# The diagonal behaviour of the one-particle Coulombic density matrix

Peter Hearnshaw  and  Alexander V. Sobolev  Department of Mathematics
University College London
Gower Street
London
WC1E 6BT UK<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We obtain bounds for all derivatives of the non-relativistic Coulombic one-
particle density matrix $\gamma(x,y)$ near the diagonal $x=y$.

###### Key words and phrases:

Multi-particle system, Schrödinger equation, one-particle density matrix

###### 2010 Mathematics Subject Classification:

Primary 35B65; Secondary 35J10, 81V55

##  1\. Introduction

Consider on $\textup{{{L}}}^{2}(\mathbb{R}^{3N})$ the Schrödinger operator

$\displaystyle\mathcal{H}=$ $\displaystyle\
\mathcal{H}_{0}+V,\quad\mathcal{H}_{0}=-\Delta=-\sum_{k=1}^{N}\Delta_{k},$
(1.1) $\displaystyle V(\mathbf{x})=$ $\displaystyle\
-Z\sum_{k=1}^{N}\frac{1}{|x_{k}|}+\sum_{1\leq j<k\leq
N}\frac{1}{|x_{j}-x_{k}|},$

describing an atom with $N$ particles (e.g. electrons) with coordinates
$\mathbf{x}=(x_{1},x_{2},\dots,x_{N})$, $x_{k}\in\mathbb{R}^{3}$,
$k=1,2,\dots,N$, and a nucleus with charge $Z>0$. The notation $\Delta_{k}$ is
used for the Laplacian w.r.t. the variable $x_{k}$. The operator $\mathcal{H}$
acts on the Hilbert space $\textup{{{L}}}^{2}(\mathbb{R}^{3N})$ and by
standard methods one proves that it is self-adjoint on the domain
$D(\mathcal{H})=\textup{{{W}}}^{2,2}(\mathbb{R}^{3N})$, see e.g. [16, Theorem
X.16]. (Here and throughout the paper we use the standard notation
$\textup{{{W}}}^{l,p}$ for the Sobolev spaces, where $l$ and $p$ indicate the
smoothness and summability respectively). Our methods allow consideration of
the molecular Schrödinger operator, but we restrict our attention to the
atomic case for simplicity. Let $\psi=\psi(\mathbf{x})$, be an eigenfunction
of the operator $\mathcal{H}$ with an eigenvalue $E\in\mathbb{R}$, i.e.
$\psi\in D(\mathcal{H})$ and

$\displaystyle(\mathcal{H}-E)\psi=0.$

For each $j=1,\dots,N$, we represent

$\displaystyle\mathbf{x}=(x_{j},\hat{\mathbf{x}}_{j}),\quad\textup{where}\
\hat{\mathbf{x}}_{j}=(x_{1},\dots,x_{j-1},x_{j+1},\dots,x_{N}),$

with obvious modifications if $j=1$ or $j=N$. The one-particle density matrix
is defined as the function

(1.2)
$\displaystyle\tilde{\gamma}(x,y)=\sum_{j=1}^{N}\int\limits_{\mathbb{R}^{3N-3}}\psi(x,\hat{\mathbf{x}}_{j})\overline{\psi(y,\hat{\mathbf{x}}_{j})}\,d\hat{\mathbf{x}}_{j},\quad(x,y)\in\mathbb{R}^{3}\times\mathbb{R}^{3}.$

This function is one of the key objects in the multi-particle quantum
mechanics, see [4], [5], [14], [15] for details and futher references. If one
assumes that all $N$ particles are spinless fermions (resp. bosons), i.e. that
the function $\psi$ is antisymmetric (resp. symmetric) under the permutations
$x_{j}\leftrightarrow x_{k}$, then the definition (1.2) simplifies:

$\displaystyle\tilde{\gamma}(x,y)=N\int_{\mathbb{R}^{3N-3}}\psi(x,\hat{\mathbf{x}})\overline{\psi(y,\hat{\mathbf{x}})}d\hat{\mathbf{x}},\
\quad\textup{where}\ \hat{\mathbf{x}}=\hat{\mathbf{x}}_{1}.$

Our main result however does not require any symmetry assumptions. We are
interested in the smoothness properties of the function (1.2). It is clear
that for this purpose it suffices to study each term in (1.2) individually.
Moreover, using permutations of the variables it is sufficient to focus just
on one term on the right-hand side of (1.2):

(1.3)
$\displaystyle\gamma(x,y)=\int_{\mathbb{R}^{3N-3}}\psi(x,\hat{\mathbf{x}})\overline{\psi(y,\hat{\mathbf{x}})}\,d\hat{\mathbf{x}},\
\quad\hat{\mathbf{x}}=(x_{2},x_{3},\dots,x_{N}).$

Throughout the paper we refer to this function as the one-particle density
matrix. In [7] the one-particle density

(1.4)
$\displaystyle\rho(x)=\gamma(x,x)=\int_{\mathbb{R}^{3N-3}}|\psi(x,\hat{\mathbf{x}})|^{2}\,d\hat{\mathbf{x}},$

was shown to be a real-analytic function of $x\not=0$. The real analyticity of
the function $\gamma(x,y)$ as a function of two variables on the domain

(1.5) $\displaystyle\mathcal{D}=\\{(x,y):|x|\,|y|\not=0,\quad
x\not=y\\}\subset\mathbb{R}^{3}\times\mathbb{R}^{3},$

was proved in [11]. As was pointed out in [11], one cannot expect analyticity
in $x$ and $y$ to hold on the diagonal $x=y$. In fact, quantum chemistry
calculations in [2] (see also [3]) show that in the neighbourhood of the
diagonal $x=y\not=0$, the function $\gamma(x,y)$ has the following behaviour:

(1.6)
$\displaystyle\operatorname{{Re}}\gamma(x,y)-\gamma(x,x)=C(x)|x-y|^{5}+o(|x-y|^{5}),\quad\textup{as}\quad
y\to x,$

with some non-zero function $C(x)$. Motivated in part by the formula (1.6), in
the current paper we aim to obtain explicit bounds for all partial derivatives
of $\gamma(x,y)$ near the diagonal $x=y$ or the points $x=0$ and $y=0$. For
the derivatives of $\gamma$ we use the standard notation
$\partial_{x}^{m}\partial_{y}^{l}\gamma(x,y)$, where
$m,l\in\mathbb{N}_{0}^{3}$, $\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$. For two
non-negative numbers (or functions) $X$ and $Y$ depending on some parameters,
we write $X\lesssim Y$ (or $Y\gtrsim X$) if $X\leq CY$ with some positive
constant $C$ independent of those parameters. The notation $B(x,R)$ is used
for the open ball of radius $R$ centred at the point $x$.

For $b\geq 0$, $t>0$ we define

(1.7) $\displaystyle h_{b}(t)=\begin{cases}1,\ \textup{if}\ b<5,\\\
\log\big{(}t^{-1}+2),\ \textup{if}\ b=5,\\\ 1+t^{5-b},\ \textup{if}\
b>5.\end{cases}$

The next theorem contains the main result.

######  Theorem 1.1.

Let $\psi$ be an eigenfunction, and let $R>0$ be fixed. Then for all
$x\not=0,y\not=0$, $x\not=y$, and all $l,m\in\mathbb{N}_{0}^{3}$ such that
$|l|\geq 1,|m|\geq 1$, we have

$\displaystyle|\partial_{x}^{l}\partial_{y}^{m}\gamma(x,y)|\lesssim\big{(}1+|x|^{2-|l|-|m|}+|y|^{2-|l|-|m|}$
$\displaystyle+h_{|l|+|m|}(|x-y|)\big{)}$ (1.8) $\displaystyle\
\times\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(x,R))}\big{)}^{\frac{1}{2}}\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(y,R))}\big{)}^{\frac{1}{2}}.$

Furthermore, for all $|l|\geq 1$,

$\displaystyle|\partial_{x}^{l}\gamma(x,y)|+|\partial_{y}^{l}\gamma(x,y)|\lesssim\big{(}1+|x|^{1-|l|}+|y|^{1-|l|}$
$\displaystyle\ +h_{|l|}(|x-y|)\big{)}$ (1.9) $\displaystyle\
\times\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(x,R))}\big{)}^{\frac{1}{2}}\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(y,R))}\big{)}^{\frac{1}{2}}.$

The implicit constants in (1.1) and (1.1) may depend on $l$, $m$ and $R$, but
are independent of $x$, $y$ and the function $\psi$.

######  Remark 1.2.

  1. (1)

Theorem 1.1 naturally extends the case of a molecule with several nuclei whose
positions are fixed. The modifications are straightforward.

  2. (2)

Observe that
$\|\rho\|_{\textup{{{L}}}^{1}(B(x,R))}\leq\|\psi\|_{\textup{{{L}}}^{2}(\mathbb{R}^{3N})}^{2}$,
$R>0$, so that the right-hand sides of (1.1) and (1.1) are finite.

  3. (3)

The bound (1.1) for $|l|=1$ implies that $\gamma(x,y)$ is a Lipschitz function
on $\mathbb{R}^{3}\times\mathbb{R}^{3}$.

  4. (4)

The bounds (1.1), (1.1) ensure that
$\gamma\in\textup{{{W}}}^{5,p}_{\textup{\tiny loc}}(\mathcal{D})$ with
arbitrary $p<\infty$, see (1.5) for the definition of $\mathcal{D}$. By
Proposition 7.1 in the Appendix, this implies that
$\gamma\in\textup{{{W}}}^{5,p}_{\textup{\tiny
loc}}\big{(}(\mathbb{R}^{3}\setminus\\{0\\})\times(\mathbb{R}^{3}\setminus\\{0\\})\big{)}$,
which means that $\gamma\in\textup{{{C}}}^{4,\theta}_{\textup{\tiny
loc}}\big{(}(\mathbb{R}^{3}\setminus\\{0\\})\times(\mathbb{R}^{3}\setminus\\{0\\})\big{)}$
for all $\theta<1$. This result just barely misses the
$\textup{{{C}}}^{4,1}$-smoothness of the factor $|x-y|^{5}$ in the formula
(1.6). In this sense Theorem 1.1 is sharp, up to a $\log$-term.

  5. (5)

The one-electron density $\rho(x)=\gamma(x,x)$ is known to be real analytic
for $x\not=0$, see [7]. As shown in [8], the function $\rho(x)$ satisfies the
bound

$\displaystyle|\partial_{x}^{l}\rho(x)|\lesssim\big{(}1+|x|^{1-|l|}\big{)}\,\|\rho\|_{\textup{{{L}}}^{1}(B(x,R))},\quad\textup{for
all}\quad l\in\mathbb{N}_{0}^{3},\quad x\not=0.$

For $|l|\leq 4$ this bound follows from (1.1) and (1.1).

  6. (6)

There is an independent (indirect) argument indicating that $\gamma(x,y)$
should have a $|x-y|^{5}$-singularity on the diagonal. More precisely, we can
show that if the function $\gamma(x,y)$ has on the diagonal a singularity of
the type $|x-y|^{b}$ with some $b>-3$, then necessarily $b=5$. This argument
is based on the analysis of spectral asymptotics of the (non-negative)
operator $\boldsymbol{\Gamma}$ with kernel $\tilde{\gamma}(x,y)$. It was shown
in [17] that the eigenvalues $\lambda_{k}(\boldsymbol{\Gamma})$ have
asymptotics of order $k^{-8/3}$ as $k\to\infty$. On the other hand, according
to the results on spectral asymptotics for integral operators with homogeneous
kernels, see [1] (and also [17] for a summary), the singularity $|x-y|^{b}$,
$b\not=0,2,4,\dots$, would produce the spectral asymptotics of order
$k^{-(1+b/3)}$. The exponent $1+b/3$ coincides with $8/3$ exactly for $b=5$,
which proves the point.

Our proofs have two main ingredients. At the heart of our method are
regularity properties of the eigenfunction $\psi$. By the standard elliptic
argument, the function $\psi$ is real analytic away from the singularities of
the potential (1.1), i.e. away from the particle coalescence points. According
to T. Kato’s seminal paper [12], at the coalescence points the function $\psi$
is Lipschitz. A detailed study of smoothness properties of $\psi$ was
conducted in the recent paper [8], to which we also refer for further
bibliography. In particular, this paper provides global pointwise bounds for
partial derivatives of $\psi$. In the study of the one-particle density (1.4)
conducted in [8] the key point was to obtain bounds for certain directional
derivatives of $\psi$. Such derivatives are also critical for our analysis in
the current paper, and we explain their importance below.

Let ${\sf{R}}=\\{1,2,\dots,N\\}$ be the set of all particle labels. A subset
${\sf P}\subset{\sf{R}}$ is called cluster. For each cluster ${\sf P}$ we
define the following cluster (directional) derivative:

(1.10) $\displaystyle{\sf D}_{{\sf P}}^{m}=$ $\displaystyle\
\biggl{(}\sum_{k\in{\sf
P}}\partial_{x_{k}^{\prime}}\biggr{)}^{m^{\prime}}\bigg{(}\sum_{k\in{\sf
P}}\partial_{x_{k}^{\prime\prime}}\bigg{)}^{m^{\prime\prime}}\bigg{(}\sum_{k\in{\sf
P}}\partial_{x_{k}^{\prime\prime\prime}}\bigg{)}^{m^{\prime\prime\prime}},$

Here $m=(m^{\prime},m^{\prime\prime},m^{\prime\prime\prime})$ with
$m^{\prime},m^{\prime\prime},m^{\prime\prime\prime}\in\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$
and $x=(x^{\prime},x^{\prime\prime},x^{\prime\prime\prime})$ with
$x^{\prime},x^{\prime\prime},x^{\prime\prime\prime}\in\mathbb{R}$. Observe
that for all $m\not=0$,

$\displaystyle{\sf D}_{\sf
P}^{m}\frac{1}{|x_{j}-x_{k}|}=0,\quad\textup{if}\quad j,k\in{\sf
P}\quad\textup{or}\quad j,k\notin{\sf P}.$

Therefore the potential (1.1) is infinitely smooth with respect to ${\sf
D}_{\sf P}$ as long as $x_{j}\not=0,j=1,2,\dots,N,$ and $x_{j}\not=x_{k}$,
where $j\in{\sf P},k\notin{\sf P}$ or $j\notin{\sf P},k\in{\sf P}$. As a
consequence, the function $\psi$ is also infinitely smooth with respect to
${\sf D}_{\sf P}$ for the same values of the coordinates, see [7] or [11]. In
particular, the cluster derivatives of $\psi$ do not have singularities at the
coalescence points $x_{j}=x_{k}$, if $j,k\in{\sf P}$ or $j,k\notin{\sf P}$.
One of the pivotal points in [8] was the pointwise bound for the cluster
derivative ${\sf D}_{\sf P}^{m}\psi(\mathbf{x})$ with explicit dependence on
the distance of $\mathbf{x}\in\mathbb{R}^{3N}$ to the coalescence set

$\displaystyle\Sigma_{{\sf
P}}=\bigg{\\{}\mathbf{x}\in\mathbb{R}^{3N}:\prod_{j\in{\sf
P}}|x_{j}|\prod_{k\in{\sf P},l\in{\sf P}^{\rm c}}|x_{k}-x_{l}|=0\bigg{\\}}.$

For our purposes we need bounds of such type for cluster derivatives involving
an arbitrary finite number $M$ of clusters ${\sf P}_{1},{\sf P}_{2},\dots,{\sf
P}_{M}$, i.e. for ${\sf D}_{{\sf P}_{1}}^{m_{1}}{\sf D}_{{\sf
P}_{2}}^{m_{2}}\cdots{\sf D}_{{\sf P}_{M}}^{m_{M}}\psi$, see Corollary 4.6.
This generalization is not immediate and requires substantial further work,
which is done in Sect. 3 and 4.

The next step of the proof is to use the bounds obtained for cluster
derivatives to estimate partial derivatives of the function (1.3). To this end
we use a partition of unity consisting of smooth functions
$\Phi(x,y,\hat{\mathbf{x}})$ of $3N+3$ variables that we call extended cut-off
functions, and study the integrals

(1.11)
$\displaystyle\gamma(x,y;\Phi)=\int_{\mathbb{R}^{3N-3}}\Phi(x,y,\hat{\mathbf{x}})\psi(x,\hat{\mathbf{x}})\overline{\psi(y,\hat{\mathbf{x}})}\,d\hat{\mathbf{x}}.$

Extended cut-offs were introduced in [11] to establish the real analyticity of
$\gamma(x,y)$ outside the diagonal.

The support of each extended cut-off divides the particles
$x_{2},x_{3},\dots,x_{N}$ into three disjoint groups: the first two groups
consist of particles that are “close” to the particles $x$ and $y$
respectively, and the third one contains the particles that are “far” from the
first two groups. This partition naturally gives rise to two clusters, denoted
${\sf P}$ and ${\sf S}$: cluster ${\sf P}$ labels the particles close to $x$,
and cluster ${\sf S}$ – the particles close to $y$. Thus, when differentiating
the integral (1.11) w.r.t. $x$ and $y$, under the integral these derivatives
convert into the cluster derivatives ${\sf D}_{\sf P}$ and ${\sf D}_{\sf S}$
respectively. The actual calculation is more involved, but we can illustrate
this conversion using the following simplified example. Let us differentiate
with respect to $x$ the integral

$\displaystyle F(x)=\int f(x,\hat{\mathbf{x}})d\hat{\mathbf{x}},$

where the integration is conducted over the space $\mathbb{R}^{3N-3}$, and we
assume for simplicity that $f\in\textup{{{C}}}^{\infty}_{0}(\mathbb{R}^{3N})$.
To this end under the integral we make the change of variables
$\hat{\mathbf{x}}=\hat{\mathbf{w}}+\hat{\mathbf{z}}$, where
$\hat{\mathbf{z}}=(z_{2},z_{3},\dots,z_{N})\in\mathbb{R}^{3N-3}$ is defined by
$z_{j}=x,j\in{\sf P},$ and $z_{j}=0,j\notin{\sf P}$, for some cluster ${\sf
P}$. Thus $F(x)$ rewrites as

$\displaystyle F(x)=\int
f(x,\hat{\mathbf{w}}+\hat{\mathbf{z}})\,d\hat{\mathbf{w}}.$

Consequently, for all $l\in\mathbb{N}_{0}^{3}$, we have

$\displaystyle\partial_{x}^{l}F(x)=\int\big{(}{\sf D}_{\sf
P}^{l}f\big{)}(x,\hat{\mathbf{w}}+\hat{\mathbf{z}})\,d\hat{\mathbf{w}}=\int{\sf
D}_{\sf P}^{l}f(x,\hat{\mathbf{x}})\,d\hat{\mathbf{x}}.$

In order to estimate the derivatives of the integral (1.11), we use the bounds
for cluster derivatives of $\psi$ obtained in the first stage of the proof.
Integrating these bounds in $\hat{\mathbf{x}}$ leads to (1.1) and (1.1)
thereby completing the proof. One should say that the application these bounds
is not immediate, but we defer the discussion of this technical step until
Section 6, see Remark 6.1.

