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Pauli operator, singular magnetic field
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\title{Pauli operator and Aharonov Casher theorem for
measure valued magnetic fields}
\author{L\'aszl\'o Erd\H os
 \thanks{Email address: {\tt lerdos@math.gatech.edu}.
Partially supported by NSF grant DMS-9970323}
\\ School of Mathematics, Georgiatech, Atlanta GA 30332 \\
and \\
Vitali Vougalter  \thanks{Email address: {\tt vitali@math.ubc.ca}}
\\ Department of Mathematics,
 University of British Columbia \\
Vancouver, B.C., Canada V6T 1Z2}



\date{May 3, 2001}



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\begin{document}
\maketitle

\begin{abstract}
We define the two dimensional Pauli operator 
and identify its core for 
magnetic fields that are regular Borel measures. The magnetic
field is generated by a scalar potential hence we bypass the usual
$\bA\in L^2_{loc}$ condition on the vector potential which
does not allow to consider such singular fields. 
We extend  Aharonov-Casher theorem for magnetic fields that are
measures with finite total variation and we present a counterexample
in case of infinite total variation. One
of the key technical tools is a weighted $L^2$ estimate on
a singular integral operator.
\end{abstract}

\medskip\noindent
{\bf AMS 2000 Subject Classification:} 81Q10

\medskip\noindent
{\it Running title:} Pauli Operator for  Measure Valued Fields.


\section{Introduction}




We consider the usual Pauli operator in $d=2$ dimensions with a magnetic
field $B$
$$
	 H = \big[\bsigma \cdot (-i\nabla + \bA) \big]^2
	=  (-i\nabla + \bA)^2 + \sigma_3B \qquad \mbox{on}
	\quad L^2(\bR^2, \bC^2),
$$
$B:=\mbox{curl}(\bA) = \nabla^\perp\!\cdot\! \bA$
with $\nabla^\perp : = (-\partial_2, \partial_1)$.
Here $\bsigma \cdot (-i\nabla + \bA)$
is the two dimensional Dirac operator on the trivial spinorbundle
over $\bR^2$ with vector potential $\bA$
and $\bsigma = (\sigma_1, \sigma_2, \sigma_3)$ are
the Pauli matrices.  Precise conditions on $\bA$ and $B$
will be specified later.


The Aharonov-Casher Theorem \cite{AC} states that
the dimension of the kernel of $H$ is given
\be
	\mbox{dim}\; \mbox{Ker} (H) 
 =\lfloor |\Phi|\rfloor,
\label{ACeq}\ee
where
$$
	\Phi:= \frac{1}{2\pi}\int_{\bR^2} B(x) \rd x
$$
(possibly $\pm \infty$)
is the flux (divided by $2\pi$)
 and $\lfloor \quad \rfloor$ denotes the lower integer
part ($\lfloor n \rfloor = n-1$ for $n\ge 1$ integer and $\lfloor 0\rfloor=0$).
Moreover, $\sigma_3 \psi= -s\psi$ for any $\psi\in \mbox{Ker} (H)$,
where $s=\mbox{sign}(\Phi)$.




On a $Spin^c$-bundle over
 $S^2$ with a smooth magnetic field the analogous theorem 
is equivalent to the index theorem (for a short direct proof 
see \cite{ES}).  From topological reasons the analogue of $\Phi$, the total curvature of a
connection,
is an integer (the Chern number of the determinant line bundle), and the number of zero modes
of the corresponding Dirac operator  is $\big| \Phi\big|$.




\medskip


In the present paper we investigate two related questions:

\medskip

(i) What is the most general class of magnetic fields for which
the Pauli operator can be properly defined on $\bR^2$?

\medskip

(ii) What is the most general class of magnetic fields
 for the Aharonov-Casher theorem to hold on $\bR^2$?


\medskip

Pauli operators are usually defined either via the magnetic
Schr\"odinger operator, $(-i\nabla +\bA)^2$, by adding
the magnetic field $\sigma_3 B$ as an external potential,
or directly by the  quadratic form of the Dirac operator
$\bsigma \cdot (-i\nabla + \bA)$  (see Section \ref{stand}).
In both ways, the standard condition  $\bA\in L^2_{loc}$ is
necessary.

On the other hand, the statement of the Aharonov-Casher theorem
uses only that $B\in L^1$, and in fact
$B$ can even be a measure.
It is therefore a natural question to extend the Pauli operator
for such magnetic fields and investigate the validity of
the Aharonov-Casher theorem.
 However, even if $B\in L^1$, it
might not be generated by an $\bA\in L^2_{loc}$. For example,
 any gauge $\bA$ generating 
the radial field $B(x)= |x|^{-2}| \log |x| \; |^{-3/2} {\bf 1}(|x|\leq
\sfrac{1}{2}) \in L^1$
satisfies $\int_{|x|\leq 1/2} |\bA(x)|^2 \rd x \ge \int_0^{1/2} 
(r|\log r|)^{-1} \rd r =\infty$ (here ${\bf 1}$ is the characteristic
function).
Hence the Pauli operator cannot be defined in the usual
way on  $C_0^\infty$ as its core. In case of a point singularity
at $p\in\bR^2$ one can study the extensions from $C_0^{\infty}(\bR^2\setminus
\{ p\})$, but such approach may not be possible
 for $B$ with a more complicated
singular set.

In this paper we present an alternative method which enables
us to define the Pauli operator for any magnetic field that is
a regular Borel measure (Theorem \ref{bm}).



\bigskip


The basic idea is to define the Pauli operator via a real generating potential
function $h$, satisfying 
\be
	\Delta h = B
\label{poisson}\ee
 instead of the
usual vector potential $\bA$. This
potential function appears in the original proof of the
Aharonov-Casher theorem.
The key identity is the following
\be
	\int\Big|\bsigma \cdot (-i\nabla + \bA) \psi\Big|^2
	= 4 \int \Big|\partial_{\bar z} (e^{-h}\psi_+)\Big|^2e^{2h}
	+ \Big|\partial_{z} (e^{h}\psi_-)\Big|^2e^{-2h}
\label{quad}\ee
for regular data, with
 $\bA := \nabla^\perp h$ (integrals without specified domains
are understood on $\bR^2$ with respect to the Lebesgue measure).
 We will {\it define} the Pauli quadratic form
by the right hand side even for less regular data.
It turns out that any magnetic field that is a regular Borel measure can be 
handled by an $h$-potential.

The main technical tool is that for an appropriate choice of $h$,
the weight function $e^{\pm 2h}$ (locally) belongs to
the Muckenhoupt $A_2$ class (\cite{GR}, \cite{St}). Therefore the
maximal operator and certain singular integral operators
are bounded on the weighted $L^2$ spaces. This will
be essential to identify the core of the operators.


We point out that this approach does {\it not} apply to the
magnetic Schr\"odinger operator $(-i\nabla +\bA)^2$.








\bigskip







The Aharonov-Casher theorem has been rigorously proven
only for a restricted class of magnetic fields on $\bR^2$.
The conditions involve some control on the decay 
at infinity  and on local singularities.
 In fact, to our knowledge,
the optimal conditions have never been investigated.
The original paper \cite{AC} does not focus on 
 conditions. The exposition \cite{CFKS} assumes
 compactly supported bounded magnetic
field $B(x)$. The Ph.D. thesis by K. Miller \cite{Mi} assumes boundedness,
and assumes that
$\int |B(x)| \log |x|  \rd x < \infty$.
The boundedness condition is clearly too strong, and
it can be easily replaced
with the assumption that $B\in \cK(\bR^2)$ Kato class.


Miller also observes that in case of integer $\Phi\neq0$
there could be either $|\Phi|$ or $|\Phi|-1$ zero states,
but if the field is compactly supported then the number
of states is always $|\Phi|-1$ \cite{CFKS}.

\medskip


The idea behind each proof is
to construct a potential function $h$ satisfying (\ref{poisson}).
{\it Locally}, $H\psi =0$ is equivalent to $\psi = (e^hg_+, e^{-h}g_-)$
with $\partial_{\bar z}g_+=0$, $\partial_{ z}g_-=0$, where
we identify $\bR^2$ with $\bC$ and use the
notations $x =(x_1, x_2)\in \bR^2$ and $z=x_1+ix_2 \in \bC$
simultaneously.
 The condition $\psi\in L^2(\bR^2,\bC^2)$ together with the explicit
growth (or decay) 
rate of $h$ at infinity determines the {\it global} solution space
by identifying the space of (anti)holomorphic functions $g_\pm$
with a controlled growth rate at infinity.


For bounded magnetic fields
decaying fast enough at infinity,
a solution to (\ref{poisson}) is given by
\be
	h(x) = {1\over 2\pi} \int_{\bR^2} \log |x-y| B(y) \rd y 
\label{trivsol}
\ee
and $h(x)$ behaves as $ \approx\Phi \log |x|$ for
large $x$. If $\Phi\ge 0$, then
$e^hg_+$ is never in $L^2$, and $e^{-h}g_-\in L^2$
if  $|g_-|$  grows at most as the
 $\Big(\lfloor\Phi\rfloor -1\Big)$-th power of $|x|$.
If $\Phi <\infty$, then $g_-$ must be a polynomial
of degree at most $\lfloor\Phi\rfloor -1$.
If $\Phi= \infty$, then the integral in (\ref{trivsol}) 
is not absolutely convergent. If the radial behavior of $B$
is regular enough, then $h$ may still be defined via (\ref{trivsol})
as a conditionally convergent integral and we then
have a solution space of infinite dimension. 



\medskip



Conditions on local regularity and decay at infinity are
used to establish bounds on the auxiliary function 
$h$ given by (\ref{trivsol}), but they are not a priori
needed for   the Aharonov-Casher
 Theorem (\ref{ACeq}). We show that local regularity conditions
are irrelevant by proving the Aharonov-Casher
theorem for any measure valued magnetic fields with finite
total variation (Theorem \ref{ACprec}). Many fields with
infinite total variation can also be covered; some
regular behavior at infinity is sufficient (Corollary \ref{reg}).
However, some control is needed in general, as we
present a counterexample
to the Aharonov-Casher theorem for a magnetic field
with infinite total variation.
\begin{counterexample}\label{cont}
There exists a continuous bounded magnetic field $B$ such that
$\int_{\bR^2} |B| =\infty$ and
\be
	\Phi := \lim_{r\to\infty} \Phi(r)
	= \lim_{r\to\infty} {1\over 2\pi} \int_{|x|\leq r} B(x) \rd x
\label{impr}\ee
exists and $\Phi >1$,
but $\mbox{dim}\; \mbox{Ker} H = 0$.
\end{counterexample}







\medskip

Finally, we recall a conjecture from \cite{Mi}:

\begin{conjecture}\label{millconj1}
Let $B(x)\ge 0$ with $\Phi := \sfrac{1}{2\pi}\int B$, which may be infinite.
Then the dimension of $\mbox{Ker}(H)$ is at least
$\lfloor \Phi\rfloor$.
\end{conjecture}

The proof in \cite{Mi} failed because it would have relied on the 
conjecture that for  any continuous function $B\ge 0$
there exists a positive
solution $h$ to (\ref{poisson}). This is false.
A counterexample (even with finite $\Phi$)
was given by C. Fefferman and B. Simon and it
was presented in \cite{Mi}. However, the same magnetic field 
does {\it not} yield a counterexample to Conjecture \ref{millconj1}.


Theorem \ref{ACprec} settles this conjecture for $\Phi<\infty$,
but the case  $\Phi =\infty$ remains open. The magnetic field
in our counterexample does not have a definite sign, in fact
 $\Phi$ is defined only as an improper integral.



\section{Definition of the Pauli operator}\label{sec:def}



\subsection{Standard definition for $\bA\in L^2_{loc}$}\label{stand}

The standard definition
of the {\it magnetic Schr\"odinger operator},
 $(-i\nabla +\bA)^2$,  or the {\it Pauli operator},
$[\bsigma\cdot(-i\nabla +\bA)]^2$, as a quadratic
form, requires $\bA \in L^2_{loc}$ 
(see e.g. \cite{LS, LL} and  for the Pauli operator
\cite{Sob}). We define
$\Pi_k := -i\partial_k +A_k$, $Q_\pm: = \Pi_1 \pm i\Pi_2$,
or with complex notation 
$Q_+ = -2i\partial_{\bar z} + a$,
$Q_- = -2i\partial_{z} + \bar a$
with $a: = A_1 + iA_2$.
These are closable operators, originally defined on $C_0^\infty(\bR^2)$.
Their closures are denoted by the same letter on the minimal
domains $\cD(\Pi_j)$ and $\cD(Q_\pm)$.

