text:
\begin{document}
\maketitle

\begin{abstract}
	In this work we prove that the eigenvalues of the $n$-dimensional massive Dirac operator $\D_0 + V$, $n\ge2$, perturbed by a possibly non-Hermitian potential $V$, are localized in the union of two disjoint disks of the complex plane, provided that $V$ is sufficiently small with respect to the mixed norms $L^1_{x_j} L^\infty_{\widehat{x}_j}$, for $j\in\{1,\dots,n\}$. In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on $V$, and in particular the spectrum is the same of the unperturbed operator, namely $\s(\D_0+V)=\s(\D_0)=\R$.

	The main tools we employ are an abstract version of the Birman-Schwinger principle, which include also the study of embedded eigenvalues, and suitable resolvent estimates for the Schr\"odinger operator.
\end{abstract}

\section{Introduction}
In recent years, non-selfadjoint operators are attracting increasing attention, not only in view of applications in quantum mechanics and other branches of physics, but also for the interesting mathematical challenges they present.
While the theory of selfadjoint operators is consolidated in a vast literature, references on the study of non-selfadjoint operators are more sparse, so much that, quoting E.B. Davies \cite{Dav07}, \textit{\lq\lq it can hardly be called a theory''}. We refer to the books \cite{BGSZ15, Tre08} for milestones on the history of the argument, and to \cite{Dav02} for some physical applications.

In this paper, we deal with the free Dirac operator $\D_0$ perturbed by a potential, formally defined by
\begin{equation*}\label{DV}
	\D_{V} = \D_{0} + V.
\end{equation*}
The relevance of these kind of operators in quantum physics is common knowledge, since in the $2$-dimensional case the operator $\D_{V}$ is related to the quantum theory of graphene, while in the $3$-dimensional case the Hamiltonian $\D_{V}$ determines the dynamic of a relativistic quantum particle of spin $\frac{1}{2}$, subject to an external electric field described by the potential $V$.

We consider the operator $\D_{V}$ acting on the Hilbert space of spinors $\H = L^2(\R^n; \C^{N} )$, where $n\ge2$, $N := 2^{\lceil n/2 \rceil}$ and ${\lceil \cdot \rceil}$ is the ceiling function.
The free Dirac operator $\D_{0}$ with non-negative mass $m$ is defined as
\begin{equation*}
	\D_{0} = -i c \hbar \, \vect{\a} \cdot \nabla + m c^2 \a_0
		 = -i c \hbar \sum_{k=1}^{n} \a_k \frac{\partial}{\partial x_k} + m c^2 \a_0,

\end{equation*}
where $c$ is the speed of light, $\hbar$ is the reduced Planck constant and the matrices $\a_k \in \C^{N \times N}$, for $k \in \{0,\dots,n\}$, are elements of the Clifford algebra (see \cite{Obo99}) satisfying the anti-commutation relations
\begin{equation}\label{clifford}
	\a_j \a_k + \a_k \a_j = 2\delta_{j,k} I_{\C^N}, \quad\text{for $j,k \in \{0,\dots,n\}$,}
\end{equation}
where $\delta_{j,k}$ is the Kronecker symbol. Without loss of generality we can take
\begin{equation*}
\a_0 =
\begin{pmatrix}
I_{\C^{N/2 \times N/2}} & \vect{0}\\
\vect{0} & -I_{\C^{N/2 \times N/2}}
\end{pmatrix}
\end{equation*}
and renormalize the unit measures such that $c=\hbar=1$.
The free Dirac operator has domain
\begin{equation*}
	\dom(\D_0) = \{ \psi \in \H \colon \nabla\psi \in \H^n \}
\end{equation*}
and it is selfadjoint with core $C_0^\infty(\R^n; \C^N)$.

\begin{comment}
In particular, for the $3$-dimensional case, they are defined as

\begin{equation*}
\a_0 =
\begin{pmatrix}
I_2 & \v{0}\\
\v{0} & -I_2
\end{pmatrix},
\quad
\a_i=
\begin{pmatrix}
\v{0} & \s_i \\
\s_i & \v{0}
\end{pmatrix},
\end{equation*}
where $\s_i$, for $i=1,2,3$, are the Pauli matrices
\begin{equation*}
\s_1=
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix},
\quad
\s_2 =
\begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix},
\quad
\s_3 =
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}.
\end{equation*}
\end{comment}

The potential $V \colon \R^n \to \C^{N \times N}$ is allowed to be any non-Hermitian matrix-valued function with $|V|\in L_{\text{loc}}^2(\R^n;\R)$, where $|V|$ is the operator norm of $V$.
Thus the resulting operator may be non-selfadjoint.
With an abuse of notation, we use the same symbol $V$ to indicate the multiplication operator generated by the matrix $V$ in $\H$ with initial domain $\dom(V)=C_0^\infty(\R^n;\C^N)$.

In this work, we are interested in location of eigenvalues for $\D_V$ in the complex plane. For the non-selfadjoint Schr\"odinger operator $-\Delta+V$, we refer to the works by Frank \cite{Fra11,Fra18}, Frank and Sabin \cite{FSab17}, Frank and Simon \cite{FSim15}. In particular, we have that the eigenvalues of $-\Delta+V$ satisfy the bound
\begin{equation*}
	|z|^{\gamma} \le D_{\gamma,n} \int_{\R_n} |V(x)|^{\gamma+n/2}dx,
	\qquad
	0 < \gamma
	\begin{cases}
	=\frac{1}{2}, &\text{if $n=1$,}
	\\
	\le \frac{1}{2}, &\text{if $n\ge2$,}
	\end{cases}
\end{equation*}
where the constant $D_{\gamma,n}>0$ is independent of $z$ and $V$.
This localization estimate was proven by Abramov, Aslanyan and Davies \cite{AAD01} for $n=1$ with the sharp constant $D_{1/2,1}=1/2$, and for larger $n$ by Frank \cite{Fra11}, combining the Birman-Schwinger principle with the  uniform Sobolev inequalities by Kenig, Ruiz and Sogge \cite{KRS87}, i.e.
\begin{equation*}
	\n{(-\Delta-z)^{-1}}_{L^p \to L^{p'}}
	\le C
	|z|^{-n/2+n/p-1},
	\qquad
	\frac{2}{n+1} \le \frac{1}{p}-\frac{1}{p'} \le \frac{2}{n},

\end{equation*}
where $p'= p/(p-1)$ is the dual exponent of $p$.
In \cite{LS09,Saf10}, Laptev and Safronov conjectured that the range of $\gamma$ for $n\ge2$ can be extended to $0<\gamma<n/2$, and Frank and Simon \cite{FSim15} proved the conjecture to be true for radial symmetric potentials.

Let us return to the Dirac operator $\D_{V}$. If we suppose $V\colon \R^n\to\C^{N \times N}$ is an Hermitian matrix-valued function, such that the operator $\D_{V}$ is selfadjoint, we have an extensive literature on its spectral properties, see for example the monograph by Thaller \cite{Tha13}. In the non-selfadjoint case, the study of the spectrum of $\D_{V}$ was initiated by Cuenin, Laptev and Tretter in \cite{CLT14} in the $1$-dimensional case, followed by
\cite{Cue14,CS18,Enb18}.

For the higher dimensional case, we refer to the works
\cite{Cue17,Dub14,FK19,Sam16}.

In \cite{CLT14}, the authors proved that, for $n=1$, if $V=(V_{ij})_{i,j \in \{1,2\}}$ with $V_{ij} \in L^1(\R)$ and
\begin{equation*}
	\n{V}_{L^1(\R)} = \int_{\R} |V(x)|dx \le 1,
\end{equation*}

then every non-embedded eigenvalue $z\in \rho(\D_{0})$ of $\D_{V}$ lies in the disjoint union
\begin{equation*}
	z \in \overline{B}_{R_0}(x^-_0) \cup \overline{B}_{R_0}(x^+_0)
\end{equation*}
of the two closed disks in the complex plane with centers and radius respectively
\begin{equation*}
	x^\pm_0 = \pm \sqrt{\frac{\n{V}_1^4-2\n{V}_1^2+2}{4(1-\n{V}_1^2)} + \frac{1}{2}},
	\qquad
	R_0 =
	\sqrt{\frac{\n{V}_1^4-2\n{V}_1^2+2}{4(1-\n{V}_1^2)} - \frac{1}{2}}.
\end{equation*}

In particular, in the massless case ($m=0$), the spectrum of $\D_{V}$ is $\R$. They also showed that this inclusion is sharp.
Their proof is essentially based on the combination of the Birman-Schwinger principle with the resolvent estimate for the free Dirac operator
\begin{equation*}
	\n{(\D_{0}-z)^{-1}}_{L^\infty(\R) \to L^1(\R)}
	\le
	\sqrt{\frac{1}{2}
		+\frac{1}{4} \left\lvert\frac{z+m}{z-m}\right\rvert
		+\frac{1}{4} \left\lvert\frac{z-m}{z+m}\right\rvert
	},
	\quad
	\text{$z \in \rho(\D_{0})$}.
\end{equation*}

It should be remarked that, in higher dimensions $n\ge2$, $L^p(\R^n) \to L^{p'}(\R^n)$ estimates for $(\D_0-z)^{-1}$ do not exist, as observed in the Introduction of \cite{Cue14}. Indeed, Cuenin points out that, due to the Stein-Thomas restriction theorem and standard estimates for Bessel potentials, the resolvent $(\D_{0}-z)^{-1} \colon L^p(\R^n) \to L^{p'}(\R^n)$ is bounded uniformly in $|z|>1$ if and only if
\begin{equation*}
	\frac{2}{n+1}
	\le
	\frac{1}{p} + \frac{1}{p'}
	\le
	\frac{1}{n},
\end{equation*}
and thus we are forced to choose $n=1$. The situation for the Schr\"odinger operator is better since the right-hand side of the above range is $2/n$, as stated in the Kenig-Ruiz-Sogge estimates.

In \cite{Cue17}, Cuenin localized the eigenvalues of the perturbed Dirac operator in terms of the $L^p$-norm of the potential $V$, but in an unbounded region of the complex plane. Indeed, Theorem\til6.1.b of \cite{Cue17} states that, if $n\ge2$ and $|V|\in L^p$, with $p\ge n$, then any eigenvalue $z \in \rho(\D_0)$ of $\D_V$ satisfies
\begin{equation*}
	\left\lvert {\Im z}/{\Re z} \right\rvert^{(n-1)/p}
	|\Im z|^{1-n/p}
	\le C
	\n{V}_{L^p(\R^n)},
\end{equation*}
where $C$ is a constant independent on $z$ and $V$.
A similar result was proved by Fanelli and Krej\v ci\v r\'ik in \cite{FK19}, where they show that, for dimension $n=3$, $|V|\in L^3(\R^3)$ and $z \in \rho(\D_0) \cap \sigma_p(\R^n)$, then
\begin{equation*}
	\left( 1 + \frac{(\Re z)^2}{(\Re\sqrt{m^2-z^2})^2} \right)^{-1/2}
	<
	\left( {\pi}/{2} \right)^{1/3}
	\sqrt{1+e^{-1}+2e^{-2}} \n{V}_{L^3(\R^3)}.
\end{equation*}
The advantage of the last result lies in the explicit condition which is easy to check in applications. However, still the eigenvalues are localized in an unbounded region around the continuous spectrum $\sigma(\D_0)=(-\infty,-m] \cup [m,+\infty)$ of the free Dirac operator $\D_0$.

Here, our main results try to generalize the one by Cuenin, Laptev and Tretter \cite{CLT14} in higher dimensions, recovering the enclosure of the eigenvalues of the massive ($m>0$) Dirac operator $\D_V$ in a compact region of the complex plane, imposing the smallness of the potential $V$ with respect to suitable mixed Lebesgue norms. In the case of the massless ($m=0$) Dirac operator, we obtain that the point spectrum of the perturbed operator $\D_{V}$ is empty, and then $\sigma(\D_{V})=\sigma(\D_0)=\R$. We also mention the recent paper \cite{CFK}, in which similar results are obtained by multiplication techniques.

Before to formalize our results in Theorems\til\ref{thm1}\til\&\til\ref{thm2}, we need to introduce the following notations used throughout the paper.

\begin{notations}\normalfont

We use the symbols $\sigma(H)$, $\sigma_p(H)$, $\sigma_e(H)$ and $\rho(H)$ respectively for the spectrum, the point spectrum, the essential spectrum and the resolvent of an operator $H$. Since for a non-selfadjoint closed operator there are various definitions for essential spectrum, we define
\begin{align*}
\sigma_e(H) =
\{z \in \C \colon \text{ $H-z$ is not a Fredholm operator} \},
\end{align*}
whereas the discrete spectrum is defined as
\begin{equation*}
\sigma_d(H) =
\{z \in \C \colon \text{$z$ is an isolated eigenvalue of $H$ of finite multiplicity}\}.
\end{equation*}
For $z\in \rho(H)$, we denote with $R_H(z) := (H-z)^{-1}$ the resolvent operator of $H$.

We recall also that
\begin{gather*}
	\sigma(-\Delta)
	=\sigma_e(-\Delta)
	=[0,+\infty),
	\\
	\sigma(\D_0)
	=\sigma_e(\D_0)
	=(-\infty,-m]\cup[m,+\infty) .
\end{gather*}

We use the symbol $(\cdot,\cdot)_{\H}$ for the inner product on the Hilbert space $\H=L^2(\R^n;\C^N)$, that is
\begin{equation*}
(\phi,\psi)_\H = \int_{\R^n} \phi^{\dagger} \cdot \psi \, dx
\end{equation*}
where $\cdot$ is the scalar product.

Fixed $j\in\{1,\dots,n\}$ and $x=(x_1,\dots,x_n)\in\R^n$, we denote
\begin{align*}
{\widehat{x}_j} &:= (x_1,\dots,x_{j-1},x_{j+1},\dots,x_n) \in \R^{n-1},
\\
(\o{x},{\widehat{x}_j}) &:=  (x_1,\dots,x_{j-1},\o{x},x_{j+1},\dots,x_n) \in\R^n.
\end{align*}
Define the mixed Lebesgue spaces $L^p_{x_j}L^q_{{\widehat{x}_j}}(\R^n)$ as the spaces of the functions with finite mixed-norm
\begin{equation*}
\n{f}_{L^p_{x_j}L^q_{{\widehat{x}_j}}} := \left( \int_{\R} \left(\int_{\R^{n-1}} |f(x_j,{\widehat{x}_j})|^q d{\widehat{x}_j} \right)^{p/q} dx_j \right)^{1/p},
\end{equation*}
where the obvious modifications occur for $p=\infty$ or $q=\infty$.

For any matrix--valued function $M \colon \R^n \to \C^{N\times N}$,

we set
\begin{equation*}
\n{M}_{L^p_{x_j} L^q_{{\widehat{x}_j}}} := \n{|M|}_{L^p_{x_j} L^q_{{\widehat{x}_j}}}
\end{equation*}
where the function $|M|\colon \R^n\to\R$ is obtained considering the operator norm $|M(x)|$ of $M(x)$ for almost every fixed $x\in\R^n$.

We also indicate with
\begin{align*}
[f *_{x_j} g](x) &:= \int_\R f(y_j,{\widehat{x}_j}) g(x_j-y_j,{\widehat{x}_j})dy_j,
\\
[\F_{x_j} f] (\xi_j, {\widehat{x}_j}) &:= \frac{1}{\sqrt{2\pi}} \int_{\R} e^{-i x_j \xi_j} f(x_j,{\widehat{x}_j}) dx_j,
\\
[\F^{-1}_{\xi_j} f] (x_j, {\widehat{x}_j}) &:= \frac{1}{\sqrt{2\pi}} \int_{\R} e^{i x_j \xi_j} f(\xi_j,{\widehat{x}_j}) d\xi_j,
\end{align*}
respectively the partial convolution respect to $x_j$, the partial Fourier transform respect to $x_j$ and its inverse. In a similar way one can define the partial (inverse) Fourier transform respect to ${\widehat{x}_j}$ and the complete (inverse) Fourier transform respect to $x$.