The paper is organized as follows. In Sect. 2 we gather information about cut-
off functions and clusters associated with them. Most of the required facts
are borrowed from [11]. Sect. 3 considers the general elliptic equation of the
form (3.1) in a ball
$B(\mathbf{x}_{0},\ell)=\\{\mathbf{x}:|\mathbf{x}-\mathbf{x}_{0}|<\ell\\}\subset\mathbb{R}^{3N}$.
Here we study cluster derivatives of solutions of (3.1), and the main focus is
on the explicit dependence of these estimates on the radius $\ell$, see
Theorem 3.2. In Sect. 4 these estimates are applied to the Schrödinger
equation to derive the bounds for the cluster derivatives of $\psi$,
summarized in Corollary 4.6. Some estimates for integrals emerging in the
proof of Theorem 1.1 are gathered in Section 5. The proof itself is completed
in Sect. 6. The appendix (Section 7) contains an elementary extension property
for the Sobolev spaces that was proved in [18].

Notation. We conclude the introduction with some general notational
conventions.

Coordinates. As mentioned earlier, we use the following standard notation for
the coordinates: $\mathbf{x}=(x_{1},x_{2},\dots,x_{N})$, where
$x_{j}\in\mathbb{R}^{3}$, $j=1,2,\dots,N$. As a rule we represent $\mathbf{x}$
in the form $\mathbf{x}=(x_{1},\hat{\mathbf{x}})$ with
$\hat{\mathbf{x}}=(x_{2},x_{3},\dots,x_{N})\in\mathbb{R}^{3N-3}$.

For $N\geq 3$ it is also useful to introduce the notation for
$\hat{\mathbf{x}}$ with $x_{j}$, $j\geq 2$, taken out. Let

(1.12)
$\displaystyle\tilde{\mathbf{x}}_{j}=(x_{2},\dots,x_{j-1},x_{j+1},\dots,x_{N}),$

so that $\hat{\mathbf{x}}=(x_{j},\tilde{\mathbf{x}}_{j})$ and
$\mathbf{x}=(x_{1},x_{j},\tilde{\mathbf{x}}_{j})$.

Clusters. Let ${\sf{R}}=\\{1,2,\dots,N\\}$. A subset ${\sf P}\subset{\sf{R}}$
is called cluster. We denote $|{\sf P}|=\operatorname{{card}}{\sf P}$, ${\sf
P}^{\rm c}={\sf{R}}\setminus{\sf P}$, ${\sf P}^{*}={\sf P}\setminus\\{1\\}$.
If ${\sf P}=\varnothing$, then $|{\sf P}|=0$ and ${\sf P}^{\rm c}={\sf{R}}$.

For $M$ clusters ${\sf P}_{1},\dots,{\sf P}_{M}$ we write $\mathbf{P}=\\{{\sf
P}_{1},{\sf P}_{2},\dots,{\sf P}_{M}\\}$, and call $\mathbf{P}$ cluster set.
Clusters ${\sf P}_{1},{\sf P}_{2},\dots,{\sf P}_{M}$ in a cluster set are not
assumed to be all disjoint or distinct.

Derivatives. Let $\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$. If
$x=(x^{\prime},x^{\prime\prime},x^{\prime\prime\prime})\in\mathbb{R}^{3}$ and
$m=(m^{\prime},m^{\prime\prime},m^{\prime\prime\prime})\in\mathbb{N}_{0}^{3}$,
then the derivative $\partial_{x}^{m}$ is defined in the standard way:

$\displaystyle\partial_{x}^{m}=\partial_{x^{\prime}}^{m^{\prime}}\partial_{x^{\prime\prime}}^{m^{\prime\prime}}\partial_{x^{\prime\prime\prime}}^{m^{\prime\prime\prime}}.$

This notation extends to $x\in\mathbb{R}^{d}$ with an arbitrary dimension
$d\geq 1$ in the obvious way. Denote also

$\displaystyle\partial^{\mathbf{m}}=\partial_{x_{1}}^{m_{1}}\partial_{x_{2}}^{m_{2}}\cdots\partial_{x_{N}}^{m_{N}},\quad\mathbf{m}=(m_{1},m_{2},\dots,m_{N})\in\mathbb{N}_{0}^{3N}.$

Let $\mathbf{P}=\\{{\sf P}_{1},{\sf P}_{2},\dots,{\sf P}_{M}\\}$ be a cluster
set, and let $\mathbf{m}=(m_{1},m_{2},\dots,m_{M})$,
$m_{k}\in\mathbb{N}_{0}^{3}$, $k=1,2,\dots,M$. Then we denote

$\displaystyle{\sf D}_{\mathbf{P}}^{\mathbf{m}}={\sf D}_{{\sf
P}_{1}}^{m_{1}}{\sf D}_{{\sf P}_{2}}^{m_{2}}\cdots{\sf D}_{{\sf
P}_{M}}^{m_{M}},$

where each individual cluster derivative is defined as in (1.10). It is easy
to see that the cluster derivatives satisfy the Leibniz rule. We use this fact
without further comments throughout the paper.

Supports. For any smooth function $f=f(\mathbf{x})$, we define
$\operatorname{{supp}}_{0}f=\\{\mathbf{x}:f(\mathbf{x})\not=0\\}$. It is clear
that the closure $\overline{\operatorname{{supp}}_{0}f}$ coincides with the
support $\operatorname{{supp}}f$ defined in the standard way. With this
definition we immediately get the useful property that

$\displaystyle\operatorname{{supp}}_{0}(fg)=\operatorname{{supp}}_{0}f\cap\operatorname{{supp}}_{0}g.$

Bounds. As explained earlier, for two non-negative numbers (or functions) $X$
and $Y$ depending on some parameters, we write $X\lesssim Y$ (or $Y\gtrsim X$)
if $X\leq CY$ with some positive constant $C$ independent of those parameters.
If $X\lesssim Y$ and $Y\lesssim X$, then $X\asymp Y$. To avoid confusion we
often make explicit comments on the nature of (implicit) constants in the
bounds. In particular, all constants (implicit or explicit) may depend on the
eigenvalue $E$, the number of particles $N$ and the charge $Z$.

##  2\. Cut-off functions and clusters

###  2.1. Admissible cut-off functions

Let

(2.1) $\displaystyle\xi\in\textup{{{C}}}^{\infty}_{0}(\mathbb{R}):\
0\leq\xi(t)\leq 1,\ \xi(t)=\begin{cases}1,\ {\rm if}\ |t|\leq
1,\\\\[5.69046pt] 0,\ {\rm if}\ |t|\geq 2.\end{cases}$

Now for $\varepsilon>0$ we define two radially-symmetric functions
$\zeta\in\textup{{{C}}}^{\infty}_{0}(\mathbb{R}^{3})$,
$\theta\in\textup{{{C}}}^{\infty}(\mathbb{R}^{3})$ as follows:

(2.2)
$\displaystyle\zeta(x)=\zeta_{\varepsilon}(x)=\xi\bigg{(}\frac{4N}{\varepsilon}|x|\bigg{)},\quad\theta(x)=\theta_{\varepsilon}(x)=1-\zeta_{\varepsilon}(x),\quad
x\in\mathbb{R}^{3},$

so that

$\displaystyle\zeta(x)=0\quad\textup{for}\quad x\notin
B\big{(}0,\varepsilon(2N)^{-1}\big{)},\qquad\qquad\theta(x)=0\quad\textup{for}\quad
x\in B\big{(}0,\varepsilon(4N)^{-1}\big{)}.$

The dependence of the cut-offs on the parameter $\varepsilon$ is important,
but it is not always reflected in the notation.

Our next step is to build out of the functions $\zeta_{\varepsilon}$ and
$\theta_{\varepsilon}$ cut-off functions of $3N$ variables. Let
$\\{f_{jk}\\},1\leq j,k\leq N$, be a set of functions such that each of them
is one of the functions $\zeta_{\varepsilon}$ or $\theta_{\varepsilon}$, and
$f_{jk}=f_{kj}$. We call functions of the form

(2.3) $\displaystyle\phi(\mathbf{x})=\prod\limits_{1\leq j<k\leq
N}f_{jk}(x_{j}-x_{k}).$

admissible cut-off functions or simply admissible cut-offs. Such cut-offs (or,
more precisely, a slightly more general version thereof) were used in [6], [7]
and also in [11]. We need only a subset of their properties established in [7]
and [11].

We associate with the function $\phi$ a cluster ${\sf Q}(\phi)$ defined next.

######  Definition 2.1.

For an admissible cut-off $\phi$, let
$I(\phi)\subset\\{(j,k)\in{\sf{R}}\times{\sf{R}}:j\not=k\\}$ be the index set
such that $(j,k)\in I(\phi)$, iff $f_{jk}=\zeta$. We say that two indices
$j,k\in{\sf{R}}$, are $\phi$-linked to each other if either $j=k$, or
$(j,k)\in I(\phi)$, or there exists a sequence of pairwise distinct indices
$j_{1},j_{2},\dots,j_{s}$, $1\leq s\leq N-2$, all distinct from $j$ and $k$,
such that $(j,j_{1}),(j_{s},k)\in I(\phi)$ and $(j_{p},j_{p+1})\in I(\phi)$
for all $p=1,2,\dots,s-1$.

The cluster ${\sf Q}(\phi)$ associated with the cut-off $\phi$ is defined as
the set of all indices that are $\phi$-linked to index $1$.

It follows from the above definition that ${\sf Q}(\phi)$ always contains
index $1$. Note also that the notion of being linked defines an equivalence
relation on ${\sf{R}}$, and the cluster ${\sf Q}(\phi)$ is nothing but the
equivalence class of index $1$. On the support of the admissible cut-off
$\phi$ the variables $x_{j}$, indexed by $j\in{\sf Q}(\phi)$, are “close” to
each other and are “far” from the remaining variables. In order to quantify
these facts below we define a number of subsets in $\mathbb{R}^{3N}$ and
$\mathbb{R}^{3N-3}$.

For any cluster ${\sf P}$ we introduce the following sets depending on the
parameter $\varepsilon>0$:

$\displaystyle X_{{\sf P}}(\varepsilon)=\begin{cases}\mathbb{R}^{3N}\quad{\rm
for}\ |{\sf P}|=0\ \textup{or}\ N,\\\\[5.69046pt]
\\{\mathbf{x}\in\mathbb{R}^{3N}:\ |x_{j}-x_{k}|>\varepsilon,\forall j\in{\sf
P},k\in{\sf P}^{\rm c}\\},\quad{\rm for}\ 0<|{\sf P}|<N,\end{cases}$

The set $X_{\sf P}(\varepsilon)$ separates the points $x_{k}$ and $x_{j}$
labeled by the clusters ${\sf P}$ and ${\sf P}^{\rm c}$ respectively. Note
that $X_{{\sf P}}(\varepsilon)=X_{{\sf P}^{\rm c}}(\varepsilon)$. Define also
the sets separating $x_{k}$’s from the origin:

$\displaystyle T_{\sf P}(\varepsilon)=\begin{cases}\mathbb{R}^{3N},\quad{\rm
for}\ |{\sf P}|=0,\\\\[5.69046pt]
\\{\mathbf{x}\in\mathbb{R}^{3N}:|x_{j}|>\varepsilon,\ \forall j\in{\sf P}\\},\
\quad{\rm for}\ |{\sf P}|>0.\end{cases}$

It is also convenient to introduce corresponding sets in the space
$\mathbb{R}^{3N-3}$:

(2.4) $\displaystyle\widehat{X}_{{\sf
P}}(x,\varepsilon)=\\{\hat{\mathbf{x}}\in\mathbb{R}^{3N-3}:(x,\hat{\mathbf{x}})\in
X_{{\sf P}}(\varepsilon)\\},\quad\textup{for all}\quad x\in\mathbb{R}^{3},$

and

(2.5) $\displaystyle\widehat{T}_{{\sf
P}}(\varepsilon)=\begin{cases}\mathbb{R}^{3N-3},\quad{\rm for}\ |{\sf
P}^{*}|=0,\\\\[5.69046pt]
\\{\hat{\mathbf{x}}\in\mathbb{R}^{3N-3}:|x_{j}|>\varepsilon,\ \forall j\in{\sf
P}^{*}\\},\ \quad{\rm for}\ |{\sf P}^{*}|>0.\end{cases}$

Observe that $\widehat{T}_{{\sf P}}(\varepsilon)=\widehat{T}_{{\sf
P}^{*}}(\varepsilon)$.

The support of the admissible cut-off $\phi$ is easily described with the help
of the sets introduced above. The next proposition is adapted from [7, Lemma
4.3(i)] and [11, Lemmata 4.2, 4.3].

######  Proposition 2.2.

For ${\sf P}={\sf Q}(\phi)$ the inclusion

$\displaystyle\operatorname{{supp}}_{0}\phi\subset X_{{\sf
P}}\big{(}\varepsilon(4N)^{-1}\big{)}$

holds.

Moreover, if $j\in{\sf Q}(\phi)$, then $|x_{1}-x_{j}|<\varepsilon/2$ for all
$\mathbf{x}\in\operatorname{{supp}}_{0}\phi$. If $|x_{1}|>\varepsilon$, then

$\displaystyle\operatorname{{supp}}_{0}\phi(x_{1},\ \cdot\
)\subset\widehat{T}_{{\sf P}}(\varepsilon/2).$

We do not use Proposition 2.2 directly in this paper, but present it in order
to demonstrate the relevance of the associated cluster ${\sf Q}(\phi)$.

###  2.2. Extended cut-offs

In our analysis the central role is played by another class of cut-off
functions. These cut-offs are functions of $3N+3$ variables and they are
defined as follows: for each
$x,y\in\mathbb{R}^{3},\hat{\mathbf{x}}\in\mathbb{R}^{3N-3}$ let

(2.6) $\displaystyle\Phi(x,y,\hat{\mathbf{x}})=\prod_{2\leq j\leq
N}g_{j}(x-x_{j})\prod_{2\leq j\leq N}h_{j}(y-x_{j})\prod_{2\leq k<l\leq
N}f_{kl}(x_{k}-x_{l}),$

where each of the functions $g_{j},h_{j}$ and $f_{jk}=f_{kj}$ is one of the
cut-offs $\theta$ or $\zeta$ defined in (2.2). We call such functions extended
cut-offs. Each extended cut-off uniquely defines two admissible cut-offs:

$\displaystyle\phi(x,\hat{\mathbf{x}})=$ $\displaystyle\ \prod_{2\leq j\leq
N}g_{j}(x-x_{j})\prod_{2\leq k<l\leq N}f_{kl}(x_{k}-x_{l}),$
$\displaystyle\mu(y,\hat{\mathbf{x}})=$ $\displaystyle\ \prod_{2\leq j\leq
N}h_{j}(y-x_{j})\prod_{2\leq k<l\leq N}f_{kl}(x_{k}-x_{l}),$

see definition (2.3). We say that the pair $\phi$, $\mu$ and the extended cut-
off $\Phi$ are associated to each other. We denote by ${\sf P}={\sf Q}(\phi)$
and ${\sf S}={\sf Q}(\mu)$ the clusters associated with $\phi$ and $\mu$
respectively.

We remind that the functions $\phi,\mu$ and $\Phi$ all depend on the parameter
$\varepsilon$. Thus whenever necessary we include $\varepsilon$ in the
notation and write, for example, $\Phi(x,y,\hat{\mathbf{x}};\varepsilon)$.
Note however that the clusters ${\sf P}$ and ${\sf S}$ do not depend on
$\varepsilon$.

Below we list some useful properties of the extended cut-offs $\Phi$ and
associated admissible $\phi$, $\mu$ adapted from [11, Lemmata 4.6, 4.8].

######  Proposition 2.3.

If ${\sf P}^{*}\cap{\sf S}$ is non-empty and $|x-y|>\varepsilon$, then
$\Phi(x,y,\hat{\mathbf{x}};\varepsilon)=0$ for all
$\hat{\mathbf{x}}\in\mathbb{R}^{3N-3}$.

In the estimates further on we assume, as a rule, that $|x-y|>\varepsilon$, so
due to Proposition 2.3 from now on we may suppose that ${\sf P}^{*}\subset{\sf
S}^{\rm c}$ (which is equivalent to ${\sf S}^{*}\subset{\sf P}^{\rm c}$).
Under this condition we obtain the following information about the support of
the extended cut-off $\Phi$:

######  Proposition 2.4.

If ${\sf P}^{*}\subset{\sf S}^{\rm c}$, then for all $x,y\in\mathbb{R}^{3}$ we
have

$\displaystyle\operatorname{{supp}}_{0}\Phi(x,y,\ \cdot\
;\varepsilon)\subset\widehat{X}_{{\sf
P}}\big{(}x,\varepsilon(4N)^{-1}\big{)}\cap$ $\displaystyle\widehat{X}_{{\sf
S}^{*}}\big{(}x,\varepsilon(4N)^{-1}\big{)}$ (2.7)
$\displaystyle\cap\widehat{X}_{{\sf
S}}\big{(}y,\varepsilon(4N)^{-1}\big{)}\cap\widehat{X}_{{\sf
P}^{*}}\big{(}y,\varepsilon(4N)^{-1}\big{)}.$

If, in addition $|x|>\varepsilon,|y|>\varepsilon$, then

(2.8) $\displaystyle\operatorname{{supp}}_{0}\Phi(x,y,\ \cdot\
;\varepsilon)\subset\widehat{T}_{{\sf
P}^{*}}(\varepsilon/2)\cap\widehat{T}_{{\sf S}^{*}}(\varepsilon/2).$

To complete this section we need to make a remark on the cluster derivatives
of the extended cut-offs. For any cluster ${\sf Q}\subset{\sf{R}}$ we denote
by ${\sf D}_{x,{\sf Q}}^{m}\Phi(x,y,\hat{\mathbf{x}})$ the cluster derivative
of $\Phi$ as a function of the variables $(x,\hat{\mathbf{x}})$, i.e.

