# Boundedness of the fifth off-diagonal derivative for the one-particle
Coulombic density matrix

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

Boundedness is demonstrated for the fifth derivative of the one-particle
reduced density matrix for non-relativistic Coulombic wavefunctions in the
vicinity of the diagonal.

###### Key words and phrases:

Multi-particle quantum system, Coulombic wavefunction, Schrödinger equation,
one-particle density matrix

###### 2010 Mathematics Subject Classification:

Primary 35B65; Secondary 35J10, 81V55

##  1\. Introduction and results

We consider the non-relativistic quantum systems of $N\geq 2$ electrons among
$N_{0}$ nuclei which represents the system of an atom or molecule. For
simplicity we restrict ourselves to the case of an atom ($N_{0}=1$), although
all results readily generalise to the molecular case. The electrons have
coordinates $\mathbf{x}=(x_{1},\dots,x_{N}),x_{k}\in{}^{3}$, $k=1,\dots,N$,
and the nucleus has charge $Z>0$ and its position fixed at the origin. The
corresponding Schrödinger operator is

(1.1) $H=-\Delta+V$

where $\Delta=\sum_{k=1}^{N}\Delta_{x_{k}}$ is the Laplacian in 3N, i.e.
$\Delta_{x_{k}}$ refers to the Laplacian applied to the variable $x_{k}$, and
$V$ is the Coulomb potential given by

(1.2) $V(\mathbb{x})=-\sum_{k=1}^{N}\frac{Z}{|x_{k}|}+\sum_{1\leq j<k\leq
N}\frac{1}{|x_{j}-x_{k}|}$

for $\mathbb{x}\in{}^{3N}$. This operator acts in $L^{2}({}^{3N})$ and is
self-adjoint on $H^{2}({}^{3N})$, see for example [1, Theorem X.16]. We
consider solutions to the eigenvalue problem for $H$ in the operator sense,
namely

(1.3) $H\psi=E\psi$

for $\psi\in H^{2}({}^{3N})$ and $E\in\real$. For each $j=1,\dots,N$, we
represent

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

for $x\in{}^{3}$. We define the one-particle reduced density matrix, or simply
density matrix, by

(1.4)
$\displaystyle\gamma(x,y)=\int_{\mathbb{R}^{3N-3}}\psi(x,\hat{\mathbf{x}})\overline{\psi(y,\hat{\mathbf{x}})}\,d\hat{\mathbf{x}},\
\quad\hat{\mathbf{x}}=\mathbb{\hat{x}}_{1}.$

for $x,y\in{}^{3}$. More commonly, the one-particle reduced density matrix is
defined as the function

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

However, since we are interested only in regularity properties we need only
study one term of (1.5), hence the definition (1.4). In fact we have
$g(x,y)=N\gamma(x,y)$ whenever $\psi$ is totally antisymmetric. An important
related function is the one-particle density, or simply density, which is
defined here as

(1.6)
$\displaystyle\rho(x)=\gamma(x,x)=\int_{\mathbb{R}^{3N-3}}|\psi(x,\hat{\mathbf{x}})|^{2}\,d\hat{\mathbf{x}},\quad
x\in{}^{3}.$

Now suppose $\psi$ is any eigenfunction obeying (1.3) with
$\left\lVert\psi\right\rVert_{L^{2}({}^{3N})}=1$, and let $\rho$ and $\gamma$
be the corresponding functions as defined in (1.6) and (1.4) respectively. In
[2], real analyticity is proven for $\gamma(x,y)$ as a function of two
variables in the set

$\mathcal{D}=\\{(x,y)\in{}^{3}\times{}^{3}:x\neq 0,y\neq 0,x\neq y\\}.$

In particular, real analyticity was not shown across the diagonal, that is
where $x=y$. Previously, real analyticity was shown to hold for the density
$\rho(x)$ on the set ${}^{3}\backslash\\{0\\}$ in [3], see also [4]. In other
words, this shows real analyticity of $\gamma$ in one variable along the
diagonal $x=y$, excluding the point $x=y=0$. Despite this, there is no
smoothness of $\gamma$ across the diagonal which is discussed in [5].
Pointwise derivative estimates for $\gamma$ on the set $\mathcal{D}$ were then
given in this paper, namely [5, Theorem 1.1], and are restated as follows. For
$b\geq 0$, $t>0$ define

$\displaystyle h_{b}(t)=\begin{cases}t^{\min\\{0,5-b\\}}&\text{if }b\neq 5\\\
\log(t^{-1}+2)&\text{if }b=5.\end{cases}$

We denote $\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$. For all $R>0$ and
$\alpha,\beta\in\mathbb{N}_{0}^{3}$ with $|\alpha|,|\beta|\geq 1$ there exists
$C$ such that

(1.7) $|\partial^{\alpha}_{x}\partial_{y}^{\beta}\gamma(x,y)|\leq
C\big{(}1+|x|^{2-|\alpha|-|\beta|}+|y|^{2-|\alpha|-|\beta|}\\\
+h_{|\alpha|+|\beta|}(|x-y|)\big{)}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(x,R))}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(y,R))}$

and for all $|\alpha|\geq 1$ there exists $C$ such that

(1.8)
$|\partial^{\alpha}_{x}\gamma(x,y)|+|\partial^{\alpha}_{y}\gamma(x,y)|\leq
C\big{(}1+|x|^{1-|\alpha|}+|y|^{1-|\alpha|}\\\
+h_{|\alpha|}(|x-y|)\big{)}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(x,R))}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(y,R))}$

for all $x,y\in{}^{3}$ with $x\neq 0$, $y\neq 0$ and $x\neq y$. The notation
$\partial_{x}^{\alpha}$ refers to the $\alpha$-partial derivative in the $x$
variable. The constant $C$ depends on $\alpha,\beta,R,N$ and $Z$. The right-
hand side is finite because $\psi$ is normalised and hence $\rho\in
L^{1}({}^{3})$. The bounded first derivative at the nucleus reflects local
Lipschitz continuity of $\gamma$ on 6. In addition, there is local boundedness
of up to four derivatives at the diagonal with at worst a logarithmic
singularity for the fifth derivative. The purpose of this current paper is to
show local boundedness of the fifth derivative at the diagonal.

The existence of the fifth-order cusp at the diagonal was previously
demonstrated in [6] (see also [7]). In this paper, quantum chemistry
calculations show that for $x,r\in{}^{3}$, $x\neq 0$ and small $r$ we have

(1.9)
$\displaystyle\operatorname{{Re}}[\gamma(x+r,x-r)]=\gamma(x,x)+C(x)|r|^{5}+R(x,r)$

for some functions $C(x)$ and $R(x,r)$, the latter having no contribution from
$|r|^{k}$ for $k=0,1,3,5$ in the small $|r|$ expansion at $r=0$.

Our main result is as follows.

######  Theorem 1.1.

Let $\psi$ be an eigenfunction of (1.3). Define $m(x,y)=\min\\{1,|x|,|y|\\}$.
Then for all $|\alpha|+|\beta|=5$ and $R>0$ we have $C$, depending on $R$ and
also on $Z,N$ and $E$, such that

(1.10) $|\partial_{x}^{\alpha}\partial_{y}^{\beta}\gamma(x,y)|\leq
Cm(x,y)^{-4}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(x,R))}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(y,R))}$

for all $x,y\in{}^{3}$ obeying $0<|x-y|\leq(2N)^{-1}m(x,y)$.

######  Remark 1.2.

  1. (1)

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

  2. (2)

The bound (1.10) naturally complements (1.7) and (1.8) for
$|\alpha|+|\beta|\neq 5$.

  3. (3)

As a consequence of [5, Proposition 7.1], the inequality (1.10) shows that
$\gamma\in
C^{4,1}_{loc}\big{(}(\mathbb{R}^{3}\backslash\\{0\\})\times(\mathbb{R}^{3}\backslash\\{0\\})\big{)}$.

Notation. As mentioned earlier, we use standard notation whereby
$\mathbb{x}=(x_{1},\dots,x_{N})\in{}^{3N}$, $x_{j}\in{}^{3}$, j=1, …, N, and
where $N$ is the number of electrons. In addition, define for $1\leq j,k\leq
N$, $j\neq k$,

(1.11) $\displaystyle\mathbb{\hat{x}}_{j}$
$\displaystyle=(x_{1},\dots,x_{j-1},x_{j+1},\dots,x_{N})$ (1.12)
$\displaystyle\mathbb{\hat{x}}_{jk}$
$\displaystyle=(x_{1},\dots,x_{j-1},x_{j+1},\dots,x_{k-1},x_{k+1},\dots,x_{N})$

with obvious modifications if either $j,k$ equals $1$ or $N$, and if $k<j$. We
define $\mathbb{\hat{x}}=\mathbb{\hat{x}}_{1}$, which will be used throughout.
Variables placed before $\mathbb{\hat{x}}_{j}$ and $\mathbb{\hat{x}}_{jk}$
will be placed in the removed slots as follows, for any $x,y\in{}^{3}$ we have

(1.13) $\displaystyle(x,\mathbb{\hat{x}}_{j})$
$\displaystyle=(x_{1},\dots,x_{j-1},x,x_{j+1},\dots,x_{N}),$ (1.14)
$\displaystyle(x,y,\mathbb{\hat{x}}_{jk})$
$\displaystyle=(x_{1},\dots,x_{j-1},x,x_{j+1},\dots,x_{k-1},y,x_{k+1},\dots,x_{N}).$

In this way,
$\mathbb{x}=(x_{j},\mathbb{\hat{x}}_{j})=(x_{j},x_{k},\mathbb{\hat{x}}_{jk})$.

We define a cluster to be any subset $P\subset\\{1,\dots,N\\}$. Denote
$P^{c}=\\{1,\dots,N\\}\backslash P$, $P^{*}=P\backslash\\{1\\}$. We will also
need cluster sets, $\mathbb{P}=(P_{1},\dots,P_{M})$, where $M\geq 1$ and
$P_{1},\dots,P_{M}$ are clusters.

First-order cluster derivatives are defined, for a non-empty cluster $P$, by

(1.15) $D^{\alpha}_{P}=\sum_{j\in P}\partial_{x_{j}}^{\alpha}\quad\text{for
}\alpha\in\mathbb{N}_{0}^{3},\,|\alpha|=1.$

For $P=\emptyset$, $D^{\alpha}_{P}$ is defined as the identity. Higher order
cluster derivatives, for
$\alpha=(\alpha^{\prime},\alpha^{\prime\prime},\alpha^{\prime\prime\prime})\in\mathbb{N}_{0}^{3}$
with $|\alpha|\geq 2$, are defined by successive application of first-order
cluster derivatives as follows,

(1.16)
$D^{\alpha}_{P}=(D_{P}^{e_{1}})^{\alpha^{\prime}}(D_{P}^{e_{2}})^{\alpha^{\prime\prime}}(D_{P}^{e_{3}})^{\alpha^{\prime\prime\prime}}$

where $e_{1},e_{2},e_{3}$ are the standard unit basis vectors of 3. Let
$\mathbb{P}=(P_{1},\dots,P_{M})$ and
$\boldsymbol{\alpha}=(\alpha_{1},\dots,\alpha_{M})$,
$\alpha_{j}\in\mathbb{N}_{0}^{3}$, $1\leq j\leq M$, then we define the
multicluster derivative (often simply referred to as cluster derivative) by

(1.17) $D_{\mathbb{P}}^{\boldsymbol{\alpha}}=D_{P_{1}}^{\alpha_{1}}\dots
D_{P_{M}}^{\alpha_{M}}.$

It can readily be seen that cluster derivatives obey the Leibniz rule.

Throughout, the letter $C$ refers to a positive constant whose value is
unimportant but may depend on $Z$, $N$ and the eigenvalue $E$.

Distance function notation and elementary results. For non-empty cluster $P$,
define

(1.18) $\displaystyle\Sigma_{P}=\Big{\\{}\mathbb{x}\in{}^{3N}:\prod_{j\in
P}|x_{j}|\prod_{\begin{subarray}{c}l\in P\\\ m\in
P^{c}\end{subarray}}|x_{l}-x_{m}|=0\Big{\\}}.$

For $P=\emptyset$ we set $\Sigma_{P}:=\emptyset$. Denote
$\Sigma_{P}^{c}={}^{3N}\backslash\Sigma_{P}$. For each $P$ we have
$\Sigma_{P}\subset\Sigma$ where

(1.19) $\Sigma=\Big{\\{}\mathbb{x}\in{}^{3N}:\prod_{1\leq j\leq
N}|x_{j}|\prod_{1\leq l<m\leq N}|x_{l}-x_{m}|=0\Big{\\}}$

is the set of singularities of the Coulomb potential $V$, (1.2). For any
cluster $P$ we can define the following two distances

(1.20) $\displaystyle d_{P}(\mathbb{x})$
$\displaystyle:=\text{dist}(\mathbb{x},\Sigma_{P})=\min\big{\\{}|x_{j}|,\,2^{-1/2}|x_{j}-x_{k}|:j\in
P,k\in P^{c}\big{\\}}$ (1.21) $\displaystyle\lambda_{P}(\mathbb{x})$
$\displaystyle:=\min\\{1,\,d_{P}(\mathbb{x})\\}$

for all $\mathbb{x}\in{}^{3N}$.

Using the formula (1.20), see for example [5, Lemma 4.2], it can be shown that
both $d_{P}$ and $\lambda_{P}$ are Lipschitz and obey

(1.22)
$|d_{P}(\mathbb{x})-d_{P}(\mathbb{y})|,|\lambda_{P}(\mathbb{x})-\lambda_{P}(\mathbb{y})|\leq|\mathbb{x}-\mathbb{y}|$

for all $\mathbb{x},\mathbb{y}\in{}^{3N}$.

Let $\mathbb{P}=(P_{1},\dots,P_{M})$ be a cluster set and
$\boldsymbol{\alpha}=(\alpha_{1},\dots,\alpha_{M})\in\mathbb{N}_{0}^{3M}$ be a
multiindex. Define

(1.23) $\Sigma_{\boldsymbol{\alpha}}=\bigcup_{j\,:\,\alpha_{j}\neq
0}\Sigma_{P_{j}}$

for non-zero $\boldsymbol{\alpha}$ and when $\boldsymbol{\alpha}=0$ we set
$\Sigma_{\boldsymbol{\alpha}}=\emptyset$. Denote
$\Sigma_{\boldsymbol{\alpha}}^{c}={}^{3N}\backslash\Sigma_{\boldsymbol{\alpha}}$.
For non-zero $\boldsymbol{\alpha}$ we can also define the two distances

(1.24) $\displaystyle d_{\boldsymbol{\alpha}}(\mathbb{x})$
$\displaystyle=\min\\{d_{P_{j}}(\mathbb{x}):\alpha_{j}\neq
0,\,j=1,\dots,M\\},$ (1.25)
$\displaystyle\lambda_{\boldsymbol{\alpha}}(\mathbb{x})$
$\displaystyle=\min\\{\lambda_{P_{j}}(\mathbb{x}):\alpha_{j}\neq
0,\,j=1,\dots,M\\}$

for all $\mathbb{x}\in{}^{3N}$. Notice we also have the identity

$\displaystyle
d_{\boldsymbol{\alpha}}(\mathbb{x})=\text{dist}(\mathbb{x},\Sigma_{\boldsymbol{\alpha}}).$

Indeed, since $\Sigma_{P_{j}}\subset\Sigma_{\boldsymbol{\alpha}}$ whenever $j$
is such that $\alpha_{j}\neq 0$, we have
$\text{dist}(\mathbb{x},\Sigma_{\boldsymbol{\alpha}})\leq
d_{P_{j}}(\mathbb{x})$ and hence
$\text{dist}(\mathbb{x},\Sigma_{\boldsymbol{\alpha}})\leq
d_{\boldsymbol{\alpha}}(\mathbb{x})$. Conversely for each
$\boldsymbol{\xi}\in\Sigma_{\boldsymbol{\alpha}}$ we have
$\boldsymbol{\xi}\in\Sigma_{P_{j}}$ for some $j$ with $\alpha_{j}\neq 0$, and
hence $|\mathbb{x}-\boldsymbol{\xi}|\geq d_{P_{j}}(\mathbb{x})$. Therefore,
$|\mathbb{x}-\boldsymbol{\xi}|\geq d_{\boldsymbol{\alpha}}(\mathbb{x})$.
Taking infimum over $\boldsymbol{\xi}\in\Sigma_{\boldsymbol{\alpha}}$ we
obtain the reverse inequality.

####

We will also use the following related quantities involving maxima of relevant
distances for non-zero $\boldsymbol{\alpha}$,

(1.26) $\displaystyle q_{\boldsymbol{\alpha}}(\mathbb{x})$
$\displaystyle=\max\\{d_{P_{j}}(\mathbb{x}):\alpha_{j}\neq 0,\,j=1,\dots,M\\}$
(1.27) $\displaystyle\mu_{\boldsymbol{\alpha}}(\mathbb{x})$
$\displaystyle=\max\\{\lambda_{P_{j}}(\mathbb{x}):\alpha_{j}\neq
0,\,j=1,\dots,M\\}.$

Finally, for $\boldsymbol{\alpha}=0$ we set
$d_{\boldsymbol{\alpha}},q_{\boldsymbol{\alpha}}\equiv 0$ and
$\lambda_{\boldsymbol{\alpha}},\mu_{\boldsymbol{\alpha}}\equiv 1$.

In order to state the results we first define the following for an arbitrary
function $u$ and any $r>0$,

(1.28) $\displaystyle f_{\infty}(\mathbb{x};r;u):=\left\lVert\nabla
u\right\rVert_{L^{\infty}(B(\mathbb{x},r))}+\left\lVert
u\right\rVert_{L^{\infty}(B(\mathbb{x},r))}$

for $\mathbb{x}\in{}^{3N}$. The ball $B(\mathbb{x},r)$ is considered in 3N.
Largely, this notation will be used for $u=\psi$ and in this case we have the
notation,

(1.29) $\displaystyle
f_{\infty}(\mathbb{x};r):=f_{\infty}(\mathbb{x};r;\psi).$

A pointwise cluster derivative bound. To prove Theorem 1.1 we will state and
prove a new pointwise bound to cluster derivatives of eigenfunctions $\psi$,
which may itself be of independent interest. It will be shown by elliptic
regularity that for all $\boldsymbol{\alpha}$ the weak cluster derivatives
$D^{\boldsymbol{\alpha}}_{\mathbb{P}}\psi$ exist in the set
$\Sigma_{\boldsymbol{\alpha}}^{c}$. It is therefore interesting to consider
how such cluster derivatives behave as the set $\Sigma_{\boldsymbol{\alpha}}$
is approached.

Previously, S. Fournais and T. Ø. Sørensen have given bounds to local
$L^{p}$-norms of cluster derivatives of $\psi$ for a single cluster $P$.
Indeed, in [8, Proposition 1.10] it is shown that for any multiindex
$0\neq\alpha\in\mathbb{N}_{0}^{3}$, $p\in(1,\infty]$ and any $0<r<R<1$ there
exists $C$, depending on $r,R,p$ and $\alpha$, such that

(1.30) $\left\lVert
D_{P}^{\alpha}\psi\right\rVert_{L^{p}(B(\mathbb{x},r\lambda_{P}(\mathbb{x})))}\leq
C\lambda_{P}(\mathbb{x})^{1-|\alpha|}\big{(}\left\lVert\nabla\psi\right\rVert_{L^{p}(B(\mathbf{x},R\lambda_{P}(\mathbf{x})))}+\left\lVert\psi\right\rVert_{L^{p}(B(\mathbf{x},R\lambda_{P}(\mathbf{x})))}\big{)}$

for all $\mathbb{x}\in\Sigma^{c}_{P}$. Notice that for every
$\mathbf{x}\in\Sigma_{P}^{c}$, we have
$B(\mathbf{x},r\lambda_{P}(\mathbf{x}))\subset\Sigma^{c}_{P}$ by the
definition of $\lambda_{P}(\mathbf{x})$. Therefore, the set $\Sigma_{P}$ is
avoided when evaluating $D_{P}^{\alpha}\psi$ in the $L^{p}$-norms.

The objective of the following theorem is to extend the bounds (1.30) in the
case of $p=\infty$ and for cluster derivatives for cluster sets $\mathbb{P}$.
In particular, estimates are obtained which depend on the order of derivative
for each of the respective clusters in $\mathbb{P}$. In the following,
$\nabla$ denotes the gradient operator in 3N.

######  Theorem 1.3.

For every cluster set $\mathbb{P}=(P_{1},\dots,P_{M})$, multiindex
$\boldsymbol{\alpha}\in\mathbb{N}_{0}^{3M}$ and any $0<r<R<1$ there exists
$C$, depending on $\boldsymbol{\alpha},r$ and $R$, such that for $k=0,1$,

(1.31) $\left\lVert
D_{\mathbb{P}}^{\boldsymbol{\alpha}}\nabla^{k}\psi\right\rVert_{L^{\infty}(B(\mathbb{x},r\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}\leq
C\lambda_{\boldsymbol{\alpha}}(\mathbb{x})^{1-k}\lambda_{P_{1}}(\mathbb{x})^{-|\alpha_{1}|}\dots\lambda_{P_{M}}(\mathbb{x})^{-|\alpha_{M}|}f_{\infty}(\mathbb{x};R)$

for all $\mathbb{x}\in\Sigma^{c}_{\boldsymbol{\alpha}}$.

Furthermore, for each $|\boldsymbol{\alpha}|\geq 1$ there exists a function
$G_{\mathbb{P}}^{\boldsymbol{\alpha}}:\Sigma^{c}_{\boldsymbol{\alpha}}\to\mathbb{C}^{3N}$
such that

(1.32)
$D_{\mathbb{P}}^{\boldsymbol{\alpha}}\nabla\psi=G_{\mathbb{P}}^{\boldsymbol{\alpha}}+\psi
D_{\mathbb{P}}^{\boldsymbol{\alpha}}\nabla F_{c}$

and for every $b\in[0,1)$ there exists $C$, depending on
$\boldsymbol{\alpha},r,R$ and $b$, such that

(1.33) $\left\lVert
G_{\mathbb{P}}^{\boldsymbol{\alpha}}\right\rVert_{L^{\infty}(B(\mathbb{x},r\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}\\\
\leq
C\mu_{\boldsymbol{\alpha}}(\mathbb{x})^{b}\lambda_{P_{1}}(\mathbb{x})^{-|\alpha_{1}|}\dots\lambda_{P_{M}}(\mathbb{x})^{-|\alpha_{M}|}f_{\infty}(\mathbb{x};R)$

for all $\mathbb{x}\in\Sigma^{c}_{\boldsymbol{\alpha}}$.

######  Remark 1.4.

  1. (1)

In the case of a single cluster and $k=0$, the bound (1.31) reestablishes
(1.30) in the case of $p=\infty$, albeit with a slightly larger radius in the
$L^{\infty}$-norms on the right-hand side.