Let
$$
	s_\bA(u,u) : = \| \Pi_1 u\|^2 +  \| \Pi_2 u\|^2 
	= \int | (-i\nabla +\bA)u|^2 \; ,\qquad 
	u\in C_0^\infty(\bR^2)
$$
be the closable quadratic form associated with the
magnetic Schr\"odinger operator on the minimal form 
domain $\cD (s_\bA)$. It is known \cite{Si} that the minimal domain
coincides with the maximal domain which consists of
all $u\in L^2$ such that $s_\bA(u,u)<\infty$.

The closable quadratic form associated with the {\it Pauli operator} is
$$
	p_\bA(\psi, \psi): =  \| Q_+\psi_+\|^2+
	\| Q_-\psi_-\|^2 =
	\int | \bsigma\cdot(-i\nabla +\bA)\psi|^2 \; , 
	\qquad \psi = \pmatrix{\psi_+\cr \psi_-}
	\in C_0^\infty(\bR^2, \bC^2) \; ,
$$
again on the minimal form domain $\cD (p_\bA) = \cD(Q_+)\otimes\cD(Q_-)$.
The condition $\bA\in L^2_{loc}$ is obviously necessary.
In both cases there exists a unique self-adjoint operator, $S_\bA$
and $P_\bA$, associated with these quadratic forms.


A simple calculation shows that  $\cD (s_\bA)\otimes\cD (s_\bA)
  \subset \cD (p_\bA)$.
 This inclusion already indicates that the
Pauli operator may be defined in a more general setup.
In case of $B=\nabla^\perp\!\cdot\! \bA \in L^\infty$, these
two domains are equal and 
$P_\bA = S_\bA\otimes I_2 + \sigma_3 B$.
If $B\in L^\infty_{loc}$ only, then the form domains coincide
locally. For more details on these statements, see Section 2 of \cite{Sob}.


\subsection{Measures and integer point fluxes}



Let $\cM$ be the set of signed real 
Borel measures $\mu(\rd x)$ on $\bR^2$ with finite
total variation, $|\mu|(\bR^2)=\int_{\bR^2} |\mu|(\rd x)<\infty$.
Let $\overline\cM$ be the
set of signed real regular  Borel measures $\mu$ on $\bR^2$,
in particular they have
$\sigma$-finite total variation.
If $\mu(\rd x) = B(x)\rd x$ is absolutely continuous, then $\mu\in\cM$ is
equivalent to $B\in L^1$.
Let  $\overline{\cM^*}$
 be the set of all measures $\mu\in \overline{\cM}$ such that
$\mu(\{x \})\in (-2\pi,2\pi)$ for any point $x\in \bR^2$,
and  $\cM^*: = \cM \cap \overline{\cM^*}$.


\begin{definition}\label{mustdef}
Two measures $\mu, \mu'\in \overline{\cM}$
 are said to be {\bf equivalent} if $\mu-\mu' = 2\pi\sum_j n_j \delta_{z_j}$,
where $n_j\in \bZ$, $z_j\in\bR^2$.
The equivalence class of any measure $\mu\in  \overline{\cM}$ contains
a unique measure, denoted by $\mu^*$,
 such that $\mu^*(\{x\})\in [-\pi, \pi)$ for any $x\in \bR^2$.
In particular, $\mu^*\in \overline{\cM^*}$. 
\end{definition}


The Pauli operator associated with  $\mu\in \overline{\cM}$
will depend only on the equivalence class of $\mu$
up to a gauge transformation, so we can work with
  $\mu\in \overline{\cM^*}$.
This just reflects the  physical expectation 
that any magnetic point flux $2\pi n\delta_z$, with integer
$n$, is removable by the gauge transformation
$\psi(x) \to e^{in \varphi}\psi(x)$, where $\varphi = \mbox{arg}(x-z)$.
In case of several point fluxes, $2\pi\sum_j n_j \delta_{z_j}$,
the phase factor should be $\exp\Big( i\sum_j n_j \mbox{arg}(x-z_j)\Big)$,
but it may not converge for an infinite set of points $\{ z_j \}$.
However, any $\mu\in \overline{\cM}$
can be uniquely written as $\mu = \mu^* + 2\pi\sum_j n_j \delta_{z_j}$
 with a set of points $z_j$
which do not accumulate in $\bR^2 \equiv \bC$. Hence there exist
 analytic functions $F_\mu(x)$  (recall $x=x_1+ix_2$)
that has zeros exactly at $z_j$
with multiplicity $n_j$ for $n_j>0$ and 
$G_{\mu}(x)$ having zeros at $z_{j}$ with multiplicity $-n_{j}$ for $ n_{j}<0$
 by Weierstrass theorem. Let $L_\mu(x): =F_\mu(x) \overline{G}_{\mu}(x)$.
 Then the
integer point fluxes can be removed by the unitary 
gauge transformation
\be
	U_\mu: \psi(x) \to {L_\mu(x)\over |L_\mu(x)|} 
	\, \psi(x)
\label{umudef}\ee
and clearly $L_\mu(x)/|L_\mu(x)| = \exp\Big(
i n_j \mbox{arg}(x-z_j) + i H_j(x)\Big)$,
where $H_j$ is a real harmonic function for $x$ near $z_j$.
In particular
$U_\mu^* (-i\nabla)U_\mu \psi
= \Big(-i\nabla + n_j \bA_j + \nabla H_j\Big)\psi$
for $\psi$ whose support contains only $z_j$ among the flux points,
and here $\nabla^\perp\cdot \bA_j = 2\pi \delta_{z_j}$.








\subsection{Potential function}\label{sec:pot}


The Pauli quadratic form for magnetic fields $\mu\in \overline{\cM^*}$
 will be defined via the right hand side of
 (\ref{quad}), where $h$ is a solution to $\Delta h =\mu$.
The following theorem shows that for $\mu\in \cM^*$ one can
always choose a good potential function $h$.
Later we will extend it for $\mu\in \overline{\cM^*}$.




\begin{theorem}\label{thm:hdef}
 Let $\mu\in \cM^*$ and $\Phi: = {1\over 2\pi}\int \mu(\rd x)$
 be the total flux (divided by $2\pi$). There is  $0<\ep(\mu)\leq 1$
such that for any
$0< \ep < \ep(\mu)$  there exists a real valued function $h=h_\ep
\in \cap_{p<2}W^{1,p}_{loc}$ with $\Delta h= \mu$
(in distributional sense), such that

(i) For any compact set $K\subset \bR^2$ and any
square $Q\subset K$
\be
	\Big( {1\over |Q|}\int_Q e^{2h}\Big)
	\Big( {1\over |Q|}\int_Q e^{-2h}\Big) \leq C_1(K,\ep,\mu)
\label{A2}\ee

(ii) $e^{\pm 2h}\in L_{loc}^{1+\ep}$.

(iii)  $h$ can be split as 
$h = h_1 + h_2$ with the following estimates:
\be
	\Bigg| \; {  h_1(x)\over \log |x|} - \Phi \; \Bigg|\leq \ep
	\qquad \mbox{for} \quad |x|\ge R(\ep,\mu)
\label{mainh}\ee
and
\be
	\int_{Q(u)} e^{\pm 2 h_2} \leq C_2(\ep) \langle u\rangle^{2\ep}
\label{errorh}
\ee
with some  constants $C_1(K,\ep,\mu), C_2(\ep)$ and $R(\ep,\mu)$.
Here $Q(u)= [u-\sfrac{1}{2}, u+\sfrac{1}{2}]^2$ 
denotes the unit square about 
$u\in\bR^2$ and $\langle u\rangle = (u^2+1)^{1/2}$.
\end{theorem}



{\it Remark.} The property (i) means that $e^{2h}$ satisfies
a certain   reversed H\"older inequality locally; in fact 
the constant in (\ref{A2}) is independent of $K$ if the scale of $Q$ is
big enough.  If  (\ref{A2}) 
were true for any square, then $e^{2h}$ would be in
the weight-class $A_2$ used in harmonic analysis (see \cite{GR,St}). 
Nevertheless, this property will allow us to use
weighted $L^2$-bounds on a certain singular integral operator
locally (Lemma \ref{local}).

We also remark  that property (ii) follows from the local analog
of the well-known fact that  $\omega \in A_2 \Longrightarrow \omega 
\in A_{p}$ for some $p<2$.

\begin{corollary}\label{w11}
If $h\in L^1_{loc}$ satisfies $\Delta h \in \overline{\cM^*}$, then
$h\in  W^{1,p}_{loc}$ for all $p<2$ and $e^{\pm 2h}\in L^{1}_{loc}$.
If, in addition, $\Delta h \in \cM^*$, then $e^{\pm 2h}\in L^{1+\ep}_{loc}$
with some $\ep>0$. 
\end{corollary}

{\it Proof.} Suppose first that $\mu: = \Delta h
\in  \cM^*$.  Choose $\ep < \ep(\mu)$
and consider $h_\ep \in \bigcap_{p<2}W^{1,p}_{loc}$ 
 constructed in Theorem \ref{thm:hdef} with 
$e^{\pm 2h_\ep} \in L^{1+\ep}_{loc}$.
Since $\Delta (h-h_\ep)=0$, we have
$h=h_\ep + \varphi$ with a smooth function $\varphi$
so the statements follow for $h$ as well.


If $\mu = \Delta h$ has infinite total variation, then
Theorem \ref{thm:hdef} cannot be applied directly.
But for any compact set $K$ one can find another compact
set $K^*$ with $K\subset \mbox{int}( K^*)$ and then
the measure $\Delta h \in \overline{\cM^*}$ restricted to $K^*$
has finite total variation. Therefore  one can find 
a function $h^*\in  \bigcap_{p<2}W^{1,p}_{loc}$,
$e^{\pm h^*}\in L^2_{loc}$ with $\Delta  h^* =\mu$ on
$K^*$, i.e., $h-h^*$ is harmonic on $K^*$, hence it is smooth and
bounded on $K$. So $h\in \bigcap_{p<2} W^{1,p}(K)$
 and $e^{\pm h} \in L^{2}(K)$ follows from the same properties
of $h^*$. $\,\,\Box$

\bigskip




{\it Proof of Theorem \ref{thm:hdef}.} Step 1.
First we write 
$\mu = \mu_d + \mu_c$,
where $\mu_d := 2\pi \sum_j C_j\delta_{z_j}$
 ($C_j\in (-1,1)$, $z_j\in \bC \equiv \bR^2$)
 is the discrete part of the measure $\mu$, and $\mu_c$ is continuous,
i.e., $\mu_c(\{ x \})=0$ for any point $x\in \bR^2$.  
The summation can be infinite, finite or empty, but
$\sum_j |C_j|<\infty$. We also assume that $z_j$'s are distinct.
Let
$$
	\ep(\mu):= {1\over 10}\min_j \Big\{ 1 - |C_j| \Big\}
$$
then clearly $\ep(\mu)>0$.
 We fix an $0<\ep <\ep(\mu)$. All objects defined
below will depend on $\ep$, but we will neglect this fact in
the notations.  



We split the
measure $\mu_d = \mu_{d,1} +  \mu_{d,2}$ 
such that 
$$
	 \mu_{d,1}
	 := 2\pi \sum_{j=1}^N  C_j\delta_{z_j}  \; ,
	\qquad \mu_{d,2}
	 := 2\pi\sum_{j=N+1}^\infty  C_j\delta_{z_j}  \; ,
$$
where $N$ is chosen such that
$2\pi\sum_{j=N+1}^\infty |C_j| < \ep/2$. 
In particular $|\mu_{d, 2}|(\bR^2)< \ep/2$.


We define
\be
	h_{d,j}(x) : = {1\over 2\pi}\int 
	\log {|x-y|\over \langle y\rangle} \mu_{d,j}(\rd y)\; ,
	 \qquad j=1,2 \; ,
\label{hddef}\ee
so that   $\Delta h_{d,j} = \mu_{d,j}$.
Notice that $ h_{d,j}(x)$ is well defined  for a.e.  $x$,
moreover $ h_{d,j} \in W^{1,p}_{loc}$ for all $p<2$ by Jensen's
inequality.