(2.9) $\displaystyle{\sf D}_{x,{\sf Q}}^{m}\Phi(x,y,\hat{\mathbf{x}})={\sf
D}_{{\sf
Q}}^{m}\tilde{\Phi}_{y}(x,\hat{\mathbf{x}}),\quad\textup{where}\quad\tilde{\Phi}_{y}(x,\hat{\mathbf{x}}):=\Phi(x,y,\hat{\mathbf{x}}).$

Similarly we define the derivative ${\sf D}_{y,{\sf
Q}}^{m}\Phi(x,y,\hat{\mathbf{x}})$. It immediately follows from the definition
of $\Phi(x,y,\hat{\mathbf{x}})$ that for any clusters ${\sf Q}_{1},{\sf
Q}_{2}$ and all $\mathbf{m}=(m_{1},m_{2})\in\mathbb{N}_{0}^{6}$, the bound
holds:

(2.10) $\displaystyle|{\sf D}_{x,{\sf Q}_{1}}^{m_{1}}{\sf D}_{y,{\sf
Q}_{2}}^{m_{2}}\Phi(x,y,\hat{\mathbf{x}};\varepsilon)|\lesssim\begin{cases}1,\quad\textup{if}\quad|\mathbf{m}|=0,\\\\[5.69046pt]
\varepsilon^{-|\mathbf{m}|}\mathcal{M}_{\varepsilon}(x,y,\hat{\mathbf{x}}),\quad\textup{if}\quad|\mathbf{m}|\geq
1,\end{cases}$

where

$\displaystyle\mathcal{M}_{\varepsilon}(x,y,\hat{\mathbf{x}})=$
$\displaystyle\ \sum_{2\leq j\leq
N}\xi\big{(}N\varepsilon^{-1}|x-x_{j}|\big{)}$ (2.11) $\displaystyle\
+\sum_{2\leq j\leq N}\xi\big{(}N\varepsilon^{-1}|y-x_{j}|\big{)}+\sum_{2\leq
j<k\leq N}\xi\big{(}N\varepsilon^{-1}|x_{j}-x_{k}|\big{)},$

see (2.1) for the definition of the function $\xi$.

##  3\. Regularity estimates

###  3.1. $\textup{{{C}}}^{1}$-regularity for elliptic equations

In what follows we rely on the well-known $\textup{{{C}}}^{1}$-regularity
bounds for solutions of second order elliptic equations on bounded domains.
This type of regularity is discussed e.g. in [13, Ch. 3] and [9, Ch. 8]. To be
precise, in [13, Ch. 3] and [9, Ch. 8] one can find bounds even in the space
$\textup{{{C}}}^{1,\theta}$ with a suitable $\theta\in(0,1)$, but we are not
concerned with the Hölder properties. Moreover, in this paper we do not need
the most general form of the equation. For our purposes it suffices to
consider the equation

(3.1) $\displaystyle\big{(}-\Delta+\mathbf{a}(x)\cdot\nabla+b(x)\big{)}u=g,$

on an open ball $B\subset\mathbb{R}^{d}$, where all the coefficients are
$\textup{{{L}}}^{\infty}(B)$-functions. The proposition below provides some
convenient $\max$-bounds for the weak solution and its first derivatives.
Since the proof is quite short, we provide it for the sake of completeness.
Note that the proof of [8, Proposition A.2] uses similar ideas.

######  Proposition 3.1.

Let $B_{R}=B(x_{0},R)\subset\mathbb{R}^{d}$ for some $x_{0}\in\mathbb{R}^{d}$
and $R>0$. Suppose that $u\in\textup{{{W}}}^{1,2}(B_{R})$ is a weak solution
of the equation (3.1), where
$\mathbf{a},b,g\in\textup{{{L}}}^{\infty}(B_{R})$, and

$\displaystyle\|\mathbf{a}\|_{\textup{{{L}}}^{\infty}(B_{R})}+\|b\|_{\textup{{{L}}}^{\infty}(B_{R})}\leq
M,$

with some constant $M>0$. Then for any $r\in(0,R)$ the function $u$ belongs to
$\textup{{{W}}}^{2,2}(B_{r})\cap\textup{{{C}}}^{1}(\overline{B_{r}})$ and

(3.2)
$\displaystyle\|u\|_{\textup{{{C}}}^{1}(\overline{B_{r}})}\lesssim\|u\|_{\textup{{{L}}}^{2}(B_{R})}+\|g\|_{\textup{{{L}}}^{\infty}(B_{R})},$

with an implicit constant that depends only on the constant $M$, dimension $d$
and the radii $r$ and $R$.

###### Proof.

The inclusion $u\in\textup{{{W}}}^{2,2}(B_{r})$ is a direct consequence of the
interior regularity.

In order to prove that the weak solution $u\in\textup{{{W}}}^{1,2}(B_{R})$ has
the $\textup{{{C}}}^{1}$-regularity in $B_{r}$ we repeatedly apply the
following elementary fact.

Assume that $u\in\textup{{{W}}}^{1,p}(B_{\rho})$ with some $p\in(1,\infty)$
and $\rho\leq R$. Then for any $\nu<\rho$ the following is true:

  1. (1)

If $p\leq d$, then $u\in\textup{{{W}}}^{1,q}(B_{\nu})$ with $q=p(1+d^{-1})$
and

$\displaystyle\|u\|_{\textup{{{W}}}^{1,q}(B_{\nu})}\lesssim$ $\displaystyle\
\|u\|_{\textup{{{L}}}^{p}(B_{\rho})}+\|g\|_{\textup{{{L}}}^{p}(B_{\rho})}$
(3.3) $\displaystyle\lesssim$ $\displaystyle\
\|u\|_{\textup{{{W}}}^{1,p}(B_{\rho})}+\|g\|_{\textup{{{L}}}^{\infty}(B_{\rho})}.$

  2. (2)

If $p>d$, then $u\in\textup{{{C}}}^{1}(\overline{B_{\nu}})$ and

$\displaystyle\|u\|_{\textup{{{C}}}^{1}(\overline{B_{\nu}})}\lesssim$
$\displaystyle\
\|u\|_{\textup{{{L}}}^{p}(B_{\rho})}+\|g\|_{\textup{{{L}}}^{p}(B_{\rho})}$
(3.4) $\displaystyle\lesssim$ $\displaystyle\
\|u\|_{\textup{{{W}}}^{1,p}(B_{\rho})}+\|g\|_{\textup{{{L}}}^{\infty}(B_{\rho})}.$

Indeed, under the assumption $u\in\textup{{{W}}}^{1,p}(B_{\rho})$, by the
interior regularity we have $u\in\textup{{{W}}}^{2,p}(B_{\nu})$ and the
standard bound holds (see, e.g. [9, Theorem 9.11]):

(3.5)
$\displaystyle\|u\|_{\textup{{{W}}}^{2,p}(B_{\nu})}\lesssim\|u\|_{\textup{{{L}}}^{p}(B_{\rho})}+\|g\|_{\textup{{{L}}}^{p}(B_{\rho})}.$

If $p<d$, then we use the bounded embedding
$\textup{{{W}}}^{2,p}\subset\textup{{{W}}}^{1,q}$, for all $q\in[p,p^{*}]$,
$p^{*}=dp(d-p)^{-1}$. In particular, the value $q=p(1+d^{-1})$ belongs to the
interval $[p,p^{*}]$, which proves (1). If $p=d$, then
$\textup{{{W}}}^{2,p}\subset\textup{{{W}}}^{1,q}$ for all $q\in[p,\infty)$,
and hence for $q=p(1+d^{-1})$ in particular. Hence (1) holds again.

If $p>d$, then we use the embedding
$\textup{{{W}}}^{2,p}\subset\textup{{{C}}}^{1}$, so that (3.5) leads to (2).

Let us proceed with the proof of (3.2). If $d=1$, then the solution
$u\in\textup{{{W}}}^{1,2}(R)$ satisfies (2) with $p=2$, which immediately
implies (3.2). Suppose that $d\geq 2$, and define the sequence

$\displaystyle q_{n}=2(1+d^{-1})^{n},\ n=0,1,\dots.$

Let $k\geq 1$ be the index such that $q_{k-1}\leq d$ and $q_{k}>d$. Pick
finitely many numbers $r_{n}>0$ such that

$\displaystyle r<r_{k}<r_{k-1}<\dots<r_{1}<r_{0}=R.$

Since $u\in\textup{{{W}}}^{1,2}(B_{R})$ we can apply the bound (1) with
$\nu=r_{1},\rho=R$ and $p=q_{0}=2$, $q=q_{1}$. Repeating this step
successively for $\rho=r_{n},\nu=r_{n+1}$ and $p=q_{n}$, $q=q_{n+1}$ for all
$n=1,\dots,k$, we arrive at the bound

$\displaystyle\|u\|_{\textup{{{W}}}^{1,q_{k}}(B_{r_{k}})}\lesssim$
$\displaystyle\
\|u\|_{\textup{{{W}}}^{1,q_{k-1}}(B_{r_{k-1}})}+\|g\|_{\textup{{{L}}}^{\infty}(B_{r_{k-1}})}\lesssim\dots$
$\displaystyle\lesssim$ $\displaystyle\
\|u\|_{\textup{{{W}}}^{1,q_{{\scaleto{1}{3pt}}}}(B_{r_{{\scaleto{1}{3pt}}}})}+\|g\|_{\textup{{{L}}}^{\infty}(B_{r_{{\scaleto{1}{3pt}}}})}\lesssim\|u\|_{\textup{{{L}}}^{2}(B_{R})}+\|g\|_{\textup{{{L}}}^{\infty}(B_{R})}$

As $q_{k}>d$, we can now use (2) which gives

$\displaystyle\|u\|_{\textup{{{C}}}^{1}(\overline{B_{r}})}\lesssim$
$\displaystyle\
\|u\|_{\textup{{{W}}}^{1,q_{k}}(B_{r_{k}})}+\|g\|_{\textup{{{L}}}^{\infty}(B_{r_{k}})}$
$\displaystyle\lesssim$ $\displaystyle\
\|u\|_{\textup{{{L}}}^{2}(B_{R})}+\|g\|_{\textup{{{L}}}^{\infty}(B_{R})}.$

This completes the proof of (3.2). ∎

###  3.2. Cluster derivatives

The next result is tailored for later use with the multi-particle Schrödinger
equation. We assume that in the equation (3.1) $d=3N$ and the variable
$\mathbf{x}$ is given by $\mathbf{x}=(x_{1},x_{2},\dots,x_{N})$.

Now we obtain better regularity properties of the weak solution of (3.1)
assuming some additional smoothness of the coefficients with respect to
cluster derivatives. Precisely, consider a weak solution $u$ of the equation
(3.1) in the ball $B(\mathbf{x}_{0},R\ell)$ with some
$\mathbf{x}_{0}\in\mathbb{R}^{3N}$, $R>0$, $\ell\in(0,1]$. Suppose that for a
cluster set $\boldsymbol{\sf P}=\\{{\sf P}_{1},{\sf P}_{2},\dots,{\sf
P}_{M}\\}$ the coefficients $\mathbf{a}$ and $b$ satisfy the bounds

(3.6) $\displaystyle|{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{m}}\mathbf{a}(\mathbf{x})|+|{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{m}}b(\mathbf{x})|\lesssim\ell^{-|\mathbf{m}|},\quad\mathbf{x}\in
B(\mathbf{x}_{0},R\ell),$

for all $\mathbf{m}\in\mathbb{N}_{0}^{3M}$, with constants potentially
depending on $\mathbf{m}$, $R$ and $\mathbf{x}_{0}$, but not on $\ell$. In the
next theorem we obtain bounds for the cluster derivatives ${\sf
D}_{\boldsymbol{\sf P}}^{\mathbf{m}}u$ with explicit dependence on the
parameter $\ell\in(0,1]$.

######  Theorem 3.2.

Assume the conditions (3.6), and let $u$ be a weak solution of the equation
(3.1) in $B(\mathbf{x}_{0},R\ell)$ with the right-hand side $g=0$. Then for
all $\mathbf{m}\in\mathbb{N}_{0}^{3M}$ and all $r<R$ the cluster derivatives
${\sf D}_{\boldsymbol{\sf P}}^{\mathbf{m}}u$ belong to
$\textup{{{C}}}^{1}\big{(}\overline{B(\mathbf{x}_{0},r\ell)}\big{)}$.
Furthermore, if $|\mathbf{m}|+k\geq 1$, where $k=0,1$, then for all
$\nu\in(r,R)$ we have

(3.7) $\displaystyle\|\nabla^{k}{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{m}}u\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},r\ell))}\lesssim\ell^{1-|\mathbf{m}|-k}\,\big{(}\ell\|u\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},\nu\ell))}+\|\nabla
u\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},\nu\ell))}\big{)}.$

If $|\mathbf{m}|\geq 2$, then also

$\displaystyle\|{\sf D}_{\boldsymbol{\sf P}}^{\mathbf{m}}$ $\displaystyle
u\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},r\ell))}$ (3.8)
$\displaystyle\lesssim$ $\displaystyle\
\ell^{2-|\mathbf{m}|}\,\big{(}\max_{\mathbf{l}:|\mathbf{l}|=2}\|{\sf
D}_{\boldsymbol{\sf
P}}^{\mathbf{l}}u\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},\nu\ell))}+\|\nabla
u\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},\nu\ell))}+\|u\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},\nu\ell))}\big{)}.$

The implicit constants in (3.7) and (3.2) may depend on the constants
$r,\nu,R$, order $\mathbf{m}$ of the derivative, cluster set $\boldsymbol{\sf
P}$, and the constants in (3.6). In particular, if the constants in (3.6) are
independent of $\mathbf{x}_{0}$, then so are the constants in (3.7) and (3.2).

In the present paper we need the bound (3.7) only. Nevertheless, the bound
(3.2) may be of independent interest.

The idea of the proof follows a similar argument in [8]. Namely, we rescale
the problem by introducing the function

$\displaystyle
w(\mathbf{y})=u(\mathbf{x}_{0}+\ell\mathbf{y}),\quad\mathbf{y}\in
B(\mathbb{0},R),$

and the coefficients

$\displaystyle\boldsymbol{\alpha}(\mathbf{y})=\mathbf{a}(\mathbf{x}_{0}+\ell\mathbf{y}),\quad\beta(\mathbf{y})=b(\mathbf{x}_{0}+\ell\mathbf{y}).$

After the scaling the equation (3.1) takes the form

(3.9) $\displaystyle-\Delta w+2\ell\ \boldsymbol{\alpha}\cdot\nabla
w+\ell^{2}\beta w=0.$

Note that

(3.10) $\displaystyle|{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{m}}\boldsymbol{\alpha}(\mathbf{y})|+|{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{m}}\beta(\mathbf{y})|\lesssim 1,\quad\mathbf{y}\in
B(\mathbb{0},R),$

for all $\mathbf{m}\in\mathbb{N}_{0}^{3M}$.

We are interested in the bounds for the function $w_{\mathbf{m}}={\sf
D}_{\boldsymbol{\sf P}}^{\mathbf{m}}w$ for $\mathbf{y}\in B(\mathbb{0},R)$.
For the sake of brevity throughout the proof we use the notation
$B_{r}=B(\mathbb{0},r)$ for $r>0$.

######  Lemma 3.3.

Let $\ell\leq 1$. Suppose that $w$ is a weak solution of the equation (3.9).
Then for all $\mathbf{m}\in\mathbb{N}_{0}^{3M}$ and all $r<R$ the cluster
derivatives $w_{\mathbf{m}}$ belong to
$\textup{{{C}}}^{1}\big{(}\overline{B_{r}})\big{)}$. Furthermore, if
$|\mathbf{m}|\geq 1$, then for all $\nu\in(r,R)$ we have

(3.11)
$\displaystyle\|w_{\mathbf{m}}\|_{\textup{{{C}}}^{1}(\overline{B_{r}})}\lesssim\ell^{2}\,\|w\|_{\textup{{{L}}}^{\infty}(B_{\nu})}+\|\nabla
w\|_{\textup{{{L}}}^{\infty}(B_{\nu})}.$

If $|\mathbf{m}|\geq 2$, then also

(3.12)
$\displaystyle\|w_{\mathbf{m}}\|_{\textup{{{C}}}^{1}(\overline{B_{r}})}\lesssim\max_{\mathbf{l}:|\mathbf{l}|=2}\|{\sf
D}_{\boldsymbol{\sf
P}}^{\mathbf{l}}w\|_{\textup{{{L}}}^{\infty}(B_{\nu})}+\ell\|\nabla
w\|_{\textup{{{L}}}^{\infty}(B_{\nu})}+\ell^{2}\|w\|_{\textup{{{L}}}^{\infty}(B_{\nu})}.$

###### Proof.

We begin the proof with a formal manipulation assuming that all the cluster
derivatives $w_{\mathbf{m}}$ exist and are as smooth as necessary. Applying
the operator ${\sf D}_{\boldsymbol{\sf P}}^{\mathbf{m}}$ to the equation (3.9)
and using (3.10) we obtain the following equation for the function
$w_{\mathbf{m}}$:

(3.13) $\displaystyle-\Delta w_{\mathbf{m}}+2\ell\
\boldsymbol{\alpha}\cdot\nabla
w_{\mathbf{m}}+\ell^{2}\beta\,w_{\mathbf{m}}=g_{\mathbf{m}},$

with

$\displaystyle
g_{\mathbf{m}}=-2\ell\sum\limits_{\begin{subarray}{c}\mathbb{0}\leq\mathbf{q}\leq\mathbf{m}\\\
|\mathbf{q}|\leq|\mathbf{m}|-1\end{subarray}}{\mathbf{m}\choose\mathbf{q}}\big{(}{\sf
D}_{\boldsymbol{\sf
P}}^{\mathbf{m}-\mathbf{q}}\boldsymbol{\alpha}\big{)}\cdot\nabla
w_{\mathbf{q}}-\ell^{2}\sum\limits_{\begin{subarray}{c}\mathbb{0}\leq\mathbf{q}\leq\mathbf{m}\\\
|\mathbf{q}|\leq|\mathbf{m}|-1\end{subarray}}{\mathbf{m}\choose\mathbf{q}}\big{(}{\sf
D}_{\boldsymbol{\sf P}}^{\mathbf{m}-\mathbf{q}}\beta\big{)}\,w_{\mathbf{q}}.$

It follows from (3.10) that

(3.14)
$\displaystyle\|g_{\mathbf{m}}\|_{\textup{{{L}}}^{\infty}(B_{\rho})}\lesssim\sum_{\mathbf{q}:0\leq|\mathbf{q}|\leq|\mathbf{m}|-1}T_{\mathbf{q}}(\ell,\rho),\quad
T_{\mathbf{q}}(\ell,\rho)=\ell\|\nabla
w_{\mathbf{q}}\|_{\textup{{{L}}}^{\infty}(B_{\rho})}+\ell^{2}\|w_{\mathbf{q}}\|_{\textup{{{L}}}^{\infty}(B_{\rho})},$

for all $\rho<R$.

