+[\Bbb{D}^*,\psi_j]\Bbb{D}
+[\Bbb{D},\psi_j]\Bbb{D}^*
\tag 4.12
$$
where $\pp_0=\Bbb{D}^*\Bbb{D}$ and 
$\Bbb{D}=\sigma\cdot (\bold{p}-\bold{A})$
($\pp_0$ is defined as the operator associated with
the quadratic form $\|\Bbb{D}\psi\|_{L^2}^2$.
We do not need that $\Bbb{D}$ is self-adjoint).
Since
$$
\align
& |\, [\Bbb{D}^*,\psi_j]\psi |
+|\, [\Bbb{D},\psi_j]\psi |\le 2|\nabla \psi_j|\, |\psi|
\le \frac{C}{\ell_j}|\psi|,
\tag 4.13 \\
& 
|\, \big[\Bbb{D}^*,[\Bbb{D},\psi_j]\big]\psi|\le |\nabla^2\psi_j|\, |\psi |
\le \frac{C}{\ell_j^2}|\psi|,
\tag 4.14
\endalign
$$
we have 
$$
\|\, [\pp_0,\psi_j]
(\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}\psi\|_{L^2}
\le C\| \psi\|_{L^2}.
\tag 4.15
$$
where we also used (3.10)--(3.11), $\Bbb{D}^*
=\Bbb{D}$ on the domain of $\Bbb{D}$, and
$\|(\pp_0+\Phi +\lambda)^{-1}\|_{L^2\to L^2}
\le \lambda^{-1}$.

To prove (4.10), we use the resolvent identity (4.2) 
and $\pp_0 =\hh_0 -\sigma\cdot \bb$ to write
$$
\aligned
&\psi_j
(\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}=
(\hh_0+\frac{1}{\ell_j^2})^{-1}\psi_j\\
&\ \ \ \ \ \ 
+(\hh_0+\frac{1}{\ell_j^2})^{-1}
\big\{ (\sigma\cdot\bb -\Phi)\psi_j
+[\pp_0,\psi_j]\big\}
(\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}.
\endaligned
\tag  4.16
$$
Hence, by the diamagnetic inequality (2.10) and (2.16), we get
$$
\aligned
&|\psi_j(\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}\psi|\\
&\le 
(-\Delta +\frac{1}{\ell_j^2})^{-1}
|\psi_j\psi| 
+(-\Delta +\frac{1}{\ell_j^2})^{-1}|\bb| \psi_j
|(\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}\psi|\\
& \ \ \ \ \ \ \ 
+(-\Delta +\frac{1}{\ell_j^2})^{-1}
\big| 
(-\Phi\psi_j +[\pp_0, \psi_j])
(\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}\psi\big|\\
&
\le C\ell_j^{\frac12}\|\psi\|_{L^2}
+C\ell_j^{2\delta}
\| (-\Delta +\frac{1}{\ell_j^2})^{\delta-\frac14}
|\bb|\psi_j
|(\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}\psi|\, \|_{L^2}
\endaligned
\tag 4.17
$$
where $0<\delta <\min\big(1/4, (p-(3/2))/2\big)$ and we also used (4.15)
in the second inequality.

To deal with the second term
 in r.h.s. of  (4.17), we apply (2.14) to obtain
$$
\aligned
&\| (-\Delta +\frac{1}{\ell_j^2})^{\delta-\frac14}
|\bb|\psi_j
|(\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}\psi|\, \|_{L^2}\\
&\le 
\|(-\Delta +\frac{1}{\ell_j^2})^{\delta-\frac14}
|\bb|^{-\delta +\frac14}\chi_{3Q_j}\|_{L^2\to L^2}
\|\, |\bb|^{\delta +\frac34}\psi_j
|(\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}\psi|\,
\|_{L^2}\\
&\le C
\| \, |\bb|^{-\delta +\frac14}\|_{L^{\frac{6}{1-4\delta}}(3Q_j)}
\|\, |\bb|^{\delta +\frac34}\|_{L^2(3Q_j)}
\|\psi_j
(\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}\psi \|_{L^\infty}\\
&\le 
C\ell_j^{2-2\delta-\frac{3}{p}}
\left\{ \int_{3Q_j} |\bb|^pdx\right\}^{1/p}
\|
\psi_j(\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}\psi \|_{L^\infty}\\
&\le
C\ell_j^{-2\delta}\varepsilon 
\|\psi_j(\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}\psi\|_{L^\infty}
\endaligned
\tag 4.18
$$
where we also used  H\"older's inequality, $(3/2)+2\delta <p$
and (3.1).
This, together with (4.17), implies that
$$
\|\psi_j (\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}\psi\|_{L^\infty}
\le C\ell_j^{1/2} \|\psi \|_{L^2}
+
C\varepsilon 
\|\psi_j(\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}\psi\|_{L^\infty}.
$$
(4.10) then follows by choosing $\varepsilon_0$ small
so that
$C\varepsilon_0<1/2$, provided that, for any $\psi\in C^\infty_0
(\Bbb{R}^3,\Bbb{C}^2)$, we know
$$
\|\psi_j(\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}\psi\|_{L^\infty}
<\infty.
\tag 4.19
$$
(4.19) can be shown by a bootstrap argument. Indeed, by (4.9) 
and (4.16)--(4.17),
$$
| \psi_j (\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}\psi|
\le 
C\ell_j^{1/2} \|\psi\|_{L^2}
+(-\Delta +\frac{1}{\ell_j^2})^{-1}
|\bb|\,
| \psi_j (\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}\psi|.
$$
Using the well-known fractional integral estimates
for $(-\Delta +\frac{1}{\ell_j^2})^{-1}$
and H\"older's inequality, we see that,
if the function $\psi_j (\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}\psi$
is in $L^{p_1}$, then it is in $L^{p_2}$ where
$(1/p_2)=(1/p_1)+(1/p)-(2/3)$. Starting with $L^2$, since $p>3/2$,
we may deduce that the function is in $L^q$ for any $q<\infty$.
It then follows that it is in $L^\infty$ by the same argument.

Finally
(4.11) follows from (4.18) and (4.10).
\enddemo

We now are ready to estimate $n(1/3, I_2)$ and $n(1/3, I_3)$.

