Boundedness of the Fifth Derivative for the One-Particle Coulombic Density Matrix at the Diagonal

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. To prove this result, improved pointwise bounds are obtained for cluster derivatives of wavefunctions involving multiple clusters.

Introduction and Results

We consider the non-relativistic quantum system of $N\ge 2$ electrons among $N_0\ge 1$ nuclei, of fixed position, which represents the system of an atom or molecule. For simplicity, we suppress spin coordinates and restrict ourselves to the case of an atom ($N_0=1$), although all results are applicable to the case where spin is included and also readily generalise to the molecular case. The electrons therefore have coordinates $\mathbf{x}=(x_1,\cdots ,x_N),x_k\in {\mathbb{R}}^3$, $k=1,\cdots ,N$, and the nucleus has charge $Z>0$ whose position we choose fixed at the origin in ${\mathbb{R}}^3$. The corresponding Schrödinger operator is

$$\begin{array}{c}\hfill H=-\mathrm{\Delta }+V\end{array}$$ 1.1

where $\mathrm{\Delta }={\sum }_{k=1}^N{\mathrm{\Delta }}_{x_k}$ is the Laplacian in ${\mathbb{R}}^{3N}$, i.e. ${\mathrm{\Delta }}_{x_k}$ refers to the Laplacian applied to the variable $x_k$, and V is the Coulomb potential given by

$$\begin{array}{c}\hfill V(\mathbf{x})=-\sum _{k=1}^N\frac{Z}{|x_k|}+\sum _{1\le j<k\le N}\frac{1}{|x_j-x_k|}\end{array}$$ 1.2

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

$$\begin{array}{c}\hfill H\psi =E\psi \end{array}$$ 1.3

for $\psi \in H^2({\mathbb{R}}^{3N})$ and $E\in \mathbb{R}$.

There is a long history of regularity and differentiablity properties of wavefunctions $\psi$; a few selected references include [2–8]. In recent years, there has been growing interest in the regularity of the reduced density matrices, which are objects derived from such wavefunctions. Of particular interest is the non-smoothness at the “diagonal”, which is a signature of electron–electron interaction. In previous work by A. V. Sobolev and the current author, it was shown that the density matrix remains bounded at the diagonal upon taking up to four derivatives [9], and it was also suggested that this should remain true for five derivatives, see also [10, 11]. The current work demonstrates this to be correct.

We now introduce notation and define precisely our objects of study. For each $j=1,\cdots ,N$, we represent

$$\begin{array}{c}\hfill {\widehat{\mathbf{x}}}_j=(x_1,\cdots ,x_{j-1},x_{j+1},\cdots ,x_N),(x,{\widehat{\mathbf{x}}}_j)=(x_1,\cdots ,x_{j-1},x,x_{j+1},\cdots ,x_N)\end{array}$$

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

$$\begin{array}{c}\hfill \gamma (x,y)={\int }_{{\mathbb{R}}^{3N-3}}\psi (x,\widehat{\mathbf{x}})\overline{\psi (y,\widehat{\mathbf{x}})}\text{d}\widehat{\mathbf{x}},\widehat{\mathbf{x}}={\widehat{\mathbf{x}}}_1.\end{array}$$ 1.4

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

$$\begin{array}{c}\hfill \tilde{\gamma }(x,y)=\sum _{j=1}^N\underset{{\mathbb{R}}^{3N-3}}{\int }\psi (x,{\widehat{\mathbf{x}}}_j)\overline{\psi (y,{\widehat{\mathbf{x}}}_j)}\text{d}{\widehat{\mathbf{x}}}_j,x,y\in {\mathbb{R}}^3.\end{array}$$ 1.5

However, since we are interested only in regularity properties, we need only study one term of (1.5), hence our use of the definition (1.4). In fact, we have $\tilde{\gamma }(x,y)=N\gamma (x,y)$ whenever $\psi$ is totally symmetric or antisymmetric, although in this work we will assume no symmetry of $\psi$. An important related function is the one-particle density, or simply the density, which is defined here as

$$\begin{array}{c}\hfill \rho (x)=\gamma (x,x)={\int }_{{\mathbb{R}}^{3N-3}}{|\psi (x,\widehat{\mathbf{x}})|}^2\text{d}\widehat{\mathbf{x}},x\in {\mathbb{R}}^3.\end{array}$$ 1.6

Now suppose $\psi$ is any eigenfunction obeying (1.3) with ${∥\psi ∥}_{L^2({\mathbb{R}}^{3N})}=1$, and let $\rho$ and $\gamma$ be the corresponding functions as defined in (1.6) and (1.4), respectively. In [12], see also [13]; real analyticity was proven for $\gamma (x,y)$ as a function of two variables in the set

$$\begin{array}{c}\hfill \mathcal{D}=\{(x,y)\in {\mathbb{R}}^3\times {\mathbb{R}}^3:x\ne 0,y\ne 0,x\ne y\}.\end{array}$$

In particular, real analyticity was not shown across the diagonal, that is, where $x=y$. Despite this, real analyticity holds for the density $\rho$ on the set ${\mathbb{R}}^3\backslash \{0\}$ as shown in [14], see also [15]. Therefore, there is real analyticity of $\gamma$ in one variable along the diagonal $x=y$, excluding the point $x=y=0$. However, there is no smoothness of $\gamma$ across the diagonal, as discussed in [9, Remark 1.2(7)] in relation to the polynomial decay of eigenvalues of the density matrix operator in [16] and [17], see also [18]. We also note that non-smoothness has also been demonstrated directly for the ($N-1$)-particle reduced density matrix in [19].

The first indication of non-smoothness at the diagonal seems to be from [10], see also [11]. Here, quantum chemistry calculations indicated the existence of a fifth-order cusp, of the form ${|x-y|}^5$, at the diagonal. To elucidate the structure at the diagonal rigorously, pointwise derivative estimates for $\gamma$ on the set $\mathcal{D}$ were then given in [9]. Due to its importance in the context of the current result, we state the following bounds which arise from [9, Theorem 1.1]. For $b\ge 0$, $t>0$, first define

$$\begin{array}{c}\hfill h_b(t)=\left\{\begin{array}{cc}{t}^{min\{0,5-b\}}\hfill & \text{if}b\ne 5\hfill \\ log(t^{-1}+2)\hfill & \text{if}b=5.\hfill \end{array}\right)\end{array}$$

We denote ${\mathbb{N}}_0=\mathbb{N}\cup \{0\}$. For all $R>0$ and $\alpha ,\beta \in {\mathbb{N}}_0^3$ with $|\alpha |,|\beta |\ge 1$ there exists C such that

$$\begin{array}{cc}\hfill |{\partial }_x^{\alpha }{\partial }_y^{\beta }\gamma (x,y)|& \le C(1+{|x|}^{2-|\alpha |-|\beta |}+{|y|}^{2-|\alpha |-|\beta |}\hfill \\ \hfill & +h_{|\alpha |+|\beta |}(|x-y|)){∥\rho ∥}_{L^1(B(x,R))}^{1/2}{∥\rho ∥}_{L^1(B(y,R))}^{1/2},\hfill \end{array}$$ 1.7

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

$$\begin{array}{cc}& |{\partial }_x^{\alpha }\gamma (x,y)|+|{\partial }_y^{\alpha }{\gamma (x,y)|\le C(1+|x|}^{1-|\alpha |}+{|y|}^{1-|\alpha |}\hfill \\ \hfill & +h_{|\alpha |}(|x-y|)){∥\rho ∥}_{L^1(B(x,R))}^{1/2}{∥\rho ∥}_{L^1(B(y,R))}^{1/2},\hfill \end{array}$$ 1.8

for all $x,y\in {\mathbb{R}}^3$ with $x\ne 0$, $y\ne 0$ and $x\ne y$. 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 $\rho \in L^1({\mathbb{R}}^3)$. The density matrix $\gamma$ can be seen to be locally Lipschitz continuous in view of the local boundedness of the first derivative of $\gamma$ on ${\mathbb{R}}^6$. In addition, there is local boundedness of up to four derivatives at the diagonal.

Finally, we remark upon a recent result by Jecko [20], where it is shown that in the special case of the ($N-1$)-particle density matrix, a formula can be produced which involves an explicit characterisation of the dominant term of the non-smoothness at the diagonal. In the most general case of eigenfunctions $\psi$, a fifth-order cusp is observed along with a smoother remainder term. The comparison is directly relevant to the current work only in the case of $N=2$, see Remark 1.2(4).

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 $|\alpha |+|\beta |=5$ we have C, depending on Z, N and E, such that

$$\begin{array}{c}\hfill |{\partial }_x^{\alpha }{\partial }_y^{\beta }\gamma (x,y)|\le \left\{\begin{array}{cc}Cm{(x,y)}^{-3}{∥\rho ∥}_{L^1(B(x,1))}^{1/2}{∥\rho ∥}_{L^1(B(y,1))}^{1/2}\hfill & \text{if}|\alpha |,|\beta |\ge 1\hfill \\ Cm{(x,y)}^{-4}{∥\rho ∥}_{L^1(B(x,1))}^{1/2}{∥\rho ∥}_{L^1(B(y,1))}^{1/2}\hfill & \text{otherwise}\hfill \end{array}\right)\end{array}$$ 1.9

for all $x,y\in {\mathbb{R}}^3$ obeying $0<|x-y|\le m(x,y)/2$.

Remark 1.2

1. The bound (1.9) improves (1.7) and (1.8) in the case of $|\alpha |+|\beta |=5$ (and $R=1$) since there is no longer a contribution from a logarithm at $x=y$.

2. As a consequence of [9, Proposition A.1] and Sobolev embeddings, the inequality (1.9) shows that $\gamma \in C^{4,1}(({\mathbb{R}}^3\backslash \{0\})\times ({\mathbb{R}}^3\backslash \{0\}))$. The definition of this space is given after Remark 1.4.

3. In view of [9, Remark 1.2(7)], the boundedness of the fifth derivative at the diagonal is consistent with the asymptotics of eigenvalue decay of the density matrix operator given in [17]. In addition, such analysis indicates the sixth derivative will in general be unbounded.

4. It was shown in [20] that for $N=2$ and for general eigenfunctions $\psi$ the fifth derivative of the corresponding one-particle density matrix is only bounded, therefore showing that (1.9) is essentially optimal at the diagonal in this case. Certain eigenfunctions will lead to greater smoothness, such as totally antisymmetric eigenfunctions, but we do not consider this here. We also remark that, unlike in [20], (1.9) also gives an bound to the fifth derivative as the nucleus is approached by either x or y.

Idea of the proof. The proof proceeds by taking partial derivatives of the density matrix $\gamma (x,y)$, with a suitable cutoff, and converting these into directional derivatives, known as cluster derivatives, under the integral. The resulting integrals can then be estimated using pointwise bounds to cluster derivatives of $\psi$. This method builds upon similar strategies employed in [8, 14, 21] for differentiating the density $\rho$, and [9, 12] for differentiating the density matrix $\gamma$. The proof largely follows [9] but with certain refinements. We describe the main steps, and differences with [9], as follows:

Step 1.

We obtain pointwise bounds to cluster derivatives of $\psi$ involving multiple clusters. The result is Theorem 1.3 and is proved in Sect. 2. It is similar to [9, Theorem 4.3] but with two refinements. The first is that the bound distinguishes between the order of the derivative corresponding to each cluster. Secondly, the cluster derivatives are expressed as a “good” term and a “bad” term. The former has greater smoothness than the cluster derivative as a whole, whereas the latter has an explicit expression which will be exploited in Step 5.

Step 2.

In Sect. 3, cutoffs are then constructed on ${\mathbb{R}}^{3N+3}$. These are similar to those in [9] but now have three corresponding clusters rather than two. This will facilitate differentiating the density matrix in both x, y, as before, but now also in the direction $x+y$.

Step 3.

We now differentiate the density matrix. First, we consider the case of the partial derivatives ${\partial }_x^{\alpha }\gamma$ and ${\partial }_y^{\alpha }\gamma$ with $|\alpha |=5$. It is possible to express these with terms involving derivatives of $x+y$ and terms of the form ${\partial }_x^{\beta }{\partial }_y^{\sigma }\gamma$ where $|\beta |,|\sigma |\ge 1$. We first consider the former type. We use that the pointwise bounds of Step 1 distinguish between the order of the derivative corresponding to each cluster to show that derivatives in $x+y$ do not contribute to the singularity at the diagonal for the density matrix.

Step 4.

We now consider derivatives ${\partial }_x^{\beta }{\partial }_y^{\sigma }\gamma$ for $|\beta |,|\sigma |\ge 1$. As in Step 3 and in [9], we evaluate these using the pointwise bounds to cluster derivatives of $\psi$. However, now we use the decomposition of the cluster derivatives into a “good” and “bad” term, as described in Step 1. The former term is sufficiently integrable to show boundedness across the $x=y$ diagonal.

Step 5.

It remains to consider the expressions involving the “bad” terms. These cannot be bounded appropriately for our purposes and doing so produces the logarithm singularity present in [9, Theorem 1.1], see (1.7) and (1.8). Instead, we use that these terms have an explicit form involving derivatives of the function |x| in ${\mathbb{R}}^3$. The relevant integrals can be handled using a series of steps involving integration by parts.

Notation

As mentioned earlier, we use a standard notation whereby $\mathbf{x}=(x_1,\cdots ,x_N)\in {\mathbb{R}}^{3N}$, $x_j\in {\mathbb{R}}^3$, j=1, ..., N, and where N is the number of electrons. In addition, define for $1\le j,k\le N$, $j\ne k$,

$$\begin{array}{cc}\hfill {\widehat{\mathbf{x}}}_j& =(x_1,\cdots ,x_{j-1},x_{j+1},\cdots ,x_N)\hfill \end{array}$$ 1.10

$$\begin{array}{cc}\hfill {\widehat{\mathbf{x}}}_{j,k}& =(x_1,\cdots ,x_{j-1},x_{j+1},\cdots ,x_{k-1},x_{k+1},\cdots ,x_N)\hfill \end{array}$$ 1.11

with obvious modifications if either j, k equals 1 or N, and if $k<j$. We define $\widehat{\mathbf{x}}={\widehat{\mathbf{x}}}_1$, which will be used throughout. Variables placed before ${\widehat{\mathbf{x}}}_j$ and ${\widehat{\mathbf{x}}}_{j,k}$ will be placed in the removed slots as follows, for any $x,y\in {\mathbb{R}}^3$ we have

$$\begin{array}{cc}\hfill (x,{\widehat{\mathbf{x}}}_j)& =(x_1,\cdots ,x_{j-1},x,x_{j+1},\cdots ,x_N),\hfill \end{array}$$ 1.12

$$\begin{array}{cc}\hfill (x,y,{\widehat{\mathbf{x}}}_{j,k})& =(x_1,\cdots ,x_{j-1},x,x_{j+1},\cdots ,x_{k-1},y,x_{k+1},\cdots ,x_N).\hfill \end{array}$$ 1.13

In this way, $\mathbf{x}=(x_j,{\widehat{\mathbf{x}}}_j)=(x_j,x_k,{\widehat{\mathbf{x}}}_{j,k})$.

For a variable $x\in {\mathbb{R}}^3$ and multiindex $\alpha \in {\mathbb{N}}_0^3$, we denote by ${\partial }_x^{\alpha }$ the $\alpha$-partial derivative in the variable x.

For variables $x,y\in {\mathbb{R}}^3$, the directional derivative ${\partial }_{x+y}$ will be defined by

$$\begin{array}{c}\hfill {\partial }_{x+y}^{\alpha }={\partial }_x^{\alpha }+{\partial }_y^{\alpha }\text{for}\alpha \in {\mathbb{N}}_0^3,|\alpha |=1.\end{array}$$ 1.14

Higher-order derivatives ${\partial }_{x+y}^{\alpha }$ are defined by successive first-order derivatives.

We define a cluster to be any subset $P\subset \{1,\cdots ,N\}$. Denote $P^c=\{1,\cdots ,N\}\backslash P$, $P^{\ast }=P\backslash \{1\}$. We will also need cluster sets, $\mathbf{P}=(P_1,\cdots ,P_M)$, where $M\ge 1$ and $P_1,\cdots ,P_M$ are clusters.