  2. (2)

When $k=0$ and $\boldsymbol{\alpha}\neq 0$, the presence of a single power of
$\lambda_{\boldsymbol{\alpha}}(\mathbf{x})$ in the bound will cancel a single
negative power of $\lambda_{P_{j}}(\mathbf{x})$ for $j$ such that
$\lambda_{P_{j}}(\mathbf{x})\leq\lambda_{P_{i}}(\mathbf{x})$ for all
$i=1,\dots,M$ with $\alpha_{i}\neq 0$. Notice that the appropriate $j$ will
depend on $\mathbf{x}$.

  3. (3)

The bound in (1.33) is stronger than that of (1.31) with $k=1$. This is
because a positive power of $\mu_{\boldsymbol{\alpha}}(\mathbf{x})$ will
partially cancel a single negative power of $\lambda_{P_{j}}(\mathbf{x})$ for
$j$ such that $\lambda_{P_{j}}(\mathbf{x})\geq\lambda_{P_{i}}(\mathbf{x})$ for
all $i=1,\dots,M$ with $\alpha_{i}\neq 0$.

The proof of Theorem 1.3 will follow a similar strategy to that of [8,
Proposition 1.10]. An additional result will be required to prove (1.33). This
result, [9], shows that $\psi$ can be made $C^{1,1}({}^{3N})$ upon
multiplication by a factor, universal in the sense that the factor depends
only on $N$ and $Z$.

We will require an elliptic regularity result, stated below, which will be
used in the proofs. Beforehand, we clarify the precise form of definitions
which we will be using. Let $\Omega$ be open, $\theta\in(0,1]$ and
$k=\mathbb{N}_{0}$. We formally define the $\theta$-Hölder seminorms for a
function $f$ by

$\displaystyle[f]_{\theta,\Omega}=\sup_{\begin{subarray}{c}x,y\in\Omega\\\
x\neq y\end{subarray}}\frac{|f(x)-f(y)|}{|x-y|^{\theta}},$
$\displaystyle[\nabla^{k}f]_{\theta,\Omega}=\sup_{|\alpha|=k}[\partial^{\alpha}f]_{\theta,\Omega}.$

The space $C^{k,\theta}(\Omega)$ is defined as all $f\in C^{k}(\Omega)$ where
$[\nabla^{k}f]_{\theta,\Omega^{\prime}}$ is finite for each $\Omega^{\prime}$
compactly contained in $\Omega$. In addition, the space
$C^{k,\theta}(\overline{\Omega})$ is defined as all $f\in
C^{k}(\overline{\Omega})$ where $[\nabla^{k}f]_{\theta,\Omega}$ is finite.
This space has a norm given by

$\displaystyle\left\lVert
f\right\rVert_{C^{k,\theta}(\overline{\Omega})}=\left\lVert
f\right\rVert_{C^{k}(\overline{\Omega})}+[\nabla^{k}f]_{\theta,\Omega}.$

For open $\Omega\subset{}^{n}$ we can consider the following elliptic
equation,

(1.34) $Lu:=-\Delta u+\mathbb{c}\cdot\nabla u+du=g$

for some $\mathbb{c}:\Omega\to\mathbb{C}^{n}$ and $d,g:\Omega\to\mathbb{C}$.
The corresponding bilinear form for operator $L$ is defined formally as

$\mathcal{L}(u,\chi)=\int_{\Omega}\big{(}\nabla
u\cdot\nabla\chi+(\mathbb{c}\cdot\nabla u)\chi+du\chi\big{)}\,dx$

for all $u\in H^{1}_{loc}(\Omega)$ and $\chi\in C^{\infty}_{c}(\Omega)$. We
say that a function $u\in H^{1}_{loc}(\Omega)$ is a weak solution to the
equation (1.34) in $\Omega$ if $\mathcal{L}(u,\chi)=\int_{\Omega}g\chi\,dx$
for every $\chi\in C^{\infty}_{c}(\Omega)$.

The following theorem is a restatement of [5, Proposition 3.1] ([8,
Proposition A.2] is similar), with additional Hölder regularity which follows
from the proof.

######  Theorem 1.5.

Let $x_{0}\in{}^{n}$, $R>0$ and $\mathbb{c},d,g\in L^{\infty}(B(x_{0},R))$ and
$u\in H^{1}(B(x_{0},R))$ be a weak solution to (1.34) then for each
$\theta\in[0,1)$ we have $u\in C^{1,\theta}(B(x_{0},R))\cap
H_{loc}^{2}(B(x_{0},R))$, and for any $r\in(0,R)$ we have

(1.35) $\left\lVert u\right\rVert_{C^{1,\theta}(\overline{B(x_{0},r)})}\leq
C(\left\lVert u\right\rVert_{L^{2}(B(x_{0},R))}+\left\lVert
g\right\rVert_{L^{\infty}(B(x_{0},R))})$

for $C=C(n,K,r,R,\theta)$ where

$\left\lVert\mathbb{c}\right\rVert_{L^{\infty}(B(x_{0},R))}+\left\lVert
d\right\rVert_{L^{\infty}(B(x_{0},R))}\leq K.$

##  2\. Proof of Theorem 1.3

Our strategy for the proof will be to choose a suitable function
$F=F(\mathbf{x})$, dependent only on $N$ and $Z$, such that the function
$e^{-F}\psi$ has greater regularity than $\psi$ itself. Such a multiplicative
factor is frequently called a Jastrow factor in mathematical literature, and
this strategy has been used successfully in, for example, [10], [11] to
elucidate regularity properties of $\psi$. The function $e^{-F}\psi$ will
solve an elliptic equation with bounded coefficients which behave suitably
well under the action of cluster derivatives. Elliptic regularity will then
produce bounds to the cluster derivatives of $e^{-F}\psi$. Such bounds can
then be used to obtain bounds to the cluster derivatives of $\psi$ itself.

###  2.1. Jastrow factors

We begin by defining the function

(2.1) $F(\mathbf{x})=F_{c}(\mathbf{x})-F_{s}(\mathbf{x}),$

for $\mathbb{x}\in{}^{3N}$, where

(2.2) $\displaystyle F_{c}(\mathbb{x})$ $\displaystyle=-\frac{Z}{2}\sum_{1\leq
j\leq N}|x_{j}|+\frac{1}{4}\sum_{1\leq l<k\leq N}|x_{l}-x_{k}|,$ (2.3)
$\displaystyle F_{s}(\mathbb{x})$ $\displaystyle=-\frac{Z}{2}\sum_{1\leq j\leq
N}\sqrt{|x_{j}|^{2}+1}+\frac{1}{4}\sum_{1\leq l<k\leq
N}\sqrt{|x_{l}-x_{k}|^{2}+1}.$

The function $F$ was used in [8]. We now detail some basic facts regarding
these functions. Firstly,

(2.4) $\Delta F_{c}=V$

where $V$ is the Coulomb potential, (1.2). The function $F_{s}$ has the same
behaviour as $F_{c}$ at infinity. However, $F_{s}\in C^{\infty}({}^{3N})$ and

(2.5) $\partial^{\alpha}F_{s}\in L^{\infty}({}^{3N})\quad\text{for all
}\,\alpha\in\mathbb{N}_{0}^{3N},|\alpha|\geq 1.$

Finally, the function $F$ obeys

(2.6) $F,\nabla F\in L^{\infty}({}^{3N}).$

The function $F$ is used to define the following object, which will be used
throughout the proof:

(2.7) $\phi=e^{-F}\psi.$

Using (1.3) the following elliptic equation can be shown to hold for $\phi$ a
weak solution,

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

Since all coefficients are bounded in 3N we can see from Theorem 1.5 that
$\phi\in C^{1,\theta}({}^{3N})$ for each $\theta\in[0,1)$. Furthermore, by the
same theorem, for any pair $0<r<R$ we can find constants $C,C^{\prime}$,
dependent only on $N,Z,E,r,R$ and $\theta$, such that

(2.9)
$\left\lVert\phi\right\rVert_{C^{1,\theta}(\overline{B(\mathbb{x},r)})}\leq
C\left\lVert\phi\right\rVert_{L^{2}(B(\mathbb{x},R))}\leq
C^{\prime}\left\lVert\phi\right\rVert_{L^{\infty}(B(\mathbb{x},R))}$

for all $\mathbb{x}\in{}^{3N}$.

From the definition of $\phi$, (2.7), along with (2.6) there exists new
constants $C<C^{\prime}$, dependent only on $Z$ and $N$ (hence independent of
$R>0$), such that

(2.10)
$C\left\lVert\phi\right\rVert_{L^{\infty}(B(\mathbb{x},R))}\leq\left\lVert\psi\right\rVert_{L^{\infty}(B(\mathbb{x},R))}\leq
C^{\prime}\left\lVert\phi\right\rVert_{L^{\infty}(B(\mathbb{x},R))}$

for all $\mathbb{x}\in{}^{3N}$.

###  2.2. Derivatives of F

Informally, our objective is to take cluster derivatives of the elliptic
equation (2.8) and apply elliptic regularity. To do so, we require bounds to
the cluster derivatives of the coefficients present in this equation. This is
the objective of the current section. To begin, we state and prove the
following preparatory lemma involving the distances introduced in (1.21),
(1.25) and (1.27).

######  Lemma 2.1.

For any
$\boldsymbol{\sigma}=(\sigma_{1},\dots,\sigma_{M})\in\mathbb{N}_{0}^{3M}$ we
have for $k=0,1$,

(2.11)
$\mu_{\boldsymbol{\sigma}}(\mathbb{y})^{k-|\boldsymbol{\sigma}|}\leq\lambda_{\boldsymbol{\sigma}}(\mathbb{y})^{k}\lambda_{P_{1}}(\mathbb{y})^{-|\sigma_{1}|}\dots\lambda_{P_{M}}(\mathbb{y})^{-|\sigma_{M}|}$

for all $\mathbb{y}\in{}^{3N}$. Furthermore, let
$\boldsymbol{\beta}^{(1)},\dots,\boldsymbol{\beta}^{(n)}$ be an arbitrary
collection of multiindices in $\mathbb{N}_{0}^{3M}$ such that
$\boldsymbol{\beta}^{(1)}+\dots+\boldsymbol{\beta}^{(n)}=\boldsymbol{\sigma}$
then

(2.12)
$\prod_{j=1}^{n}\lambda_{\boldsymbol{\beta}^{(j)}}(\mathbb{y})\leq\lambda_{\boldsymbol{\sigma}}(\mathbb{y})$

for all $\mathbb{y}\in{}^{3N}$.

###### Proof.

The results are trivial in the case of $\boldsymbol{\sigma}=0$ therefore we
assume in the following that $\boldsymbol{\sigma}$ is non-zero. First, observe
for all $j=1,\dots,M$,

$\begin{split}\mu_{\boldsymbol{\sigma}}(\mathbb{y})^{-|\sigma_{j}|}&\leq\lambda_{P_{j}}(\mathbb{y})^{-|\sigma_{j}|}\\\
\mu_{\boldsymbol{\sigma}}(\mathbb{y})^{1-|\sigma_{j}|}&\leq\lambda_{P_{j}}(\mathbb{y})^{1-|\sigma_{j}|}\quad\text{
if }\sigma_{j}\neq 0\end{split}$

by the definition of $\mu_{\boldsymbol{\sigma}}$. Now perform the following
trivial expansion of the product,

$\begin{split}\mu_{\boldsymbol{\sigma}}(\mathbb{y})^{-|\boldsymbol{\sigma}|}&=\mu_{\boldsymbol{\sigma}}(\mathbb{y})^{-|\sigma_{1}|}\dots\mu_{\boldsymbol{\sigma}}(\mathbb{y})^{-|\sigma_{M}|}\\\
&\leq\lambda_{P_{1}}(\mathbb{y})^{-|\sigma_{1}|}\dots\lambda_{P_{M}}(\mathbb{y})^{-|\sigma_{M}|}\end{split}$

which proves (2.11) for $k=0$. For $k=1$, consider that for each $\mathbb{y}$
we can find $l=1,\dots,M$ such that
$\lambda_{\boldsymbol{\sigma}}(\mathbb{y})=\lambda_{P_{l}}(\mathbb{y})$ and
$\sigma_{l}\neq 0$. Then,

$\begin{split}\mu_{\boldsymbol{\sigma}}(\mathbb{y})^{1-|\boldsymbol{\sigma}|}&=\mu_{\boldsymbol{\sigma}}(\mathbb{y})^{-|\sigma_{1}|}\dots\mu_{\boldsymbol{\sigma}}(\mathbb{y})^{1-|\sigma_{l}|}\dots\mu_{\boldsymbol{\sigma}}(\mathbb{y})^{-|\sigma_{M}|}\\\
&\leq\lambda_{P_{1}}(\mathbb{y})^{-|\sigma_{1}|}\dots\lambda_{P_{l}}(\mathbb{y})^{1-|\sigma_{l}|}\dots\lambda_{P_{M}}(\mathbb{y})^{-|\sigma_{M}|}\\\
&=\lambda_{\boldsymbol{\sigma}}(\mathbb{y})\lambda_{P_{1}}(\mathbb{y})^{-|\sigma_{1}|}\dots\lambda_{P_{M}}(\mathbb{y})^{-|\sigma_{M}|}\end{split}$

as required.

####

Finally we prove (2.12). As above, take arbitrary $\mathbb{y}$ and a
corresponding $l$ such that
$\lambda_{\boldsymbol{\sigma}}(\mathbb{y})=\lambda_{P_{l}}(\mathbb{y})$ with
$\sigma_{l}\neq 0$. For each $1\leq j\leq n$ we denote the
$\mathbb{N}_{0}^{3}$-components of $\boldsymbol{\beta}^{(j)}$ as
$\boldsymbol{\beta}^{(j)}=(\beta^{(j)}_{1},\dots,\beta^{(j)}_{M})$. We know
$\boldsymbol{\beta}^{(1)}+\dots+\boldsymbol{\beta}^{(n)}=\boldsymbol{\sigma}$
so in particular, $\beta_{l}^{(1)}+\dots+\beta_{l}^{(n)}=\sigma_{l}$. Since
$\sigma_{l}\neq 0$ there exists at least one $1\leq r\leq n$ such that
$\beta^{(r)}_{l}\neq 0$. Hence by the definition of
$\lambda_{\boldsymbol{\beta}^{(r)}}$,

$\lambda_{\boldsymbol{\beta}^{(r)}}(\mathbb{y})=\min\\{\lambda_{P_{s}}(\mathbb{y}):\beta_{s}^{(r)}\neq
0\text{,
}s=1,\dots,M\\}\leq\lambda_{P_{l}}(\mathbb{y})=\lambda_{\boldsymbol{\sigma}}(\mathbb{y}).$

The remaining factors $\lambda_{\boldsymbol{\beta}^{(j)}}(\mathbb{y})$, for
$j\neq r$, can each be bounded above by one.

∎

The following lemma will be useful in proving results about taking cluster
derivatives of $F$, as defined in (2.1). Later, we will apply it using $f$ as
the function $|x|$ for $x\in{}^{3}$, or derivatives thereof.

######  Lemma 2.2.

Let $f\in C^{\infty}({}^{3}\backslash\\{0\\})$ and $k\in\mathbb{N}_{0}$ be
such that for each $\sigma\in\mathbb{N}_{0}^{3}$ there exists $C$ such that

(2.13) $|\partial^{\sigma}f(x)|\leq C|x|^{k-|\sigma|}\text{ for all }x\neq 0.$

Then for any $\boldsymbol{\alpha}\neq 0$ with $|\boldsymbol{\alpha}|\geq k$
there exists some new $C$ such that for any $l,m=1,\dots,N,$ the weak
derivatives $D_{\mathbb{P}}^{\boldsymbol{\alpha}}(f(x_{l}))$ and
$D_{\mathbb{P}}^{\boldsymbol{\alpha}}(f(x_{l}-x_{m}))$ are both smooth in
$\Sigma_{\boldsymbol{\alpha}}^{c}$ and obey

$|D_{\mathbb{P}}^{\boldsymbol{\alpha}}(f(x_{l}))|,\,|D_{\mathbb{P}}^{\boldsymbol{\alpha}}(f(x_{l}-x_{m}))|\leq
Cq_{\boldsymbol{\alpha}}(\mathbb{x})^{k-|\boldsymbol{\alpha}|}$

for all $\mathbb{x}\in\Sigma_{\boldsymbol{\alpha}}^{c}$.

###### Proof.

Take any $j=1,\dots,M$ with $\alpha_{j}\neq 0$, then we have
$D^{\alpha_{j}}_{P_{j}}(f(x_{l}))\equiv 0$ for each $l\in P_{j}^{c}$.
Therefore, for $D^{\boldsymbol{\alpha}}_{\mathbb{P}}(f(x_{l}))$ to not be
identically zero we require that $l\in P_{j}$ for each $j$ with
$\alpha_{j}\neq 0$. For such $l$ we have $x_{l}\neq 0$ since
$\mathbb{x}\in\Sigma^{c}_{\boldsymbol{\alpha}}$, and

$|x_{l}|\geq d_{P_{j}}(\mathbb{x})$

for each $j$ with $\alpha_{j}\neq 0$ by (1.20). Therefore, for constant $C$ in
(2.13), we have

$|D^{\boldsymbol{\alpha}}_{\mathbb{P}}(f(x_{l}))|=|\partial^{\alpha_{1}+\dots+\alpha_{M}}f(x_{l})|\leq
C|x_{l}|^{k-|\boldsymbol{\alpha}|}\leq
Cq_{\boldsymbol{\alpha}}(\mathbb{x})^{k-|\boldsymbol{\alpha}|}$

because $|\boldsymbol{\alpha}|\geq k$. Similarly, for each $j=1,\dots,M$, with
$\alpha_{j}\neq 0$ we have $D^{\alpha_{j}}_{P_{j}}(f(x_{l}-x_{m}))\equiv 0$ if
either $l,m\in P_{j}$ or $l,m\in P_{j}^{c}$. Therefore, for
$D^{\boldsymbol{\alpha}}_{\mathbb{P}}(f(x_{l}-x_{m}))$ to not be identically
zero we require that

$(l,m)\in\bigcap_{j\,:\,\alpha_{j}\neq 0}\big{(}(P_{j}\times
P_{j}^{c})\cup(P_{j}^{c}\times P_{j})\big{)}.$

For such $(l,m)$ we have $x_{l}\neq x_{m}$ since
$\mathbb{x}\in\Sigma^{c}_{\boldsymbol{\alpha}}$ and

$|x_{l}-x_{m}|\geq\sqrt{2}\,d_{P_{j}}(\mathbb{x})$

for each $j$ with $\alpha_{j}\neq 0$ by (1.20). Therefore, for some constant
$C^{\prime}$,

$|D^{\boldsymbol{\alpha}}_{\mathbb{P}}(f(x_{l}-x_{m}))|=|\partial^{\alpha_{1}+\dots+\alpha_{M}}f(x_{l}-x_{m})|\leq
C|x_{l}-x_{m}|^{k-|\boldsymbol{\alpha}|}\leq
C^{\prime}q_{\boldsymbol{\alpha}}(\mathbb{x})^{k-|\boldsymbol{\alpha}|}$

because $|\boldsymbol{\alpha}|\geq k$. ∎

The following lemma provides pointwise bounds to cluster derivatives of
functions involving $F$.

######  Lemma 2.3.

For any cluster set $\mathbb{P}$ and any $|\boldsymbol{\sigma}|\geq 1$ there
exists $C$, which depends on $\boldsymbol{\sigma}$, such that for $k=0,1$,

(2.14)
$\big{|}D_{\mathbb{P}}^{\boldsymbol{\sigma}}\nabla^{k}F(\mathbb{y})\big{|},\,\big{|}D_{\mathbb{P}}^{\boldsymbol{\sigma}}\nabla^{k}(e^{F})(\mathbb{y})\big{|}\leq
C\lambda_{\boldsymbol{\sigma}}(\mathbb{y})^{1-k}\lambda_{P_{1}}(\mathbb{y})^{-|\sigma_{1}|}\dots\lambda_{P_{M}}(\mathbb{y})^{-|\sigma_{M}|}$
(2.15) $\big{|}D_{\mathbb{P}}^{\boldsymbol{\sigma}}|\nabla
F(\mathbb{y})|^{2}\big{|}\leq
C\lambda_{P_{1}}(\mathbb{y})^{-|\sigma_{1}|}\dots\lambda_{P_{M}}(\mathbb{y})^{-|\sigma_{M}|}$

for all $\mathbb{y}\in\Sigma^{c}_{\boldsymbol{\sigma}}$. The bound to the
first object in (2.14) also holds when $F$ is replaced by $F_{c}$.

###### Proof.

Let $\tau$ be the function defined as $\tau(x)=|x|$ for $x\in{}^{3}$. Then, by
definition (2.2) we can write

(2.16) $F_{c}(\mathbb{y})=-\frac{Z}{2}\sum_{1\leq j\leq
N}\tau(y_{j})+\frac{1}{4}\sum_{1\leq l<m\leq N}\tau(y_{l}-y_{m}).$

For each $r=1,\dots,N$, denote by $\nabla_{y_{r}}$ the three-dimensional
gradient in the variable $y_{r}$. We then have

(2.17)
$\nabla_{y_{r}}F_{c}(\mathbb{y})=-\frac{Z}{2}\nabla\tau(y_{r})+\frac{1}{4}\sum_{\begin{subarray}{c}m=1\\\
m\neq r\end{subarray}}^{N}\nabla\tau(y_{r}-y_{m}).$

We apply the $D^{\boldsymbol{\sigma}}_{\mathbb{P}}$-derivative to each of
(2.16) and (2.17), using Lemma 2.2 with $f=\tau$ and $f=\nabla\tau$
respectively. This shows that for $k=0,1$ the functions
$D_{\mathbb{P}}^{\boldsymbol{\sigma}}\nabla^{k}F_{c}$ exist and are smooth on
the set $\Sigma^{c}_{\boldsymbol{\sigma}}$. Furthermore, there exists $C$,
depending on $\boldsymbol{\sigma}$, such that

(2.18)
$\displaystyle|D_{\mathbb{P}}^{\boldsymbol{\sigma}}\nabla^{k}F_{c}(\mathbb{y})|$
$\displaystyle\leq
Cq_{\boldsymbol{\sigma}}(\mathbb{y})^{1-k-|\boldsymbol{\sigma}|}\leq
C\mu_{\boldsymbol{\sigma}}(\mathbb{y})^{1-k-|\boldsymbol{\sigma}|}$

for all $\mathbb{y}\in\Sigma^{c}_{\boldsymbol{\sigma}}$. In the second
inequality we used that $\mu_{\boldsymbol{\sigma}}(\mathbb{y})\leq
q_{\boldsymbol{\sigma}}(\mathbb{y})$. Recall that $\nabla F_{s}$ is smooth
with all derivatives bounded, see (2.5). Since $F=F_{c}-F_{s}$ and
$\mu_{\boldsymbol{\sigma}}\leq 1$, we have that for each
$\boldsymbol{\sigma}\neq 0$ and $k=0,1$, there exists $C$, depending on
$\boldsymbol{\sigma}$, such that

(2.19) $|D_{\mathbb{P}}^{\boldsymbol{\sigma}}\nabla^{k}F(\mathbb{y})|\leq
C\mu_{\boldsymbol{\sigma}}(\mathbb{y})^{1-k-|\boldsymbol{\sigma}|}$

for all $\mathbb{y}\in\Sigma^{c}_{\boldsymbol{\sigma}}$. By a straightforward
application of Lemma 2.1, this proves the first bound in (2.14), for $k=0,1$.