\proclaim {\bf Lemma 4.3} There exists a constant $C>0$ such that
$$
\int_0^\infty n(1/3,I_2)\, d\lambda
\le
C\int_{\Bbb{R}^3}|V(x)|^{5/2}dx
+C\int_{\Bbb{R}^3} \Phi(x)^{3/2} |V(x)|\, dx.
$$
\endproclaim
\demo{\bf Proof}
We first note that, since $\| (\pp_0+\lambda)^{-1/2}\|_{L^2\to L^2}
\le 1/\sqrt{\lambda}$,
$$
\align
n(1/3,I_2)
&=n\big
(1/3,\sum_j Y\phi_j (\pp_0 +\Phi +\frac{1}{\ell_j^2} +\lambda)^{-1}
(\Phi+\frac{1}{\ell_j^2})
\phi_j (\pp_0 +\lambda)^{-1/2}\big)\\
&\le n\big(\sqrt{\lambda}/3,
\sum_j Y\phi_j (\pp_0 +\Phi +\frac{1}{\ell_j^2} +\lambda)^{-1}
(\Phi+\frac{1}{\ell_j^2})
\phi_j\big)\\
&\le
n\big(\sqrt{\lambda}/6,
\sum_j Y\phi_j \big\{ (\pp_0 +\Phi +\frac{1}{\ell_j^2} +\lambda)^{-1}
-(\pp_0 +\Phi +\frac{1}{\ell_j^2} )^{-1}\big\}
(\Phi+\frac{1}{\ell_j^2})
\phi_j\big)\\
&\ \ \ \ \ \ +
n\big(\sqrt{\lambda}/6,
\sum_j Y\phi_j (\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}
(\Phi+\frac{1}{\ell_j^2})
\phi_j\big)\\
&
=K_1+K_2.
\endalign
$$

Using
$$
(\pp_0 +\Phi +\frac{1}{\ell_j^2} +\lambda)^{-1}
-(\pp_0 +\Phi +\frac{1}{\ell_j^2} )^{-1}
=-\lambda
(\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}
(\pp_0 +\Phi +\frac{1}{\ell_j^2} +\lambda)^{-1}
$$
and (2.4), 
we have
$$
\aligned
K_1
&=n\big(\sqrt{\lambda}/6,
\sum_j Y\phi_j  \lambda
 (\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}
(\pp_0 +\Phi +\frac{1}{\ell_j^2} +\lambda)^{-1}
(\Phi+\frac{1}{\ell_j^2})
\phi_j\big)\\
&\le
C{\lambda^2}
\|
\sum_j Y\phi_j  (\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}
(\pp_0 +\Phi +\frac{1}{\ell_j^2} +\lambda)^{-1}
(\Phi+\frac{1}{\ell_j^2})
\phi_j\|_4^4\\
&\le C\lambda^2
\sum_j
\| Y\phi_j  (\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}
(\pp_0 +\Phi +\frac{1}{\ell_j^2} +\lambda)^{-1}
(\Phi+\frac{1}{\ell_j^2})
\phi_j\|_4^4\\
&\le C\lambda^2
\sum_j
\| Y\phi_j  (\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}
(\pp_0 +\Phi +\frac{1}{\ell_j^2} +\lambda)^{-1}
\phi_j\|_4^4\cdot\frac{1}{\ell_j^8}
\endaligned
$$
where we have used the finite intersection property of the supports of
$\phi_j$ as in the proof of Lemma 4.1.

We now use (2.6) and 
the $(L^2, L^2)$ bound of $(\pp_0 +\Phi +\frac{1}{\ell_j^2}
+\lambda)^{-\alpha}$ ($\alpha =1, 1/2$) to obtain
$$
\align
\| Y\phi_j   (\pp_0 &+\Phi +\frac{1}{\ell_j^2})^{-1}
(\pp_0 +\Phi +\frac{1}{\ell_j^2} +\lambda)^{-1}
\phi_j\|_4\\
&\le
\| Y\phi_j  (\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}\|_4
\cdot\frac{1}{\lambda +\frac{1}{\ell_j^2}}\\
&\le
\| Y\phi_j (\pp_0 +\Phi+\frac{1}{\ell_j^2})^{-1/2}\|_4
\cdot\frac{\ell_j}{\lambda +\frac{1}{\ell_j^2}}\\
&\le 
\|Y\phi_j (\hh_0 +\frac{1}{\ell_j^2})^{-1/2}\|_4
\| (\hh_0 +\frac{1}{\ell_j^2})^{1/2}
(\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1/2}\|_{L^2\to L^2}
\cdot\frac{\ell_j}{\lambda +\frac{1}{\ell_j^2}}\\
&\le
C\|Y\phi_j (\hh_0 +\frac{1}{\ell_j^2})^{-1/2}\|_4
\cdot\frac{\ell_j}{\lambda +\frac{1}{\ell_j^2}}\\
&\le C\|Y\phi_j\|_{L^4}
\cdot\frac{\ell_j^2}{\lambda^{5/8}}
\endalign
$$
where we also used (3.8) and (2.12)
in the last two inequality. It then follows that
$$
K_1\le \frac{C}{\sqrt{\lambda}}
\sum_j \int_{\Bbb{R}^3}
|Y\phi_j|^4dx
\le \frac{C}{\sqrt{\lambda}}\int_{\{x\in \Bbb{R}^3:V(x)< -\lambda\} } |V|^2dx.
\tag 4.20
$$

It remains to estimate $K_2$.

Since $Y\le |V|^{1/2}$, we have
$$
K_2
\le n\big(\sqrt{\lambda}/6, \sum_j
|V|^{1/2}\phi_j (\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}
(\Phi+\frac{1}{\ell_j^2})\phi_j\big).
$$
It follows that
$$
\aligned
\int_0^\infty K_2\, d\lambda
&\le
C\| \sum_j
|V|^{1/2}\phi_j (\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}
(\Phi+\frac{1}{\ell_j^2})\phi_j\|_2^2\\
&\le C\sum_j
\|\,
|V|^{1/2}\phi_j (\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}
\phi_j\|_2^2\cdot\frac{1}{\ell_j^4}.
\endaligned
\tag 4.21
$$