Finally, let us consider the function spaces
\begin{equation*}
	X \equiv X(\R^n) := \bigcap_{j=1}^{n} L^1_{x_j} L^2_{{\widehat{x}_j}}(\R^n),
	\qquad
	Y \equiv Y(\R^n) := \bigcap_{j=1}^{n} L^1_{x_j} L^\infty_{{\widehat{x}_j}}(\R^n),
\end{equation*}
with norms defined as
\begin{equation*}\label{Xnorm}
	\n{f}_X =
	\max_{j\in\{1,\dots,n\}}
	\n{ f }_{L^1_{x_j} L^2_{{\widehat{x}_j}}},
	\qquad
	\n{f}_Y =
	\max_{j\in\{1,\dots,n\}}
	\n{ f }_{L^1_{x_j} L^\infty_{{\widehat{x}_j}}}.
\end{equation*}
The dual space of $X$ is given (see e.g. \cite{BL12}) by
\begin{equation*}
	X^* \equiv X^*(\R^n) := \sum_{j=1}^{n} L^\infty_{x_j} L^2_{{\widehat{x}_j}}(\R^n),
\end{equation*}
with the norm
\begin{equation}\label{normX*}
	\n{f}_{X^*}
	:=
	\inf
	\left\{ \sum_{j=1}^{n} \n{f_j}_{L^\infty_{x_j} L^2_{{\widehat{x}_j}}}
	\colon
	f=\sum_{j=1}^n f_j
	\right\}.
\end{equation}

\end{notations}

We can announce now our results.

\begin{theorem}
	\label{thm1}
	Let $m>0$.
	There exists a constant $C_0>0$, independent on $V$, such that if
	\begin{equation*}
	\n{V}_Y	< C_0,
	\end{equation*}
	then every eigenvalues $z \in \sigma_p(\D_{V})$ of $\D_{V}$ lies in the union
	\begin{equation*}
	z \in \overline{B}_{R_0} (x^-_0) \cup \overline{B}_{R_0} (x^+_0)
	\end{equation*}
	of the two closed disks in $\C$ with centers in $x_0^-,x_0^+$ and radius $R_0$, with
	\begin{equation*}
	x^\pm_0 := \pm m\,\frac{\V^2+1}{\V^2-1},
	\quad
	R_0 := m\,\frac{2\V}{\V^2-1},
	\quad
	\V \equiv \V(V) := \left[\frac{(n+1)C_0}{\n{V}_Y} - n \right]^2 > 1.
	\end{equation*}
\end{theorem}

\begin{theorem}\label{thm2}
	Let $m=0$.
	There exists a constant $C_0>0$, independent on $V$, such that if
	\begin{equation*}
	\n{V}_Y	< C_0,
	\end{equation*}
	then $\D_V$ has no eigenvalues.

	In particular, we have
	$\sigma(\D_{V})=\sigma_{e}(\D_{V})=\R$.
\end{theorem}

\begin{remark}
The crucial tool in our proof is a uniform resolvent estimate for the resolvent of the  free Dirac operator. This approach is inspired by \cite{Fra11}, where the result by Kenig, Ruiz and Sogge \cite{KRS87} was used for the same purpose. In our case, we prove in Section\til\ref{sec2} the following estimates, of independent interest:
\begin{equation*}
  \n{R_{-\Delta}(z)}_{X \to X^*} \le C |z|^{-1/2},
  \qquad
	\n{\partial_k R_{-\Delta}(z)}_{X \to X^*}  \le C.
\end{equation*}
and
\begin{equation*}
\n{ R_{\D_{0}}(z) }_{X \to X^*}
\le C \left[n + \left|\frac{z+m}{z-m}\right|^{\sgn(\Re z)/2}\right].
\end{equation*}
These can be regarded as precised resolvent estimates of Agmon--H\"{o}rmander type. Note also that similar uniform estimates, in less sharp norms, were proved earlier by the first and second Authors in \cite{DAnconaFanelli07-a,DAnconaFanelli08-a,EGG19}.

In Section\til\ref{sec3}, we combine our uniform estimates with the Birman-Schwinger principle, enabling in Section\til\ref{sec4} to complete the proof of Theorems\til\ref{thm1}\til\&\til\ref{thm2}.
\end{remark}

\begin{remark}
	The following embedding for the space $Y$ under consideration hold:
	\begin{equation}\label{YinLn}
	Y \hookrightarrow L^{n,1}(\R^n) \hookrightarrow L^n(\R^n),
	\end{equation}
	where $L^{p,q}(\R^n)$ is the Lorentz space. Moreover, we have
	\begin{equation*}
	W^{1,1}(\R^n)
	\hookrightarrow
	\bigcap_{j=1}^{n}  L^1_{{\widehat{x}_j}}
	L^\infty_{x_j}(\R^n)
	\hookrightarrow L^{n/(n-1),1}(\R^n),
	\end{equation*}
	where $W^{m,p}(\R^n)$ is the Sobolev space, and so, for the $2$-dimensional case, we get in particular
	\begin{equation*}
	W^{1,1}(\R^2)
	\hookrightarrow
	Y =
	L^1_{x_1} L^\infty_{x_2}(\R^2)
	\cap
	L^1_{x_2} L^\infty_{x_1}(\R^2)
	\hookrightarrow
	L^{2,1}(\R^2)
	\hookrightarrow
	L^{2}(\R^2).
	\end{equation*}
	We refer to the papers by Fournier \cite{Fou87}, Blei and Fournier \cite{BF89} and Milman \cite{Mil91} for these inclusions.

\end{remark}

\section{Uniform resolvent estimates}\label{sec2}

Let us start
fixing constants $r,R,\delta>0$ such that
\begin{equation*}\label{cond_const}
1<r<R,
\qquad
\sqrt{R^2-1} < \delta <1,
\end{equation*}
and consider
the open cover $\S=\{\S^+_{ j}, \S^-_{ j}, \S_\infty \}_{j\in\{1,\dots,n\}}$ of the space $\R^n$ defined by
\begin{equation*}
\S^\pm_{ j} = \{ \xi\in\R^n \colon \pm \xi_j > \delta|\widehat{\xi}_j|, \,\, |\xi|<R \},
\qquad
\S_\infty = \{ \xi\in\R^n \colon |\xi|>r \}.
\end{equation*}

We can find a smooth partition of unity $\{ \chi^+_{ j}, \chi^-_{ j} , \chi_\infty \}_{j\in\{1,\dots,n\}}$ subordinate to $\S$, i.e. a family of smooth positive functions such that
\begin{equation*}
\supp \chi^\pm_{ j} \subset \S^\pm_{ j},
\quad
\supp \chi_{\infty} \subset \S_{\infty},
\quad
\chi_{\infty} + \sum_{j=1}^n [ \chi^+_{ j} + \chi^-_{ j} ] \equiv 1.
\end{equation*}
From these, define the smooth partition of unity $\chi=\{\chi_j\}_{j\in\{1,\dots,n\}}$, with
\begin{equation}\label{def_chij}
\chi_j := \chi_j^+ +\chi_j^- + \frac{1}{n}\chi_{ \infty},
\end{equation}
and hence, for $j\in\{1,\dots,n\}$, the Fourier multipliers
\begin{equation*}\label{fj}
\chi_j(|z|^{-1/2} D) f =
\F^{-1}_\xi[\chi_j(|z|^{-1/2}\xi) \, \F_x f].

\end{equation*}

Note in particular that
\begin{equation}\label{sumchij}
	\sum_{j=1}^n \chi_j(|z|^{-1/2} D) f = f.
\end{equation}

Therefore, the following estimates hold.

\begin{lemma}\label{lem1}
	For every $z\in \rho(-\Delta) = \C\setminus[0,+\infty)$,
	$f \in L_{x_j}^1 L_{{\widehat{x}_j}}^2$ and $j,k \in\{1,\dots,n\}$, we have that

	\begin{align*}
	\n{ \chi_j \left( |z|^{-1/2} D \right) R_{-\Delta}(z) f }_{L_{x_j}^\infty L_{{\widehat{x}_j}}^2}
	&\le C \, |z|^{-1/2}
	\n{f}_{L_{x_j}^1 L_{{\widehat{x}_j}}^2},
	\\

	\n{ \chi_j \left( |z|^{-1/2} D \right) \partial_k  R_{-\Delta}(z) f }_{L_{x_j}^\infty L_{{\widehat{x}_j}}^2}
	&\le C
	\n{f}_{L_{x_j}^1 L_{{\widehat{x}_j}}^2},
	\end{align*}
	where $\{\chi_j\}_{j\in \{1,\dots,n\}}$ are defined in \eqref{def_chij} and $C>0$ does not depend on $z$.

	In particular, it follows that
	\begin{align*}
	\n{R_{-\Delta}(z)}_{X \to X^*} & \le C |z|^{-1/2},
	\\
	\n{\partial_k R_{-\Delta}(z)}_{X \to X^*} & \le C.
	\end{align*}
\end{lemma}

\begin{lemma}\label{lem2}
	For every $z\in\rho(\D_0) = \C\setminus\{\zeta\in\R \colon |\zeta| \ge m\}$, $f \in L_{x_j}^1 L_{{\widehat{x}_j}}^2$ and $j\in \{1,\dots,n\}$ we have that

	\begin{equation*}\label{RDest}
	\n{ \chi_j \left( |z^2-m^2|^{-1/2} D \right) R_{\D_{0}}(z) f }_{L_{x_j}^\infty L_{{\widehat{x}_j}}^2}
	\le C \left[n + \left|\frac{z+m}{z-m}\right|^{\sgn(\Re z)/2}\right]
	\n{f}_{L_{x_j}^1 L_{{\widehat{x}_j}}^2},
	\end{equation*}
	where $\{\chi_j\}_{j\in \{1,\dots,n\}}$ are defined in \eqref{def_chij}
	and $C>0$ is the same as in Lemma\til\ref{lem1}. In particular, it follows that
	\begin{equation*}
	\n{ R_{\D_{0}}(z) }_{X \to X^*}
	\le C \left[n + \left|\frac{z+m}{z-m}\right|^{\sgn(\Re z)/2}\right].
	\end{equation*}
\end{lemma}

\begin{remark}
	Before to proceed further, we give an heuristic explanation for the choice of the localization in the frequency domain via the Fourier multiplier $\chi_j(|z|^{-1/2}D)$. Since the symbol $(|\xi|^2-z)^{-1}$ of the resolvent $R_{-\Delta}(z)$ blows-up as $z \to \zeta\ge0$, our trick is to use the norms $L^\infty_{x_j} L^2_{{\widehat{x}_j}}$ for $j\in\{1,\dots,n\}$, which allows us to restrict the problem from the spherical surface $\{\xi\in\R^n \colon |\xi|=|z|^{-1/2}\}$ to the \lq\lq equator'' given by $\{ \xi\in\R^n \colon \xi_j=0, |\widehat{\xi}_j|=|z|^{-1/2} \}$. We then avoid these regions thanks to the smooth functions $\chi_j$.
\end{remark}

\begin{proof}[Proof of Lemma\til\ref{lem1}.]
The \lq\lq in particular'' part trivially follows from \eqref{sumchij}
and from the definitions of the norms on $X$ and $X^*$.

For the simplicity, from now on $C>0$ will stand for a generic positive constant independent on $z$ and which can change from line to line. Clearly, by scaling, we can consider only unitary $z\in\C\setminus \{1\}$.
Thus it is sufficient to prove that
\begin{equation*}
\n{ \chi_j(D) \partial_k^s  R_{-\Delta}(z) f }_{L_{x_j}^\infty L_{{\widehat{x}_j}}^2}
\le C
\n{f}_{L_{x_j}^1 L_{{\widehat{x}_j}}^2},
\end{equation*}
where $|z|=1$, $s\in \{0,1\}$, $\partial^0_k=1, \partial^1_k=\partial_k$ and $j,k\in\{1,\dots,n\}$.
This is equivalent to
	\begin{equation}\label{estz=1}
	\n{ \F^{-1}_{\xi} \left( \frac{\xi_k^s  \, \chi_j(\xi) }{|\xi|^2-\l-i\e} \F_{x}f \right) }_{L_{x_j}^\infty L_{{\widehat{x}_j}}^2}
	\le C
	\n{f}_{L_{x_j}^1 L_{{\widehat{x}_j}}^2},
	\end{equation}
where we have settled $z=\l+i\e$, $\l^2+\e^2=1$ and $z\neq1$. We proceed splitting $\chi_j$ in the functions which appear in its definition \eqref{def_chij}, localizing ourselves in the regions of the frequency domain near the unit sphere, i.e. $\S^\pm_j$, and far from it, i.e. $\S_\infty$.

\textit{Estimate on $\S^\pm_j$.}
We want to prove
\begin{equation}\label{eq_Sj}
\n{ \F^{-1}_{\xi} \left( \frac{\xi_k^s \chi_j^\pm(\xi)}{|\xi|^2-\l-i\e} \F_{x}f \right) }_{L_{x_j}^\infty L_{{\widehat{x}_j}}^2}
\le C
\n{f}_{L^2_{{\widehat{x}_j}} L^1_{x_j}}.
\end{equation}

Let us define the family of operators
\begin{equation*}
T^\pm_j \colon L^p_{x_j}L^2_{{\widehat{x}_j}} \to  L^p_{x_j}L^2_{{\widehat{x}_j}}, \quad
f \mapsto T^\pm_j f := \F_\xi^{-1} \left(\hat{f} \circ \Phi \right),
\end{equation*}
where
\begin{equation*}
\Phi(\xi) := (\xi_j+\varphi(\widehat{\xi}_j), \widehat{\xi}_j),
\quad
\vp(\widehat{\xi}_j) := \pm\sqrt{|1-|\widehat{\xi}_j|^2|}.
\end{equation*}

Roughly speaking, the operators $T^\pm_j$ \lq\lq flatten'' in the frequency domain the hemisphere of the unitary sphere, namely $\{ \xi\in\R^n \colon |\xi|=1, \pm\xi_j > 0 \}$.

Writing more explicitly the introduced operators, we have that
\begin{equation*}
\begin{split}
T^\pm_j f(x)
&= \frac{1}{(2\pi)^{n/2}} \int_{\R^n} e^{i x\cdot\xi} \hat{f}(\xi_j+\varphi(\widehat{\xi}_j), \widehat{\xi}_j) d\xi \\
&= \frac{1}{(2\pi)^{n}} \int_{\R^n} e^{i x\cdot\xi} \int_{\R^n} f(y) e^{-iy \cdot(\xi_j+\varphi(\widehat{\xi}_j), \widehat{\xi}_j)} dy d\xi \\
&= \frac{1}{(2\pi)^{n}}
\int_{\R^{n-1}} e^{i {\widehat{x}_j}\cdot\widehat{\xi}_j}
\int_{\R^{n-1}} e^{-i y'\cdot\widehat{\xi}_j}
\int_{\R}\int_{\R} f(y)
e^{i (x_j-y_j)\xi_j -i y_j\varphi(\widehat{\xi}_j)}
dy_j d\xi_j dy'd\widehat{\xi}_j \\
&= \frac{1}{2\pi}
\F^{-1}_{\widehat{\xi}_j} \F_{y'} \left(
e^{-i x_j \varphi(\widehat{\xi}_j)}
\int_{\R}\int_{\R} f(y)
e^{i (x_j-y_j)\xi_j}
dy_j d\xi_j \right)\\
&= \F^{-1}_{\widehat{\xi}_j} \F_{y'} \left(
e^{-i x_j \varphi(\widehat{\xi}_j)}
f(x_j,y')
\right)
\end{split}
\end{equation*}
where we exploited the substitution $\xi_j\mapsto\xi_j-\varphi(\widehat{\xi}_j)$ in the fourth equality. Hence, applying the Plancherel Theorem two times, we obtain that $T^\pm_j$ are isometries respect to the norm $L^p_{x_j}L^2_{{\widehat{x}_j}}$, i.e. for $p \in [1,+\infty]$ it holds the relation
\begin{equation}\label{iso}
\n{T^\pm_j f}_{L^p_{x_j}L^2_{{\widehat{x}_j}}} = \n{f}_{L^p_{x_j}L^2_{{\widehat{x}_j}}}	.
\end{equation}

Thanks to this equality, we can write
\begin{equation*}\label{est_conv}
\begin{split}
\n{ \F^{-1}_{\xi} \left( \frac{\xi_k^s \chi_j^\pm(\xi)}{|\xi|^2-\l-i\e} \F_{x}f \right) }_{L_{x_j}^\infty L_{{\widehat{x}_j}}^2}
&= \n{ T^\pm_j \F^{-1}_{\xi} \left( \frac{\xi_k^s \chi_j^\pm(\xi)}{|\xi|^2-\l-i\e} \F_{x}f \right) }_{L_{x_j}^\infty L_{{\widehat{x}_j}}^2}
\\