####

We proceed to prove the second bound in (2.14). For every
$\boldsymbol{\eta}\leq\boldsymbol{\sigma}$ we have that
$\Sigma_{\boldsymbol{\eta}}\subset\Sigma_{\boldsymbol{\sigma}}$ and therefore
$D_{\mathbb{P}}^{\boldsymbol{\eta}}\nabla^{k}F$ exists and is smooth in
$\Sigma_{\boldsymbol{\sigma}}^{c}$, for $k=0,1$. We use the chain rule for
weak derivatives to show that $D^{\boldsymbol{\sigma}}_{\mathbb{P}}(e^{F})$
exists in $\Sigma_{\boldsymbol{\sigma}}^{c}$ and is equal to a sum of terms,
each of the form

(2.20) $e^{F}\prod_{1\leq j\leq n}D^{\boldsymbol{\beta}^{(j)}}_{\mathbb{P}}F$

for some $1\leq n\leq|\boldsymbol{\sigma}|$ and some collection
$0\neq\boldsymbol{\beta}^{(j)}\in\mathbb{N}_{0}^{3M}$ for $j=1,\dots,n$, where
$\boldsymbol{\beta}^{(1)}+\dots+\boldsymbol{\beta}^{(n)}=\boldsymbol{\sigma}$.
For each $j=1,\dots,n$, we write
$\boldsymbol{\beta}^{(j)}=(\beta^{(j)}_{1},\dots,\beta^{(j)}_{M})$ to denote
the $\mathbb{N}_{0}^{3}$-components of $\boldsymbol{\beta}^{(j)}$. Similarly,
the weak derivative $D^{\boldsymbol{\sigma}}_{\mathbb{P}}\nabla(e^{F})$ exists
in $\Sigma_{\boldsymbol{\sigma}}^{c}$ and the gradient of the general term
(2.20) is equal to

(2.21) $e^{F}\,\nabla F\prod_{1\leq j\leq
n}D_{\mathbb{P}}^{\boldsymbol{\beta}^{(j)}}F+e^{F}\sum_{r=1}^{n}\Big{(}D_{\mathbb{P}}^{\boldsymbol{\beta}^{(r)}}\nabla
F\,\prod_{\begin{subarray}{c}1\leq j\leq n\\\ j\neq
r\end{subarray}}D_{\mathbb{P}}^{\boldsymbol{\beta}^{(j)}}F\Big{)}.$

We will now find bounds for the above expressions. To do this we will use
(2.19) to bound cluster derivatives of $F$. This, along with (2.11) and (2.12)
of Lemma 2.1, allows us to bound the following product

$\displaystyle\Big{|}\prod_{j=1}^{n}D_{\mathbb{P}}^{\boldsymbol{\beta}^{(j)}}F(\mathbb{y})\Big{|}$
$\displaystyle\leq
C\prod_{j=1}^{n}\mu_{\boldsymbol{\beta}^{(j)}}(\mathbb{y})^{1-|\boldsymbol{\beta}^{(j)}|}$
$\displaystyle\leq
C\prod_{j=1}^{n}\lambda_{\boldsymbol{\beta}^{(j)}}(\mathbb{y})\lambda_{P_{1}}(\mathbb{y})^{-|\beta^{(j)}_{1}|}\dots\lambda_{P_{M}}(\mathbb{y})^{-|\beta^{(j)}_{M}|}$
$\displaystyle=C\Big{(}\prod_{j=1}^{n}\lambda_{\boldsymbol{\beta}^{(j)}}(\mathbb{y})\Big{)}\Big{(}\prod_{l=1}^{M}\lambda_{P_{l}}(\mathbb{y})^{-|\beta^{(1)}_{l}|}\dots\lambda_{P_{l}}(\mathbb{y})^{-|\beta^{(n)}_{l}|}\Big{)}$
$\displaystyle\leq
C\lambda_{\boldsymbol{\sigma}}(\mathbb{y})\lambda_{P_{1}}(\mathbb{y})^{-|\sigma_{1}|}\dots\lambda_{P_{M}}(\mathbb{y})^{-|\sigma_{M}|}$

where $C$ is some constant, depending on $\boldsymbol{\sigma}$, and all
$\mathbb{y}\in\Sigma^{c}_{\boldsymbol{\sigma}}$. We used that for each
$l=1,\dots,M$, we have
$|\beta^{(1)}_{l}|+\dots+|\beta^{(n)}_{l}|=|\sigma_{l}|$ as a consequence of
$\boldsymbol{\beta}^{(1)}+\dots+\boldsymbol{\beta}^{(n)}=\boldsymbol{\sigma}$.
We now bound the following product in a similar manner using (2.19) and Lemma
2.1. For any $r=1,\dots,n$,

$\displaystyle\Big{|}D^{\boldsymbol{\beta}^{(r)}}_{\mathbb{P}}\nabla
F(\mathbb{y})\prod_{\begin{subarray}{c}1\leq j\leq n\\\ j\neq
r\end{subarray}}D^{\boldsymbol{\beta}^{(j)}}_{\mathbb{P}}F(\mathbb{y})\Big{|}$
$\displaystyle\leq
C\lambda_{P_{1}}(\mathbb{y})^{-|\sigma_{1}|}\dots\lambda_{P_{M}}(\mathbb{y})^{-|\sigma_{M}|}$

for some constant $C$, depending on $\boldsymbol{\sigma}$, and all
$\mathbb{y}\in\Sigma^{c}_{\boldsymbol{\sigma}}$. Using the fact that $e^{F}$
and $\nabla F$ are bounded in 3N, it is now straightforward to bound (2.20)
and (2.21) appropriately. This completes the proof of (2.14).

####

Finally, we prove the inequality (2.15). It can be shown that the following
Leibniz rule holds

$D_{\mathbb{P}}^{\boldsymbol{\sigma}}|\nabla
F|^{2}=\sum_{\boldsymbol{\beta}\leq\boldsymbol{\sigma}}{\boldsymbol{\sigma}\choose\boldsymbol{\beta}}D^{\boldsymbol{\beta}}_{\mathbb{P}}\nabla
F\cdot D^{\boldsymbol{\sigma}-\boldsymbol{\beta}}_{\mathbb{P}}\nabla F$

in $\Sigma^{c}_{\boldsymbol{\sigma}}$. For every
$\boldsymbol{\beta}\leq\boldsymbol{\sigma}$ we can bound
$D^{\boldsymbol{\beta}}_{\mathbb{P}}\nabla F$ by a constant if
$\boldsymbol{\beta}=0$, and by (2.19) otherwise. This gives some constant $C$,
depending on $\boldsymbol{\sigma}$, such that

$\displaystyle\big{|}D_{\mathbb{P}}^{\boldsymbol{\sigma}}|\nabla
F(\mathbb{y})|^{2}\big{|}$ $\displaystyle\leq
C\sum_{\boldsymbol{\beta}\leq\boldsymbol{\sigma}}\mu_{\boldsymbol{\beta}}(\mathbb{y})^{-|\boldsymbol{\beta}|}\mu_{\boldsymbol{\sigma}-\boldsymbol{\beta}}(\mathbb{y})^{-|\boldsymbol{\sigma}|+|\boldsymbol{\beta}|}$

for all $\mathbb{y}\in\Sigma^{c}_{\boldsymbol{\sigma}}$. The required bound is
obtained after an application of (2.11) of Lemma 2.1.

∎

####

The following result extends the bounds given in Lemma 2.3 to give bounds to
$L^{\infty}$-norms in balls. It is useful to note that if
$\mathbb{x}\in\Sigma_{\boldsymbol{\sigma}}^{c}$, for $\boldsymbol{\sigma}\neq
0$, then
$B(\mathbb{x},\nu\lambda_{\boldsymbol{\sigma}}(\mathbb{x}))\subset\Sigma_{\boldsymbol{\sigma}}^{c}$
for all $\nu\in(0,1)$.

######  Lemma 2.4.

For any $|\boldsymbol{\sigma}|\geq 1$ and any $\nu\in(0,1)$ there exists $C$,
depending on $\boldsymbol{\sigma}$ and $\nu$, such that for $k=0,1$,

(2.22) $\left\lVert
D_{\mathbb{P}}^{\boldsymbol{\sigma}}\nabla^{k}F\right\rVert_{L^{\infty}(B(\mathbb{x},\nu\lambda_{\boldsymbol{\sigma}}(\mathbb{x})))},\,\left\lVert
D_{\mathbb{P}}^{\boldsymbol{\sigma}}\nabla^{k}(e^{F})\right\rVert_{L^{\infty}(B(\mathbb{x},\nu\lambda_{\boldsymbol{\sigma}}(\mathbb{x})))}\\\
\leq
C\lambda_{\boldsymbol{\sigma}}(\mathbb{x})^{1-k}\lambda_{P_{1}}(\mathbb{x})^{-|\sigma_{1}|}\dots\lambda_{P_{M}}(\mathbb{x})^{-|\sigma_{M}|}$
(2.23) $\left\lVert D_{\mathbb{P}}^{\boldsymbol{\sigma}}|\nabla
F|^{2}\right\rVert_{L^{\infty}(B(\mathbb{x},\nu\lambda_{\boldsymbol{\sigma}}(\mathbb{x})))}\leq
C\lambda_{P_{1}}(\mathbb{x})^{-|\sigma_{1}|}\dots\lambda_{P_{M}}(\mathbb{x})^{-|\sigma_{M}|}$

for all $\mathbb{x}\in\Sigma_{\boldsymbol{\sigma}}^{c}$. The bound to the
first norm in (2.22) also holds when $F$ is replaced by $F_{c}$.

###### Proof.

Take some $\mathbb{x}\in\Sigma_{\boldsymbol{\sigma}}^{c}$. By (1.22), for each
$\mathbb{y}\in B(\mathbb{x},\nu\lambda_{\boldsymbol{\sigma}}(\mathbb{x}))$ we
have

$|\lambda_{P_{j}}(\mathbb{x})-\lambda_{P_{j}}(\mathbb{y})|\leq|\mathbb{x}-\mathbb{y}|\leq\nu\lambda_{\boldsymbol{\sigma}}(\mathbb{x})\leq\nu\lambda_{P_{j}}(\mathbb{x})$

for each $j=1,\dots,M$ with $\sigma_{j}\neq 0$. The final inequality above
uses the definition of $\lambda_{\boldsymbol{\sigma}}$ in (1.25). By
rearrangement, we have

(2.24) $(1-\nu)\lambda_{P_{j}}(\mathbb{x})\leq\lambda_{P_{j}}(\mathbb{y}).$

Therefore,

(2.25)
$\lambda_{P_{1}}(\mathbb{y})^{-|\sigma_{1}|}\dots\lambda_{P_{M}}(\mathbb{y})^{-|\sigma_{M}|}\leq(1-\nu)^{-|\boldsymbol{\sigma}|}\lambda_{P_{1}}(\mathbb{x})^{-|\sigma_{1}|}\dots\lambda_{P_{M}}(\mathbb{x})^{-|\sigma_{M}|}$

for all $\mathbb{y}\in
B(\mathbb{x},\nu\lambda_{\boldsymbol{\sigma}}(\mathbb{x}))$. We now prove an
analogous inequality. First, we see that
$\lambda_{\boldsymbol{\sigma}}(\mathbb{x})=\lambda_{P_{l}}(\mathbb{x})$ for
some $1\leq l\leq M$ with $\sigma_{l}\neq 0$ which will depend on the choice
of $\mathbb{x}$. We also note that
$\lambda_{\boldsymbol{\sigma}}(\mathbb{y})\leq\lambda_{P_{l}}(\mathbb{y})$ for
all $\mathbb{y}\in{}^{3N}$ which follows from $\sigma_{l}\neq 0$ and the
definition of $\lambda_{\boldsymbol{\sigma}}$. Therefore,

$\displaystyle\lambda_{\boldsymbol{\sigma}}(\mathbb{y})\lambda_{P_{1}}(\mathbb{y})^{-|\sigma_{1}|}\dots\lambda_{P_{M}}(\mathbb{y})^{-|\sigma_{M}|}$
$\displaystyle\leq\lambda_{P_{l}}(\mathbb{y})^{1-|\sigma_{l}|}\prod_{\begin{subarray}{c}j=1\\\
j\neq l\end{subarray}}^{M}\lambda_{P_{j}}(\mathbb{y})^{-|\sigma_{j}|}$
$\displaystyle\leq(1-\nu)^{1-|\boldsymbol{\sigma}|}\lambda_{P_{l}}(\mathbb{x})^{1-|\sigma_{l}|}\prod_{\begin{subarray}{c}j=1\\\
j\neq l\end{subarray}}^{M}\lambda_{P_{j}}(\mathbb{x})^{-|\sigma_{j}|}$ (2.26)
$\displaystyle=(1-\nu)^{1-|\boldsymbol{\sigma}|}\lambda_{\boldsymbol{\sigma}}(\mathbb{x})\lambda_{P_{1}}(\mathbb{x})^{-|\sigma_{1}|}\dots\lambda_{P_{M}}(\mathbb{x})^{-|\sigma_{M}|}$

for all $\mathbb{y}\in
B(\mathbb{x},\nu\lambda_{\boldsymbol{\sigma}}(\mathbb{x}))$. In the second
step we applied (2.24). The required bounds then arise from Lemma 2.3 followed
by either (2.25) or (2.26) as appropriate.

∎

###  2.3. Cluster derivatives of $\phi$ and $\psi$

####

We introduce the following notation. For cluster set
$\mathbb{P}=(P_{1},\dots,P_{M})$ and
$\boldsymbol{\alpha}\in\mathbb{N}_{0}^{3M}$ define

$\phi_{\boldsymbol{\alpha}}=D_{\mathbb{P}}^{\boldsymbol{\alpha}}\phi.$

We have previously obtained estimates for the coefficients, and their
derivatives, in equation (2.8). By taking cluster derivatives of this
equation, we obtain elliptic equations whose weak solutions are
$\phi_{\boldsymbol{\alpha}}$. By a process of rescaling we will apply elliptic
regularity to give quantative bounds to the $L^{\infty}$-norms of
$\phi_{\boldsymbol{\alpha}}$ and $\nabla\phi_{\boldsymbol{\alpha}}$. From here
it is straightforward to obtain corresponding bounds for
$D_{\mathbb{P}}^{\boldsymbol{\alpha}}\psi$ and
$D_{\mathbb{P}}^{\boldsymbol{\alpha}}\nabla\psi$.

######  Lemma 2.5.

For every $|\boldsymbol{\alpha}|\geq 1$ we have $\phi_{\boldsymbol{\alpha}}$
is a weak solution to

(2.27) $-\Delta\phi_{\boldsymbol{\alpha}}-2\nabla
F\cdot\nabla\phi_{\boldsymbol{\alpha}}+(\Delta F_{s}-|\nabla
F|^{2}-E)\phi_{\boldsymbol{\alpha}}\\\
=\sum_{\begin{subarray}{c}\boldsymbol{\sigma}\leq\boldsymbol{\alpha}\\\
\boldsymbol{\sigma}\neq
0\end{subarray}}{\boldsymbol{\alpha}\choose\boldsymbol{\sigma}}\big{(}2D_{\mathbb{P}}^{\boldsymbol{\sigma}}\nabla
F\cdot\nabla\phi_{\boldsymbol{\alpha}-\boldsymbol{\sigma}}-D_{\mathbb{P}}^{\boldsymbol{\sigma}}(\Delta
F_{s}-|\nabla
F|^{2})\phi_{\boldsymbol{\alpha}-\boldsymbol{\sigma}}\big{)}=:g_{\boldsymbol{\alpha}}$

in $\Sigma^{c}_{\boldsymbol{\alpha}}$, and therefore
$\phi_{\boldsymbol{\alpha}}\in C^{1}(\Sigma_{\boldsymbol{\alpha}}^{c})\cap
H^{2}_{loc}(\Sigma_{\boldsymbol{\alpha}}^{c})$.

###### Proof.

We prove by induction, starting with $|\boldsymbol{\alpha}|=1$. Let
$\mathcal{L}(\cdot,\cdot)$ be the bilinear form corresponding to the operator
acting on $\phi_{\boldsymbol{\alpha}}$ on the left-hand side of (2.27), as
defined in (1.34). Since $\phi\in H^{2}_{loc}({}^{3N})$ we have
$\phi_{\boldsymbol{\alpha}}\in H^{1}_{loc}({}^{3N})$. We use integration by
parts to show that, for any $\chi\in
C^{\infty}_{c}(\Sigma^{c}_{\boldsymbol{\alpha}})$,

(2.28)
$\mathcal{L}(\phi_{\boldsymbol{\alpha}},\chi)=-\mathcal{L}(\phi,D_{\mathbb{P}}^{\boldsymbol{\alpha}}\chi)+\int_{\Sigma^{c}_{\boldsymbol{\alpha}}}g_{\boldsymbol{\alpha}}\chi\,d\mathbb{x}.$

Now, $\mathcal{L}(\phi,D_{\mathbb{P}}^{\boldsymbol{\alpha}}\chi)=0$ since
$\phi$ is a weak solution to (2.8), hence (2.27) holds. By Lemma 2.3 and that
$\phi\in C^{1}({}^{3N})$ we see that $g_{\boldsymbol{\alpha}}\in
L_{loc}^{\infty}(\Sigma^{c}_{\boldsymbol{\alpha}})$. Hence, by Theorems 1.5,
$\phi_{\boldsymbol{\alpha}}\in C^{1}(\Sigma^{c}_{\boldsymbol{\alpha}})\cap
H_{loc}^{2}(\Sigma^{c}_{\boldsymbol{\alpha}})$

Assume the hypothesis holds for all multiindices
$1\leq|\boldsymbol{\alpha}|\leq k-1$ for some $k\geq 2$. Take some arbitrary
multiindex $|\boldsymbol{\alpha}|=k$. We will prove the induction hypothesis
for $\boldsymbol{\alpha}$. First, we state the useful fact that for any
$\boldsymbol{\sigma}\leq\boldsymbol{\alpha}$ we have
$\Sigma_{\boldsymbol{\sigma}}\subset\Sigma_{\boldsymbol{\alpha}}$. Take some
$\boldsymbol{\eta}\leq\boldsymbol{\alpha}$ with $|\boldsymbol{\eta}|=1$.
Notice, then, that $\phi_{\boldsymbol{\alpha}-\boldsymbol{\eta}}\in
H^{2}_{loc}(\Sigma^{c}_{\boldsymbol{\alpha}})$ by the induction hypothesis,
and hence $\phi_{\boldsymbol{\alpha}}\in
H^{1}_{loc}(\Sigma^{c}_{\boldsymbol{\alpha}})$. This allows
$\mathcal{L}(\phi_{\boldsymbol{\alpha}},\chi)$ to be defined for any $\chi\in
C_{c}^{\infty}(\Sigma^{c}_{\boldsymbol{\alpha}})$. Integration by parts is
then used on the $D^{\boldsymbol{\eta}}_{\mathbb{P}}$-derivative, in a similar
way to (2.28). Along with the induction hypothesis and further applications of
integration by parts, we find that (2.27) holds for
$\phi_{\boldsymbol{\alpha}}$ as a weak solution. Regularity for
$\phi_{\boldsymbol{\alpha}}$ is shown in a similar way to the
$|\boldsymbol{\alpha}|=1$ case. Indeed, the induction hypothesis can be used
to show $g_{\boldsymbol{\alpha}}\in
L^{\infty}_{loc}(\Sigma^{c}_{\boldsymbol{\alpha}})$.

∎

####

We now apply the $C^{1}$-elliptic regularity estimates of Theorem 1.5, via a
scaling procedure, to the equations (2.27).

######  Lemma 2.6.

For all $|\boldsymbol{\alpha}|\geq 1$ and $0<r<R<1$ there exists $C$,
dependent only on $E,Z,N,r$ and $R$, such that

(2.29)
$\left\lVert\nabla\phi_{\boldsymbol{\alpha}}\right\rVert_{L^{\infty}(B(\mathbb{x},r\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}\\\
\leq
C\lambda_{\boldsymbol{\alpha}}(\mathbb{x})^{-1}\big{(}\left\lVert\phi_{\boldsymbol{\alpha}}\right\rVert_{L^{\infty}(B(\mathbb{x},R\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}+\lambda_{\boldsymbol{\alpha}}(\mathbb{x})^{2}\left\lVert
g_{\boldsymbol{\alpha}}\right\rVert_{L^{\infty}(B(\mathbb{x},R\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}\big{)}$

for all $\mathbb{x}\in\Sigma_{\boldsymbol{\alpha}}^{c}$. The function
$g_{\boldsymbol{\alpha}}$ was given by (2.27).

###### Proof.

Fix any $\mathbb{x}\in\Sigma_{\boldsymbol{\alpha}}^{c}$ and denote
$\lambda=\lambda_{\boldsymbol{\alpha}}(\mathbb{x})$. The proof proceeds via a
rescaling. Define

$\begin{split}w(\mathbb{y})&=\phi_{\boldsymbol{\alpha}}(\mathbb{x}+\lambda\mathbb{y})\\\
\mathbb{c}(\mathbb{y})&=-2\nabla F(\mathbb{x}+\lambda\mathbb{y})\\\
d(\mathbb{y})&=\Delta F_{s}(\mathbb{x}+\lambda\mathbb{y})-|\nabla
F(\mathbb{x}+\lambda\mathbb{y})|^{2}-E\\\
f(\mathbb{y})&=g_{\boldsymbol{\alpha}}(\mathbb{x}+\lambda\mathbb{y})\end{split}$

for all $\mathbb{y}\in B(0,1)$. Then by Lemma 2.5 we have that $w$ is a weak
solution to

(2.30) $-\Delta w+\lambda\mathbb{c}\cdot\nabla w+\lambda^{2}dw=\lambda^{2}f$

in $B(0,1)$. By (2.6) and (2.5), and that $\lambda\leq 1$ by definition, we
obtain

$\lambda\left\lVert\mathbb{c}\right\rVert_{L^{\infty}(B(0,1))}+\lambda^{2}\left\lVert
d\right\rVert_{L^{\infty}(B(0,1))}\leq K$

for some $K$, dependent only on $E,Z$ and $N$ and in particular is independent
of our choice of $\mathbb{x}$. Therefore, by Theorem 1.5, with $\theta=0$, we
get some $C$, dependent only on $N,K,r$ and $R$, such that

(2.31) $\left\lVert w\right\rVert_{C^{1}(\overline{B(0,r)})}\leq C(\left\lVert
w\right\rVert_{L^{\infty}(B(0,R))}+\lambda^{2}\left\lVert
f\right\rVert_{L^{\infty}(B(0,R))}).$

Finally, we use $\nabla
w(\mathbb{y})=\lambda\nabla\phi_{\boldsymbol{\alpha}}(\mathbb{x}+\lambda\mathbb{y})$
by the chain rule, and rewrite (2.31) to give (2.29).