Using $\phi_j\psi_j\equiv \phi_j$, (4.16), and (4.9), we have
$$
\align
&\|\,
|V|^{1/2}\phi_j (\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}
\phi_j\|_2\\
&
\le
\|\, |V|^{1/2}\phi_j (\hh_0+\frac{1}{\ell_j^2})^{-1} \phi_j\|_2
+\|\, |V|^{1/2}\phi_j (\hh_0+\frac{1}{\ell_j^2})^{-1}
(\sigma\cdot \bb)\psi_j
(\pp_0 +\Phi  +\frac{1}{\ell_j^2})^{-1}\phi_j\|_2\\
&
+\|\,
 |V|^{1/2}\phi_j (\hh_0+\frac{1}{\ell_j^2})^{-1}
(-\Phi\psi_j +[\pp_0,\psi_j])
(\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}\phi_j\|_2\\
&\le C
\|\, |V|^{1/2}\phi_j (\hh_0+\frac{1}{\ell_j^2})^{-1}\|_2\\
&\ \ \ \ 
+
\|\, |V|^{1/2}\phi_j (\hh_0+\frac{1}{\ell_j^2})^{-\frac34 -\delta} \|_2
\|(\hh_0+\frac{1}{\ell_j^2})^{-\frac14+\delta}
(\sigma\cdot \bb)\psi_j
(\pp_0 +\Phi +\frac{1}{\ell_j^2})^{-1}\phi_j\|_{L^2\to L^2}\\
&\le 
C \|\, |V|^{1/2}\phi_j (\hh_0+\frac{1}{\ell_j^2})^{-1}\|_2
+C\ell_j^{\frac12-2\delta}
\|\, |V|^{1/2}\phi_j (\hh_0+\frac{1}{\ell_j^2})^{-\frac34 -\delta} 
\|_2\\
&\le C\ell_j^{\frac12}\|\, |V|^{1/2}\phi_j\|_{L^2}
\endalign
$$
where we also used (4.11) in the third and (2.11)
in the last inequality.

Inserting the estimate above
into the r.h.s. of (4.21), we obtain
$$
\int_0^\infty
K_2 \, d\lambda
\le 
C\sum_j
\frac{1}{\ell_j^3}
\int_{\Bbb{R}^3} |V|\, \phi_j^2\, dx
\le C\int_{\Bbb{R}^3} \Phi(x)^{3/2} |V(x)|dx.
$$
This, together with (4.20), gives the desired estimate.
\enddemo

\proclaim{\bf Lemma 4.4}
There exists a constant $C>0$ such that
$$
\int_0^\infty n(1/3, I_3)\, d\lambda
\le C\int_{\Bbb{R}^3} |V(x)|^{5/2}dx
+\int_{\Bbb{R}^3} \Phi(x)^{3/2}
|V(x)|dx.
$$
\endproclaim

\demo{\bf Proof}
It follows from  (2.5) and (4.12) that
$$
\aligned
&n(1/3, I_3)
=n\big(1/3,
\sum_j Y\phi_j (\pp_0 +\Phi +\frac{1}{\ell_j^2}+\lambda)^{-1}
[\pp_0,\phi_j](\pp_0+\lambda)^{-1/2}\big)\\
&\le
n\big(1/6,
\sum_j Y\phi_j (\pp_0 +\Phi +\frac{1}{\ell_j^2}+\lambda)^{-1}
[\Bbb{D}^*, [\Bbb{D},\phi_j]](\pp_0 +\lambda)^{-1/2}\big)\\
&\ \ \ \ \ +
n\big(1/6,
2\sum_j Y\phi_j (\pp_0 +\Phi +\frac{1}{\ell_j^2}+\lambda)^{-1}
\left\{ [\Bbb{D}^*, \phi_j] \Bbb{D}
+[\Bbb{D}, \phi_j] \Bbb{D}^*\right\}
 (\pp_0 +\lambda)^{-1/2}\big)
\endaligned
\tag 4.22
$$
where $\Bbb{D}=\sigma\cdot (\bold{p}-\bold{A})$. 

Since
$|\big[\Bbb{D}^*, [\Bbb{D},\phi_j]\big]\psi|\le |\nabla^2 \phi_j|\, |\psi|$,
the first term on the r.h.s. of (4.22)
can be treated exactly as $n(1/3, I_2)$.
With $\| \Bbb{D} (\pp_0 +\lambda )^{-1/2}\|_{L^2\to L^2}\le 1$
and $\| \Bbb{D}^* (\pp_0 +\lambda )^{-1/2}\|_{L^2\to L^2}\le 1$, we may
use the same argument as in the proof of Lemma 4.1 to
bound the second term by
$$
\aligned
C\| 
\sum_j Y\phi_j (\pp_0 +\Phi & +\frac{1}{\ell_j^2}+\lambda)^{-1}
[\Bbb{D}, \phi_j]\|_4^4
+C\| 
\sum_j Y\phi_j (\pp_0 +\Phi +\frac{1}{\ell_j^2}+\lambda)^{-1}
[\Bbb{D}^*, \phi_j]\|_4^4\\
&\le
C\sum_j \|Y\phi_j (\pp_0 +\Phi +\frac{1}{\ell_j^2}+\lambda)^{-1}\|_4^4
\cdot\frac{1}{\ell_j^4}\\
&
\le C\sum_j
\|Y\phi_j (\pp_0 +\Phi +\frac{1}{\ell_j^2}+\lambda)^{-1/2}\|_4^4\\
&\le C\sum_j
\|Y\phi_j (\hh_0+\lambda)^{-1/2}\|_4^4\\
&\le\frac{C}{\sqrt{\lambda}}
\int_{\{ x\in \Bbb{R}^3:V(x)< -\lambda\} }
|V|^2dx.
\endaligned
$$
The lemma then follows by integration.
\enddemo

We need one more lemma before we carry out the proof of Theorem 1.1.

\proclaim{\bf Lemma 4.5}
There exist two constants $C_1>0$, $C_2>0$
depending on $\varepsilon$ and
$p$, such that, for every $x\in \Bbb{R}^3$,
$$
C_1 \Phi (x)\le b_p(x)\le C_2 \Phi(x).
$$
\endproclaim

\demo{\bf Proof}
Suppose $x\in Q_j$ for some $j$. Then $\Phi (x)\approx 1/\ell_j^2$.