&= \n{ \F_\xi^{-1} \left( \frac{ (\xi_k^s \chi_j^\pm ) \circ \Phi}{|\Phi|^2-\l-i\e}
	\, \wh{T^\pm_j f} \right)}_{L^\infty_{x_j}L^2_{{\widehat{x}_j}}}
\\
&= \frac{1}{\sqrt{2\pi}}
\n{ {a_{\l,\e}}(D) \psi *_{x_j}
	\F_{\xi_j}^{-1} \left( \frac{\wh{T^\pm_j f} }{\xi_j-i|\e|} \right)}_{L^\infty_{x_j}L^2_{\widehat{\xi}_j}}
\\
&\le \frac{1}{\sqrt{2\pi}}
\n{ {a_{\l,\e}}(D) \psi }_{L^1_{x_j}L^{\infty}_{\widehat{\xi}_j}}
\n{ \F_\xi^{-1} \left( \frac{\wh{T^\pm_j f} }{\xi_j-i|\e|} \right)}_{L^{\infty}_{x_j}L^2_{\widehat{\xi}_j}}
\end{split}
\end{equation*}
where the last inequality follows from the Young's Theorem and
\begin{gather*}
{a_{\l,\e}}(D) \psi
=\F^{-1}_{\xi_j} \left( a_{\l,\e} \F_{x_j}(\psi) \right) ,
\\
a_{\l,\e} (\xi) :=
\frac{(\xi_j-i|\e|) \,
\left(\xi_k \pm \delta_{k,j}\sqrt{1-|\widehat{\xi}_j|^2}\right)^s}
{\xi_j \left(\xi_j \pm 2\sqrt{1-|\widehat{\xi}_j|^2} \right)+1-\l-i\e}
\,
\sqrt{(\chi^\pm_j \circ \Phi)(\xi)} ,
\\
\psi(x_j,\widehat{\xi}_j) = \F_{\xi_j}^{-1} \left(\sqrt{(\chi^\pm_j \circ \Phi)(\xi)}\right)
.
\end{gather*}
Observe that we dropped the absolute value in the definition of $\varphi$, namely it results $\sqrt{|1-|\widehat{\xi}_j|^2|}=\sqrt{1-|\widehat{\xi}_j|^2}$, because $\supp\{\chi_j^\pm \circ \Phi\} \subset \{\xi\in\R^n \colon |\widehat{\xi}_j|\le1\}$, thanks to the definition of $\S^\pm_j$ and from the assumption $\delta\ge\sqrt{R^2-1}$. Despite of the involute definition, it is simple to see that ${a_{\l,\e}}(D) \psi \in \mathfrak{S}$, where $\mathfrak{S}$ is the space of Schwartz functions, since ${a_{\l,\e}}(D) \psi$ is the inverse Fourier transform of a smooth compactly supported function.
Moreover, considering ${a_{\l,\e}}(D) \psi$ as a pseudo-differential operator of symbol $a_{\l,\e}$ applied to the Schwartz function $\psi$, let us to observe that, since letting $\l+i\e\to1$ we have
\begin{equation*}
	\lim_{\l+i\e\to1} a_{\l,\e} (\xi) =
	\frac{\left(\xi_k \pm \delta_{k,j}\sqrt{1-|\widehat{\xi}_j|^2}\right)^s}{\xi_j \pm 2\sqrt{1-|\widehat{\xi}_j|^2}}
	\,
	\sqrt{\chi_j^\pm \left(\xi_j\pm\sqrt{1-|\widehat{\xi}_j|^2},\widehat{\xi}_j \right)} =: a(\xi) \in\mathfrak{S}
\end{equation*}
pointwisely, then ${a_{\l,\e}}(D)\psi \to {a}(D)\psi$ in $\mathfrak{S}$, and so
\begin{equation*}
\lim_{\l+i\e\to1}
\n{{a_{\l,\e}}(D) \psi}_{L^1_{x_j}L^\infty_{\widehat{\xi}_j}}
= \n{a(D)\psi}_{L^1_{x_j}L^\infty_{\widehat{\xi}_j}} < +\infty.
\end{equation*}
Thus, $\n{{a_{\l,\e}}(D) \psi}_{L^1_{x_j}L^\infty_{\widehat{\xi}_j}}$ is uniformly bounded respect to $z\in\C$ unitary.

Hence we have obtained
\begin{equation}\label{est1}
\n{ \F^{-1}_{\xi} \left( \frac{\xi_k^s \chi_j^\pm(\xi)}{|\xi|^2-\l-i\e} \F_{x}f \right) }_{L_{x_j}^\infty L_{{\widehat{x}_j}}^2}
\le C
\n{ \F_\xi^{-1} \left( \frac{\wh{T^\pm_j f} }{\xi_j-i|\e|} \right)}_{L^{\infty}_{x_j}L^2_{\widehat{\xi}_j}}.
\end{equation}
By Plancherel and Young Theorems, and by equality \eqref{iso}, we have
\begin{equation*}
\begin{split}
\sqrt{2\pi} \n{ \F_{\xi_j}^{-1} \left( \frac{\wh{T^\pm_j f} }{\xi_j-i|\e|} \right)}_{L^\infty_{x_j}L^2_{\widehat{\xi}_j}}
&= \n{ \F_{\xi_j}^{-1} \left( \frac{1 }{\xi_j-i|\e|}\right) *_{x_j} \F_{{\widehat{x}_j}} (T^\pm_j f) }_{L^\infty_{x_j}L^2_{\widehat{\xi}_j}}
\\
& = \n{ i e^{-|\e| x_j } \theta *_{x_j}
	\F_{{\widehat{x}_j}} (T^\pm_j f) }_{L^\infty_{x_j}L^2_{\widehat{\xi}_j}}
\\
& \le \n{ e^{-|\e| x_j } \theta *_{x_j}
	\n{T^\pm_j f}_{L^2_{{\widehat{x}_j}}} }_{L^\infty_{x_j}}
\\
&\le \n{ e^{-|\e| x_j } \theta }_{L^\infty_{x_j}} \n{f}_{L^1_{x_j}L^2_{{\widehat{x}_j}}}
\\
&= \n{f}_{L^1_{x_j}L^2_{{\widehat{x}_j}}}
,
\end{split}
\end{equation*}
where $\theta \equiv \theta(x_j)$ is the Heaviside function. Thus, inserting in \eqref{est1}, we get \eqref{eq_Sj}.

\textit{Estimate on $\S_\infty$.}

We want to prove now
\begin{equation}\label{eq_Sinfty}
\n{ \F^{-1}_{\xi} \left( \frac{\xi_k^s \chi_\infty(\xi) }{|\xi|^2-\l-i\e} \F_{x}f \right) }_{L_{x_j}^\infty L_{{\widehat{x}_j}}^2}
\le C
\n{f}_{L^2_{{\widehat{x}_j}} L^1_{x_j}}.
\end{equation}
We need to distinguish three cases depending on whether we are localized in the regions defined by
\begin{align*}
	\Cyl^{1}_{R,j} &:= \{ \xi\in\R^n \colon |\widehat{\xi}_j|>R \},
	\\
	\Cyl^{2}_{R,j} &:= \{ \xi\in\R^n \colon |\widehat{\xi}_j|\le R, |\xi_j|\le 2R \},
	\\
	\Cyl^{3}_{R,j} &:= \{ \xi\in\R^n \colon |\widehat{\xi}_j|\le R, |\xi_j|> 2R \}.
\end{align*}

Let us set then
\begin{align*}
	\chi^1_\infty(\xi) &:=
	\begin{cases}
	\makebox[\longtext][l]{1} &\text{if $|\widehat{\xi}_j|>R$,}\\
	\makebox[\longtext][l]{0} &\text{otherwise,}
	\end{cases}
	\\

	\chi^2_\infty(\xi) &:=
	\begin{cases}
	\chi_\infty(\xi) &\text{if $|\widehat{\xi}_j|\le R$ and $|\xi_j|\le 2R$,}\\
	\makebox[\longtext][l]{0} &\text{otherwise,}
	\end{cases}
	\\

	\chi^3_\infty(\xi) &:=
	\begin{cases}
	\makebox[\longtext][l]{1} &\text{if $|\widehat{\xi}_j|\le R$ and $|\xi_j|> 2R$,}\\
	\makebox[\longtext][l]{0} &\text{otherwise,}
	\end{cases}
\end{align*}
and observe that $\chi_\infty = \chi^1_\infty + \chi^2_\infty + \chi^3_\infty$, since $\chi_\infty \equiv 1$ for $|\xi|>R$, from the requirements on the cover $\S$ and the partition $\chi$.

By Plancherel Theorem and H\"older's, Young's and Minkowski's integral inequalities, for $h\in\{1,2,3\}$ we infer
\begin{equation*}
\n{ \F^{-1}_{\xi} \left( \frac{\xi_k^s \chi^h_\infty(\xi)}{|\xi|^2-\l-i\e} \F_{x}f \right) }_{L_{x_j}^\infty L_{{\widehat{x}_j}}^2}
\le
{C_h}
\n{f}_{L^1_{x_j} L^2_{{\widehat{x}_j}}}
\end{equation*}
with
\begin{equation}\label{Ci}
	C_h :=
	\frac{1}{\sqrt{2\pi}}
	\n{ \F^{-1}_{\xi_j} \left( \frac{\xi_k^s \, \chi^h_\infty(\xi) }{\xi_j^2 + \s^2} \right) }_{L^\infty_{x_j} L^\infty_{\widehat{\xi}_j}},
	\quad
	\s:=\sqrt{|\widehat{\xi}_j|^2-\l-i\e}.
\end{equation}
Here and below, we always consider the principal branch of the complex square root function.

Clearly, if we prove that $C_h$ for $h\in\{1,2,3\}$ is bounded uniformly respect to $\l,\e$, we recover \eqref{eq_Sinfty}.

\textit{Estimate on $\Cyl^1_{R,j}$.} Observing that $\chi_{ \infty}^1(\xi) \equiv
\chi_{ \infty}^1(\widehat{\xi}_j)$ and
noting that
\begin{align*}
\Re\sigma &= \sqrt{ \frac{|\sigma|^2+|\widehat{\xi}_j|^2-\l}{2} } >0,
\end{align*}
we can compute explicitly the Fourier transforms:
\begin{itemize}
	\item if $k \neq j$, then
	\begin{equation*}
	\begin{split}
		C_1 =
		\n{ \chi_\infty^1(\widehat{\xi}_j)
			\xi_k^s \frac{e^{-\s|x_j|}}{2\s} }_{L^\infty_{x_j} L^\infty_{\widehat{\xi}_j}}
		&\le
		\n{ \chi_\infty^1(\widehat{\xi}_j) |\widehat{\xi}_j|^s \frac{e^{- \Re\sigma |x_j|}}{2|\s|} }_{L^\infty_{x_j} L^\infty_{\widehat{\xi}_j}}
		\\&\le
		\sup_{|\widehat{\xi}_j|>R} \frac{|\widehat{\xi}_j|^s}{2(|\widehat{\xi}_j|^4-2\l|\widehat{\xi}_j|^2+1)^{1/4}}
		\\&\le
		\begin{cases}
		\frac{R^s}{2\sqrt{R^2-1}} & \text{if $\l>0$,}
		\\
		1/2 & \text{if $\l\le0$;}
		\end{cases}
	\end{split}
	\end{equation*}

	\item if $s=1$, $k = j$, then
	\begin{equation*}
		C_1 =
		\n{ \chi_\infty^1(\widehat{\xi}_j) \frac{i}{2} \sgn\{x_j\} e^{-\s|x_j|} }_{L^\infty_{x_j} L^\infty_{\widehat{\xi}_j}}
		\le \frac{1}{2}.
	\end{equation*}
\end{itemize}

\textit{Estimate on $\Cyl^2_{R,j}$.} Expliciting the definition of inverse Fourier transform in \eqref{Ci} and from the fact that $\chi^2_\infty(\xi) = 0$ when $|\xi|<r$, we can trivially see that
\begin{equation*}
C_2 \le
\frac{1}{2\pi}
\n{  \int_{-\infty}^{+\infty}
	\frac{ |e^{i x_j \xi_j}| \, |\xi_k^s| \, \chi^2_\infty(\xi)}{||\xi|^2-\l|}
	d\xi_j
}_{L^\infty_{x_j} L^\infty_{\widehat{\xi}_j}}
\le
\frac{(2R)^s}{2\pi}
\n{
	\frac{\chi^2_\infty(\xi)}{|\xi|^2-1}
}_{L^\infty_{\widehat{\xi}_j} L^1_{\xi_j}}
\end{equation*}
which is finite since $\chi^2_\infty$ is compactly supported due to its definition.

\textit{Estimate on $\Cyl^3_{R,j}$.}
Expliciting the inverse Fourier transform in \eqref{Ci}, recalling the definition of $\chi^3_\infty$ and exploiting the substitution $\xi_j \mapsto \sgn\{x_j\}\xi_j$, we have
\begin{align*}
C_3 &=
\frac{1}{2\pi}
\n{ (1-\chi^1_\infty)(\widehat{\xi}_j) \int_{|\xi_j|>R} e^{i |x_j| \xi_j} \frac{\xi_k^s }{\xi_j^2 + \s^2} d\xi_j }_{L^\infty_{x_j} L^\infty_{\widehat{\xi}_j}}
\\
&=
\frac{1}{2\pi}
\n{ \int_{|\xi_j|>R} \psi(x_j,\widehat{\xi}_j,\xi_j) d\xi_j }_{L^\infty_{x_j} L^\infty_{\widehat{\xi}_j}}
\end{align*}
where, for fixed $\widehat{\xi}_j, x_j$,

the complex function $\psi(x_j,\widehat{\xi}_j,\cdot) \colon\C\to\C$ is defined by
\begin{equation*}
	\psi(x_j,\widehat{\xi}_j,w) :=
	\left\{
	\begin{aligned}
	& (1-\chi^1_\infty)(\widehat{\xi}_j) \frac{\xi_k^s}{w^2 + \sigma^2}e^{i |x_j| w}, &&\text{if $k\neq j$,}
	\\
	& (1-\chi^1_\infty)(\widehat{\xi}_j) \frac{w}{w^2 + \sigma^2} e^{i |x_j| w}, &&\text{if $s=1$, $k=j$,}
	\end{aligned}
	\right.
\end{equation*}
which is holomorphic in $\C\setminus\{w_-,w_+\}$, where $w_\pm = \pm i\sigma$. Observe that $\psi\equiv0$ for $|\widehat{\xi}_j|>R$, and if $|\widehat{\xi}_j|\le R$ we have
\begin{equation}\label{|sigma|}
|w_\pm| = |\sigma| = \sqrt[4]{ (|\widehat{\xi}_j|^2-\l)^2 +\e^2 } < \sqrt{2}R .
\end{equation}
Define, for a radius $A>0$, the semicircle
$
	\gamma_A := \{Ae^{i\theta} \colon \theta\in [0,\pi] \}
$
in the upper half complex plane.
Fixing $\rho>R$, by the Residue Theorem, we get
\begin{equation*}
	\left(
	\int_{[-\rho,-2R]} - \int_{\gamma_{2R}} + \int_{[2R,\rho]} + \int_{\gamma_\rho}
	\right) \psi(x_j,\widehat{\xi}_j,w) dw = 0.
\end{equation*}
Observing that we can consider $x_j\neq0$, letting $\rho\to +\infty$ we can apply the Jordan's lemma to the integral on the curve $\gamma_\rho$, finally getting
\begin{align*}
	C_3 &=
	\frac{1}{2\pi} \n{ \int_{\gamma_{2R}} \psi(x_j, \widehat{\xi}_j, w) dw }_{L^\infty_{x_j} L^\infty_{\widehat{\xi}_j}}
	\\
	&\le
	\frac{(2R)^s}{2\pi} \n{ (1-\chi^1_\infty)(\widehat{\xi}_j) \int_{0}^{\pi}  \frac{d\theta }{|4R^2e^{2i\theta} + \sigma^2|} }_{L^\infty_{\widehat{\xi}_j}}
	\\
	&\le
	(2R)^{s-2}

\end{align*}
where we used the relation \eqref{|sigma|}.

Summing all up, we can finally recover the desired estimate \eqref{estz=1}, where the positive constant $C$ does not depend on $\l,\e$, but only on $R$ and the partition $\chi$.

\end{proof}

We can prove now Lemma\til\ref{lem2}, as a straightforward corollary of Lemma\til\ref{lem1}.

\begin{proof}[Proof of Lemma\til\ref{lem2}]
	Again, the \lq\lq in particular'' part follows from
	\eqref{sumchij}
	and the definitions of the $X$ and $X^*$ norms.