∎

The following proposition uses induction to write estimates for both
$\nabla\phi_{\boldsymbol{\alpha}}$ and $\phi_{\boldsymbol{\alpha}}$ with a
bound involving only the zeroth and first-order derivatives of $\phi$.
Corollary A.2 of the Appendix is used in the proof to improve the regularity
of the second derivative of $\phi$, and this benefit is passed on through
induction to all higher orders of derivative. The function
$f_{\infty}(\,\cdot\,;R;\phi)$ was defined in (1.28).

######  Proposition 2.7.

For any $|\boldsymbol{\alpha}|\geq 1$, any $0<r<R<1$ and $b\in[0,1)$ there
exists $C$, dependent on $\boldsymbol{\alpha},r,R$ and $b$, such that for
$k=0,1$, with $k+|\boldsymbol{\alpha}|\geq 2$,

(2.32)
$\left\lVert\nabla^{k}\phi_{\boldsymbol{\alpha}}\right\rVert_{L^{\infty}(B(\mathbb{x},r\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}\leq
C\lambda_{\boldsymbol{\alpha}}(\mathbb{x})^{1-k}\lambda_{P_{1}}(\mathbb{x})^{-|\alpha_{1}|}\dots\lambda_{P_{M}}(\mathbb{x})^{-|\alpha_{M}|}\mu_{\boldsymbol{\alpha}}(\mathbb{x})^{b}f_{\infty}(\mathbb{x};R;\phi)$

for all $\mathbb{x}\in\Sigma_{\boldsymbol{\alpha}}^{c}$. The inequality also
holds for $|\boldsymbol{\alpha}|=1$ and $k=0$ if we take $b=0$.

###### Proof.

Suppose $|\boldsymbol{\alpha}|=1$. Let $l$ be the index $1\leq l\leq M$ such
that $\alpha_{l}\neq 0$. Then
$\lambda_{P_{l}}=\lambda_{\boldsymbol{\alpha}}=\mu_{\boldsymbol{\alpha}}$ and
$\Sigma_{\boldsymbol{\alpha}}=\Sigma_{P_{l}}$. The required bound for $k=b=0$
follows from the definition of cluster derivative, (1.15). When $k=1$ and
$b\in[0,1)$ the bound follows directly from Corollary A.2 with $P=P_{l}$.

For $|\boldsymbol{\alpha}|\geq 2$ we prove by induction. Assume the hypothesis
holds for all multiindices $\boldsymbol{\alpha}$ with
$1\leq|\boldsymbol{\alpha}|\leq m-1$ for some $m\geq 2$. Take any
$\boldsymbol{\alpha}$ with $|\boldsymbol{\alpha}|=m$. We prove the $k=0$ and
$k=1$ cases in turn. It is useful, here, to state the following fact: for all
$\boldsymbol{\sigma}\leq\boldsymbol{\alpha}$ we have
$\Sigma_{\boldsymbol{\sigma}}\subset\Sigma_{\boldsymbol{\alpha}}$ and hence
$\lambda_{\boldsymbol{\alpha}}(\mathbb{x})\leq\lambda_{\boldsymbol{\sigma}}(\mathbb{x})$
for all $\mathbf{x}\in\Sigma^{c}_{\boldsymbol{\alpha}}$.

The $k=0$ case is a straightforward application of the induction hypothesis
and is described as follows. Firstly, for each
$\mathbf{x}\in\Sigma_{\boldsymbol{\alpha}}^{c}$ it is clear that there exists
some $1\leq l\leq M$ such that
$\lambda_{\boldsymbol{\alpha}}(\mathbb{x})=\lambda_{P_{l}}(\mathbb{x})$ and
where $\alpha_{l}\neq 0$. Therefore we can find some
$\boldsymbol{\eta}\leq\boldsymbol{\alpha}$ with $|\boldsymbol{\eta}|=1$ and
$\eta_{j}=0$ for each $j\neq l$ where $1\leq j\leq M$. Now, from the
definition of cluster derivatives (1.16),

(2.33)
$\displaystyle\left\lVert\phi_{\boldsymbol{\alpha}}\right\rVert_{L^{\infty}(B(\mathbb{x},r\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}$
$\displaystyle\leq
N\left\lVert\nabla\phi_{\boldsymbol{\alpha}-\boldsymbol{\eta}}\right\rVert_{L^{\infty}(B(\mathbb{x},r\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}.$

We can then use the induction assumption to show existence of $C$ such that

(2.34)
$\displaystyle\left\lVert\nabla\phi_{\boldsymbol{\alpha}-\boldsymbol{\eta}}\right\rVert_{L^{\infty}(B(\mathbb{x},r\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}$
$\displaystyle\leq
C\lambda_{P_{l}}(\mathbb{x})\lambda_{P_{1}}(\mathbb{x})^{-|\alpha_{1}|}\dots\lambda_{P_{M}}(\mathbb{x})^{-|\alpha_{M}|}\mu_{\boldsymbol{\alpha}-\boldsymbol{\eta}}(\mathbb{x})^{b}f_{\infty}(\mathbb{x};R;\phi),$

where we can then use
$\lambda_{\boldsymbol{\alpha}}(\mathbb{x})=\lambda_{P_{l}}(\mathbb{x})$ and
$\mu_{\boldsymbol{\alpha}-\boldsymbol{\eta}}(\mathbb{x})\leq\mu_{\boldsymbol{\alpha}}(\mathbb{x})$
to complete the bound (2.32) for our choice of $\boldsymbol{\alpha}$ in the
case of $k=0$.

The $k=1$ case follows from the induction hypothesis and Lemma 2.6. Let
$r^{\prime}=(r+R)/2$. Firstly, by the definition of $g_{\boldsymbol{\alpha}}$,
(2.27), along with (2.5) and Lemma 2.4 we can find $C$, depending on
$\boldsymbol{\alpha}$ and $r^{\prime}$, such that

$\left\lVert
g_{\boldsymbol{\alpha}}\right\rVert_{L^{\infty}(B(\mathbb{x},r^{\prime}\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}\leq
C\sum_{\begin{subarray}{c}\boldsymbol{\sigma}\leq\boldsymbol{\alpha}\\\
\boldsymbol{\sigma}\neq
0\end{subarray}}\lambda_{P_{1}}(\mathbb{x})^{-|\sigma_{1}|}\dots\lambda_{P_{M}}(\mathbb{x})^{-|\sigma_{M}|}\big{(}\left\lVert\phi_{\boldsymbol{\alpha}-\boldsymbol{\sigma}}\right\rVert_{L^{\infty}(B(\mathbb{x},r^{\prime}\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}\\\
+\left\lVert\nabla\phi_{\boldsymbol{\alpha}-\boldsymbol{\sigma}}\right\rVert_{L^{\infty}(B(\mathbb{x},r^{\prime}\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}\big{)}.$

It follows by the induction hypothesis with $b=0$, along with the definition
of $f_{\infty}$, (1.29), that we can find some $C$, depending on
$\boldsymbol{\alpha},r^{\prime}$ and $R$, such that

(2.35) $\left\lVert
g_{\boldsymbol{\alpha}}\right\rVert_{L^{\infty}(B(\mathbb{x},r^{\prime}\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}\leq
C\lambda_{P_{1}}(\mathbb{x})^{-|\alpha_{1}|}\dots\lambda_{P_{M}}(\mathbb{x})^{-|\alpha_{M}|}f_{\infty}(\mathbb{x};R;\phi)$

for all $\mathbb{x}\in\Sigma_{\boldsymbol{\alpha}}^{c}$. Now, to prove the
required bound (2.32) for $k=1$ we apply Lemma 2.6 to obtain some constant
$C$, depending on $r$ and $r^{\prime}$, such that

$\displaystyle\left\lVert\nabla\phi_{\boldsymbol{\alpha}}\right\rVert_{L^{\infty}(B(\mathbb{x},r\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}$
$\displaystyle\leq
C\big{(}\lambda_{\boldsymbol{\alpha}}(\mathbb{x})^{-1}\left\lVert\phi_{\boldsymbol{\alpha}}\right\rVert_{L^{\infty}(B(\mathbb{x},r^{\prime}\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}+\lambda_{\boldsymbol{\alpha}}(\mathbb{x})\left\lVert
g_{\boldsymbol{\alpha}}\right\rVert_{L^{\infty}(B(\mathbb{x},r^{\prime}\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}\big{)}.$

We use the $k=0$ case, proven above, to bound the $L^{\infty}$-norm of
$\phi_{\boldsymbol{\alpha}}$ in the first term on the right-hand side of the
above inequality. To the second term, we apply (2.35) and the simple
inequality
$\lambda_{\boldsymbol{\alpha}}(\mathbb{x})\leq\mu_{\boldsymbol{\alpha}}(\mathbb{x})^{b}$.
Together, these give the bound (2.32) for $k=1$ and our choice of
$\boldsymbol{\alpha}$. This completes the induction.

∎

Using the definition $\phi=e^{-F}\psi$, we now obtain bounds to the cluster
derivatives of the eigenfunction $\psi$ using those for the cluster
derivatives of $\phi$ in the above proposition.

###### Proof of Theorem 1.3.

It is clear that (1.31) holds when $\boldsymbol{\alpha}=0$. Therefore,
consider $|\boldsymbol{\alpha}|\geq 1$. We first prove (1.31) for $k=0$. Take
$\mathbb{x}\in\Sigma_{\boldsymbol{\alpha}}^{c}$, then by the Leibniz rule for
cluster derivatives we have

(2.36) $\displaystyle D_{\mathbb{P}}^{\boldsymbol{\alpha}}\psi$
$\displaystyle=\sum_{\boldsymbol{\beta}\leq\boldsymbol{\alpha}}{\boldsymbol{\alpha}\choose\boldsymbol{\beta}}D_{\mathbb{P}}^{\boldsymbol{\beta}}\big{(}e^{F}\big{)}\phi_{\boldsymbol{\alpha}-\boldsymbol{\beta}}$

in $B(\mathbb{x},r\lambda_{\boldsymbol{\alpha}}(\mathbb{x}))$. Now, for each
$\boldsymbol{\beta}\leq\boldsymbol{\alpha}$ there exists some $C$, independent
of $\mathbf{x}$, such that

$\left\lVert
D_{\mathbb{P}}^{\boldsymbol{\beta}}\big{(}e^{F}\big{)}\phi_{\boldsymbol{\alpha}-\boldsymbol{\beta}}\right\rVert_{L^{\infty}(B(\mathbb{x},r\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}\leq
C\lambda_{\boldsymbol{\alpha}}(\mathbb{x})\lambda_{P_{1}}(\mathbb{x})^{-|\alpha_{1}|}\dots\lambda_{P_{M}}(\mathbb{x})^{-|\alpha_{M}|}f_{\infty}(\mathbb{x};R;\phi).$

To prove this, we use Lemma 2.4 and Proposition 2.7 with $b=0$. In addition,
if $\boldsymbol{\beta}=0$ we use (2.6) and if
$\boldsymbol{\beta}=\boldsymbol{\alpha}$ we use the definition (1.29). If
$\boldsymbol{\beta}\notin\\{0,\boldsymbol{\alpha}\\}$ we use the inequality
$\lambda_{\boldsymbol{\beta}}(\mathbb{x})\lambda_{\boldsymbol{\alpha}-\boldsymbol{\beta}}(\mathbb{x})\leq\lambda_{\boldsymbol{\alpha}}(\mathbb{x})$,
which is obtained from Lemma 2.1. The required bound (1.31) for $k=0$ then
follows from (2.36) and (2.9)-(2.10).

Take $\mathbf{x}\in\Sigma^{c}_{\boldsymbol{\alpha}}$. By the definition
$F=F_{c}-F_{s}$, the equality $\nabla\psi=\psi\nabla F+e^{F}\nabla\phi$ and
the Leibniz rule, we have

$D_{\mathbb{P}}^{\boldsymbol{\alpha}}\nabla\psi=G_{\mathbb{P}}^{\boldsymbol{\alpha}}+\psi
D_{\mathbb{P}}^{\boldsymbol{\alpha}}\nabla F_{c}$

in $B(\mathbb{x},r\lambda_{\boldsymbol{\alpha}}(\mathbb{x}))$, where

(2.37) $\displaystyle G^{\boldsymbol{\alpha}}_{\mathbb{P}}$
$\displaystyle=\sum_{\begin{subarray}{c}\boldsymbol{\beta}\leq\boldsymbol{\alpha}\\\
|\boldsymbol{\beta}|\geq
1\end{subarray}}{\boldsymbol{\alpha}\choose\boldsymbol{\beta}}\psi_{\boldsymbol{\beta}}\,D_{\mathbb{P}}^{\boldsymbol{\alpha}-\boldsymbol{\beta}}\nabla
F+\sum_{\boldsymbol{\beta}\leq\boldsymbol{\alpha}}{\boldsymbol{\alpha}\choose\boldsymbol{\beta}}\nabla\phi_{\boldsymbol{\beta}}\,D_{\mathbb{P}}^{\boldsymbol{\alpha}-\boldsymbol{\beta}}\big{(}e^{F}\big{)}-\psi\,D_{\mathbb{P}}^{\boldsymbol{\alpha}}\nabla
F_{s}.$

We bound each term in $G^{\boldsymbol{\alpha}}_{\mathbb{P}}$ as in (1.33).
Take any $\boldsymbol{\beta}\leq\boldsymbol{\alpha}$ with
$|\boldsymbol{\beta}|\geq 1$. Notice that
$\Sigma_{\boldsymbol{\beta}}\subset\Sigma_{\boldsymbol{\alpha}}$ and hence
$\lambda_{\boldsymbol{\alpha}}(\mathbf{x})\leq\lambda_{\boldsymbol{\beta}}(\mathbf{x})$.
Now, there exists $C$, independent of $\mathbf{x}$, such that

(2.38)
$\displaystyle\left\lVert\psi_{\boldsymbol{\beta}}\,D_{\mathbb{P}}^{\boldsymbol{\alpha}-\boldsymbol{\beta}}\nabla
F\right\rVert_{L^{\infty}(B(\mathbb{x},r\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}$
$\displaystyle\leq
C\lambda_{\boldsymbol{\beta}}(\mathbb{x})\lambda_{P_{1}}(\mathbb{x})^{-|\alpha_{1}|}\dots\lambda_{P_{M}}(\mathbb{x})^{-|\alpha_{M}|}f_{\infty}(\mathbb{x};R).$

To prove this inequality, we bound derivatives of $F$ using Lemma 2.4 if
$\boldsymbol{\beta}\neq\boldsymbol{\alpha}$ and (2.6) if
$\boldsymbol{\beta}=\boldsymbol{\alpha}$. The derivatives
$\psi_{\boldsymbol{\beta}}$ are then bounded using (1.31) with $k=0$, which
was proven above. The right-hand side of (2.38) can be bounded as in (1.33) by
using the simple bound
$\lambda_{\boldsymbol{\beta}}\leq\mu_{\boldsymbol{\alpha}}$. Next, we can find
$C$, independent of $\mathbf{x}$, such that

$\displaystyle\left\lVert\nabla\phi_{\boldsymbol{\beta}}\,D_{\mathbb{P}}^{\boldsymbol{\alpha}-\boldsymbol{\beta}}\big{(}e^{F}\big{)}\right\rVert_{L^{\infty}(B(\mathbb{x},r\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}$
$\displaystyle\leq
C\mu_{\boldsymbol{\beta}}(\mathbb{x})^{b}\lambda_{\boldsymbol{\alpha}-\boldsymbol{\beta}}(\mathbb{x})\lambda_{P_{1}}(\mathbb{x})^{-|\alpha_{1}|}\dots\lambda_{P_{M}}(\mathbb{x})^{-|\alpha_{M}|}f_{\infty}(\mathbb{x};R;\phi).$

This is proven using Proposition 2.7 to bound the derivatives
$\nabla\phi_{\boldsymbol{\beta}}$, and Lemma 2.4 and (2.6) to bound
derivatives of $e^{F}$. The right-hand side of the above inequality can be
bounded as in (1.33) after use of (2.9)-(2.10) and the inequalities
$\mu_{\boldsymbol{\beta}}\leq\mu_{\boldsymbol{\alpha}}$ and
$\lambda_{\boldsymbol{\alpha}-\boldsymbol{\beta}}\leq 1$. In addition, by
Lemma 2.4 there exists $C$ such that

$\displaystyle\left\lVert\nabla\phi\,D_{\mathbb{P}}^{\boldsymbol{\alpha}}\big{(}e^{F}\big{)}\right\rVert_{L^{\infty}(B(\mathbb{x},r\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}$
$\displaystyle\leq
C\lambda_{\boldsymbol{\alpha}}(\mathbb{x})\lambda_{P_{1}}(\mathbb{x})^{-|\alpha_{1}|}\dots\lambda_{P_{M}}(\mathbb{x})^{-|\alpha_{M}|}f_{\infty}(\mathbb{x};R;\phi).$

Use of (2.9)-(2.10), as before, and the inequality
$\lambda_{\boldsymbol{\alpha}}\leq\mu_{\boldsymbol{\alpha}}$ give the correct
bound. Finally, the last term in (2.37) is readily bounded appropriately using
(2.5). In view of (2.37), this completes the proof of (1.33).

To prove (1.31) for $k=1$ we use (1.32), (1.33) and the following inequality,

$\left\lVert\psi D_{\mathbb{P}}^{\boldsymbol{\alpha}}\nabla
F_{c}\right\rVert_{L^{\infty}(B(\mathbb{x},r\lambda_{\boldsymbol{\alpha}}(\mathbb{x})))}\leq
C\lambda_{P_{1}}(\mathbb{x})^{-|\alpha_{1}|}\dots\lambda_{P_{M}}(\mathbb{x})^{-|\alpha_{M}|}f_{\infty}(\mathbb{x};R),$

which holds for some $C$ as a direct consequence of Lemma 2.4.

∎

##  3\. Biscaled cutoffs and their associated clusters

In the proof of Theorem 1.1, partial derivatives of $\gamma(x,y)$ will be
written in terms of integrals involving cluster derivatives of $\psi$. Cutoff
functions are used here to ensure these cluster derivatives are evaluated only
at points in 3N where they exist and where we can apply Theorem 1.3.

The cutoffs introduced in this section are of a similar form to those used in
[5], which themselves are based on those used in [3]. In all these cases,
cutoffs enforce bounds on the distances between pairs of particles on the
support of the cutoff. The bounds on these distances can be scaled using a
scaling parameter. The cutoffs defined here will involve the points (i.e. the
“particles”) $x,y$ and $x_{j}$, $2\leq j\leq N$. For a given cutoff, we will
define a corresponding cluster $P$ which is a collection of the particles
whose distance to $x$ is bounded above appropriately. The same is be done for
$y$, with a cluster $S$. In using these cutoffs, derivatives of $\gamma(x,y)$
in the $x$-variable can naturally be written as cluster derivatives using the
$P$ cluster, and those in the $y$-variable can be written as cluster
derivatives using the $S$ cluster.

We will also need to take derivatives of $\gamma(x,y)$ in both $x$ and $y$
simultanously (which will later be denoted by $\partial_{u}$). For this, it is
natural to consider a larger cluster $Q$ which contains both $P$ and $S$. The
particles in $Q$ are held close to both $x$ and $y$, but potentially looser
than how particles in $P$ and $S$ are held to $x$ and $y$ respectively. For
this reason, the cutoffs used here will involve two scaling parameters
$\delta$ and $\epsilon$ and hence are called biscaled.

###  3.1. Definition of $\Phi$

To begin, take some $\chi\in C_{c}^{\infty}(\real)$, $0\leq\chi\leq 1$, with

(3.1) $\displaystyle\chi(s)$ $\displaystyle=\begin{cases}1&\text{ if }|s|\leq
1\\\ 0&\text{ if }|s|\geq 2.\end{cases}$

for $s\in\real$. For each $t>0$ we can define the following two cutoff factors
$\zeta_{t}=\zeta_{t}(z)$ and $\theta_{t}=\theta_{t}(z)$ by

(3.2)
$\zeta_{t}(z)=\chi\Big{(}\frac{4N|z|}{t}\Big{)},\qquad\theta_{t}(z)=1-\zeta_{t}(z)$

for $z\in{}^{3}$. We have the following support criteria for cutoff factors.
For any $z\in{}^{3}$ and $t>0$,

  * •

If $\zeta_{t}(z)\neq 0$ then $|z|<(2N)^{-1}t$,

  * •

If $\theta_{t}(z)\neq 0$ then $|z|>(4N)^{-1}t$.

Let $0<\delta<(4N)^{-1}\epsilon$ (the use of $(4N)^{-1}$ is explained in the
following lemma). We define a biscaled cutoff, which depends on $\delta$ and
$\epsilon$ as parameters, as a function
$\Phi=\Phi_{\delta,\epsilon}(x,y,\mathbb{\hat{x}})$ defined by

(3.3) $\Phi_{\delta,\epsilon}(x,y,\mathbb{\hat{x}})=\prod_{2\leq j\leq
N}g^{(1)}_{j}(x-x_{j})\prod_{2\leq j\leq N}g^{(2)}_{j}(y-x_{j})\prod_{2\leq
k<l\leq N}f_{kl}(x_{k}-x_{l})$

for $x,y\in{}^{3}$ and $\mathbb{\hat{x}}\in{}^{3N-3}$, and where
$g^{(1)}_{j},g^{(2)}_{j},f_{kl}\in\\{\zeta_{\delta},\theta_{\delta}\zeta_{\epsilon},\theta_{\epsilon}\\}$
for $2\leq j\leq N$ and $2\leq k<l\leq N$. In addition, for $2\leq k<l\leq N$
we define $f_{lk}=f_{kl}$.

The idea is that on the support of $\Phi$ we have upper and/or lower bounds to
the distances between various pairs of particles. If we take the particles
$x_{j}$ and $x_{k}$, for example, we have that $\zeta_{\delta}(x_{j}-x_{k})$
holds the two particles close relative to a distance $\delta$,
$\theta_{\epsilon}(x_{j}-x_{k})$ holds them apart relative to a distance
$\epsilon$, and $\theta_{\delta}\zeta_{\epsilon}$ keeps them within an
intermediate distance apart.

We use the standard notation $\mathds{1}_{S}$ to denote the indicator function
on a set $S$. It is then convenient to define for each $t>0$,

(3.4) $\displaystyle\mathds{1}_{t}(z)$
$\displaystyle=\mathds{1}_{\\{(4N)^{-1}t<|z|<(2N)^{-1}t\\}}(z)$ (3.5)
$\displaystyle\mathds{1}^{\prime}_{t}(z)$
$\displaystyle=\mathds{1}_{\\{(4N)^{-1}t<|z|<1\\}}(z)$

for $z\in{}^{3}$. And for each $t>0$ we define the function
$M_{t}=M_{t}(x,y,\mathbb{\hat{x}})$ by

(3.6) $\displaystyle M_{t}(x,y,\mathbb{\hat{x}})=\sum_{2\leq j\leq
N}\mathds{1}_{t}(x-x_{j})+\sum_{2\leq j\leq
N}\mathds{1}_{t}(y-x_{j})+\sum_{2\leq k<l\leq N}\mathds{1}_{t}(x_{k}-x_{l})$

for $x,y\in{}^{3}$ and $\mathbb{\hat{x}}\in{}^{3N-3}$.