Let $\delta\in (0,1)$ and $\ell_j=\ell(Q_j)$.
Since $Q(x,\delta \ell_j)\subset 12 Q_j$, we have
$$
\align
(\delta \ell_j)^2 \bigg(\frac{1}{(\delta\ell_j)^3}
\int_{Q(x,\delta \ell_j)}&
|\bb(y)|^pdy\bigg)^{1/p}
= (\delta \ell_j)^{2-\frac{3}{p}}
\left(\int_{Q(x,\delta \ell_j)}
|\bb(y)|^pdy\right)^{1/p}\\
&\le
(\delta \ell_j)^{2-\frac{3}{p}}
\left(\int_{12 Q_j}
|\bb(y)|^pdy\right)^{1/p}\\
&\le
(\delta \ell_j)^{2-\frac{3}{p}}
\cdot
\frac{\varepsilon}{\ell_j^{2-\frac{3}{p}}}
=\delta^{2-\frac{3}{p}}
\varepsilon <1.
\endalign
$$
Hence, by the definition of $\ell_p(x)$ (see (1.9)), we have
$\ell_p(x)\ge \ell_j$. It follows that
$$
b_p(x)=\frac{1}{\{ \ell_p(x)\}^2 }
\le\frac{1}{\ell_j^2}\le C_2 \Phi (x).
$$

To show $b_p(x)\ge C_1 \Phi(x)$, we note that, if $N$ is large,
$12 Q_j^+\subset Q(x,N\ell_j)$ where $Q_j^+$ is the dyadic
parent of $Q_j$. Hence
$$
\align
(N\ell_j)^2
\bigg(\frac{1}{(N\ell_j)^3}
\int_{Q(x,N\ell_j)}&
|\bb(y)|^pdy\bigg)^{1/p}
=
(N\ell_j)^{2-\frac{3}{p}}
\left(
\int_{Q(x,N\ell_j)}
|\bb(y)|^pdy\right)^{1/p}\\
&\ge 
(N\ell_j)^{2-\frac{3}{p}}
\left(
\int_{12 Q_j^+}
|\bb(y)|^pdy\right)^{1/p}\\
&>
(N\ell_j)^{2-\frac{3}{p}}\cdot \frac{\varepsilon}
{(2\ell_j)^{2-\frac{3}{p}}}
=\varepsilon
\left(\frac{N}{2}\right)^{2-\frac{3}{p}}
>1
\endalign
$$
if $N$ is large enough. Again, by definition,
this implies that $\ell_p(x)\le N\ell_j$. Therefore
$$
b_p(x)=\frac{1}{\{\ell_p(x)\}^2 }
\ge \frac{1}{N^2\ell_j^2}
\ge C_1 \Phi (x).
$$
\enddemo

Finally we are in a position to give the 

\demo{\bf Proof of Theorem 1.1}

It suffices to prove the theorem for $\gamma=1$ (see \cite{20}).
We may also assume, without the loss of generality, that $V\le 0$ a.e..

By means of (4.1) and (4.3), we have
$$
\align
\Cal{M}_1&=\sum_{\lambda_j<0} |\lambda_j|
=2\int_0^\infty N(2\lambda, \pp_0+V)\, d\lambda\\
&\le 2\int_0^\infty n(1, Y(\pp_0+\lambda)^{-1/2})\, d\lambda\\
&\le 2\sum_{j=1}^3
\int_0^\infty
n(1/3, I_j)\, d \lambda.
\endalign
$$
The theorem now follows from (4.8) and Lemmas 4.3--4.5.
\enddemo
\bigskip

\centerline{\bf  5. Fields with Constant Directions}

In this section we study the special case where the magnetic
field $\bb$ has a constant direction. The goal is to prove
Theorem 1.2 stated in the Introduction.

Without the loss of generality, we may assume that
$\bold{A}=(A_1(x_1,x_2), A_2(x_1,x_2),0)$
and $\bb=(0,0,B)$ where $B=\frac{\partial A_2}{\partial x_1}
-\frac{\partial A_1}{\partial x_2}$.
With this assumption, we have the following identity:
$$
\int_{\Bbb{R}^3}
|\sigma\cdot (\bold{p}-\bold{A})\psi|^2dx
=\int_{\Bbb{R}^3}
| [\sigma_1(p_1-A_1)+\sigma_2(p_2-A_2)]\psi|^2dx
+\int_{\Bbb{R}^3}
|\frac{\partial \psi}{\partial x_3}|^2dx.
$$
This in particular implies that, for $\lambda>0$,
$$
\| (-\partial_{x_3}^2 +\lambda)^{1/2}
(\pp+\lambda)^{-1/2}\|_{L^2\to L^2}\le 1.
\tag 5.1
$$

We shall use $x^\prime =(x_1,x_2)$, $y^\prime=(y_1,y_2)$
to denote points in $\Bbb{R}^2$.


Our approach to the case of constant direction fields is
 similar to that in Section 4
for arbitrary fields.

We begin by observing that, if $\bb=(0,0,B(x^\prime))$,
the basic length scale $\ell_p(x)$ defined by (1.9) is reduced
to
$$
\ell_p(x^\prime)
=\sup \left\{
\ell>0:\ \ 
\ell^2
\left\{ \frac{1}{\ell^2}\int_{S(x^\prime,\ell)}
|B(y^\prime)|^p dy^\prime\right\}^{1/p}\le 1 \right\}
\tag 5.2
$$
where $S(x^\prime,\ell)$ denotes the square in $\Bbb{R}^2$
centered at $x^\prime$ with side length $\ell$.

We will assume that $B\not\equiv 0$ and
$B\in L^p_{loc}(\Bbb{R}^2)$ for some $p>1$.
With this we have $0<\ell_p(x^\prime)<\infty$ 
for any $x^\prime\in \Bbb{R}^2$.

We now sketch the construction of the partition of unity for
$\Bbb{R}^2$ associated with $B$, which is parallel
to that in Section 3.

First we write
$$
\Bbb{R}^2
=\bigcup_j S_j
\tag 5.3
$$
where $\{ S_j\}_{j=1}^\infty$ are maximal elements
in the set of all dyadic squares $S$ in $\Bbb{R}^2$ such that
$$
\left\{
\int_{12 S} |B(x^\prime)|^pd x^\prime\right\}^{1/p}
\le
\frac{\varepsilon}
{[\ell(S)]^{2-\frac{2}{p}}}
\tag 5.4
$$
where $\varepsilon\in (0,1)$ is a constant to be determined later.