	From the anticommutation relations \eqref{clifford} we infer, for every $z\in\C$,
	\begin{equation*}\label{trick}
	(\D_{0} - zI_N)(\D_{0} + zI_N) = (-\Delta + m^2 -z^2)I_N.
	\end{equation*}
	So, thanks to this well-known trick, for $z\in\rho(\D_0)$ we can write
	\begin{equation*}
	R_{\D_{0}}(z) = (\D_{0} +zI_N) R_{-\Delta}(z^2-m^2)I_N.
	\end{equation*}
	Set $f_j = \chi_j(|z^2-m^2|^{-1/2}D)f$ for the simplicity. Exploiting Lemma~\ref{lem1}, it is easy to get
	\begin{equation*}
	\begin{split}
	\n{ R_{\D_{0}}(z) f_j}_{L^{\infty}_{x_j} L^2_{\widehat{x}_j}}
	\le&\,
	\n{ \sum_{k=1}^n \alpha_k \partial_k R_{-\Delta}(z^2-m^2) f_j}_{L^{\infty}_{x_j} L^2_{\widehat{x}_j}}
	\\
	&+\n{ (m\a_0 + z I_{N}) R_{-\Delta}(z^2-m^2) f_j}_{L^{\infty}_{x_j} L^2_{\widehat{x}_j}}
	\\
	\le&\,
	\sum_{k=1}^n
	\n{ \partial_k R_{-\Delta}(z^2-m^2) f_j}_{L^{\infty}_{x_j} L^2_{\widehat{x}_j}}
	\\
	&+ \max\{|z+m|, |z-m|\} \n{ R_{-\Delta}(z^2-m^2) f_j}_{L^{\infty}_{x_j} L^2_{\widehat{x}_j}}
	\\
	\le&\,
	C \left[n + \left|\frac{z+m}{z-m}\right|^{\sgn(\Re z)/2}\right]
	\n{f}_{L^{1}_{x_j} L^2_{\widehat{x}_j}} .
	\end{split}
	\end{equation*}
	as claimed.
\end{proof}

\section{The Birman-Schwinger principle}\label{sec3}

In this section, working in great generality, we precisely define the closed extension of a perturbed operator with a factorizable potential, formally defined as $H_0+B^*A$. Next, we state an abstract version of the Birman-Schwinger principle.
We will follow the approach of Kato \cite{Kato66} and Konno and Kuroda \cite{KK66}.

	Let $\H, \H'$ be Hilbert spaces and consider the densely defined, closed, linear operators
	\begin{equation*}

	H_0 \colon \dom(H_0) \subseteq \H \to \H,
	\quad
	A \colon \dom(A) \subseteq \H \to \H',
	\quad
	B \colon \dom(B) \subseteq \H \to \H',

	\end{equation*}
	such that $\rho(H_0) \neq \varnothing$
	and
	\begin{equation*}
		\dom(H_0) \subseteq\dom(A),
		\quad
		\dom(H_0^*) \subseteq\dom(B).
	\end{equation*}
	For the simplicity, we assume also that $\sigma(H_0) \subset \R$ and $\sigma_p(H_0)=\varnothing$.
	For $z\in\rho(H_0)$, denote by $R_{H_0}(z)=(H_0-z)^{-1}$ the resolvent operator of $H_0$.

As a warm-up, to sketch the idea of the principle, let us firstly consider the case of bounded operators $A$ and $B$. Thus, $H=H_0+B^*A$ is well-defined as a sum operator. Moreover, if $z\in\rho(H_0)$, we can define the bounded operator $Q(z)=A(H_0-z)^{-1}B^*$. It is easy to check that $z\in\sigma_p(H)\cap\rho(H_0)$ implies $-1\in\sigma_p(Q(z))$, and so \mbox{$\n{Q(z)}_{\H'\to\H'} \ge 1$}. Hence, a bound for the norm of the Birman-Schwinger operator $Q(z)$ give us information on the localization of the non-embedded eigenvalues of $H$.

Let us return to the general setting of an unbounded potential $B^*A$. Furthermore, we study also the case of embedded eigenvalues. To afford these, we assume a stronger set of hypotheses respect to the ones in \cite{KK66}, which instead will be proved here in Lemma\til\ref{lemBSop}.

\begin{hypothesis}[\textbf{Hyp}]\label{hyp}
	Let $X$ a complex Banach space of function on $\R^n$ such that the following estimates hold true:
	\begin{align*}
	\n{AR_{H_0}(z)}_{X\to \H'} &\le \alpha \Lambda(z),
	\\
	\n{B^*}_{\H' \to X} &\le \beta,
	\end{align*}
	where $\alpha,\beta>0$, $\Lambda(z)$ is finite for $z\in\rho(H_0)$ and continue up to $\sigma(H_0) \setminus \Omega$, for some set $\Omega \subseteq \sigma(H_0)$, and there exists $z_0\in\rho(H_0)$ such that $\Lambda(z_0)<(\alpha\beta)^{-1}$.
\end{hypothesis}

\begin{lemma}[Birman-Schwinger operator]
	\label{lemBSop}
	Assume (Hyp). Then, for $z\in\rho(H_0)$, the operator $AR_{H_0}(z)B^*$, densely defined on $\dom(B^*)$, has a closed extension $Q(z)$ in $\H'$,
		\begin{equation*}
		Q(z) = \overline{AR_{H_0}(z)B^*}
		\end{equation*}
	with norm bounded by
		\begin{equation}\label{Qnorm}
		\n{Q(z)}_{\H'\to\H'} \le \alpha\beta\Lambda(z).
		\end{equation}

	Moreover, there exists $z_0 \in\rho(H_0)$ such that $-1 \in \rho(Q(z_0)).$
\end{lemma}

\begin{proof}
	For $z\in\rho(H_0)$ and $\varphi\in\dom(B^*)$, we have
	\begin{equation*}
	\n{AR_{H_0}(z)B^* \varphi}_{\H'\to\H'} \le \n{AR_{H_0}(z)}_{X\to \H'} \n{B^*}_{\H' \to X} \n{\varphi}_{\H'}
	\end{equation*}
	and so by density \eqref{Qnorm}.

	In particular, for $z_0\in\rho(H_0)$ such that $\Lambda(z_0)<(\alpha\beta)^{-1}$, we get $\n{Q(z_0)}_{\H'\to\H'} < 1$. Hence, from Neumann series there exists $(1+Q(z_0))^{-1}$, and so $-1\in\rho(Q(z_0))$.
\end{proof}

Let us collect some useful facts in the next lemma.

\begin{lemma}\label{lem-rel}
	Suppose (Hyp) and fix $z, z_1, z_2 \in \rho(H_0)$. Then the following relations hold true:
	\begin{enumerate}[label=(\roman*)]
		\item $AR_{H_0}(z) \in \mathcal{B}(\H,\H'),
		\quad
		\overline{R_{H_0}(z)B^*} = [B(H_0^*-\overline{z})^{-1}]^* \in \mathcal{B}(\H',\H)$,

		\item
			$\overline{R_{H_0}(z_1)B^*}-\overline{R_{H_0}(z_2)B^*}
			= (z_1-z_2) R_{H_0}(z_1) \overline{R_{H_0}(z_2)B^*}
			= (z_1-z_2) R_{H_0}(z_2) \overline{R_{H_0}(z_1)B^*}$,

		\item $Q(z) = A\overline{R_{H_0}(z)B^*}$,
		\quad
		$Q(\overline{z})^* = B\overline{R_{H_0}(\overline{z})^* A^*}$,

		\item $\ran(\overline{R_{H_0}(z)B^*}) \subseteq \dom(A),
		\quad
		\ran(\overline{R_{H_0}(\overline{z})^* A^*}) \subseteq \dom(B)$,

		\item $Q(z_1) -Q(z_2)
			= (z_1-z_2) AR_{H_0}(z_1) \overline{R_{H_0}(z_2)B^*}
			= (z_1-z_2) AR_{H_0}(z_2) \overline{R_{H_0}(z_1)B^*}$.
	\end{enumerate}
\end{lemma}

\begin{proof}
	See Lemma\til2.2 in \cite{GLMZ05}.
\end{proof}

We can construct now the extension of the perturbed operator $H_0+B^*A$.

\begin{lemma}[Extension of operators with factorizable potential]
	\label{lem_ext}
	Suppose (Hyp). Fix $z_0 \in\rho(H_0)$, given by Lemma\til\ref{lemBSop}, such that $-1\in\rho(Q(z_0))$. The operator
	\begin{equation}\label{R}
	R_{H}(z_0) = R_{H_0}(z_0) - \overline{R_{H_0}(z_0)B^*} (1+Q(z_0))^{-1} AR_{H_0}(z_0)
	\end{equation}
	defines a densely defined, closed, linear operator $H$ in $\H$, which have $R_{H}(z_0)$ as resolvent,
	\begin{equation*}
	R_{H}(z_0) = (H-z_0)^{-1},
	\end{equation*}
	and which is an extension of $H_0 + B^*A$.

\end{lemma}

\begin{proof}
	We refer to Theorem\til2.3 in \cite{GLMZ05}. See also the work by Kato \cite{Kato66}.
\end{proof}

Finally, we enunciate the Birman-Schwinger principle.

\begin{lemma}[Birman-Schwinger principle]
	\label{lemBS}
	Suppose (Hyp). Let $z_0 \in\rho(H_0)$, given by Lemma\til\ref{lemBSop}, such that $-1\in\rho(Q(z_0))$ and $H$ be the extension of $H_0+B^*A$, given by Lemma\til\ref{lem_ext}.

	Fix $z \in \sigma_p(H)$ with eigenfunction $0\neq\psi\in\dom(H)$, i.e. $H\psi=z\psi$.

	Setted $\phi := A\psi$, we have that $\phi\neq0$ and
	\begin{enumerate}[label=(\roman*)]
		\item if $z \in \rho(H_0)$ then
		\begin{equation*}
		Q(z)\phi = -\phi
		\end{equation*}
		and in particular
		\begin{equation*}
		1 \le \n{Q(z)}_{\H'\to\H'} \le \alpha\beta\Lambda(z);
		\end{equation*}

		\item
		if $z\in \sigma(H_0)\setminus\Omega$ and $\psi\in X_{\text{loc}}$, i.e. $\chrt_K \psi \in X$ for every compact set $K \subset \R^n$ where $\chrt_K$ is the indicator function of $K$,
		then
		\begin{equation}\label{embQ}
		\lim_{\e\to0^\pm} (\varphi,Q(z+i\e)\phi)_{\H'} = -(\varphi, \phi)_{\H'}
		\end{equation}
		for every $\varphi \in \H'$ compactly supported, and in particular
		\begin{equation}\label{embQest}
		1 \le \liminf_{\e\to0^\pm} \n{Q(z+i\e)}_{\H'\to\H'} \le \alpha\beta \lim_{\e\to0^\pm} \Lambda(z+i\e).
		\end{equation}
	\end{enumerate}
\end{lemma}

\begin{proof}
	Let us prove only case (ii) for the embedded eigenvalues, being case (i) similar and easier, adapting the argument of Lemma\til1 in \cite{KK66}

	for the non-embedded eigenvalues.

	Noted that $H\psi =z \psi$ is equivalent to
	\begin{equation}\label{bs0}
		\psi = (z-z_0) R_{H}(z_0) \psi,
	\end{equation}
	we obtain from \eqref{R} that
	\begin{equation}\label{bs1}
		(H_0 - z -i\e) R_{H_0}(z_0) \psi
		=
		-(z-z_0) \overline{R_{H_0}(z_0)B^*} (1+Q(z_0))^{-1} AR_{H_0}(z_0) \psi
		-i\e R_{H_0}(z_0) \psi.
	\end{equation}
	Define $\widetilde{\psi}=(1+Q(z_0))^{-1} AR_{H_0}(z_0)\psi$. If $\widetilde{\psi}=0$, by \eqref{bs1} follows $(H_0-z)R_{H_0}(z_0)\psi=0$. Since $0\neq R_{H_0}(z_0)\psi \in \dom(H_0)$, we get $z\in\sigma_p(H_0)$, which however is empty by our assumptions on $H_0$. Thus, we proved $\widetilde{\psi}\neq0$.

	Moreover, we can show the identity
	\begin{equation}\label{bs2}
		\phi = A\psi = (z-z_0) (1+Q(z_0))^{-1} AR_{H_0}(z_0)\psi = (z-z_0) \widetilde{\psi},
	\end{equation}
	from which in particular $\phi\neq 0$.
	Indeed, by \eqref{R} and (iii) of Lemma\til\ref{lem-rel}, it follows that
	\begin{equation*}
		AR_{H}(z_0) = (1+Q(z_0))^{-1}AR_{H_0}(z_0)
	\end{equation*}
	which combined with \eqref{bs0} gives us \eqref{bs2}.

	Let us prove the limit \eqref{embQ}. Multiplying by $(1+Q(z_0))^{-1} AR_{H_0}(z+i\e)$ both sides of \eqref{bs1}, we obtain
	\begin{equation*}
		\begin{split}
		\widetilde{\psi}
		=
		&-(z-z_0) (1+Q(z_0))^{-1} A R_{H_0}(z+i\e) \overline{R_{H_0}(z_0)B^*} \widetilde{\psi}
		\\
		&- i\e (1+Q(z_0))^{-1} A R_{H_0}(z+i\e) R_{H_0}(z_0) \psi
		\end{split}
	\end{equation*}
	and so, by (v) of Lemma\til\ref{lem-rel} and by the resolvent identity, we have
	\begin{equation*}
	\begin{split}
	\widetilde{\psi}
	=
	&- \frac{z-z_0}{z-z_0+i\e} (1+Q(z_0))^{-1} [Q(z+i\e) - Q(z_0)] \widetilde{\psi}
	\\
	&- \frac{i\e}{z-z_0+i\e} (1+Q(z_0))^{-1} A [R_{H_0}(z+i\e) - R_{H_0}(z_0)] \psi

	\\

	=
	& \, \widetilde{\psi} - \frac{z-z_0}{z-z_0+i\e} (1+Q(z_0))^{-1} (1+Q(z+i\e))\widetilde{\psi}
	\\
	&- \frac{i\e}{z-z_0+i\e} (1+Q(z_0))^{-1} A R_{H_0}(z+i\e)\psi,
	\end{split}
	\end{equation*}

from which, using identity \eqref{bs2}, we finally arrive to
\begin{equation*}
	Q(z+i\e) \phi
	= -\phi
	- i\e A R_{H_0}(z+i\e)\psi.
\end{equation*}

Fixed $\varphi \in \H'$ with compact support $K$, we have that
\begin{equation}\label{Qint_eq}
(\varphi, Q(z+i\e) \phi)_{\H'}
= -(\varphi, \phi)_{\H'}
- i\e (\varphi, A R_{H_0}(z+i\e)\psi)_{\H'}.
\end{equation}
Since
\begin{equation*}
	\begin{split}
	(\varphi, A R_{H_0}(z+i\e)\psi)_{\H'}
	&=
	\int_{K} \varphi^{\dagger} \cdot A R_{H_0}(z+i\e)\psi
	\\
	&=
	\int_{K} R_{H_0}(z-i\e) A^* \varphi^{\dagger} \cdot \psi
	\\
	&=
	(\varphi, A R_{H_0}(z+i\e) [\chrt_K \psi] )_{\H'},
	\end{split}
\end{equation*}
by (Hyp) we get
\begin{equation*}
	|(\varphi, A R_{H_0}(z+i\e)\psi)_{\H'}|
	\le
	\n{\varphi}_{\H'} \n{A R_{H_0}(z+i\e) [\chrt_K \psi]}_{\H'}
	\le
	\alpha\Lambda(z+i\e) \n{\varphi}_{\H'} \n{\chrt_K \psi}_X,
\end{equation*}
and so the last term in \eqref{Qint_eq} vanishes as $\e\to0^{\pm}$, proving \eqref{embQ}.

Finally, we prove the first inequality in \eqref{embQest}, being the second one given by Lemma\til\ref{lemBSop}.
For $n\in \N$, let $\chi_n(x) := \chi(x/n)$ where $\chi \in C_0^\infty(\R^n)$ is a cut-off function such that $\chi(x)=1$ for $|x|\le 1$ and $\chi(x)=0$ for $|x|\ge 2$. Since, by \eqref{embQ},
\begin{equation*}
	|(\chi_n \phi , \phi)_{\H'}|
	=
	\lim_{\e\to0^{\pm}}
	|(\chi_n \phi, Q(z+i\e)\phi)_{\H'}| \le
	\n{\chi_n \phi}_{\H'} \n{\phi}_{\H'}
	\liminf_{\e\to0^{\pm}}
	\n{Q(z+i\e)}_{\H'\to\H'}
\end{equation*}
we get \eqref{embQest} letting $n\to+\infty$.
\end{proof}

\section{Proof of the Theorems}\label{sec4}

Here we simply apply the results of the last section to our problem. In our case, $\H=\H'=L^2(\R^n;\C^N)$, $X=\bigcap_{j=1}^n L^1_{x_j}L^2_{\widetilde{x}_j}$, $H_0$ is the free Dirac operator $\D_{0}$, and the factorization of $V$ is reached considering the polar decomposition $V=UW$, i.e. $W=(V^* V)^{1/2}$ and the unitary matrix $U$ is a partial isometry, and setting $A=W^{1/2}$ and $B=W^{1/2}U^*$.