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

$$\begin{array}{c}\hfill D_P^{\alpha }=\sum _{j\in P}{\partial }_{x_j}^{\alpha }\text{for}\alpha \in {\mathbb{N}}_0^3,|\alpha |=1.\end{array}$$ 1.15

For $P=\mathrm{\varnothing }$, $D_P^{\alpha }$ 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 $|\alpha |\ge 2$, are defined by successive application of first-order cluster derivatives as follows,

$$\begin{array}{c}\hfill D_P^{\alpha }={(D_P^1)}^{\prime }{(D_P^2)}^{\prime \prime }{(D_P^3)}^{\prime \prime \prime }\end{array}$$ 1.16

where $e_1,e_2,e_3$ are the standard unit basis vectors of ${\mathbb{R}}^3$ and therefore $D_P^1$, $D_P^2$ and $D_P^3$ are defined according to (1.15). Let $\mathbf{P}=(P_1,\cdots ,P_M)$ and $\mathit{\alpha }=({\alpha }_1,\cdots ,{\alpha }_M)$, ${\alpha }_j\in {\mathbb{N}}_0^3$, $1\le j\le M$, then we define the multicluster derivative (often simply referred to as cluster derivative) by

$$\begin{array}{c}\hfill D_{\mathbf{P}}^{\mathit{\alpha }}=D_{P_1}^1\cdots D_{P_M}^M.\end{array}$$ 1.17

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

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

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

$$\begin{array}{c}\hfill {\mathrm{\Sigma }}_P=\{\mathbf{x}\in {\mathbb{R}}^{3N}:\prod _{j\in P}|x_j|\prod _{\begin{array}{c}l\in P\\ m\in P^c\end{array}}|x_l-x_m|=0\}.\end{array}$$ 1.18

For $P=\mathrm{\varnothing }$ we set ${\mathrm{\Sigma }}_P:=\mathrm{\varnothing }$. Denote ${\mathrm{\Sigma }}_P^c={\mathbb{R}}^{3N}\backslash {\mathrm{\Sigma }}_P$. For each P we have ${\mathrm{\Sigma }}_P\subset \mathrm{\Sigma }$ where

$$\begin{array}{c}\hfill \mathrm{\Sigma }=\{\mathbf{x}\in {\mathbb{R}}^{3N}:\prod _{1\le j\le N}|x_j|\prod _{1\le l<m\le N}|x_l-x_m|=0\}\end{array}$$ 1.19

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

$$\begin{array}{cc}\hfill d_P(\mathbf{x})& :=\text{dist}(\mathbf{x},{\mathrm{\Sigma }}_P)=min\{|x_j|,2^{-1/2}|x_j-x_k|:j\in P,k\in P^c\}\hfill \end{array}$$ 1.20

$$\begin{array}{cc}\hfill {\lambda }_P(\mathbf{x})& :=min\{1,d_P(\mathbf{x})\}\hfill \end{array}$$ 1.21

$$\begin{array}{cc}\hfill {\nu }_P(\mathbf{x})& :=min\{1;|x_j|:j\in P\}\hfill \end{array}$$ 1.22

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

Using the formula (1.20), see, for example, [9, Lemma 4.2], it can be shown that

$$\begin{array}{c}\hfill |d_P(\mathbf{x})-d_P(\mathbf{y})|,|{\lambda }_P(\mathbf{x})-{\lambda }_P(\mathbf{y})|,|{\nu }_P(\mathbf{x})-{\nu }_P(\mathbf{y})|\le |\mathbf{x}-\mathbf{y}|\end{array}$$ 1.23

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

Let $\mathbf{P}=(P_1,\cdots ,P_M)$ be a cluster set and $\mathit{\alpha }=({\alpha }_1,\cdots ,{\alpha }_M)\in {\mathbb{N}}_0^{3M}$ be a multiindex. Define

$$\begin{array}{c}\hfill {\mathrm{\Sigma }}_{\mathit{\alpha }}=\bigcup _{j:{\alpha }_j\ne 0}{\mathrm{\Sigma }}_{P_j}\end{array}$$ 1.24

for nonzero $\mathit{\alpha }$, and when $\mathit{\alpha }=0$, we set ${\mathrm{\Sigma }}_{\mathit{\alpha }}=\mathrm{\varnothing }$. Denote ${\mathrm{\Sigma }}_{\mathit{\alpha }}^c={\mathbb{R}}^{3N}\backslash {\mathrm{\Sigma }}_{\mathit{\alpha }}$. For nonzero $\mathit{\alpha }$, we can also define the distances

$$\begin{array}{cc}\hfill d_{\mathit{\alpha }}(\mathbf{x})& =min\{d_{P_j}(\mathbf{x}):{\alpha }_j\ne 0,j=1,\cdots ,M\},\hfill \end{array}$$ 1.25

$$\begin{array}{cc}\hfill {\lambda }_{\mathit{\alpha }}(\mathbf{x})& =min\{{\lambda }_{P_j}(\mathbf{x}):{\alpha }_j\ne 0,j=1,\cdots ,M\},\hfill \end{array}$$ 1.26

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

$$\begin{array}{c}\hfill d_{\mathit{\alpha }}(\mathbf{x})=\text{dist}(\mathbf{x},{\mathrm{\Sigma }}_{\mathit{\alpha }}).\end{array}$$

Indeed, since ${\mathrm{\Sigma }}_{P_j}\subset {\mathrm{\Sigma }}_{\mathit{\alpha }}$ whenever j is such that ${\alpha }_j\ne 0$, we have $\text{dist}(\mathbf{x},{\mathrm{\Sigma }}_{\mathit{\alpha }})\le d_{P_j}(\mathbf{x})$ and hence $\text{dist}(\mathbf{x},{\mathrm{\Sigma }}_{\mathit{\alpha }})\le d_{\mathit{\alpha }}(\mathbf{x})$. Conversely for each $\mathit{\xi }\in {\mathrm{\Sigma }}_{\mathit{\alpha }}$ we have $\mathit{\xi }\in {\mathrm{\Sigma }}_{P_j}$ for some j with ${\alpha }_j\ne 0$, and hence $|\mathbf{x}-\mathit{\xi }|\ge d_{P_j}(\mathbf{x})$. Therefore, $|\mathbf{x}-\mathit{\xi }|\ge d_{\mathit{\alpha }}(\mathbf{x})$. Taking infimum over $\mathit{\xi }\in {\mathrm{\Sigma }}_{\mathit{\alpha }}$ we obtain the reverse inequality.

We will also use the following related quantities involving maxima of relevant distances for nonzero $\mathit{\alpha }$,

$$\begin{array}{cc}\hfill q_{\mathit{\alpha }}(\mathbf{x})& =max\{d_{P_j}(\mathbf{x}):{\alpha }_j\ne 0,j=1,\cdots ,M\}\hfill \end{array}$$ 1.27

$$\begin{array}{cc}\hfill {\mu }_{\mathit{\alpha }}(\mathbf{x})& =max\{{\lambda }_{P_j}(\mathbf{x}):{\alpha }_j\ne 0,j=1,\cdots ,M\}.\hfill \end{array}$$ 1.28

For $\mathit{\alpha }=0$ we set $d_{\mathit{\alpha }},q_{\mathit{\alpha }}\equiv 0$ and ${\lambda }_{\mathit{\alpha }},{\mu }_{\mathit{\alpha }}\equiv 1$.

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

$$\begin{array}{c}\hfill f_{\infty }(\mathbf{x};r;u):={∥\mathrm{\nabla },u∥}_{L^{\infty }(B(\mathbf{x},r))}+{∥u∥}_{L^{\infty }(B(\mathbf{x},r))}\end{array}$$ 1.29

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

$$\begin{array}{c}\hfill f_{\infty }(\mathbf{x};r):=f_{\infty }(\mathbf{x};r;\psi ).\end{array}$$ 1.30

Furthermore, later we will not track the dependence on r; hence, it is convenient to define

$$\begin{array}{c}\hfill f_{\infty }(\mathbf{x}):=f_{\infty }(\mathbf{x};\frac{1}{2};\psi ).\end{array}$$ 1.31

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 itself is of independent interest. It will be shown by elliptic regularity that for all $\mathit{\alpha }$ the weak cluster derivatives $D_{\mathbf{P}}^{\mathit{\alpha }}\psi$ exist in the set ${\mathrm{\Sigma }}_{\mathit{\alpha }}^c$. We consider how such cluster derivatives behave as the set ${\mathrm{\Sigma }}_{\mathit{\alpha }}$ 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 1.12] it is shown that for any multiindex $0\ne \alpha \in {\mathbb{N}}_0^3$, $p\in (1,\infty ]$ and any $0<r<R<1$ there exists C, depending on r, R, p and $\alpha$, such that

$$\begin{array}{c}\hfill {∥D_P^{\alpha },\psi ∥}_{L^p(B(\mathbf{x},r{\lambda }_P(\mathbf{x})))}\le C{\lambda }_P{(\mathbf{x})}^{1-|\alpha |}({∥\mathrm{\nabla },\psi ∥}_{L^p(B(\mathbf{x},R{\lambda }_P(\mathbf{x})))}+{∥\psi ∥}_{L^p(B(\mathbf{x},R{\lambda }_P(\mathbf{x})))})\end{array}$$ 1.32

for all $\mathbf{x}\in {\mathrm{\Sigma }}_P^c$. Notice that for every $\mathbf{x}\in {\mathrm{\Sigma }}_P^c$, we have $B(\mathbf{x},r{\lambda }_P(\mathbf{x}))\subset {\mathrm{\Sigma }}_P^c$ by the definition of ${\lambda }_P(\mathbf{x})$. The exponent $1-|\alpha |$ is natural since both $\psi$ and $\mathrm{\nabla }\psi$ are locally bounded; therefore, we only get a singularity for $|\alpha |\ge 2$.

The objective of the following theorem is to extend the bounds (1.32) in the case of $p=\infty$ and for cluster sets $\mathbf{P}$ which may contain more than one cluster. Bounds for cluster sets are present in [9, Theorem 4.3], but here we obtain estimates which depend on the order of derivative on each cluster in $\mathbf{P}$. This will be vital in proving the boundedness of the fifth derivative of the density matrix at the diagonal.

Before stating the theorem, we introduce a function which is used to express the singular nature of the derivatives of $\psi$, namely $F_c=F_c(\mathbf{x})$ defined by

$$\begin{array}{c}\hfill F_c(\mathbf{x})=-\frac{Z}{2}\sum _{1\le j\le N}|x_j|+\frac{1}{4}\sum _{1\le l<k\le N}|x_l-x_k|.\end{array}$$ 1.33

This is described in more detail in Sect. 2.1. In the following, the symbol $\mathrm{\nabla }$ denotes the gradient operator in ${\mathbb{R}}^{3N}$.

Theorem 1.3

For every cluster set $\mathbf{P}=(P_1,\cdots ,P_M)$, multiindex $\mathit{\alpha }\in {\mathbb{N}}_0^{3M}$ and any $R>0$ there exists C, depending on $\mathit{\alpha }$ and R, such that for $k=0,1$,

$$\begin{array}{c}\hfill |D_{\mathbf{P}}^{\mathit{\alpha }}{\mathrm{\nabla }}^k\psi (\mathbf{x})|\le C{\lambda }_{\mathit{\alpha }}{(\mathbf{x})}^{1-k}{\lambda }_{P_1}{(\mathbf{x})}^{-|{\alpha }_1|}\cdots {\lambda }_{P_M}{(\mathbf{x})}^{-|{\alpha }_M|}{f}_{\infty }(\mathbf{x};R)\end{array}$$ 1.34

for all $\mathbf{x}\in {\mathrm{\Sigma }}_{\mathit{\alpha }}^c$.

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

$$\begin{array}{c}\hfill D_{\mathbf{P}}^{\mathit{\alpha }}\mathrm{\nabla }\psi =G_{\mathbf{P}}^{\mathit{\alpha }}+\psi D_{\mathbf{P}}^{\mathit{\alpha }}\mathrm{\nabla }{F}_c,\end{array}$$ 1.35

and for each $0\le b<1$ there exists C, depending on $\mathit{\alpha },R$ and b, such that

$$\begin{array}{c}\hfill |G_{\mathbf{P}}^{\mathit{\alpha }}(\mathbf{x})|\le C{\mu }_{\mathit{\alpha }}{(\mathbf{x})}^b{\lambda }_{P_1}{(\mathbf{x})}^{-|{\alpha }_1|}\cdots {\lambda }_{P_M}{(\mathbf{x})}^{-|{\alpha }_M|}{f}_{\infty }(\mathbf{x};R)\end{array}$$ 1.36

for all $\mathbf{x}\in {\mathrm{\Sigma }}_{\mathit{\alpha }}^c$.

Remark 1.4

1. In the case of a single cluster and $k=0$, the bound (1.34) reestablishes (1.32) in the case of $p=\infty$, albeit with a larger radius in the $L^{\infty }$-norms on the right-hand side, which doesn’t concern us.

2. The presence of a single power of ${\lambda }_{\mathit{\alpha }}(\mathbf{x})$ in the bound (1.34), for $k=0$, will cancel a single negative power of the smallest ${\lambda }_{P_j}(\mathbf{x})$ for j such that ${\alpha }_j\ne 0$. Notice that the appropriate j will depend on $\mathbf{x}$.

3. The bound in (1.36) is stronger than that of (1.34) with $k=1$. This is because a positive power of ${\mu }_{\mathit{\alpha }}(\mathbf{x})$ will partially cancel a single negative power of the largest ${\lambda }_{P_j}(\mathbf{x})$ for j such that ${\alpha }_j\ne 0$. The term $\psi D_{\mathbf{P}}^{\mathit{\alpha }}\mathrm{\nabla }{F}_c$ in (1.35) only obeys a bound as in (1.34) for $k=1$ and therefore represents the part of $D_{\mathbf{P}}^{\mathit{\alpha }}\mathrm{\nabla }\psi$ which inhibits an improved derivative bound.

The proof of Theorem 1.3 will adopt a similar strategy to that used originally in [8, Proposition 1.12], and developed in [9, Theorem 4.3]. To prove (1.36), we will use a result in [22], which we restate as Theorem A.1 in appendix, where it was shown that $\psi$ can be made $W_{\mathit{loc}}^{2,\infty }({\mathbb{R}}^{3N})$ 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 $\mathrm{\Omega }$ be open, $\theta \in (0,1]$ and $k={\mathbb{N}}_0$. We formally define the $\theta$-Hölder seminorms for a function f by

$$\begin{array}{cc}& {[f]}_{\theta ,\mathrm{\Omega }}=\underset{\begin{array}{c}x,y\in \mathrm{\Omega }\\ x\ne y\end{array}}{sup}\frac{|f(x)-f(y)|}{{|x-y|}^{\theta }},\hfill \\ \hfill & [{\mathrm{\nabla }}^k{f]}_{\theta ,\mathrm{\Omega }}=\underset{|\alpha |=k}{sup}{[{\partial }^{\alpha }f]}_{\theta ,\mathrm{\Omega }}.\hfill \end{array}$$

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

$$\begin{array}{c}\hfill {∥f∥}_{C^{k,\theta }(\overline{\mathrm{\Omega }})}={∥f∥}_{C^k(\overline{\mathrm{\Omega }})}+{[{\mathrm{\nabla }}^{k}f]}_{\theta ,\mathrm{\Omega }}.\end{array}$$

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

$$\begin{array}{c}\hfill Lu:=-\mathrm{\Delta }u+\mathbf{c}·\mathrm{\nabla }u+du=g\end{array}$$ 1.37

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

$$\begin{array}{c}\hfill \mathcal{L}(u,\chi )={\int }_{\mathrm{\Omega }}(\mathrm{\nabla }u·\mathrm{\nabla }\chi +(\mathbf{c}·\mathrm{\nabla }u)\chi +\text{d}u\chi )\text{d}x\end{array}$$

for all $u\in H_{\mathit{loc}}^1(\mathrm{\Omega })$ and $\chi \in C_c^{\infty }(\mathrm{\Omega })$. We say that a function $u\in H_{\mathit{loc}}^1(\mathrm{\Omega })$ is a weak solution to the equation (1.37) in $\mathrm{\Omega }$ if $\mathcal{L}(u,\chi )={\int }_{\mathrm{\Omega }}g\chi dx$ for every $\chi \in C_c^{\infty }(\mathrm{\Omega })$.

The following theorem is a restatement of [9, Proposition 3.1] ([8, Proposition A.2] is similar), with additional Hölder regularity which follows from the proof.

Theorem 1.5

Let $x_0\in {\mathbb{R}}^n$, $R>0$ and $\mathbf{c},d,g\in L^{\infty }(B(x_0,R))$ and $u\in H^1(B(x_0,R))$ be a weak solution to (1.37) then for each $\theta \in [0,1)$ we have $u\in C^{1,\theta }(B(x_0,R))\cap H_{\mathit{loc}}^2(B(x_0,R))$, and for any $r\in (0,R)$ we have

$$\begin{array}{c}\hfill {∥u∥}_{C^{1,\theta }(\overline{B(x_0,r)})}\le C({∥u∥}_{L^2(B(x_0,R))}+{∥g∥}_{L^{\infty }(B(x_0,R))})\end{array}$$ 1.38

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

$$\begin{array}{c}\hfill {∥\mathbf{c}∥}_{L^{\infty }(B(x_0,R))}+{∥d∥}_{L^{\infty }(B(x_0,R))}\le K.\end{array}$$

Proof of Theorem 1.3

The proof will involve the use of a multiplicative factor, frequently called a Jastrow factor in mathematical literature, which improves a lower bound to the regularity of $\psi$. This is a strategy that has been used successfully in, for example, [4, 21] to elucidate regularity properties of $\psi$. We start by defining a function $F=F(\mathbf{x})$, depending only on N and Z, such that the function $e^{-F}\psi$ solves an elliptic equation with bounded coefficients. These coefficients behave suitably well under the action of cluster derivatives. This allows us to use elliptic regularity to produce bounds to the cluster derivatives of $e^{-F}\psi$ and hence to $\psi$ itself.

Jastrow Factors

The function $F_c$ was introduced in (1.33). This will be the part of the Jastrow factor which captures the predominant singularities of $\psi$. It obeys the following

$$\begin{array}{c}\hfill \mathrm{\Delta }{F}_c=V\end{array}$$ 2.1

where V is the Coulomb potential, (1.2).