\bigskip

Step 2. We split $\mu_c = \mu_{c,1}+ \mu_{c,2}$
such that $\mu_{c,1}$ be compactly supported and
$|\mu_{c, 2}|(\bR^2) < \ep/2$. 
We set $\mu_j:= \mu_{d, j}+ \mu_{c,j}$, $j=1,2$.
Then we define
\be
	h_{c,j}(x): = {1\over 2\pi} \int_{\bR^2} \log
	 {|x-y|\over \langle y\rangle} \mu_{c,j}(\rd y)\; , \qquad j=1,2 \; ,
\label{hcdef}\ee
clearly  $h_{c,j} \in W^{1,p}_{loc}$ for all $p<2$, and 
 $\Delta h_{c,j}= \mu_{c,j}$ (in distributional sense).




Finally, we define
\be
	h_1 :=  h_{d,1}+h_{c,1}\; ,\qquad
	 h_2: = h_{d,2}+h_{c,2} \; ,\qquad h: =  h_1 + h_2
\label{htotdef}\ee
and clearly $\Delta h_j = \mu_j$.
Since  $\mu_{d, 1}$ and $\mu_{c,1}$ are compactly supported,
the estimate (\ref{mainh}) is straightforward.
We will also need the notation $\nu: =\mu_{d,2} + \mu_c = \mu_{c,1}+\mu_2$.



\bigskip

Step 3. For any integer $L$
we define $\Lambda_L:= (2^{-L}\bZ)^2 +
(2^{-L-1}, 2^{-L-1})$ to be  the shifted and rescaled integer lattice.
We define the {\it dyadic squares of scale $L$} to be the squares
$$
	 D^{(L)}_{k}: =\Big[ k_1- 2^{-L-1}, k_1+ 2^{-L-1}\Big)
	\times \Big[k_2- 2^{-L-1}, k_2+ 2^{-L-1}\Big)
$$
of side-length $2^{-L}$ about the lattice
points $k=(k_1, k_2)\in \Lambda_L$. 
The squares 
$$
	\wt D^{(L)}_{k} : =\Big[ k_1- 2^{-L}, k_1+ 2^{-L}\Big)
	\times \Big[k_2- 2^{-L}, k_2+ 2^{-L}\Big)
$$
of double side-length with the same center $k$
are called {\it doubled dyadic squares of scale $L$}.
Similarly, the squares 
$$
	\wh D^{(L)}_{k} : =\Big[ k_1- 3\cdot 2^{-L-1}, k_1+ 3\cdot 
	2^{-L-1}\Big)
	\times \Big[k_2- 3\cdot 2^{-L-1}, k_2+ 3\cdot 2^{-L-1}\Big)
$$
are called {\it tripled dyadic squares of scale $L$}.
For a fixed scale $L$
the collection of dyadic squares is denoted by $\cD_L$.
$\wt \cD_L$ and $\wh\cD_L$ denote the set of doubled and tripled
dyadic squares, respectively.
The elements of $\cD_L$ partition $\bR^2$ for each $L$.
Notice also that every square $Q\subset \bR^2$ can be covered by
a doubled dyadic square of area not bigger than a universal
constant times $|Q|$.



\begin{lemma}\label{Mlemma}
There exists  $1\leq M=M(\mu,\ep)<\infty$ such that
$|\mu|(Q) < 2\pi(1-\ep)$ for any $Q\in\wh\cD_M$.
\end{lemma}



{\it Proof.}  We first
 notice that the support of $\mu_{d,1}$ consists of
finitely many points,
hence for large enough $L$ each element of $\wh\cD_L$ contains at most one
point from this support.


Second, since the measure $|\nu| = |\mu_c| + |\mu_{d,2}|$
 does not charge more than $\ep/2$
to any point, we claim that
 there exists a positive integer $1\leq M=M(\mu,\ep)<\infty$ such
that $|\nu|(D) < \ep$ for any dyadic square of scale $M$. We
can choose $M(\mu,\ep)\ge L$.

This statement is clear by a dyadic decomposition; we start with
the partition of $\bR^2$ into dyadic squares of scale $L$. 
There are just finitely many squares $D\in \cD_L$ such that
$|\nu|(D)\ge \ep$. We split
these squares further into four identical dyadic squares.
 If this process stops after
finitely many steps, then we have reached our $M$
as the scale of the finest decomposition. Now
suppose on the contrary that this process never stops.
Then we could find a strictly decreasing
sequence of nested dyadic squares
$D_1 \supset D_2 \supset \ldots$ such that $|\nu|(D_j)\ge \ep$,
but  $|\nu|$
would charge at least $\ep$ weight to their intersection  which is a 
point. 


Finally, since $|\mu|= |\mu_{d,1}| + |\nu|$ and every
tripled square  can be covered by 9 dyadic
squares of the same scale, we have
$|\mu|(Q) \leq 9\ep + 2\pi\max_j |C_j| < 2\pi(1-\ep)$
for each $Q\in \wh \cD_M$ for large enough $M$. $\,\,\,\Box$.






\bigskip


Step 4. Now we turn to the proof of (\ref{A2}) and
first we prove it for any doubled dyadic square of big scale.
%Since every square $Q$ with $|Q|\leq 1$ 
% can be covered by  a doubled
%dyadic square of comparable size, it is enough to prove
%(\ref{A2}) for doubled dyadic squares of nonnegative scale.
%Moreover, it is enough to prove it for doubled
%dyadic squares of scale at least $M$
%at the expense of increasing the constant $C_1(K,\ep,\mu)$ by 
%a factor $16^{M(\mu,\ep)}$.
Let  $Q = \wt D^{(K)}_k\in \wt\cD_K$ be a doubled dyadic square
with $K\ge M$ and let $\wh Q = \wh D^{(K)}_k$ be the corresponding tripled
square with the same center $k\in \Lambda_K$.
We split the measure $\mu$ as
$$
	\mu = \mu^{int} + \mu^{ext} : = {\bf 1}_{\wh Q} \mu +
	{\bf 1}_{\wh Q^c} \mu
$$
with $|\mu| = |\mu^{int}| + |\mu^{ext}|$ and
$h$ is decomposed accordingly as $h = h^{int}+ h^{ext}$ with
$$
	h^{\#}(x) := {1\over 2\pi} \int_{\bR^2} \log
	{|x-y|\over \langle y \rangle}\mu^\# (\rd y) \; ,
$$
where $\# = \mbox{int, ext}$. We also define
$$
	\wt h^{int}(x) := {1\over 2\pi} \int_{\bR^2} \log
	|x-y|\mu^{int} (\rd y) = h^{int}(x) +  
	{1\over 2\pi} \int_{\bR^2} \log \langle y \rangle\mu^{int} (\rd y)\; .
$$


Let $\Av_{Q}h^{ext} : = |Q|^{-1}\int_{Q} h^{ext}$
be the average of $ h^{ext}$ on $Q$. A simple calculation shows
that
\be
	\Big| h^{ext}(x) -\Av_{Q} h^{ext}\Big|
	\leq C |\mu|(\bR^2)\; , \qquad \forall x\in Q
\label{av}
\ee
with a universal constant $C$
using that $\mu^{ext}$ is supported outside of the tripled square.
Therefore
$$
	\Big( {1\over | Q|}\int_{Q} e^{2h}\Big)
	\Big( {1\over | Q|}\int_{ Q} e^{-2h}\Big)
	\leq e^{4C|\mu|(\bR^2)}\Big( {1\over |Q|}\int_{ Q}
	 e^{2\wt h^{int}}\Big)
	\Big( {1\over | Q|}\int_{Q} e^{-2\wt h^{int}}\Big)
$$

We split $\mu^{int}$ into its positive and negative
parts: $\mu^{int} = \mu^{int}_+ - \mu^{int}_-$, we
let  $\phi_\pm: = {1\over 2\pi} \int_{\wh Q} \mu^{int}_\pm\ge 0$.
By Lemma \ref{Mlemma} and $K\ge M$ we have $\phi:=\phi_++\phi_- <(1-\ep)$.



Now we apply Jensen's inequality for the probability
measures $(2\pi\phi_\pm)^{-1} \mu^{int}_\pm$ (if $\phi_\pm\neq0$):
$$
	\int_{ Q} e^{2\wt h^{int}}
	= \int_{ Q} \exp\Big( {1\over2\pi\phi_+}
	 \int_{\wh Q}\log |x-y|^{2\phi_+} \mu^{int}_+(\rd y)\Big)
	 \exp\Big( {1\over2\pi\phi_-}
	 \int_{\wh Q}\log |x-y'|^{-2\phi_-} \mu^{int}_-(\rd y')\Big)
	\rd x
$$
\be
	\leq  \int_{Q} 
	\rd x {1\over 2\pi\phi_+} \int_{\wh Q} \mu^{int}_+(\rd y)
	 {1\over 2\pi\phi_-}\int_{\wh Q} \mu^{int}_-(\rd y')
	|x-y|^{2\phi_+}|x-y'|^{-2\phi_-}
	\leq C(\ep)|Q|^{1+\phi_+-\phi_-}
\label{one}\ee
with an $\ep$-dependent constant. When performing the $\rd x$
integration, we used the fact that $\phi_- < 1-\ep$, hence
the singularity is integrable. 
Similarly, we have 
$$
	\int_{Q} e^{-2\wt h^{int}} \leq C(\ep)| Q|^{1-\phi_++\phi_-}
$$
which completes the proof of (\ref{A2}) for doubled dyadic
squares of scale at least $M$ with a $K$-independent constant.





\bigskip

Step 5. Next, we prove $e^{\pm 2h}\in L_{loc}^{1+\ep}$. We can follow
the argument in Step 4. On any square $Q\in \wt\cD_M$ we can 
use that $h^{ext}$ is bounded by (\ref{av}) and we can
focus on $\exp (\pm 2 \wt h_{int})$. 
Then we use Jensen's inequality (\ref{one}) and use the fact that
$x\mapsto |x-y|^{- 2(1+\ep)\phi_\pm}$ is locally integrable
since $\phi_\pm < (1-\ep)$.


\bigskip


Step 6. Now we complete 
the proof of (\ref{A2}) for all squares $Q\subset K$. Since every
square can be covered by a doubled dyadic square of comparable
size, we can assume that $Q$ is such a square. If the scale of $Q$
is smaller than $M$, then $|Q|^{-1}\leq 4^{M(\mu,\ep)}$
and  we can simply use 
$e^{\pm 2h}\in L_{loc}^1$
to estimate the integrals. 

\bigskip


Step 7. Finally, we prove (\ref{errorh}). Let $\wh Q(u):= [u-1, u+1]^2$
and we split the measure $\mu_2$ as
$$
	\mu_2 = \mu_2^{int} +  \mu_2^{ext}
	:= {\bf 1}_{\wh Q(u)} \mu_2 + {\bf 1}_{\wh Q(u)^c}  \mu_2 
$$
and the function $h_2= h_2^{int} + h_2^{ext}$, where
$$
	h_2^\# (x) = {1\over 2\pi}\int_{\bR^2} 
	\log  {|x-y|\over \langle y\rangle } \mu_2^\#(\rd y) \; , 
	\qquad \# = \mbox{int, ext} \; .
$$
Similarly to the estimates (\ref{av}) and (\ref{one}) in Step 4,
we obtain
$$
	\int_{Q(u)} e^{\pm2 h_2} \leq C(\ep)
	\exp{ \Big(\pm 2\Av_{Q(u)} h_2^{ext}\Big)}
	\exp{ \Big(2 |\mu_2|(\wh Q(u)) \log \langle u \rangle\Big)} \; ,
$$
and a simple calculation shows
$$
	\Big|  \Av_{Q(u)} h_2^{ext}\Big|
	\leq \int_{Q(u)}
	\int_{\wh Q(u)^c}
	\Big| \log {|x-y|\over \langle y\rangle}\Big|
	\; |\mu_2^{ext}|(\rd y) \rd x
	\leq  |\mu_2|(\wh Q^c(u))\log \langle u \rangle + C(\ep) \; .
$$
From these estimates (\ref{errorh}) follows
using that $  |\mu_2|(\bR^2)\leq \ep$.
$\,\,\,\Box$
\bigskip



\subsection{Definition of the Pauli operator for measure valued
fields}\label{sec:core}