It is easy to see that hypothesis (Hyp) holds thanks to Lemma\til\ref{lem2} with
\begin{equation*}
	\alpha=\beta=\n{V}_Y^{1/2},
	\quad
	\Lambda(z)=nC \left[n + \left|\frac{z+m}{z-m}\right|^{\sgn(\Re z)/2}\right],
	\quad
	\Omega =
	\begin{cases}
	\{-m,m\} & \text{if $m\neq 0$,}
	\\
	\varnothing & \text{if $m=0$.}
	\end{cases}
\end{equation*}
Indeed, for $\varphi\in\H$,
	\begin{equation*}
		\begin{split}
		\n{A R_{\D_{0}}(z) \varphi}_{\H}
		&\le
		\sum_{j=1}^{n}
		\n{A \, \chi_j(|z^2-m^2|^{-1/2}D) R_{\D_{0}}(z) \varphi}_{\H}
		\\
		&\le
		C \left[n + \left|\frac{z+m}{z-m}\right|^{\sgn(\Re z)/2}\right]
		\sum_{j=1}^{n}
		\n{A}_{L^2_{x_j} L^\infty_{\widehat{x}_j}}
		\n{\varphi}_{L^1_{x_j} L^2_{\widehat{x}_j}}
		\\
		&
		\le nC \left[n + \left|\frac{z+m}{z-m}\right|^{\sgn(\Re z)/2}\right]
		\n{V}^{1/2}_{Y} \n{\varphi}_{X},

		\\

		\n{B^* \varphi}_X
		&\le
		\n{V}_Y \n{\varphi}_{\H},
		\end{split}
	\end{equation*}
where we used the relation
\begin{equation*}
\n{A}_{L^2_{x_j} L^\infty_{\widehat{x}_j}}
=
\n{B^*}_{L^2_{x_j} L^\infty_{\widehat{x}_j}}
=
\n{W^{1/2}}_{L^2_{x_j} L^\infty_{\widehat{x}_j}}
=
\n{V}^{1/2}_{L^1_{x_j} L^\infty_{\widehat{x}_j}}.
\end{equation*}
To see that there exists $z_0 \in \rho(\D_0)$ such that $\Lambda(z_0)<(\alpha\beta)^{-1}$, define
\begin{equation*}
	C_0=[n(n+1)C]^{-1},
	\quad
	\V = [(n+1)C_0/\n{V}_Y - n]^2.
\end{equation*}
Since from the hypothesis of Theorems\til\ref{thm1}\til\&\til\ref{thm2} we have $\n{V}_Y < C_0$ and so $\V>1$, the condition
$1 \le \alpha\beta\Lambda(z)$
is equivalent to $\V \le |z/z|$ if $m=0$, and to
\begin{equation}\label{disks}
\left( \Re z - \sgn(\Re z) m\,\frac{\V^2+1}{\V^2-1}\right)^2 + \Im z^2
\le
\left(m\,\frac{2\V}{\V^2-1}\right)^2
\end{equation}
if $m>0$. Then, if $m=0$ it is sufficient to choose $z_0\in\C\setminus\R$, whereas if $m>0$ we take $z_0 \in \rho(\D_0)$ outside the disks in the statement of Theorem\til\ref{thm1}.

Finally, it is trivial to see that if $\psi\in\H$, then $\psi\in X_{\text{loc}}$, since, for every compact set $K\in\R^n$,
we have
$
\n{\chrt_K \psi}_{L^1_{x_j} L^2_{\widehat{x}_j}}
\le
\kappa \n{\psi}_{L^2}
$,
for some constant $\kappa$ depending on $K$.

Thus, we can apply Lemma\til\ref{lemBS}, which combined with relation \eqref{disks} and with relation $\V \le |z/z|$ proves

Theorem\til\ref{thm1} and Theorem\til\ref{thm2} respectively.

For the final claim in Theorem\til\ref{thm2},
we will follow the argument in \cite{CLT14} to prove that the potential $V \in Y = \bigcap_{j=1}^n L^1_{x_j} L^\infty_{\widehat{x}_j} (\R^n)$ leaves the essential spectrum invariant and that the residual spectrum of $\D_V$ is absent.

To get the invariance of the essential spectrum, it is sufficient to prove that, fixed $z \in \rho(\D_0)$ such that $-1\in \rho(Q(z))$, the operator $A R_{\D_0}(z)$ is an Hilbert-Schmidt operator, and hence compact. Thus, identity \eqref{R} gives us
\begin{equation*}
R_{\D_V}(z) - R_{\D_0}(z)
=
- \overline{R_{\D_0}(z) B^*}
(1+Q(z))^{-1}
A R_{\D_0}(z)
\end{equation*}
from which follows that $R_{\D_V}(z) - R_{\D_0}(z)$ is compact and so, by Theorem 9.2.4 in \cite{EE},
\begin{equation*}
\sigma_e(\D_V) = \sigma_e(\D_0) =
(-\infty,m] \cup [m,\infty).
\end{equation*}

To see that $A R_{\D_0}(z)$ is an Hilbert-Schmidt operator, we need to prove that its kernel $A(x) \K(z,x-y)$ is in $L^2(\R^n\times\R^n;\C^N)$, where $\K(z,x-y)$ is the kernel of the resolvent $(\D_0-z)^{-1}$.
By the Young inequality
\begin{equation}\label{HSnorm}
\begin{split}
\n{A(\D-z)^{-1}}_{HS}^2
=
\int_{\R^n} \int_{\R^n} |A(x)|^2 |\K(z,x-y)|^2 dx dy

\le
\n{V}_{L^p}
\n{\K}_{L^{2q}}^2
\end{split}
\end{equation}
where $1/p + 1/q =2$. Hence we need to find in which Lebesgue space $L^{2q}(\R^n;\C^N)$ the kernel $\K(z,x)$ lies.

For $z\in\rho(-\Delta)=\C\setminus[0,\infty)$, it is well-know (see e.g. \cite{GS}) that the kernel $\K_0(z,x-y)$ of the resolvent operator $(-\Delta-z)^{-1}$ is given by
\begin{equation*}
\K_0(z,x-y)
=
\frac{1}{(2\pi)^{n/2}}
\left( \frac{\sqrt{-z}}{|x-y|}\right)^{\frac{n}{2}-1}
K_{\frac{n}{2}-1} (\sqrt{-z}|x-y|)
\end{equation*}
where $K_{\nu}(w)$ is the modified Bessel function of second kind and we consider the principal branch of the complex square root.
Fixed now $z\in\rho(\D_0)=\C\setminus\{\zeta\in\R \colon |\zeta|\ge m\}$, from the identity
\begin{equation*}
(\D_0-zI_N)^{-1} = (\D_0+zI_N)(-\Delta+m^2-z^2)^{-1}I_N
\end{equation*}
and relations 9.6.26 in \cite{AS} for the derivative of the modified Bessel functions, we obtain
\begin{align*}
\K(z,x-y)
&=
\frac{1}{(2\pi)^{n/2}}
\left(\frac{k(z)}{|x-y|}\right)^{\frac{n}{2}}

\vect{\alpha} \cdot (x-y)
K_{\frac{n}{2}}(k(z) |x-y|)
\\
& \quad +
\frac{1}{(2\pi)^{n/2}}
\left(\frac{k(z)}{|x-y|}\right)^{\frac{n}{2}-1}
(m\alpha_0+z)
K_{\frac{n}{2}-1} (k(z) |x-y|)
\end{align*}
where for the simplicity
$k(z) = \sqrt{m^2-z^2}$. From the limiting form for the modified Bessel functions
\begin{align*}
K_{\nu}(w)
&\sim \frac{1}{2} \Gamma(\nu) \left(\frac{w}{2}\right)^{-\nu}
&&\text{for $\Re \nu >0$ and $w \to 0$,}
\\
K_{0}(w)
&\sim -\ln w
&&\text{for $w \to 0$,}
\\
K_\nu(w)
&\sim \sqrt{\frac{\pi}{2w}} e^{-w}
&&\text{for $z\to\infty$ in $|\arg z| \le 3\pi/2-\delta$,}
\end{align*}
we obtain that
\begin{equation*}
\n{\K(z,x)}
\le C(n,m,z)
\left\{
\begin{aligned}
& \frac{1}{|x|^{n-1}}
&\text{if $|x|\le x_0(n,m,z)$}
\\
& |x|^{-(n-1)/2} e^{-\Re k(z) |x|}
&\text{if $|x|\ge x_0(n,m,z)$}
\end{aligned}
\right.
\end{equation*}
for some positive constants $C(n,m,z)$, $x_0(n,m,z)$ depending on $z$. Hence is clear that $\K(z,x) \in L^{2q}(\R^n; \C^N)$ for $2q<n/(n-1)$ and, consequently, from equation \eqref{HSnorm} we have that $A(\D_0-z)^{-1}$ is an Hilbert-Schmidt operator if $V \in L^p(\R^n;\C^N)$ for $p>n/2$.
Since, by \eqref{YinLn}, $V \in L^n(\R^n;\C^N)$, the proof of the identity $\sigma_e(\D_V)=\sigma_e(\D_0)$ is complete.

Finally, to get the absence of residual spectrum, since $\rho(\D_0)=\C\setminus\sigma_e(\D_0)$ is composed by one, or two in the massless case, connected components which intersect $\rho(\D_V)$ in a non-empty set, by Theorem XVII.2.1 in \cite{GGK} we have
$
\sigma(\D_V) \setminus \sigma_e(\D_V) = \sigma_d(\D_V).
$

\section*{Acknowledgement}
The authors are members of the Gruppo Nazionale per L'Analisi Matematica, la
Probabilit\`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di
Alta Matematica (INdAM).

The third author is partially supported by \textit{Progetti per Avvio alla Ricerca di Tipo\til1 -- Sapienza Universit\`a di Roma}.

\bibliographystyle{plain}


TITLE: The meeting of Cars
QUESTION [2 upvotes]: Three cars, A, B and C move towards north in a particular straight track (consider the length of the tract infinite). Another car D comes from a certain distance towards south. The car A meets B at 8 am., meets C at 9 am. and crosses D at 10 am. Consequently, D crosses B at 12 pm. and crosses C at 02 pm. Find the exact time when B crossed C. (Consider cars as point masses with constant velocity, refer to the following image as their initial position)

One trivial solution to this problem done by me is to draw a distance time graph of these cars and take the meeting point of any two lines for two different cars as their meeting time in respect to the time axis. (I took the origin as 8 am. and plotted the given meeting points to find the meeting points of the graph of car B and car C). My answer was 10:40 am, and it seems to be correct. But I want a stronger logic and proof (may be algebraic). Let's see a newer proof can arise or not!

REPLY [1 votes]: Since we're only interested in the time at which events happen, we may as well analyze things from car D's point of view, which is to say, may may as well assume that car D is stationary.  Since from the moment that car A meets car B it takes A $2$ hours to reach D whereas it takes B $4$ hours, we infer that car A is traveling twice as fast as B (in car D's reference frame).  Likewise, from the moment car A meets car C, it takes A only $1$ hour to reach D whereas it takes C $5$ hours, so we can infer that car is is traveling five times as fast as C.
From this we conclude that car B is traveling two and half times as fast as car C.  Put differently, car C is traveling at 40% of car B's speed.
Now let's run time backwards, with C departing from D and then B departing from D two hours "later."  In the time it takes B to catch up (or should we say "down"?) with C, C will have traveled 40% as far as B, which means that in the two hours it spent on the road alone, it must have traveled 60% of the total distance.  There are six 20-minute intervals in two hours, so car C travels 10% of the total distance every 20 minutes, which means it's been on the road for 200 minutes when B catches it.  Keeping in mind that C "started" at 2pm and we're running time backwards, 200 minutes (i.e., 3 hours and 20 minutes) translates into 10:40am.


TITLE: Show that the following process based on exponential waiting time is Markov wrt natural filtration
QUESTION [1 upvotes]: For a parameter $\lambda>0$ consider an exponential random variable: $T\sim \operatorname{Exp}(\lambda)$ and define a process $X_{t}=(0)1_{T> t}+(t-T)1_{T\leq t}$. Show that $X_{t}$ is $\mathcal{F}_{t}^{X}$ - Markov process with transition function:
$P_{t}(0,A)=\exp(-\lambda t)1_{A}(0)+\int_{0}^{t}1_{A}(t-s)\lambda\exp(-\lambda s)ds$
and for $x>0$:
$P_{t}(x,A)=1_{A}(x+t)$
Basically all I need to show is that for $t,h\geq 0, \; A \in \mathcal{B}((0,\infty))$ and $n\in \mathbb N$ with $0=t_{0}<...<t_{n}=t$ and $A_{i}\in \mathcal{B}((0,\infty))$ for all $i = 0,...,n$ that:
$E[1_{\{X_{t+h}\in A\}}\prod\limits_{i=0}^{n}1_{\{X_{t_{i}}\in A_{i}\}}]=E[P_{h}(X_{t},A)\prod\limits_{i=0}^{n}1_{\{X_{t_{i}}\in A_{i}\}}]\; (*)$
Now in order to evaluate $(*)$ I will consider the two cases $\{T\leq t\}$ and $\{T> t\}$, thus considering
$E[P_{h}(X_{t},A)\prod\limits_{i=0}^{n}1_{\{X_{t_{i}}\in A_{i}\}},T\leq t]=E[1_{A}(X_{t}+h)\prod\limits_{i=0}^{n}1_{\{X_{t_{i}}\in A_{i}\}},T\leq t]=E[1_{A}(t+h-T)\prod\limits_{i=0}^{n}1_{\{X_{t_{i}}\in A_{i}\}},T\leq t]=E[1_{A}(X_{t+h})\prod\limits_{i=0}^{n}1_{\{X_{t_{i}}\in A_{i}\}},T\leq t]$ and so we are done in the case that $T\leq t$. But now in the case that $T> t$:
$E[P_{h}(X_{t},A)\prod\limits_{i=0}^{n}1_{\{X_{t_{i}}\in A_{i}\}},T> t]=E[P_{h}(0,A)\prod\limits_{i=0}^{n}1_{\{X_{t_{i}}\in A_{i}\}},T> t]=\prod\limits_{i=0}^{n}\delta_{0}(A_{i})P_{h}(0,A)\mathbb P(T> t)$
So I supppose this problem would reduce itself to showing that $P_{h}(0,A)\mathbb P(T> t)=1_{A}(X_{t+h})$, so now clearly:
$P_{h}(0,A)\mathbb P(T> t)=\exp(-\lambda t)P_{h}(0,A)=\exp(-\lambda t)(\exp(-\lambda h)1_{A}(0)+\int_{0}^{h}1_{A}(h-s)\lambda\exp(-\lambda s)ds)$
This is where I am unsure of how to proceed, to show that the above is equal to $1_{A}(X_{t+h})$, any ideas?

REPLY [3 votes]: I think there is a typo in your formula. I feel like we should have $P_t(0,A)=e^{\lambda t}1_A(0)+\int_0^t1_A(t-s)e^{\lambda s}\textrm{d}s$ (or, indeed, the calculation below should show it!).
Well, $P_h(0,A) \mathbb{P}(T>t)$ is definitely equal to neither $0$ nor $1$, so that's not gonna be equal to an indicator function. However, we have
\begin{align}
&\mathbb{P}[X_{t+h}\in A, X_i\in A_i\; \forall i, T>t]=\mathbb{P}[X_{t+h}\in A, 0\in A_i \; \forall i, T>t]\\
=&\mathbb{E}[1_A(X_{t+h}) 1_{T>t} \prod_i 1_{A_i}(0)]=\prod_{i} 1_{A_i}(0) \mathbb{E}[1_A(X_{t+h}) 1_{T>t}]
\end{align}
And, furthermore,
\begin{align}
\mathbb{E}[1_A(X_{t+h}) 1_{T>t}] = & \mathbb{E}[ 1_A(t+h-T)1_{T\in (t,t+h]}+1_A(0)1_{T\geq t+h}] \\
=& \int_0^{\infty} e^{-\lambda s} \left(1_A(t+h-s)1_{[t,t+h]}(s)+1_A(0)1_{(t+h,\infty)}(s)\right)\textrm{d}s\\
=& \int_{t}^{t+h} e^{-\lambda s} 1_A(t+h-s)\textrm{d}s+\int_{t+h}^{\infty} e^{-\lambda s}1_A(0)\textrm{d}s \\
=& e^{-\lambda t}\int_0^h e^{-\lambda s}1_A(h-s)\textrm{d}s+1_A(0) e^{-\lambda(t+h)}\\
=& P_h(0,A)e^{-\lambda t}\\
=& P_h(0,A)\mathbb{P}(T>t),
\end{align}
which should finish the proof for you.