Next we use the same argument as in the proof of Lemma 3.2
to show that,
$$
\frac12\ell(S_k)
\le \ell(S_j)
\le 2\ell(S_k),
\ \ \ \ \text{ if }
\ \ 4S_j\cap 4S_k\neq\emptyset.
\tag 5.5
$$
It follows from (5.5) that there exists a sequence of functions
$\{ \varphi_j\}$ such that
$$
\align
&\varphi_j\in C_0^\infty (2S_j, \Bbb{R})\ \ \ \text{ and }\ \ 
0\le \varphi_j\le 1,
\tag 5.6\\
& |\nabla^\alpha \varphi_j|\le C_\alpha/\ell_j^{|\alpha|}
\ \ \ \text{ where }\ \ \ell_j=\ell(S_j),
\tag 5.7\\
& \sum_j\varphi_j^2\equiv 1\ \ \ \text{ in } \ \Bbb{R}^2.
\tag 5.8
\endalign
$$

Let
$$
\Psi =\sum_j\frac{1}{\ell_j^2}\, \varphi_j^2.
\tag 5.9
$$

\proclaim{\bf Theorem 5.1}
There exist constants $C>0$ and $\varepsilon_0>0$ such that,
if $\varepsilon$ in (5.4) is less than $\varepsilon_0$, then
$$
\hh_0\le C\left\{ \pp_0 +\Psi\right\}.
$$
\endproclaim

\demo{\bf Proof}
We will only show that, if $\varepsilon$ in (5.4) is small,
then
$$
\int_{\Bbb{R}^3}
|\, |B|^{1/2} \varphi_j\psi|^2dx
\le C\int_{\Bbb{R}^3}
|\sigma\cdot (\bold{p}-\bold{A})(\varphi_j\psi)|^2dx
\tag 5.10
$$
for any $\psi\in C^\infty_0(\Bbb{R}^3,\Bbb{C}^2)$.
The rest of the proof is exactly the same as that of Theorem 3.1.

To show (5.10), we need the following Sobolev inequality
$$
\left(\int_S |f|^qdx^\prime\right)^{1/q}
\le C_q |S|^{1/q}
\left( \int_S |\nabla_2 f|^2dx^\prime\right)^{1/2}
\tag 5.11
$$
for $f\in C_0^\infty (S,\Bbb{R})$ where $S$ is a square, $\nabla_2
=(\partial_{x_1},\partial_{x_2})$, and $1<q<\infty$.

Let $\frac1p +\frac1q =1$. By (5.11) and the inequality
$|\nabla |\psi|\,|\le | (\bold{p}-\bold{A})\psi |$, we have
$$
\aligned
\int_{\Bbb{R}^3}
|\, |B|^{1/2} \varphi_j\psi|^2dx
&=\int_{\Bbb{R}} d x_3
\int_{\Bbb{R}^2}
|B|\, |\varphi_j\psi|^2dx^\prime\\
&\le
\int_{\Bbb{R}} d x_3
\left\{ \int_{2S_j} |B|^pd x^\prime\right\}^{1/p}
\left\{ \int_{2S_j}
|\varphi_j \psi|^{2q}d x^\prime\right\}^{1/q}\\
&\le C\int_{\Bbb{R}} d x_3
\left\{ \int_{2S_j} |B|^pd x^\prime\right\}^{1/p}
\left\{ \int_{2S_j}
|\nabla_2 |\varphi_j \psi|\, |^2dx^\prime\right\}\cdot \ell_j^{2/q}\\
&\le
C\varepsilon
\int_{\Bbb{R}^3}
|\nabla |\varphi_j \psi|\, |^2dx\\
&\le 
C\varepsilon\int_{\Bbb{R}^3}
|(\bold{p}-\bold{A})(\varphi_j\psi)|^2dx\\
&\le
C\varepsilon\int_{\Bbb{R}^3}
|\sigma\cdot (\bold{p}-\bold{A})(\varphi_j\psi)|^2dx
+C\varepsilon
\int_{\Bbb{R}^3}
|\, |B|^{1/2}\varphi_j \psi|^2 dx
\endaligned
$$
where we also used (5.4) in the third inequality.
(5.10) then follows by choosing $\varepsilon$
small so that $C\varepsilon<1/2$.
\enddemo

Note that Theorem 5.1 implies that
$$
\| (\hh_0 +\lambda)^{1/2}
(\pp_0+\Psi +\lambda)^{-1/2}\|_{L^2\to L^2}
\le C.
\tag 5.12
$$
Also, same arguments as in the proof of Lemma 3.4 give
$$
\| (\pp_0 +\Psi +\frac{1}{\ell_j^2}
+\lambda)^{-1/2}
\varphi_j (\pp_0+\lambda)^{1/2}
\|_{L^2\to L^2}\le C.
\tag 5.13
$$

To prove Theorem 1.2, we will show that
$$
N(2\lambda, \pp_0 +V)
\le \frac{C}{\sqrt{\lambda}}
\int_{\{x\in \Bbb{R}^3: V(x)<-\lambda\}}
|V(x)|^2dx
+
\frac{C}{\sqrt{\lambda}}
\int_{\{x\in \Bbb{R}^3: V(x)<-\lambda\} }
b_p(x)|V(x)|dx.
$$
To this end, we use the resolvent identity (4.2) to write
$$
\aligned
Y(\pp_0+\lambda)^{-1/2}
&=
\sum_j Y\varphi_j (\pp_0+\Psi +\frac{1}{\ell_j^2}
+\lambda)^{-1} \varphi_j (\pp_0+\lambda)^{1/2}\\
&\ \ \ \ \ \ \ 
+\sum_j Y\varphi_j (\pp_0+\Psi +\frac{1}{\ell_j^2}
+\lambda)^{-1} (\Psi +\frac{1}{\ell_j^2})\varphi_j
(\pp_0+\lambda)^{-1/2}\\
&\ \ \ \ \ \ \ 
+\sum_j Y\varphi_j (\pp_0+\Psi +\frac{1}{\ell_j^2}
+\lambda)^{-1}
[\pp_0,\varphi_j](\pp_0+\lambda)^{-1/2}\\
&=J_1+J_2+J_3
\endaligned
$$
where $Y=|V+\lambda|^{1/2}_-$. As before,
$$
N(2\lambda, \pp_0 +V)
\le n(1, Y(\pp_0+\lambda)^{-1/2})
\le n(1/3, J_1)
+n(1/3, J_2)
+n(1/3, J_3).
\tag 5.14
$$

\proclaim{\bf Lemma 5.1}
There exists a constant $C>0$ such that, for any $\lambda>0$,
$$
n(1/3, J_1)
\le \frac{C}{\sqrt{\lambda}}
\int_{\{ x\in \Bbb{R}^3: V(x)<-\lambda\} }
|V(x)|^2dx.
$$
\endproclaim

The proof of Lemma 5.1, which uses (5.12)--(5.13) and (2.12),
is similar to that of Lemma 4.1.
We leave it to the reader.