For any real valued function $h\in L^1_{loc}(\bR^2)$ we define the
following symmetric quadratic form:
$$
        \cp^h(\psi, \xi): =  \cp^h_+(\psi_+, \xi_+)
	+ \cp^h_-(\psi_-, \xi_-)
$$
with
$$
	\cp^h_+(\psi_+, \xi_+):=
	4\int \overline{\partial_{\bar z} (e^{-h}\psi_+)} 
	\partial_{\bar z} (e^{-h}\xi_+) e^{2h}\; ,
	\qquad
	\cp^h_-(\psi_-, \xi_-):= 4\int
	\overline{\partial_{ z} (e^{h}\psi_-)} 
	\partial_{z} (e^{h}\xi_-) e^{-2h}
$$
on the natural maximal domains
$$
	\cD(\cp^h_\pm) = \Big\{ \psi_\pm
	\in L^2(\bR^2)\; : \; 
	\cp^h_\pm(\psi_\pm, \psi_\pm)<\infty \Big\}\; ,
$$
$$
	\cD(\cp^h) =\cD(\cp^h_+)\otimes\cD(\cp^h_-)=
	 \Big\{ \psi = \pmatrix{\psi_+\cr \psi_-}
	\in L^2(\bR^2, \bC^2)\; : \; 
	\cp^h(\psi, \psi)<\infty \Big\}
$$

We use $\| \cdot \|$ to denote the usual $L^2(\bR^2, \rd x)$
or $L^2(\bR^2, \bC^2,\rd x)$ norms.
We define the following norms on functions
$$
	\triple f \triple_{h,+} := \Big[  \| f \|^2 
	+ \| \partial_{\bar z} (e^{-h}f)
	e^{h}\|^2 \Big]^{1/2}, \qquad 
	\triple f \triple_{h,-}:= \Big[  \| f \|^2 
	+ \| \partial_{z} (e^{h}f)
	e^{-h}\|^2\Big]^{1/2} 
$$
and for a spinor $\psi$  we let
\be
	 \triple \psi \triple_h: =
	\triple \psi_+ \triple_{h,+} + \triple \psi_- \triple_{h,-} \; .
\label{triplenorm}\ee


For any real function $h \in L^1_{loc}$ with $\Delta h \in \overline{\cM}$,
 we  define the set
\be
	\cC_h: = \Big\{ \psi = \pmatrix{g_+ e^{h}\cr g_- e^{-h}} 
	\; : \; g_\pm \in C_0^\infty(\bR^2)\Big\} \; .
\label{ccdef}\ee
Notice that this set
depends only on $\mu =\Delta h$: if $h, h'$ are two functions
such that $\Delta h = \Delta h' = \mu$ in distributional sense,
then $h-h'$ is harmonic, i.e., smooth. Therefore $e^h$ and $e^{h'}$
differ by a smooth multiplicative factor, i.e. $\cC_h =\cC_{h'}$,
hence we can denote this set by $\cC_\mu$.
Moreover, by Theorem \ref{thm:hdef},
for any $\mu\in \overline{\cM^*}$ and any compact set $K$,
there exists an $h \in L^1_{loc}$ with $\Delta h =\mu$ on $K$,
and  $h$ is unique up to a smooth additive factor.
Since the support of $g_\pm$ is compact, the following 
set is well-defined for all $\mu \in \overline{\cM^*}$
\be
	\cC_\mu : = \Big\{ \psi = \pmatrix{g_+ e^{h}\cr g_- e^{-h}} \; : \;
	 g_\pm \in C_0^\infty(\bR^2), \; \Delta h = \mu
	\quad \mbox{on} \; \mbox{supp}(g_-)\cup\mbox{supp}(g_+) \Big\} \; .
\label{cmudef}\ee
 

\medskip


\begin{theorem}\label{Hdef} 
Let $h\in L^1_{loc}(\bR^2)$
 be a real valued function such that $\mu:=\Delta h \in \overline{\cM^*}$.
Then 

(i) The quadratic form $\cp^h$ is  nonnegative, symmetric and closed, hence
it defines a unique selfadjoint operator $H_h$
$$
	(H_h\psi, \xi) := \cp^h(\psi, \xi)\; , \qquad \psi\in \cD(H_h), \;
	\xi\in  \cD(\cp^h)
$$
with domain
$$
	\cD(H_h) : = \{ \psi\in \cD(\cp^h)\; : \; \cp^h(\psi, \cdot)\in 
	L^2(\bR^2, \bC^2)'\}
$$


(ii) The  set $\cC_\mu$ is dense in $\cD(\cp^h)$ with respect to
 $\triple \cdot \triple_h$,
i.e., it is a form core of $H_h$.


(iii) For any  $L^1_{loc}$-functions
 $h$ and  $h'$ with $\Delta h = \Delta h' \in \overline{\cM^*}$,
the operators $H_h$ and $H_{h'}$ are unitarily equivalent
by a $U(1)$-gauge transformation. In particular, the spectral
properties of $H_h$ depend only on  $\mu=\Delta h$.
\end{theorem}




\begin{definition}\label{deff}
 For any real function $h\in L^1_{loc}$ with $\mu=\Delta h \in 
\overline{\cM^*}$
the operator $H_h$ will be called the {\bf Pauli operator
with generating potential $h$}.
For any $\mu\in \overline{\cM^*}$ 
the unitarily equivalent operators $\{ H_h\; : \;
\Delta h =\mu\}$ are called the {\bf Pauli operators 
 with a magnetic field $\mu$}. 
The Pauli operators for any $\mu\in \overline{\cM}$ are defined
as $U_\mu^*HU_\mu$ on the core $U_\mu^*C_\mu$,
 where $H$ is a Pauli operator 
with field $\mu^*\in\overline{\cM^*}$ (see Definition \ref{mustdef})
 and $U_\mu$ is defined in (\ref{umudef}).
\end{definition}


To complete the definition of the Pauli operator for
any magnetic field $\mu\in \overline{\cM}$, we need

\begin{theorem}\label{bm}
For any $\mu\in \overline{\cM^*}$, there exists $h\in L^1_{loc}$
with $\Delta h = \mu$. Hence the above definition of 
 $H_h$ actually defines the Pauli operators
for any measure valued magnetic field $\mu\in \overline{\cM}$.
\end{theorem}





\medskip


{\it Proof of Theorem \ref{Hdef}.} From Corollary \ref{w11}
we know that $e^{\pm h}\in 
 L^2_{loc}$, and we show below that
 for any doubled dyadic square $Q_0$ the estimate
\be
	\Big( {1\over |Q|}\int_Q e^{2h}\Big)
	\Big( {1\over |Q|}\int_Q e^{-2h}\Big) \leq C_3(h,Q_0)
\label{A2loc}\ee
analogous to (\ref{A2}) is valid  on any square $Q\subset Q_0$, 
 with a $(h, Q_0)$-dependent constant. 
These are the two properties of $h$ which we use below.

For any $Q_0$ one can find a compact set $K$ such that
$Q_0\subset \mbox{int}(K)$ and $\mu = \Delta h$
restricted to $K$, $\mu|_K$, has finite total variation.
Let $\ep =\ep(\mu|_K)/2$  and we consider
$h_\ep$ defined in Theorem \ref{thm:hdef}. Since $\Delta h = \Delta h_\ep$
on $K$,
we can write $h= h_\ep + \varphi$ with a smooth real function
$\varphi$ depending on $h$. In particular, for any doubled
dyadic square $Q_0$ the estimate (\ref{A2}) for $h_\ep$ implies that
(\ref{A2loc}) is valid for $h= h_\ep + \varphi$ on any square $Q\subset Q_0$.



\medskip

{\it Part (i).} Let $\psi_n =(\psi_{n+}, \psi_{n-})$ be a Cauchy
sequence in the norm $\triple \cdot \triple_h$,
i.e., $\psi_n\to \psi$ in $L^2(\rd x)$,
 $ \partial_{\bar z} (e^{-h}\psi_{n+}) \to u_+$
in $L^2(e^{2h}\rd x)$ and $ \partial_{ z} (e^{h}\psi_{n-}) \to u_-$
in $L^2(e^{-2h}\rd x)$.
 We have to show that
$ \partial_{\bar z} (e^{-h} \psi_+) = u_+$,
$ \partial_{z} (e^{h} \psi_-) = u_-$.
For any $\phi \in C_0^\infty (\bR^2)$ 
$$
	\int \overline{\phi} u_+ = \lim_{n\to\infty} 
	\int \overline{\phi} \partial_{\bar z} (e^{-h}\psi_{n+})
	= - \lim_{n\to\infty} 
	\int \partial_{\bar z}\overline{\phi} \, e^{-h}\psi_{n+}
	= -\int \partial_{\bar z}\overline{\phi}  \, e^{-h}\psi_+
$$
hence $ \partial_{\bar z} (e^{-h}\psi_+) = u_+$ in distributional sense.
Here we used that
$$
	\Big|\int \overline{\phi} \Big( u_+
	 -  \partial_{\bar z} (e^{-h}\psi_{n+})\Big)\Big|
	\leq \|  \overline{\phi} e^{-h}\| \Big\|
	u_+ -  \partial_{\bar z} (e^{-h}\psi_{n+})\Big\|_{L^2(e^{2h})} \to 0
$$
and
$$
	\Big|\int \partial_{\bar z}\overline{\phi}\,  e^{-h}(\psi_+-\psi_{n+})
	\Big|\leq \Big\| \partial_{\bar z}\overline{\phi} e^{-h}\Big\|
	\| \psi_+-\psi_{n+}\|\to 0\; ,
$$
which follows from $e^{-h} \in L^2_{loc}$. The proof of the spin-down
component is similar. This shows that the form $\pi^h$ is closed.
The rest of the argument is standard  (see, e.g., Lemma 1 in \cite{LS}).
 


\bigskip

{\it Part (ii).} The spin-up and spin-down parts can be treated separately
and analogously, so  we focus only on the spin-up part. 

\medskip

Step 1. We first show that the set
$$
	\cC_0: = \{ f\in \cD(\pi^h_+), \;\mbox{supp}(f) \; \mbox{compact}\}
$$
is dense in $\cD(\pi^h_+)$ with respect to $\triple \cdot \triple_{h,+}$.
This is standard: 
let $\chi(x)$ be a compactly supported smooth cutoff function,
 $0\leq \chi \leq 1$, $\chi(x) \equiv 1$
for $|x|\leq 1$, and
let $\chi_n(x): =\chi(x/n)$. For any $f\in \cD(\pi^h_+)$ we consider
$f_n=\chi_n f$, then clearly $\triple f-f_n\triple_{h,+}\to 0$.



\bigskip


Step 2. We need the following

\begin{lemma}\label{nabla}
Let  $f\in\cC_0$ then 
 $\nabla (fe^{-h})\in L^2(e^{2h})$.
\end{lemma}


{\it Proof of Lemma \ref{nabla}.} Let $g:=fe^{-h}$.
Let $Q_1$ be a doubled dyadic square that contains a neighborhood of $K:=
\mbox{supp}\, (g)$, and let $Q_0$ be a doubled dyadic square
that strictly contains $Q_1$ and $|Q_0|=4|Q_1|$.
We define
\be
	\omega (x): = \left\{ 
	\begin{array}{cr} e^{2h(x)}& \mbox{for}\;  x\in Q_1 \cr
	1 & \mbox{for} \; x\in Q_1^c \; .
	\end{array}
	\right.
\label{omgive}\ee
\begin{lemma}\label{local}
The function $\omega (x)$ satisfies the  inequality
\be
	\Big( {1\over |Q|}\int_Q \omega\Big)
	\Big( {1\over |Q|}\int_Q \omega^{-1}\Big) \leq C_4(h, Q_0)
\label{hold}\ee
for any square $Q\subset \bR^2$, i.e.,
$\omega$ is an $A_2$-weight (see \cite{GR, St}).
\end{lemma}

{\it Proof of Lemma \ref{local}.} It is sufficient to prove (\ref{hold}) for
all doubled dyadic squares $Q$. 
It is easy to see that one of the following cases occurs:
(i) $Q$ is disjoint from $Q_1$,
(ii) $Q\subset Q_0$, (iii) $|Q_1| \leq 9 |Q|$.
In the first case
 (\ref{hold}) is trivial, in case (ii)
it follows from (\ref{A2loc}).
Finally, in case (iii) we have
$$
	\Big( {1\over |Q|}\int_Q \omega\Big)
	\Big( {1\over |Q|}\int_Q \omega^{-1}\Big)
	\leq 36^2
        \Big( 1 + {1\over |Q_0|}\int_{Q_0} e^{2h}\Big)
	\Big( 1 + {1\over |Q_0|}\int_{Q_0} e^{-2h}\Big)
$$
hence (\ref{hold}) holds with an appropriate constant.
 $\,\,\Box$.