The cutoffs, $\Phi$, defined in (3.3) form a partition of unity as shown in
the following lemma.

######  Lemma 3.1.

There exists a finite collection $\\{\Phi^{(j)}\\}_{j=1}^{J}$ of biscaled
cutoffs (3.3), with integer $J$ depending only on $N$, such that whenever
$0<\delta\leq(4N)^{-1}\epsilon$ we have

$\sum_{j=1}^{J}\Phi^{(j)}(x,y,\mathbb{\hat{x}})=1$

for all $x,y\in{}^{3}$, $\mathbb{\hat{x}}\in{}^{3N-3}$.

###### Proof.

First, we claim that for all $z\in{}^{3}$,

(3.7) $\displaystyle\theta_{\delta}(z)\theta_{\epsilon}(z)$
$\displaystyle=\theta_{\epsilon}(z),$ (3.8)
$\displaystyle\zeta_{\delta}(z)\theta_{\epsilon}(z)$ $\displaystyle=0,$ (3.9)
$\displaystyle\zeta_{\delta}(z)\zeta_{\epsilon}(z)$
$\displaystyle=\zeta_{\delta}(z).$

This claim, along with the definitions (3.2), shows that

(3.10)
$1=\big{(}\zeta_{\delta}(z)+\theta_{\delta}(z)\big{)}\big{(}\zeta_{\epsilon}(z)+\theta_{\epsilon}(z)\big{)}=\zeta_{\delta}(z)+\theta_{\delta}(z)\zeta_{\epsilon}(z)+\theta_{\epsilon}(z)$

for all $z\in{}^{3}$. To prove (3.7) and (3.8) we need only consider $z$ such
that $\theta_{\epsilon}(z)\neq 0$, in which case
$|z|>(4N)^{-1}\epsilon\geq\delta$. For such $z$ we therefore have
$\theta_{\delta}(z)=1$, by the definition of $\theta_{\delta}$, which proves
(3.7). For (3.8), if we also have $\zeta_{\delta}(z)\neq 0$, then
$|z|<(2N)^{-1}\delta$ by the support criteria for $\zeta_{\delta}$, giving a
contradiction. For (3.9) we need only consider $z$ such that
$\zeta_{\delta}(z)\neq 0$, in which case $|z|<(2N)^{-1}\delta$. This gives
$4N|z|\epsilon^{-1}<(2N)^{-1}$ and hence, by definition,
$\zeta_{\epsilon}(z)=1$.

####

For any $A,B\subset\\{2,\dots,N\\}$ with $A\cap B=\emptyset$ we define

$\tau_{A,B}(\mathbb{x})=\prod_{j\in A}\zeta_{\delta}(x_{1}-x_{j})\prod_{j\in
B}(\theta_{\delta}\zeta_{\epsilon})(x_{1}-x_{j})\prod_{j\in\\{2,\dots,N\\}\backslash(A\cup
B)}\theta_{\epsilon}(x_{1}-x_{j})$

and therefore

$\sum_{\begin{subarray}{c}A\subset\\{2,\dots,N\\}\\\
B\subset\\{2,\dots,N\\}\backslash
A\end{subarray}}\tau_{A,B}(\mathbb{x})=\prod_{2\leq j\leq
N}\big{(}\zeta_{\delta}(x_{1}-x_{j})+(\theta_{\delta}\zeta_{\epsilon})(x_{1}-x_{j})+\theta_{\epsilon}(x_{1}-x_{j})\big{)}=1$

for all $\mathbb{x}\in{}^{3N}$. Let $\Xi=\\{(j,k):2\leq k<l\leq N\\}$. For
each subset $Y,Z\subset\Xi$ with $Y\cap Z=\emptyset$ we define

$T_{Y,Z}(\mathbb{\hat{x}})=\prod_{(j,k)\in
Y}\zeta_{\delta}(x_{j}-x_{k})\prod_{(j,k)\in
Z}(\theta_{\delta}\zeta_{\epsilon})(x_{j}-x_{k})\prod_{(j,k)\,\in\,\Xi\backslash(Y\cup
Z)}\theta_{\epsilon}(x_{j}-x_{k})$

and therefore

$\sum_{\begin{subarray}{c}Y\,\subset\,\Xi\\\ Z\,\subset\,\Xi\backslash
Y\end{subarray}}T_{Y,Z}(\mathbb{\hat{x}})=\prod_{2\leq j<k\leq
N}\big{(}\zeta_{\delta}(x_{j}-x_{k})+(\theta_{\delta}\zeta_{\epsilon})(x_{j}-x_{k})+\theta_{\epsilon}(x_{j}-x_{k})\big{)}=1$

for all $\mathbb{\hat{x}}\in{}^{3N-3}$. Overall,

$\sum_{\begin{subarray}{c}A\subset\\{2,\dots,N\\}\\\
B\subset\\{2,\dots,N\\}\backslash
A\end{subarray}}\quad\sum_{\begin{subarray}{c}C\subset\\{2,\dots,N\\}\\\
D\subset\\{2,\dots,N\\}\backslash
C\end{subarray}}\quad\sum_{\begin{subarray}{c}Y\,\subset\,\Xi\\\
Z\,\subset\,\Xi\backslash
Y\end{subarray}}\tau_{A,B}(x,\mathbb{\hat{x}})\tau_{C,D}(y,\mathbb{\hat{x}})T_{Y,Z}(\mathbb{\hat{x}})=1,$

which is a sum of biscaled cutoffs.

∎

###  3.2. Clusters corresponding to $\Phi$

####

We now introduce three clusters corresponding to each $\Phi$, defined in
(3.3). To define these clusters we first introduce index sets based solely on
the choice of cutoff factors $g^{(1)}_{j},g^{(2)}_{j}$ and $f_{kl}$ in the
definition of $\Phi$, (3.3). The index sets and clusters are therefore not
dependent on $x,y,\mathbb{\hat{x}}$ nor on the scaling parameters $\delta$ and
$\epsilon$.

####

We define the index set $L\subset\\{(j,k)\in\\{1,\dots,N\\}^{2}:j\neq k\\}$ as
follows.

  * •

We have $(j,k)\in L$ if $f_{jk}\neq\theta_{\epsilon}$ for $j,k=2,\dots,N$.
Also $(1,j),(j,1)\in L$ if
$(g^{(1)}_{j},g^{(2)}_{j})\neq(\theta_{\epsilon},\theta_{\epsilon})$ for
$j=2,\dots,N$.

Furthermore, we define two more index sets
$J,K\subset\\{(j,k)\in\\{1,\dots,N\\}^{2}:j\neq k\\}$ as follows,

  * •

We have $(j,k)\in J$ if $f_{jk}=\zeta_{\delta}$ for $j,k=2,\dots,N$. Also
$(1,j),(j,1)\in J$ if $g^{(1)}_{j}=\zeta_{\delta}$ for $j=2,\dots,N$.

  * •

We have $(j,k)\in K$ if $f_{jk}=\zeta_{\delta}$ for $j,k=2,\dots,N$. Also
$(1,j),(j,1)\in K$ if $g^{(2)}_{j}=\zeta_{\delta}$ for $j=2,\dots,N$.

These index sets obey $J,K\subset L$. For an arbitrary index set
$I\subset\\{(j,k)\in\\{1,\dots,N\\}^{2}:j\neq k\\}$ we say that two indices
$j,k\in\\{1,\dots,N\\}$ are $I$-linked if either $j=k$, or $(j,k)\in I$, or
there exist pairwise distinct indices $j_{1},\dots,j_{s}$ for $1\leq s\leq
N-2$, all distinct from $j$ and $k$ such that
$(j,j_{1}),(j_{1},j_{2}),\dots,(j_{s},k)\in I$.

####

The cluster $Q=Q(\Phi)$ is defined as the set of all indices $L$-linked to
$1$. The cluster $P=P(\Phi)$ is defined as the set of all indices $J$-linked
to $1$. The cluster $S=S(\Phi)$ is defined as the set of all indices
$K$-linked to $1$. Since $J,K\subset L$ we see that $P,S\subset Q$.

###  3.3. Support of $\Phi$

Continuing, the clusters $P,S$ and $Q$ will be understood as $P(\Phi),S(\Phi)$
and $Q(\Phi)$, respectively, when the biscaled cutoff $\Phi$ is unambiguous.
The following lemma shows that on the support of $\Phi$, the cluster
$P^{*}=P\backslash\\{1\\}$ represents a set of particles $x_{j}$, $j\in
P^{*}$, which are close to $x$, and the cluster $S^{*}=S\backslash\\{1\\}$
represents a set of particles $x_{j}$, $j\in S^{*}$, close to $y$. In both
cases this closeness is with respect to the parameter $\delta$. The cluster
$Q^{*}=Q\backslash\\{1\\}$ (recall $P,S\subset Q$) represents a set of
particles $x_{j}$, $j\in Q^{*}$, close to both $x$ and $y$, albeit held
potentially looser since this closeness is with respect to the larger
parameter $\epsilon$.

######  Lemma 3.2.

Let $\delta\leq(4N)^{-1}\epsilon$ and $\delta\leq|x-y|\leq 2\delta$. Then
$\Phi(x,y,\hat{\mathbb{x}})=0$ unless

(3.11) $|x-x_{k}|<\delta/2<|y-x_{k}|\quad\text{ for }\quad k\in P^{*}$ (3.12)
$|y-x_{k}|<\delta/2<|x-x_{k}|\quad\text{ for }\quad k\in S^{*}$ (3.13)
$|x-x_{k}|,\,|y-x_{k}|<\epsilon/2\quad\text{ for }\quad k\in Q^{*}.$

Therefore, if $\Phi(x,y,\,\cdot\,)$ is not identically zero for each
$x,y\in{}^{3}$ with $\delta\leq|x-y|\leq 2\delta$, then $P^{*}\cap
S^{*}=\emptyset$.

###### Proof.

By definition, if $k\in P^{*}$ either $g^{(1)}_{k}=\zeta_{\delta}$ or there
exist pairwise distinct $j_{1},\dots,j_{s}\in\\{2,\dots,N\\}$ with $1\leq
s\leq N-2$ such that $g^{(1)}_{j_{1}}=\zeta_{\delta}$ and
$f_{j_{1},j_{2}},f_{j_{2},j_{3}}\,\dots,\,f_{j_{s},k}=\zeta_{\delta}$. In the
former case, support criteria for $\zeta_{\delta}(x-x_{k})$ gives
$|x-x_{k}|<(2N)^{-1}\delta$. In the latter case, support conditions give
$|x-x_{j_{1}}|,\,|x_{j_{1}}-x_{j_{2}}|,\,\dots,\,|x_{j_{s}}-x_{k}|<(2N)^{-1}\delta$
and so by the triangle inequality $|x-x_{k}|<\delta/2$. Now,
$|y-x_{k}|\geq|x-y|-|x-x_{k}|>\delta/2$ by the reverse triangle inequality.
The case of $k\in S^{*}$ is analogous.

####

Now let $k\in Q^{*}$. First, we consider the case where either
$g^{(1)}_{k}\neq\theta_{\epsilon}$ or $g^{(2)}_{k}\neq\theta_{\epsilon}$, or
both. Without loss, assume $g^{(1)}_{k}\neq\theta_{\epsilon}$. Then either
$g^{(1)}_{k}=\zeta_{\delta}$ or $g^{(1)}_{k}=\theta_{\delta}\zeta_{\epsilon}$.
Hence by support criteria we have the inequalities,

$|x-x_{k}|<(2N)^{-1}\epsilon,$ $|y-x_{k}|\leq|x-y|+|x-x_{k}|\leq
2\delta+(2N)^{-1}\epsilon\leq\epsilon/N,$

which gives the required inequality since $N\geq 2$. Now, suppose that
$g^{(1)}_{k}=g^{(2)}_{k}=\theta_{\epsilon}$. Then there exist pairwise
distinct $j_{1},\dots,j_{s}\in\\{2,\dots,N\\}$ with $1\leq s\leq N-2$ such
that $f_{j_{1},j_{2}},f_{j_{2},j_{3}},\dots,f_{j_{s},k}\neq\theta_{\epsilon}$
and either $g^{(1)}_{j_{1}}\neq\theta_{\epsilon}$ or
$g^{(2)}_{j_{1}}\neq\theta_{\epsilon}$. As before, we see that

$|x-x_{j_{1}}|,\,|y-x_{j_{1}}|\leq\epsilon/N$

regardless of which (or both) of $g^{(1)}_{j_{1}}$ and $g^{(2)}_{j_{1}}$ are
not $\theta_{\epsilon}$. It’s also clear that by support criteria,
$|x_{j_{1}}-x_{j_{2}}|,\,\dots,\,|x_{j_{s}}-x_{k}|\leq(2N)^{-1}\epsilon$.
Therefore, by the triangle inequality,

$|x-x_{k}|\leq|x-x_{j_{1}}|+|x_{j_{1}}-x_{j_{2}}|+\dots+|x_{j_{s}}-x_{k}|\leq\frac{\epsilon}{N}+\frac{\epsilon(N-2)}{2N}=\frac{\epsilon}{2},$

and similarly we can show $|y-x_{k}|\leq\epsilon/2$, completing the proof. ∎

###  3.4. Factorisation of biscaled cutoffs

Let $\Phi$ be given by (3.3). We can define a partial product of $\Phi$ as a
function of the form

(3.14) $\Phi^{\prime}(x,y,\mathbb{\hat{x}})=\prod_{j\in
T_{1}}g^{(1)}_{j}(x-x_{j})\prod_{j\in
T_{2}}g^{(2)}_{j}(y-x_{j})\prod_{(k,l)\in R_{1}}f_{kl}(x_{k}-x_{l})$

where $T_{1},T_{2}\subset\\{2,\dots,N\\}$, $R_{1}\subset\\{(k,l):2\leq k<l\leq
N\\}$.

We now define classes of partial products of $\Phi$ which corresponding to a
cluster. Let $T$ be an arbitrary cluster with $1\in T$.

(3.15) $\displaystyle\Phi(x,y,\mathbb{\hat{x}};T)$ $\displaystyle=\prod_{j\in
T^{*}}g^{(1)}_{j}(x-x_{j})\prod_{j\in
T^{*}}g^{(2)}_{j}(y-x_{j})\prod_{\begin{subarray}{c}k,l\in T^{*}\\\
k<l\end{subarray}}f_{kl}(x_{k}-x_{l}),$ (3.16)
$\displaystyle\Phi(x,y,\mathbb{\hat{x}};T^{c})$
$\displaystyle=\prod_{\begin{subarray}{c}k,l\in T^{c}\\\
k<l\end{subarray}}f_{kl}(x_{k}-x_{l}),$ (3.17)
$\displaystyle\Phi(x,y,\mathbb{\hat{x}};T,T^{c})$ $\displaystyle=\prod_{j\in
T^{c}}g^{(1)}_{j}(x-x_{j})\prod_{j\in
T^{c}}g^{(2)}_{j}(y-x_{j})\prod_{\begin{subarray}{c}k\in T^{*}\\\ l\in
T^{c}\end{subarray}}f_{kl}(x_{k}-x_{l}).$

Formulae (3.15) and (3.16) are consistent since the former concerns clusters
containing $1$ and the latter concerns clusters not containing $1$.

If we consider both $x$ and $y$ to adopt the role of particle $1$ then these
functions can be interpreted as $\Phi(\,\cdot\,;T)$ involving pairs of
particles in $T$, $\Phi(\,\cdot\,;T^{c})$ involving pairs in $T^{c}$, and
$\Phi(\,\cdot\,;T,T^{c})$ involving pairs where one lies in $T$ and another
lies in $T^{c}$.

######  Lemma 3.3.

Given a biscaled cutoff $\Phi$ and any cluster $T$ with $1\in T$ we have

(3.18)
$\Phi(x,y,\mathbb{\hat{x}})=\Phi(x,y,\mathbb{\hat{x}};T)\Phi(x,y,\mathbb{\hat{x}};T,T^{c})\Phi(x,y,\mathbb{\hat{x}};T^{c}).$

###### Proof.

This identity follows from the definitions (3.15)-(3.17) and the equality

$\prod_{2\leq k<l\leq N}f_{kl}(x_{k}-x_{l})=\prod_{\begin{subarray}{c}k,l\in
T^{*}\\\ k<l\end{subarray}}f_{kl}(x_{k}-x_{l})\prod_{\begin{subarray}{c}k\in
T^{*}\\\ l\in
T^{c}\end{subarray}}f_{kl}(x_{k}-x_{l})\prod_{\begin{subarray}{c}k,l\in
T^{c}\\\ k<l\end{subarray}}f_{kl}(x_{k}-x_{l}),$

since for any $2\leq k<l\leq N$ we have $f_{kl}=f_{lk}$. ∎

Let $Q=Q(\Phi)$. Then the partial product $\Phi(\,\cdot\,;Q,Q^{c})$ consists
of only $\theta_{\epsilon}$ cutoff factors, as shown in the following lemma.

######  Lemma 3.4.

For $Q=Q(\Phi)$,

(3.19) $\Phi(x,y,\mathbb{\hat{x}};Q,Q^{c})=\prod_{j\in
Q^{c}}\theta_{\epsilon}(x-x_{j})\prod_{j\in
Q^{c}}\theta_{\epsilon}(y-x_{j})\prod_{\begin{subarray}{c}k\in Q^{*}\\\ l\in
Q^{c}\end{subarray}}\theta_{\epsilon}(x_{k}-x_{l}).$

###### Proof.

By the definition of the cluster $Q(\Phi)$ we have
$g^{(1)}_{j}=g^{(2)}_{j}=\theta_{\epsilon}$ for each $j\in Q^{c}$, and
$f_{kl}=\theta_{\epsilon}$ for each $k\in Q^{*}$ and $l\in Q^{c}$. ∎

###  3.5. Derivatives of biscaled cutoffs

The definition of cluster derivatives, (1.16), is modestly extended to allow
action on biscaled cutoffs. For any cluster $T$ we define the following three
cluster derivatives which can act on functions of $x$, $y$ and
$\mathbb{\hat{x}}$, such as $\Phi$. We set

(3.20) $\displaystyle D^{\alpha}_{x,T}=\partial^{\alpha}_{x}+\sum_{j\in
T^{*}}\partial^{\alpha}_{x_{j}}\quad\text{and}\quad
D^{\alpha}_{y,T}=\partial^{\alpha}_{y}+\sum_{j\in
T^{*}}\partial^{\alpha}_{x_{j}}\quad\text{for
}\alpha\in\mathbb{N}_{0}^{3},\,|\alpha|=1$ (3.21) $\displaystyle
D^{\alpha}_{x,y,T}=\partial^{\alpha}_{x}+\partial^{\alpha}_{y}+\sum_{j\in
T^{*}}\partial^{\alpha}_{x_{j}}\quad\text{for
}\alpha\in\mathbb{N}_{0}^{3},\,|\alpha|=1.$

which are extended to higher order multiindices $\alpha\in\mathbb{N}_{0}^{3}$
by successive application of first-order derivatives, as in (1.16).

The following lemma gives partial derivative estimates of the cutoff factors.
We will require the following elementary result. For each
$\sigma\in\mathbb{N}_{0}^{3}$ and $s\in\real$ there exists $C>0$ such that for
any $z_{0}\in{}^{3}$ we get $\partial_{z}^{\sigma}|z+z_{0}|^{s}\leq
C|z+z_{0}|^{s-|\sigma|}$ for all $z\in{}^{3}$, $z\neq-z_{0}$. Recall the
function $\mathds{1}_{t}$ was defined in (3.4).

######  Lemma 3.5.

For any $\sigma\in\mathbb{N}_{0}^{3}$ with $|\sigma|\geq 1$ and any $t>0$
there exists $C$, depending on $\sigma$ but independent of $t$, such that

(3.22) $|\partial^{\sigma}\zeta_{t}(z)|,\,|\partial^{\sigma}\theta_{t}(z)|\leq
Ct^{-|\sigma|}\mathds{1}_{t}(z)$

for all $z\in{}^{3}$.

###### Proof.

Without loss we consider the case of $\theta_{t}$, the case of $\zeta_{t}$
being similar. In the following, $\chi^{(j)}$ refers to the $j$-th
(univariate) derivative of the function $\chi$ defined in (3.1). Now, since
$|\sigma|\geq 1$ the chain rule shows that $\partial^{\sigma}\theta_{t}(z)$
can be written as a sum of terms of the form

(3.23)
$\Big{(}\frac{4N}{t}\Big{)}^{m}\,\chi^{(m)}\Big{(}\frac{4N|z|}{t}\Big{)}\,\partial_{z}^{\sigma_{1}}|z|\,\dots\,\partial_{z}^{\sigma_{m}}|z|$

where $1\leq m\leq|\sigma|$, and
$\sigma_{1},\dots,\sigma_{m}\in\mathbb{N}_{0}^{3}$ are non-zero multiindices
obeying

$\sigma_{1}+\dots+\sigma_{m}=\sigma.$

Since $m\geq 1$ we have that if $\chi^{(m)}(s)\neq 0$ then $s\in(1,2)$, and
therefore for any term (3.23) to be non-zero we require that

(3.24) $(4N)^{-1}t<|z|<(2N)^{-1}t.$

By the remark preceeding the current lemma, there exists $C$, dependent on
$\sigma_{1},\dots,\sigma_{m}$, such that

$\displaystyle\partial_{z}^{\sigma_{1}}|z|\,\dots\,\partial_{z}^{\sigma_{m}}|z|\leq
C|z|^{m-|\sigma|}\leq C(4N)^{|\sigma|-m}t^{m-|\sigma|},$

using (3.24). Therefore, the terms (3.23) can readily be bounded to give the
desired result. ∎

We now give bounds for the cluster derivatives (3.20)-(3.21) acting on
cutoffs.

######  Lemma 3.6.

Let $\Phi$ be any biscaled cutoff and let $Q=Q(\Phi)$. Then

(3.25) $\displaystyle D^{\alpha}_{x,y,Q}\Phi(\,\cdot\,;Q)\equiv 0$

for all $\alpha\in\mathbb{N}_{0}^{3}$ with $|\alpha|\geq 1$.

###### Proof.

By the chain rule, each function in the product (3.15) for $\Phi(\,\cdot\,;Q)$
has zero derivative upon action of $D^{\alpha}_{x,y,Q}$. ∎

Recall that the notion of partial products was defined in (3.14) and the
function $M_{t}$ for $t>0$ was defined in (3.6).