Let $\xi_j\in C_0^\infty (3S_j,\Bbb{R})$ such that
$0\le \xi_j\le 1$, $\xi_j\varphi_j\equiv \varphi_j$,
and $|\nabla^\alpha\xi_j|\le C_\alpha/\ell_j^{|\alpha|}$.

The following lemma is needed to estimate $n(1/3,J_2)$
and $n(1/3,J_3)$.

\proclaim{\bf Lemma 5.2}
Let $0<\delta \le \frac12 (1-\frac1p)$. There exists a constant
$C_\delta>0$ such that, for any $\lambda >0$,
$$
\| (\hh_0 +\frac{1}{\ell_j^2} +\lambda)^{-\frac12 +\delta}
(\sigma\cdot \bb)\xi_j
(\pp_0+\Psi +\frac{1}{\ell_j^2}+\lambda)^{-1}
\|_{L^2\to L^2}
\le C_\delta \, \ell_j^{1-2\delta}.
$$
\endproclaim

\demo{\bf Proof}
Note that, by (5.12),
$$
\aligned
 \| (\hh_0 +&\frac{1}{\ell_j^2} +\lambda)^{-\frac12 +\delta}
(\sigma\cdot \bb)\xi_j
(\pp_0+\Psi +\frac{1}{\ell_j^2}+\lambda)^{-1}
\|_{L^2\to L^2}\\
&\le
\| (\hh_0 +\frac{1}{\ell_j^2} +\lambda)^{-\frac12 +\delta}
(\sigma\cdot \bb)\xi_j
(\pp_0+\Psi +\frac{1}{\ell_j^2}+\lambda)^{-1/2}
\|_{L^2\to L^2}\cdot \ell_j\\
&\le 
\| (\hh_0 +\frac{1}{\ell_j^2} +\lambda)^{-\frac12 +\delta}
(\sigma\cdot \bb)\xi_j
(\hh_0+\frac{1}{\ell_j^2}+\lambda)^{-1/2}
\|_{L^2\to L^2}\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \cdot
\|(\hh_0+\frac{1}{\ell_j^2}+\lambda)^{1/2}
(\pp_0+\Psi +\frac{1}{\ell_j^2}+\lambda)^{-1/2}
\|_{L^2\to L^2}\cdot \ell_j\\
&\le
C\ell_j
\| (\hh_0 +\frac{1}{\ell_j^2} +\lambda)^{-\frac12 +\delta}
(\sigma\cdot \bb)\xi_j
(\hh_0+\frac{1}{\ell_j^2}+\lambda)^{-\frac12}
\|_{L^2\to L^2}\\
&\le
C\ell_j^{1+2\delta}
\| (\hh_0 +\frac{1}{\ell_j^2} +\lambda)^{-\frac12 +\delta}
(\sigma\cdot \bb)\xi_j
(\hh_0+\frac{1}{\ell_j^2}+\lambda)^{-\frac12 +\delta}
\|_{L^2\to L^2}.
\endaligned
$$
Thus, by the diamagnetic inequality (2.10) and duality, 
it suffices to show that
$$
\|\, |B|^{1/2}\chi_{3S_j}
(-\Delta +\frac{1}{\ell_j^2}
+\lambda )^{-\frac12 +\delta}
\|_{L^2\to L^2}
\le C_\delta\, \ell_j^{-2\delta}.
\tag 5.15
$$

We claim that (5.15) follows from Lemma 2.3. Indeed, 
by Lemma 2.3, the l.h.s. of (5.15) is bounded by
$$
C_\delta \, \|\, |B|^{1/2}\|_{L^{\frac{2}{1-2\delta}}(3S_j)}
\le C_\delta \ell_j^{-2\delta}
$$
where we  used $p\ge 1/(1-2\delta)$, H\"older's inequality, and
(5.4) in the inequality.

The proof is complete.
\enddemo

We are now ready to deal with $n(1/3,J_2)$ and
$n(1/3,J_3)$.

\proclaim{\bf Lemma 5.3}
There exists a constant $C>0$ such that
$$
n(1/3, J_2)
\le \frac{C}{\sqrt{\lambda}}
\int_{\{ x\in\Bbb{R}^3: V(x)<-\lambda\} }
\Psi (x) |V(x)|dx.
$$
\endproclaim

\demo{\bf Proof}
First we note that, by (2.4), (2.6), and (5.1),
$$
\aligned
n(1/3, J_2)
&=n\big(1/3,
\sum_j Y\varphi_j (\pp_0 +\Psi +\frac{1}{\ell_j^2}
+\lambda)^{-1} (\Psi +\frac{1}{\ell_j^2})\varphi_j
 (\pp_0+\lambda)^{-1/2}\big)\\
&\le C
\| \sum_j Y\varphi_j (\pp_0 +\Psi +\frac{1}{\ell_j^2}
+\lambda)^{-1} (\Psi +\frac{1}{\ell_j^2})\varphi_j (\pp_0+\lambda)^{-1/2}
\|_2^2\\
&\le C
\|
\sum_j Y\varphi_j(-\partial_{x_3}^2+\lambda)^{-1/2}
 (\pp_0 +\Psi +\frac{1}{\ell_j^2}
+\lambda)^{-1} (\Psi +\frac{1}{\ell_j^2})\varphi_j
\|_2^2\\
&\ \ \ \ \ \ \ \ \cdot
\| (-\partial_{x_3}^2+\lambda)^{1/2} (\pp_0+\lambda)^{-1/2}\|^2_{L^2\to L^2}\\
&\le C
\|\sum_j Y\varphi_j(-\partial_{x_3}^2+\lambda)^{-1/2}
 (\pp_0 +\Psi +\frac{1}{\ell_j^2}
+\lambda)^{-1} (\Psi +\frac{1}{\ell_j^2})\varphi_j
\|_2^2\\
&\le C
\sum_j \|Y\varphi_j(-\partial_{x_3}^2+\lambda)^{-1/2}
 (\pp_0 +\Psi +\frac{1}{\ell_j^2}
+\lambda)^{-1} (\Psi +\frac{1}{\ell_j^2})\varphi_j
\|_2^2\\
&\le C
\sum_j \|Y\varphi_j(-\partial_{x_3}^2+\lambda)^{-1/2}
 (\pp_0 +\Psi +\frac{1}{\ell_j^2}
+\lambda)^{-1}\varphi_j
\|_2^2\cdot\frac{1}{\ell_j^4}
\endaligned
$$
where we used the finite intersection property of supp$\varphi_j$ in
the fourth inequality. The fact that $\pp_0+\Psi$ commutes with
$\partial_{x_3}$ is essential in the estimate above.