\medskip

Since
$|\nabla g|^2 = 2(|\partial_z g|^2 + |\partial_{\bar z} g|^2)$
and $\omega = e^{2h}$ on $\mbox{supp}\, (g)$,
Lemma \ref{nabla} follows immediately from 
\be
	\int_{\bR^2} |\partial_{z}g|^2\omega
	\leq C_5(h, Q_0)
	 \int_{\bR^2} |\partial_{\bar z}g|^2\omega \, .
\label{key1}
\ee
Notice that
$$
	\wh{\partial_{ z}g} (\xi) = m(\xi)\wh{\partial_{ \bar z}g} (\xi)
	\quad \mbox{with} \quad m(\xi):=
	{(\xi_1 -i\xi_2)^2\over |\xi|^2}\; ,
$$
where hat stands for Fourier transform, $\xi\in \bR^2$,
and $m(\xi)$ is a homogeneous
multiplier of degree 0. Hence
(\ref{key1}) is just the weighted $L^2$-inequality
for the regular singular integral operator $T_m$ with Fourier multiplier 
$m(\xi)$ and with weight $\omega\in A_2$ \cite{GR, St}. $\,\,\Box$


\bigskip

Step 3. To conclude that $\cC_0\cap e^h C_0^\infty$ is dense
in $\cC_0$ with respect to $\triple \, \cdot\, \triple_{h,+}$,
 we  use the fact that $C_0^\infty$ is dense in the
weighted Sobolev space $W^{1, 2}(\omega)$
with the $A_2$-weight $\omega$ (see e.g. \cite{K}).
Here we only recall the key point of the proof.
Let $g\in W^{1, 2}(\omega)$ compactly supported and 
 $g_\ep := J_\ep \ast g \in C_0^\infty$ where
$J_\ep(x): = \ep^{-2} J(x/\ep)$ is a standard mollifier: $0\leq J \leq 1$,
$\int J =1$, $J$ smooth, compactly supported. 
Then the functions $|\nabla g_\ep| \leq   J_\ep \ast |\nabla g|$, 
 have an $L^2$-integrable majorant by the weighted
maximal inequality \cite{St} applied to $|\nabla g| \in L^2(\omega)$,
hence $g_\ep\to g$ in $W^{1, 2}(\omega)$ as $\ep\to0$.
Notice that every $g_\ep$ is supported on a common
compact neighborhood of the support of $g$.



\bigskip

{\it Part (iii).}
Since $\Delta h = \Delta h'$, we can write $h' = h +\varphi$
with a smooth real function $\varphi$.
We  define $\lambda$ as the harmonic conjugate of $\varphi$,
 $\nabla \lambda = \nabla^\perp\varphi$,
which exists and is smooth by $\Delta \varphi =0$. By
 $\partial_{\bar z} (\varphi + i\lambda)=0$ we have
$$
	\pi^h(\psi, \psi) = \pi^{h'}\Big( e^{-i\lambda}\psi,
	 e^{-i\lambda}\psi\Big) \; ,\qquad \psi \in \cC_\mu \; ,
$$
 and then by the density of $\cC_\mu$
we obtain the same relation for all $\psi\in \cD(\pi^h)$.
$\,\,\,\Box$


{\it Proof of Theorem \ref{bm}.} 
Since $|\mu|$ is finite on every bounded set, we can find
a sequence of disjoint
rings, $R_j := \{ x\; : \; r_j \leq |x| \leq r_j+2\delta_j\}$,
 $j=1, 2,\ldots$,
with appropriate widths $2\delta_j>0$ and
radii $r_j\to\infty$ (as $j\to\infty$), such
that $\sum_j |\mu|(R_j) < \infty$. For  $j=0$ we set $r_j=\delta_j=0$.
 Let $0\leq \chi_j\leq 1$ ($j=0,1,\ldots$)
 be smooth functions such that
$\chi_j(x)\equiv 1$ for $r_j +2\delta_j \leq |x|\leq r_{j+1}$ and
$\chi_j(x) \equiv 0$ for $|x|\leq r_j +\delta_j$
 or $|x|\ge r_{j+1}+\delta_{j+1}$.
Notice that the supports of $\chi_j$ are disjoint.

We define $\mu_j:=\mu \cdot {\bf 1}\{ x\; : 
r_j+2\delta_j \leq |x|\leq r_{j+1}\}$,
 By Theorem \ref{thm:hdef} 
there exist $h_j \in \bigcap_{p<2}W^{1,p}_{loc}$, 
$e^{\pm h_j}\in L^2_{loc}$
 with  $\Delta h_j = \mu_j$. 
We notice that
$$
	\Delta \Big(\sum_j \chi_j h_j) =
	\nu+\sum_j \chi_j \mu_j
$$
where $\nu$ is absolutely
continuous, $\nu = N(x)\rd x$ with 
$$
	N =  \sum_j \Big[ 2\nabla \chi_j\cdot \nabla h_j
	+ h_j \Delta \chi_j\Big] \in L^1_{loc}
$$
We can find a decomposition $N= N_1 + N_2$, $N_1\in L^\infty_{loc}$ and
$N_2\in L^1(\bR^2)$.



Let $\kappa : = \mu - \sum_j \chi_j \mu_j - N_2(x) \rd x$,
then $\kappa\in \cM$ since $N_2\in L^1(\bR^2)$ and
the measure $\mu - \sum_j \chi_j \mu_j$ belongs to $\cM$ since it vanishes
on the complement of $\bigcup_j R_j$, it
has a total variation  smaller than $|\mu|$ on  each $R_j$
and $\sum_j |\mu|(R_j)<\infty$. It is also clear that
$\kappa$ does not charge more to any point than $\mu$ does
since $0\leq\chi_j\leq 1$ and they have disjoint supports,
hence $\kappa\in \cM^*$.


By Theorem \ref{thm:hdef} there is
 $k  \in \bigcap_{p<2}W^{1,p}_{loc}$, 
$e^{\pm k}\in L^2_{loc}$ such that $\Delta k =\kappa$.
We define $h^*: = k + \sum_j \chi_j h_j$, clearly 
$h^* \in \bigcap_{p<2}W^{1,p}_{loc}$ and
$\mu = \Delta  h^* -N_1(x)\rd x$.
%Since $\mu\in \overline{\cM^*}$ and $N_1\in L^1_{loc}$,
%we have $\Delta h \in \overline{\cM^*}$, hence
% $e^{\pm h^*}\in L^2_{loc}$ by Corollary \ref{w11}.



By the Poincar\'e formula
there exists $\bA\in L^\infty_{loc}$
with $\nabla^\perp\!\cdot \! \bA = - N_1$.
For any fixed $p<\infty$,  one can find $\wt\bA\in L^p_{loc}$
with $\nabla^\perp\!\cdot \! \wt\bA  = - N_1$, $\nabla\cdot \wt\bA =0$
by Lemma 1.1. (ii) \cite{L}. But then $\wt\bA^\perp = (-\wt A_2, \wt A_1)$
is curl-free, hence $\wt\bA^\perp = \nabla \wt h$ for some $\wt h \in
W^{1,p}_{loc}$ by Lemma 1.1. (i) \cite{L}. Then $\Delta \wt h = N_1$,
hence $h:= h^* - \wt h \in L^1_{loc}$ satisfies $\Delta h =\mu$.
$\;\;\Box$

\medskip


 

Finally, we have to verify that the Pauli operator $H_h$
defined in this Section coincides with the standard Pauli
operator if $\bA\in L^2_{loc}$, modulo a gauge transformation.


\begin{proposition}\label{coinc}
Let $\bA\in L^2_{loc}$ and let $P_\bA$ be the operator 
defined in Section \ref{stand}.
We assume that $\nabla^\perp\!\cdot\!  \bA$ (in distributional sense)
is a measure and that  $\mu:= \nabla^\perp\!\cdot\! \bA\in \overline{\cM}$.
Then $\mu\in \overline{\cM^*}$, in fact $\mu$ has no discrete component.
Moreover,  if $\Delta h =\mu$ with some $h\in L^1_{loc}$, then
the operator $H_h$ defined in Theorem \ref{Hdef}
is unitarily equivalent to $P_\bA$.
\end{proposition}


{\it Remark:}  $\bA\in L^2_{loc}$ does not imply that
$\nabla^\perp\!\cdot\! \bA$ is even locally a measure of finite variation.
One example is the radial gauge $\bA (x): = \Phi(|x|)|x|^{-2} x^\perp$,
$x^\perp:=(-x_2, x_1)$,  that generates the
radial field 
$$
	B(x): = \sum_{n=1}^\infty {(-4)^{n}\over n}\cdot
	{\bf 1}( 2^{-n} \leq |x| < 2^{-n+1}).
$$
with flux $\Phi(r): = \int_{|x|\leq r} B(x) \rd x$.
 One can easily check that
$\int_{|x|\leq 1} |B(x)| \rd x = \infty$ but
$\bA\in L^2_{loc}$.

However, if $\nabla^\perp\!\cdot\!   \bA\ge 0$ as a distribution, then
it is a (positive) Borel measure $\mu \in \overline{\cM^*}$
 (see \cite{LL}).



\bigskip

{\it Proof.} First we show that $\mu = \nabla^\perp\!\cdot\!  \bA$
has no discrete component. 
Suppose, on the contrary, that $\mu(\{ x\})\neq 0$  for some $x$,
and we can assume $x=0$, $\mu(\{ 0 \})>0$. 
Let $\chi$ be a radially symmetric smooth function on $\bR^2$,
$0\leq \chi \leq 1$, $\mbox{supp}\chi \subset \{ |x|\leq 2\}$,
$\chi (x)\equiv 1$  for $|x|\leq 1$,
$|\nabla \chi|\leq 2$, and let $\chi_n(x) : = \chi(2^n x)$.
Clearly $-\int \bA\cdot\nabla^\perp \chi_n  =\int \chi_n \rd \mu \to 
\mu(\{ 0\})$ as $n\to\infty$. Using polar coordinates,
we have,  for large enough $n$,
$$
	{1\over 2}\mu(\{ 0 \}) \leq - \int \bA\cdot \nabla^\perp \chi_n
	\leq 4\int_{2^{-n}}^{2^{-n+1}} \int_0^{2\pi}|\bA(s, \theta)| 
	\rd \theta\, \rd s
$$
$$
	\leq 4\sqrt{2\pi}
	 \Bigg(\int_{2^{-n}}^{2^{-n+1}} \int_0^{2\pi}|\bA(s, \theta)|^2 
	\rd \theta\, s \, \rd s\Bigg)^{1/2} 
	\leq  4\sqrt{2\pi}
	 \Bigg(\int |\bA(x)|^2  \cdot {\bf 1}(2^{-n}\leq |x|\leq 2^{-n+1})
	\rd x\Bigg)^{1/2} 
$$
hence $\int_{|x|\leq 1} |\bA|^2=\infty$. 
The proof also works if we assume only $\mu \in \overline{\cM}$
instead of $\mu \in {\cM}$.


\medskip
Now we prove the unitarity. Without loss of generality
we can assume that $\nabla\cdot \bA =0$
by part (ii) Lemma 1.1. of \cite{L}.
 Let $\bA_h: = \nabla^\perp h$, 
then $\nabla^\perp\!\cdot\!   \bA_h = \mu$,
 $\nabla\cdot \bA_h =0$ and $\bA_h\in L^1_{loc}$  by Corollary \ref{w11}. 
Since  $\nabla^\perp\!\cdot\!  (\bA- \bA_h) = 0$,
 there exists $\lambda \in W^{1,1}_{loc}$ such that $\bA = \bA_h + \nabla
\lambda$ by part (i)  Lemma 1.1. of \cite{L}. Taking the divergence,
we see that $\Delta \lambda =0$. Let $\varphi$ be a smooth
harmonic conjugate of $\lambda$, $\nabla\lambda = \nabla^\perp \varphi$.
 A simple calculation shows that
$$
	\int |\bsigma\cdot (-i\nabla +\bA)\psi|^2
	= \pi^{h+\varphi}(\psi, \psi) 
	=\pi^h \Big( e^{i\lambda}\psi, e^{i\lambda}\psi\Big)
$$
for all $\psi\in \cD(p_\bA) = \cD(\pi^{h+\varphi})$,
i.e., the quadratic forms $p_\bA$ and $\pi^h$ are unitarily equivalent.
$\;\;\Box$



\subsection{Pauli operator generated by both potentials}\label{sec:inf}



 Theorem \ref{bm}  showed that every measure $\mu\in \overline{\cM^*}$
can be generated by an $h$-potential, $\Delta h =\mu$,
and we defined the Pauli operators.
However, it may be useful to combine the scalar potential
with the usual vector potential $\bA\in L^2_{loc}$ to generate
the given magnetic field. In this way one has more freedom in
choosing the potentials.
Typically, the singularities can be easier handled by
the $h$-potential, and the standard $h=\sfrac{1}{2\pi}
\log |\, \cdot \, |\ast\mu$
formula is (locally) available. But this formula exhibits a strong 
non-locality of $h$, and the truncation method of the proof
of Theorem \ref{bm} is not particularly convenient in practice.
Large distance behavior of the bulk magnetic field is better
described by a vector potential. 
In this section we give such a unified definition of the Pauli
operator.