Now we need to justify the above formal calculations. First note that since
$w$ is a weak solution of (3.9), by Proposition 3.1 it belongs to
$\textup{{{W}}}^{2,2}(B_{\rho})\cap\textup{{{C}}}^{1}(\overline{B_{\rho}})$
for all $\rho<R$, so that $T_{\mathbb{0}}(\ell,\rho)=\ell\|\nabla
w\|_{\textup{{{L}}}^{\infty}(B_{\rho})}+\ell^{2}\|w\|_{\textup{{{L}}}^{\infty}(B_{\rho})}$
is finite, and as a result, for $|\mathbf{m}|=1$ we have

(3.15)
$\displaystyle\|g_{\mathbf{m}}\|_{\textup{{{L}}}^{\infty}(B_{\rho})}\lesssim
T_{\mathbb{0}}(\ell,\rho)=\ell\|\nabla
w\|_{\textup{{{L}}}^{\infty}(B_{\rho})}+\ell^{2}\,\|w\|_{\textup{{{L}}}^{\infty}(B_{\rho})},\quad
0<\rho<R.$

By Proposition 3.1, for all $\mathbf{m}$ such that $|\mathbf{m}|=1$ and all
$r<\rho$, we conclude that
$w_{\mathbf{m}}\in\textup{{{W}}}^{2,2}(B_{r})\cap\textup{{{C}}}^{1}(\overline{B_{r}})$
and

$\displaystyle\|w_{\mathbf{m}}\|_{\textup{{{C}}}^{1}(\overline{B_{r}})}\lesssim$
$\displaystyle\
\|w_{\mathbf{m}}\|_{\textup{{{L}}}^{\infty}(B_{\rho})}+\|g_{\mathbf{m}}\|_{\textup{{{L}}}^{\infty}(B_{\rho})}$
$\displaystyle\lesssim$ $\displaystyle\ \|\nabla
w\|_{\textup{{{L}}}^{\infty}(B_{\rho})}+\|g_{\mathbf{m}}\|_{\textup{{{L}}}^{\infty}(B_{\rho})}\lesssim\|\nabla
w\|_{\textup{{{L}}}^{\infty}(B_{\rho})}+\ell^{2}\,\|w\|_{\textup{{{L}}}^{\infty}(B_{\rho})}.$

This proves (3.11) for $|\mathbf{m}|=1$.

Using the above observation as the induction base, we now prove that
$w_{\mathbf{m}}$ is indeed well-defined for all
$\mathbf{m}\in\mathbb{N}_{0}^{3M}$ and that it is a weak solution of the
equation (3.13). To provide the induction step assume that for some
$k=1,\dots,$ and all $\mathbf{m},1\leq|\mathbf{m}|\leq k$, the function
$w_{\mathbf{m}}$ is a weak solution of (3.13) in $B_{\rho}$ for all $\rho<R$
(and hence belongs to
$\textup{{{W}}}^{2,2}(B_{r})\cap\textup{{{C}}}^{1}(\overline{B_{r}}),r<\rho,$
by Proposition 3.1), and that it satisfies (3.11). Let us prove that the same
is true for the function $w_{\mathbf{n}}$ for all $\mathbf{n}$ such that
$|\mathbf{n}|=k+1$.

First note that for all $\rho<\nu<R$ the function $g_{\mathbf{n}}$ satisfies
the bound

(3.16)
$\displaystyle\|g_{\mathbf{n}}\|_{\textup{{{L}}}^{\infty}(B_{\rho})}\lesssim\ell\|\nabla
w\|_{\textup{{{L}}}^{\infty}(B_{\nu})}+\ell^{2}\,\|w\|_{\textup{{{L}}}^{\infty}(B_{\nu})}.$

Indeed, in the bound (3.14) for $T_{\mathbb{0}}(\ell,\rho)$ we use the
estimate (3.15). For $T_{\mathbf{q}}(\ell,\rho)$ with $1\leq|\mathbf{q}|\leq
k$ we use (3.11) to obtain the estimate

$\displaystyle
T_{\mathbf{q}}(\ell,\rho)\lesssim\ell\|w_{\mathbf{q}}\|_{\textup{{{C}}}^{1}(\overline{B_{\rho}})}\lesssim\ell\,\|\nabla
w\|_{\textup{{{L}}}^{\infty}(B_{\nu})}+\ell^{3}\,\|w\|_{\textup{{{L}}}^{\infty}(B_{\nu})}.$

In view of (3.14), this gives (3.16) for $g_{\mathbf{n}}$, as required.
Furthermore, as $w_{\mathbf{m}}\in\textup{{{W}}}^{2,2}(B_{\rho})$ for all
$|\mathbf{m}|\leq k$, we have
$w_{\mathbf{n}}\in\textup{{{W}}}^{1,2}(B_{\rho})$. Now, integrating the
equation (3.9) against the function $(-1)^{k+1}{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{n}}\eta$, with an arbitrary
$\eta\in\textup{{{C}}}^{\infty}_{0}(B_{\rho})$ we obtain

$\displaystyle 0=(-1)^{k+1}\int\nabla w\cdot\nabla\ {\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{n}}\eta\,d\mathbf{y}+(-1)^{k+1}2\ell$ $\displaystyle\
\int\boldsymbol{\alpha}\cdot\nabla w\,{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{n}}\eta\,d\mathbf{y}$ $\displaystyle+$ $\displaystyle\
(-1)^{k+1}\ell^{2}\int\beta w\,{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{n}}\eta\,d\mathbf{y}$ $\displaystyle=\int\nabla
w_{\mathbf{n}}\cdot\nabla\ \eta\,d\mathbf{y}+2\ell$ $\displaystyle\
\int\boldsymbol{\alpha}\cdot\nabla w_{\mathbf{n}}\,\eta\,d\mathbf{y}$
$\displaystyle+$ $\displaystyle\ \ell^{2}\int\beta
w_{\mathbf{n}}\eta\,d\mathbf{y}-\int g_{\mathbf{n}}\,\eta\,d\mathbf{y},$

and hence $w_{\mathbf{n}}$ is a weak solution of (3.13) in $B_{\rho}$. Since
the coefficients and the right-hand side of (3.13) are bounded uniformly in
$\ell$, by Proposition 3.1,
$w_{\mathbf{n}}\in\textup{{{W}}}^{2,2}(B_{r})\cap\textup{{{C}}}^{1}(\overline{B_{r}})$
for all $r<\rho$ and

$\displaystyle\|w_{\mathbf{n}}\|_{\textup{{{C}}}^{1}(\overline{B_{r}})}\lesssim$
$\displaystyle\
\|w_{\mathbf{n}}\|_{\textup{{{L}}}^{\infty}(B_{\rho})}+\|g_{\mathbf{n}}\|_{\textup{{{L}}}^{\infty}(B_{\rho})}$
(3.17) $\displaystyle\lesssim$ $\displaystyle\
\|w_{\mathbf{n}}\|_{\textup{{{L}}}^{\infty}(B_{\rho})}+\ell\|\nabla
w\|_{\textup{{{L}}}^{\infty}(B_{\nu})}+\ell^{2}\|w\|_{\textup{{{L}}}^{\infty}(B_{\nu})},$

where we have also used (3.16). Let $\mathbf{m},|\mathbf{m}|=k$, be such that
$|\mathbf{n}-\mathbf{m}|=1$. Therefore, by (3.11),

$\displaystyle\|w_{\mathbf{n}}\|_{\textup{{{L}}}^{\infty}(B_{\rho})}\lesssim\|\nabla
w_{\mathbf{m}}\|_{\textup{{{L}}}^{\infty}(B_{\rho})}\lesssim\|\nabla
w\|_{\textup{{{L}}}^{\infty}(B_{\nu})}+\ell^{2}\,\|w\|_{\textup{{{L}}}^{\infty}(B_{\nu})}.$

Together with (3.2) this leads to

$\displaystyle\|w_{\mathbf{n}}\|_{\textup{{{C}}}^{1}(\overline{B_{r}})}\lesssim\|\nabla
w\|_{\textup{{{L}}}^{\infty}(B_{\nu})}+\ell^{2}\|w\|_{\textup{{{L}}}^{\infty}(B_{\nu})},$

as required. It remains to conclude that by induction, the bound (3.11) holds
for all $\mathbf{m}\in\mathbb{N}_{0}^{3M}$.

Proof of (3.12) is also conducted by induction. Assume that $|\mathbf{m}|=2$.
From (3.2) we get that

$\displaystyle\|w_{\mathbf{m}}\|_{\textup{{{C}}}^{1}(\overline{B_{r}})}\lesssim$
$\displaystyle\
\|w_{\mathbf{m}}\|_{\textup{{{L}}}^{\infty}(B_{\rho})}+\ell\|\nabla
w\|_{\textup{{{L}}}^{\infty}(B_{\nu})}+\ell^{2}\|w\|_{\textup{{{L}}}^{\infty}(B_{\nu})}$
(3.18) $\displaystyle\lesssim$ $\displaystyle\
\max_{\mathbf{l}:|\mathbf{l}|=2}\|{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{l}}w\|_{\textup{{{L}}}^{\infty}(B_{\nu})}+\ell\|\nabla
w\|_{\textup{{{L}}}^{\infty}(B_{\nu})}+\ell^{2}\|w\|_{\textup{{{L}}}^{2}(B_{\nu})},$

for all $r<\rho<\nu<R$, which gives (3.12). The bound (3.2) serves as the
induction base. Let us now provide the induction step. Suppose that (3.12)
holds for all $\mathbf{m}$ such that $2\leq|\mathbf{m}|\leq k$ with some
$k=2,3,\dots$. Let us prove that it holds for all $w_{\mathbf{n}}$ where
$|\mathbf{n}|=k+1$. Let $\mathbf{m},|\mathbf{m}|=k$, be such that
$|\mathbf{n}-\mathbf{m}|=1$. Thus (3.12) for $w_{\mathbf{m}}$ implies that

$\displaystyle\|w_{\mathbf{n}}\|_{\textup{{{L}}}^{\infty}(B_{\rho})}\lesssim\|\nabla
w_{\mathbf{m}}\|_{\textup{{{L}}}^{\infty}(B_{\rho})}\lesssim\max_{\mathbf{l}:|\mathbf{l}|=2}\|{\sf
D}_{\boldsymbol{\sf
P}}^{\mathbf{l}}w\|_{\textup{{{L}}}^{\infty}(B_{\nu})}+\ell\|\nabla
w\|_{\textup{{{L}}}^{\infty}(B_{\nu})}+\ell^{2}\|w\|_{\textup{{{L}}}^{2}(B_{\nu})}.$

Substituting this inequality in (3.2), we obtain (3.12) for $w_{\mathbf{n}}$.

Consequently, (3.12) holds for all $\mathbf{m}\in\mathbb{N}_{0}^{3M}$,
$|\mathbf{m}|\geq 2$. ∎

We would like to point out one fact which was not needed in the above proof
but deserves mentioning. The bound (3.11) for $|\mathbf{m}|=1$ and bound
(3.12) for $|\mathbf{m}|\geq 2$ imply (3.11) for all $|\mathbf{m}|\geq 2$.
Indeed, by (3.11) with $|\mathbf{q}|=1$, we have

$\displaystyle\max_{\mathbf{l}:|\mathbf{l}|=2}\|{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{l}}w\|_{\textup{{{L}}}^{\infty}(B_{\rho})}\lesssim\sum_{|\mathbf{q}|=1}\|\nabla
w_{\mathbf{q}}\|_{\textup{{{L}}}^{\infty}(B_{\rho})}\lesssim\|\nabla
w\|_{\textup{{{L}}}^{\infty}(B_{\nu})}+\ell^{2}\|w\|_{\textup{{{L}}}^{\infty}(B_{\nu})},$

for all $\rho<\nu<R$. After substitution in (3.12) this gives (3.11) for all
$|\mathbf{m}|\geq 2$, as claimed.

###### Proof of Theorem 3.2.

Since

$\displaystyle\nabla^{k}w_{\mathbf{m}}(\mathbf{y})=\ell^{|\mathbf{m}|+k}\big{(}\nabla^{k}{\sf
D}_{\boldsymbol{\sf P}}^{\mathbf{m}}u\big{)}(\mathbf{x}_{0}+\ell\mathbf{y}),\
k=0,1,\quad\mathbf{m}\in\mathbb{N}_{0}^{3M},$

the bound (3.11) for $|\mathbf{m}|\geq 1$ rewrites as (3.7). If
$\mathbf{m}=\mathbb{0}$ and $|k|=1$, then (3.7) is trivial.

The bound (3.2) is obtained from (3.12) in the same way. ∎

##  4\. The Schrödinger equation

###  4.1. Reduction of the Schrödinger equation

Here we apply the bounds obtained in the previous section to the Schrödinger
equation

(4.1) $\displaystyle-\Delta\psi+(V-E)\psi=0,$

where the potential $V$ is defined in (1.1). Throughout this section we do not
impose the condition $\psi\in\textup{{{L}}}^{2}(\mathbb{R}^{3N})$, but
consider local solutions of the equation (4.1). In order to reduce (4.1) to
the equation of the form (3.1) we use the representation $\psi=e^{F}\phi$ with
a function $F$ such that

(4.2) $\displaystyle F,\nabla F\in\textup{{{L}}}^{\infty}(\mathbb{R}^{3N}).$

In the mathematical physics literature the function $e^{F}$ is often called
the Jastrow factor, see e.g. [8], [10]. After the substitution the equation
(4.1) rewrites as

(4.3) $\displaystyle-\Delta\phi-2\nabla F\cdot\nabla\phi+(V-\Delta F-|\nabla
F|^{2}-E)\phi=0.$

More precisely, if $\psi\in\textup{{{W}}}^{1,2}_{\tiny{\rm
loc}}(\mathbb{R}^{3N})$ is a weak solution of (4.1), then we also have
$\phi\in\textup{{{W}}}^{1,2}_{\tiny{\rm loc}}(\mathbb{R}^{3N})$ and $\phi$ is
a weak solution of (4.3). We make a standard choice of the function $F$ which
ensures that the factor $\phi$ has better smoothness properties than $\psi$:

$\displaystyle F(\mathbf{x})=\sum_{j=1}^{N}\bigg{(}-\frac{Z}{2}|x_{j}|+$
$\displaystyle\ \frac{1}{4}\sum_{j<k\leq N}|x_{j}-x_{k}|\bigg{)}$ (4.4)
$\displaystyle+$ $\displaystyle\
\sum_{k=1}^{N}\bigg{(}\frac{Z}{2}\sqrt{|x_{j}|^{2}+1}-\frac{1}{4}\sum_{j<k\leq
N}\sqrt{|x_{j}-x_{k}|^{2}+1}\bigg{)}.$

In fact, the conventional choice of $F$ is given by the first sum on the
right-hand side, which has a bounded gradient, whereas the second term is
added to ensure that $F$ itself is bounded on $\mathbb{R}^{3N}$. Thus both
conditions (4.2) are satisfied. Due to the straightforward identities

$\displaystyle\Delta|x_{j}-x_{k}|=\frac{4}{|x_{j}-x_{k}|},\quad\Delta|x_{j}|=\frac{2}{|x_{j}|},$

we have

$\displaystyle\Delta\bigg{[}\sum_{j=1}^{N}\bigg{(}-\frac{Z}{2}|x_{j}|+\frac{1}{4}\sum_{j<k\leq
N}|x_{j}-x_{k}|\bigg{)}\bigg{]}=V(\mathbf{x}).$

Therefore

(4.5) $\displaystyle\nabla^{k}(-\Delta
F+V)\in\textup{{{L}}}^{\infty}(\mathbb{R}^{3N}),\ \quad\textup{for all}\quad
k=0,1,\dots.$

The following fact was proved in [8, Proposition A.2]. We provide a somewhat
different (shorter) proof that uses only Proposition 3.1.

######  Lemma 4.1.

Let $\psi\in\textup{{{W}}}^{1,2}_{\tiny{\rm loc}}(\mathbb{R}^{3N})$ be a weak
solution of (4.1). Then $\psi\in\textup{{{W}}}^{1,\infty}_{\tiny{\rm
loc}}(\mathbb{R}^{3N})$ and for all numbers $r>0$ and $R>0$ such that $r<R$ we
have

(4.6)
$\displaystyle\|\psi\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},r))}+\|\nabla\psi\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},r))}\lesssim\|\psi\|_{\textup{{{L}}}^{2}(B(\mathbf{x}_{0},R))},$

where the implicit constant does not depend on $\psi$ and
$\mathbf{x}_{0}\in\mathbb{R}^{3N}$, but depends on $r$ and $R$.