The function $F_c$ is unbounded on ${\mathbb{R}}^{3N}$ and it will be convenient to subtract a smooth term $F_s$, defined below, to offset the growth of $F_c$. We also split up $F_c$ into terms involving electron–nucleus interactions, $F_c^{(en)}$ and those involving electron–electron interactions, $F_c^{(ee)}$. More precisely, set

$$\begin{array}{cc}\hfill F(\mathbf{x})& =F_c(\mathbf{x})-F_s(\mathbf{x}),F_c(\mathbf{x})=F_c^{(en)}(\mathbf{x})+F_c^{(ee)}(\mathbf{x}),\hfill \end{array}$$ 2.2

where

$$\begin{array}{c}\hfill F_c^{(en)}(\mathbf{x})=-\frac{Z}{2}\sum _{1\le j\le N}|x_j|,F_c^{(ee)}(\mathbf{x})=\frac{1}{4}\sum _{1\le l<k\le N}|x_l-x_k|,\end{array}$$ 2.3

$$\begin{array}{c}\hfill F_s(\mathbf{x})=-\frac{Z}{2}\sum _{1\le j\le N}\sqrt{|x_j{|}^2+1}+\frac{1}{4}\sum _{1\le l<k\le N}\sqrt{|x_l-x_k{|}^2+1},\end{array}$$ 2.4

for $\mathbf{x}\in {\mathbb{R}}^{3N}$. The function F was used in [8]. As remarked above, $F_s\in C^{\infty }({\mathbb{R}}^{3N})$ and in fact

$$\begin{array}{c}\hfill {\partial }^{\alpha }{F}_s\in L^{\infty }({\mathbb{R}}^{3N})\text{for all}\alpha \in {\mathbb{N}}_0^{3N},|\alpha |\ge 1.\end{array}$$ 2.5

Furthermore, the function F obeys

$$\begin{array}{c}\hfill F,\mathrm{\nabla }F\in L^{\infty }({\mathbb{R}}^{3N}).\end{array}$$ 2.6

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

$$\begin{array}{c}\hfill \varphi =e^{-F}\psi .\end{array}$$ 2.7

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

$$\begin{array}{c}\hfill -\mathrm{\Delta }\varphi -2\mathrm{\nabla }F·\mathrm{\nabla }\varphi +(\mathrm{\Delta }{F}_s-|\mathrm{\nabla }F{|}^2-E)\varphi =0.\end{array}$$ 2.8

Since all coefficients are bounded in ${\mathbb{R}}^{3N}$, we can see from Theorem 1.5 that $\varphi \in C^{1,\theta }({\mathbb{R}}^{3N})$ 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

$$\begin{array}{c}\hfill {∥\varphi ∥}_{C^{1,\theta }(\overline{B(\mathbf{x},r)})}\le C{∥\varphi ∥}_{L^2(B(\mathbf{x},R))}\le C^{\prime }{∥\varphi ∥}_{L^{\infty }(B(\mathbf{x},R))}\end{array}$$ 2.9

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

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

$$\begin{array}{c}\hfill C{∥\varphi ∥}_{L^{\infty }(B(\mathbf{x},R))}\le {∥\psi ∥}_{L^{\infty }(B(\mathbf{x},R))}\le C^{\prime }{∥\varphi ∥}_{L^{\infty }(B(\mathbf{x},R))}\end{array}$$ 2.10

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

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), (1.26) and (1.28).

Lemma 2.1

For any $\mathit{\sigma }=({\sigma }_1,\cdots ,{\sigma }_M)\in {\mathbb{N}}_0^{3M}$, we have for $k=0,1$,

$$\begin{array}{c}\hfill {\mu }_{\mathit{\sigma }}{(\mathbf{y})}^{k-|\mathit{\sigma }|}\le {\lambda }_{\mathit{\sigma }}{(\mathbf{y})}^k{\lambda }_{P_1}{(\mathbf{y})}^{-|{\sigma }_1|}\cdots {\lambda }_{P_M}{(\mathbf{y})}^{-|{\sigma }_M|}\end{array}$$ 2.11

for all $\mathbf{y}\in {\mathbb{R}}^{3N}$. Furthermore, for some $n\ge 1$ let ${\mathit{\beta }}^{(1)},\cdots ,{\mathit{\beta }}^{(n)}$ be an arbitrary collection of multiindices in ${\mathbb{N}}_0^{3M}$ such that ${\mathit{\beta }}^{(1)}+\cdots +{\mathit{\beta }}^{(n)}=\mathit{\sigma }$, then

$$\begin{array}{c}\hfill \prod _{j=1}^n{\lambda }_{{\mathit{\beta }}^{(j)}}(\mathbf{y})\le {\lambda }_{\mathit{\sigma }}(\mathbf{y})\end{array}$$ 2.12

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

Proof

The results are trivial in the case of $\mathit{\sigma }=0$; therefore, we assume in the following that $\mathit{\sigma }$ is nonzero. First, observe for all $j=1,\cdots ,M$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mu }_{\mathit{\sigma }}{(\mathbf{y})}^{-|{\sigma }_j|}& \le {\lambda }_{P_j}{(\mathbf{y})}^{-|{\sigma }_j|}\hfill \\ \hfill {\mu }_{\mathit{\sigma }}{(\mathbf{y})}^{1-|{\sigma }_j|}& \le {\lambda }_{P_j}{(\mathbf{y})}^{1-|{\sigma }_j|}\text{if}{\sigma }_j\ne 0\hfill \end{array}\end{array}$$

by the definition of ${\mu }_{\mathit{\sigma }}$. Now perform the following trivial expansion of the product,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mu }_{\mathit{\sigma }}{(\mathbf{y})}^{-|\mathit{\sigma }|}& ={\mu }_{\mathit{\sigma }}{(\mathbf{y})}^{-|{\sigma }_1|}\cdots {\mu }_{\mathit{\sigma }}{(\mathbf{y})}^{-|{\sigma }_M|}\hfill \\ \hfill & \le {\lambda }_{P_1}{(\mathbf{y})}^{-|{\sigma }_1|}\cdots {\lambda }_{P_M}{(\mathbf{y})}^{-|{\sigma }_M|}\hfill \end{array}\end{array}$$

which proves (2.11) for $k=0$. For $k=1$, consider that for each $\mathbf{y}$ we can find $l=1,\cdots ,M$ such that ${\lambda }_{\mathit{\sigma }}(\mathbf{y})={\lambda }_{P_l}(\mathbf{y})$ and ${\sigma }_l\ne 0$. Then,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mu }_{\mathit{\sigma }}{(\mathbf{y})}^{1-|\mathit{\sigma }|}& ={\mu }_{\mathit{\sigma }}{(\mathbf{y})}^{-|{\sigma }_1|}\cdots {\mu }_{\mathit{\sigma }}{(\mathbf{y})}^{1-|{\sigma }_l|}\cdots {\mu }_{\mathit{\sigma }}{(\mathbf{y})}^{-|{\sigma }_M|}\hfill \\ \hfill & \le {\lambda }_{P_1}{(\mathbf{y})}^{-|{\sigma }_1|}\cdots {\lambda }_{P_l}{(\mathbf{y})}^{1-|{\sigma }_l|}\cdots {\lambda }_{P_M}{(\mathbf{y})}^{-|{\sigma }_M|}\hfill \\ \hfill & ={\lambda }_{\mathit{\sigma }}(\mathbf{y}){\lambda }_{P_1}{(\mathbf{y})}^{-|{\sigma }_1|}\cdots {\lambda }_{P_M}{(\mathbf{y})}^{-|{\sigma }_M|}\hfill \end{array}\end{array}$$

as required.

Finally, we prove (2.12). As above, take arbitrary $\mathbf{y}$ and a corresponding l such that ${\lambda }_{\mathit{\sigma }}(\mathbf{y})={\lambda }_{P_l}(\mathbf{y})$ with ${\sigma }_l\ne 0$. For each $1\le j\le n$ we denote the ${\mathbb{N}}_0^3$-components of ${\mathit{\beta }}^{(j)}$ as ${\mathit{\beta }}^{(j)}=({\beta }_1^{(j)},\cdots ,{\beta }_M^{(j)})$. We know ${\mathit{\beta }}^{(1)}+\cdots +{\mathit{\beta }}^{(n)}=\mathit{\sigma }$ so in particular, ${\beta }_l^{(1)}+\cdots +{\beta }_l^{(n)}={\sigma }_l$. Since ${\sigma }_l\ne 0$, there exists at least one $1\le r\le n$ such that ${\beta }_l^{(r)}\ne 0$. Hence, by the definition of ${\lambda }_{{\mathit{\beta }}^{(r)}}$,

$$\begin{array}{c}\hfill {\lambda }_{{\mathit{\beta }}^{(r)}}(\mathbf{y})=min\{{\lambda }_{P_s}(\mathbf{y}):{\beta }_s^{(r)}\ne 0\text{,}s=1,\cdots ,M\}\le {\lambda }_{P_l}(\mathbf{y})={\lambda }_{\mathit{\sigma }}(\mathbf{y}).\end{array}$$

The remaining factors ${\lambda }_{{\mathit{\beta }}^{(j)}}(\mathbf{y})$, for $j\ne r$, can each be bounded above by one. $\square$

The following lemma will be useful in proving results about taking cluster derivatives of F, as defined in (2.2). It is a similar, but stronger version of [9, Lemma 4.4]. Later, we will apply it using f(x) as the function |x| for $x\in {\mathbb{R}}^3$, or derivatives thereof.

Lemma 2.2

Let $f\in C^{\infty }({\mathbb{R}}^3\backslash \{0\})$ and $k\in {\mathbb{N}}_0$ be such that for each $\sigma \in {\mathbb{N}}_0^3$, there exists C such that

$$\begin{array}{c}\hfill |{\partial }^{\sigma }{f(x)|\le C|x|}^{k-|\sigma |}\text{for all}x\ne 0.\end{array}$$ 2.13

Then for any $\mathit{\alpha }\ne 0$ with $|\mathit{\alpha }|\ge k$ there exists some new C such that for any $l,m=1,\cdots ,N,$ the weak derivatives $D_{\mathbf{P}}^{\mathit{\alpha }}(f(x_l))$ and $D_{\mathbf{P}}^{\mathit{\alpha }}(f(x_l-x_m))$ are both smooth in ${\mathrm{\Sigma }}_{\mathit{\alpha }}^c$ and obey

$$\begin{array}{c}\hfill |D_{\mathbf{P}}^{\mathit{\alpha }}(f(x_l))|,|D_{\mathbf{P}}^{\mathit{\alpha }}(f(x_l-x_m))|\le C{q}_{\mathit{\alpha }}{(\mathbf{x})}^{k-|\mathit{\alpha }|}\end{array}$$

for all $\mathbf{x}\in {\mathrm{\Sigma }}_{\mathit{\alpha }}^c$.

Proof

Take any $j=1,\cdots ,M$ with ${\alpha }_j\ne 0$, then we have $D_j^j(f(x_l))\equiv 0$ for each $l\in P_j^c$. Therefore, for $D_{\mathbf{P}}^{\mathit{\alpha }}(f(x_l))$ to not be identically zero we require that $l\in P_j$ for each j with ${\alpha }_j\ne 0$. For such l, we have $x_l\ne 0$ since $\mathbf{x}\in {\mathrm{\Sigma }}_{\mathit{\alpha }}^c$, and

$$\begin{array}{c}\hfill |x_l|\ge d_{P_j}(\mathbf{x})\end{array}$$

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

$$\begin{array}{c}\hfill |D_{\mathbf{P}}^{\mathit{\alpha }}(f(x_l))|=|{\partial }^{{\alpha }_1+\cdots +{\alpha }_M}f(x_l)|\le C|x_l{|}^{k-|\mathit{\alpha }|}\le C{q}_{\mathit{\alpha }}{(\mathbf{x})}^{k-|\mathit{\alpha }|}\end{array}$$

because $|\mathit{\alpha }|\ge k$. Similarly, for each $j=1,\cdots ,M$, with ${\alpha }_j\ne 0$ we have $D_j^j(f(x_l-x_m))\equiv 0$ if either $l,m\in P_j$ or $l,m\in P_j^c$. Therefore, for $D_{\mathbf{P}}^{\mathit{\alpha }}(f(x_l-x_m))$ to not be identically zero we require that

$$\begin{array}{c}\hfill (l,m)\in \bigcap _{j:{\alpha }_j\ne 0}((P_j\times P_j^c)\cup (P_j^c\times P_j)).\end{array}$$

For such (l, m), we have $x_l\ne x_m$ since $\mathbf{x}\in {\mathrm{\Sigma }}_{\mathit{\alpha }}^c$ and

$$\begin{array}{c}\hfill |x_l-x_m|\ge \sqrt{2}{d}_{P_j}(\mathbf{x})\end{array}$$

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

$$\begin{array}{c}\hfill |D_{\mathbf{P}}^{\mathit{\alpha }}(f(x_l-x_m))|=|{\partial }^{{\alpha }_1+\cdots +{\alpha }_M}f(x_l-x_m)|\le C|x_l-x_m{|}^{k-|\mathit{\alpha }|}\le C^{\prime }{q}_{\mathit{\alpha }}{(\mathbf{x})}^{k-|\mathit{\alpha }|}\end{array}$$

because $|\mathit{\alpha }|\ge k$. $\square$

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

Lemma 2.3

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

$$\begin{array}{cc}& |D_{\mathbf{P}}^{\mathit{\sigma }}{\mathrm{\nabla }}^{k}F(\mathbf{y})|,|D_{\mathbf{P}}^{\mathit{\sigma }}{\mathrm{\nabla }}^k(e^F)(\mathbf{y})|\le C{\lambda }_{\mathit{\sigma }}{(\mathbf{y})}^{1-k}{\lambda }_{P_1}{(\mathbf{y})}^{-|{\sigma }_1|}\cdots {\lambda }_{P_M}{(\mathbf{y})}^{-|{\sigma }_M|}\hfill \\ \hfill \end{array}$$ 2.14

$$\begin{array}{cc}& |D_{\mathbf{P}}^{\mathit{\sigma }}{|\mathrm{\nabla }F(\mathbf{y})|}^2|\le C{\lambda }_{P_1}{(\mathbf{y})}^{-|{\sigma }_1|}\cdots {\lambda }_{P_M}{(\mathbf{y})}^{-|{\sigma }_M|}\hfill \end{array}$$ 2.15

for all $\mathbf{y}\in {\mathrm{\Sigma }}_{\mathit{\sigma }}^c$. The bound to the first quantity in (2.14) also holds when F is replaced by $F_c$.

Proof

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

$$\begin{array}{c}\hfill F_c(\mathbf{y})=-\frac{Z}{2}\sum _{1\le j\le N}\tau (y_j)+\frac{1}{4}\sum _{1\le l<m\le N}\tau (y_l-y_m).\end{array}$$ 2.16

For each $r=1,\cdots ,N$, denote by ${\mathrm{\nabla }}_{y_r}$ the three-dimensional gradient in the variable $y_r$. We then have

$$\begin{array}{c}\hfill {\mathrm{\nabla }}_{y_r}{F}_c(\mathbf{y})=-\frac{Z}{2}\mathrm{\nabla }\tau (y_r)+\frac{1}{4}\sum _{\begin{array}{c}m=1\\ m\ne r\end{array}}^N\mathrm{\nabla }\tau (y_r-y_m).\end{array}$$ 2.17

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

$$\begin{array}{cc}\hfill |D_{\mathbf{P}}^{\mathit{\sigma }}{\mathrm{\nabla }}^k{F}_c(\mathbf{y})|& \le C{q}_{\mathit{\sigma }}{(\mathbf{y})}^{1-k-|\mathit{\sigma }|}\le C{\mu }_{\mathit{\sigma }}{(\mathbf{y})}^{1-k-|\mathit{\sigma }|}\hfill \end{array}$$ 2.18

for all $\mathbf{y}\in {\mathrm{\Sigma }}_{\mathit{\sigma }}^c$. In the second inequality, we used that ${\mu }_{\mathit{\sigma }}(\mathbf{y})\le q_{\mathit{\sigma }}(\mathbf{y})$. Recall that $\mathrm{\nabla }{F}_s$ is smooth with all derivatives bounded, see (2.5). Since $F=F_c-F_s$ and ${\mu }_{\mathit{\sigma }}\le 1$, we have that for each $\mathit{\sigma }\ne 0$ and $k=0,1$, there exists C, depending on $\mathit{\sigma }$, such that

$$\begin{array}{c}\hfill |D_{\mathbf{P}}^{\mathit{\sigma }}{\mathrm{\nabla }}^{k}F(\mathbf{y})|\le C{\mu }_{\mathit{\sigma }}{(\mathbf{y})}^{1-k-|\mathit{\sigma }|}\end{array}$$ 2.19

for all $\mathbf{y}\in {\mathrm{\Sigma }}_{\mathit{\sigma }}^c$. 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 $\mathit{\eta }\le \mathit{\sigma }$, we have that ${\mathrm{\Sigma }}_{\mathit{\eta }}\subset {\mathrm{\Sigma }}_{\mathit{\sigma }}$, and therefore, $D_{\mathbf{P}}^{\mathit{\eta }}{\mathrm{\nabla }}^{k}F$ exists and is smooth in ${\mathrm{\Sigma }}_{\mathit{\sigma }}^c$, for $k=0,1$. We use the chain rule for weak derivatives to show that $D_{\mathbf{P}}^{\mathit{\sigma }}(e^F)$ exists in ${\mathrm{\Sigma }}_{\mathit{\sigma }}^c$ and is equal to a sum of terms, each of the form

$$\begin{array}{c}\hfill e^F\prod _{1\le j\le n}{D}_{\mathbf{P}}^{(j)}F\end{array}$$ 2.20

for some $1\le n\le |\mathit{\sigma }|$ and some collection $0\ne {\mathit{\beta }}^{(j)}\in {\mathbb{N}}_0^{3M}$ for $j=1,\cdots ,n$, where ${\mathit{\beta }}^{(1)}+\cdots +{\mathit{\beta }}^{(n)}=\mathit{\sigma }$. For each $j=1,\cdots ,n$, we write ${\mathit{\beta }}^{(j)}=({\beta }_1^{(j)},\cdots ,{\beta }_M^{(j)})$ to denote the ${\mathbb{N}}_0^3$-components of ${\mathit{\beta }}^{(j)}$. Similarly, the weak derivative $D_{\mathbf{P}}^{\mathit{\sigma }}\mathrm{\nabla }(e^F)$ exists in ${\mathrm{\Sigma }}_{\mathit{\sigma }}^c$ and the gradient of the general term (2.20) is equal to

$$\begin{array}{c}\hfill e^F\mathrm{\nabla }F\prod _{1\le j\le n}{D}_{\mathbf{P}}^{(j)}F+e^F\sum _{r=1}^n\left(D_{\mathbf{P}}^{(r)},\mathrm{\nabla },F,,\prod _{\begin{array}{c}1\le j\le n\\ j\ne r\end{array}},D_{\mathbf{P}}^{(j)},F\right).\end{array}$$ 2.21