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\numberwithin{equation}{section}

\title[Perturbation of Baum-Bott residues]{Perturbation of Baum-Bott residues}
\author[F. Bracci]{Filippo Bracci}
\author[T. Suwa]{Tatsuo Suwa}
\address{F. Bracci: Dipartimento Di Matematica\\
Universit\`{a} di Roma \textquotedblleft Tor Vergata\textquotedblright\ \\
Via Della Ricerca Scientifica 1, 00133 \\
Roma, Italy} \email{fbracci@mat.uniroma2.it}
\address{T. Suwa: Department of Mathematics\\
Hokkaido University\\
Sapporo 060-0810\\
Japan} \email{tsuwa@sci.hokudai.ac.jp}

\keywords{Holomorphic foliations; Baum-Bott residues;
deformations}

\thanks{Research partially supported by  a project PRIN2007 and a grant of JSPS}

\begin{abstract}
We prove that Baum-Bott residues vary continuously under smooth
deformations of holomorphic foliations. This provides an
effective way to compute residues.
\end{abstract}

\maketitle

\section{Introduction}

A holomorphic foliation $\mF$ on a complex manifold $M$ is
known to produce a ``holomorphic action'', as discovered by P.
Baum and R. Bott in \cite{BB2}, on the virtual bundle $TM/\mF$.
Such a partial holomorphic action provides a holomorphic
connection for the bundle $TM/\mF$ along $\mF$ outside the
singularities of $\mF$ and thus produces localization of
sufficiently high degree classes of $TM/\mF$ around the
singularities of $\mF$. Such localizations are called
``Baum-Bott residues'' (see \cite[Thm. 2]{BB2}, \cite[Ch.VI,
Thm. 3.7]{Su2}). When the singularity is isolated the Baum-Bott
residue can be expressed in terms of a Grothendieck residue
(see \cite[(0.6)]{BB2}). When the singular set is non-isolated
in some cases some formulas are available (see \cite[Thm.
3]{BB2} and \cite{D}) but, in general, explicit computation of
the residues is rather difficult.

The aim of the present paper is to study the behavior of the
Baum-Bott residues under smooth deformations. This provides an
effective tool for computing residues  explicitly.

More in details, we consider a smooth deformation of a complex
manifold. This is essentially a smooth fibration over a smooth
manifold, whose fibers are complex manifolds (see Section
\ref{def}). On each such a fiber we consider a holomorphic
foliation which varies smoothly  (see Section \ref{def-fol}).
We prove that the Baum-Bott residues (when taken together
suitably) vary continuously under smooth deformations.

We state here a simple consequence of our main Theorem
\ref{main} for the case of classes of top degree, referring the
reader to Section \ref{Bottres} for the general case. Thus, let
$P$ be a real manifold, the ``parameter space''. Let
$\tM:=\{M_t\}_{t\in P}$, be a deformation of complex manifolds
of dimension $n$. Let $\tmF:=\{\mF_t\}$ be a deformation of
holomorphic foliations on $M_t$. Then $\tmF$ defines naturally
a smooth foliation on $\tM$ (see Section \ref{def-fol}).

Suppose the singular set $S_{t_0}$ of $\mF_{t_0}$ in $M_{t_0}$
is compact and connected. The analytic set $S_{t_0}$ is
contained in a connected component in $\tM$ of the singular set
of the smooth foliation $\tmF$, and we denote by $S_t$ the
intersection of such component with $M_t$. The set $S_t$ is
contained in the singular set of $\mF_t$ but in general may not
be connected. Thus, we let $S_t=\cup S_t^\lambda$ be the
connected components decomposition of $S_t$.

\begin{theorem}\label{main-intro}
Suppose that $S_t$ is compact for all $t\in P$. Let $\varphi$
be a homogeneous symmetric polynomial of degree $n$ and denote
by $\hbox{BB}_\va(\mF_t;S^\lambda_t)$ the Baum-Bott residue of
$\mF_t$ at $S^\lambda_t$. Then
\[
\lim_{t\to t_0}\sum_\lambda \hbox{BB}_\va(\mF_t;S^\lambda_t)
=\hbox{BB}_\va(\mF_{t_0};S_{t_0}).
\]
\end{theorem}

A general version of the previous theorem is Theorem
\ref{main}, whose proof is contained in Sections \ref{Bottvan}
and \ref{Bottres}. The rough idea of the proof is to define a
special connection on the regular part of the bundle
$T\tM/\tmF$ such that on each fiber $M_t$ defines the special
connection given by the Baum-Bott action and see the residues
as the integral of a smooth form on $\tM$ along the fibers.

In Section \ref{examples} we give an explicit example of the
previous result. In particular, aside from explicit
computation, the example shows that if the residues in the same
connected component of $\tM$ are not taken together, continuity
is lost.

\medbreak

Part of this work was done while the first named author was
visiting the University of Tokyo. We would like to thank Prof.
J. Noguchi for providing us inspiring environment for research.

\section{Deformation of manifolds}\label{def}

The theory of deformation of complex structures was first
systematically developed by K. Kodaira and D. C. Spencer
\cite{KS}, here we recall the basic material relevant for our
needs.

\begin{definition}
A {\sl  deformation of manifolds} is a triple $(\tM, P, \pi)$,
where $P$ is a $C^\infty$ manifold of real dimension $s$,
called the {\sl parameter space},  $\tM$ is  a $C^\infty$
manifold of real dimension $2n+s$, called the {\sl ambient
manifold}, and $\pi: \tM\to P$ is a surjective $C^\infty$ map
such that there exists a covering $\{U_\al\}$ (called an {\sl
adapted deformation coordinates covering} of $\tM$) with the
following properties:
\begin{enumerate}
\item for each $\al$, the open set $U_\al$ is   diffeomorphic
to $D\times V$, where $D$ is an open set of $\C^n$ and $V$
is an open set of $\R^s$, with coordinates
$(z^\al_1,\ldots, z^\al_n, t_1^\al,\ldots, t_s^\al)$,
\item $\pi(U_\al)$ is diffeomorphic to $V$ and $\pi$ is
compatible with the projection $D\times V\to V$,
\item on $U_\al\cap U_\beta\neq \emptyset$ we may express as
\begin{equation}\label{cambio}
\begin{cases}
z^\beta_i=z^\beta_j(z^\al, t^\al)\quad i=1,\ldots, n\\
t^\beta_j=t^\beta_j(t^\al)\quad j=1,\ldots, s
\end{cases}
\end{equation}
and, for each fixed $t^\al$, the map $z^\al\mapsto
z^\beta(z^\al, t^\al)$ is holomorphic.
\end{enumerate}
\end{definition}

For $t\in P$ we let $M_t:=\pi^{-1}(t)$ be the fiber over $t$.
By definition the fibers $M_t$, for $t\in P$, are complex
manifolds. In particular we can define the sheaf $\OM$ of
$C^\infty$ functions on $\tM$ such that $f\in \OM(U)$ if for
all $x\in U$, $f|_{\pi^{-1}(\pi(x))}  \in
\Ol_{\pi^{-1}(\pi(x))}(U\cap {\pi^{-1}(\pi(x))})$.

\begin{remark}
Let $U_\al\subset \tM$ be a coordinate chart of an adapted
coordinate covering for $\tM$. A function $f$ belongs to
$\OM(U_\al)$ if and only if $f(z_\al, t_\al)$ is a $C^\infty$
function such that $f(\cdot, t_\al)$ is holomorphic (note that
this
 is well defined by \eqref{cambio}).
\end{remark}

\begin{definition}
Let $E$ be a $C^\infty$ complex vector bundle of rank $r$ over
$\tM$. We say that $E$ is an {\sl $\OM$-(vector) bundle} if
there exists a trivializing atlas $\{U_\al\}$ for $E$, with
frames $\{e_1^\al, \ldots, e_r^\al\}$ for $E|_{U_\al}$,  such
that the transition matrices with respect to those frames have
entries which are  local sections of $\OM$. Such frames
$\{e_1^\al, \ldots, e_r^\al\}$ are called {\sl $\OM$-frames}.
\end{definition}

Given an $\OM$-bundle $E$, we denote by $\OM(E)$ the
$\OM$-module of $\OM$ sections of $E$. Namely, $s\in \OM(E)(U)$
is a $C^\infty$ section of $E$ over the open set $U\subset \tM$
such that in any $\OM$-frame $\{e_1^\al, \ldots, e_r^\al\}$
over $U_\al$ with $U_\al\cap U\neq \emptyset$ the section $s$
is given by
\[
s(z^\al, t^\al)=\sum_{j=1}^r f^\al_j(z^\al, t^\al) e_j^\al, \quad f^\al_j \in \OM(U_\al\cap U).
\]

Let $T_{\R}\pi:=\ker \pi_\ast$. Since the fibers of the
fibration $\pi: \tM \to P$ are holomorphic, we can define the
complex vector bundles
\[
T\pi:=\bigcup_{x\in \tM} T_x \pi^{-1}(\pi(x)), \quad \overline{T\pi}:=\bigcup_{x\in \tM} \overline{T_x \pi^{-1}(\pi(x))}.
\]
Local frames for $T\pi$ and $\overline{T\pi}$ in an adapted
deformation coordinates covering are given respectively by
$\{\frac{\de}{\de z^\al_j}\}_{j=1,\ldots, n}$ and
$\{\frac{\de}{\de \overline{z}^\al_j}\}_{j=1,\ldots, n}$ and
\[
T_{\R}\pi\otimes \C=T\pi \oplus \overline{T\pi}.
\]
Using an adapted deformation coordinates covering, by
\eqref{cambio}, it is easy to see that $T\pi$ is an $\OM$-vector
bundle over $\tM$. Moreover, it has a natural structure of {\sl
$\OM$-Lie algebra}, namely, using local coordinates, one can
easily see that if $v, w \in \OM(T\pi)(U)$ then
\[
[v,w]\in \OM(T\pi)(U).
\]

\section{Deformation of foliations}\label{def-fol}

Deformations of holomorphic foliations, especially from the
point of view of moduli spaces, have been studied by a number
of authors ({\sl e.g.} \cite{G}, \cite{P}, \cite{R}). Here we
consider  $C^\infty$ families of singular holomorphic
foliations.

Let ${\mathcal S}$ be an $\OM$-module. We say that ${\mathcal
S}$ is {\sl coherent}   if, for each point $x\in \tM$ there
exists an open neighborhood $U\subset \tM$ and two integers
$p,q \geq 0$ such that
\begin{equation}\label{coer-S}
\OM|_U^p\stackrel{\va}{\longrightarrow} \OM|_U^q\longrightarrow {\mathcal S}|_U\to 0,
\end{equation}
is an exact sequence of $\OM|_U$-modules.

\begin{definition}
Let $(\tM, P, \pi)$ be a deformation of manifolds . A coherent
$\OM$-submodule $\tmF$ of $\OM(T\pi)$ such that $[\tmF,
\tmF]\subset \tmF$ is called a {\sl deformation of foliations}.
\end{definition}

Given a deformation of foliations $\tmF$ on a deformation of
manifolds $(\tM, P, \pi)$, we denote by ${\mathcal C}^\infty_P$
the sheaf of germs of complex valued smooth functions on $P$,
and for each $t\in P$, by $\mI_t:=\{f\in {\mathcal
C}^\infty_{P} : f(t)=0\}$ the ideal sheaf of smooth functions
vanishing at $t$. The set $\mR:=\pi^\ast{\mathcal
C}^\infty_{P}$  is the sheaf of smooth functions on $\tM$ that
are constant along the fibers, and it is naturally a subsheaf
of $\OM$. Noting that $\mR/\pi^*\mI_t$ is supported on
$M_t=\pi^{-1}(t)$, we define
\[
\mF_{t}:=\tmF\otimes_{\mR} \mR/\pi^\ast\mI_t.
\]
Note that $\OM\otimes_{\mR} \mR/\pi^\ast\mI_t=\Ol_{M_t}$, the sheaf of holomorphic
functions on $M_t$. Hence, if $\mathcal E$ is an $\OM$-module
over $\tM$, then $\mathcal E\otimes_{\mR} \mR/\pi^\ast\mI_t$ is an $\Ol_{M_t}$-module over $M_t$.

In particular, the sheaf $\mF_t$ is an $\Ol_{M_t}$-module. In
adapted deformation coordinates, if $X_1,\ldots, X_r$ are local
generators of $\tmF$, given by
\[
X_j(z^\al, t^\al)=\sum f_{ij}(z^\al, t^\al)\frac{\de}{\de z^\al_i},
\]
then $\mF_{t_0}$ is locally generated by the $X_j(z^\al,
t_0^\al)$'s. Namely it is generated by the vector fields
\begin{equation}\label{gen-F}
X_j(z^\al, t_0^\al)=\sum f_{ij}(z^\al,t_0^\al)\frac{\de}{\de z^\al_i}
\end{equation}
obtained by evaluating $f_{ij}(z^\al,t^\al)$ at $t=t_0$. From
this remark, it follows easily:

\begin{lemma}
For all $t\in P$, the sheaf $\mF_t$ defines a holomorphic
foliation on $M_t$.
\end{lemma}

We have the following exact sequence of $\OM$-modules on $\tM$:
\begin{equation}\label{ex-OM}
0\longrightarrow \tmF \longrightarrow \OM(T\pi)\longrightarrow \mNtF \longrightarrow 0.
\end{equation}
The {\sl singular set} of $\tmF$ is by definition
\[
S(\tmF):=\{x \in \tM: {\mNtF}_{,x} \hbox{\ is not\ } {\mathcal O}_{\tM,x}-\hbox{free}\}.
\]

\begin{lemma}\label{lemmaA}
For each point $x\in \tM$ there exists an open neighborhood
$U\subset \tM$ and two integers $p,q \geq 0$ such that
\begin{equation}\label{coer-NF}
\OM|_U^p\stackrel{\va}{\longrightarrow} \OM|_U^q\longrightarrow \mNtF|_U\to 0,
\end{equation}
is an exact sequence of $\OM|_U$-modules. Moreover,
\[
S(\tmF)|_U=\{x\in U : \rk \va_x \hbox{\ is not maximal}\}.
\]
\end{lemma}

\begin{proof}
Since $\tmF$  is $\OM$-coherent and $\OM(T\pi)$ is $\OM$-locally free,  from
(\ref{ex-OM}) it follows that $\mNtF$ is $\OM$-coherent as well,
so that \eqref{coer-NF} holds. The final statement follows from
standard commutative algebra and \eqref{coer-NF}.
\end{proof}

\begin{lemma}\label{lemmaB}
For each  $t\in P$ such that $M_t\not\subset S(\tmF)$ the
following sequence of $\Ol_{M_t}$-modules over $M_t$ is exact:
\begin{equation}\label{long-Ex}
0\to \tmF\otimes_{\mR} \mR/\pi^\ast \mI_t
\to \OM(T\pi)\otimes_{\mR} \mR/\pi^\ast \mI_t\to
\mNtF\otimes_{\mR} \mR/\pi^\ast \mI_t \to 0.
\end{equation}
\end{lemma}

\begin{proof} Since taking tensor products is right exact, it suffices to prove that the second map
from the left is injective.