Next we use the resolvent identity (4.2) to obtain
$$
\aligned
&\|Y\varphi_j(-\partial_{x_3}^2+\lambda)^{-1/2}
 (\pp_0 +\Psi +\frac{1}{\ell_j^2}
+\lambda)^{-1}\varphi_j
\|_2\\
&=
\|Y\varphi_j(-\partial_{x_3}^2+\lambda)^{-1/2}
 \xi_j(\pp_0 +\Psi +\frac{1}{\ell_j^2}
+\lambda)^{-1}\varphi_j
\|_2\\
&\le
\|Y\varphi_j(-\partial_{x_3}^2+\lambda)^{-1/2}
 (\hh_0+\frac{1}{\ell_j^2}
+\lambda)^{-1}\varphi_j
\|_2\\
&\ \ \ \ \ 
+\|Y\varphi_j(-\partial_{x_3}^2+\lambda)^{-1/2}
 (\hh_0+\frac{1}{\ell_j^2}
+\lambda)^{-1}(\sigma\cdot \bb)\xi_j
(\pp_0 +\Psi +\frac{1}{\ell_j^2}
+\lambda)^{-1}\varphi_j
\|_2\\
&\ \ \ \ \ 
+\|Y\varphi_j(-\partial_{x_3}^2+\lambda)^{-1/2}
 (\hh_0+\frac{1}{\ell_j^2}
+\lambda)^{-1}
\left\{
-\Psi\xi_j +[\pp_0,\xi_j]\right\}
(\pp_0 +\Psi +\frac{1}{\ell_j^2}
+\lambda)^{-1}\varphi_j
\|_2\\
&=K_1+K_2+K_3.
\endaligned
$$

By (2.13),
$$
K_1\le C\lambda^{-1/4}
(\frac{1}{\ell_j^2}+\lambda)^{-1/2}
\|Y\varphi_j\|_{L^2}
\le C\lambda^{-1/4}\ell_j \|Y\varphi_j\|_{L^2}. \tag 5.16
$$

To estimate $K_2$, we use (2.6), Lemma 5.2, and (2.13) to obtain
$$
\aligned
K_2&\le 
\|Y\varphi_j (-\partial_{x_3}^2+\lambda)^{-1/2}
(\hh_0+\frac{1}{\ell_j^2}+\lambda)^{-\frac12 -\delta}\|_2\\
&\ \ \ \ \ \ \ \ \cdot
\| (\hh_0+\frac{1}{\ell_j^2}+\lambda)^{-\frac12 +\delta}
(\sigma\cdot\bb )\xi_j
(\pp_0+\Psi +\frac{1}{\ell_j^2}+\lambda)^{-1}\|_{L^2\to L^2}\\
&\le
C\, 
\|Y\varphi_j (-\partial_{x_3}^2+\lambda)^{-1/2}
(\hh_0+\frac{1}{\ell_j^2}+\lambda)^{-\frac12 -\delta}\|_2
\cdot \ell_j^{1-2\delta}\\
&\le C\,
\lambda^{-1/4} (\frac{1}{\ell_j^2}+\lambda)^{-\delta}
\|Y\varphi_j\|_{L^2}\cdot \ell_j^{1-2\delta}\\
&\le C\, 
\lambda^{-1/4}\ell_j \|Y\varphi_j\|_{L^2}
\endaligned
\tag 5.17
$$
where $\delta=\frac12 (1-\frac1p)$.

To bound $K_3$, we observe that
$$
\| \left\{ -\Psi\xi_j
+[\pp_0,\xi_j]\right\}
(\pp_0+\Psi +\frac{1}{\ell_j^2}
+\lambda)^{-1}
\|_{L^2\to L^2}\le C
$$
using the same argument as in the proof of (4.9). It follows that
$$
\aligned
K_3
&\le
\| Y\varphi_j (-\partial_{x_3}^2 +\lambda)^{-1/2}
(\hh_0+\frac{1}{\ell_j^2}+\lambda)^{-1}\|_2\\
&\ \ \ \ \ \ \ \ \ \ \cdot
\| \left\{ -\Psi\xi_j
+[\pp_0,\xi_j]\right\}
(\pp_0+\Psi +\frac{1}{\ell_j^2}
+\lambda)^{-1}
\|_{L^2\to L^2}\\
&\le C\lambda^{-1/4}\ell_j \|Y\varphi_j\|_{L^2}.
\endaligned
\tag 5.18
$$

Finally, we add (5.16), (5.17) and (5.18) to conclude that
$$
\|Y\varphi_j (-\partial_{x_3}^2+\lambda)^{-1/2}
(\pp_0+\Psi +\frac{1}{\ell_j^2}
+\lambda)^{-1}
\varphi_j
\|_2
\le K_1+K_2 +K_3
\le C\lambda^{-1/4}\ell_j \|Y\varphi_j\|_{L^2}.
$$
It follows that
$$
\aligned
n(1/3,J_2)
&\le
C\sum_j 
\|Y\varphi_j (-\partial_{x_3}^2+\lambda)^{-1/2}
(\pp_0+\Psi +\frac{1}{\ell_j^2}
+\lambda)^{-1}
\varphi_j
\|_2^2\cdot\frac{1}{\ell_j^4}\\
&\le\frac{C}{\sqrt{\lambda}}
\sum_j
\frac{1}{\ell_j^2}
\int_{\Bbb{R}^3} |Y\varphi_j|^2dx\\
&\le \frac{C}{\sqrt{\lambda}}
\int_{\{ x\in\Bbb{R}^3: V(x)< -\lambda\} }
\Psi(x) |V(x)|dx
\endaligned
$$
since $|Y|\le |V|^{1/2}\chi_{\{ x\in\Bbb{R}^3: V(x)< -\lambda\} }$.
The proof is finished.
\enddemo

\proclaim{\bf Lemma 5.4}
There exists a constant $C>0$ such that
$$
n(1/3,J_3)
\le\frac{C}{\sqrt{\lambda}}
\int_{\{ x\in\Bbb{R}^3: V(x)< -\lambda\} }
|V(x)|^2dx
+
\frac{C}{\sqrt{\lambda}}
\int_{\{ x\in\Bbb{R}^3: V(x)< -\lambda\} }
\Psi(x)
|V(x)|dx.
$$
\endproclaim