\bigskip






For any  $h\in L^1_{loc}$, $\bA\in L^2_{loc}$ we define
the quadratic form
$$
	\pi^{h,\bA}(\psi, \psi)
	: = \int \Big| (-2i\partial_{\bar z} + a)(e^{-h}\psi_+)\Big|^2
	e^{2h}
	+ \int \Big| (-2i\partial_{z} + \bar{a})(e^{h}\psi_+)\Big|^2e^{-2h}
$$
on the maximal domain
$$
	\cD(\pi^{h, \bA}) : = \Big\{ \psi \in L^2(\bR^2, \bC^2)
	\; : \; \triple \psi \triple_{h,\bA} <\infty\Big\}\; ,
$$
where $a= A_1+ iA_2$ and
$$
	\triple \psi \triple_{h,\bA}: =
	\Big[ \| \psi \|^2 + \pi^{h,\bA}(\psi, \psi)\Big]^{1/2} \; .
$$

Let
$$
	\cP^*: = \Big\{ (h, \bA) \; : \; h\in L^1_{loc}, 
	\; e^{\pm h}\bA \in L^2_{loc}, \;
	\Delta h \in \overline{\cM^*},
	 \nabla^\perp\!\cdot\!  \bA \in \overline{\cM}\Big\}
$$
be the set of admissible potential pairs. The measure
$\mu : = \Delta h + \nabla^\perp\!\cdot\!  \bA \in \overline{\cM}$ is
called the magnetic field generated by $ (h, \bA)$. 
We recall from Corollary \ref{w11} 
that $(h, \bA)\in\cP^*$ implies $h\in \bigcap_{p<2} W^{1,p}_{loc}$
 and $e^{\pm h} \in L^{2}_{loc}$,
moreover, $e^{\pm h} \bA\in L^2_{loc}$ implies $\bA\in L^2_{loc}$.

Since $\nabla^\perp\!\cdot\!  \bA $ has no discrete component
(Proposition \ref{coinc}), the measure $\mu$ generated by
 $ (h, \bA)\in \cP^*$ is in $\overline{\cM^*}$.
In particular, the set of measures generated by 
a potential pair from $\cP^*$ is the same as the 
set of measures generated by only $L^1_{loc}$ $h$-potentials
(Theorem \ref{bm}).





\medskip










\begin{theorem}\label{thm:inf}
(i) (Self-adjointness). Assume that  $ (h, \bA)\in \cP^*$
and let $\mu: =\Delta h + \nabla^\perp\cdot \bA$.
Then  $\pi^{\bA, h}$ is a nonnegative symmetric
 closed form, hence it defines a
unique self-adjoint operator $H_{h,\bA}$. 

(ii)  (Core). The set 
$\cC_{\mu}$ (see (\ref{cmudef}))  is dense in $\cD(\pi^{h,\bA})$
with respect to $ \triple \, \cdot \, \triple_{h,\bA}$, i.e.,
it is a form core for  $H_{h,\bA}$.



(iii) (Consistency). If  $ (h, \bA)\in \cP^*$ and $\wt h\in L^1_{loc}$
such that $\Delta h + \nabla^\perp\!\cdot\!\bA = \Delta \wt h$, then
$H_{h,\bA}$ is unitary equivalent to $H_{\wt h}$ defined
in Theorem \ref{Hdef}. 




\end{theorem}


\begin{definition} For any $(h,\bA)\in \cP^*$ the operator
$H_{h,\bA}$ is called the {\bf Pauli operator with a potential 
pair $(h, \bA)$}.
\end{definition}

Notice that Proposition \ref{coinc} and (iii) of  Theorem \ref{thm:inf}
guarantees that the Pauli operators with the same magnetic
field are unitarily equivalent, irrespectively which definition
we use.







{\it Proof of Theorem \ref{thm:inf}.} {\it Part (i).}
The proof that $\pi^{h, \bA}$ is closed   is very similar
 to  the proof
of part (i)  of Theorem \ref{Hdef}.
 The operators $\partial_{\bar z}$
and $\partial_{z}$ should be replaced by  $\partial_{\bar z}+ia$
and $\partial_{z}-i\overline{a}$, but the extra terms with 
$a$ can always be estimated by the local $L^2$ norm of $e^{\pm h}\bA$.


{\it Part (ii).} Step 1.
We need the following preliminary observation. Since
$\bA\in L^2_{loc}$,
we can consider the decomposition $\bA= \wt\bA + \nabla\wt\lambda$,
$\nabla\cdot\wt\bA=0$, $\wt\bA\in L^2_{loc}$, $\wt\lambda\in W^{1,2}_{loc}$
(see Lemma 1.1 \cite{L}).
 $\wt\bA$ and $\wt\lambda$ are called the divergence-free
and the gradient component of $\bA\in L^2_{loc}$, and notice
that  $\wt \bA$ is  unique up to a smooth gradient,  since
if $\wt \bA + \nabla\wt\lambda =\wt \bA' + \nabla\wt\lambda'$
then $0= \nabla\cdot(\bA-\bA')= \Delta(\wt\lambda' - \wt\lambda)$,
i.e., $\wt\lambda' - \wt\lambda$ is smooth.  

Moreover, if $\bA e^{\pm h}\in L^2_{loc}$, then $\wt\bA 
e^{\pm h}\in L^2_{loc}$ as well. To see this,
we fix a compact set $K$ and  a compact set $K^*$
whose interior contains $K$, then we
 choose a cutoff function $0\leq\varphi_K\leq 1$
with $\varphi_K\equiv 1$ on $K$ and $\mbox{supp}\, \varphi_K \subset K^*$.
We let $\bA_K: = \varphi_K\bA \in L^2$
and let $\lambda$ be defined via its Fourier transform
$$
	\wh \lambda (\xi): = {\xi \cdot \wh\bA_K(\xi)\over |\xi|^2}\; ,
	\qquad\xi\in \bR^2
$$
i.e., $-\Delta \lambda = \nabla\cdot \bA$.
Then $\nabla \lambda$ is obtained from $\bA_K$ by
the action of a  singular integral operator
whose multiplier is $\xi\otimes \xi/|\xi|^2$.
Choose $\omega$ as in (\ref{omgive}),
 where $Q_0$ is a dyadic square containing $K^*$,
then $\omega\in A_2$.
Hence, by the weighted $L^2$-inequality we have
$$
	\int |\nabla \lambda|^2 \omega \leq C(\omega) \int |\bA_K|^2 \omega
	= C(\omega)\int |\bA_K|^2  e^{2h}
$$
with some
$\omega$-dependent constant. In particular 
$\nabla\lambda \in L^2_{loc}(e^{ 2h})$. The proof
of $\nabla\lambda \in L^2_{loc}(e^{ -2h})$ is identical.
Now $\wt\lambda$ satisfies $\Delta \wt \lambda = -\Delta \lambda$
on $K$, i.e. $\wt\lambda$ and $\lambda$ differ by an additive
smooth function, hence $\nabla\wt\lambda \in L^2_{loc}(e^{\pm 2h})$,
which means that $\wt\bA 
e^{\pm h}\in L^2_{loc}$.

\medskip

Step 2. We show  that $\cC_\mu \subset\cD(\pi^{h, \bA})$
if $(h, \bA)\in \cP^*$, $\mu=\Delta h +\nabla^\perp\cdot \bA$.
Let $\psi \in \cC_\mu$ be compactly supported on $K$ and let $K^*$ be
a compact set whose interior contains $K$.
 Since
$\mu$ restricted to $K^*$ has finite total variation, we can apply
 Theorem \ref{thm:hdef} for the restricted measure to construct a
function $h^*\in L^1_{loc}$ such that $\Delta h^* = \mu = \Delta h
+ \nabla^\perp\cdot\wt\bA$ on $K^*$. Then
there is a real function $\chi$ such that $\wt\bA = \nabla^\perp (h^*-h) +
\nabla\chi$ (Lemma 1.1. of \cite{L}).
After taking the divergence, we see that
$\chi$ is harmonic on $K^*$.
 Let $\varphi\in C^\infty$ be its harmonic conjugate,
$\nabla\chi=\nabla^\perp\varphi$. We have the identity
\be
	\pi^{h, \bA}(e^{-i\wt\lambda}\psi, e^{-i\wt\lambda} \psi) = 
	\pi^{h, \wt\bA}(\psi, \psi)
	=\pi^{h^*+\varphi}\Big( \psi,  \psi\Big)
\label{gaid1}\ee
for any $\psi$ supported on $K^*$. Since $\psi  \in \cC_\mu$,
we can write $\psi_\pm = g_\pm e^{\pm h^*}$ and we see that
the right hand side of (\ref{gaid1}) 
is finite, hence $e^{-i\wt\lambda}\psi \in
\cD(\pi^{h, \bA})$. But by Schwarz inequality
$$
	\pi^{h, \bA}(e^{-i\wt\lambda}\psi, e^{-i\wt\lambda} \psi)
	 \ge {1\over 2} \pi^{h, \bA}(\psi, \psi)
	- 4 \sum_\pm
	\|g_\pm\|_\infty^2\int_{K^*} |\nabla\wt\lambda|^2 e^{\pm 2h}
$$
hence $\psi \in \cD(\pi^{h, \bA})$ by Step 1.



\medskip

Step 3. We now show that $\cC_\mu$ is dense in $\cD(\pi^{h, \bA})$
with respect to $\triple \, \cdot \,\triple_{h, \bA}$
if $(h, \bA)\in \cP^*$, $\mu=\Delta h +\nabla^\perp\cdot \bA$.
We first notice that it is sufficient to show that 
$\cC_{\mu}$ is dense in the set 
$$
	\overline{\cC}_0:=\{ \psi \in \cD(\pi^{h, \bA}), \;
	: \;\mbox{supp}(\psi) \;\mbox{compact}\}
$$ 
similarly to Step 1
of the proof of Theorem \ref{Hdef} (ii). So let $\psi\in \cD(\pi^{h, \bA})$
be supported on a compact set $K$. As in Step 2, we let $h^*\in L^1_{loc}$ 
be a function such that $\Delta h^* = \mu$ on a compact neighborhood $K^*$
of $K$, $K\subset \mbox{int}(K^*)$. As before, we have 
 $\wt\bA = \nabla^\perp (h^*-h) + \nabla\chi$ with a harmonic $\chi$
and let  $\varphi\in C^\infty(K^*)$ be its harmonic conjugate,
$\nabla\chi=\nabla^\perp\varphi$. The identity (\ref{gaid1})
is now written as
\be
	\pi^{h, \bA}(\psi, \psi)  
	=\pi^{h^*+\varphi}\Big( e^{i\wt\lambda}\psi,  e^{i\wt\lambda}\psi\Big)
\label{gaid}\ee
for any $\psi$ supported on $K^*$. In particular, 
$\psi\in \cD(\pi^{h, \bA})$ implies $e^{i\wt\lambda}\psi\in\cD(\pi^{h^*
+\varphi})$.

We define the set
$$
	\wt\cC_\mu: = \Bigg\{ \psi =\pmatrix{g_+ e^h\cr g_-e^{-h}}
	\; : \; g_\pm \in L^{\infty}_0, \; \Delta h = \mu
	\quad \mbox{on}\; \mbox{supp} (g_-) \cup
	\mbox{supp} (g_+) \Bigg\}
$$
where $L_0^\infty$ denotes the set of bounded, compactly supported
functions. The  set $\wt\cC_\mu$ is well defined, see the remark before
the definition (\ref{cmudef}).