###### Proof.

The function $\phi=e^{-F}\psi$ is a weak solution of the equation (4.3) in
$B(\mathbf{x}_{0},R)$ with arbitrary $\mathbf{x}_{0}\in\mathbb{R}^{3N}$ and
$R>0$. As established above, the coefficients in this equation are uniformly
bounded on $\mathbb{R}^{3N}$. According to Proposition 3.1,
$\phi\in\textup{{{C}}}^{1}(\mathbb{R}^{3N})$ and for any pair of radii $r,R$,
$r<R$, we have the bound

(4.7)
$\displaystyle\|\phi\|_{\textup{{{L}}}^{\infty}(B_{r})}+\|\nabla\phi\|_{\textup{{{L}}}^{\infty}(B_{r})}\lesssim\|\phi\|_{\textup{{{L}}}^{2}(B_{R})},$

where $B_{R}=B(\mathbf{x}_{0},R)$, and the implicit constant is independent of
$\mathbf{x}_{0}$. Since $F$ and $\nabla F$ satisfy (4.2), we can write for
$\psi$ that

$\displaystyle\|\psi\|_{\textup{{{L}}}^{\infty}(B_{r})}\lesssim\|\phi\|_{\textup{{{L}}}^{\infty}(B_{r})},\quad\|\phi$
$\displaystyle\|_{\textup{{{L}}}^{2}(B_{R})}\lesssim\|\psi\|_{\textup{{{L}}}^{2}(B_{R})}\quad\textup{and}$
$\displaystyle\|\nabla\psi\|_{\textup{{{L}}}^{\infty}(B_{r})}\lesssim\|\nabla\phi\|_{\textup{{{L}}}^{\infty}(B_{r})}$
$\displaystyle+\|\phi\|_{\textup{{{L}}}^{\infty}(B_{r})}.$

Therefore (4.7) implies (4.6), as required. ∎

###  4.2. Cluster derivatives of $\psi$: application of Theorem 3.2

For a cluster ${\sf P}\subset\\{1,2,\dots,N\\}$ introduce the set

$\displaystyle\Sigma_{{\sf
P}}=\bigg{\\{}\mathbf{x}\in\mathbb{R}^{3N}:\prod_{j\in{\sf
P}}|x_{j}|\prod_{k\in{\sf P},l\in{\sf P}^{\rm c}}|x_{k}-x_{l}|=0\bigg{\\}},$

and the distance

(4.8) $\displaystyle\operatorname{d}_{{\sf
P}}(\mathbf{x})=\min\\{|x_{j}|,\frac{1}{\sqrt{2}}|x_{j}-x_{k}|:j\in{\sf P},\
k\in{\sf P}^{\rm c}\\},\quad\lambda_{{\sf
P}}(\mathbf{x})=\min\\{1,\operatorname{d}_{{\sf P}}(\mathbf{x})\\}.$

The paper [8] contains bounds for cluster derivatives ${\sf D}_{\sf
P}^{m}\psi$ (see e.g. [8, Proposition 1.10]) outside the set $\Sigma_{\sf P}$,
depending explicitly on the distance $\operatorname{d}_{\sf P}$. We need a
generalization of this result to cluster sets $\boldsymbol{\sf P}=\\{{\sf
P}_{1},{\sf P}_{2},\dots,{\sf P}_{M}\\}$. Define

$\displaystyle\Sigma_{\boldsymbol{\sf P}}=\cup_{j=1}^{M}\Sigma_{{\sf P}_{j}}$

and

(4.9) $\displaystyle\operatorname{d}_{\boldsymbol{\sf
P}}(\mathbf{x})=\min_{j}\operatorname{d}_{{\sf
P}_{j}}(\mathbf{x}),\quad\lambda_{\boldsymbol{\sf
P}}(\mathbf{x})=\min\\{1,\operatorname{d}_{\boldsymbol{\sf
P}}(\mathbf{x})\\}.$

The presence of the factor $1/\sqrt{2}$ in the definition (4.8) is convenient
since it ensures that the function $\operatorname{d}_{\boldsymbol{\sf P}}$ is
Lipschitz with Lipschitz constant $=1$:

######  Lemma 4.2.

For all $\mathbf{x},\mathbf{y}\in\mathbb{R}^{3N}$ we have

(4.10) $\displaystyle|\operatorname{d}_{\boldsymbol{\sf
P}}(\mathbf{x})-\operatorname{d}_{\boldsymbol{\sf
P}}(\mathbf{y})|\leq|\mathbf{x}-\mathbf{y}|.$

The same inequality holds for the function $\lambda_{\boldsymbol{\sf
P}}(\mathbf{x})$.

###### Proof.

It suffices to prove the inequality just for one cluster ${\sf P}$. Let
$j\in{\sf P}$, $k\in{\sf P}^{\rm c}$, and estimate:

$\displaystyle|x_{j}|\leq$ $\displaystyle\
|y_{j}|+|x_{j}-y_{j}|\leq|y_{j}|+|\mathbf{x}-\mathbf{y}|,$
$\displaystyle|x_{j}-x_{k}|\leq$ $\displaystyle\
|y_{j}-y_{k}|+|y_{j}-x_{j}|+|y_{k}-x_{k}|\leq|y_{j}-y_{k}|+\sqrt{2}\,|\mathbf{x}-\mathbf{y}|.$

For the last inequality we used the elementary fact that if $a^{2}+b^{2}\leq
c^{2}$ for some positive $a,b$ and $c$, then $a+b\leq\sqrt{2}\,c$. Taking the
minimum (4.8), we get $\operatorname{d}_{\sf
P}(\mathbf{x})\leq\operatorname{d}_{{\sf
P}}(\mathbf{y})+|\mathbf{x}-\mathbf{y}|$, which gives (4.10). The function
$\lambda_{\boldsymbol{\sf P}}$ trivially satisfies the same inequality. ∎

Our objective is to find $\textup{{{L}}}^{\infty}$-bounds for the cluster
derivatives ${\sf D}_{\boldsymbol{\sf P}}^{\mathbf{m}}\psi$ outside the set
$\Sigma_{\boldsymbol{\sf P}}$. Let us fix
$\mathbf{x}_{0}\in\mathbb{R}^{3N}\setminus\Sigma_{\boldsymbol{\sf P}}$, and
denote

$\displaystyle\lambda:=\lambda_{\boldsymbol{\sf P}}(\mathbf{x}_{0}).$

Observe that for any $R\in(0,1)$ the inclusion holds:
$B(\mathbf{x}_{0},R\lambda)\subset\mathbb{R}^{3N}\setminus\Sigma_{\boldsymbol{\sf
P}}$. Indeed, by Lemma 4.2,

$\displaystyle|\lambda_{\boldsymbol{\sf
P}}(\mathbf{x})-\lambda|\leq|\mathbf{x}-\mathbf{x}_{0}|<R\lambda,\quad\mathbf{x}\in
B(\mathbf{x}_{0},R\lambda),$

which implies that

(4.11) $\displaystyle 0<(1-R)\lambda\leq\lambda_{\boldsymbol{\sf
P}}(\mathbf{x})\leq(1+R)\lambda,\quad\textup{for all}\quad\mathbf{x}\in
B(\mathbf{x}_{0},R\lambda),$

and hence proves the claim. In the next theorem we estimate the derivatives
${\sf D}_{\boldsymbol{\sf P}}^{\mathbf{m}}\psi$ in the ball
$B(\mathbf{x}_{0},R\lambda)$.

######  Theorem 4.3.

Let $\mathbf{x}_{0}\in\mathbb{R}^{3N}\setminus\Sigma_{\boldsymbol{\sf P}}$ and
let $\lambda=\lambda_{\boldsymbol{\sf P}}(\mathbf{x}_{0})$. Then for any $r,R$
such that $0<r<R<1$, and for all $\mathbf{m}\in\mathbb{N}_{0}^{3M}$, the
cluster derivatives ${\sf D}_{\boldsymbol{\sf P}}^{\mathbf{m}}\psi$ belong to
$\textup{{{C}}}^{1}\big{(}\overline{B(\mathbf{x}_{0},R\lambda)}\big{)}$.
Moreover, if $|\mathbf{m}|+k\geq 1$ with $k=0,1$, then

(4.12) $\displaystyle\|{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{m}}\nabla^{k}\psi\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},r\lambda))}\lesssim\lambda^{1-|\mathbf{m}|-k}\big{(}\|\psi\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},R\lambda))}+\|\nabla\psi\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},R\lambda))}\big{)},$

with an implicit constant depending on $r,R$, but independent of
$\mathbf{x}_{0}\in\mathbb{R}^{3N}\setminus\Sigma_{\boldsymbol{\sf P}}$.

For a single cluster ${\sf P}$ the bound (4.12), even for arbitrary
$\textup{{{L}}}^{p}$-norms with $p\in(1,\infty]$ was found in [8, Proposition
1.10]. The proof in [8] does not immediately generalize to arbitrary cluster
sets. As in [8] we use the reduction to the equation (4.3). In [8] the choice
of the Jastrow factor depended on the cluster ${\sf P}$, whereas we use the
standard function (4.1) independent of the cluster set $\boldsymbol{\sf P}$.
To check that the coefficients in the equation (4.3) satisfy the conditions of
Theorem 3.2, we start with the following elementary observation.

######  Lemma 4.4.

Let $g\in\textup{{{C}}}^{\infty}(\mathbb{R}^{3}\setminus\\{0\\})$ be such that

$\displaystyle|\partial^{m}g(x)|\lesssim|x|^{-|m|},\quad\textup{for all}\quad
m\in\mathbb{N}_{0}^{3}.$

Then for any cluster set $\boldsymbol{\sf P}$ and any $j,k=1,2,\dots,N$, we
have

$\displaystyle|{\sf D}_{\boldsymbol{\sf P}}^{\mathbf{m}}g(x_{j})|+|{\sf
D}_{\boldsymbol{\sf
P}}^{\mathbf{m}}g(x_{j}-x_{k})|\lesssim\operatorname{d}_{\boldsymbol{\sf
P}}(\mathbf{x})^{-|\mathbf{m}|},$

for all $\mathbf{x}\in\mathbb{R}^{3N}\setminus\Sigma_{\boldsymbol{\sf P}}$.

###### Proof.

For simplicity we prove the lemma for one cluster, which is denoted ${\sf P}$.
Assume that $m\in\mathbb{N}_{0}^{3}$, $|m|\geq 1$. It is clear that $D_{\sf
P}^{m}g(x_{j}-x_{k})=0$ if $j,k\in{\sf P}$ or $j,k\in{\sf P}^{\rm c}$. If
$j\in{\sf P},k\in{\sf P}^{\rm c}$ or $k\in{\sf P},j\in{\sf P}^{\rm c}$, then

$\displaystyle|{\sf D}_{\sf
P}^{m}g(x_{j}-x_{k})|=|\partial_{x}^{m}g(x)|_{x=x_{j}-x_{k}}\lesssim|x_{j}-x_{k}|^{-|m|}\lesssim\operatorname{d}_{\sf
P}(\mathbf{x})^{-|m|},$

as claimed. In the same way we get the required estimate for ${\sf D}_{{\sf
P}}^{m}g(x_{j})$. ∎

######  Corollary 4.5.

Let $F$ be the function (4.1). Then for any cluster set $\boldsymbol{\sf P}$
we have for all $\mathbf{x}\notin\Sigma_{\boldsymbol{\sf P}}$ and all
$\mathbf{m}$ that

(4.13) $\displaystyle\big{|}{\sf D}_{\boldsymbol{\sf P}}^{\mathbf{m}}\nabla
F(\mathbf{x})\big{|}+\big{|}{\sf D}_{\boldsymbol{\sf P}}^{\mathbf{m}}|\nabla
F(\mathbf{x})|^{2}\big{|}\lesssim\operatorname{d}_{\boldsymbol{\sf
P}}(\mathbf{x})^{-|\mathbf{m}|}.$

If $|\mathbf{m}|\geq 1$, then

(4.14) $\displaystyle\big{|}{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{m}}e^{F(\mathbf{x})}\big{|}\lesssim\lambda_{\boldsymbol{\sf
P}}(\mathbf{x})^{1-|\mathbf{m}|}.$

###### Proof.

Since $F$ is a sum of terms of the type $|x_{j}-x_{k}|$ and $|x_{j}|$, the
required bound for $\nabla F$ immediately follows from the definition (4.1)
and Lemma 4.4. For $|\nabla F|^{2}$ we use the Leibniz rule, which leads again
to (4.13).

To prove the bound (4.14) observe that the derivative ${\sf
D}_{\boldsymbol{\sf P}}^{\mathbf{m}}e^{F}$ represents as a sum of finitely
many terms of the form

$\displaystyle({\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{k}_{1}}F\big{)}^{n_{1}}\,({\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{k}_{2}}F\big{)}^{n_{2}}\cdots({\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{k}_{s}}F\big{)}^{n_{s}}\,e^{F},$

where
$1\leq|\mathbf{k}_{1}|<|\mathbf{k}_{2}|<\cdots<|\mathbf{k}_{s}|\leq|\mathbf{m}|$,
$n_{j}\geq 1$, and
$|\mathbf{k}_{1}|\,n_{1}+|\mathbf{k}_{2}|\,n_{2}+\cdots+|\mathbf{k}_{s}|\,n_{s}=|\mathbf{m}|$.
Each such term estimates by

$\displaystyle\operatorname{d}_{\boldsymbol{\sf
P}}(\mathbf{x})^{n_{1}(1-|\mathbf{k}_{1}|)+n_{2}(1-|\mathbf{k}_{2}|)+\cdots
n_{s}(1-|\mathbf{k}_{1}|)}=\operatorname{d}_{\boldsymbol{\sf
P}}(\mathbf{x})^{n_{1}+n_{2}+\dots+n_{s}-|\mathbf{m}|}\leq\lambda_{\boldsymbol{\sf
P}}(\mathbf{x})^{1-|\mathbf{m}|},$

as required. ∎

###### Proof of Theorem 4.3.

In view of (4.3), the function $\phi=e^{-F}\psi$ satisfies the equation (3.1)
where

$\displaystyle\mathbf{a}=-2\nabla F,\quad b=V-\Delta F-|\nabla F|^{2}-E.$

Let us fix a number $R_{1}\in(R,1)$. By virtue of (4.5) and (4.13), thus
defined coefficients $\mathbf{a}$ and $b$ satisfy the bound

$\displaystyle|{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{m}}\mathbf{a}(\mathbf{x})|+|{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{m}}b(\mathbf{x})|\lesssim 1+\operatorname{d}_{\boldsymbol{\sf
P}}(\mathbf{x})^{-|\mathbf{m}|}\lesssim\lambda_{\boldsymbol{\sf
P}}(\mathbf{x})^{-|\mathbf{m}|}\lesssim\lambda^{-|\mathbf{m}|},\quad\mathbf{x}\in
B(\mathbf{x}_{0},R_{1}\lambda),$

where $\lambda=\lambda_{\boldsymbol{\sf P}}(\mathbf{x}_{0})$. For the last
inequality we have used (4.11). Thus the condition (3.6) is fulfilled with
$\ell=\lambda\leq 1$. Consequently, by Theorem 3.2, ${\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{m}}\phi\in\textup{{{C}}}^{1}(\overline{B(\mathbf{x}_{0},r\lambda)})$
for all $r<R_{1}$. Moreover, if $|\mathbf{m}|+k\geq 1$, where $k=0,1$, then
for $r<R$ we have

(4.15) $\displaystyle\|\nabla^{k}{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{m}}\phi\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},r\lambda))}\lesssim\lambda^{1-|\mathbf{m}|-k}\,\big{(}\lambda\,\|\phi\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},R\lambda))}+\|\nabla\phi\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},R\lambda))}\big{)}.$

Now we need to replace $\phi$ with the function $\psi=e^{F}\phi$. Let us prove
(4.12) with $k=0$, $|\mathbf{m}|\geq 1$. By the Leibniz rule,

$\displaystyle{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{m}}\big{(}e^{F}\phi\big{)}=$ $\displaystyle\
\sum_{0\leq\mathbf{q}\leq\mathbf{m}}{\mathbf{m}\choose\mathbf{q}}{\sf
D}_{\boldsymbol{\sf P}}^{\mathbf{m}-\mathbf{q}}\big{(}e^{F}\big{)}{\sf
D}_{\boldsymbol{\sf P}}^{\mathbf{q}}\phi$ $\displaystyle=$ $\displaystyle\
\sum_{\begin{subarray}{c}0\leq\mathbf{q}\leq\mathbf{m}\\\
0<|\mathbf{q}|<|\mathbf{m}|\end{subarray}}{\mathbf{m}\choose\mathbf{q}}{\sf
D}_{\boldsymbol{\sf P}}^{\mathbf{m}-\mathbf{q}}\big{(}e^{F}\big{)}{\sf
D}_{\boldsymbol{\sf P}}^{\mathbf{q}}\phi+{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{m}}\big{(}e^{F}\big{)}\phi+e^{F}{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{m}}\phi.$

By (4.14) and (4.15), for $\mathbf{x}\in B(\mathbf{x}_{0},r\lambda)$ the first
sum is bounded by

$\displaystyle\lambda^{2-|\mathbf{m}|}\,\big{(}\lambda\|\phi\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},R\lambda))}+\|\nabla\phi\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},R\lambda))}\big{)}.$

The second term is estimated with the help of (4.14) by

$\displaystyle\lambda^{1-|\mathbf{m}|}\,\|\phi\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},R\lambda))}.$

Using (4.15) the third term is estimated by

$\displaystyle\lambda^{1-|\mathbf{m}|}\,\big{(}\lambda\,\|\phi\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},R\lambda))}+\|\nabla\phi\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},R\lambda))}\big{)}.$

Consequently,

(4.16) $\displaystyle\|{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{m}}\psi\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},r\lambda))}\lesssim\lambda^{1-|\mathbf{m}|}\,\big{(}\|\phi\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},R\lambda))}+\|\nabla\phi\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x}_{0},R\lambda))}\big{)}.$

It remains to note that in view of (4.2),

$\displaystyle|\phi|\lesssim|\psi|,\quad|\nabla\phi|\lesssim|\psi|+|\nabla\psi|,$

so that the right-hand side of (4.16) is estimated by the right-hand side of
(4.12) with $k=0$.

The case $k=1$ is done in the same way. ∎

Later on we use Theorem 4.3 in a slightly different form:

######  Corollary 4.6.

For all $\mathbf{m}\in\mathbb{N}_{0}^{3M}$ the cluster derivatives ${\sf
D}_{\boldsymbol{\sf P}}^{\mathbf{m}}\psi$ belong to
$\textup{{{C}}}^{1}\big{(}\mathbb{R}^{3N}\setminus\Sigma_{\boldsymbol{\sf
P}}\big{)}$, and under the condition $|\mathbf{m}|+k\geq 1$ with $k=0,1$, for
all $\mathbf{x}\in\mathbb{R}^{3N}\setminus\Sigma_{\boldsymbol{\sf P}}$ and all
$R>0$, we have

(4.17) $\displaystyle|{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{m}}\nabla^{k}\psi(\mathbf{x})|\lesssim$ $\displaystyle\
\lambda_{\boldsymbol{\sf
P}}(\mathbf{x})^{1-|\mathbf{m}|-k}f_{\infty}(\mathbf{x};R),$ (4.18)
$\displaystyle f_{\infty}(\mathbf{x};R)=$ $\displaystyle\
\|\psi\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x},R))}+\|\nabla\psi\|_{\textup{{{L}}}^{\infty}(B(\mathbf{x},R))}.$

The implicit constant in (4.17) is independent of $\psi$ and $\mathbf{x}$, but
may depend on $R$.