\demo{\bf Proof}
Using
$[\pp_0,\varphi_j]=\big[\Bbb{D}^*,[\Bbb{D},\varphi_j]\big]
+[\Bbb{D},\varphi_j]\Bbb{D}^*
+[\Bbb{D}^*,\varphi_j]\Bbb{D}
$, we have
$$
\aligned
J_3&=
\sum_jY\varphi_j (\pp_0+\Psi +\frac{1}{\ell_j^2}+\lambda)^{-1}
[\pp_0,\varphi_j] (\pp_0 +\lambda)^{-1/2}\\
&=
\sum_jY\varphi_j (\pp_0+\Psi +\frac{1}{\ell_j^2}+\lambda)^{-1}
\big[\Bbb{D}^*,[\Bbb{D},\varphi_j]\big] (\pp_0 +\lambda)^{-1/2}\\
&\ \ \ \ 
+\sum_jY\varphi_j (\pp_0+\Psi +\frac{1}{\ell_j^2}+\lambda)^{-1}
\left\{
[\Bbb{D},\varphi_j] \Bbb{D}^*
+
[\Bbb{D}^*,\varphi_j] \Bbb{D}
\right\}
(\pp_0 +\lambda)^{-1/2}\\
&=J_{31}
+J_{32}.
\endaligned
$$
Thus,
$$
n(1/3,J_3)\le n(1/6, J_{31}) +n(1/6, J_{32}).
$$

Since 
$| [\Bbb{D}^*,[\Bbb{D},\varphi_j]]\psi|\le |\nabla^2 \varphi_j|\, |\psi|$,
we may use the same argument as in the proof of Lemma 5.3 to show that
$$
n(1/6, J_{31})
\le\frac{C}{\sqrt{\lambda}}
\int_{\{ x\in\Bbb{R}^3: V(x)< -\lambda\} }
\Psi (x) |V(x)|dx.
$$
On the other hand, using $\| \Bbb{D} (\pp_0+\lambda)^{-1/2}
\|_{L^2\to L^2}\le 1 $,
$\| \Bbb{D}^* (\pp_0+\lambda)^{-1/2}
\|_{L^2\to L^2}\le 1 $,
and (5.12), we may bound $n(1/6, J_{32})$ by
$$
\aligned
&C\| \sum_jY\varphi_j (\pp_0+\Psi +\frac{1}{\ell_j^2}+\lambda)^{-1}
\left\{
[\Bbb{D}^*,\varphi_j] \Bbb{D}
+
[\Bbb{D},\varphi_j] \Bbb{D}^*\right\}
(\pp_0 +\lambda)^{-1/2}\|_4^4\\
&\le C\| \sum_jY\varphi_j (\pp_0+\Psi +\frac{1}{\ell_j^2}+\lambda)^{-1}
[\Bbb{D}^*,\varphi_j]\|_4^4
+
C\| \sum_jY\varphi_j (\pp_0+\Psi +\frac{1}{\ell_j^2}+\lambda)^{-1}
[\Bbb{D},\varphi_j]\|_4^4\\
&\le C \sum_j\|Y\varphi_j (\pp_0+\Psi +\frac{1}{\ell_j^2}+\lambda)^{-1}
[\Bbb{D}^*,\varphi_j]\|_4^4
+
C \sum_j\|Y\varphi_j (\pp_0+\Psi +\frac{1}{\ell_j^2}+\lambda)^{-1}
[\Bbb{D},\varphi_j]\|_4^4\\
&\le C
 \sum_j\|Y\varphi_j (\pp_0+\Psi +\frac{1}{\ell_j^2}+\lambda)^{-1/2}\|_4^4
\cdot\frac{1}{(\frac{1}{\ell_j^2}+\lambda)^2}
\cdot \frac{1}{\ell_j^4}\\
&\le C
\sum_j\|Y\varphi_j (\hh_0+\frac{1}{\ell_j^2}+\lambda)^{-1/2}\|_4^4
\| (\hh_0+\frac{1}{\ell_j^2}+\lambda)^{1/2}
(\pp_0+\Psi +\frac{1}{\ell_j^2}+\lambda)^{-1/2}\|_{L^2\to L^2}^4\\
&\le C
\cdot
\sum_j\|Y\varphi_j (\hh_0+\frac{1}{\ell_j^2}+\lambda)^{-1/2}\|_4^4\\
&\le
\frac{C}{\sqrt{\lambda}}
\int_{\{ x\in\Bbb{R}^3: V(x)< -\lambda\} }
|V(x)|^2dx
\endaligned
$$
where (2.12) is used in the last inequality.

The proof is complete.

Finally we are in a position to give the
\demo{\bf Proof of Theorem 1.2}

It follows from (5.14), Lemmas 5.1, 5.3, and 5.4 that
$$
\aligned
N(2\lambda, \pp_0 +V)
&\le n(1/3, J_1) +n(1/3, J_2) +n(1/3, J_3)\\
&\le 
\frac{C}{\sqrt{\lambda}}
\int_{\{ x\in\Bbb{R}^3: V(x)< -\lambda\} }
|V(x)|^2dx
+\frac{C}{\sqrt{\lambda}}
\int_{\{ x\in\Bbb{R}^3: V(x)< -\lambda\} }
\Psi (x)|V(x)|dx.
\endaligned
$$
Thus, if $\gamma>1/2$,
$$
\aligned
\Cal{M}_\gamma
&=\sum_{\lambda_j<0}
|\lambda|^\gamma
=\gamma \int_0^\infty
\lambda^{\gamma -1} N(\lambda,\pp_0+V)\, d\lambda\\
&=
\gamma 2^\gamma 
\int_0^\infty
\lambda^{\gamma -1} N(2\lambda,\pp_0+V)\, d\lambda\\
&\le
C_\gamma \int_{\Bbb{R}^3}
|V(x)|_-^{\gamma+3/2}dx
+C_\gamma
\int_0^\infty
\Psi(x) |V(x)|_-^{\gamma+1/2}dx.
\endaligned
$$

Noting that $\Psi(x)\approx b_p(x)$ for $p>1$  by an argument similar
to that in the proof of Lemma 4.5,
we are done.
\enddemo

\Refs
\widestnumber\key{20}

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\ref
\key 2
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J.~Stubbe
\paper
A Lieb-Thirring bound for a magnetic Pauli Hamiltonian
\jour Commun.~Math.~Phys.
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\yr 1997
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\ref
\key 3
\by
L.~Erd\"os
\paper
Magnetic Lieb-Thirring inequalities
\jour Commun.~Math.~Phys
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\ref 
\key 4
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