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

$$\begin{array}{cc}\hfill |\prod _{j=1}^n{D}_{\mathbf{P}}^{(j)}F(\mathbf{y})|& \le C\prod _{j=1}^n{\mu }_{{\mathit{\beta }}^{(j)}}{(\mathbf{y})}^{1-|{\mathit{\beta }}^{(j)}|}\hfill \\ \hfill & \le C\prod _{j=1}^n{\lambda }_{{\mathit{\beta }}^{(j)}}(\mathbf{y}){\lambda }_{P_1}{(\mathbf{y})}^{-|{\beta }_1^{(j)}|}\cdots {\lambda }_{P_M}{(\mathbf{y})}^{-|{\beta }_M^{(j)}|}\hfill \\ \hfill & =C\left(\prod _{j=1}^n,{\lambda }_{{\mathit{\beta }}^{(j)}},(\mathbf{y})\right)\left(\prod _{l=1}^M,{\lambda }_{P_l},{(\mathbf{y})}^{-|{\beta }_l^{(1)}|},\cdots ,{\lambda }_{P_l},{(\mathbf{y})}^{-|{\beta }_l^{(n)}|}\right)\hfill \\ \hfill & \le C{\lambda }_{\mathit{\sigma }}(\mathbf{y}){\lambda }_{P_1}{(\mathbf{y})}^{-|{\sigma }_1|}\cdots {\lambda }_{P_M}{(\mathbf{y})}^{-|{\sigma }_M|}\hfill \end{array}$$

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

$$\begin{array}{cc}\hfill |D_{\mathbf{P}}^{(r)}\mathrm{\nabla }F(\mathbf{y})\prod _{\begin{array}{c}1\le j\le n\\ j\ne r\end{array}}{D}_{\mathbf{P}}^{(j)}F(\mathbf{y})|& \le C{\lambda }_{P_1}{(\mathbf{y})}^{-|{\sigma }_1|}\cdots {\lambda }_{P_M}{(\mathbf{y})}^{-|{\sigma }_M|}\hfill \end{array}$$

for some constant C, depending on $\mathit{\sigma }$, and all $\mathbf{y}\in {\mathrm{\Sigma }}_{\mathit{\sigma }}^c$. Using the fact that $e^F$ and $\mathrm{\nabla }F$ are bounded in ${\mathbb{R}}^{3N}$, 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

$$\begin{array}{c}\hfill D_{\mathbf{P}}^{\mathit{\sigma }}{|\mathrm{\nabla }F|}^2=\sum _{\mathit{\beta }\le \mathit{\sigma }}\left(\genfrac{}{}{0pt}{}{\mathit{\sigma }}{\mathit{\beta }}\right)D_{\mathbf{P}}^{\mathit{\beta }}\mathrm{\nabla }F·D_{\mathbf{P}}^{\mathit{\sigma }-\mathit{\beta }}\mathrm{\nabla }F\end{array}$$

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

$$\begin{array}{cc}\hfill |D_{\mathbf{P}}^{\mathit{\sigma }}{|\mathrm{\nabla }F(\mathbf{y})|}^2|& \le C\sum _{\mathit{\beta }\le \mathit{\sigma }}{\mu }_{\mathit{\beta }}{(\mathbf{y})}^{-|\mathit{\beta }|}{\mu }_{\mathit{\sigma }-\mathit{\beta }}{(\mathbf{y})}^{-|\mathit{\sigma }|+|\mathit{\beta }|}\hfill \end{array}$$

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

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 $\mathbf{x}\in {\mathrm{\Sigma }}_{\mathit{\sigma }}^c$, for $\mathit{\sigma }\ne 0$, then $B(\mathbf{x},\nu {\lambda }_{\mathit{\sigma }}(\mathbf{x}))\subset {\mathrm{\Sigma }}_{\mathit{\sigma }}^c$ for all $\nu \in (0,1)$.

Lemma 2.4

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

$$\begin{array}{cc}& {∥D_{\mathbf{P}}^{\mathit{\sigma }},{\mathrm{\nabla }}^k,F∥}_{L^{\infty }(B(\mathbf{x},\nu {\lambda }_{\mathit{\sigma }}(\mathbf{x})))},{∥D_{\mathbf{P}}^{\mathit{\sigma }},{\mathrm{\nabla }}^k,(e^F)∥}_{L^{\infty }(B(\mathbf{x},\nu {\lambda }_{\mathit{\sigma }}(\mathbf{x})))}\hfill \\ \hfill & \le C{\lambda }_{\mathit{\sigma }}{(\mathbf{x})}^{1-k}{\lambda }_{P_1}{(\mathbf{x})}^{-|{\sigma }_1|}\cdots {\lambda }_{P_M}{(\mathbf{x})}^{-|{\sigma }_M|}\hfill \end{array}$$ 2.22

$$\begin{array}{cc}& {∥D_{\mathbf{P}}^{\mathit{\sigma }},{|\mathrm{\nabla }F|}^2∥}_{L^{\infty }(B(\mathbf{x},\nu {\lambda }_{\mathit{\sigma }}(\mathbf{x})))}\le C{\lambda }_{P_1}{(\mathbf{x})}^{-|{\sigma }_1|}\cdots {\lambda }_{P_M}{(\mathbf{x})}^{-|{\sigma }_M|}\hfill \end{array}$$ 2.23

for all $\mathbf{x}\in {\mathrm{\Sigma }}_{\mathit{\sigma }}^c$. The bound to the first norm in (2.22) also holds when F is replaced by $F_c$.

Proof

Take some $\mathbf{x}\in {\mathrm{\Sigma }}_{\mathit{\sigma }}^c$. By (1.23), for each $\mathbf{y}\in B(\mathbf{x},\nu {\lambda }_{\mathit{\sigma }}(\mathbf{x}))$ we have

$$\begin{array}{c}\hfill |{\lambda }_{P_j}(\mathbf{x})-{\lambda }_{P_j}(\mathbf{y})|\le |\mathbf{x}-\mathbf{y}|\le \nu {\lambda }_{\mathit{\sigma }}(\mathbf{x})\le \nu {\lambda }_{P_j}(\mathbf{x})\end{array}$$

for each $j=1,\cdots ,M$ with ${\sigma }_j\ne 0$. The final inequality above uses the definition of ${\lambda }_{\mathit{\sigma }}$ in (1.26). By rearrangement, we have

$$\begin{array}{c}\hfill (1-\nu ){\lambda }_{P_j}(\mathbf{x})\le {\lambda }_{P_j}(\mathbf{y}).\end{array}$$ 2.24

Therefore,

$$\begin{array}{c}\hfill {\lambda }_{P_1}{(\mathbf{y})}^{-|{\sigma }_1|}\cdots {\lambda }_{P_M}{(\mathbf{y})}^{-|{\sigma }_M|}\le {(1-\nu )}^{-|\mathit{\sigma }|}{\lambda }_{P_1}{(\mathbf{x})}^{-|{\sigma }_1|}\cdots {\lambda }_{P_M}{(\mathbf{x})}^{-|{\sigma }_M|}\end{array}$$ 2.25

for all $\mathbf{y}\in B(\mathbf{x},\nu {\lambda }_{\mathit{\sigma }}(\mathbf{x}))$. We now prove an analogous inequality. First, we see that ${\lambda }_{\mathit{\sigma }}(\mathbf{x})={\lambda }_{P_l}(\mathbf{x})$ for some $1\le l\le M$ with ${\sigma }_l\ne 0$ which will depend on the choice of $\mathbf{x}$. We also note that ${\lambda }_{\mathit{\sigma }}(\mathbf{y})\le {\lambda }_{P_l}(\mathbf{y})$ for all $\mathbf{y}\in {\mathbb{R}}^{3N}$ which follows from ${\sigma }_l\ne 0$ and the definition of ${\lambda }_{\mathit{\sigma }}$. Therefore,

$$\begin{array}{cc}\hfill {\lambda }_{\mathit{\sigma }}(\mathbf{y}){\lambda }_{P_1}{(\mathbf{y})}^{-|{\sigma }_1|}\cdots {\lambda }_{P_M}{(\mathbf{y})}^{-|{\sigma }_M|}& \le {\lambda }_{P_l}{(\mathbf{y})}^{1-|{\sigma }_l|}\prod _{\begin{array}{c}j=1\\ j\ne l\end{array}}^M{\lambda }_{P_j}{(\mathbf{y})}^{-|{\sigma }_j|}\hfill \\ \hfill & \le {(1-\nu )}^{1-|\mathit{\sigma }|}{\lambda }_{P_l}{(\mathbf{x})}^{1-|{\sigma }_l|}\prod _{\begin{array}{c}j=1\\ j\ne l\end{array}}^M{\lambda }_{P_j}{(\mathbf{x})}^{-|{\sigma }_j|}\hfill \\ \hfill & ={(1-\nu )}^{1-|\mathit{\sigma }|}{\lambda }_{\mathit{\sigma }}(\mathbf{x}){\lambda }_{P_1}{(\mathbf{x})}^{-|{\sigma }_1|}\cdots {\lambda }_{P_M}{(\mathbf{x})}^{-|{\sigma }_M|}\hfill \end{array}$$ 2.26

for all $\mathbf{y}\in B(\mathbf{x},\nu {\lambda }_{\mathit{\sigma }}(\mathbf{x}))$. 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. $\square$

Cluster Derivatives of $\varphi$ and $\psi$

We introduce the following notation. For cluster set $\mathbf{P}=(P_1,\cdots ,P_M)$ and $\mathit{\alpha }\in {\mathbb{N}}_0^{3M}$, we define

$$\begin{array}{c}\hfill {\varphi }_{\mathit{\alpha }}=D_{\mathbf{P}}^{\mathit{\alpha }}\varphi .\end{array}$$

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 ${\varphi }_{\mathit{\alpha }}$. By a process of rescaling, using a method developed in [8], we will apply elliptic regularity to give quantative bounds to the $L^{\infty }$-norms of ${\varphi }_{\mathit{\alpha }}$ and $\mathrm{\nabla }{\varphi }_{\mathit{\alpha }}$. From here, it is straightforward to obtain corresponding bounds for $D_{\mathbf{P}}^{\mathit{\alpha }}\psi$ and $D_{\mathbf{P}}^{\mathit{\alpha }}\mathrm{\nabla }\psi$.

Lemma 2.5

For every $|\mathit{\alpha }|\ge 1$, we have ${\varphi }_{\mathit{\alpha }}$ is a weak solution to

$$\begin{array}{cc}& -\mathrm{\Delta }{\varphi }_{\mathit{\alpha }}-2\mathrm{\nabla }F·\mathrm{\nabla }{\varphi }_{\mathit{\alpha }}+(\mathrm{\Delta }{F}_s-{|\mathrm{\nabla }F|}^2-E){\varphi }_{\mathit{\alpha }}\hfill \\ \hfill & =\sum _{\begin{array}{c}\mathit{\sigma }\le \mathit{\alpha }\\ \mathit{\sigma }\ne 0\end{array}}\left(\genfrac{}{}{0pt}{}{\mathit{\alpha }}{\mathit{\sigma }}\right)(2D_{\mathbf{P}}^{\mathit{\sigma }}\mathrm{\nabla }F·\mathrm{\nabla }{\varphi }_{\mathit{\alpha }-\mathit{\sigma }}-D_{\mathbf{P}}^{\mathit{\sigma }}(\mathrm{\Delta }{F}_s-{|\mathrm{\nabla }F|}^2){\varphi }_{\mathit{\alpha }-\mathit{\sigma }})=:g_{\mathit{\alpha }}\hfill \end{array}$$ 2.27

in ${\mathrm{\Sigma }}_{\mathit{\alpha }}^c$, and therefore, ${\varphi }_{\mathit{\alpha }}\in C^1({\mathrm{\Sigma }}_{\mathit{\alpha }}^c)\cap H_{\mathit{loc}}^2({\mathrm{\Sigma }}_{\mathit{\alpha }}^c)$.

Proof

We prove by induction, starting with $|\mathit{\alpha }|=1$. Let $\mathcal{L}(·,·)$ be the bilinear form corresponding to the operator acting on ${\varphi }_{\mathit{\alpha }}$ on the left-hand side of (2.27), as defined in (1.37). Since $\varphi \in H_{\mathit{loc}}^2({\mathbb{R}}^{3N})$, we have ${\varphi }_{\mathit{\alpha }}\in H_{\mathit{loc}}^1({\mathbb{R}}^{3N})$. We use integration by parts to show that, for any $\chi \in C_c^{\infty }({\mathrm{\Sigma }}_{\mathit{\alpha }}^c)$,

$$\begin{array}{c}\hfill \mathcal{L}({\varphi }_{\mathit{\alpha }},\chi )=-\mathcal{L}(\varphi ,D_{\mathbf{P}}^{\mathit{\alpha }}\chi )+{\int }_{{\mathrm{\Sigma }}_{\mathit{\alpha }}^c}{g}_{\mathit{\alpha }}\chi \text{d}\mathbf{x}.\end{array}$$ 2.28

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

Assume the hypothesis holds for all multiindices $1\le |\mathit{\alpha }|\le k-1$ for some $k\ge 2$. Take some arbitrary multiindex $|\mathit{\alpha }|=k$. We will prove the induction hypothesis for $\mathit{\alpha }$. First, we state the useful fact that for any $\mathit{\sigma }\le \mathit{\alpha }$ we have ${\mathrm{\Sigma }}_{\mathit{\sigma }}\subset {\mathrm{\Sigma }}_{\mathit{\alpha }}$. Take some $\mathit{\eta }\le \mathit{\alpha }$ with $|\mathit{\eta }|=1$. Notice, then, that ${\varphi }_{\mathit{\alpha }-\mathit{\eta }}\in H_{\mathit{loc}}^2({\mathrm{\Sigma }}_{\mathit{\alpha }}^c)$ by the induction hypothesis, and hence ${\varphi }_{\mathit{\alpha }}\in H_{\mathit{loc}}^1({\mathrm{\Sigma }}_{\mathit{\alpha }}^c)$. This allows $\mathcal{L}({\varphi }_{\mathit{\alpha }},\chi )$ to be defined for any $\chi \in C_c^{\infty }({\mathrm{\Sigma }}_{\mathit{\alpha }}^c)$. Integration by parts is then used on the $D_{\mathbf{P}}^{\mathit{\eta }}$-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 ${\varphi }_{\mathit{\alpha }}$ as a weak solution. Regularity for ${\varphi }_{\mathit{\alpha }}$ is shown in a similar way to the $|\mathit{\alpha }|=1$ case. Indeed, the induction hypothesis can be used to show $g_{\mathit{\alpha }}\in L_{\mathit{loc}}^{\infty }({\mathrm{\Sigma }}_{\mathit{\alpha }}^c)$. $\square$

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

$$\begin{array}{cc}& {∥\mathrm{\nabla },{\varphi }_{\mathit{\alpha }}∥}_{L^{\infty }(B(\mathbf{x},r{\lambda }_{\mathit{\alpha }}(\mathbf{x})))}\hfill \\ \hfill & \le C{\lambda }_{\mathit{\alpha }}{(\mathbf{x})}^{-1}({∥{\varphi }_{\mathit{\alpha }}∥}_{L^{\infty }(B(\mathbf{x},R{\lambda }_{\mathit{\alpha }}(\mathbf{x})))}+{\lambda }_{\mathit{\alpha }}{(\mathbf{x})}^2{∥g_{\mathit{\alpha }}∥}_{L^{\infty }(B(\mathbf{x},R{\lambda }_{\mathit{\alpha }}(\mathbf{x})))})\hfill \end{array}$$ 2.29

for all $\mathbf{x}\in {\mathrm{\Sigma }}_{\mathit{\alpha }}^c$. The function $g_{\mathit{\alpha }}$ was given by (2.27).