It  is  true on the stalk over
each $x\in M_t$ such that $x\not\in S(\tmF)$, since $\mN_{{\mathcal F},x}$ is ${\mathcal{\widetilde{O}}_{{\widetilde{M}},x}}$-free.
We  note that according
to Lemma \ref{lemmaA}, $S(\tmF)|_{U\cap M_t}=\{x\in U\cap M_t :
\rk \va_x \hbox{\ is not maximal}\}$. Hence, for $t$ fixed,
these equations give rise to an analytic subset $S(\tmF)\cap
M_t$ of $M_t$, provided $M_t\not\subset S(\tmF)$. As a
consequence, $S(\tmF)\cap M_t$ is thin in $M_t$.
This shows that, since $\tmF$ is a subsheaf of $\OM(T\pi)$, it is also true on the stalk
over $x\in  S(\tmF)\cap M_t$.
\end{proof}

For each $t\in P$ we have the following exact sequence of
$\Ol_{M_t}$-modules:
\begin{equation}\label{exNF}
0 \longrightarrow \mF_t \longrightarrow \Ol_{M_t}(T M_t) \longrightarrow \mN_{\mF_t}\longrightarrow 0.
\end{equation}

\begin{definition}
Let $t\in P$. If $M_t\subset S(\tmF)$, we let
$S(\mF_t):=M_t$. Otherwise we let
\[
S(\mF_t):=\{x \in M_t : \mN_{\mF_t,x} \hbox{\ is not } \Ol_{M_t}-\hbox{free}\}.
\]
\end{definition}

\begin{proposition}\label{singu}
For all $t\in P$ it holds
\[
S(\mF_t)=S(\tmF)\cap M_t.
\]
\end{proposition}

\begin{proof}
If $M_t\subset S(\tmF)$ there is nothing to prove.

Thus, assume $M_t\not\subset S(\tmF)$. Since
$\OM(T\pi)\otimes_{\mR} \mR/\pi^\ast\mI_t=\Ol_{M_t}(TM_t)$, comparing
\eqref{long-Ex} and \eqref{exNF} we see that
\begin{equation}\label{ugua}
\mN_{\mF_t}=\mNtF\otimes_{\mR} \mR/\pi^\ast \mI_t,
\end{equation}
from which the statement follows at once.
\end{proof}

\section{Relative Bott vanishing for a deformation of
foliations}\label{Bottvan}

In this section we discuss a  Bott type vanishing theorem for
deformations of foliations. Thus, we let  $(\tM, P, \pi)$ be a
deformation of manifolds  and $\tmF$ a deformation of foliations on
$\tM$. In this section we assume
\[
S(\tmF)=\emptyset
\]
so that there exists an $\OM$-subbundle $\tilde F$ of $T\pi$ such that $\tmF=\OM(\tilde F)$.

We refer to \cite{BB2} for the notion of partial connections (see also \cite{ABST}, \cite{ABT}, \cite{Su2}).
As an example, given an $\OM$-bundle $E$ over $\tM$, we can define a ``relative
$\debar$-connection'' for $E$ along $\overline{T\pi}$ as
follows. We let
\[
\debar_E : C^\infty_{\tM}(E) \to C^\infty_{\tM}(\overline{T^\ast\pi}\otimes E),
\]
imposing that, given an $\OM$-frame
$\{\sigma_1^\al,\ldots,\sigma_r^\al\}$, and a $C^\infty$
section of $E$, $\sigma^\al:=\sum f_j^\al \sigma_j^\al$, it
holds
\[
\debar_E (\sigma^\al)=\sum_{j=1}^r\sum_{k=1}^n \frac{\de f_j^\al}{\de \overline{z}^\al_k}d\overline{z}^\al_k \otimes \sigma^\al_j.
\]
Since the transition matrices for $E$ with respect to
$\OM$-frames contains only entries in $\OM$, it is easy to see
that such a definition is well given and it is a partial
connection for $E$ along $\overline{T\pi}$.

\begin{definition}
Let $E$ be an $\OM$-bundle over $\tM$ and let $\mE$ be the sheaf
of its $\OM$-sections. A {\sl flat partial $\OM$-connection}
for $E$ along $\tmF$ is a $\C$-linear map
\[
\delta: \mE\to\tmF^\ast \otimes \mE
\]
with the properties that for all $X\in \tmF$, $f,g\in \OM$ and
$\sigma\in \mE$
\[
\delta_{(f X)}( g\sigma)=f\left(g\delta_X (\sigma)+ dg(X)\sigma\right)
\]
and
\[
\delta_X \circ \delta_Y=0, \quad \forall X,Y \in \tmF.
\]
\end{definition}

If $\delta$ is as above, it induces a ($C^\infty$) partial connection
\[
\delta : C^\infty_{\tM}(E) \to C^\infty_{\tM}( F^\ast\otimes E)
\]
such that, for $X\in \tmF$ and
$\sigma\in \mE$, we have $\delta_X(\sigma)\in \mE$.
Thus
\[
\delta\oplus\bar\partial_E : C^\infty_{\tM}(E) \to C^\infty_{\tM}( (F^\ast\oplus \overline{T^\ast\pi})\otimes E)
\]
is a partial connection. We say that a connection $\nabla:
C^\infty_{\tM}(E) \to
C^\infty_{\tM}((T^\ast\tM\otimes\C)\otimes E)$ extends
$\delta\oplus\bar\partial_E$ if
$\nabla_X=(\delta\oplus\bar\partial_E)_X$ for all sections $X$
of $F\oplus \overline{T\pi}$. Such a connection $\nabla$ always
exists (cf. \cite{BB2}).

We have the following ``relative Bott vanishing'' theorem for
actions of deformations of foliations:

\begin{theorem}\label{relative Bott vanishing}
let  $(\tM, P, \pi)$ be a deformation of manifolds  and $\tmF$ a
deformation of foliations on $\tM$ of rank $p$. Assume that
$S(\mF)=\emptyset$. Let $\mE$ be the sheaf of $\OM$-sections of
an $\OM$-bundle $E$ over $\tM$. Assume there exists a flat
partial $\OM$-connection $\delta$ for $\mE$ along $\tmF$. Then, for  any connection $\nabla$ for $E$ extending $\delta\oplus\bar\partial_E$, denoting by
$\iota_t: M_t \hookrightarrow \tM$ the natural embedding, it
follows
\[
\iota_t^\ast (\varphi(\nabla))=0,
\]
for all $t\in P$ and all symmetric homogeneous polynomials
$\varphi$ of degree $d>n-p$.
\end{theorem}

\begin{proof}
Let $\tF$ be the $\OM$-bundle whose associated sheaf of
sections is $\tmF$. Write
\[
T \tM\otimes \C= \tF \oplus F_1\oplus \overline{T\pi} \oplus \pi^\ast(TP),
\]
where $F_1$ is any $C^\infty$ complement of $\tF$ in $T\pi$.

Let $K$ be the curvature  of $\nabla$. Let $\{s_1, \ldots,
s_p\}$ be a local $\OM$-frame for $\tF$, and $\{\frac{\de}{\de
\overline{z}_1},\ldots, \frac{\de}{\de \overline{z}_n}\}$ the
natural frame for $\overline{T\pi}$ in adapted deformation
coordinates. Since $\tF$ is an $\OM$-subbundle of $T\pi$, we
can write $s_j=\sum_{k=1}^n a_k(z,t)\frac{\de}{\de z_k}$ for
$j=1,\ldots, p$ and $a_k\in \OM$.  Hence, $[s_j,\frac{\de}{\de
\overline{z}_k}]=0$ for $j=1,\ldots, p$ and $k=1,\ldots, n$.

Arguing similarly as in the proof of \cite[Prop. 3.27]{BB2}
(see also \cite[Thm. 6.1]{ABT}) since $\OM$-sections of $E$
generate as $C^\infty_{\tM}$-module the sheaf of
$C^\infty$-sections of $E$, one can see that
\[
K(s_j,s_k)=K(s_j,\frac{\de}{\de \overline{z}_h})=K(\frac{\de}{\de \overline{z}_h}, \frac{\de}{\de \overline{z}_l})=0
\]
for all $j,k=1,\ldots, p$ and $h,l=1,\ldots, n$. For instance,
given $\sigma$ an $\OM$-section of $E$, we have
\[
K(s_j,\frac{\de}{\de \overline{z}_h})(\sigma)=\nabla_{s_j}(\nabla_{\frac{\de}{\de \overline{z}_h}} \sigma)-
\nabla_{\frac{\de}{\de \overline{z}_h}}(\nabla_{s_j} \sigma)-\nabla_{[s_j,\frac{\de}{\de \overline{z}_h}]}\sigma=0,
\]
because $\nabla_{\frac{\de}{\de \overline{z}_h}}
\sigma=(\debar_E)_{\frac{\de}{\de \overline{z}_h}}\sigma=0$ by
definition, since $\sigma$ is an $\OM$-section; $\nabla_{s_j}
\sigma$ is another $\OM$-section of $E$, hence
$\nabla_{\frac{\de}{\de \overline{z}_h}}(\nabla_{s_j}
\sigma)=(\debar_E)_{\frac{\de}{\de
\overline{z}_h}}(\nabla_{s_j} \sigma)=0$ and
$[s_j,\frac{\de}{\de \overline{z}_h}]=0$.

As a consequence, the entries of the matrix representing $K$
are $2$-forms belonging to the ideal generated by a dual basis
of $F_1$ (which has dimension $n-p$) and by $dt_1,\ldots,
dt_s$, where these latter are a basis of $\pi^\ast(T^\ast P)$.
Therefore, if $\varphi$ has degree greater than $n-p$, it
follows that
\[
\varphi(\nabla)=\sum \omega_j \wedge dt_j,
\]
for some $2d-1$ forms $\omega_j$, hence,
$\iota^\ast(\varphi(\nabla))=0$.
\end{proof}

We recall that if $M$ is a complex manifold and $\mF$ is a
non-singular holomorphic foliation on $M$ then there exists a
natural holomorphic partial connection $\delta$ for the normal
bundle of the foliation $\mN_{\mF}$ along $\mF$ given by the so
called {\sl Baum-Bott action} (see \cite{BB2}, \cite{Su2}).
Such a connection is {\sl flat}, in the sense that $\delta\circ
\delta=0$. It is defined as follows:
\begin{equation}\label{Bott}
\delta_X(\sigma):=\rho([X,\tilde{\sigma}])
\end{equation}
where $\sigma\in \mN_{\mF}$ is a holomorphic section of the
normal bundle to the foliation, $\tilde{\sigma}\in \Ol_M(TM)$
is a holomorphic section of the tangent bundle to $M$ such that
$\rho(\tilde{\sigma})=\sigma$, where $\rho: \Ol_M(TM)\to
\mN_{\mF}$ is the natural projection, and $X\in \mF$.

We are going to show that a deformation of foliations gives
rise to a flat partial $\OM$-connection for $\mNtF$ along
$\tmF$ such that its ``restriction'' to each fiber $M_t$ is the
holomorphic flat partial connection for the normal bundle to
$\mF_t$ given by the Baum-Bott action:

\begin{proposition}\label{relative action}
Let  $(\tM, P, \pi)$ be a deformation of manifolds  and $\tmF$ a
deformation of foliations on $\tM$. Assume that
$S(\tmF)=\emptyset$. Then there exists a flat partial
$\OM$-connection $\tilde{\delta}$ for $\mNtF$ along $\tmF$.
Moreover, if $\iota_t: M_t\hookrightarrow \tM$ is the natural
embedding, then $\iota_t^\ast(\tilde{\delta})$ is the
holomorphic flat partial connection for $\mN_{\mF}$ along
$\mF_t$ given by the Baum-Bott action.
\end{proposition}

\begin{proof}
Let $\tilde{\rho}: \OM(T\pi)\to \mNtF$ be the natural
projection. For $X\in \tmF$ and $\sigma\in\mNtF$ we define
\begin{equation}\label{relative-Bott}
\tilde{\delta}_X(\sigma):=\tilde{\rho}([X,\tilde{\sigma}]),
\end{equation}
where $\tilde{\sigma}\in \OM(T\pi)$ is such that
$\tilde{\rho}(\tilde{\sigma})=\sigma$. Involutivity of $\tmF$
shows that $\tilde{\delta}$ is well-defined and  flatness
follows from the Jacobi identity, so that $\tilde{\delta}$ is a
partial $\OM$-connection  for $\mNtF$ along $\tmF$.

Comparing \eqref{relative-Bott} with \eqref{Bott}, it is easy
to see that $\iota^\ast_t(\tilde{\delta})$ is the  flat partial
$\Ol_{M_t}$-connection for $\mN_{\mF_t}$ along $\mF_t$ given by
the Baum-Bott action.
\end{proof}

In particular, Theorem \ref{relative Bott vanishing} and
Proposition \ref{relative action} imply the following:

\begin{corollary}\label{vBott}
Let  $(\tM, P, \pi)$ be a deformation of manifolds  and $\tmF$ a
deformation of foliations on $\tM$. Assume that
$S(\tmF)=\emptyset$. Then there exists a connection $\nabla$
for $\mNtF$ such that, denoting by $\iota_t: M_t
\hookrightarrow \tM$ the natural embedding, it follows
\[
\iota_t^\ast (\varphi(\nabla))=0,
\]
for all $t\in P$ and all symmetric homogeneous polynomials
$\varphi$ of degree $d>n-p$.
\end{corollary}

\section{Residues of Baum-Bott types on  deformations of manifolds}\label{Bottres}

In this section we assume $(\tM, P, \pi)$ is a deformations of
manifolds and $\tmF$ is a deformation of foliations on $\tM$.
We also assume that $\mNtF$ admits a {\sl $C^\infty$ locally
free resolution}, namely, there exists an  exact sequence of
${\mathcal C}^\infty_{\tM}$-modules:
\begin{equation}\label{fin-siz}
0\to \mE_q\to \cdots \to \mE_0 \to \mNtF\otimes_{\OM} {\mathcal C}^\infty_{\tM} \to 0,
\end{equation}
such that each $\mE_j$ is locally ${\mathcal
C}_{\tM}^\infty$-free.

\begin{remark}
If $\tmF$ is locally $\OM$-free then such a condition is
satisfied with $q=1$ and $\mE_1=\tmF\otimes_{\OM} {\mathcal
C}^\infty_{\tM} $, $\mE_0=\OM(T\pi)\otimes_{\OM} {\mathcal
C}^\infty_{\tM} $.
\end{remark}

Let $E_j$ be the vector bundle over $\tM$ whose sheaf of
$C^\infty$ sections is $\mE_j$. Then $\mNtF$ is a virtual bundle in the
$K$-group $K(\tM)$ and its total Chern class is defined as
\[
c(\mNtF)=\prod_{i=0}^q c(E_i)^{(-1)^i}.
\]
We briefly sketch here the theory we need, and refer the reader
to \cite[Section 4]{BB2}, \cite{L} and \cite[Ch.II,  8]{Su2}
for details.

Let $\tU_1$ be an open neighborhood of $S(\tmF)$ and let
$\tU_0:=\tM\setminus S(\tmF)$. We denote by $(\nabla_0^\bullet,
\nabla_1^\bullet)$ the family of $q+1$ connections compatible
with \eqref{fin-siz} and adapted to the covering ${\mathcal
\tU}:=\{\tU_0, \tU_1\}$ of $\tM$. Namely,
$\nabla_l^\bullet=(\nabla_l^q,\ldots, \nabla_l^0)$, $l=0,1$ is
a family
 such that $\nabla^j_l$ is a connection for
$E_j|_{\tU_l}$, $j=0,\ldots, q$, $l=0,1$ and the following
diagram is commutative for $i=1,\ldots, q$ and $l=0,1$:
\begin{equation}
\begin{CD}
 & E_i|_{\tU_l} @>\nabla_l^i>> C^\infty_{\tM}(T^\ast \tM \otimes E_i|_{\tU_l}) \\
 & @VVV  @VVV \\
 & E_{i-1}|_{\tU_l} @>\nabla_l^{i-1}>>
C^\infty_{\tM}(T^\ast \tM \otimes E_{i-1}|_{\tU_l})\\
\end{CD}
\end{equation}
Moreover, let $N_{\tF}$ be the vector bundle on $\tU_0$ whose
sheaf of sections is $\mNtF\otimes_{\OM} {\mathcal C}^\infty_{\tM} |_{\tU_0}$. Let $\nabla$ be an
extension of the  flat partial $\OM$-connection
$\tilde{\delta}$ for $\mNtF|_{\tU_0}$ along $\tmF$ given by
Proposition \ref{relative action}. It is then possible to
choose $\nabla_0^\bullet$ to be compatible with $\nabla$ (in
the sense explained before).