######  Lemma 3.7.

Let $\delta\leq(4N)^{-1}\epsilon$ and $\Phi=\Phi_{\delta,\epsilon}$ be a
biscaled cutoff. Let $Q=Q(\Phi)$. Then for any multiindex
$\boldsymbol{\alpha}\in\mathbb{N}_{0}^{3N+3}$ there exists $C$, dependent on
$\boldsymbol{\alpha}$ but independent of $\delta$ and $\epsilon$, such that
for any partial products $\Phi^{\prime}$ of $\Phi$ we have

(3.26)
$\displaystyle|\partial^{\boldsymbol{\alpha}}\Phi^{\prime}(x,y,\mathbb{\hat{x}})|\leq\begin{cases}C\epsilon^{-|\boldsymbol{\alpha}|}&\text{
if }\Phi^{\prime}=\Phi(\,\cdot\,;Q,Q^{c})\\\
C\big{(}\epsilon^{-|\boldsymbol{\alpha}|}+\delta^{-|\boldsymbol{\alpha}|}M_{\delta}(x,y,\mathbb{\hat{x}})\big{)}&\text{
otherwise.}\\\ \end{cases}$

###### Proof.

Lemma 3.5 gives bounds for the partial derivatives of the function
$\zeta_{\delta},\theta_{\delta},\zeta_{\epsilon},\theta_{\epsilon}$.
Considering $\theta_{\delta}\zeta_{\epsilon}$, we apply the Leibniz rule with
$\sigma\in\mathbb{N}_{0}^{3}$, $|\sigma|\geq 1$, to obtain

$\displaystyle\partial^{\sigma}(\theta_{\delta}\zeta_{\epsilon})(z)$
$\displaystyle=\sum_{\mu\leq\sigma}{\sigma\choose\mu}\partial^{\mu}\theta_{\delta}(z)\partial^{\sigma-\mu}\zeta_{\epsilon}(z)$
(3.27)
$\displaystyle=\partial^{\sigma}\theta_{\delta}(z)+\partial^{\sigma}\zeta_{\epsilon}(z),$

since, for each $\mu\leq\sigma$ with $\mu\neq 0$ and $\mu\neq\sigma$ we have

$\partial^{\mu}\theta_{\delta}(z)\,\partial^{\sigma-\mu}\zeta_{\epsilon}(z)\equiv
0$

by Lemma 3.5 and the definition (3.4).

####

Now, to evaluate $\partial^{\boldsymbol{\alpha}}\Phi^{\prime}$ we apply the
Leibniz rule to the product (3.14) using (3.27) where appropriate.
Differentiated cutoff factors are bounded by (3.22) and any remaining
undifferentiated cutoff factors are bounded above by 1. If
$\Phi^{\prime}=\Phi(\,\cdot\,;Q,Q^{c})$, all cutoff factors are of the form
$\theta_{\epsilon}$ by Lemma 3.4 and therefore we need only use the bounds in
(3.22) with $t=\epsilon$.

∎

The derivative $D_{x,y,Q}$ acting on $\Phi$ is special in that it contributes
only powers of $\epsilon$ (and not $\delta$) to the bounds. This is shown in
the next lemma.

######  Lemma 3.8.

Let $\delta\leq(4N)^{-1}\epsilon$ and $\Phi=\Phi_{\delta,\epsilon}$ be a
biscaled cutoff. Let $Q=Q(\Phi)$. For any multiindices
$\alpha\in\mathbb{N}_{0}^{3}$ and
$\boldsymbol{\sigma}\in\mathbb{N}_{0}^{3N+3}$ there exists $C$, independent of
$\epsilon$ and $\delta$, such that

$|\partial^{\boldsymbol{\sigma}}D^{\alpha}_{x,y,Q}\Phi(x,y,\mathbb{\hat{x}})|\leq
C\epsilon^{-|\alpha|}\big{(}\epsilon^{-|\boldsymbol{\sigma}|}+\delta^{-|\boldsymbol{\sigma}|}M_{\delta}(x,y,\mathbb{\hat{x}})\big{)}$

for all $x,y,\mathbb{\hat{x}}$.

###### Proof.

First, set $\Phi^{\prime}=\Phi(\,\cdot\,;Q)\,\Phi(\,\cdot\,;Q^{c})$ and
$\Phi^{\prime\prime}=\Phi(\,\cdot\,;Q,Q^{c})$. We then have

$D^{\alpha}_{x,y,Q}\Phi=\Phi^{\prime}\,D^{\alpha}_{x,y,Q}\Phi^{\prime\prime}$

which follows from Lemmas 3.3 and 3.6 and that $\Phi(\,\cdot\,;Q^{c})$ is not
dependent on variables involved in the $D^{\alpha}_{x,y,Q}$-derivative. By the
definition (3.21), the derivative $D^{\alpha}_{x,y,Q}\Phi^{\prime\prime}$ can
be written as a sum of partial derivatives of the form
$\partial^{\boldsymbol{\alpha}}\Phi^{\prime\prime}$ where
$\boldsymbol{\alpha}\in\mathbb{N}_{0}^{3N+3}$ obeys
$|\boldsymbol{\alpha}|=|\alpha|$. Now, by the Leibniz rule and Lemma 3.7 there
exists some constants $C$ and $C^{\prime}$, independent of $\delta$ and
$\epsilon$, such that

$\displaystyle|\partial^{\boldsymbol{\sigma}}(\Phi^{\prime}\,\partial^{\boldsymbol{\alpha}}\Phi^{\prime\prime})|$
$\displaystyle\leq\sum_{\boldsymbol{\tau}\leq\boldsymbol{\sigma}}{\boldsymbol{\sigma}\choose\boldsymbol{\tau}}|\partial^{\boldsymbol{\tau}}\Phi^{\prime}|\,|\partial^{\boldsymbol{\sigma}-\boldsymbol{\tau}+\boldsymbol{\alpha}}\Phi^{\prime\prime}|$
$\displaystyle\leq
C\sum_{\boldsymbol{\tau}\leq\boldsymbol{\sigma}}(\epsilon^{-|\boldsymbol{\tau}|}+\delta^{-|\boldsymbol{\tau}|}M_{\delta})\epsilon^{-|\boldsymbol{\sigma}|+|\boldsymbol{\tau}|-|\boldsymbol{\alpha}|}$
$\displaystyle\leq
C^{\prime}\epsilon^{-|\boldsymbol{\alpha}|}(\epsilon^{-|\boldsymbol{\sigma}|}+\delta^{-|\boldsymbol{\sigma}|}M_{\delta}),$

completing the proof. ∎

##  4\. Proof of Theorem 1.1

The idea of the proof is to turn partial derivatives of an integral, such as
the density matrix $\gamma(x,y)$ weighted with a suitable cutoff, into cluster
derivatives under the integral. For the density matrix we then estimate the
resulting integrals involving cluster derivatives of $\psi$ using the
pointwise bounds of Theorem 1.3.

Although Theorem 1.1 is stated using partial derivatives of the density matrix
$\gamma(x,y)$ in the $x$\- and $y$-variables, it is more appropriate to
consider directional derivatives. Indeed, we define new variables

(4.1) $\displaystyle u$ $\displaystyle=(x+y)/2,$ (4.2) $\displaystyle v$
$\displaystyle=(x-y)/2,$

and consider the $\partial_{u}^{\alpha}\partial_{v}^{\beta}$-derivatives of
the density matrix. The $\partial_{u}$-derivatives act along the direction
parallel to the diagonal and it is found that they do not affect the non-
smoothness we get at the diagonal, regardless of how many of these derivatives
are taken. The $\partial_{v}$-derivatives act in the direction perpendicular
to the diagonal and these derivatives are found to contribute to worsening the
non-smoothness at the diagonal.

Consider derivatives of the form
$\partial_{x}^{\alpha}\partial_{y}^{\beta}\gamma(x,y)$ where
$|\alpha|+|\beta|=2$. As discussed in [5], the case where $|\alpha|=|\beta|=1$
(the mixed derivatives) is particularly well-behaved in contrast to the other
cases. The reason behind this is that when $|\alpha|=|\beta|=1$,
differentiation under the integral leads to both $\psi$ factors being
differentiated exactly once. The greater regularity in this case follows from
the well known fact that $\psi,\nabla\psi\in L_{loc}^{\infty}({}^{3N})$, first
proven in [12], whereas higher order derivatives of $\psi$ do not have this
locally boundedness. This is used in the proof in the following way. In (4.34)
below, it will be shown that derivatives of the form $\partial_{v}^{\sigma}$
for $|\sigma|=2$ can be written in terms of $\partial_{u}^{\sigma}$ and the
mixed derivatives $\partial_{x}^{\alpha}\partial_{y}^{\beta}\gamma(x,y)$ for
some $|\alpha|=|\beta|=1$. The benefit of this identity is that two
$v$-derivatives (which act to worsen the singularity at the diagonal) have
been transformed into two $u$-derivatives (which do not worsen the
singularity) along with mixed derivatives which have good regularity. This
method only works for two $v$-derivatives, and the strategy of using cluster
derivatives, as described above, must be used in conjunction. The difficulties
encountered in the fifth derivative of the density matrix are described in a
later section.

###  4.1. Density matrix notation

The proof will require auxiliary functions related to the density matrix which
we introduce now. For $l,m\in\mathbb{N}_{0}^{3}$ with $|l|,|m|\leq 1$ define,

(4.3)
$\gamma_{l,m}(x,y)=\int_{{}^{3N-3}}\partial_{x}^{l}\psi(x,\mathbb{\hat{x}})\overline{\partial_{y}^{m}\psi(y,\mathbb{\hat{x}})}\,d\mathbb{\hat{x}}.$

In this notation, it is clear that $\gamma=\gamma_{0,0}$. For any biscaled
cutoff $\Phi$, defined in (3.3), we set

$\displaystyle\gamma_{l,m}(x,y;\Phi)$
$\displaystyle=\int_{{}^{3N-3}}\partial_{x}^{l}\psi(x,\mathbb{\hat{x}})\overline{\partial_{y}^{m}\psi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}},$

and define $\gamma(\,\cdot\,;\Phi)=\gamma_{0,0}(\,\cdot\,;\Phi)$. We will
consider the above functions in the variables (4.1) and (4.2). It is then
natural to define for all $u,v\in{}^{3}$,

(4.4) $\displaystyle\tilde{\gamma}_{l,m}(u,v)$
$\displaystyle=\gamma_{l,m}(u+v,u-v),$ (4.5)
$\displaystyle\tilde{\gamma}_{l,m}(u,v;\Phi)$
$\displaystyle=\gamma_{l,m}(u+v,u-v;\Phi).$

###  4.2. Integrals involving $f_{\infty}$

The following proposition is a restatement of [5, Lemma 5.1] and is proven in
that paper. Notice that the function $M_{t}$ was defined in (3.6) and has a
slightly different form to the corresponding function used in the paper.

######  Proposition 4.1.

Given $R>0$, there exists $C$ such that

(4.6)
$\int_{{}^{3N-3}}f_{\infty}(x,\mathbb{\hat{x}};R)f_{\infty}(y,\mathbb{\hat{x}};R)\,d\mathbb{\hat{x}}\leq
C\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(x,2R))}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(y,2R))}$

for all $x,y\in{}^{3}$. In addition, given $G\in L^{1}({}^{3})$ there exists
$C$, independent of $G$, such that

(4.7)
$\int_{{}^{3N-3}}\big{(}|G(x_{j}-x_{k})|+|G(z-x_{k})|+|G(x_{j})|\big{)}f_{\infty}(x,\mathbb{\hat{x}};R)f_{\infty}(y,\mathbb{\hat{x}};R)\,d\mathbb{\hat{x}}\\\
\leq C\left\lVert
G\right\rVert_{L^{1}({}^{3})}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(x,2R))}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(y,2R))}$

for all $x,y,z\in{}^{3}$, and $j,k=2,\dots,N$, $j\neq k$. In particular, for
any $t>0$ there exists $C$, independent of $t$, such that

(4.8)
$\int_{{}^{3N-3}}M_{t}(x,y,\mathbb{\hat{x}})f_{\infty}(x,\mathbb{\hat{x}};R)f_{\infty}(y,\mathbb{\hat{x}};R)\,d\mathbb{\hat{x}}\leq
Ct^{3}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(x,2R))}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(y,2R))}$

for all $x,y\in{}^{3}$.

We introduce the following quantities based on the bounded distances (1.21).
For any $x,y\in{}^{3}$ and $\mathbb{\hat{x}}\in{}^{3N-3}$ define

(4.9) $\displaystyle\lambda(x,y,\mathbb{\hat{x}})$
$\displaystyle=\min\\{\lambda_{P}(x,\mathbb{\hat{x}}),\,\lambda_{S^{*}}(x,\mathbb{\hat{x}}),\,\lambda_{P^{*}}(y,\mathbb{\hat{x}}),\,\lambda_{S}(y,\mathbb{\hat{x}})\\},$
(4.10) $\displaystyle\pi(x,y,\mathbb{\hat{x}})$
$\displaystyle=\min\\{\lambda_{Q}(x,\mathbb{\hat{x}}),\,\lambda_{Q}(y,\mathbb{\hat{x}})\\}.$

Later, we will see that these quantities appear to negative powers when we
apply Theorem 1.3 to clusters derivatives of $\psi$ involving the clusters
$P,S$ and $Q$. The next lemma will give conditions for when these quantities
can be bounded away from zero on the support of $\Phi$.

Beforehand, we consider an alternative formulae for (4.10) and (4.9). Recall
that $1\in P,S,Q$ by definition. Using (1.20) we find that

(4.11) $\pi(x,y,\mathbb{\hat{x}})=\min\\{1,\,\,|x|,\,\,|y|,\,\,|x_{j}|:j\in
Q^{*},\,\,2^{-1/2}|x-x_{k}|:k\in Q^{c},\\\ 2^{-1/2}|y-x_{k}|:k\in
Q^{c},\,\,2^{-1/2}|x_{j}-x_{k}|:j\in Q^{*},k\in Q^{c}\\}.$

In the case of $P^{*}\cap S^{*}=\emptyset$ we similarly find that

(4.12)
$\lambda(x,y,\mathbb{\hat{x}})=\min\\{1,\,\,|x|,\,\,|y|,\,\,|x_{j}|:j\in
P^{*}\cup S^{*},\,\,2^{-1/2}|x-x_{k}|:k\in P^{c},\\\ 2^{-1/2}|y-x_{k}|:k\in
S^{c},\,\,2^{-1/2}|x_{j}-x_{k}|:(j,k)\in(P^{*}\times P^{c})\cup(S^{*}\times
S^{c})\\}.$

######  Lemma 4.2.

Let $\delta\leq(4N)^{-1}\epsilon$, and $\epsilon\leq 1$ and let
$\Phi=\Phi_{\delta,\epsilon}$ be an arbitrary biscaled cutoff. Let
$x,y\in{}^{3}$ be such that $\delta\leq|x-y|\leq 2\delta$, and
$|x|,|y|\geq\epsilon$. Then there exists a constant $C$, dependent only on
$N$, such that

(4.13) $\displaystyle\pi(x,y,\mathbb{\hat{x}})$ $\displaystyle\geq C\epsilon$
(4.14) $\displaystyle\lambda(x,y,\mathbb{\hat{x}})$ $\displaystyle\geq
C\delta$

whenever $\Phi(x,y,\mathbb{\hat{x}})\neq 0$. In addition, for all $b\geq 0$
with $b\neq 3$ and $R>0$ there exists $C$, depending on $b$ and $R$ but
independent of $\delta,\epsilon,x$ and $y$ such that

(4.15)
$\int_{\textnormal{supp}\,\Phi(x,y,\,\cdot\,)}\lambda(x,y,\mathbb{\hat{x}})^{-b}f_{\infty}(x,\mathbb{\hat{x}};R)f_{\infty}(y,\mathbb{\hat{x}};R)\,d\mathbb{\hat{x}}\\\
\leq
C(\epsilon^{-b}+h_{b}(\delta))\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(x,2R))}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(y,2R))}$

where, for all $t>0$, we define

$\displaystyle h_{b}(t)=\begin{cases}0&\text{ if }b<3\\\ t^{3-b}&\text{ if
}b>3.\end{cases}$

The following corollary will be useful later.

######  Corollary 4.3.

There exists $C$, depending on $R$ but independent of $\delta$ and $\epsilon$,
such that

(4.16)
$\sum_{r=1}^{2}\int_{{}^{3N-3}}\lambda(x,y,\mathbb{\hat{x}})^{-2}f_{\infty}(x,\mathbb{\hat{x}};R)f_{\infty}(y,\mathbb{\hat{x}};R)|\nabla^{r}\Phi(x,y,\mathbb{\hat{x}})|\,d\mathbb{\hat{x}}\\\
\leq
C\epsilon^{-3}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(x,2R))}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(y,2R))}$

for all $x,y\in{}^{3}$ with $\delta\leq|x-y|\leq 2\delta$ and
$|x|,|y|\geq\epsilon$.

###### Proof of Corollary 4.3..

When $r=1$ the bound for the integral is immediate. Using Lemma 3.8 and
(4.14), the integral in (4.16) for $r=2$ can be bounded by some constant
multiplying

$\int_{\textnormal{supp}\,\Phi(x,y,\,\cdot\,)}\big{(}\epsilon^{-1}\lambda(x,y,\mathbb{\hat{x}})^{-2}+\delta^{-3}M_{\delta}(x,y,\mathbb{\hat{x}})\big{)}f_{\infty}(x,\mathbb{\hat{x}};R)f_{\infty}(y,\mathbb{\hat{x}};R)\,d\mathbb{\hat{x}}$

The required bound follows from (4.15) of Lemma 4.2, (4.8) of Proposition 4.1
and that $\epsilon<1$.

∎

###### Proof of Lemma 4.2..

By Lemma 3.2 we need only consider $\Phi$ with $P^{*}\cap S^{*}=\emptyset$. By
the definition of the cluster $Q$, if $j\in Q^{*}$ and $k\in Q^{c}$ then
$f_{jk}=\theta_{\epsilon}$ in the formula (3.3) defining $\Phi$. Similarly, if
$k\in Q^{c}$ then $g^{(1)}_{k}=g^{(2)}_{k}=\theta_{\epsilon}$. Using the
support criteria of $\theta_{\epsilon}$ we therefore get

(4.17) $\displaystyle|x-x_{k}|,\,|y-x_{k}|,\,|x_{j}-x_{k}|$
$\displaystyle\geq(4N)^{-1}\epsilon\qquad j\in Q^{*},k\in Q^{c}.$

In addition, if $j\in Q^{*}$ we get

(4.18) $|x_{j}|\geq|x|-|x-x_{j}|\geq\epsilon/2\qquad j\in Q^{*}$

since for such $j$ we have $|x-x_{j}|\leq\epsilon/2$ by Lemma 3.2. The lower
bound (4.13) then follows from the formula (4.11) for
$\pi(x,y,\mathbb{\hat{x}})$.

####

In a similar way we consider the clusters $P$ and $S$. Indeed, let $j\in
P^{*}$, $k\in P^{c}$ or $j\in S^{*}$, $k\in S^{c}$, then
$f_{jk}\neq\zeta_{\delta}$. Similarly, if $k\in P^{c}$ then
$g^{(1)}_{k}\neq\zeta_{\delta}$, and if $k\in S^{c}$ then
$g^{(2)}_{k}\neq\zeta_{\delta}$. Notice that if a cutoff factor is not
$\zeta_{\delta}$ then it must either be $\theta_{\delta}\zeta_{\epsilon}$ or
$\theta_{\epsilon}$, both of which are only supported away from zero.
Therefore by the support criteria of these factors we obtain the following
inequalities,

(4.19) $\displaystyle|x-x_{k}|,|y-x_{l}|$
$\displaystyle\geq(4N)^{-1}\delta\qquad k\in P^{c},\,l\in S^{c}$ (4.20)
$\displaystyle|x_{j}-x_{k}|$ $\displaystyle\geq(4N)^{-1}\delta\qquad j\in
P^{*},\,k\in P^{c}\text{ or }j\in S^{*},\,k\in S^{c}.$

We also have (4.18) for $j\in P^{*}\cup S^{*}$ since $P,S\subset Q$. The lower
bound (4.14) then follows from the formula (4.12) for
$\lambda(x,y,\mathbb{\hat{x}})$.

####

We recall the function $\mathds{1}^{\prime}_{\delta}$ was defined in (3.5). By
the formula (4.12) we can use (4.18) to obtain some $C$, depending on $b$,
such that

(4.21) $\lambda(x,y,\mathbb{\hat{x}})^{-b}\leq
C\Big{(}\epsilon^{-b}+\sum_{k\in
P^{c}}\mathds{1}^{\prime}_{\delta}(x-x_{k})\,|x-x_{k}|^{-b}+\sum_{k\in
S^{c}}\mathds{1}^{\prime}_{\delta}(y-x_{k})\,|y-x_{k}|^{-b}\\\
+\sum_{\begin{subarray}{c}(j,k)\in(P^{*}\times P^{c})\\\ \cup(S^{*}\times
S^{c})\end{subarray}}\mathds{1}^{\prime}_{\delta}(x_{j}-x_{k})\,|x_{j}-x_{k}|^{-b}\Big{)},$

where the upper bounds in the indicator functions (3.5) can be included
because $1$ lies in the minimum (4.12) and the lower bounds follow from
(4.18)-(4.20). The bound (4.15) then follows from the above inequality along
with both (4.6) and (4.7) of Proposition 4.1, where we choose $G$ to be the
function $G(z)=\mathds{1}^{\prime}_{\delta}(z)|z|^{-b}$ for $z\in{}^{3}$.

∎

###  4.3. Differentiating the density matrix - some required bounds

We collect certain results which will be used throughout this section. Let
$\delta\leq(4N)^{-1}\epsilon$ and $\epsilon\leq 1$ and suppose
$\Phi=\Phi_{\delta,\epsilon}$ is a biscaled cutoff as defined in (3.3). As
usual, we denote $Q=Q(\Phi)$, $P=P(\Phi)$ and $S=S(\Phi)$. Let
$\eta\in\mathbb{N}_{0}^{3}$ and
$\boldsymbol{\nu}=(\nu_{1},\nu_{2})\in\mathbb{N}_{0}^{6}$ be arbitrary.
Firstly, we define

(4.22)
$\Phi^{(\eta,\boldsymbol{\nu})}(x,y,\mathbb{\hat{x}})=D^{\eta}_{x,y,Q}D^{\nu_{1}}_{x,P}D^{\nu_{2}}_{y,S}\Phi(x,y,\mathbb{\hat{x}})$

where in the notation it is implicit the clusters used are $Q,P$ and $S$
corresponding to $\Phi$. By Lemma 3.8 and the definition of cluster
derivatives (1.16) it can be shown that for each $\eta$ and $\boldsymbol{\nu}$
there exists $C$, independent of $\delta$ and $\epsilon$, such that

(4.23) $|\Phi^{(\eta,\boldsymbol{\nu})}(x,y,\mathbb{\hat{x}})|\leq
C\epsilon^{-|\eta|}\big{(}\epsilon^{-|\boldsymbol{\nu}|}+\delta^{-|\boldsymbol{\nu}|}M_{\delta}(x,y,\mathbb{\hat{x}})\big{)}$

for all $x,y\in{}^{3}$ and $\mathbb{\hat{x}}\in{}^{3N-3}$.