Proof

Fix any $\mathbf{x}\in {\mathrm{\Sigma }}_{\mathit{\alpha }}^c$ and denote $\lambda ={\lambda }_{\mathit{\alpha }}(\mathbf{x})$. The proof proceeds via a rescaling. Define

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill w(\mathbf{y})& ={\varphi }_{\mathit{\alpha }}(\mathbf{x}+\lambda \mathbf{y})\hfill \\ \hfill \mathbf{c}(\mathbf{y})& =-2\mathrm{\nabla }F(\mathbf{x}+\lambda \mathbf{y})\hfill \\ \hfill d(\mathbf{y})& =\mathrm{\Delta }{F}_s(\mathbf{x}+\lambda \mathbf{y})-{|\mathrm{\nabla }F(\mathbf{x}+\lambda \mathbf{y})|}^2-E\hfill \\ \hfill f(\mathbf{y})& =g_{\mathit{\alpha }}(\mathbf{x}+\lambda \mathbf{y})\hfill \end{array}\end{array}$$

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

$$\begin{array}{c}\hfill -\mathrm{\Delta }w+\lambda \mathbf{c}·\mathrm{\nabla }w+{\lambda }^{2}dw={\lambda }^{2}f\end{array}$$ 2.30

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

$$\begin{array}{c}\hfill \lambda {∥\mathbf{c}∥}_{L^{\infty }(B(0,1))}+{\lambda }^2{∥d∥}_{L^{\infty }(B(0,1))}\le K\end{array}$$

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

$$\begin{array}{c}\hfill {∥w∥}_{C^1(\overline{B(0,r)})}\le C({∥w∥}_{L^{\infty }(B(0,R))}+{\lambda }^2{∥f∥}_{L^{\infty }(B(0,R))}).\end{array}$$ 2.31

Finally, we use $\mathrm{\nabla }w(\mathbf{y})=\lambda \mathrm{\nabla }{\varphi }_{\mathit{\alpha }}(\mathbf{x}+\lambda \mathbf{y})$ by the chain rule, and rewrite (2.31) to give (2.29). $\square$

The following proposition uses induction to write estimates for both $\mathrm{\nabla }{\varphi }_{\mathit{\alpha }}$ and ${\varphi }_{\mathit{\alpha }}$ with a bound involving only the zeroth and first-order derivatives of $\varphi$. Corollary A.3 of appendix is used in the proof to improve the regularity of the second derivative of $\varphi$, and this benefit is passed on through induction to all higher orders of derivative. The function $f_{\infty }(·;R;\varphi )$ was defined in (1.29).

Proposition 2.7

For any $|\mathit{\alpha }|\ge 1$, any $0<r<R<1$ and any $0\le b<1$ there exists C, dependent on $\mathit{\alpha },r,R$ and b, such that for $k=0,1$, with $k+|\mathit{\alpha }|\ge 2$,

$$\begin{array}{cc}& {∥{\mathrm{\nabla }}^k,{\varphi }_{\mathit{\alpha }}∥}_{L^{\infty }(B(\mathbf{x},r{\lambda }_{\mathit{\alpha }}(\mathbf{x})))}\hfill \\ \hfill & \le C{\lambda }_{\mathit{\alpha }}{(\mathbf{x})}^{1-k}{\lambda }_{P_1}{(\mathbf{x})}^{-|{\alpha }_1|}\cdots {\lambda }_{P_M}{(\mathbf{x})}^{-|{\alpha }_M|}{\mu }_{\mathit{\alpha }}{(\mathbf{x})}^b{f}_{\infty }(\mathbf{x};R;\varphi )\hfill \end{array}$$

for all $\mathbf{x}\in {\mathrm{\Sigma }}_{\mathit{\alpha }}^c$. The inequality also holds for $|\mathit{\alpha }|=1$ and $k=0$ if we take $b=0$.

Proof

Suppose $|\mathit{\alpha }|=1$ and $k=1$. Let l be the index $1\le l\le M$ such that ${\alpha }_l\ne 0$. Then ${\mu }_{\mathit{\alpha }}={\lambda }_{P_l}$ and ${\mathrm{\Sigma }}_{\mathit{\alpha }}={\mathrm{\Sigma }}_{P_l}$. The bound then follows from Corollary A.3 with $P=P_l$. Indeed, for each $s>0$ we can find some constant $C_s$ such that $log(t)\le C_s{t}^s$ for all $t>0$. We use this fact along with the simple inequality ${\lambda }_{P_l}\le {\nu }_{P_l}$.

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

The $k=0$ case is a straightforward application of the induction hypothesis and is described as follows. Firstly, for each $\mathbf{x}\in {\mathrm{\Sigma }}_{\mathit{\alpha }}^c$ it is clear that there exists some $1\le l\le M$ such that ${\lambda }_{\mathit{\alpha }}(\mathbf{x})={\lambda }_{P_l}(\mathbf{x})$ and where ${\alpha }_l\ne 0$. Therefore, we can find some $\mathit{\eta }\le \mathit{\alpha }$ with $|\mathit{\eta }|=1$ and ${\eta }_j=0$ for each $j\ne l$ where $1\le j\le M$. Now, from the definition of cluster derivatives (1.16),

$$\begin{array}{cc}\hfill {∥{\varphi }_{\mathit{\alpha }}∥}_{L^{\infty }(B(\mathbf{x},r{\lambda }_{\mathit{\alpha }}(\mathbf{x})))}& \le N{∥\mathrm{\nabla },{\varphi }_{\mathit{\alpha }-\mathit{\eta }}∥}_{L^{\infty }(B(\mathbf{x},r{\lambda }_{\mathit{\alpha }}(\mathbf{x})))}.\hfill \end{array}$$ 2.32

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

$$\begin{array}{cc}& {∥\mathrm{\nabla },{\varphi }_{\mathit{\alpha }-\mathit{\eta }}∥}_{L^{\infty }(B(\mathbf{x},r{\lambda }_{\mathit{\alpha }}(\mathbf{x})))}\hfill \\ \hfill & \le C{\lambda }_{P_l}(\mathbf{x}){\lambda }_{P_1}{(\mathbf{x})}^{-|{\alpha }_1|}\cdots {\lambda }_{P_M}{(\mathbf{x})}^{-|{\alpha }_M|}{\mu }_{\mathit{\alpha }-\mathit{\eta }}{(\mathbf{x})}^b{f}_{\infty }(\mathbf{x};R;\varphi ),\hfill \end{array}$$ 2.33

where we can then use ${\lambda }_{P_l}(\mathbf{x})={\lambda }_{\mathit{\alpha }}(\mathbf{x})$ and ${\mu }_{\mathit{\alpha }-\mathit{\eta }}(\mathbf{x})\le {\mu }_{\mathit{\alpha }}(\mathbf{x})$ to complete the required bound for our choice of $\mathit{\alpha }$ 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 $g_{\mathit{\alpha }}$, (2.27), along with (2.5) and Lemma 2.4 we can find C, depending on $\mathit{\alpha }$ and $r^{\prime }$, such that

$$\begin{array}{cc}\hfill {∥g_{\mathit{\alpha }}∥}_{L^{\infty }(B(\mathbf{x},r^{\prime }{\lambda }_{\mathit{\alpha }}(\mathbf{x})))}& \le C\sum _{\begin{array}{c}\mathit{\sigma }\le \mathit{\alpha }\\ \mathit{\sigma }\ne 0\end{array}}{\lambda }_{P_1}{(\mathbf{x})}^{-|{\sigma }_1|}\cdots {\lambda }_{P_M}{(\mathbf{x})}^{-|{\sigma }_M|}({∥{\varphi }_{\mathit{\alpha }-\mathit{\sigma }}∥}_{L^{\infty }(B(\mathbf{x},r^{\prime }{\lambda }_{\mathit{\alpha }}(\mathbf{x})))}\hfill \\ \hfill & +{∥\mathrm{\nabla },{\varphi }_{\mathit{\alpha }-\mathit{\sigma }}∥}_{L^{\infty }(B(\mathbf{x},r^{\prime }{\lambda }_{\mathit{\alpha }}(\mathbf{x})))}).\hfill \end{array}$$

It follows by the induction hypothesis, using $b=0$, along with the definition of $f_{\infty }$, (1.30), that we can find some C, depending on $\mathit{\alpha },r^{\prime }$ and R, such that

$$\begin{array}{cc}& {∥g_{\mathit{\alpha }}∥}_{L^{\infty }(B(\mathbf{x},r^{\prime }{\lambda }_{\mathit{\alpha }}(\mathbf{x})))}\hfill \\ \hfill & \le C{\lambda }_{P_1}{(\mathbf{x})}^{-|{\alpha }_1|}\cdots {\lambda }_{P_M}{(\mathbf{x})}^{-|{\alpha }_M|}{f}_{\infty }(\mathbf{x};R;\varphi )\hfill \end{array}$$ 2.34

for all $\mathbf{x}\in {\mathrm{\Sigma }}_{\mathit{\alpha }}^c$. Next, we apply Lemma 2.6 to obtain some constant C, depending on r and $r^{\prime }$, such that

$$\begin{array}{cc}\hfill {∥\mathrm{\nabla },{\varphi }_{\mathit{\alpha }}∥}_{L^{\infty }(B(\mathbf{x},r{\lambda }_{\mathit{\alpha }}(\mathbf{x})))}& \le C({\lambda }_{\mathit{\alpha }}{(\mathbf{x})}^{-1}{∥{\varphi }_{\mathit{\alpha }}∥}_{L^{\infty }(B(\mathbf{x},r^{\prime }{\lambda }_{\mathit{\alpha }}(\mathbf{x})))}\hfill \\ \hfill & +{\lambda }_{\mathit{\alpha }}(\mathbf{x}){∥g_{\mathit{\alpha }}∥}_{L^{\infty }(B(\mathbf{x},r^{\prime }{\lambda }_{\mathit{\alpha }}(\mathbf{x})))}).\hfill \end{array}$$

We use the $k=0$ case, proven above, to bound the $L^{\infty }$-norm of ${\varphi }_{\mathit{\alpha }}$ in the first term on the right-hand side of the above inequality. To the second term, we apply (2.34) and the simple inequality ${\lambda }_{\mathit{\alpha }}(\mathbf{x})\le {\mu }_{\mathit{\alpha }}(\mathbf{x})$. Together, these give the required bound for $k=1$ and our choice of $\mathit{\alpha }$. This completes the induction. $\square$

We now obtain bounds to the cluster derivatives of the eigenfunction $\psi$ using those for $\varphi$ in the above proposition.

Proof of Theorem 1.3

It is clear that (1.34) holds when $\mathit{\alpha }=0$. Therefore, consider $|\mathit{\alpha }|\ge 1$. We first prove (1.34) for $k=0$. Take $\mathbf{x}\in {\mathrm{\Sigma }}_{\mathit{\alpha }}^c$, then by the Leibniz rule for cluster derivatives we have

$$\begin{array}{cc}\hfill D_{\mathbf{P}}^{\mathit{\alpha }}\psi & =\sum _{\mathit{\beta }\le \mathit{\alpha }}\left(\genfrac{}{}{0pt}{}{\mathit{\alpha }}{\mathit{\beta }}\right)D_{\mathbf{P}}^{\mathit{\beta }}(e^F){\varphi }_{\mathit{\alpha }-\mathit{\beta }}\hfill \end{array}$$ 2.35

in $B(\mathbf{x},r{\lambda }_{\mathit{\alpha }}(\mathbf{x}))$. Now, for each $\mathit{\beta }\le \mathit{\alpha }$ there exists some C, independent of $\mathbf{x}$, such that

$$\begin{array}{c}\hfill {∥D_{\mathbf{P}}^{\mathit{\beta }},(,e^F,),{\varphi }_{\mathit{\alpha }-\mathit{\beta }}∥}_{L^{\infty }(B(\mathbf{x},r{\lambda }_{\mathit{\alpha }}(\mathbf{x})))}\le C{\lambda }_{\mathit{\alpha }}(\mathbf{x}){\lambda }_{P_1}{(\mathbf{x})}^{-|{\alpha }_1|}\cdots {\lambda }_{P_M}{(\mathbf{x})}^{-|{\alpha }_M|}{f}_{\infty }(\mathbf{x};R;\varphi ).\end{array}$$

To prove this, we use Lemma 2.4 and Proposition 2.7 with $b=0$. In addition, if $\mathit{\beta }=0$ we use (2.6), and if $\mathit{\beta }=\mathit{\alpha }$, we use the definition (1.30). If $\mathit{\beta }\notin \{0,\mathit{\alpha }\}$, we use the inequality ${\lambda }_{\mathit{\beta }}(\mathbf{x}){\lambda }_{\mathit{\alpha }-\mathit{\beta }}(\mathbf{x})\le {\lambda }_{\mathit{\alpha }}(\mathbf{x})$, which is obtained from Lemma 2.1. The required bound (1.34) for $k=0$ then follows from (2.35) and (2.9)–(2.10).

Take $\mathbf{x}\in {\mathrm{\Sigma }}_{\mathit{\alpha }}^c$. By the definition $F=F_c-F_s$, the equality $\mathrm{\nabla }\psi =\psi \mathrm{\nabla }F+e^F\mathrm{\nabla }\varphi$ and the Leibniz rule, we have (1.35) with

$$\begin{array}{cc}\hfill G_{\mathbf{P}}^{\mathit{\alpha }}& =\sum _{\begin{array}{c}\mathit{\beta }\le \mathit{\alpha }\\ |\mathit{\beta }|\ge 1\end{array}}\left(\genfrac{}{}{0pt}{}{\mathit{\alpha }}{\mathit{\beta }}\right)D_{\mathbf{P}}^{\mathit{\beta }}\psi D_{\mathbf{P}}^{\mathit{\alpha }-\mathit{\beta }}\mathrm{\nabla }F+\sum _{\mathit{\beta }\le \mathit{\alpha }}\left(\genfrac{}{}{0pt}{}{\mathit{\alpha }}{\mathit{\beta }}\right)\mathrm{\nabla }{\varphi }_{\mathit{\beta }}{D}_{\mathbf{P}}^{\mathit{\alpha }-\mathit{\beta }}(e^F)-\psi D_{\mathbf{P}}^{\mathit{\alpha }}\mathrm{\nabla }{F}_s.\hfill \end{array}$$ 2.36

We bound each term in $G_{\mathbf{P}}^{\mathit{\alpha }}$ as in (1.36). Take any $\mathit{\beta }\le \mathit{\alpha }$ with $|\mathit{\beta }|\ge 1$. Notice that ${\mathrm{\Sigma }}_{\mathit{\beta }}\subset {\mathrm{\Sigma }}_{\mathit{\alpha }}$ and hence ${\lambda }_{\mathit{\alpha }}(\mathbf{x})\le {\lambda }_{\mathit{\beta }}(\mathbf{x})$. Now, there exists C, independent of $\mathbf{x}$, such that

$$\begin{array}{cc}\hfill {∥D_{\mathbf{P}}^{\mathit{\beta }},\psi ,,D_{\mathbf{P}}^{\mathit{\alpha }-\mathit{\beta }},\mathrm{\nabla },F∥}_{L^{\infty }(B(\mathbf{x},r{\lambda }_{\mathit{\alpha }}(\mathbf{x})))}& \le C{\lambda }_{\mathit{\beta }}(\mathbf{x}){\lambda }_{P_1}{(\mathbf{x})}^{-|{\alpha }_1|}\cdots {\lambda }_{P_M}{(\mathbf{x})}^{-|{\alpha }_M|}{f}_{\infty }(\mathbf{x};R).\hfill \end{array}$$ 2.37

To prove this inequality, we bound derivatives of F using Lemma 2.4 if $\mathit{\beta }\ne \mathit{\alpha }$ and (2.6) if $\mathit{\beta }=\mathit{\alpha }$. The derivatives $D_{\mathbf{P}}^{\mathit{\beta }}\psi$ are then bounded using (1.34) with $k=0$, which was proven above. The right-hand side of (2.37) can be bounded as in (1.36) by using the simple bound ${\lambda }_{\mathit{\beta }}\le {\mu }_{\mathit{\alpha }}$. Next, still considering $|\mathit{\beta }|\ge 1$ we can find C, independent of $\mathbf{x}$, such that

$$\begin{array}{cc}& {∥\mathrm{\nabla },{\varphi }_{\mathit{\beta }},,D_{\mathbf{P}}^{\mathit{\alpha }-\mathit{\beta }},(,e^F,)∥}_{L^{\infty }(B(\mathbf{x},r{\lambda }_{\mathit{\alpha }}(\mathbf{x})))}\hfill \\ \hfill & \le C{\mu }_{\mathit{\beta }}{(\mathbf{x})}^b{\lambda }_{\mathit{\alpha }-\mathit{\beta }}(\mathbf{x}){\lambda }_{P_1}{(\mathbf{x})}^{-|{\alpha }_1|}\cdots {\lambda }_{P_M}{(\mathbf{x})}^{-|{\alpha }_M|}{f}_{\infty }(\mathbf{x};R;\varphi ).\hfill \end{array}$$

This is proven using Proposition 2.7 to bound the derivatives $\mathrm{\nabla }{\varphi }_{\mathit{\beta }}$, 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 (1.36) after use of (2.9)–(2.10) and the inequalities ${\mu }_{\mathit{\beta }}\le {\mu }_{\mathit{\alpha }}$ and ${\lambda }_{\mathit{\alpha }-\mathit{\beta }}\le 1$. In addition, by Lemma 2.4 there exists C such that

$$\begin{array}{cc}\hfill {∥\mathrm{\nabla },\varphi ,,D_{\mathbf{P}}^{\mathit{\alpha }},(,e^F,)∥}_{L^{\infty }(B(\mathbf{x},r{\lambda }_{\mathit{\alpha }}(\mathbf{x})))}& \le C{\lambda }_{\mathit{\alpha }}(\mathbf{x}){\lambda }_{P_1}{(\mathbf{x})}^{-|{\alpha }_1|}\cdots {\lambda }_{P_M}{(\mathbf{x})}^{-|{\alpha }_M|}{f}_{\infty }(\mathbf{x};R;\varphi ).\hfill \end{array}$$

Use of (2.9)-(2.10), as before, and the inequality ${\lambda }_{\mathit{\alpha }}\le {\mu }_{\mathit{\alpha }}$ give the correct bound. Finally, the last term in (2.36) is readily bounded appropriately using (2.5). In view of (2.36), this completes the proof of (1.36).