###### Proof.

Let $R_{1}=\min\\{1/2,R\\}$. Then it follows from (4.12) that

$\displaystyle|{\sf D}_{\boldsymbol{\sf
P}}^{\mathbf{m}}\nabla^{k}\psi(\mathbf{x})|\lesssim$ $\displaystyle\
\lambda(\mathbf{x})^{1-|\mathbf{m}|-k}f_{\infty}(\mathbf{x};R_{1}\lambda(\mathbf{x}))\leq\lambda(\mathbf{x})^{1-|\mathbf{m}|-k}f_{\infty}(\mathbf{x};R).$

Here we have used the fact that $\lambda(\mathbf{x})\leq 1$. This completes
the proof. ∎

##  5\. Auxiliary integral bounds

Here we derive several integral bounds that are instrumental in the proof of
the main result in Section 6.

Let $\psi\in\textup{{{L}}}^{2}(\mathbb{R}^{3N})$ be an eigenfunction. Along
with the notation (4.18) it is convenient to introduce for arbitrary $R>0$
also

$\displaystyle
f_{2}(\mathbf{x})=f_{2}(\mathbf{x};R)=\|\psi\|_{\textup{{{L}}}^{2}(B(\mathbf{x},R))}.$

According to (4.6), for any $R>0$ we have

(5.1) $\displaystyle f_{\infty}(\mathbf{x};R)\lesssim f_{2}(\mathbf{x};2R),$

with an implicit constant depending on $R$.

The next lemma provides bounds for integrals involving the functions
$f_{\infty}(\mathbf{x})$ and $f_{2}(\mathbf{x})$. Recall that the function
$\mathcal{M}_{\varepsilon}(x,y,\hat{\mathbf{x}})$ is defined in (2.2), and the
density $\rho(x)$ – in (1.4)

######  Lemma 5.1.

Let $x,y\in\mathbb{R}^{3}$. Then

(5.2) $\displaystyle\int
f_{\infty}(x,\hat{\mathbf{x}};R)f_{\infty}(y,\hat{\mathbf{x}};R)d\hat{\mathbf{x}}\lesssim\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(x,2R))}\big{)}^{\frac{1}{2}}\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(y,2R))}\big{)}^{\frac{1}{2}}.$

For any $G\in\textup{{{L}}}^{1}(\mathbb{R}^{3})$ we have

$\displaystyle\int\big{[}|G(x_{j}-x_{k})|+|G(t-x_{k})|+$ $\displaystyle\
|G(x_{k})|\big{]}f_{\infty}(x,\hat{\mathbf{x}};R)f_{\infty}(y,\hat{\mathbf{x}};R)\,d\hat{\mathbf{x}}$
(5.3) $\displaystyle\
\lesssim\|G\|_{\textup{{{L}}}^{1}(\mathbb{R}^{3})}\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(x,2R))}\big{)}^{\frac{1}{2}}\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(y,2R))}\big{)}^{\frac{1}{2}},$

for all $j,k=2,3,\dots,N,j\not=k$, and $t\in\mathbb{R}^{3}$. In particular,

(5.4)
$\displaystyle\int\mathcal{M}_{\varepsilon}(x,y,\hat{\mathbf{x}})f_{\infty}(x,\hat{\mathbf{x}};R)f_{\infty}(y,\hat{\mathbf{x}};R)\,d\hat{\mathbf{x}}\lesssim\varepsilon^{3}\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(x,2R))}\big{)}^{\frac{1}{2}}\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(y,2R))}\big{)}^{\frac{1}{2}}.$

The implicit constants in the bounds (5.2), (5.1) and (5.4) depend on $R$, but
are independent of $x,y,t\in\mathbb{R}^{3}$.

###### Proof.

Due to the Schwarz inequality, it suffices to estimate the integrals for
$x=y$. For $u\in\mathbb{R}^{d}$ and $R>0$ denote by
$\mathbbm{1}^{(d)}_{u,R}(x),x\in\mathbb{R}^{d},$ the indicator function of the
ball $B(u,R)\subset\mathbb{R}^{d}$.

Using (5.1) we get

$\displaystyle\int\big{(}f_{\infty}(x,\hat{\mathbf{x}};R)\big{)}^{2}d\hat{\mathbf{x}}\lesssim$
$\displaystyle\
\int\big{(}f_{2}(x,\hat{\mathbf{x}};2R)\big{)}^{2}d\hat{\mathbf{x}}$
$\displaystyle=$ $\displaystyle\
\int\int|\psi(\mathbf{z})|^{2}\mathbbm{1}^{(3N)}_{(x,\hat{\mathbf{x}}),2R}(\mathbf{z})\,d\mathbf{z}\,d\hat{\mathbf{x}}$
$\displaystyle=$ $\displaystyle\
\int|\psi(\mathbf{z})|^{2}\int\mathbbm{1}^{(3N)}_{\mathbf{z},2R}(x,\hat{\mathbf{x}})\,d\hat{\mathbf{x}}\,d\mathbf{z}.$

Observe that

$\displaystyle\mathbbm{1}^{(3N)}_{\mathbf{z},2R}(x,\hat{\mathbf{x}})\leq\mathbbm{1}^{(3)}_{z_{1},2R}(x)\mathbbm{1}^{(3N-3)}_{\hat{\mathbf{z}},2R}(\hat{\mathbf{x}})=\mathbbm{1}^{(3)}_{x,2R}(z_{1})\mathbbm{1}^{(3N-3)}_{\hat{\mathbf{z}},2R}(\hat{\mathbf{x}}),$

so the integral does not exceed

$\displaystyle\int|\psi(\mathbf{z})|^{2}\mathbbm{1}^{(3)}_{x,2R}(z_{1})\bigg{(}\int\mathbbm{1}^{(3N-3)}_{\hat{\mathbf{z}},2R}(\hat{\mathbf{x}})\,d\hat{\mathbf{x}}\bigg{)}\,d\mathbf{z}\lesssim
R^{3N-3}\int\limits_{|z-x|<2R}\rho(z)dz.$

This proves (5.2).

Proof of (5.1). Again it suffices to estimate the integral for $x=y$:

$\displaystyle\int|G(x_{j}-x_{k})|$ $\displaystyle\
\big{(}f_{\infty}(x,\hat{\mathbf{x}};R)\big{)}^{2}\,d\hat{\mathbf{x}}\lesssim\int|G(x_{j}-x_{k})|\big{(}f_{2}(x,\hat{\mathbf{x}};2R)\big{)}^{2}\,d\hat{\mathbf{x}}$
$\displaystyle=$ $\displaystyle\
\int|\psi(\mathbf{z})|^{2}\int|G(x_{j}-x_{k})|\mathbbm{1}^{(3N)}_{\mathbf{z},2R}(x,\hat{\mathbf{x}})\,d\hat{\mathbf{x}}\,d\mathbf{z}.$

Represent $\hat{\mathbf{x}}=(x_{j},\tilde{\mathbf{x}}_{j})$ with
$\tilde{\mathbf{x}}_{j}\in\mathbb{R}^{3N-6}$, as defined in (1.12), and
estimate:

$\displaystyle\mathbbm{1}^{(3N)}_{\mathbf{z},2R}(x,\hat{\mathbf{x}})\leq\mathbbm{1}^{(3)}_{z_{1},2R}(x)\,\mathbbm{1}^{(3N-6)}_{\tilde{\mathbf{z}}_{j},2R}(\tilde{\mathbf{x}}_{j})=\mathbbm{1}^{(3)}_{x,2R}(z_{1})\,\mathbbm{1}^{(3N-6)}_{\tilde{\mathbf{z}}_{j},2R}(\tilde{\mathbf{x}}_{j}).$

Consequently, the integral estimates by

$\displaystyle\int|\psi(\mathbf{z})|^{2}\mathbbm{1}^{(3)}_{x,2R}(z_{1})$
$\displaystyle\
\int\mathbbm{1}_{B(\tilde{\mathbf{z}}_{j},2R)}(\tilde{\mathbf{x}}_{j})\bigg{[}\int|G(x_{j}-x_{k})|\,dx_{j}\bigg{]}\,d\tilde{\mathbf{x}}_{j}\,d\mathbf{z}$
$\displaystyle=$ $\displaystyle\
\|G\|_{\textup{{{L}}}^{1}(\mathbb{R}^{3})}\int\limits_{|z_{1}-x|<2R}|\psi(\mathbf{z})|^{2}\int\mathbbm{1}^{(3N-6)}_{\tilde{\mathbf{z}}_{j},2R}(\tilde{\mathbf{x}}_{j})\,d\tilde{\mathbf{x}}_{j}\,d\mathbf{z}$
$\displaystyle\lesssim$ $\displaystyle\
\|G\|_{\textup{{{L}}}^{1}(\mathbb{R}^{3})}R^{3N-6}\int\limits_{|z-x|<2R}\rho(z)dz,$

as claimed. The bounds with $G(t-x_{k})$ and $G(x_{k})$ are proved in the same
way.

The bound (5.4) follows from (5.1) with $G(s)=\xi(N\varepsilon^{-1}|s|)$, see
(2.1) for the definition of the function $\xi$. ∎

Let us now apply the obtained bounds to integrals involving the distance
function $\lambda_{{\sf Q}}(\mathbf{x})$ with an arbitrary cluster ${\sf Q}$.
Recall that $\lambda_{{\sf Q}}(\mathbf{x})$, $\widehat{X}_{\sf
Q}(t;\varepsilon)$ and $\widehat{T}_{{\sf Q}}(\delta)$ are defined in (4.8),
(2.4) and (2.5) respectively, and the function $h_{a}$ is defined in (1.7).

######  Lemma 5.2.

Let $a\geq 0$ and $\varepsilon>0$. Let ${\sf Q}$ be an arbitrary cluster. Then
for any $R>0$,

$\displaystyle\int\limits_{\widehat{X}_{\sf
Q}(t;\varepsilon)\cap\widehat{T}_{{\sf Q}}(\varepsilon)}\lambda_{{\sf
Q}}(t,\hat{\mathbf{x}})^{-a}f_{\infty}(x,\hat{\mathbf{x}};R)$ $\displaystyle\
f_{\infty}(y,\hat{\mathbf{x}};R)d\hat{\mathbf{x}}$ (5.5)
$\displaystyle\lesssim$ $\displaystyle\
\big{(}1+|t|^{-a}+h_{a+2}(\varepsilon)\big{)}\,\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(x,2R))}\big{)}^{\frac{1}{2}}\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(y,2R))}\big{)}^{\frac{1}{2}},$

uniformly in $x,y,t\in\mathbb{R}^{3}$. If $1\notin{\sf Q}$, then the term
$|t|^{-a}$ is absent. The implicit constant in (5.2) may depend on $R$, but is
independent of $\varepsilon$.

###### Proof.

By the definition of $\lambda_{\sf Q}$ we have

$\displaystyle\lambda_{\sf Q}(t,\hat{\mathbf{x}})^{-a}\leq$ $\displaystyle\
1+\sum_{k\in{\sf Q}}|x_{k}|^{-a}$ (5.6) $\displaystyle\ \sum_{k\in{\sf
Q}}\,|t-x_{k}|^{-a}+\sum_{j\in{\sf Q},k\in({\sf Q}^{\rm
c})^{*}}|x_{j}-x_{k}|^{-a},\quad\textup{if}\quad 1\in{\sf Q}^{\rm c},$

and

$\displaystyle\lambda_{\sf Q}(t,\hat{\mathbf{x}})^{-a}\leq 1+$ $\displaystyle\
|t|^{-a}+\sum_{k\in{\sf Q}^{*}}|x_{k}|^{-a}$ (5.7) $\displaystyle\
+\sum_{k\in{\sf Q}^{\rm c}}|t-x_{k}|^{-a}+\sum_{j\in{\sf Q}^{*},k\in{\sf
Q}^{\rm c}}|x_{j}-x_{k}|^{-a},\quad\textup{if}\quad 1\in{\sf Q}.$

Let us estimate the contributions from each of the summands. Assume first that
$1\notin{\sf Q}$ (i.e. $\lambda_{\sf Q}$ satisfies (5)) and estimate the
integral

(5.8) $\displaystyle F_{jk}(t,x,y)=\int\limits_{\widehat{X}_{\sf
Q}(t;\varepsilon)\cap\widehat{T}_{{\sf
Q}}(\varepsilon)}|x_{j}-x_{k}|^{-a}f_{\infty}(x,\hat{\mathbf{x}};R)$
$\displaystyle\ f_{\infty}(y,\hat{\mathbf{x}};R)\,d\hat{\mathbf{x}},$

for an arbitrary fixed pair $j\in{\sf Q},k\in({\sf Q}^{\rm c})^{*}$. Since
$j,k\geq 2$, in view of the definition (2.4) of $\widehat{X}_{\sf
Q}(t,\varepsilon)$, we have

$\displaystyle\widehat{X}_{\sf
Q}(t;\varepsilon)\subset\\{\hat{\mathbf{x}}\in\mathbb{R}^{3N-3}:|x_{j}-x_{k}|>\varepsilon\\}.$

Therefore,

$\displaystyle
F_{jk}(t,x,y)\leq\int\limits_{|x_{j}-x_{k}|>\varepsilon}|x_{j}-x_{k}|^{-a}\,f_{\infty}(x,\hat{\mathbf{x}};R)\,f_{\infty}(y,\hat{\mathbf{x}};R)\,d\hat{\mathbf{x}}.$

If $\varepsilon\geq 1$, then by (5.2) we have

(5.9) $\displaystyle
F_{jk}(t,x,y)\lesssim\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(x,2R))}\big{)}^{\frac{1}{2}}\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(y,2R))}\big{)}^{\frac{1}{2}}.$

If $\varepsilon<1$, then

$\displaystyle F_{jk}(t,x,y)\leq$ $\displaystyle\
\int\limits_{|x_{j}-x_{k}|>1}f_{\infty}(x,\hat{\mathbf{x}};R)\,f_{\infty}(y,\hat{\mathbf{x}};R)\,d\hat{\mathbf{x}}$
$\displaystyle\qquad+\int
G(x_{j}-x_{k})\,f_{\infty}(x,\hat{\mathbf{x}};R)\,f_{\infty}(y,\hat{\mathbf{x}};R)\,d\hat{\mathbf{x}},$

where

(5.10) $\displaystyle
G(s)=\mathbbm{1}_{\\{\varepsilon<|s|<1\\}}(s)|s|^{-a},\quad
s\in\mathbb{R}^{3}.$

For the first integral use (5.2) again. Since
$\|G\|_{\textup{{{L}}}^{1}(\mathbb{R}^{3})}\lesssim h_{a+2}(\varepsilon)$ (see
(1.7)) the second integral is bounded by

$\displaystyle
h_{a+2}(\varepsilon)\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(x,2R))}\big{)}^{\frac{1}{2}}\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(y,2R))}\big{)}^{\frac{1}{2}},$

in view of of (5.1). Thus the integral (5.8), and hence the contribution from
the last term in (5) as well, satisfies the bound (5.2).

In the same way, using (5.1) we derive the bound (5.2) for the integrals
containing the remaining terms in (5). Let us estimate, for example, the
integral

$\displaystyle F_{k}(t,x,y)=\int\limits_{\widehat{X}_{\sf
Q}(t;\varepsilon)\cap\widehat{T}_{{\sf
Q}}(\varepsilon)}|x_{k}|^{-a}\,f_{\infty}(x,\hat{\mathbf{x}};R)\,f_{\infty}(y,\hat{\mathbf{x}};R)\,d\hat{\mathbf{x}},$

for an arbitrary fixed $k\in{\sf Q}$. By the definition (2.5),

$\displaystyle\widehat{T}_{\sf
Q}(\varepsilon)\subset\\{\hat{\mathbf{x}}\in\mathbb{R}^{3N-3}:|x_{k}|>\varepsilon\\},$

so that

$\displaystyle
F_{k}(t,x,y)\leq\int\limits_{|x_{k}|>\varepsilon}|x_{k}|^{-a}\,f_{\infty}(x,\hat{\mathbf{x}};R)\,f_{\infty}(y,\hat{\mathbf{x}};R)\,d\hat{\mathbf{x}},$

As in the case of the integral (5.8), if $\varepsilon\geq 1$, then $F_{k}$
satisfies the bound (5.9). If $\varepsilon<1$, then

$\displaystyle F_{k}(t,x,y)\leq$ $\displaystyle\
\int\limits_{|x_{k}|>1}f_{\infty}(x,\hat{\mathbf{x}};R)\,f_{\infty}(y,\hat{\mathbf{x}};R)\,d\hat{\mathbf{x}}$
$\displaystyle\qquad+\int
G(x_{k})\,f_{\infty}(x,\hat{\mathbf{x}};R)\,f_{\infty}(y,\hat{\mathbf{x}};R)\,d\hat{\mathbf{x}},$

where $G$ is given by (5.10). Arguing as for $F_{jk}$ above, we conclude that
$F_{k}$ satisfies (5.2) as well. This proves (5.2) for the case $1\notin{\sf
Q}$.