Now take any $x,y\in{}^{3}$ with $\delta\leq|x-y|\leq 2\delta$ and
$|x|,|y|\geq\epsilon$, and suppose $\Phi(x,y,\mathbb{\hat{x}})\neq 0$. Then,
as a consequence of Lemma 4.2 we have both $\pi(x,y,\mathbb{\hat{x}})$ and
$\lambda(x,y,\mathbb{\hat{x}})$ are positive. Therefore, by the definitions
(4.9), (4.10) and (1.20), (1.21),

(4.24)
$\displaystyle(x,\mathbb{\hat{x}})\in\Sigma_{Q}^{c}\cap\Sigma_{P}^{c}\cap\Sigma_{S^{*}}^{c}\quad\text{and}\quad(y,\mathbb{\hat{x}})\in\Sigma_{Q}^{c}\cap\Sigma_{P^{*}}^{c}\cap\Sigma_{S}^{c}.$

This allows us to apply Theorem 1.3. Indeed, for every
$\eta\in\mathbb{N}_{0}^{3}$,
$\boldsymbol{\nu}=(\nu_{1},\nu_{2})\in\mathbb{N}_{0}^{6}$ and all $R>0$ there
exists $C$, independent of our choice of $x,y$ and $\mathbb{\hat{x}}$, such
that for $k=0,1$,

(4.25)
$\displaystyle\big{|}D_{Q}^{\eta}D^{\boldsymbol{\nu}}_{\\{P,S^{*}\\}}\nabla^{k}\psi(x,\mathbb{\hat{x}})\big{|}$
$\displaystyle\leq
C\pi(x,y,\mathbb{\hat{x}})^{-|\eta|}\lambda(x,y,\mathbb{\hat{x}})^{-|\boldsymbol{\nu}|}f_{\infty}(x,\mathbb{\hat{x}};R),$
(4.26)
$\displaystyle\big{|}D_{Q}^{\eta}D^{\boldsymbol{\nu}}_{\\{P^{*},S\\}}\nabla^{k}\psi(y,\mathbb{\hat{x}})\big{|}$
$\displaystyle\leq
C\pi(x,y,\mathbb{\hat{x}})^{-|\eta|}\lambda(x,y,\mathbb{\hat{x}})^{-|\boldsymbol{\nu}|}f_{\infty}(y,\mathbb{\hat{x}};R).$

For convenience, we choose a bound which holds for both values of $k$. The
functions $\lambda$ and $\pi$ are defined in (4.9) and (4.10) respectively.

Now, let $|\boldsymbol{\nu}|\geq 1$. By Theorem 1.3 with $b=1/2$ we can write

(4.27) $\displaystyle
D_{\\{P,S^{*}\\}}^{\boldsymbol{\nu}}\nabla\psi(x,\mathbb{\hat{x}})$
$\displaystyle=G_{\\{P,S^{*}\\}}^{\boldsymbol{\nu}}(x,\mathbb{\hat{x}})+\psi(x,\mathbb{\hat{x}})\big{(}D_{\\{P,S^{*}\\}}^{\boldsymbol{\nu}}\nabla
F_{c}(x,\mathbb{\hat{x}})\big{)}$ (4.28) $\displaystyle
D_{\\{P^{*},S\\}}^{\boldsymbol{\nu}}\nabla\psi(y,\mathbb{\hat{x}})$
$\displaystyle=G_{\\{P^{*},S\\}}^{\boldsymbol{\nu}}(y,\mathbb{\hat{x}})+\psi(y,\mathbb{\hat{x}})\big{(}D_{\\{P^{*},S\\}}^{\boldsymbol{\nu}}\nabla
F_{c}(y,\mathbb{\hat{x}})\big{)}$

for functions $G_{\\{P,S^{*}\\}}^{\boldsymbol{\nu}}$ and
$G_{\\{P^{*},S\\}}^{\boldsymbol{\nu}}$ which obey

(4.29)
$\displaystyle\big{|}G^{\boldsymbol{\nu}}_{\\{P,S^{*}\\}}(x,\mathbb{\hat{x}})\big{|}$
$\displaystyle\leq
C\lambda(x,y,\mathbb{\hat{x}})^{1/2-|\boldsymbol{\nu}|}f_{\infty}(x,\mathbb{\hat{x}};R),$
(4.30)
$\displaystyle\big{|}G^{\boldsymbol{\nu}}_{\\{P^{*},S\\}}(y,\mathbb{\hat{x}})\big{|}$
$\displaystyle\leq
C\lambda(x,y,\mathbb{\hat{x}})^{1/2-|\boldsymbol{\nu}|}f_{\infty}(y,\mathbb{\hat{x}};R),$

for some $C$, dependent on $\boldsymbol{\nu}$ but independent of the choice of
$x,y$ and $\mathbb{\hat{x}}$.

By Lemma 2.3 and (2.2), for each $\boldsymbol{\nu}\in\mathbb{N}_{0}^{6}$ there
exists $C$, independent of $x,y$ and $\mathbb{\hat{x}}$, such that

(4.31) $\displaystyle\big{|}D^{\boldsymbol{\nu}}_{\\{P,S^{*}\\}}\nabla
F_{c}(x,\mathbb{\hat{x}})\big{|}+\big{|}D^{\boldsymbol{\nu}}_{\\{P^{*},S\\}}\nabla
F_{c}(y,\mathbb{\hat{x}})\big{|}$ $\displaystyle\leq
C\lambda(x,y,\mathbb{\hat{x}})^{-|\boldsymbol{\nu}|}.$

###  4.4. Differentiating the density matrix - first and second derivatives

In the following, let $l,m\in\mathbb{N}_{0}^{3}$ obey $|l|=|m|=1$. In standard
notation, by $\partial^{l}_{x_{1}}\psi$ we mean the $l$-partial derivative in
the first $\mathbb{R}^{3}$ component of $\psi$. Then by differentiation under
the integral,

(4.32) $\displaystyle\partial^{l}_{u}\tilde{\gamma}(u,v)$
$\displaystyle=\int_{{}^{3N-3}}\partial_{x_{1}}^{l}\psi(u+v,\mathbb{\hat{x}})\overline{\psi(u-v,\mathbb{\hat{x}})}\,d\mathbb{\hat{x}}+\int_{{}^{3N-3}}\psi(u+v,\mathbb{\hat{x}})\overline{\partial_{x_{1}}^{l}\psi(u-v,\mathbb{\hat{x}})}\,d\mathbb{\hat{x}}$
$\displaystyle=\tilde{\gamma}_{l,0}(u,v)+\tilde{\gamma}_{0,l}(u,v),$

and similarly,

(4.33) $\displaystyle\partial^{l}_{v}\tilde{\gamma}(u,v)$
$\displaystyle=\tilde{\gamma}_{l,0}(u,v)-\tilde{\gamma}_{0,l}(u,v)$

for all $u,v\in{}^{3}$. The above equalities are used to obtain a formula
relating second order $u$-derivatives to second order $v$-derivatives of
$\tilde{\gamma}$. Omitting the argument $(u,v)$ we get

$\displaystyle\big{(}\partial_{u}^{l+m}-\partial_{v}^{l+m}\big{)}\tilde{\gamma}$
$\displaystyle=\partial_{u}^{l}(\tilde{\gamma}_{m,0}+\tilde{\gamma}_{0,m})-\partial_{v}^{l}(\tilde{\gamma}_{m,0}-\tilde{\gamma}_{0,m})$
$\displaystyle=\big{(}\partial_{u}^{l}+\partial_{v}^{l}\big{)}\tilde{\gamma}_{0,m}+\big{(}\partial_{u}^{l}-\partial_{v}^{l}\big{)}\tilde{\gamma}_{m,0}$
(4.34) $\displaystyle=2(\tilde{\gamma}_{l,m}+\tilde{\gamma}_{m,l})$

where the final equality is obtained by differentiation under the integral, as
in (4.32).

###  4.5. Differentiating the density matrix - general derivatives

Partial derivatives of $\gamma$ are written as linear combinations of
integrals involving cluster derivatives of $\psi$. One such integral is
bounded in the following lemma. Since it is more involved than the other such
integrals, the proof is postponed until later in the section.

######  Lemma 4.4.

Let $\delta\leq(4N)^{-1}\epsilon$ and $\epsilon\leq 1$ and let
$\Phi=\Phi_{\delta,\epsilon}$ be an arbitrary biscaled cutoff. Let $P=P(\Phi)$
and $S=S(\Phi)$ obey $P^{*}\cap S^{*}=\emptyset$. For any
$\boldsymbol{\alpha},\boldsymbol{\beta}\in\mathbb{N}_{0}^{6}$ with
$|\boldsymbol{\alpha}|+|\boldsymbol{\beta}|=3$, any $l,m\in\mathbb{N}_{0}^{3}$
with $|l|=|m|=1$, and any $R>0$ there exists $C$ such that

(4.35)
$\Big{|}\int_{{}^{3N-3}}\big{(}D_{\\{P,S^{*}\\}}^{\boldsymbol{\alpha}}\partial_{x}^{l}\psi(x,\mathbb{\hat{x}})\big{)}\big{(}D_{\\{P^{*},S\\}}^{\boldsymbol{\beta}}\partial_{y}^{m}\psi(y,\mathbb{\hat{x}})\big{)}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}\Big{|}\\\
\leq
C\epsilon^{-3}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(x,R))}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(y,R))}$

for all $\delta\leq|x-y|\leq 2\delta$ and $|x|,|y|\geq\epsilon$. The constant
$C$ depends on $R$ but is independent of $\delta,\epsilon$.

In part two of the following lemma the conditions on $\eta,\mu,l$ and $m$ are
not the most general, but for simplicity we restrict ourselves to these
assumptions. We use notation of cluster derivatives of $\Phi$ from (4.22).

######  Lemma 4.5.

Let $\delta\leq(4N)^{-1}\epsilon$ and $\epsilon\leq 1$ and
$\Phi=\Phi_{\delta,\epsilon}$ be an arbitrary biscaled cutoff. Let
$\eta,\mu\in\mathbb{N}_{0}^{3}$ be arbitrary and $l,m\in\mathbb{N}_{0}^{3}$ be
such that $|l|,|m|\leq 1$.

  1. (i)

On the set of $u,v$ such that $\delta/2\leq|v|\leq\delta$ and
$|u+v|,|u-v|\geq\epsilon$, the derivative
$\partial_{u}^{\eta}\partial_{v}^{\mu}\tilde{\gamma}_{l,m}(u,v;\Phi)$ is equal
to a linear combination of integrals of the form

(4.36)
$\int_{{}^{3N-3}}\big{(}D_{Q}^{\chi_{1}}D_{\\{P,S^{*}\\}}^{\boldsymbol{\alpha}}\partial_{x_{1}}^{l}\psi(u+v,\mathbb{\hat{x}})\big{)}\big{(}D_{Q}^{\chi_{2}}D_{\\{P^{*},S\\}}^{\boldsymbol{\beta}}\partial_{x_{1}}^{m}\psi(u-v,\mathbb{\hat{x}})\big{)}\,\cdot\\\
\Phi^{(\chi_{3},\boldsymbol{\sigma})}(u+v,u-v,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}$

where
$\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{\sigma}\in\mathbb{N}_{0}^{6}$
and $\boldsymbol{\chi}=(\chi_{1},\chi_{2},\chi_{3})\in\mathbb{N}_{0}^{9}$ obey
$|\boldsymbol{\chi}|=|\eta|$ and
$|\boldsymbol{\alpha}|+|\boldsymbol{\beta}|+|\boldsymbol{\sigma}|=|\mu|$.

  2. (ii)

Furthermore, suppose either

     * •

$|\mu|\leq 2$, or

     * •

$|\mu|=3$, $\eta=0$ and $|l|=|m|=1$.

Then for all $R>0$ we have some $C_{0}$ such that

(4.37)
$|\partial_{u}^{\eta}\partial_{v}^{\mu}\tilde{\gamma}_{l,m}(u,v;\Phi)|\leq
C_{0}\epsilon^{-|\eta|-|\mu|}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(u+v,R))}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(u-v,R))}$

for all $\delta/2\leq|v|\leq\delta$ and $|u+v|,|u-v|\geq\epsilon$. The
constant $C_{0}$ depends on $R$ but is independent of $\delta,\epsilon$.

###### Proof of i)..

By Lemma 3.2, we need only consider $\Phi$ such that $P^{*}\cap
S^{*}=\emptyset$. For each choice of $u$ and $v$ we define a $u$\- and
$v$-dependent change of variables for the integral
$\tilde{\gamma}_{lm}(u,v;\Phi)$ defined in (4.5). To start, we define two
vectors
$\mathbb{\hat{a}}=(a_{2},\dots,a_{N}),\mathbb{\hat{b}}=(b_{2},\dots,b_{N})\in{}^{3N-3}$
by

(4.38) $\displaystyle a_{k}$ $\displaystyle=\begin{cases}u\quad&\text{ if
}k\in Q^{*}\\\ 0\quad&\text{ if }k\in Q^{c}\end{cases}$ $\displaystyle b_{k}$
$\displaystyle=\begin{cases}v\quad&\text{ if }k\in P^{*}\\\ -v\quad&\text{ if
}k\in S^{*}\\\ 0\quad&\text{ if }k\in(P\cup S)^{c},\end{cases}$

and define
$\boldsymbol{\hat{\omega}}_{u,v}=\mathbb{\hat{a}}+\mathbb{\hat{b}}$. We then
apply a translational change of variables which allows us to write

(4.39)
$\tilde{\gamma}_{lm}(u,v;\Phi)=\int_{{}^{3N-3}}\partial_{x_{1}}^{l}\psi(u+v,\mathbb{\hat{z}}+\boldsymbol{\hat{\omega}}_{u,v})\overline{\partial_{x_{1}}^{m}\psi(u-v,\mathbb{\hat{z}}+\boldsymbol{\hat{\omega}}_{u,v})}\Phi(u+v,u-v,\mathbb{\hat{z}}+\boldsymbol{\hat{\omega}}_{u,v})\,d\mathbb{\hat{z}}.$

We will then apply differentiation under the integral. Beforehand, we show how
such derivatives will act on each function within the integrand. For a
function $f$ and any $r\in\mathbb{N}_{0}^{3}$ with $|r|=1$ we see that by the
chain rule

$\displaystyle\partial^{r}_{u}[f(u\pm
v,\mathbb{\hat{z}}+\boldsymbol{\hat{\omega}}_{u,v})]$
$\displaystyle=D^{r}_{Q}f(u\pm
v,\mathbb{\hat{z}}+\boldsymbol{\hat{\omega}}_{u,v})$
$\displaystyle\partial^{r}_{v}[f(u+v,\mathbb{\hat{z}}+\boldsymbol{\hat{\omega}}_{u,v})]$
$\displaystyle=D^{r}_{P}f(u+v,\mathbb{\hat{z}}+\boldsymbol{\hat{\omega}}_{u,v})-D^{r}_{S^{*}}f(u+v,\mathbb{\hat{z}}+\boldsymbol{\hat{\omega}}_{u,v})$
$\displaystyle\partial^{r}_{v}[f(u-v,\mathbb{\hat{z}}+\boldsymbol{\hat{\omega}}_{u,v})]$
$\displaystyle=D^{r}_{P^{*}}f(u-v,\mathbb{\hat{z}}+\boldsymbol{\hat{\omega}}_{u,v})-D^{r}_{S}f(u-v,\mathbb{\hat{z}}+\boldsymbol{\hat{\omega}}_{u,v}).$

Applying repeatedly, we obtain for arbitrary
$\sigma,\nu\in\mathbb{N}_{0}^{3}$,

$\displaystyle\partial^{\sigma}_{u}\partial^{\nu}_{v}[f(u+v,\mathbb{\hat{z}}+\boldsymbol{\hat{\omega}}_{u,v})]$
$\displaystyle=\sum_{\tau\leq\nu}c_{\tau,\nu}\big{(}D^{\sigma}_{Q}D^{\tau}_{P}D^{\nu-\tau}_{S^{*}}f(u+v,\mathbb{\hat{z}}+\boldsymbol{\hat{\omega}}_{u,v})\big{)}$
$\displaystyle\partial^{\sigma}_{u}\partial^{\nu}_{v}[f(u-v,\mathbb{\hat{z}}+\boldsymbol{\hat{\omega}}_{u,v})]$
$\displaystyle=\sum_{\tau\leq\nu}c_{\tau,\nu}\big{(}D^{\sigma}_{Q}D^{\tau}_{P^{*}}D^{\nu-\tau}_{S}f(u-v,\mathbb{\hat{z}}+\boldsymbol{\hat{\omega}}_{u,v})\big{)}.$

where $c_{\tau,\nu}=(-1)^{|\nu|-|\tau|}{\nu\choose\tau}$. In a similar manner,
for the cutoff we use the definitions (3.20)-(3.21) and (4.22) to write

$\partial_{u}^{\sigma}\partial_{v}^{\nu}[\Phi(u+v,u-v,\mathbb{\hat{z}}+\boldsymbol{\hat{\omega}}_{u,v})]=\sum_{\tau\leq\nu}c_{\tau,\nu}\Phi^{(\sigma,\tau,\nu-\tau)}(u+v,u-v,\mathbb{\hat{z}}+\boldsymbol{\hat{\omega}}_{u,v}).$

By differentiating (4.39) under the integral, applying the Leibniz rule and
reversing the change of variables, we find that
$\partial^{\eta}_{u}\partial^{\mu}_{v}\tilde{\gamma}_{lm}(u,v;\Phi)$ is a
linear combination of terms of the required form.

Proof of ii). By part (i) it suffices to prove the required bound for
integrals of the form (4.36) with $|\boldsymbol{\chi}|=|\eta|$ and
$|\boldsymbol{\alpha}|+|\boldsymbol{\beta}|+|\boldsymbol{\sigma}|=|\mu|$.
Rewriting such integrals in the variables $x=u+v$ and $y=u-v$ we get

(4.40)
$\int_{{}^{3N-3}}\big{(}D_{Q}^{\chi_{1}}D_{\\{P,S^{*}\\}}^{\boldsymbol{\alpha}}\partial_{x_{1}}^{l}\psi(x,\mathbb{\hat{x}})\big{)}\big{(}D_{Q}^{\chi_{2}}D_{\\{P^{*},S\\}}^{\boldsymbol{\beta}}\partial_{x_{1}}^{m}\psi(y,\mathbb{\hat{x}})\big{)}\Phi^{(\chi_{3},\boldsymbol{\sigma})}(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}.$

Notice that the variables $x$ and $y$ must obey $\delta\leq|x-y|\leq 2\delta$
and $|x|,|y|\geq\epsilon$. Firstly, we bound the above integral in the case
where $|\boldsymbol{\alpha}|+|\boldsymbol{\beta}|\leq 2$ and
$|\boldsymbol{\alpha}|+|\boldsymbol{\beta}|+|\boldsymbol{\sigma}|\leq 3$ for
any $|l|,|m|\leq 1$. By (4.23), (4.25), (4.26) followed by (4.13) of Lemma 4.2
we can bound this integral in absolute value by some constant multiplied by

$\displaystyle\epsilon^{-|\boldsymbol{\chi}|}\int_{\text{supp}\,\Phi(x,y,\cdot)}\big{(}\epsilon^{-|\boldsymbol{\sigma}|}+\delta^{-|\boldsymbol{\sigma}|}M_{\delta}(x,y,\mathbb{\hat{x}})\big{)}\lambda(x,y,\mathbb{\hat{x}})^{-|\boldsymbol{\alpha}|-|\boldsymbol{\beta}|}f_{\infty}(x,\mathbb{\hat{x}};R)f_{\infty}(y,\mathbb{\hat{x}};R)\,d\mathbb{\hat{x}}.$

Selective use of (4.14) of Lemma 4.2 allows us to bound this quantity by some
constant multiplied by

$\displaystyle\epsilon^{-|\boldsymbol{\chi}|}\int_{\text{supp}\,\Phi(x,y,\cdot)}\big{(}\epsilon^{-|\boldsymbol{\sigma}|}\lambda(x,y,\mathbb{\hat{x}})^{-|\boldsymbol{\alpha}|-|\boldsymbol{\beta}|}+\delta^{-|\boldsymbol{\alpha}|-|\boldsymbol{\beta}|-|\boldsymbol{\sigma}|}M_{\delta}(x,y,\mathbb{\hat{x}})\big{)}f_{\infty}(x,\mathbb{\hat{x}};R)f_{\infty}(y,\mathbb{\hat{x}};R)\,d\mathbb{\hat{x}}.$

We can use (4.15) of Lemma 4.2, using that
$|\boldsymbol{\alpha}|+|\boldsymbol{\beta}|\leq 2$ by assumption, and (4.8) of
Proposition 4.1 to bound this quantity by some constant multiplied by

$\displaystyle\epsilon^{-|\boldsymbol{\alpha}|-|\boldsymbol{\beta}|-|\boldsymbol{\sigma}|-|\boldsymbol{\chi}|}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(x,2R))}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(y,2R))}$

where it was used that $\delta\leq 1\leq\epsilon^{-1}$ and
$|\boldsymbol{\alpha}|+|\boldsymbol{\beta}|+|\boldsymbol{\sigma}|\leq 3$.
After a return to $u,v$-variables, this proves the bound for integrals (4.36)
in the case where $|\boldsymbol{\alpha}|+|\boldsymbol{\beta}|\leq 2$ and
$|\boldsymbol{\alpha}|+|\boldsymbol{\beta}|+|\boldsymbol{\sigma}|\leq 3$. It
remains to bound (4.40) in the case where
$|\boldsymbol{\alpha}|+|\boldsymbol{\beta}|=3$,
$|\boldsymbol{\sigma}|=|\boldsymbol{\chi}|=0$ and $|l|=|m|=1$. This follows
directly from Lemma 4.4. ∎

######  Lemma 4.6.

Take any $\eta,\mu,l,m\in\mathbb{N}_{0}^{3}$ as in Lemma 4.5(ii). Then for all
$R>0$ we have $C$ such that

(4.41) $|\partial_{u}^{\eta}\partial_{v}^{\mu}\tilde{\gamma}_{l,m}(u,v)|\leq
C\min\\{1,|u+v|,|u-v|\\}^{-|\eta|-|\mu|}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(u+v,R))}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(u-v,R))}$

for all $u,v\in{}^{3}$ obeying $0<|v|\leq(4N)^{-1}\min\\{1,|u+v|,|u-v|\\}$.

###### Proof.

Firstly, by Lemma 3.1 there exists a finite collection of biscaled cutoffs,
$\Phi^{(j)}$, $j=1,\dots,J$, such that

(4.42)
$\tilde{\gamma}_{l,m}=\sum_{j=1}^{J}\tilde{\gamma}_{l,m}\big{(}\,\cdot\,;\Phi_{\delta,\epsilon}^{(j)}\big{)}$

holds for all choices of $0<\delta\leq(4N)^{-1}\epsilon$.