To prove (1.34) for $k=1$, we use (1.35), (1.36) and the following inequality

$$\begin{array}{c}\hfill {∥\psi ,D_{\mathbf{P}}^{\mathit{\alpha }},\mathrm{\nabla },F_c∥}_{L^{\infty }(B(\mathbf{x},r{\lambda }_{\mathit{\alpha }}(\mathbf{x})))}\le C{\lambda }_{P_1}{(\mathbf{x})}^{-|{\alpha }_1|}\cdots {\lambda }_{P_M}{(\mathbf{x})}^{-|{\alpha }_M|}{f}_{\infty }(\mathbf{x};R)\end{array}$$

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

Cutoffs

In the proof of Theorem 1.1, partial derivatives of the density matrix will become cluster derivatives of $\psi$ under the integral. In this section, we introduce cutoff functions which will included in the density matrix. These can be made to form a partition of unity and are present to ensure that the cluster derivatives of $\psi$ are supported only away from their singularities.

The cutoffs introduced in this section are an extension of similar cutoffs used in [12, 14] and [9]. They involve the variables (i.e. “particles”) x, y and $x_j$, $2\le j\le N$. For a given cutoff, we will define corresponding clusters P and S. Particles in P (or S) will be close to x (or y, respectively) on the support of the cutoff. The cluster P (or S) will eventually facilitate taking x- (or y-)derivatives of the density matrix.

We will also need to take derivatives of the density matrix in the variable $x+y$. For this, it will be natural to consider a larger cluster Q which contains both P and S, but can include other particles. 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.

Definition of $\mathrm{\Phi }$

To begin, take some $\xi \in C_c^{\infty }(\mathbb{R})$, $0\le \xi (s)\le 1$, with

$$\begin{array}{cc}\hfill \xi (s)& =\left\{\begin{array}{cc}1\hfill & \text{if}|s|\le 1\hfill \\ 0\hfill & \text{if}|s|\ge 2.\hfill \end{array}\right)\hfill \end{array}$$ 3.1

for $s\in \mathbb{R}$. For each $t>0$, we can define the following two cutoff factors ${\zeta }_t=C_c^{\infty }({\mathbb{R}}^3)$ and ${\theta }_t\in C^{\infty }({\mathbb{R}}^3)$ by

$$\begin{array}{c}\hfill {\zeta }_t(z)=\xi (\frac{4N|z|}{t}),{\theta }_t(z)=1-{\zeta }_t(z),z\in {\mathbb{R}}^3.\end{array}$$ 3.2

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

• if ${\zeta }_t(z)\ne 0$ then ${|z|<(2N)}^{-1}t$,

• if ${\theta }_t(z)\ne 0$ then ${|z|>(4N)}^{-1}t$.

Take any $\delta ,ϵ$ with $0<2\delta \le ϵ$. We define our cutoff, which depends on $\delta$ and $ϵ$ as parameters, as a function $\mathrm{\Phi }={\mathrm{\Phi }}_{\delta ,ϵ}(x,y,\widehat{\mathbf{x}})$ given by

$$\begin{array}{c}\hfill {\mathrm{\Phi }}_{\delta ,ϵ}(x,y,\widehat{\mathbf{x}})=\prod _{2\le j\le N}{g}_j^{(1)}(x-x_j)\prod _{2\le j\le N}{g}_j^{(2)}(y-x_j)\prod _{2\le k<l\le N}{f}_{k,l}(x_k-x_l)\end{array}$$ 3.3

for $x,y\in {\mathbb{R}}^3$ and $\widehat{\mathbf{x}}\in {\mathbb{R}}^{3N-3}$, and where $g_j^{(1)},g_j^{(2)},f_{k,l}\in \{{\zeta }_{\delta },{\theta }_{\delta }{\zeta }_{ϵ},{\theta }_{ϵ}\}$ for $2\le j\le N$ and $2\le k<l\le N$. In addition, for $2\le k<l\le N$ we define $f_{l,k}=f_{k,l}$.

The idea is that on the support of $\mathrm{\Phi }$ 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 on the support of ${\zeta }_{\delta }(x_j-x_k)$ the two particles are separated by a distance bounded above by a quantity proportional to $\delta$. Similarly, on the support of ${\theta }_{ϵ}(x_j-x_k)$ they are separated by a distance bounded below by a quantity proportional to $ϵ$. Finally, on the support of $({\theta }_{\delta }{\zeta }_{ϵ})(x_j-x_k)$ the distance will be in an intermediate range between two quantities scaling as $\delta$ and $ϵ$, respectively.

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

Lemma 3.1

There exists a collection ${\{{\mathrm{\Phi }}^{(j)}\}}_{j=1}^J$ of cutoffs (3.3), with integer J depending only on N, such that whenever $0<2\delta \le ϵ$ we have

$$\begin{array}{c}\hfill \sum _{j=1}^J{\mathrm{\Phi }}^{(j)}(x,y,\widehat{\mathbf{x}})=1\end{array}$$

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

Proof

From the definitions (3.2) and support criteria, it is straightforward to verify that

$$\begin{array}{c}\hfill {\theta }_{\delta }(z){\theta }_{ϵ}(z)={\theta }_{ϵ}(z),{\zeta }_{\delta }(z){\theta }_{ϵ}(z)=0,{\zeta }_{\delta }(z){\zeta }_{ϵ}(z)={\zeta }_{\delta }(z)\end{array}$$ 3.4

for all $z\in {\mathbb{R}}^3$. It follows that on ${\mathbb{R}}^3$,

$$\begin{array}{c}\hfill 1\equiv ({\zeta }_{\delta }+{\theta }_{\delta })({\zeta }_{ϵ}+{\theta }_{ϵ})={\zeta }_{\delta }+{\theta }_{\delta }{\zeta }_{ϵ}+{\theta }_{ϵ}.\end{array}$$ 3.5

For any $A,B\subset \{2,\cdots ,N\}$ with $A\cap B=\mathrm{\varnothing }$, we define

$$\begin{array}{c}\hfill {\tau }_{A,B}(\mathbf{x})=\prod _{j\in A}{\zeta }_{\delta }(x_1-x_j)\prod _{j\in B}({\theta }_{\delta }{\zeta }_{ϵ})(x_1-x_j)\prod _{j\in \{2,\cdots ,N\}\backslash (A\cup B)}{\theta }_{ϵ}(x_1-x_j)\end{array}$$

for $\mathbf{x}=(x_1,\cdots ,x_N)\in {\mathbb{R}}^{3N}$. Therefore,

$$\begin{array}{c}\hfill \sum _{\begin{array}{c}A\subset \{2,\cdots ,N\}\\ B\subset \{2,\cdots ,N\}\backslash A\end{array}}{\tau }_{A,B}(\mathbf{x})=\prod _{2\le j\le N}\{{\zeta }_{\delta }(x_1-x_j)+({\theta }_{\delta }{\zeta }_{ϵ})(x_1-x_j)+{\theta }_{ϵ}(x_1-x_j)\}=1\end{array}$$

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

$$\begin{array}{c}\hfill T_{Y,Z}(\widehat{\mathbf{x}})=\prod _{(j,k)\in Y}{\zeta }_{\delta }(x_j-x_k)\prod _{(j,k)\in Z}({\theta }_{\delta }{\zeta }_{ϵ})(x_j-x_k)\prod _{(j,k)\in \mathrm{\Xi }\backslash (Y\cup Z)}{\theta }_{ϵ}(x_j-x_k)\end{array}$$

for $\widehat{\mathbf{x}}=(x_2,\cdots ,x_N)\in {\mathbb{R}}^{3N-3}$. Therefore,

$$\begin{array}{c}\hfill \sum _{\begin{array}{c}Y\subset \mathrm{\Xi }\\ Z\subset \mathrm{\Xi }\backslash Y\end{array}}{T}_{Y,Z}(\widehat{\mathbf{x}})=\prod _{2\le j<k\le N}({\zeta }_{\delta }(x_j-x_k)+({\theta }_{\delta }{\zeta }_{ϵ})(x_j-x_k)+{\theta }_{ϵ}(x_j-x_k))=1\end{array}$$

for all $\widehat{\mathbf{x}}\in {\mathbb{R}}^{3N-3}$. Overall, for all $x,y\in {\mathbb{R}}^3$, $\widehat{\mathbf{x}}\in {\mathbb{R}}^{3N-3}$,

$$\begin{array}{c}\hfill \sum _{\begin{array}{c}A\subset \{2,\cdots ,N\}\\ B\subset \{2,\cdots ,N\}\backslash A\end{array}}\sum _{\begin{array}{c}C\subset \{2,\cdots ,N\}\\ D\subset \{2,\cdots ,N\}\backslash C\end{array}}\sum _{\begin{array}{c}Y\subset \mathrm{\Xi }\\ Z\subset \mathrm{\Xi }\backslash Y\end{array}}{\tau }_{A,B}(x,\widehat{\mathbf{x}}){\tau }_{C,D}(y,\widehat{\mathbf{x}})T_{Y,Z}(\widehat{\mathbf{x}})=1,\end{array}$$

which is a sum of cutoffs of the form (3.3). $\square$

Clusters Corresponding to $\mathrm{\Phi }$

We start with a definition. We use the term index set for a subset $\mathsf{I}\subset \{(j,k)\in {\{1,\cdots ,N\}}^2:j\ne k\}$. Given one such $\mathsf{I}$, we say that two indices $j,k\in \{1,\cdots ,N\}$ are $\mathsf{I}$-linked if either $j=k$, or $(j,k)\in \mathsf{I}$, or there exist pairwise distinct indices $j_1,\cdots ,j_s$ for $1\le s\le N-2$, all distinct from j and k such that $(j,j_1),(j_1,j_2),\cdots ,(j_s,k)\in \mathsf{I}$.

To each $\mathrm{\Phi }$, we now introduce three corresponding clusters. To define these, we first introduce index sets $\mathsf{L}$, $\mathsf{J}$ and $\mathsf{K}$ based solely on the choice of cutoff factors $g_j^{(1)},g_j^{(2)}$ and $f_{k,l}$ in the definition (3.3) of $\mathrm{\Phi }$. The index sets and clusters are therefore not dependent on $x,y,\widehat{\mathbf{x}}$ nor on the scaling parameters $\delta$ and $ϵ$. In the following, we fix some $\mathrm{\Phi }$.

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

• We have $(j,k)\in \mathsf{L}$ if $f_{j,k}\ne {\theta }_{ϵ}$ for $j,k=2,\cdots ,N$. Also $(1,j),(j,1)\in \mathsf{L}$ if $(g_j^{(1)},g_j^{(2)})\ne ({\theta }_{ϵ},{\theta }_{ϵ})$ for $j=2,\cdots ,N$.

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

• We have $(j,k)\in \mathsf{J}$ if $f_{j,k}={\zeta }_{\delta }$ for $j,k=2,\cdots ,N$. Also $(1,j),(j,1)\in \mathsf{J}$ if $g_j^{(1)}={\zeta }_{\delta }$ for $j=2,\cdots ,N$.

• We have $(j,k)\in \mathsf{K}$ if $f_{j,k}={\zeta }_{\delta }$ for $j,k=2,\cdots ,N$. Also $(1,j),(j,1)\in \mathsf{K}$ if $g_j^{(2)}={\zeta }_{\delta }$ for $j=2,\cdots ,N$.

These index sets obey $\mathsf{J},\mathsf{K}\subset \mathsf{L}$.

The clusters corresponding to $\mathrm{\Phi }$ are then defined as follows. The cluster $Q=Q(\mathrm{\Phi })$ is the set of all indices $\mathsf{L}$-linked to 1, the cluster $P=P(\mathrm{\Phi })$ is the set of all indices $\mathsf{J}$-linked to 1, and finally, the cluster $S=S(\mathrm{\Phi })$ is the set of all indices $\mathsf{K}$-linked to 1. Since $\mathsf{J},\mathsf{K}\subset \mathsf{L}$ we see that $P,S\subset Q$. We remark that if $g_j^{(1)}$, $g_j^{(2)}$ and $f_{j,k}$ all equal ${\theta }_{ϵ}$, then $\mathsf{L}$, $\mathsf{J}$ and $\mathsf{K}$ are all empty and therefore $P=S=Q=\{1\}$. In fact, the clusters P, S and Q always contain 1.

Support of $\mathrm{\Phi }$

The following lemma shows that on the support of $\mathrm{\Phi }$, the cluster $P^{\ast }:=P\backslash \{1\}$ represents a set of particles whose positions $x_j$, $j\in P^{\ast }$, are close to x, and the cluster $S^{\ast }:=S\backslash \{1\}$ represents a set of particles whose positions $x_j$, $j\in S^{\ast }$, are close to y. In both cases, this closeness is with respect to the parameter $\delta$. On the support of $\mathrm{\Phi }$, the cluster $Q^{\ast }:=Q\backslash \{1\}$ (recall $P,S\subset Q$) represents a set of particles whose positions $x_j$, $j\in Q^{\ast }$, are close to both x and y, albeit held potentially looser since this closeness is with respect to the larger parameter $ϵ$.

Lemma 3.2

We have $\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})=0$ unless

$$\begin{array}{ccc}\hfill |x-x_k|,|y-x_k|,|x_j-x_k|& >{(4N)}^{-1}ϵ\hfill & \hfill j\in Q^{\ast },k\in Q^c,\end{array}$$ 3.6

$$\begin{array}{ccc}\hfill |x-x_j|,|y-x_k|& >{(4N)}^{-1}\delta \hfill & \hfill j\in P^c,k\in S^c,\end{array}$$ 3.7

$$\begin{array}{ccc}\hfill |x_r-x_s|& >{(4N)}^{-1}\delta \hfill & \hfill r\in P_{}^{\ast },s\in P^c\text{or}r\in S_{}^{\ast },s\in S^c.\end{array}$$ 3.8

Proof

Take $j\in Q^{\ast }$ and $k\in Q^c$. By the definition of the cluster Q, we have $f_{j,k}={\theta }_{ϵ}$ and $g_k^{(1)}=g_k^{(2)}={\theta }_{ϵ}$ in the formula (3.3) defining $\mathrm{\Phi }$. If we assume $\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\ne 0$, then the support criteria for ${\theta }_{ϵ}(x-x_k),{\theta }_{ϵ}(y-x_k)$ and ${\theta }_{ϵ}(x_j-x_k)$ give (3.6). The inequalities (3.7) and (3.8) are proven in a similar way. $\square$

Lemma 3.3

Let $0<2\delta \le ϵ$. Then $\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})=0$ unless

$$\begin{array}{c}\hfill |x-x_k|<\delta /2\text{for}k\in P^{\ast },\end{array}$$ 3.9

$$\begin{array}{c}\hfill |y-x_k|<\delta /2\text{for}k\in S^{\ast },\end{array}$$ 3.10

and if $|x-y|\ge \delta$ then $\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})=0$ unless

$$\begin{array}{c}\hfill |y-x_k|>\delta /2\text{for}k\in P^{\ast },\end{array}$$ 3.11

$$\begin{array}{c}\hfill |x-x_k|>\delta /2\text{for}k\in S^{\ast }.\end{array}$$ 3.12

Therefore, if $P^{\ast }\cap S^{\ast }\ne \mathrm{\varnothing }$ then $\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})=0$ for all $|x-y|\ge \delta$ and $\widehat{\mathbf{x}}\in {\mathbb{R}}^{3N-3}$.

Furthermore, if $|x|,|y|\ge ϵ$, then $\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})=0$ unless

$$\begin{array}{c}\hfill min\{|x-x_k|,|y-x_k|\}<ϵ/2\text{for}k\in Q^{\ast },\end{array}$$ 3.13

$$\begin{array}{c}\hfill |x_k|>ϵ/2\text{for}k\in Q^{\ast }.\end{array}$$ 3.14

Proof

By definition, if $k\in P_{}^{\ast }$ either $g_k^{(1)}={\zeta }_{\delta }$ or there exist pairwise distinct $j_1,\cdots ,j_s\in \{2,\cdots ,N\}$ with $1\le s\le N-2$ such that $g_1^{(1)}={\zeta }_{\delta }$ and $f_{j_1,j_2},f_{j_2,j_3},\cdots ,f_{j_s,k}={\zeta }_{\delta }$. 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 $|x-x_{j_1}|,|x_{j_1}-x_{j_2}|,\cdots ,|x_{j_s}-x_k{|<(2N)}^{-1}\delta$ and so by the triangle inequality $|x-x_k|<\delta /2$. Now, $|y-x_k|\ge |x-y|-|x-x_k|>\delta /2$ by the reverse triangle inequality. The case of $k\in S^{\ast }$ is analogous.