Suppose now that $1\in{\sf Q}$ so that $\lambda_{{\sf Q}}$ satisfies (5). The
sums on the right-hand side of (5) are similar to those in (5) and are treated
as in the first part of the proof, and hence they lead to the estimate (5.2).
The only term which is new compared to (5) is $|t|^{-a}$. Using (5.2) we
estimate its contribution by

$\displaystyle|t|^{-a}\int
f_{\infty}(x,\hat{\mathbf{x}};R)\,f_{\infty}(y,\hat{\mathbf{x}};R)\,d\hat{\mathbf{x}}\lesssim\,|t|^{-a}\,\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(x,2R))}\big{)}^{\frac{1}{2}}\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(y,2R))}\big{)}^{\frac{1}{2}}.$

This completes the proof of (5.2). ∎

Lemma 5.2 has a useful corollary that will be crucial in the proof of Theorem
1.1. Let $\Phi=\Phi(x,y,\hat{\mathbf{x}};\varepsilon)$ be an extended cut-off
as defined in (2.6), and let ${\sf P}={\sf P}(\varepsilon)$ and ${\sf S}={\sf
S}(\varepsilon)$ be the clusters for the admissible cut-offs associated with
$\Phi$. For all $a\geq 0$ define

(5.11)
$\displaystyle\begin{cases}{\mathcal{J}}_{a}^{(1)}(x,y;\varepsilon,R)=&\
\int\limits_{\operatorname{{supp}}_{0}\Phi(x,y,\ \cdot\
;\varepsilon)}\big{(}\lambda_{\\{{\sf S}^{*},{\sf
P}\\}}(x,\hat{\mathbf{x}})\big{)}^{-a}f_{\infty}(x,\hat{\mathbf{x}};R)f_{\infty}(y,\hat{\mathbf{x}};R)\,d\hat{\mathbf{x}},\\\\[14.22636pt]
{\mathcal{J}}_{a}^{(2)}(x,y;\varepsilon,R)=&\
\int\limits_{\operatorname{{supp}}_{0}\Phi(x,y,\ \cdot\
;\varepsilon)}\big{(}\lambda_{\\{{\sf S},{\sf
P}^{*}\\}}(y,\hat{\mathbf{x}})\big{)}^{-a}f_{\infty}(x,\hat{\mathbf{x}};R)f_{\infty}(y,\hat{\mathbf{x}};R)\,d\hat{\mathbf{x}},\\\\[14.22636pt]
{\mathcal{J}}_{a}^{(3)}(x,y;\varepsilon,R)=&\
\varepsilon^{-a}\int\,\mathcal{M}_{\varepsilon}(x,y;\hat{\mathbf{x}})f_{\infty}(x,\hat{\mathbf{x}};R)f_{\infty}(y,\hat{\mathbf{x}};R)\,d\hat{\mathbf{x}}.\end{cases}$

######  Lemma 5.3.

Suppose that for some $\varepsilon>0$ we have $|x|>\varepsilon$,
$|y|>\varepsilon$ and $|x-y|>\varepsilon$. Then for all $R>0$,

$\displaystyle{\mathcal{J}}_{a}^{(1)}(x,y;\varepsilon,R)+$ $\displaystyle\
{\mathcal{J}}_{a}^{(2)}(x,y;\varepsilon,R)+{\mathcal{J}}_{a}^{(3)}(x,y;\varepsilon,R)$
(5.12) $\displaystyle\lesssim$ $\displaystyle\
\big{(}1+|x|^{-a}+|y|^{-a}+h_{a+2}(\varepsilon)\big{)}\,\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(x,2R))}\big{)}^{\frac{1}{2}}\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(y,2R))}\big{)}^{\frac{1}{2}}.$

The implicit constant in (5.2) may depend on $R$, but is independent of
$\varepsilon$.

###### Proof.

By Proposition 2.3 we may assume that ${\sf P}^{*}\subset{\sf S}^{\rm c}$.

First estimate $\mathcal{J}_{a}^{(1)}(x,y;\varepsilon,R)$. By the definition
(4.9), we have

$\displaystyle\lambda_{\\{{\sf S}^{*},{\sf
P}\\}}(x,\hat{\mathbf{x}})^{-1}\leq\lambda_{{\sf
S}^{*}}(x,\hat{\mathbf{x}})^{-1}+\lambda_{{\sf P}}(x,\hat{\mathbf{x}})^{-1}.$

Furthermore, since ${\sf P}^{*}\subset{\sf S}^{\rm c}$, we can use (2.4) and
(2.8):

$\displaystyle\operatorname{{supp}}_{0}\Phi(x,y;\ \cdot\
;\varepsilon)\subset\widehat{X}_{\sf
P}(x;\varepsilon(4N)^{-1})\cap\widehat{T}_{{\sf
P}^{*}}(\varepsilon/2)\quad\textup{and}$
$\displaystyle\operatorname{{supp}}_{0}\Phi(x,y,\ \cdot\
;\varepsilon)\subset\widehat{X}_{{\sf
S}^{*}}(x;\varepsilon(4N)^{-1})\cap\widehat{T}_{{\sf S}^{*}}(\varepsilon/2).$

These lead to the bound

$\displaystyle{\mathcal{J}}_{a}^{(1)}(x,y;\varepsilon,R)\lesssim$
$\displaystyle\ \int\limits_{\widehat{X}_{\sf
P}(x;\varepsilon(4N)^{-1})\cap\widehat{T}_{{\sf
P}}(\varepsilon/2)}\lambda_{{\sf
P}}(x,\hat{\mathbf{x}})^{-a}f_{\infty}(x,\hat{\mathbf{x}};R)f_{\infty}(y,\hat{\mathbf{x}};R)d\hat{\mathbf{x}}$
$\displaystyle\ +\int\limits_{\widehat{X}_{{\sf
S}^{*}}(x;\varepsilon(4N)^{-1})\cap\widehat{T}_{{\sf
S}^{*}}(\varepsilon/2)}\lambda_{{\sf
S}^{*}}(x,\hat{\mathbf{x}})^{-a}f_{\infty}(x,\hat{\mathbf{x}};R)f_{\infty}(y,\hat{\mathbf{x}};R)d\hat{\mathbf{x}}.$

By (5.2) with $t=x$, each of these integrals is bounded by

$\displaystyle\big{(}1+|x|^{-a}+h_{a+2}(\varepsilon)\big{)}\,\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(x,2R))}\big{)}^{\frac{1}{2}}\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(y,2R))}\big{)}^{\frac{1}{2}},$

which implies (5.3). The integral ${\mathcal{J}}_{a}^{(2)}(x,y;\varepsilon,R)$
is estimated in the same way.

The integral $\mathcal{J}_{a}^{(3)}(x,y;\varepsilon,R)$ is estimated with the
help of (5.4):

$\displaystyle\mathcal{J}_{a}^{(3)}(x,y;\varepsilon,R)\lesssim$
$\displaystyle\
\varepsilon^{3-a}\,\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(x,2R))}\big{)}^{\frac{1}{2}}\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(y,2R))}\big{)}^{\frac{1}{2}}$
$\displaystyle\lesssim$ $\displaystyle\
h_{a+2}(\varepsilon)\,\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(x,2R))}\big{)}^{\frac{1}{2}}\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(y,2R))}\big{)}^{\frac{1}{2}}.$

Combining the obtained bounds we complete the proof of (5.3). ∎

Now we are in a position to prove Theorem 1.1.

##  6\. Estimates for the density matrix: proof of Theorem 1.1

To cover both inequalities (1.1) and (1.1) in one proof we study the following
function:

(6.1)
$\displaystyle\gamma_{r,p}(x,y):=\partial_{x}^{r}\partial_{y}^{p}\gamma(x,y)=\int\partial_{x}^{r}\psi(x,\hat{\mathbf{x}})\overline{\partial_{y}^{p}\psi(y,\hat{\mathbf{x}})}\,d\hat{\mathbf{x}},$

where $r,p\in\mathbb{N}_{0}^{3}$ are such that $|r|\leq 1$, $|p|\leq 1$. By
Corollary 4.6 the integrand in (6.1) does not exceed
$f_{\infty}(x,\hat{\mathbf{x}};R)f_{\infty}(y,\hat{\mathbf{x}};R)$, see (4.18)
for the definition of $f_{\infty}(\mathbf{x};R)$. Therefore, the estimate
(5.2) entails the bound

(6.2)
$\displaystyle|\gamma_{r,p}(x,y)|\lesssim\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(x,2R))}\big{)}^{\frac{1}{2}}\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(y,2R))}\big{)}^{\frac{1}{2}}.$

In order to estimate the derivatives of $\gamma_{r,p}(x,y)$ we begin by
studying “local” quantities. For a fixed $\varepsilon>0$ let
$\Phi=\Phi(x,y,\hat{\mathbf{x}};\varepsilon)$ be an extended cut-off as
defined in (2.6). Define

(6.3)
$\displaystyle\gamma_{r,p}(x,y;\Phi,\varepsilon)=\int\Phi(x,y,\hat{\mathbf{x}};\varepsilon)\partial_{x}^{r}\psi(x,\hat{\mathbf{x}})\overline{\partial_{y}^{p}\psi(y,\hat{\mathbf{x}})}\,d\hat{\mathbf{x}}.$

######  Remark 6.1.

The motivation to study the function (6.1) instead of the original one-
particle density matrix (1.3) can be explained by the following intuitive
argument. As we saw earlier (see Corollary 4.6) higher order cluster
derivatives of $\psi$ become ever more singular near the coalescence set
$\Sigma_{\boldsymbol{\sf P}}$ (i.e. for small $\lambda_{\boldsymbol{\sf P}}$)
as the order of derivative grows. This however does not apply to first order
derivatives – together with the function $\psi$ itself, they remain bounded
near $\Sigma_{\boldsymbol{\sf P}}$. Thus, taking derivatives of first order
under the integral (1.3) does not make the integral “worse”. At the same time,
by virtue of the formula

(6.4)
$\displaystyle\partial_{x}^{m}\partial_{y}^{l}\gamma(x,y)=\partial_{x}^{m-r}\partial_{y}^{l-p}\gamma_{r,p}(x,y),$

this reduces the order of the remaining derivatives. In particular, when
proving (1.1) further in this section, where $\gamma$ is differentiated with
respect to both variables $x$ and $y$, we use (6.4) with $|p|=|r|=1$, thereby
reducing the order of the derivatives with respect to $x$ and $y$ by one each.
On the other hand, in the proof of (1.1) where the differentiation is
conducted with respect to one of the variables $x$ or $y$ only, we cannot take
full advantage of this order reduction. Rather, we derive (1.1) from (1.1)
using the fundamental theorem of calculus which introduces the “missing”
derivative artificially.

Let us estimate the derivatives of $\gamma_{r,p}$ for arbitrary $|r|\leq 1$,
$|p|\leq 1$.

######  Lemma 6.2.

Let $|r|\leq 1$, $|p|\leq 1$. Suppose that for some $\varepsilon$ we have
$|x|>\varepsilon$, $|y|>\varepsilon$ and $|x-y|>\varepsilon$. Then for all
$\alpha,\beta\in\mathbb{N}_{0}^{3}$, and all $R>0$,

$\displaystyle|\partial_{x}^{\alpha}\partial_{y}^{\beta}\gamma_{r,p}(x,$
$\displaystyle y,\Phi;\varepsilon)|$ (6.5) $\displaystyle\lesssim$
$\displaystyle\
\big{(}1+|x|^{-|\alpha|-|\beta|}+|y|^{-|\alpha|-|\beta|}+h_{|\alpha|+|\beta|+2}(\varepsilon)\big{)}\,\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(x,R))}\big{)}^{\frac{1}{2}}\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(y,R))}\big{)}^{\frac{1}{2}}.$

with an implicit constant independent of $\psi$, $\varepsilon$, but dependent
on $R$.

###### Proof.

Throughout the proof for the brevity of notation we often omit the dependence
on $\varepsilon$. Let ${\sf P}$ and ${\sf S}$ be the clusters for the
admissible cut-offs $\phi$ and $\mu$ associated with $\Phi$. By Proposition
2.3 we may assume that ${\sf S}^{*}\subset{\sf P}^{\rm c}$, which is
equivalent to ${\sf P}^{*}\subset{\sf S}^{\rm c}$.

If $\alpha=\beta=0$, then (6.2) holds because of (6.2). Thus we may assume
that $|\alpha|+|\beta|\geq 1$. Let us make the following change of variables
under the integral $\gamma_{r,p}(x,y;\Phi)$. Define
$\hat{\mathbf{z}}=(z_{2},z_{3},\dots,z_{N})\in\mathbb{R}^{3N-3}$ by

$\displaystyle z_{j}=\begin{cases}x,\ j\in{\sf P}^{*},\\\ y,\ j\in{\sf
S}^{*},\\\ 0,\ j\in{\sf P}^{\rm c}\cap{\sf S}^{\rm c}.\end{cases}$

Change the variable in (6.3):
$\hat{\mathbf{x}}=\hat{\mathbf{w}}+\hat{\mathbf{z}}$, so that (6.3) rewrites
as

$\displaystyle\gamma_{r,p}(x,y;\Phi)=$ $\displaystyle\
\int\Phi(x,y,\hat{\mathbf{w}}+\hat{\mathbf{z}})\partial_{x}^{r}\psi(x,\hat{\mathbf{w}}+\hat{\mathbf{z}})\overline{\partial_{y}^{p}\psi(y,\hat{\mathbf{w}}+\hat{\mathbf{z}})}\,d\hat{\mathbf{w}}.$

For any function $g=g(x,\hat{\mathbf{x}})$ and all
$l,m_{1},m_{2}\in\mathbb{N}_{0}^{3}$, we have

$\displaystyle\partial_{x}^{l}\big{(}g(x,\hat{\mathbf{w}}+\hat{\mathbf{z}})\big{)}=({\sf
D}_{{\sf
P}}^{l}g)(x,\hat{\mathbf{w}}+\hat{\mathbf{z}}),\quad\partial_{x}^{l}\big{(}g(y,\hat{\mathbf{w}}+\hat{\mathbf{z}})\big{)}=({\sf
D}_{{\sf P}^{*}}^{l}g)(y,\hat{\mathbf{w}}+\hat{\mathbf{z}})$

and

$\displaystyle\partial_{x}^{m_{1}}\partial_{y}^{m_{2}}\big{(}\Phi(x,y,\hat{\mathbf{w}}+\hat{\mathbf{z}})\big{)}=\big{(}{\sf
D}_{x,{\sf P}}^{m_{1}}{\sf D}_{y,{\sf
S}}^{m_{2}}\Phi\big{)}(x,y,\hat{\mathbf{w}}+\hat{\mathbf{z}}),$

where we have used the notation (2.9) for the cluster derivatives of the
function $\Phi$. Denote

$\displaystyle Z(x,y)=\operatorname{{supp}}_{0}\,\Phi(x,y,\,\cdot\,).$

Thus we conclude that
$\partial_{x}^{\alpha}\partial_{y}^{\beta}\gamma_{r,p}(x,y;\Phi),\alpha,\beta\in\mathbb{N}_{0}^{3},$
is a linear combination of terms of the form

$\displaystyle{\mathcal{J}}_{\mathbf{m},\mathbf{n},\mathbf{k}}(x,y;\varepsilon)=\int_{Z(x,y)}\big{(}{\sf
D}_{x,{\sf P}}^{m_{1}}{\sf D}_{y,{\sf S}}^{m_{2}}$ $\displaystyle\
\Phi(x,y,\hat{\mathbf{x}})\big{)}$ (6.6) $\displaystyle\ \times\big{(}{\sf
D}_{\\{{\sf P},{\sf
S}^{*}\\}}^{\mathbf{n}}\partial_{x}^{r}\psi(x,\hat{\mathbf{x}})\big{)}\overline{\big{(}{\sf
D}_{\\{{\sf P}^{*},{\sf
S}\\}}^{\mathbf{k}}\partial_{y}^{p}\psi(y,\hat{\mathbf{x}})\big{)}}\,d\hat{\mathbf{x}}$

with

$\displaystyle\mathbf{m}=(m_{1},m_{2}),\ \mathbf{n}=$ $\displaystyle\
(n_{1},n_{2}),\mathbf{k}=(k_{1},k_{2}),\quad\textup{where}$
$\displaystyle|m_{1}|+|n_{1}|+|k_{1}|=|\alpha|,\quad$ $\displaystyle\
|m_{2}|+|n_{2}|+|k_{2}|=|\beta|.$

The cluster derivatives of $\Phi$ are defined in (2.9). For these derivatives
we use the bound (2.10):

(6.7) $\displaystyle|{\sf D}_{x,{\sf P}}^{m_{1}}{\sf D}_{y,{\sf
S}}^{m_{2}}\Phi(x,y,\hat{\mathbf{x}})|\lesssim\begin{cases}1,\quad\textup{if}\quad|\mathbf{m}|=0,\\\\[5.69046pt]
\varepsilon^{-|\mathbf{m}|}\mathcal{M}_{\varepsilon}(x,y,\hat{\mathbf{x}}),\quad\textup{if}\quad|\mathbf{m}|\geq
1,\end{cases}$

For the derivatives of $\psi$ we use Corollary 4.6:

(6.8) $\displaystyle\begin{cases}|{\sf D}_{\\{{\sf S}^{*},{\sf
P}\\}}^{\mathbf{n}}\partial_{x}^{r}\psi(x,\hat{\mathbf{x}})|\lesssim&\
\big{(}\lambda_{\\{{\sf S}^{*},{\sf
P}\\}}(x,\hat{\mathbf{x}})\big{)}^{-|\mathbf{n}|}f_{\infty}(x,\hat{\mathbf{x}};R/2),\\\\[5.69046pt]
|{\sf D}_{\\{{\sf S},{\sf
P}^{*}\\}}^{\mathbf{k}}\partial_{y}^{p}\psi(y,\hat{\mathbf{x}})|\lesssim&\
\big{(}\lambda_{\\{{\sf S},{\sf
P}^{*}\\}}(y,\hat{\mathbf{x}})\big{)}^{-|\mathbf{k}|}f_{\infty}(y,\hat{\mathbf{x}};R/2).\end{cases}$

In order to avoid cumbersome expressions, in the following calculations we use
the notation

$\displaystyle\mu(x,\hat{\mathbf{x}})=\lambda_{\\{{\sf S}^{*},{\sf
P}\\}}(x,\hat{\mathbf{x}}),\,\tilde{\mu}(y,\hat{\mathbf{x}})=\lambda_{\\{{\sf
S},{\sf P}^{*}\\}}(y,\hat{\mathbf{x}}).$

Assume first that $\mathbf{m}=\mathbb{0}$. In this case, by virtue of (6.7)
and (6.8), the integral (6) satisfies the estimate

$\displaystyle|{\mathcal{J}}_{\mathbb{0},\mathbf{n},\mathbf{k}}(x,y;\varepsilon)|\lesssim\int_{Z(x,y)}\big{(}\mu(x,\hat{\mathbf{x}})\big{)}^{-|\mathbf{n}|}\,\big{(}\tilde{\mu}(y,\hat{\mathbf{x}})\big{)}^{-|\mathbf{k}|}f_{\infty}(x,\hat{\mathbf{x}};R/2)\,f_{\infty}(y,\hat{\mathbf{x}};R/2)\,d\hat{\mathbf{x}}.$

By Young’s inequality, for all $\mathbf{n},\mathbf{k}$, such that
$|\mathbf{n}|+|\mathbf{k}|=|\alpha|+|\beta|\geq 1$, we have

$\displaystyle\big{(}\mu(x,\hat{\mathbf{x}})\big{)}^{-|\mathbf{n}|}$
$\displaystyle\ \big{(}\tilde{\mu}(y,\hat{\mathbf{x}})\big{)}^{-|\mathbf{k}|}$
$\displaystyle\leq\frac{|\mathbf{n}|}{|\alpha|+|\beta|}\big{(}\mu(x,\hat{\mathbf{x}})\big{)}^{-|\alpha|-|\beta|}+\frac{|\mathbf{k}|}{|\alpha|+|\beta|}\big{(}\tilde{\mu}(y,\hat{\mathbf{x}})\big{)}^{-|\alpha|-|\beta|}.$

Consequently, using the notation (5.11) we can estimate,

$\displaystyle|{\mathcal{J}}_{\mathbb{0},\mathbf{n},\mathbf{k}}(x,y;\varepsilon)|\lesssim{\mathcal{J}}_{|\alpha|+|\beta|}^{(1)}(x,y;\varepsilon,R/2)+{\mathcal{J}}_{|\alpha|+|\beta|}^{(2)}(x,y;\varepsilon,R/2),$

with a constant independent of $\varepsilon$. By (5.3), the right-hand side
satisfies (6.2), as required.