Since  $\cC_\mu$ is dense in  $\cD(\pi^{h^*+\varphi})$
with respect to  $\triple\, \cdot \, \triple_{h^*+\varphi}$
by  part (ii)  of Theorem \ref{Hdef}, we 
can find a sequence of spinors $\xi_n \in 
\cC_{\mu}$ such that $\triple \xi_n -  e^{i\wt\lambda}\psi
\triple_{h^*+\varphi}\to0$. We can assume that all $\xi_n$ are supported
in $K^*$ (see remark at the end of Step 3 of the proof Theorem 
\ref{thm:hdef} (ii)).  But then 
$\triple e^{-i\wt\lambda} \xi_n -  \psi
\triple_{h,\bA}\to0$ again by (\ref{gaid}), in particular the
set $\overline{\cC}_1 :=  \cD(\pi^{h, \bA})\bigcap \wt\cC_\mu$
is dense in  $\Big( \cD(\pi^{h, \bA}), 
\triple\, \cdot \, \triple_{h, \bA}\Big)$. 

Finally, we show that $\cC_\mu$ is dense in $\Big( \overline{\cC}_1,
\triple\, \cdot \, \triple_{h, \bA}\Big)$. 
 Let $\chi \in  \overline{\cC}_1$, i.e., $\chi_\pm = g_\pm e^{\pm h}$
with some compactly supported bounded functions $g_\pm$.

Notice that if $g$ is a bounded function, then $ge^{\pm h}\bA \in L^2_{loc}$
since $e^{\pm h}\bA\in L^2_{loc}$. In particular
$(\partial_{\bar z} + ia)g_+   \in L^2(e^{2h})$ 
implies  $\partial_{\bar z}g_+  \in  L^2(e^{2h})$.
But then $\nabla g_+  \in  L^2(e^{2h})$ by Lemma \ref{nabla}
and similarly for $g_-$.
We focus only on the spin-up part, 
the spin-down part is similar.



Let $g^{(\ep)} : = J_\ep\ast  g_+$,
where $J_\ep$ is a standard mollifier (see Step 3. of the proof 
of Theorem \ref{Hdef} (ii)). Recall that $\|g^{(\ep)}  \|_\infty
\leq \|g_+\|_\infty$ and the functions
$|\nabla g^{(\ep)}|\leq J_\ep\ast |\nabla g_+|$
have an $L^2(e^{2h})$-integrable majorant using the weighted
maximal inequality.
By passing to a subsequence
$g^{(\ep)}\to g_+$, $\nabla g^{(\ep)} \to\nabla g_+$
in $L^2(e^{2h})$ and $g^{(\ep)}\to g_+$  a.e. as $\ep\to0$.
Therefore
$$
	 \int \Big|(\partial_{\bar z}+ia) (g^{(\ep)}  - g_+)\Big|^2
	e^{2h} 
	 + \Big\| (g^{(\ep)}  - g_+)e^{h}\Big\|^2 
$$
$$
	\leq  2\int \Big|\nabla (g^{(\ep)}  - g_+)\Big|^2
	e^{2h} + 2\int |\bA|^2 |g^{(\ep)}  - g_+|^2
	e^{2h} 
	 + \Big\| (g^{(\ep)}  - g_+)e^{h}\Big\|^2  \to 0
$$
as $\ep\to0$ since
 $e^{ h}\in L^2_{loc}$ and $e^{h}\bA\in L^2_{loc}$.












\medskip

{\it Part (iii).} Since $\Delta \wt h \in \overline{\cM^*}$,
we know that $\wt h\in L^2_{loc}$ (Corollary \ref{w11}).
 Since $\nabla^\perp\cdot (\nabla^\perp h -
\nabla^\perp\wt h + \bA)=0$, there exists $\lambda\in W^{1,2}_{loc}$
with $\nabla^\perp(h-\wt h) + \bA = \nabla\lambda$.
A simple calculation shows that
$$
	\pi^{h, \bA}(\psi, \psi) = \pi^{\wt h}( e^{i\lambda}\psi,
	 e^{i\lambda}\psi) \; . \qquad \qquad \Box
$$







\section{Aharonov-Casher theorem}


We prove the following extension of the  Aharonov-Casher
theorem:

\begin{theorem}\label{ACprec} Let $\mu\in \overline{\cM}$ and 
we assume that  $\mu^*\in \cM^*$.
Let $\Phi: = \sfrac{1}{2\pi} \int \mu^*$.
The dimension of the kernel of any Pauli operator $H$
with magnetic field $\mu$ is given
$$
	\mbox{dim} \mbox{Ker} \, H = \left\{
	\begin{array}{ccl}
	[ |\Phi| ]  & \mbox{if} & \Phi\not\in\bZ 
	\;\; \mbox{or} \;\; \Phi=0\cr
	[ |\Phi| ] \; \mbox{or} \;  [ |\Phi| ]-1 & \mbox{if} & 
	\Phi\in\bZ \setminus\{ 0\}
	\end{array}
	\right.
$$
(here $[a]$ denotes the integer part of $a$).
In case of nonzero integer $\Phi$ both cases can
occur, but if, additionally, $\mu^*$ has a compact support or
has a definite sign, then always
 $\mbox{dim} \mbox{Ker} \, H = [ |\Phi| ]-1$.
In all cases the kernel is in the eigenspace of $\sigma_3$:
$\mbox{Ker}(H)\subset\{ \psi\; : \; \sigma_3 \psi = -s\psi\}$ 
with $s=\mbox{sign}(\Phi)$.
\end{theorem}


\bigskip

{\it Proof.} 
We can assume that $\mu\in \cM^*$ and we
choose $h:= h_\ep$ with some $\ep < \ep(\mu)$.
By (iii) of Theorem (\ref{Hdef}) it is sufficient to
consider the operator $H_h$. We can also assume that
$\Phi\ge 0$.

Suppose first that $\Phi$ is not integer, let $\{ \Phi \} = \Phi - [\Phi]$
be its fractional part.
Choose $\ep <\min\Big\{ \ep (\mu), \{ \Phi \}/3, (1-\{ \Phi \})/3 \Big\}$. 
Any  normalized eigenspinor $\psi$ with $\pi^h (\psi, \psi)=0$
 must be in the form $\psi = ( e^{h} g_+, e^{-h} g_-)$ where
$g_+$ is holomorphic and $g_-$ is antiholomorphic.

First we show that $g_+ =0$. Let $u\in\bR^2$ with $|u|\ge R(\ep, \mu)+1$,
$$
	1 = \|\psi\|^2 \ge 
	\int_{Q(u)} e^{2h}|g_+|^2 \ge
	\Big( \int_{Q(u)} e^{h_1}|g_+| \Big)^2
	\Big( \int_{Q(u)} e^{-2h_2} \Big)^{-1}
$$
then by Theorem \ref{thm:hdef} and subharmonicity of $|g_+|$
 we see that $|g_+(u)|\leq C\int_{Q(u)}|g_+| \leq
C\langle u \rangle^{-\Phi+2\ep} \to 0$ as $u\to\infty$, hence $g_+=0$.


A similar calculation shows that 
$|g_-(u)| \leq C\langle u \rangle^{\Phi+2\ep}$, i.e., $g_-$
must be a polynomial of degree  at most $[\Phi]$ since $\Phi + 2\ep < 
[\Phi]+1$. However, a polynomial of degree $[\Phi]$ would give
$$
	\int e^{-2h} |g_-|^2 \ge C\sum_{k\in \Lambda_0, |k|\ge R}
	|k|^{2[\Phi]}	
	\int_{Q(k)} e^{-2h} \ge C\sum_{k\in \Lambda_0, |k|\ge R}
	|k|^{2[\Phi]} \Big(\int_{Q(k)} e^{2h}\Big)^{-1}
$$
$$ 
	\ge  C\sum_{k\in \Lambda_0, |k|\ge R}
	|k|^{2([\Phi]-\Phi -2\ep)} =\infty
$$
 for some large enough $R$ and various
constants $C$.

On the other hand, the functions $g_-(z) = 1, z, \ldots , z^{[\Phi]-1}$ all
give normalizable spinors since for these choices
\be
	\int e^{-2h}|g_-|^2 \leq C\sum_{k\in \Lambda_0}
	|k|^{2([\Phi]-\Phi-1)}	\int_{Q(k)} e^{-2h_2} 
	\leq  C\sum_{k\in \Lambda_0}
	|k|^{2([\Phi]-\Phi-1+\ep)} <\infty \; .
\label{thres}\ee


If $\Phi$ is integer, then the same arguments work except
(\ref{thres}); in fact $g_-(z)=z^{\Phi-1}$ may or may not give
normalizable solutions.

If $\mu$ is compactly supported and $\Phi\ge 0$ 
is integer then from the definition
of $h$ (\ref{hddef})-(\ref{htotdef}) we see that
$\Big|h(x) - \Phi \log|x|\Big|$ is bounded for all large enough $|x|$.
Similarly one can easily verify that if $\mu\ge 0$, then
$h(x) \leq \Phi \log \langle x \rangle + C$ since
$\log |x-y|\leq \log  \langle x \rangle  +\log  \langle y \rangle +C$.
In both cases $z^{\Phi-1}e^{-h}$ is not $L^2$-normalizable at infinity.

However, if $\mu$ can change sign and not compactly supported
then there could be $\Phi\in \bZ$ zero energy states.
For example the radial field (with $\beta >0$, $N\in {\bf N}$) 
$$
	B(x)= \left\{ 
	\begin{array}{ccc}
	2(N+\beta) e^{-2} & \mbox{for} & |x|\leq e \cr
	-\beta (|x|\log |x|)^{-2} & \mbox{for} & e <|x|
	\end{array}
	\right. \; ,
$$
with $\Phi = \sfrac{1}{2\pi} \int B =N$,
is generated by a radial potential $h(x)$
such that  $h(x) = N\log |x| + \beta \log\log |x|$
for large $x$ and is regular for small $x$. 
The threshold state
$z^{\Phi-1}e^{-h}$ is normalizable only for $\beta >\sfrac{1}{2}$,
so in this case the dimension of the kernel  is $\Phi$, otherwise $\Phi -1$.
$\,\,\,\,\Box$



\bigskip

This theorem requires $\mu\in\cM$. We will see in Section
\ref{count} that the Aharonov-Casher theorem need not be
true for magnetic fields with infinite total variation.
However, the proof above still works for magnetic fields
that can be decomposed into the sum of a component in $\cM$ and
a component with  a regularly behaving generating potential.
We just remark one possible extension:

\begin{corollary}\label{reg} Suppose that $\mu\in \overline{\cM}$ 
can be written as $\mu = \mu_{rad} + \wt\mu$ such that
 $\wt \mu\in \cM$ and $\mu_{rad}$ is a rotationally symmetric Borel
measure (i.e., $\mu_{rad} = \mu_{rad}\circ R$ for any rotation $R$ in $\bR^2$
around the origin). We can assume that $\mu_{rad}(\{ 0 \}) =0$
by including the possible delta function at the origin into $\wt\mu$.
 We assume that
that $\Phi_{rad}: =\lim_{R\to\infty} \Phi(R):=
\lim_{R\to\infty}\sfrac{1}{2\pi} \int_{|x|\leq R} \mu_{rad}(\rd x)$ exists 
(possibly infinite) and $h_{rad}(x): = \Phi(|x|) \log |x|$
satisfies $\nabla h_{rad}\in L^\infty_{loc}$.
 Then all statements of
Theorem \ref{ACprec} for the Pauli operator with magnetic field $\mu$
are valid with $\Phi : = \Phi_{rad}+
\sfrac{1}{2\pi}\int \wt\mu^*(\rd x)$. 
\end{corollary}

{\it Proof.}
Let $\bA_{rad}:= \nabla^\perp h_{rad} \in L^\infty_{loc}$
and let $\wt h$ be the generating function of $\wt\mu$
given in Theorem \ref{thm:hdef} for some $\ep<\ep(\wt\mu)$.
Then $(\wt h, \bA_{rad})\in \cP^*$ with a magnetic field $\mu$,
hence the Pauli operators are well defined and unitarily equivalent.
Clearly $\pi^{\wt h,\bA_{rad}}(\psi,\psi) = \pi^h(\psi,\psi)$
with $h=h_{rad}+\wt h$, hence any  zero energy state $\psi$
must be in the form $\psi = (e^h g_+, e^{-h}g_-)$
where $g_\pm$ are (anti)holomorphic. 
Now we can follow the proof of Theorem \ref{ACprec}.
We use the estimates (\ref{A2}), (\ref{errorh}) for
the $\wt h$ part of the generating potential and
we estimate $h_{rad}(x): = \Phi(|x|) \log |x|$ 
by $\lim_{R\to\infty}\Phi(R)$ for large $R$, $\lim_{R\to0}|\Phi(R)|=0$
for small $R$ and by $\Phi \in L^\infty_{loc}(\bR)$ for intermediate $R$.
$\;\;\Box$








\section{Counterexample}\label{count}
 
In this section we present the construction of the
Counterexample \ref{cont}. For simplicity, the magnetic
field will  be only bounded and not continuous,
but it will be easy to see that a small mollification does
not modify the estimates.