Let $C_{0}$ be the constant from Lemma 4.5 such that (4.40) holds for each
$\Phi^{(j)}$, $j=1,\dots,J$. Fix any $v_{0}\neq 0$ and $u_{0}$ such that
$|v_{0}|\leq(4N)^{-1}\min\\{1,|u_{0}+v_{0}|,|u_{0}-v_{0}|\\}$ and set
$\delta_{0}=|v_{0}|$ and
$\epsilon_{0}=\min\\{1,|u_{0}+v_{0}|,|u_{0}-v_{0}|\\}$. Then by (4.42),

$\displaystyle|\partial_{u}^{\eta}\partial_{v}^{\mu}\tilde{\gamma}_{l,m}(u_{0},v_{0})|$
$\displaystyle\leq\sum_{j=1}^{J}\big{|}\partial_{u}^{\eta}\partial_{v}^{\mu}\tilde{\gamma}_{l,m}\big{(}u_{0},v_{0};\Phi_{\delta_{0},\epsilon_{0}}^{(j)}\big{)}\big{|}$
$\displaystyle\leq
JC_{0}\min\\{1,|u_{0}+v_{0}|,|u_{0}-v_{0}|\\}^{-|\eta|-|\mu|}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(u_{0}+v_{0},R))}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(u_{0}-v_{0},R))}.$

Since $C_{0}$ does not depend on the choice of $\delta_{0}$ and
$\epsilon_{0}$, the constant $JC_{0}$ does not depend on the choice of $u_{0}$
and $v_{0}$.

∎

######  Proposition 4.7.

For all $\alpha,\beta\in\mathbb{N}_{0}^{3}$ with $|\alpha|+|\beta|=5$ and all
$R>0$ there exists $C$, depending on $R$, such that

(4.43) $|\partial_{u}^{\alpha}\partial_{v}^{\beta}\tilde{\gamma}(u,v)|\leq
C\min\\{1,|u+v|,|u-v|\\}^{-4}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(u+v,R))}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(u-v,R))}$

for all $u,v\in{}^{3}$ obeying $0<|v|\leq(4N)^{-1}\min\\{1,|u+v|,|u-v|\\}$.

###### Proof.

First, we consider the case where $|\beta|\leq 3$. We use (4.32) and (4.33)
for when $|\beta|\leq 2$ and $|\beta|=3$ respectively. The bound then follows
from Lemma 4.6 with $|\eta|+|\mu|=4$ and $|\mu|\leq 2$. Now, consider
$4\leq|\beta|\leq 5$. Take $l,m\in\mathbb{N}_{0}^{3}$, $|l|=|m|=1$ be such
that $l+m\leq\beta$. Then by (4.34),

$\partial_{u}^{\alpha}\partial_{v}^{\beta}\tilde{\gamma}(u,v)=\partial_{u}^{\alpha+l+m}\partial_{v}^{\beta-
l-m}\tilde{\gamma}(u,v)-2\big{(}\partial_{u}^{\alpha}\partial_{v}^{\beta-
l-m}\tilde{\gamma}_{l,m}(u,v)+\partial_{u}^{\alpha}\partial_{v}^{\beta-
l-m}\tilde{\gamma}_{m,l}(u,v)\big{)}.$

The first term on the right-hand side has $|\beta|-|l|-|m|\leq 3$ hence the
required bound follows from the previous step. The remaining terms can be
bounded using Lemma 4.6

∎

The proof of our main theorem is an immediate consequence of this proposition.

###### Proof of Theorem 1.1..

The proof follows from Proposition 4.7 with $u=(x+y)/2$ and $v=(x-y)/2$, along
with the definition (4.4). ∎

###  4.6. Proof of Lemma 4.4

To prove Lemma 4.4, we will examine the cluster derivatives of $\psi$ present
in (4.35) more closely. In particular, Theorem 1.3 allows us to write such
derivatives in terms of derivatives of $F_{c}$ and a function,
$G^{\boldsymbol{\alpha}}_{\mathbb{P}}$, of higher regularity near certain
singularities. Sign cancellation allows uniform boundedness of the integral
(4.35) as $x$ and $y$ approach each other, and is more easily handled via
derivatives of $F_{c}$ rather than those of $\psi$ itself. Indeed, due to the
simple formula definining $F_{c}$, it is possible to characterise all its
cluster derivatives explicitly. A series of steps, mostly involving
integration by parts, will complete the proof.

To begin, we use definition (2.2) to write

(4.44) $\displaystyle\nabla_{x}F_{c}(x,\mathbb{\hat{x}})$
$\displaystyle=-\frac{Z}{2}\nabla_{x}|x|+\frac{1}{4}\sum_{2\leq j\leq
N}\nabla_{x}|x-x_{j}|$

with the formula also holding when $x$ is replaced by $y$. Let $P$ and $S$ be
arbitrary clusters with $1\in P,S$ and $P^{*}\cap S^{*}=\emptyset$. Let
$\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2})\in\mathbb{N}_{0}^{6}$,
$\boldsymbol{\beta}=(\beta_{1},\beta_{2})\in\mathbb{N}_{0}^{6}$ and let
$l,m\in\mathbb{N}_{0}^{3}$ obey $|l|=|m|=1$. Then,

(4.45) $\displaystyle
D_{\\{P,S^{*}\\}}^{\boldsymbol{\alpha}}\partial_{x}^{l}|x|$
$\displaystyle=\begin{cases}\partial_{x}^{\alpha_{1}+l}|x|&\quad\text{if
}\alpha_{2}=0\\\ 0&\quad\text{if }|\alpha_{2}|\geq 1,\end{cases}$ (4.46)
$\displaystyle D_{\\{P^{*},S\\}}^{\boldsymbol{\beta}}\partial_{y}^{m}|y|$
$\displaystyle=\begin{cases}\partial_{y}^{\beta_{2}+m}|y|&\quad\text{if
}\beta_{1}=0\\\ 0&\quad\text{if }|\beta_{1}|\geq 1.\end{cases}$

In the following, the cluster derivatives in (4.47) are understood to act with
respect to the ordered variables $(x,x_{2},\dots,x_{N})$ and the cluster
derivatives in (4.48) are understood to act with respect to the ordered
variables $(y,x_{2},\dots,x_{N})$. For later convenience, on the right-hand
side of both formulae, all derivatives in the $x$\- or $y$-variable are
rewritten to act on $x_{j}$. Now assume
$|\boldsymbol{\alpha}|,|\boldsymbol{\beta}|\geq 1$, then

(4.47) $\displaystyle
D_{\\{P,S^{*}\\}}^{\boldsymbol{\alpha}}\partial_{x}^{l}|x-x_{j}|$
$\displaystyle=\begin{cases}(-1)^{|\alpha_{1}|+1}\partial_{x_{j}}^{\alpha_{1}+\alpha_{2}+l}|x-x_{j}|&\text{
if }|\alpha_{2}|\geq 1\text{ and }j\in S^{*}\\\ 0&\text{ if }|\alpha_{2}|\geq
1\text{ and }j\in S^{c}\\\
(-1)^{|\alpha_{1}|+1}\partial_{x_{j}}^{\alpha_{1}+\alpha_{2}+l}|x-x_{j}|&\text{
if }\alpha_{2}=0\text{ and }j\in P^{c}\\\ 0&\text{ if }\alpha_{2}=0\text{ and
}j\in P^{*}\end{cases}$ (4.48) $\displaystyle
D_{\\{P^{*},S\\}}^{\boldsymbol{\beta}}\partial_{y}^{m}|y-x_{j}|$
$\displaystyle=\begin{cases}(-1)^{|\beta_{2}|+1}\partial_{x_{j}}^{\beta_{1}+\beta_{2}+m}|y-x_{j}|&\text{
if }|\beta_{1}|\geq 1\text{ and }j\in P^{*}\\\ 0&\text{ if }|\beta_{1}|\geq
1\text{ and }j\in P^{c}\\\
(-1)^{|\beta_{2}|+1}\partial_{x_{j}}^{\beta_{1}+\beta_{2}+m}|y-x_{j}|&\text{
if }\beta_{1}=0\text{ and }j\in S^{c}\\\ 0&\text{ if }\beta_{1}=0\text{ and
}j\in S^{*}.\end{cases}$

In particular, since $P^{*}\cap S^{*}=\emptyset$, we have
$D_{\\{P,S^{*}\\}}^{\boldsymbol{\alpha}}\partial_{x}^{l}|x-x_{j}|\equiv 0$
unless $j\in P^{c}$ and
$D_{\\{P^{*},S\\}}^{\boldsymbol{\beta}}\partial_{y}^{m}|y-x_{j}|\equiv 0$
unless $j\in S^{c}$.

We will often use the following elementary fact. For each $z_{0}\in{}^{3}$ and
$\eta\in\mathbb{N}_{0}^{3}$ there exists $C$, dependent on $\eta$ but
independent of $z_{0}$, such that

(4.49) $\big{|}\partial_{z}^{\eta}|z_{0}-z|\big{|}\leq
C|z_{0}-z|^{1-|\eta|}\quad\text{for all }z\neq z_{0}.$

Therefore, by (4.12) we have for each $|\eta|\geq 1$ some $C$ and $C^{\prime}$
such that

(4.50)
$\displaystyle\big{|}\partial_{x_{j}}^{\eta}|x-x_{j}|\big{|}+\big{|}\partial_{x_{k}}^{\eta}|y-x_{k}|\big{|}$
$\displaystyle\leq
C\big{(}|x-x_{j}|^{1-|\eta|}+|y-x_{k}|^{1-|\eta|}\big{)}\leq
C^{\prime}\lambda(x,y,\mathbb{\hat{x}})^{1-|\eta|}$

for all $2\leq j,k\leq N$ if $|\eta|=1$, and all $j\in P^{c}$, $k\in S^{c}$ if
$|\eta|\geq 2$. Here, $\lambda(x,y,\mathbb{\hat{x}})$ is defined in (4.9)
using the clusters $P$ and $S$.

######  Lemma 4.8.

Let $P=P(\Phi)$ and $S=S(\Phi)$ obey $P^{*}\cap S^{*}=\emptyset$, and take any
$R>0$. Consider $|\boldsymbol{\alpha}|+|\boldsymbol{\beta}|=3$ and
$|l|=|m|=1$. For $|\boldsymbol{\alpha}|,|\boldsymbol{\beta}|\geq 1$, the
integral

(4.51)
$\int_{{}^{3N-3}}\big{(}D_{\\{P,S^{*}\\}}^{\boldsymbol{\alpha}}\partial_{x}^{l}F_{c}(x,\mathbb{\hat{x}})\big{)}\psi(x,\mathbb{\hat{x}})\big{(}D_{\\{P^{*},S\\}}^{\boldsymbol{\beta}}\partial_{y}^{m}F_{c}(y,\mathbb{\hat{x}})\big{)}\overline{\psi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}},$

can be bounded as in (4.35). For $|\boldsymbol{\alpha}|=3$, the integrals

(4.52)
$\displaystyle\int_{{}^{3N-3}}\big{(}D_{\\{P,S^{*}\\}}^{\boldsymbol{\alpha}}\partial_{x}^{l}F_{c}(x,\mathbb{\hat{x}})\big{)}\psi(x,\mathbb{\hat{x}})\partial_{y}^{m}F(y,\mathbb{\hat{x}})\overline{\psi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}},$
(4.53)
$\displaystyle\int_{{}^{3N-3}}\big{(}D_{\\{P,S^{*}\\}}^{\boldsymbol{\boldsymbol{\alpha}}}\partial_{x}^{l}F_{c}(x,\mathbb{\hat{x}})\big{)}\psi(x,\mathbb{\hat{x}})e^{F}(y,\mathbb{\hat{x}})\overline{\partial_{y}^{m}\phi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}},$

and, for $|\boldsymbol{\beta}|=3$, the integrals

(4.54)
$\displaystyle\int_{{}^{3N-3}}\partial_{x}^{l}F(x,\mathbb{\hat{x}})\psi(x,\mathbb{\hat{x}})\big{(}D_{\\{P^{*},S\\}}^{\boldsymbol{\beta}}\partial_{y}^{m}F_{c}(y,\mathbb{\hat{x}})\big{)}\overline{\psi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}},$
(4.55)
$\displaystyle\int_{{}^{3N-3}}e^{F}(x,\mathbb{\hat{x}})\partial_{x}^{l}\phi(x,\mathbb{\hat{x}})\big{(}D_{\\{P^{*},S\\}}^{\boldsymbol{\beta}}\partial_{y}^{m}F_{c}(y,\mathbb{\hat{x}})\big{)}\overline{\psi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}$

can each be bounded as in (4.35).

###### Proof.

We first prove the bound for (4.51). Use of (4.44) and (4.45)-(4.48) to expand
the integral will produce a linear combination of the following terms, where
$j\in P^{c}$ and $k\in S^{c}$,

$\displaystyle\partial_{x}^{\alpha_{1}+l}|x|\partial_{y}^{\beta_{2}+m}|y|$
$\displaystyle\int_{{}^{3N-3}}\psi(x,\mathbb{\hat{x}})\overline{\psi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}\quad\text{if
}\alpha_{2}=\beta_{1}=0,$ $\displaystyle\partial_{x}^{\alpha_{1}+l}|x|$
$\displaystyle\int_{{}^{3N-3}}\partial_{x_{k}}^{\beta_{1}+\beta_{2}+m}|y-x_{k}|\psi(x,\mathbb{\hat{x}})\overline{\psi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}\quad\text{if
}\alpha_{2}=0,$ $\displaystyle\partial_{y}^{\beta_{2}+m}|y|$
$\displaystyle\int_{{}^{3N-3}}\partial_{x_{j}}^{\alpha_{1}+\alpha_{2}+l}|x-x_{j}|\psi(x,\mathbb{\hat{x}})\overline{\psi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}\quad\text{if
}\beta_{1}=0,$
$\displaystyle\int_{{}^{3N-3}}\partial_{x_{j}}^{\alpha_{1}+\alpha_{2}+l}|x-x_{j}|\partial_{x_{k}}^{\beta_{1}+\beta_{2}+m}|y-x_{k}|\psi(x,\mathbb{\hat{x}})\overline{\psi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}.$

Derivatives of $|x|$ and $|y|$ are bounded using (4.49) and
$|x|,|y|\geq\epsilon$. Now we bound each integral in absolute value. The
first, second and third integrals are readily bounded by (4.50) and Lemma 4.2.
Finally, for the fourth integral we use (4.59) of Lemma 4.9.

Now suppose $|\boldsymbol{\alpha}|=3$. First, we prove the bound for the
integral (4.52). Recall $F=F_{c}-F_{s}$. Using this, along with (4.44) and
(4.45)-(4.48) we can expand the integral as linear combination of the
following terms, where $j\in P^{c}$ and $k\in\\{2,\dots,N\\}$,

$\displaystyle\partial_{x}^{\alpha_{1}+l}|x|\partial_{y}^{m}|y|$
$\displaystyle\int_{{}^{3N-3}}\psi(x,\mathbb{\hat{x}})\overline{\psi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}\quad\text{if
}\alpha_{2}=0,$ $\displaystyle\partial_{x}^{\alpha_{1}+l}|x|$
$\displaystyle\int_{{}^{3N-3}}\partial_{x_{k}}^{m}|y-x_{k}|\psi(x,\mathbb{\hat{x}})\overline{\psi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}\quad\text{if
}\alpha_{2}=0,$ $\displaystyle\partial_{x}^{\alpha_{1}+l}|x|$
$\displaystyle\int_{{}^{3N-3}}\partial_{x_{k}}^{m}F_{s}(y,\mathbb{\hat{x}})\psi(x,\mathbb{\hat{x}})\overline{\psi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}\quad\text{if
}\alpha_{2}=0,$ $\displaystyle\partial_{y}^{m}|y|$
$\displaystyle\int_{{}^{3N-3}}\partial_{x_{j}}^{\alpha_{1}+\alpha_{2}+l}|x-x_{j}|\psi(x,\mathbb{\hat{x}})\overline{\psi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}},$
$\displaystyle\int_{{}^{3N-3}}\partial_{x_{j}}^{\alpha_{1}+\alpha_{2}+l}|x-x_{j}|\partial_{x_{k}}^{m}|y-x_{k}|\psi(x,\mathbb{\hat{x}})\overline{\psi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}$
$\displaystyle\int_{{}^{3N-3}}\partial_{x_{j}}^{\alpha_{1}+\alpha_{2}+l}|x-x_{j}|\partial_{x_{k}}^{m}F_{s}(y,\mathbb{\hat{x}})\psi(x,\mathbb{\hat{x}})\overline{\psi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}$

Derivatives of $|x|$ and $|y|$ are bounded using (4.49) and
$|x|,|y|\geq\epsilon$. The first three integrals above are then readily
bounded using (4.6) of Proposition 4.1 and that $\nabla F_{s}\in
L^{\infty}({}^{3N})$. For the fourth and sixth integral we use (4.57) of Lemma
4.9 with $\chi\equiv 1$ and
$\chi(x,y,\mathbb{\hat{x}})=\partial_{x_{k}}^{m}F_{s}(y,\mathbb{\hat{x}})$
respectively. Finally, for the fifth integral we use the same lemma,
specifically (4.59). This proves (4.52). The proof of (4.54) is similar in the
case where $|\boldsymbol{\beta}|=3$.

Next, using (4.44), (4.45) and (4.47) we can rewrite the integral (4.53) as a
linear combination of the following two terms, where $k\in P^{c}$,

$\displaystyle\partial_{x}^{\alpha_{1}+l}|x|$
$\displaystyle\int_{{}^{3N-3}}\psi(x,\mathbb{\hat{x}})e^{F}(y,\mathbb{\hat{x}})\overline{\partial_{y}^{m}\phi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}\quad\text{if
}\alpha_{2}=0,$
$\displaystyle\int_{{}^{3N-3}}\partial_{x_{k}}^{\alpha_{1}+\alpha_{2}+l}|x-x_{k}|\psi(x,\mathbb{\hat{x}})e^{F}(y,\mathbb{\hat{x}})\overline{\partial_{y}^{m}\phi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}.$

Derivatives of $|x|$ are bounded using (4.49) and $|x|\geq\epsilon$. The first
integral is then bounded by (2.6), (2.9), (2.10) and finally (4.6) of
Proposition 4.1. The second integral is bounded immediately from (4.69) of
Lemma 4.10. This proves (4.53). The proof of (4.55) is similar in the case
where $|\boldsymbol{\beta}|=3$. ∎

###### Proof of Lemma 4.4.

To prove the required inequality, first consider the case where
$|\boldsymbol{\alpha}|,|\boldsymbol{\beta}|\geq 1$. We can then use both
(4.27) and (4.28) to write

$\displaystyle\int_{{}^{3N-3}}\big{(}D_{\\{P,S^{*}\\}}^{\boldsymbol{\alpha}}\partial_{x}^{l}\psi(x,\mathbb{\hat{x}})\big{)}\overline{\big{(}D_{\\{P^{*},S\\}}^{\boldsymbol{\beta}}\partial_{y}^{m}\psi(y,\mathbb{\hat{x}})\big{)}}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}$
$\displaystyle\quad=\int_{{}^{3N-3}}\big{(}G_{\\{P,S^{*}\\}}^{\boldsymbol{\alpha},l}(x,\mathbb{\hat{x}})\big{)}\overline{\big{(}D_{\\{P^{*},S\\}}^{\boldsymbol{\beta}}\partial_{y}^{m}\psi(y,\mathbb{\hat{x}})\big{)}}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}$
$\displaystyle\qquad+\int_{{}^{3N-3}}\big{(}D_{\\{P,S^{*}\\}}^{\boldsymbol{\alpha}}\partial_{x}^{l}F_{c}(x,\mathbb{\hat{x}})\big{)}\psi(x,\mathbb{\hat{x}})\overline{\big{(}G_{\\{P^{*},S\\}}^{\boldsymbol{\beta},m}(y,\mathbb{\hat{x}})\big{)}}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}$
$\displaystyle\qquad+\int_{{}^{3N-3}}\big{(}D_{\\{P,S^{*}\\}}^{\boldsymbol{\alpha}}\partial_{x}^{l}F_{c}(x,\mathbb{\hat{x}})\big{)}\psi(x,\mathbb{\hat{x}})\big{(}D_{\\{P^{*},S\\}}^{\boldsymbol{\beta}}\partial_{y}^{m}F_{c}(y,\mathbb{\hat{x}})\big{)}\overline{\psi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}$

We bound each of these integrals. We start with the final integral on the
right-hand side which is just (4.51) of Lemma 4.8. Next, by using
(4.25)-(4.26) and (4.29)-(4.31), the first and second integrals on the right-
hand side can be bounded in absolute value by some constant multiplied by

(4.56)
$\int_{{}^{3N-3}}\lambda(x,y,\mathbb{\hat{x}})^{-5/2}f_{\infty}(x,\mathbb{\hat{x}};R)f_{\infty}(y,\mathbb{\hat{x}};R)\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}\\\
\leq
C\epsilon^{-5/2}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(x,2R))}\left\lVert\rho\right\rVert^{1/2}_{L^{1}(B(y,2R))}$

where the inequality above holds for some $C$ by (4.15) of Lemma 4.2. This
proves the required bound when $|\boldsymbol{\alpha}|,|\boldsymbol{\beta}|\geq
1$.

Now we consider the case where $|\boldsymbol{\alpha}|=3$, and hence
$\boldsymbol{\beta}=0$. We use that $\nabla\psi=\psi\nabla F+e^{F}\nabla\phi$
and (4.27) to give

$\displaystyle\int_{{}^{3N-3}}\big{(}D_{\\{P,S^{*}\\}}^{\boldsymbol{\alpha}}\partial_{x}^{l}\psi(x,\mathbb{\hat{x}})\big{)}\overline{\partial_{y}^{m}\psi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}$
$\displaystyle\quad=\int_{{}^{3N-3}}\big{(}G_{\\{P,S^{*}\\}}^{\boldsymbol{\alpha},l}(y,\mathbb{\hat{x}})\big{)}\overline{\partial_{y}^{m}\psi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}$
$\displaystyle\qquad+\int_{{}^{3N-3}}\big{(}D_{\\{P,S^{*}\\}}^{\boldsymbol{\alpha}}\partial_{x}^{l}F_{c}(x,\mathbb{\hat{x}})\big{)}\psi(x,\mathbb{\hat{x}})\partial_{y}^{m}F(y,\mathbb{\hat{x}})\overline{\psi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}$
$\displaystyle\qquad+\int_{{}^{3N-3}}\big{(}D_{\\{P,S^{*}\\}}^{\boldsymbol{\alpha}}\partial_{x}^{l}F_{c}(x,\mathbb{\hat{x}})\big{)}\psi(x,\mathbb{\hat{x}})e^{F}(y,\mathbb{\hat{x}})\overline{\partial_{y}^{m}\phi(y,\mathbb{\hat{x}})}\Phi(x,y,\mathbb{\hat{x}})\,d\mathbb{\hat{x}}.$

As before, the first integral on the right-hand side can be bounded in
absolute value by some constant multiplying (4.56). The second and third
integrals on the right-hand side are just (4.52) and (4.53) respectively. The
proof of the $|\boldsymbol{\beta}|=3$ case is similar to the
$|\boldsymbol{\alpha}|=3$ case.

∎

We now prove two lemmas which were used to prove Lemma 4.8.

######  Lemma 4.9.

Let $P=P(\Phi)$ and $S=S(\Phi)$ obey $P^{*}\cap S^{*}=\emptyset$, and take any
$R>0$.

  1. (i)