Now, we let $\varphi$ be a homogeneous symmetric polynomial of
degree $d>n-p$. One can define the class $\va(\mNtF )$ in the
\v{C}ech-de Rham cohomology \v{H}$^{2d}({\mathcal \tU})$ which
is represented by
\[
\va(\nabla_\ast^\bullet):=(\va(\nabla_0^\bullet),
\va(\nabla_1^\bullet), \va(\nabla_0^\bullet,
\nabla_1^\bullet)),
\]
where, by the compatibility condition,
$\va(\nabla_0^\bullet)=\va(\nabla)$ is a $2d$ form on $\tU_0$,
$\va(\nabla_1^\bullet)$ is the $2d$ form on $\tU_1$ associated
to the family $\nabla_1^\bullet$ and $\va(\nabla_0^\bullet,
\nabla_1^\bullet)$ is a $(2d-1)$-form on $\tU_0\cap \tU_1$ such
that $d \va(\nabla_0^\bullet,
\nabla_1^\bullet)=\va(\nabla_1^\bullet)-\va(\nabla_0^\bullet)$.
The \v{C}ech-de Rham cohomology \v{H}$^*({\mathcal \tU})$ is
naturally isomorphic to the de Rham cohomology $H^*_{dR}(\tM,
\C)$.

If $M_t\not\subset S(\tmF)$, tensorizing \eqref{fin-siz} with
$\mR/\pi^\ast \mI_t$ we obtain the following exact sequence of
${\mathcal C}^\infty_{M_t}$-modules (cf. the proof of Lemma
\ref{lemmaB}):
\begin{equation}\label{fin-sizF} 0\to \mE_q\otimes_{\mR} \mR/\pi^\ast \mI_t\to
\cdots \to \mE_0\otimes_{\mR} \mR/\pi^\ast \mI_t\to \mNtF\otimes_{\OM} {\mathcal C}^\infty_{\tM} \otimes_{\mR} \mR/\pi^\ast \mI_t \to 0,
\end{equation}
where $\mE_j\otimes_{\mR} \mR/\pi^\ast \mI_t$ is the sheaf of
$C^\infty$ sections of the restriction of the bundle $E_j$ to
$M_t$. By \eqref{ugua}, it is then easy to see the following:

\begin{lemma}
Let  $t\in P$ and let $\iota_t: M_t\to \tM$ be the natural
embedding. If $M_t\not\subset S(\tmF)$ then
$(\iota_t^\ast(\nabla_0^\bullet),\iota_t^\ast(\nabla_1^\bullet))$
is a family of connections for the virtual bundle $\mN_{\mF_t}$
compatible with \eqref{fin-sizF}.
\end{lemma}

By Corollary \ref{vBott} and by the compatibility condition, it
follows that for all homogeneous symmetric polynomials $\va$ of
degree $d>n-p$, the class $\va(\mN_{\mF_t})$ is represented in
the \v{C}ech-de~Rham cohomology associated to the covering
$\mathcal\tU\cap M_t$ of $M_t$ by the cocyle
\begin{equation*}
\begin{split}
 \va(\iota_t^\ast\nabla^\bullet_\ast)&=(\iota_t^\ast\va(\nabla_0^\bullet),
\iota_t^\ast\va(\nabla_1^\bullet), \iota_t^\ast\va(\nabla_0^\bullet,
\nabla_1^\bullet))=(\iota_t^\ast\va(\nabla),
\iota_t^\ast\va(\nabla_1^\bullet), \iota_t^\ast\va(\nabla_0^\bullet,
\nabla_1^\bullet))\\&=(0,
\iota_t^\ast\va(\nabla_1^\bullet), \iota_t^\ast\va(\nabla_0^\bullet,
\nabla_1^\bullet)).
\end{split}
\end{equation*}
Hence, since by Proposition \ref{singu}, $\tU_0\cap
M_t=M_t\setminus S(\mF_t)$, the previous cocycle defines a
localization of $\va(\mN_{\mF_t})$, call it $\va(\mN_{\mF_t},
\mF_t)$, in the relative \v{C}ech-de Rham cohomology
\v{H}$^{2d}(\mathcal\tU\cap M_t, M_t\setminus S(\mF_t))$. The
{\sl Baum-Bott residue}  is the image of $\va(\mN_{\mF_t},
\mF_t)$ by the Alexander homomorphism $A:
\hbox{\v{H}}^{2d}(\mathcal\tU\cap M_t, M_t\setminus
S(\mF_t))\to H_{dR}^{2n-2d}(\tU_1\cap M_t)^\ast$. If $S(\mF_t)$
is made of $k$ connected components than
$H_{dR}^{2n-2d}(\tU_1\cap M_t)^\ast$ is a direct sum of $k$
addends, and we can consider the Baum-Bott residue at each
connected component of $S(\mF_t)$.  If $\tU_1\cap M_t$ is a
regular neighborhood of $S(\mF_t)$ then the above residue can
be thought of as being in $H_{2n-2d}(S(\mF_t), \C)$.

Now, let  $S'(\tmF)\subseteq S(\tmF)$ be a connected component.
We assume that
\[
\hbox{$S_t:=M_t\cap S'(\tmF)$ is compact}\quad \forall t\in P.
\]
\begin{remark}
Note that even if $S(\tmF)$ is connected by assumption,
$S(\mF_t)$ might not.
\end{remark}
Let $\tilde{R}$ be a real manifold of dimension $2n+s$ with
boundary such that  $S'(\tmF)$ is contained in the interior of
$\tilde{R}$, no other components of $S(\tmF)$ intersect
$\tilde{R}$ and $\de \tilde{R}$ is transverse to $M_t$ for all
$t\in P$. Moreover, we can take $\tilde{R}$ in such a way that
$R_t:= \tilde{R}\cap M_t$ is compact for all $t\in P$.

 We let
$U_t:=\tU_1\cap M_t$ and denote by $H_{dR}^{\ast}(U_t)$ the de
Rham cohomology of $U_t$. By the previous construction, we can
express the {\sl Baum-Bott residue}
$\hbox{BB}_\va(\mF_t;S_t)\in H_{dR}^{2n-2d}(U_t)^\ast$ as
follows:
\begin{equation}\label{expre}
\hbox{BB}_\va(\mF_t;S_t): H_{dR}^{2n-2d}(U_t) \ni [\tau]\mapsto \int_{R_t} \iota_t^\ast\va(\nabla_1^\bullet)\wedge \tau -
\int_{\de R_t}\iota_t^\ast\va(\nabla_0^\bullet,\nabla_1^\bullet)\wedge \tau.
\end{equation}

\begin{remark}
Note that there exists a natural morphism
$H_{dR}^{2n-2d}(U_t)^\ast\rightarrow H_{dR}^{2n-2d}(M_t)^\ast$.
Therefore one can remove the dependence on $\tU_1$ in this
construction. Moreover, if $M_t$ is compact, then
$H_{dR}^{2n-2d}(M_t)^\ast=H_{2n-2d}(M_t)$.
\end{remark}

Now we are in good shape to prove our main result:

\begin{theorem}\label{main}
Let  $(\tM, P, \pi)$ be a deformation of manifolds  and $\tmF$
a deformation of foliations on $\tM$ of rank $p$. Suppose that
$\mNtF$ admits a  $C^\infty$ locally free resolution.  Let
$S'(\tmF)\subseteq S(\tmF)$ be a connected component of the
singular set of $\tmF$ and let $S_t:=M_t\cap S'(\tmF)$. Assume
that for all $t\in P$ the set $S_t$ is compact and $S_t\neq
M_t$. Let $\varphi$ be a homogeneous symmetric polynomial of
degree $d>n-p$. Under these assumptions, the Baum-Bott residue
$\hbox{BB}_\va(\mF_t;S_t)$ is continuous in $t\in P$. Namely,
for any $C^\infty$ $(2n-2d)$-form $\tilde{\tau}$ on $\tM$ such
that $\iota^\ast_t(\tilde{\tau})$ is closed for all $t\in P$,
\[
\lim_{t\to t_0}\hbox{BB}_\va(\mF_t;S_t)\left(\iota_t^\ast(\tilde{\tau})\right)
=\hbox{BB}_\va(\mF_{t_0};S_{t_0})\left(\iota_{t_0}^\ast(\tilde{\tau})\right).
\]
\end{theorem}

\begin{proof}
From the previous construction and \eqref{expre} it follows
that the Baum-Bott residues on $M_t$ are expressed by means of
smooth forms on $\tM$. Hence, they vary continuously.
\end{proof}

Note that, if $S_t$ is not connected and $S_t=\cup_\lambda
S_t^\lambda$ is its connected components decomposition, then
\[
\hbox{BB}_\va(\mF_t;S_t)=\sum_\lambda
\hbox{BB}_\va(\mF_t;S^\lambda_t).
\]

\section{An example}\label{examples}

In $\Po^3$ with homogeneous coordinates $[x_1:x_2:x_3:x_4]$ we
consider the vector field which is defined in the affine chart
$x_4\neq 0$  with coordinates $x=x_1/x_4, y=x_2/x_4, z=x_3/x_4$
by
\[
X(x,y,z):=x\frac{\de}{\de x}+x\frac{\de}{\de y}+y\frac{\de}{\de z}.
\]
The singularities are the line $L$ given by $x_1=x_2=0$ and the
point at infinity given by $Q:=[1:1:1:0]$ (see the next
expression \eqref{otherc}).

The vector field $X$ generates a one-dimensional foliation
$\mF$ given by $X: \Po^3\times \C \to T\Po^3$ on $\Po^3$. By
the Baum-Bott theorem, we can localize $\varphi(T\Po^3/\mF)$
for homogeneous symmetric polynomials $\varphi$ of degree $3$.
Such polynomials are essentially given by $c_1^3$, $c_1c_2$ and
$c_3$. Moreover, since $\mF$ is trivial, we obtain that
$\varphi(T\Po^3/\mF)=\varphi(T\Po^3)$. Let $\Ol(1)$ be the
hyperplane bundle on $\Po^3$ and let $\xi:=c_1(\Ol(1))\in
H^2_{dR}(\Po^3)$. From the Euler exact sequence, it follows
that $c(T\Po^3)=(1+\xi)^4$, from which
\begin{equation}\label{ctp}
\int_{\Po^3} c_1^3(T\Po^3)=64, \quad \int_{\Po^3} c_1c_2(T\Po^3)=24, \quad \int_{\Po^3} c_3(T\Po^3)=4.
\end{equation}
Changing coordinates, in the affine chart $x_3\neq 0$ with
coordinates $\tx=x_1/x_3, \ty=x_2/x_3, \tz=x_4/x_3$ the vector
field $X$ has the expression:
\begin{equation}\label{otherc}
X(\tx,\ty,\tz)=(\tx-\tx\ty)\frac{\de}{\de \tx}+(\tx-\ty^2)\frac{\de}{\de \ty}-\ty\tz \frac{\de}{\de \tz}.
\end{equation}
From this it follows that the first jet of $X$ at $Q$ is given
by the non-degenerate matrix
\[
A:=\left(
              \begin{array}{ccc}
                0 & -1 & 0 \\
                1 & -2 & 0 \\
                0 & 0 & -1 \\
              \end{array}
            \right).
\]
Hence since $Q$ is a non-degenerate isolated singularity for
$X$ it follows (see, {\sl e.g.} \cite[(0.7)]{BB2} or
\cite{Su2})
\begin{equation}\label{resi-nondeg}
\hbox{BB}_{\varphi}(X;Q)=\frac{\varphi(A)}{\det A},
\end{equation}
that is
\begin{equation}\label{expl-BB}
\hbox{BB}_{c_1^3}(X;Q)=27 \quad \hbox{BB}_{c_1c_2}(X;Q)=9 \quad \hbox{BB}_{c_3}(X;Q)=1.
\end{equation}
By the Baum-Bott theorem,
\[
\int_{\Po^3} \va(T\Po^3)=\hbox{BB}_{\varphi}(X;Q)+\hbox{BB}_{\varphi}(X;L).
\]
From this and by \eqref{ctp} and \eqref{expl-BB} we obtain
\begin{equation}\label{expl}
\hbox{BB}_{c_1^3}(X;L)=37 \quad \hbox{BB}_{c_1c_2}(X;L)=15 \quad \hbox{BB}_{c_3}(X;L)=3.
\end{equation}
However, computing such residues directly without using the
Baum-Bott theorem seem to be very complicated because the
singular set is not isolated.

We present now a deformation procedure which allows to compute
the previous residues and explain in practice how our Theorem
\ref{main-intro} works.

Let $\tM:=\Po^3\times (-1,1)$ and let  $\tmF$  be the
deformation of foliations defined by the vector fields $X_t$,
$t\in (-1,1)$, which on the chart $x_4\neq 0$ are defined as
\[
X_t(x,y,z)=(x+tz)\frac{\de}{\de x}+x\frac{\de}{\de y}+y\frac{\de}{\de z}.
\]
On the chart $x_3\neq 0$ the vector field $X_t$ is given by
\[
X(\tx,\ty,\tz)=(\tx-\tx\ty+t)\frac{\de}{\de \tx}+(\tx-\ty^2)\frac{\de}{\de \ty}-\ty\tz \frac{\de}{\de \tz}.
\]
The singularities of $X_t$ for $t\neq 0$ are given by
$O:=[0:0:0:1]$ and $P_j(t):=[u_{t,j}^2:u_{t,j}: 1:0]$ for
$j=1,2,3$, where the $u_{t,j}$'s are the three roots of the
equation $\lambda^3-\lambda^2-t=0$.

At the point $O$ the first jet of $X_t$, $t\neq 0$, is
non-degenerate and it is given by the matrix
\[
\left(
              \begin{array}{ccc}
                1 & 0 & t \\
                1 & 0 & 0 \\
                0 & 1 & 0 \\
              \end{array}
            \right).
\]
From this it and from \eqref{resi-nondeg},
\begin{equation}\label{XsO}
\hbox{BB}_{c_1^3}(X_t;O)=\frac{1}{t} \quad \hbox{BB}_{c_1c_2}(X_t;O)=0 \quad \hbox{BB}_{c_3}(X_t;O)=1.
\end{equation}

\begin{remark}
It is interesting to note that $\lim_{t\to
0}\hbox{BB}_{c_1^3}(X_t;O)=\infty$, namely, the residue by
itself is not continuous, but it is so when taken the sum of
the residues for the singularities which belong to the same
connected components in the ambient space $\tM$.
\end{remark}

At the point $P_j(t)$ the vector field $X_t$ has first jet
given by the matrix
\[
B(t,j):=\left(
              \begin{array}{ccc}
                1-u_{t,j} & -u_{t,j}^2 & 0 \\
                1 & -2u_{t,j} & 0 \\
                0 & 0 & -u_{t,j} \\
              \end{array}
            \right),
\]
with determinant $\det B(t,j)=  u_{t,j}^2(2-3 u_{t,j})$. Thus,
for $t\to 0$, $t\neq 0$ the points $P_j(t)$ are isolated
non-degenerate singularities for $X_t$ and one can use
\eqref{resi-nondeg} to compute the residues:
\begin{equation}\label{XsP}
\begin{split}
&\hbox{BB}_{c_1^3}(X_t;P_j(t))=\frac{(1-4u_{t,j})^3}{u_{t,j}^2(2-3 u_{t,j})} \quad \hbox{BB}_{c_1c_2}(X_t;P_j(t))=\frac{3(6u_{t,j}-1-8u_{t,j})}{u_{t,j}^2(2-3 u_{t,j})} \\& \hbox{BB}_{c_3}(X_t;P_j(t))=1.
\end{split}
\end{equation}

Now, as $t\to 0$, it follows that two of the roots of of the
equation $\lambda^3-\lambda^2-t=0$ tend to $0$ and one tends to
$1$. We assume that $u_{t,1}, u_{t,2}\to 0$ and $u_{t,3}\to1$.
Hence, if $S'(\tmF)$ is the connected component which contains
the line $L$ in the manifold deformation $M\times (-1,1)$, the
intersection of $S'(\tmF)$ with $M\times \{t\}$ is given by the
points $O, P_1(t), P_2(t)$. While, the connected component in
$M\times (-1,1)$ which contains $Q$ contains all the points
$P_3(t)$.

A direct computation---taking into account that
$u_{t,1}+u_{t,2}+u_{t,3}=1$,
$u_{t,1}u_{t,2}+u_{t,1}u_{t,3}+u_{t,2}u_{t,3}=0$ and
$u_{t,1}u_{t,2}u_{t,3}=t$---shows that for $\varphi=c_1^3,
c_1c_2, c_3$
\begin{equation*}
\begin{split}
&\lim_{t\to 0}\hbox{BB}_{\varphi}(X_t;P_3(t))=\hbox{BB}_{\varphi}(X;Q),\\
&\lim_{t\to 0}[\hbox{BB}_{\varphi}(X_t;P_1(t))+\hbox{BB}_{\varphi}(X_t;P_2(t))+\hbox{BB}_{\varphi}(X_t;O)]=\hbox{BB}_{\varphi}(X;L).
\end{split}
\end{equation*}

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