Let $\delta < 1/10$ be a fixed small number
 and $N_k = 10k$ for $k=1, 2, \ldots$.
We denote the $N_k$-th roots of unity by $\zeta_{k,j}: = \exp(2\pi i j/N_k)$,
$j=1, 2, \ldots N_k$.
Let $D_{k,n,j}: =\{ x \; : \; |x-n\zeta_{k,j}|\leq\delta\}$
 be the disk of radius $\delta$ about $n\zeta_{k,j}$,
let  $\overline{D}_{k,n,j}: =\{ x \; : \; |x-n\zeta_{k,j}|\leq 2\delta\}$
be the twice bigger disk.



Let $0<\ep< 1/4$ be fixed. The magnetic field $B$  is given as
$B: = B_0 + \wt B - \wh B$
with
$$
	B_0(x) : = 2(1+\ep)\delta^{-2} {\bf 1}( |x|\leq \delta)
$$
$$
	\wt B := \sum_{k=1}^\infty \wt B_k, \qquad
	\wt B_k: =\sum_{n=4^k+1}^{4^k+2^k}\wt  B_{k,n},
	\qquad\wt  B_{k,n}(x): = \sum_{j=1}^{N_k} 2
	 \delta^{-2} {\bf 1}(x\in D_{k,n,j})
$$
and
$$
	 \wh B : =  \sum_{k=1}^\infty \wh  B_k, \qquad
	 \wh B_k: =\sum_{n=4^k+1}^{ 4^k+2^k} \wh  B_{k,n},
	\qquad  \wh B_{k,n}(x): = {1\over 2\pi}\int_0^{2\pi}
	\wt  B_{k,n}(|x|e^{i\theta})\rd \theta
$$

The field $\wt B$  consists
of uniform field "bumps" with flux $2\pi$ localized on the disks $D_{k,n,j}$
 around points $n\zeta_{k,j}$
that are located on  concentric circles of radius $n$.
The field $\wh B$  is 
 the radial average of $\wt B$.
The field $B_k=\wt B_k - \wh B_k$ is called the $k$-th {\it band}.
The relation (\ref{impr}) is straightforward by construction.





We define the  potential function
$h : = h_0 + \wt h - \wh h$, with
$$
	h_0(x):= {1\over 2\pi} \int_{\bR^2} \log|x-y|B_0(y)\rd y
$$
$$
	\wt h := \sum_{k=1}^\infty \wt h_k, \qquad
	\wt h_k: =\sum_{n=4^k+1}^{4^k+2^k}\wt  h_{k,n},
	\qquad\wt  h_{k,n}(x): =
	 {1\over 2\pi} \int_{\bR^2} \log|x-y|\wt B_{k,n}(y)\rd y -N_k \log n
$$
and
$$
 	\wh h := \sum_{k=1}^\infty \wh h_k, \qquad
	\wh h_k: =\sum_{n=4^k+1}^{ 4^k+2^k}\wh h_{k,n},
	\qquad\wh  h_{k,n}(x): =
	  {1\over 2\pi}  \int_0^{|x|} {\rd r\over  r} \int_{|y|\leq r}
	  \wh B_{k,n}(y)\rd y \; .
$$
Clearly $\Delta \wt h_k=\wt B_k$,  $\Delta \wh h_k=\wh B_k$.
Easy computations yield the following relations:
$$
	h_0(x) = (1+\ep) \log |x| \qquad \mbox{for} \quad |x|\ge \delta\; ,
$$
$$
	\wt  h_{k,n}(x) =  \sum_{j=1}^{N_k}\log \Big| \zeta_{k,j}-
	{x_1+ix_2\over n}\Big|
	=  \log\Big| 1- \Big( {x_1+ix_2\over n}\Big)^{N_k}\Big|
	\qquad \mbox{for} \quad x \not\in \bigcup_{j=1}^{N_k} D_{k,n,j}\; ,
$$
$$
	\wh h_{k,n}(x) = 
	\left\{ 
	\begin{array}{ccc}
	0 & \mbox{for} & |x|\leq n-\delta \cr
	N_k \Big[ \log {|x|\over n} + O(n^{-1}) \Big] & 
	 \mbox{for} & |x|\ge n-\delta \; .\cr
	\end{array}
	\right. \; 
$$


The infinite sums in the definition of $\wt h$ and
$\wh h$ are absolutely convergent, hence $h\in L^\infty_{loc}$.
The sum of  the $\wt h_{k}(x)$'s converges since
$$
	\sum_{k=1}^\infty \sum_{n=4^k+1}^{4^k+2^k} 
	 \Big| {x\over n}\Big|^{N_k} <\infty
$$
for each fixed $x$ and $\wh h_k(x)$ is actually zero for all
but finite $k$.
 Therefore we know that $\Delta h = B$
in distributional  sense.
Moreover,  we can rearrnge the sums and write
$$
	h = h_0 + \sum_{k=1}^\infty h_k ,
	\qquad  h_k: = \sum_{n=4^k+1}^{4^k+2^k} h_{k,n} \; ,
	\qquad  h_{k,n} := \wt h_{k,n} - \wh h_{k,n} \; .
$$

A short calculation shows that
for each $k_0$
\be
	\sum_{k=1\atop k\neq k_0}^\infty |h_k(x)| =O(1)
	\qquad \mbox{for} \quad 3\cdot 4^{k_0-1}-1 \leq |x|\leq 3\cdot 4^{k_0}
	+1 \; .
\label{out}\ee
%$$
%	\sum_{k> k_0}
%	\sum_{n=4^k+1}^{ 4^k+2^k} \Big| {x\over n}\Big|^{N_k} \leq C
%$$
%if $|x|\leq 3\cdot 4^{k_0}+1$ and
%$$
%	\sum_{k <k_0}
%	\sum_{n=4^k+1}^{ 4^k+2^k} \Big| {n\over x}\Big|^{N_k} \leq C
%$$
%if $|x|\ge 3\cdot 4^{k_0-1}-1$, by the choice $N_k =10k$.
Hence the size of $h(x)$ is determined by $h_0(x)$ and
the band nearest to $x$.




\bigskip

Now we show that $\int e^{-2h}=\infty$, in fact $\int_{D} e^{-2h}=\infty$
where $D: = \bigcup_{k,n,j} \Big( \overline{D}_{k,n,j}\setminus 
D_{k,n,j}\Big)$. The proof of $\int e^{2h}=\infty$ is similar but easier.
 We fix $k\ge 1$, $4^k + 1\leq m \leq 4^k+2^k$,
$1\leq \ell \leq N_k$ and let $x\in \overline{D}_{k,m,\ell}\setminus 
D_{k,m,\ell}$. Then
$$
	h_{k,n}(x) = \left\{ 
	\begin{array}{ccc}
	\log \Big| 1 - \Big({n\over x_1+ix_2}\Big)^{N_k}\Big| + N_k O(n^{-1})
	 & \mbox{for} & n<m \cr
	\log \Big| 1 - \Big({x_1+ix_2\over n}\Big)^{N_k}\Big|
	 & \mbox{for} & n>m \; .
	\end{array}
	\right. \; 
$$
Writing $x_1+ix_2= m\zeta_{k,\ell}+ (\varrho_1+i\varrho_2)$, 
$\varrho= (\varrho_1, \varrho_2)\in \bR^2$,  
$\delta \leq |\varrho|\leq 2\delta$
and expanding $h_{k,n}(x)$ around $m\zeta_{k,\ell}$ up
to second order in $\varrho$ we easily obtain that $h_{k,n}(x)\leq
N_k O(n^{-1})$ for each $n\neq m$ if $\delta$ is small enough, hence
 $h_k(x)\leq h_{k,m}(x) + O(1)$.


Moreover, $h_{k,m}(x) = \log \Big| 1- [(x_1+ix_2)/m]^{N_k}\Big| + O(1)$
since $|\wh h_{k,m}(x)|=O(1)$ for any $m-2\delta \leq |x|\leq m+2\delta$.
Hence
$$
	\int e^{-2h} \ge C \sum_{k=1}^\infty \sum_{m=4^k+1}^{ 4^k+2^k}
	{1\over m^{2(1+\ep)}}
	\sum_{\ell=1}^{N_k} \int_{ \overline{D}_{k,m,\ell}\setminus 
	D_{k,m,\ell}} \Big| 1 - \Big( {x\over m}\Big)^{N_k}\Big|^{-2}
	 \rd x
$$
$$
	= C \sum_{k=1}^\infty \sum_{m=4^k+1}^{ 4^k+2^k}
	{N_k \over m^{2(1+\ep)}}  \int_{\delta \leq |\varrho|\leq 2\delta}
	\Big| 1 - \Big( 1 + {\varrho_1+i\varrho_2\over m}\Big)^{N_k}\Big|^{-2}
	\rd \varrho
	\ge C 	 \sum_{k=1}^\infty {1\over N_k}\, 2^{k(1-4\ep)} =\infty\; .
$$



\medskip

Finally, we have to show that $e^{-h}{\bar f} \not\in L^2(\bR^2)$ for any
entire function $f$. 

First we show  that $f$ cannot have zeros. Suppose that
 $a$ is (one of) its zero closest to the origin, i.e.,
$f(z) = (z-a)^m g(z)$, $g$ is entire,
$g(0)\neq 0$, $m\ge 1$. Let $A_k:=\{ x \; : \;
3\cdot 4^k -1\leq |x|\leq 3\cdot 4^k +1\}$, then $h(x)=h_0(x) + O(1)$
for all $x\in A_k$ by (\ref{out}). Hence for a large enough $K$
$$
	\int e^{-2h}|f|^2 \ge C\sum_{k=K}^\infty
	\int_{A_k} e^{-2h_0(x)} |x-a|^{2m} |g(x)|^2 \rd x
$$
$$
	\ge C\sum_{k=K}^\infty 4^{2k(m-1-\ep)} \int_{A_k}|g(x)|^2 \rd x
	\ge  C|g(0)|^2\sum_{k=K}^\infty 4^{2k(m-1-\ep)}\cdot 4^k =\infty
$$
using that $|g|^2$ is subharmonic and the area of $A_k$ is 
of order $4^k$.


Now, since $f$ has no zeros, we can write $f=e^\varphi$ and we would
like to show that $\varphi$ is constant. It is enough to show that
$R:= \mbox{Re}\, \varphi$ is constant and we can assume $R(a)=0$. Suppose that
$\nabla R(a)\neq 0$ for some $a\in \bC$. Let $z_k$ be the point
where the maximum of  $R$ over the closed disk
$D_k:= \{ |x|\leq 3\cdot 4^k\}$ is attained. Since $R$
 is harmonic, $|z_k| = 3\cdot 4^k$.
Using (\ref{out}) and the  subharmonicity of $|e^{2\varphi}|$,  we have
\be
	\int e^{-2h}|f|^2 \ge C\sum_{k=1}^\infty 4^{-2k(1+\ep)}
	\int_{|x-z_k|\leq 1} |e^{2\varphi(x)}|\rd x
	\ge   C\sum_{k=1}^\infty 4^{-2k(1+\ep)} e^{2 R(z_k)}\; .
\label{pois}\ee
From the Poisson formula we easily obtain 
$|\nabla R(a)|\leq 4^{-k} \max_{D_k} R = 4^{-k} R(z_k)$
for large enough $k$. Hence $R(z_k) \ge 4^k|\nabla R(a)|$
and the integral in (\ref{pois}) is infinite. $\,\,\,\Box$.








\medskip


{\it Acknowledgements.} This work started during the first
author's visit at the Erwin Schr\"odinger Institute, Vienna.
Valuable discussions with T. Hoffmann-Ostenhof and M. Loss 
are gratefully acknowledged. 




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\bigskip

L\'aszl\'o ERD\H OS 

\vskip-10pt
 
School of Mathematics 
\vskip-10pt
Georgia Institute of Technology 
\vskip-10pt
Atlanta, GA 30332, USA 
\vskip-10pt
E-mail: {\tt lerdos@math.gatech.edu}

\bigskip

Vitali VOUGALTER  
\vskip-10pt
Department of Mathematics
\vskip-10pt
University of British Columbia
\vskip-10pt
Vancouver, B.C. Canada V6T 1Z2
\vskip-10pt
E-mail: {\tt vitali@math.ubc.ca}




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