Now assume that $|\mathbf{m}|\geq 1$. Using again (6.7) and (6.8) we get the
estimate

$\displaystyle|{\mathcal{J}}_{\mathbf{m},\mathbf{n},\mathbf{k}}(x,y;\varepsilon)|\lesssim\varepsilon^{-|\mathbf{m}|}\,\int_{Z(x,y)}\mathcal{M}_{\varepsilon}(x,y,\hat{\mathbf{x}})$
$\displaystyle\
\big{(}\mu(x,\hat{\mathbf{x}})\big{)}^{-|\mathbf{n}|}\,\big{(}\tilde{\mu}(y,\hat{\mathbf{x}})\big{)}^{-|\mathbf{k}|}$
$\displaystyle\
f_{\infty}(x,\hat{\mathbf{x}};R/2)\,f_{\infty}(y,\hat{\mathbf{x}};R/2)\,d\hat{\mathbf{x}}.$

By Young’s inequality again, for all $\mathbf{m},\mathbf{n},\mathbf{k}$ such
that $|\mathbf{m}|+|\mathbf{n}|+|\mathbf{k}|=|\alpha|+|\beta|\geq 1$, we have

$\displaystyle\varepsilon^{-|\mathbf{m}|}\big{(}\mu(x,\hat{\mathbf{x}})\big{)}^{-|\mathbf{n}|}$
$\displaystyle\
\big{(}\tilde{\mu}(y,\hat{\mathbf{x}})\big{)}^{-|\mathbf{k}|}\leq\frac{|\mathbf{m}|}{|\alpha|+|\beta|}\varepsilon^{-|\alpha|-|\beta|}$
$\displaystyle+\frac{|\mathbf{n}|}{|\alpha|+|\beta|}\big{(}\mu(x,\hat{\mathbf{x}})\big{)}^{-|\alpha|-|\beta|}+\frac{|\mathbf{k}|}{|\alpha|+|\beta|}\big{(}\tilde{\mu}(y,\hat{\mathbf{x}})\big{)}^{-|\alpha|-|\beta|}.$

Therefore,

$\displaystyle|{\mathcal{J}}_{\mathbf{m},\mathbf{n},\mathbf{k}}(x,y;\varepsilon)|\lesssim$
$\displaystyle\
\varepsilon^{-|\alpha|-|\beta|}\,\int_{Z(x,y)}\mathcal{M}_{\varepsilon}(x,y,\hat{\mathbf{x}})f_{\infty}(x,\hat{\mathbf{x}};R/2)\,f_{\infty}(y,\hat{\mathbf{x}};R/2)\,d\hat{\mathbf{x}}$
$\displaystyle\
+\int_{Z(x,y)}\big{(}\mu(x,\hat{\mathbf{x}})\big{)}^{-|\alpha|-|\beta|}\mathcal{M}_{\varepsilon}(x,y,\hat{\mathbf{x}})\,f_{\infty}(x,\hat{\mathbf{x}};R/2)\,f_{\infty}(y,\hat{\mathbf{x}};R/2)\,d\hat{\mathbf{x}}$
$\displaystyle\
+\int_{Z(x,y)}\big{(}\tilde{\mu}(y,\hat{\mathbf{x}})\big{)}^{-|\alpha|-|\beta|}\mathcal{M}_{\varepsilon}(x,y,\hat{\mathbf{x}})\,f_{\infty}(x,\hat{\mathbf{x}};R/2)\,f_{\infty}(y,\hat{\mathbf{x}};R/2)\,d\hat{\mathbf{x}}.$

In the second and in the third integral estimate
$|\mathcal{M}_{\varepsilon}(x,y,\hat{\mathbf{x}})|\lesssim 1$ and use the
notation (5.11):

$\displaystyle|{\mathcal{J}}_{\mathbf{m},\mathbf{n},\mathbf{k}}(x,y;\varepsilon)|\lesssim$
$\displaystyle\ {\mathcal{J}}_{|\alpha|+|\beta|}^{(1)}(x,y;\varepsilon,R/2)$
$\displaystyle\
+{\mathcal{J}}_{|\alpha|+|\beta|}^{(2)}(x,y;\varepsilon,R/2)+{\mathcal{J}}_{|\alpha|+|\beta|}^{(3)}(x,y;\varepsilon,R/2),$

with a constant independent of $\varepsilon$. By (5.3), the right-hand side
satisfies (6.2).

Putting the estimates for $|\mathbf{m}|=0$ and $|\mathbf{m}|\geq 1$ together,
and summing over $\mathbf{m},\mathbf{n},\mathbf{k}$, we arrive at (6.2),
thereby completing the proof. ∎

######  Corollary 6.3.

Suppose that for some $\varepsilon>0$ we have $|x|>\varepsilon$,
$|y|>\varepsilon$ and $|x-y|>\varepsilon$. Then for all $\alpha,\beta$ and
$|p|\leq 1,|r|\leq 1,$ and all $R>0$ the bound holds:

$\displaystyle|\partial_{x}^{\alpha}\partial_{y}^{\beta}\gamma_{r,p}(x,$
$\displaystyle y)|$ (6.9) $\displaystyle\lesssim$
$\displaystyle\big{(}1+|x|^{-|\alpha|-|\beta|}+|y|^{-|\alpha|-|\beta|}+h_{|\alpha|+|\beta|+2}(\varepsilon)\big{)}\,\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(x,R))}\big{)}^{\frac{1}{2}}\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(y,R))}\big{)}^{\frac{1}{2}},$

with an implicit constant that is independent of $x,y$ and $\varepsilon$ but
may depend on $R$.

###### Proof.

In order to use Lemma 6.2 we build a partition of unity consisting of extended
cut-offs. Recall the notation ${\sf{R}}=\\{1,2,\dots,N\\}$. Let
$\Xi=\\{(j,k)\in{\sf{R}}\times{\sf{R}}:j<k\\}$. For each subset
$\Upsilon\subset\Xi$ introduce the admissible cut-off (see (2.3) for the
definition of admissible cut-offs)

$\displaystyle\phi_{\Upsilon}(\mathbf{x};\varepsilon)=\prod_{(j,k)\in\Upsilon}\zeta_{\varepsilon}(x_{j}-x_{k})\prod_{(j,k)\in\Upsilon^{\rm
c}}\theta_{\varepsilon}(x_{j}-x_{k}).$

It is clear that

$\displaystyle\sum_{\Upsilon\subset{\Xi}}\phi_{\Upsilon}(\mathbf{x};\varepsilon)=\prod_{(j,k)\in\Xi}\big{(}\zeta_{\varepsilon}(x_{j}-x_{k})+\theta_{\varepsilon}(x_{j}-x_{k})\big{)}=1.$

Furthermore, for every cluster ${\sf S}\subset{\sf{R}}^{*}$ define

$\displaystyle\tau_{{\sf S}}(y,\hat{\mathbf{x}};\varepsilon)=\prod_{j\in{\sf
S}}\zeta_{\varepsilon}(y-x_{j})\prod_{j\in({\sf S}^{\rm
c})^{*}}\theta_{\varepsilon}(y-x_{j}).$

It is clear that

$\displaystyle\sum_{{\sf S}\subset{\sf{R}}^{*}}\tau_{{\sf
S}}(y,\hat{\mathbf{x}};\varepsilon)=\prod_{j\in{\sf{R}}^{*}}\big{(}\zeta_{\varepsilon}(x_{1}-x_{j})+\theta_{\varepsilon}(x_{1}-x_{j})\big{)}=1.$

Define

$\displaystyle\Phi_{\Upsilon,{\sf
S}}(x,y,\hat{\mathbf{x}};\varepsilon)=\phi_{\Upsilon}(x,\hat{\mathbf{x}};\varepsilon)\tau_{{\sf
S}}(y,\hat{\mathbf{x}};\varepsilon),\quad(x,y)\in\mathbb{R}^{3}\times\mathbb{R}^{3},\,\hat{\mathbf{x}}\in\mathbb{R}^{3N-3},$

so that

$\displaystyle\sum\limits_{\Upsilon\subset\Xi,\ {\sf
S}\subset{\sf{R}}^{*}}\Phi_{\Upsilon,{\sf
S}}(x,y,\hat{\mathbf{x}};\varepsilon)=1.$

Each function $\Phi_{\Upsilon,{\sf S}}$ is an extended cut-off function, as
defined in Subsect. 2.2. Using the definition (6.3), the function (6.1) can be
represented as

$\displaystyle\gamma_{r,p}(x,y)=\sum_{\Upsilon\subset\Xi,\ {\sf
S}\subset{\sf{R}}^{*}}\gamma(x,y;\Phi_{\Upsilon,{\sf S}},\varepsilon).$

Applying Lemma 6.2 to each summand we arrive at (6.3). ∎

###### Proof of the bound (1.1).

Assume that $x\not=0,y\not=0,x\not=y$ and that $|l|\geq 1$, $|m|\geq 1$.
Represent $l=\alpha+r$, $m=\beta+p$ with $|r|=|p|=1$, so that
$|l|=|\alpha|+1,|m|=|\beta|+1$. Furthermore, denote

$\displaystyle\varepsilon=\frac{1}{2}\min\\{|x|,|y|,|x-y|\\},$

so that $|x-y|>\varepsilon$ and $|x|>\varepsilon,|y|>\varepsilon$. Thus it
follows from (6.3) that

$\displaystyle|\partial_{x}^{l}\partial_{y}^{m}\gamma(x,$ $\displaystyle y)|$
(6.10) $\displaystyle\lesssim$
$\displaystyle\big{(}|x|^{2-|l|-|m|}+|y|^{2-|l|-|m|}+h_{|l|+|m|}(\varepsilon)\big{)}\,\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(x,R))}\big{)}^{\frac{1}{2}}\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(y,R))}\big{)}^{\frac{1}{2}}.$

Since for all $a\geq 0$ we have

$\displaystyle h_{a}(\varepsilon)\lesssim$ $\displaystyle\
h_{a}(|x-y|)+h_{a}(|x|)+h_{a}(|y|)$ $\displaystyle\lesssim$ $\displaystyle\
1+|x|^{2-a}+|y|^{2-a}+h_{a}(|x-y|),$

the bound (6) implies (1.1). ∎

###### Proof of (1.1).

For convenience denote

$\displaystyle
A(x,y;R_{1},R_{2})=\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(x,R_{1}))}\big{)}^{\frac{1}{2}}\big{(}\|\rho\|_{\textup{{{L}}}^{1}(B(y,R_{2}))}\big{)}^{\frac{1}{2}}.$

Assume that $|l|\geq 1$. Representing $l=\alpha+r$, with some $r:|r|=1$, and
taking $\beta=p=0$, we obtain from (6.3) that

$\displaystyle|\partial_{x}^{l}\gamma(x,y)|+$ $\displaystyle\
|\partial_{y}^{l}\gamma(x,y)|$ (6.11) $\displaystyle\lesssim$ $\displaystyle\
\big{(}1+|x|^{1-|l|}+|y|^{1-|l|}+h_{|l|+1}(|x-y|)\big{)}\,A(x,y;R/2,R/2).$

To ”upgrade” this bound to (1.1) we use the Fundamental Theorem of Calculus.
Let $z\in\mathbb{R}^{3}$, $z\not=x,$ be such that the segment

$\displaystyle\\{y_{s}=y+s(z-y),\,s\in[0,1]\\},$

does not contain the point $x$. Then

(6.12)
$\displaystyle\partial_{x}^{l}\gamma(x,y)-\partial_{x}^{l}\gamma(x,z)=-\int_{0}^{1}(z-y)\cdot\nabla_{y}(\partial_{x}^{l}\gamma)(x,y_{s})\,ds.$

A similar formula holds for the derivative $\partial_{y}^{l}\gamma(x,y)$, but
we omit this part of the argument and complete the proof for the derivative
$\partial_{x}^{l}\gamma(x,y)$ only. Since the integrand in (6.12) contains
derivatives both w.r.t. $x$ and $y$, we can use the bound (1.1) proved
previously. First we make a convenient choice of $z$. Denote

$\displaystyle|x-y|=d,\quad\delta=\frac{1}{4}\min\\{|x|,|y|\\},\quad\delta_{1}=\min\\{1,\delta,R/2\\},\quad{\sf
e}=\frac{x-y}{|x-y|},$

so

$\displaystyle x=y+|x-y|\,{\sf e}=y+d\,{\sf e}.$

Take $z=y-\delta_{1}\,{\sf e}$, so that

$\displaystyle|y-y_{s}|=s\delta_{1},\,|z-y|=\delta_{1},\
|z-x|=\delta_{1}+d,\quad|z|\geq|y|-\delta_{1}\geq\frac{3}{4}|y|,$

and

$\displaystyle|y_{s}|\geq|y|-s\delta_{1}\geq\frac{3}{4}|y|,\quad|x-y_{s}|=d+s\delta_{1},\quad
s\in[0,1].$

Now apply (1.1) with the radius $R/2$ to estimate the integrand in (6.12):

$\displaystyle|(z-y)\cdot\nabla_{y}($
$\displaystyle\partial_{x}^{l}\gamma)(x,y_{s})|$ $\displaystyle\lesssim$
$\displaystyle\
|z-y|\big{(}1+|y_{s}|^{1-|l|}+|x|^{1-|l|}+h_{|l|+1}(|x-y_{s}|)\big{)}A(x,y_{s};R/2,R/2)$
(6.13) $\displaystyle\lesssim$ $\displaystyle\
\delta_{1}\big{(}1+|y|^{1-|l|}+|x|^{1-|l|}+h_{|l|+1}(d+s\delta_{1})\big{)}A(x,y;R/2,R).$

An elementary calculation shows that for all $a\geq 0$ the bound holds:

$\displaystyle\delta_{1}\int_{0}^{1}h_{a+1}(d+s\delta_{1})ds\lesssim
h_{a}(d).$

Consequently, integrating the bound (6) in $s\in[0,1]$ we obtain from (6.12)
that

(6.14)
$\displaystyle|\partial_{x}^{l}\gamma(x,y)|\lesssim|\partial_{x}^{l}\gamma(x,z)|+\big{(}1+|y|^{1-|l|}+|x|^{1-|l|}+h_{|l|}(|x-y|)\big{)}A(x,y;R,R).$

To estimate $|\partial_{x}^{l}\gamma(x,z)|$ we use (6):

$\displaystyle|\partial_{x}^{l}\gamma(x,z)|\lesssim\big{(}1+|x|^{1-|l|}+|z|^{1-|l|}+h_{|l|+1}(|x-z|)\big{)}\,A\big{(}x,z;R/2,R/2\big{)}.$

As $|z-x|=d+\delta_{1}\geq\delta_{1}$ we can estimate:

$\displaystyle h_{|l|+1}(|x-z|)\leq h_{|l|+1}(\delta_{1})\lesssim
1+|x|^{1-|l|}+|y|^{1-|l|}.$

Furthermore, as $|y-z|=\delta_{1}\leq R/2$, we have
$A\big{(}x,z;R/2,R/2\big{)}\leq A\big{(}x,y;R/2,R\big{)}$. Consequently,

$\displaystyle|\partial_{x}^{l}\gamma(x,z)|\lesssim\big{(}1+|x|^{1-|l|}+|y|^{1-|l|}\big{)}\,A\big{(}x,y;R/2,R\big{)}.$

Together with (6.14) this bound entails (1.1). ∎

This completes the proof of Theorem 1.1.

##  7\. Appendix

Here we provide an elementary fact concerning extensions of Sobolev spaces.
The proof can be found in the appendix to [18].

Consider spaces of functions that depend either on one variable
$x\in\mathbb{R}^{d}$ or on two variables
$(t,x)\in\mathbb{R}^{l}\times\mathbb{R}^{d}$. Denote
$K=\\{t\in\mathbb{R}^{l}:|t|<1\\}$, $B=\\{x\in\mathbb{R}^{d}:|x|<1\\}$ and
$B_{0}=B\setminus\\{0\\}$.

######  Proposition 7.1.

Let the dimension $l$ be arbitrary, let $d\geq 2$, $m\geq 1$, and
$p\in[d(d-1)^{-1},\infty]$. Then
$\textup{{{W}}}^{m,p}(B_{0})=\textup{{{W}}}^{m,p}(B)$ and
$\textup{{{W}}}^{m,p}\big{(}K\times
B_{0}\big{)}=\textup{{{W}}}^{m,p}\big{(}K\times B\big{)}$.

We use this fact in Remark 1.2 to describe smoothness properties of the one-
particle density matrix $\gamma(x,y)$.

Acknowledgments. The authors are grateful to A. Nazarov for his help regarding
Proposition 3.1, and to D. Edmunds, V. Kozlov, V. Maz’ya, G. Rozenblum, D.
Vassiliev for their advice on Sobolev spaces.

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