Now let $k\in Q^{\ast }$. First, we consider the case where either $g_k^{(1)}\ne {\theta }_{ϵ}$ or $g_k^{(2)}\ne {\theta }_{ϵ}$, or both. Without loss, assume $g_k^{(1)}\ne {\theta }_{ϵ}$. Then either $g_k^{(1)}={\zeta }_{\delta }$ or $g_k^{(1)}={\theta }_{\delta }{\zeta }_{ϵ}$, but both give the inequality

$$\begin{array}{c}\hfill |x-x_k{|<(2N)}^{-1}ϵ\end{array}$$

by support criteria. Hence, (3.13) holds in this case. Now suppose that $g_k^{(1)}=g_k^{(2)}={\theta }_{ϵ}$. Then there must exist pairwise distinct $j_1,\cdots ,j_s\in \{2,\cdots ,N\}$ with $1\le s\le N-2$ such that $f_{j_1,j_2},f_{j_2,j_3},\cdots ,f_{j_s,k}\ne {\theta }_{ϵ}$ and either $g_1^{(1)}\ne {\theta }_{ϵ}$ or $g_1^{(2)}\ne {\theta }_{ϵ}$. Without loss, take $g_1^{(1)}\ne {\theta }_{ϵ}$, so that, as before, $|x-x_{j_1}{|<(2N)}^{-1}ϵ$. Similarly, we get $|x_{j_1}-x_{j_2}|,\cdots ,|x_{j_s}-x_k{|<(2N)}^{-1}ϵ$. Therefore, by the triangle inequality, $|x-x_k|<ϵ/2$, hence completing the proof of (3.13). Finally, (3.14) follows from (3.13) and $|x|,|y|\ge ϵ$. $\square$

Factorisation of Cutoffs

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

$$\begin{array}{c}\hfill {\mathrm{\Phi }}^{\prime }(x,y,\widehat{\mathbf{x}})=\prod _{j\in R_1}{g}_j^{(1)}(x-x_j)\prod _{j\in R_2}{g}_j^{(2)}(y-x_j)\prod _{(k,l)\in R_3}{f}_{k,l}(x_k-x_l)\end{array}$$ 3.15

where $R_1,R_2\subset \{2,\cdots ,N\}$, $R_3\subset \{(k,l):2\le k<l\le N\}$.

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

$$\begin{array}{cc}\hfill \mathrm{\Phi }(x,y,\widehat{\mathbf{x}};T)& =\prod _{j\in T^{\ast }}{g}_j^{(1)}(x-x_j)\prod _{j\in T^{\ast }}{g}_j^{(2)}(y-x_j)\prod _{\begin{array}{c}k,l\in T^{\ast }\\ k<l\end{array}}{f}_{k,l}(x_k-x_l),\hfill \end{array}$$ 3.16

$$\begin{array}{cc}\hfill \mathrm{\Phi }(x,y,\widehat{\mathbf{x}};T^c)& =\prod _{\begin{array}{c}k,l\in T^c\\ k<l\end{array}}{f}_{k,l}(x_k-x_l),\hfill \end{array}$$ 3.17

$$\begin{array}{cc}\hfill \mathrm{\Phi }(x,y,\widehat{\mathbf{x}};T,T^c)& =\prod _{j\in T^c}{g}_j^{(1)}(x-x_j)\prod _{j\in T^c}{g}_j^{(2)}(y-x_j)\prod _{\begin{array}{c}k\in T^{\ast }\\ l\in T^c\end{array}}{f}_{k,l}(x_k-x_l).\hfill \end{array}$$ 3.18

Formulae (3.16) and (3.17) are separate definitions in the cases where the cluster contains 1 and does not contain 1, respectively.

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

Lemma 3.4

Given a cutoff $\mathrm{\Phi }$ of the form (3.3) and any cluster T with $1\in T$, we have

$$\begin{array}{c}\hfill \mathrm{\Phi }(x,y,\widehat{\mathbf{x}})=\mathrm{\Phi }(x,y,\widehat{\mathbf{x}};T)\mathrm{\Phi }(x,y,\widehat{\mathbf{x}};T,T^c)\mathrm{\Phi }(x,y,\widehat{\mathbf{x}};T^c).\end{array}$$ 3.19

Proof

This identity follows from the definitions (3.16)–(3.18) and the equality

$$\begin{array}{c}\hfill \prod _{2\le k<l\le N}{f}_{k,l}(x_k-x_l)=\prod _{\begin{array}{c}k,l\in T^{\ast }\\ k<l\end{array}}{f}_{k,l}(x_k-x_l)\prod _{\begin{array}{c}k\in T^{\ast }\\ l\in T^c\end{array}}{f}_{k,l}(x_k-x_l)\prod _{\begin{array}{c}k,l\in T^c\\ k<l\end{array}}{f}_{k,l}(x_k-x_l),\end{array}$$

since for any $2\le k<l\le N$ we have $f_{k,l}=f_{l,k}$ by definition. $\square$

Let $Q=Q(\mathrm{\Phi })$. Then the partial product $\mathrm{\Phi }(·;Q,Q^c)$ consists of only ${\theta }_{ϵ}$ cutoff factors, as shown in the following lemma.

Lemma 3.5

For $Q=Q(\mathrm{\Phi })$,

$$\begin{array}{c}\hfill \mathrm{\Phi }(x,y,\widehat{\mathbf{x}};Q,Q^c)=\prod _{j\in Q^c}{\theta }_{ϵ}(x-x_j)\prod _{j\in Q^c}{\theta }_{ϵ}(y-x_j)\prod _{\begin{array}{c}k\in Q^{\ast }\\ l\in Q^c\end{array}}{\theta }_{ϵ}(x_k-x_l).\end{array}$$ 3.20

Proof

By the definition of the cluster $Q(\mathrm{\Phi })$, we have $g_j^{(1)}=g_j^{(2)}={\theta }_{ϵ}$ for each $j\in Q^c$, and $f_{k,l}={\theta }_{ϵ}$ for each $k\in Q^{\ast }$ and $l\in Q^c$. The formula then follows from (3.18). $\square$

Derivatives

The definition of cluster derivatives, (1.15), is modestly extended to allow action on cutoffs. For any cluster T, we define the following three cluster derivatives which can act on functions of x, y and $\widehat{\mathbf{x}}$, such as $\mathrm{\Phi }$. We set

$$\begin{array}{cc}& D_{x,T}^{\alpha }={\partial }_x^{\alpha }+\sum _{j\in T^{\ast }}{\partial }_j^{\alpha }\text{and}{D}_{y,T}^{\alpha }={\partial }_y^{\alpha }+\sum _{j\in T^{\ast }}{\partial }_j^{\alpha }\text{for}\alpha \in {\mathbb{N}}_0^3,|\alpha |=1,\hfill \end{array}$$ 3.21

$$\begin{array}{cc}& D_{x,y,T}^{\alpha }={\partial }_x^{\alpha }+{\partial }_y^{\alpha }+\sum _{j\in T^{\ast }}{\partial }_j^{\alpha }\text{for}\alpha \in {\mathbb{N}}_0^3,|\alpha |=1,\hfill \end{array}$$ 3.22

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 $s\in \mathbb{R}$, there exists $C>0$ such that for any $x_0\in {\mathbb{R}}^3$ we get $|{\partial }_x^{\sigma }|x+x_0{|}^s|\le C{|x+x_0|}^{s-|\sigma |}$ for all $x\in {\mathbb{R}}^3$, $x\ne -x_0$. We use the standard notation ${\mathbb{1}}_S$ to denote the indicator function on a set S.

Lemma 3.6

For any $\sigma \in {\mathbb{N}}_0^3$ with $|\sigma |\ge 1$ and any $t>0$ there exists C, depending on $\sigma$ but independent of t, such that

$$\begin{array}{c}\hfill |{\partial }^{\sigma }{\zeta }_t(x)|,|{\partial }^{\sigma }{\theta }_t(x)|\le C{t}^{-|\sigma |}{\mathbb{1}}_{\{{(4N)}^{-1}t<|x|<{(2N)}^{-1}t\}}(x)\end{array}$$ 3.23

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

Proof

Without loss, we consider the case of ${\zeta }_t$, the case of ${\theta }_t$ being similar. Recall $\xi =\xi (s)$ was defined in (3.1) and we denote by ${\xi }^{(m)}$ the m-th (univariate) derivative of $\xi$. Since $|\sigma |\ge 1$ the chain rule shows that ${\partial }^{\sigma }{\zeta }_t(x)$ can be written as a sum of terms of the form

$$\begin{array}{c}\hfill {(4N{t}^{-1})}^m{\xi }^{(m)}(4N|x|t^{-1}){\partial }^{{\sigma }_1}|x|\cdots {\partial }^{{\sigma }_m}|x|\end{array}$$ 3.24

where $1\le m\le |\sigma |$, and ${\sigma }_1,\cdots ,{\sigma }_m\in {\mathbb{N}}_0^3$ are nonzero multiindices obeying

$$\begin{array}{c}\hfill {\sigma }_1+\cdots +{\sigma }_m=\sigma .\end{array}$$

Since $m\ge 1$, we have that if ${\xi }^{(m)}(s)\ne 0$ then $s\in (1,2)$. Therefore, for any term (3.24) to be nonzero we require that

$$\begin{array}{c}\hfill {(4N)}^{-1}t<|x|<{(2N)}^{-1}t.\end{array}$$ 3.25

By the remark preceeding the current lemma, there exists C, dependent on ${\sigma }_1,\cdots ,{\sigma }_m$, such that

$$\begin{array}{c}\hfill {\partial }^{{\sigma }_1}|x|\cdots {\partial }^{{\sigma }_m}|x|\le {C|x|}^{m-|\sigma |}\le C{(4N)}^{|\sigma |-m}{t}^{m-|\sigma |},\end{array}$$

using (3.25). Therefore, the terms (3.24) can readily be bounded to give the desired result. $\square$

We now give bounds for the cluster derivatives (3.21)–(3.22) acting on cutoffs.

Lemma 3.7

Let $Q=Q(\mathrm{\Phi })$. Then $D_{x,y,Q}^{\alpha }\mathrm{\Phi }(·;Q)\equiv 0$ for all $\alpha \in {\mathbb{N}}_0^3$ with $|\alpha |\ge 1$.

Proof

By the chain rule, each function in the product (3.16) for $\mathrm{\Phi }(·;Q)$ has zero derivative upon action of $D_{x,y,Q}^{\alpha }$. $\square$

For the next result, we will use the following function, defined for all $t>0$ by

$$\begin{array}{cc}\hfill M_t(x,y,\widehat{\mathbf{x}})& =\sum _{2\le j\le N}{\mathbb{1}}_{\{{(4N)}^{-1}t<|x-x_j|<{(2N)}^{-1}t\}}(x-x_j)\hfill \\ \hfill & +\sum _{2\le j\le N}{\mathbb{1}}_{\{{(4N)}^{-1}t<|y-x_j|<{(2N)}^{-1}t\}}(y-x_j)\hfill \\ \hfill & +\sum _{2\le k<l\le N}{\mathbb{1}}_{\{{(4N)}^{-1}t<|x_k-x_l|<{(2N)}^{-1}t\}}(x_k-x_l)\hfill \end{array}$$ 3.26

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

Lemma 3.8

Let $0<2\delta \le ϵ$ and $\mathrm{\Phi }={\mathrm{\Phi }}_{\delta ,ϵ}$ be a cutoff of the form (3.3). Let $Q=Q(\mathrm{\Phi })$. Then for any multiindex $\mathit{\alpha }\in {\mathbb{N}}_0^{3N+3}$ there exists C, dependent on $\mathit{\alpha }$ but independent of $\delta$ and $ϵ$, such that for any partial products ${\mathrm{\Phi }}^{\prime }$ of $\mathrm{\Phi }$ we have

$$\begin{array}{c}\hfill |{\partial }^{\mathit{\alpha }}{\mathrm{\Phi }}^{\prime }(x,y,\widehat{\mathbf{x}})|\le \left\{\begin{array}{cc}C{ϵ}^{-|\mathit{\alpha }|}\hfill & \text{if}{\mathrm{\Phi }}^{\prime }=\mathrm{\Phi }(·;Q,Q^c)\hfill \\ C({ϵ}^{-|\mathit{\alpha }|}+{\delta }^{-|\mathit{\alpha }|}{M}_{\delta }(x,y,\widehat{\mathbf{x}}))\hfill & \text{otherwise.}\hfill \end{array}\right)\end{array}$$ 3.27

Proof

Lemma 3.6 gives bounds for the partial derivatives of the functions ${\zeta }_{\delta },{\theta }_{\delta },{\zeta }_{ϵ},{\theta }_{ϵ}$. Considering ${\theta }_{\delta }{\zeta }_{ϵ}$, we apply the Leibniz rule with $\sigma \in {\mathbb{N}}_0^3$, $|\sigma |\ge 1$, to obtain, for $z\in {\mathbb{R}}^3$

$$\begin{array}{cc}\hfill {\partial }^{\sigma }({\theta }_{\delta }{\zeta }_{ϵ})(z)& =\sum _{\mu \le \sigma }\left(\genfrac{}{}{0pt}{}{\sigma }{\mu }\right){\partial }^{\mu }{\theta }_{\delta }(z){\partial }^{\sigma -\mu }{\zeta }_{ϵ}(z)={\partial }^{\sigma }{\theta }_{\delta }(z)+{\partial }^{\sigma }{\zeta }_{ϵ}(z),\hfill \end{array}$$ 3.28

since, for each $\mu \le \sigma$ with $\mu \ne 0$ and $\mu \ne \sigma$ we have

$$\begin{array}{c}\hfill {\partial }^{\mu }{\theta }_{\delta }(z){\partial }^{\sigma -\mu }{\zeta }_{ϵ}(z)\equiv 0\end{array}$$

by Lemma 3.6 and that $2\delta \le ϵ$.

Now, to evaluate ${\partial }^{\mathit{\alpha }}{\mathrm{\Phi }}^{\prime }$ for a general ${\mathrm{\Phi }}^{\prime }$ we apply the Leibniz rule to the product (3.15) using (3.28) where appropriate. Differentiated cutoff factors ${\zeta }_{\delta },{\theta }_{\delta },{\zeta }_{ϵ},{\theta }_{ϵ}$ are bounded by (3.23). The indicator function in this bound is not required in the case of ${\zeta }_{ϵ}$ or ${\theta }_{ϵ}$, whereas in the case of ${\zeta }_{\delta }$ or ${\theta }_{\delta }$ the indicator function is bounded above by $M_t$. Any remaining undifferentiated cutoff factors are bounded above by 1. If ${\mathrm{\Phi }}^{\prime }=\mathrm{\Phi }(·;Q,Q^c)$, all cutoff factors are of the form ${\theta }_{ϵ}$ by Lemma 3.5, and therefore, we need only use the bounds in (3.23) with $t=ϵ$. $\square$

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

Lemma 3.9

Let $0<2\delta \le ϵ$ and $\mathrm{\Phi }={\mathrm{\Phi }}_{\delta ,ϵ}$ be a cutoff of the form (3.3). Let $Q=Q(\mathrm{\Phi })$. For any multiindices $\alpha \in {\mathbb{N}}_0^3$ and $\mathit{\sigma }\in {\mathbb{N}}_0^{3N+3}$, there exists C, independent of $ϵ$ and $\delta$, such that

$$\begin{array}{c}\hfill |{\partial }^{\mathit{\sigma }}{D}_{x,y,Q}^{\alpha }\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})|\le C{ϵ}^{-|\alpha |}({ϵ}^{-|\mathit{\sigma }|}+{\delta }^{-|\mathit{\sigma }|}{M}_{\delta }(x,y,\widehat{\mathbf{x}}))\end{array}$$

for all $x,y,\widehat{\mathbf{x}}$.

Proof

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

$$\begin{array}{c}\hfill D_{x,y,Q}^{\alpha }\mathrm{\Phi }={\mathrm{\Phi }}^{\prime }{D}_{x,y,Q}^{\alpha }{\mathrm{\Phi }}^{\prime \prime }\end{array}$$

which follows from Lemmas 3.4 and 3.7, and that $\mathrm{\Phi }(·;Q^c)$ is not dependent on variables involved in the $D_{x,y,Q}^{\alpha }$-derivative. By the definition (3.22), the derivative $D_{x,y,Q}^{\alpha }{\mathrm{\Phi }}^{\prime \prime }$ can be written as a sum of partial derivatives of the form ${\partial }^{\mathit{\alpha }}{\mathrm{\Phi }}^{\prime \prime }$ where $\mathit{\alpha }\in {\mathbb{N}}_0^{3N+3}$ obeys $|\mathit{\alpha }|=|\alpha |$. Now, by the Leibniz rule and Lemma 3.8 there exist some constants C and $C^{\prime }$, independent of $\delta$ and $ϵ$, such that

$$\begin{array}{cc}\hfill |{\partial }^{\mathit{\sigma }}({\mathrm{\Phi }}^{\prime }{\partial }^{\mathit{\alpha }}{\mathrm{\Phi }}^{\prime \prime })|& \le \sum _{\mathit{\tau }\le \mathit{\sigma }}\left(\genfrac{}{}{0pt}{}{\mathit{\sigma }}{\mathit{\tau }}\right)|{\partial }^{\mathit{\tau }}{\mathrm{\Phi }}^{\prime }||{\partial }^{\mathit{\sigma }-\mathit{\tau }+\mathit{\alpha }}{\mathrm{\Phi }}^{\prime \prime }|\hfill \\ \hfill & \le C\sum _{\mathit{\tau }\le \mathit{\sigma }}({ϵ}^{-|\mathit{\tau }|}+{\delta }^{-|\mathit{\tau }|}{M}_{\delta }){ϵ}^{-|\mathit{\sigma }|+|\mathit{\tau }|-|\mathit{\alpha }|}\hfill \\ \hfill & \le C^{\prime }{ϵ}^{-|\alpha |}({ϵ}^{-|\mathit{\sigma }|}+{\delta }^{-|\mathit{\sigma }|}{M}_{\delta }),\hfill \end{array}$$

completing the proof. $\square$

Integrals Involving $f_{\infty }$

The following proposition is a restatement of [9, Lemma 5.1] and is proved in that paper. The function $M_t$ defined in (3.26) has a slightly different form to the corresponding function used in the paper but this does not affect the proof. Recall $f_{\infty }$ was defined in (1.29).

Proposition 3.10

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

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^{3N-3}}{f}_{\infty }(x,\widehat{\mathbf{x}};R)f_{\infty }(y,\widehat{\mathbf{x}};R)\text{d}\widehat{\mathbf{x}}\le C{∥\rho ∥}_{L^1(B(x,2R))}^{1/2}{∥\rho ∥}_{L^1(B(y,2R))}^{1/2}\end{array}$$ 3.29

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

$$\begin{array}{cc}& {\int }_{{\mathbb{R}}^{3N-3}}(|G(x_j-x_k)|+|G(z-x_k)|+|G(x_j)|)f_{\infty }(x,\widehat{\mathbf{x}};R)f_{\infty }(y,\widehat{\mathbf{x}};R)\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & \le C{∥G∥}_{L^1({\mathbb{R}}^3)}{∥\rho ∥}_{L^1(B(x,2R))}^{1/2}{∥\rho ∥}_{L^1(B(y,2R))}^{1/2}\hfill \end{array}$$ 3.30

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

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^{3N-3}}{M}_t(x,y,\widehat{\mathbf{x}})f_{\infty }(x,\widehat{\mathbf{x}};R)f_{\infty }(y,\widehat{\mathbf{x}};R)\text{d}\widehat{\mathbf{x}}\le C{t}^3{∥\rho ∥}_{L^1(B(x,2R))}^{1/2}{∥\rho ∥}_{L^1(B(y,2R))}^{1/2}\end{array}$$ 3.31

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

Recall from (1.31) that $f_{\infty }(·)$ was, for convenience, defined as $f_{\infty }(·;R)$ with $R=1/2$.

Lemma 3.11

Let $\mathrm{\Phi }={\mathrm{\Phi }}_{\delta ,ϵ}$ be a cutoff of the form (3.3), and let j, k, r, s be as in (3.7) and (3.8). Then for any $b=b_1+b_2+b_3$, $b\ne 3$, $b_1,b_2,b_3\ge 0$ there exists C, independent of x, y and $\delta ,ϵ$, such that

$$\begin{array}{cc}& {\int }_{\text{supp}\mathrm{\Phi }(x,y,·)}|x-x_j{|}^{-b_1}|y-x_k{|}^{-b_2}{|x_r-x_s|}^{-b_3}{f}_{\infty }(x,\widehat{\mathbf{x}})f_{\infty }(y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & \le C(1+{\delta }^{min\{0,3-b\}}){∥\rho ∥}_{L^1(B(x,1))}^{1/2}{∥\rho ∥}_{L^1(B(y,1))}^{1/2}.\hfill \end{array}$$ 3.32

Proof

By Young’s inequality,

$$\begin{array}{cc}& |x-x_j{|}^{-b_1}|y-x_k{|}^{-b_2}{|x_r-x_s|}^{-b_3}\hfill \\ \hfill & \le \frac{1}{b}(b_1|x-x_j{|}^{-b}+b_2|y-x_k{|}^{-b}+b_3{|x_r-x_s|}^{-b}).\hfill \end{array}$$

It therefore suffices to prove (3.32) where only one of $b_1,b_2$ and $b_3$ is nonzero. We consider the case where this is $b_1$, the other cases are similar. We split the integral into one where $|x-x_j|\ge 1$ and one where $|x-x_j|<1$. In the former case, we can bound using (3.29) of Proposition 3.10, whereas in the latter we use (3.30) of the same proposition with $G(x-x_j)={\mathbb{1}}_{\{{(4N)}^{-1}\delta <|x-x_j|<1\}}(x-x_j){|x-x_j|}^{-b}$. The lower bound can be included in the indicator function due to (3.7) of Lemma 3.2. $\square$

Proof of Theorem 1.1

We begin by introducing auxiliary functions related to the density matrix, $\gamma (x,y)$, defined in (1.4). For $l,m\in {\mathbb{N}}_0^3$ with $|l|,|m|\le 1$, define

$$\begin{array}{c}\hfill {\gamma }_{l,m}(x,y)={\int }_{{\mathbb{R}}^{3N-3}}{\partial }_x^l\psi (x,\widehat{\mathbf{x}})\overline{{\partial }_y^m\psi (y,\widehat{\mathbf{x}})}\text{d}\widehat{\mathbf{x}}.\end{array}$$ 4.1

In this notation, it is clear that $\gamma ={\gamma }_{0,0}$. By differentiation under the integral, we have

$$\begin{array}{cc}\hfill {\partial }_x^l{\partial }_y^m\gamma (x,y)& ={\gamma }_{l,m}(x,y).\hfill \end{array}$$ 4.2

Furthermore, for any cutoff $\mathrm{\Phi }$ of the form (3.3), we set

$$\begin{array}{cc}\hfill {\gamma }_{l,m}(x,y;\mathrm{\Phi })& ={\int }_{{\mathbb{R}}^{3N-3}}{\partial }_x^l\psi (x,\widehat{\mathbf{x}})\overline{{\partial }_y^m\psi (y,\widehat{\mathbf{x}})}\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}},\hfill \end{array}$$

and define $\gamma (·;\mathrm{\Phi })={\gamma }_{0,0}(·;\mathrm{\Phi })$.

To produce the required bound for derivatives ${\partial }_x^{\alpha }{\partial }_y^{\beta }\gamma (x,y)$ for $|\alpha |+|\beta |=5$ we consider separately the cases where $|\alpha |,|\beta |\ge 1$ and where at least one of $\alpha ,\beta$ is zero. In the former case, we will immediately differentiate under the integral once in both x and y. This is because Proposition 4.1 is not affected by whether l and m in the proposition are zero or represent single derivatives. This itself is a consequence of the fact that $\psi$ and $\mathrm{\nabla }\psi$ are both bounded functions, the function $\psi$ only becoming singular after two derivatives.

When one of $\alpha ,\beta$ is zero, we can rewrite the derivatives into an expression involving $(x+y)$-derivatives as follows. Without loss, suppose $\beta =0$ and hence $|\alpha |=5$. Let $l,m\in {\mathbb{N}}_0^3$, $|l|=|m|=1$ be such that $l+m\le \alpha$. Then we can use ${\partial }_{x+y}^{l+m}=({\partial }_x^l+{\partial }_y^l)({\partial }_x^m+{\partial }_y^m)$, see (1.14), to write the identity

$$\begin{array}{c}\hfill {\partial }_x^{l+m}={\partial }_{x+y}^{l+m}-{\partial }_y^{l+m}-{\partial }_x^l{\partial }_y^m-{\partial }_x^m{\partial }_y^l,\end{array}$$ 4.3

which will hold on sufficiently smooth functions. We note that rearrangement would give us the corresponding expression for ${\partial }_y^{l+m}$. The final three terms above introduce y-derivatives, and since we already have the three remaining derivatives of $\alpha$, we can differentiate once in both x and y, as in the previous case. The first term in the equality above involves an $(x+y)$-derivative. It will be shown that such derivatives do not contribute to the singularity at the diagonal for the density matrix.

Proof of Theorem 1.1.

We first state the following proposition which bounds particular derivatives of ${\gamma }_{l,m}$. As described above, this will be sufficient to bound an arbitrary fifth partial derivative of the density matrix $\gamma$ and hence prove Theorem 1.1. Here, as before, $m(x,y)=min\{1,|x|,|y|\}$.

Proposition 4.1

Take any $l,m\in {\mathbb{N}}_0^3$, $|l|,|m|\le 1$, and all ${\mu }_1,{\mu }_2,{\mu }_3\in {\mathbb{N}}_0^3$ which obey either

• (i)$|{\mu }_2|+|{\mu }_3|\ne 3$ with ${\mu }_1$ arbitrary, or
• (ii)$|{\mu }_2|+|{\mu }_3|=3$ and ${\mu }_1=0$.

Then there exists C such that

$$\begin{array}{cc}& |{\partial }_{x+y}^1{\partial }_x^2{\partial }_y^3{\gamma }_{l,m}(x,y)|\hfill \\ \hfill & \le Cm{(x,y)}^{-|{\mu }_1|-|{\mu }_2|-|{\mu }_3|}{|x-y|}^{min\{0,3-|{\mu }_2|-|{\mu }_3|\}}{∥\rho ∥}_{L^1(B(x,1))}^{1/2}{∥\rho ∥}_{L^1(B(y,1))}^{1/2}\hfill \end{array}$$ 4.4

for all $x,y\in {\mathbb{R}}^3$ obeying $0<|x-y|\le m(x,y)/2$.

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

Proof of Theorem 1.1

If $|\alpha |,|\beta |\ge 1$ then take any $l\le \alpha$ and $m\le \beta$ with $|l|=|m|=1$. We can then write

$$\begin{array}{c}\hfill {\partial }_x^{\alpha }{\partial }_y^{\beta }\gamma ={\partial }_x^{\alpha -l}{\partial }_y^{\beta -m}{\gamma }_{l,m}.\end{array}$$

It is then straightforward to obtain the required bound by Proposition 4.1 with ${\mu }_1=0$, ${\mu }_2=\alpha -l$ and ${\mu }_3=\beta -m$.

We now consider the case where either $\alpha$ or $\beta$ is zero. Without loss, assume $\beta =0$. Let $l\le \alpha$ and $m\le \alpha -l$ obey $|l|=|m|=1$. We then have by (4.3),

$$\begin{array}{cc}\hfill {\partial }_x^{\alpha }\gamma & ={\partial }_{x+y}^{l+m}{\partial }_x^{\alpha -l-m}\gamma -{\partial }_x^{\alpha -l-m}{\partial }_y^{l+m}\gamma -{\partial }_x^{\alpha -m}{\partial }_y^m\gamma -{\partial }_x^{\alpha -l}{\partial }_y^l\gamma \hfill \end{array}$$

It suffices to bound each term separately. The final three terms have at least one derivative on each x and y, hence can be treated as before. Next, the first term can be rewritten as ${\partial }_{x+y}^{l+m}{\partial }_x^{\alpha -l-m-r}{\gamma }_{r,0}$ for some $r\le \alpha -l-m$ with $|r|=1$. Use of Proposition 4.1 with ${\mu }_1=l+m$, ${\mu }_2=\alpha -l-m-r$ and ${\mu }_3=0$ then gives the required bound. $\square$

Proof of Proposition 4.1

The following lemma gives bounds to derivatives of ${\gamma }_{l,m}(·;\mathrm{\Phi })$, where the cutoffs $\mathrm{\Phi }$ are those introduced in Sect. 3. We will use that these cutoffs form a partition of unity to obtain bounds for the derivatives of ${\gamma }_{l,m}$ and hence prove Proposition 4.1. As before, for a given $\mathrm{\Phi }$ we will set $P=P(\mathrm{\Phi })$, $S=S(\mathrm{\Phi })$, $Q=Q(\mathrm{\Phi })$ to be the corresponding clusters, as defined in Sect. 3.

Lemma 4.2

Let $\mathrm{\Phi }={\mathrm{\Phi }}_{\delta ,ϵ}$ be a cutoff of the form (3.3) with $P^{\ast }\cap S^{\ast }=\mathrm{\varnothing }$. For all $l,m\in {\mathbb{N}}_0^3$, $|l|,|m|\le 1$, and any ${\mu }_1,{\mu }_2,{\mu }_3$ as in Proposition 4.1, there exists a constant C such that for all $0<2\delta \le ϵ\le 1$ we have

$$\begin{array}{cc}& |{\partial }_{x+y}^1{\partial }_x^2{\partial }_y^3{\gamma }_{l,m}(x,y;{\mathrm{\Phi }}_{\delta ,ϵ})|\hfill \\ \hfill & \le C{ϵ}^{-|{\mu }_1|-|{\mu }_2|-|{\mu }_3|}{\delta }^{min\{0,3-|{\mu }_2|-|{\mu }_3|\}}{∥\rho ∥}_{L^1(B(x,1))}^{1/2}{∥\rho ∥}_{L^1(B(y,1))}^{1/2}\hfill \end{array}$$ 4.5

for all $|x|,|y|\ge ϵ$ and $|x-y|\le 2\delta$.

After the following proof, the remainder of this section is dedicated to proving the above lemma.

Proof of Proposition 4.1

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

$$\begin{array}{c}\hfill {\gamma }_{l,m}=\sum _{j=1}^J{\gamma }_{l,m}(·;{\mathrm{\Phi }}_{\delta ,ϵ}^{(j)})\end{array}$$ 4.6

holds everywhere for all choices of $0<2\delta \le ϵ$.

Let K be the number of cutoffs ${\mathrm{\Phi }}^{(j)}$, $j=1,\cdots ,J$, such that $P{({\mathrm{\Phi }}^{(j)})}^{\ast }\cap S{({\mathrm{\Phi }}^{(j)})}^{\ast }=\mathrm{\varnothing }$. We then label the cutoffs with this property by ${\mathrm{\Phi }}^{(j_k)}$ for $k=1,\cdots ,K$.

Fix any x, y such that $0<|x-y|\le m(x,y)/2$. Then set $\delta =|x-y|/2$ and $ϵ=m(x,y)/2$. By Lemma 3.3, we can use (4.6) to write

$$\begin{array}{c}\hfill {\gamma }_{l,m}(x^{\prime },y^{\prime })=\sum _{k=1}^K{\gamma }_{l,m}(x^{\prime },y^{\prime };{\mathrm{\Phi }}_{\delta ,ϵ}^{(j_k)})\end{array}$$ 4.7

for all $|x^{\prime }-y^{\prime }|\ge |x-y|/2$. Then by Lemma 4.2, for each $k=1,\cdots ,K$, we have

$$\begin{array}{cc}& |{\partial }_{x^{\prime }+y^{\prime }}^1{\partial }_{x^{\prime }}^2{\partial }_{y^{\prime }}^3{\gamma }_{l,m}(x^{\prime },y^{\prime })|\le \sum _{k=1}^K|{\partial }_{x^{\prime }+y^{\prime }}^1{\partial }_{x^{\prime }}^2{\partial }_{y^{\prime }}^3{\gamma }_{l,m}(x^{\prime },y^{\prime };{\mathrm{\Phi }}_{\delta ,ϵ}^{(j_k)})|\hfill \\ \hfill & \le Cm{(x,y)}^{-|{\mu }_1|-|{\mu }_2|-|{\mu }_3|}{|x-y|}^{min\{0,3-|{\mu }_2|-|{\mu }_3|\}}{∥\rho ∥}_{L^1(B(x^{\prime },1))}^{1/2}{∥\rho ∥}_{L^1(B(y^{\prime },1))}^{1/2}\hfill \end{array}$$

for all $x^{\prime },y^{\prime }$ such that $|x-y|/2<|x^{\prime }-y^{\prime }|\le |x-y|$ and $|x^{\prime }|,|y^{\prime }|\ge m(x,y)/2$. In particular, it holds for $x^{\prime }=x$ and $y^{\prime }=y$. The constant C does not depend on the choice of $\delta$ and $ϵ$; therefore, the bound holds for all required x and y. $\square$

The cutoffs introduced in Sect. 3 facilitate taking simultaneous derivatives in x, y and $x+y$. The strategy in proving Lemma 4.2 will be to turn partial derivatives ${\gamma }_{l,m}(·;\mathrm{\Phi })$ into cluster derivatives under the integral. Theorem 1.3 then gives the required pointwise bounds for these cluster derivatives and in fact distinguishes between derivatives of different clusters

Partial derivatives in x and y will produce cluster derivatives for clusters $P,S,P^{\ast }$ and $S^{\ast }$, whereas derivatives in $x+y$ will produce cluster derivatives for the cluster Q. We wish to separate the contributions from x- and y-derivatives and from ($x+y$)-derivatives. To do this, it will be convenient to group together the distances ${\lambda }_P$ etc. which will later appear when Theorem 1.3 is applied to the differentiated density matrix. This is what we define now as the functions $\lambda$ and $\pi$.

First, we fix a cutoff $\mathrm{\Phi }={\mathrm{\Phi }}_{\delta ,ϵ}$ to be used throughout this section, and let P, S, Q be the corresponding clusters. We will assume that $P^{\ast }\cap S^{\ast }=\mathrm{\varnothing }$ and $0<2\delta \le ϵ\le 1$.

For any $x,y\in {\mathbb{R}}^3$ and $\widehat{\mathbf{x}}\in {\mathbb{R}}^{3N-3}$, we define

$$\begin{array}{cc}\hfill \lambda (x,y,\widehat{\mathbf{x}})& =min\{{\lambda }_P(x,\widehat{\mathbf{x}}),{\lambda }_{S^{\ast }}(x,\widehat{\mathbf{x}}),{\lambda }_{P^{\ast }}(y,\widehat{\mathbf{x}}),{\lambda }_S(y,\widehat{\mathbf{x}})\},\hfill \end{array}$$ 4.8

$$\begin{array}{cc}\hfill \pi (x,y,\widehat{\mathbf{x}})& =min\{{\lambda }_Q(x,\widehat{\mathbf{x}}),{\lambda }_Q(y,\widehat{\mathbf{x}})\}.\hfill \end{array}$$ 4.9

Recall that $1\in P,S,Q$ by definition. Using (1.20), we find that

$$\begin{array}{cc}\hfill \pi (x,y,\widehat{\mathbf{x}})& =min\{1,|x|,|y|,|x_j|:j\in Q^{\ast },2^{-1/2}|x-x_k|:k\in Q^c,\hfill \\ \hfill & 2^{-1/2}|y-x_k|:k\in Q^c,2^{-1/2}|x_j-x_k|:j\in Q^{\ast },k\in Q^c\}.\hfill \end{array}$$ 4.10

Using that $P^{\ast }\cap S^{\ast }=\mathrm{\varnothing }$, we similarly find that

$$\begin{array}{cc}& \lambda (x,y,\widehat{\mathbf{x}})=min\{1,|x|,|y|,|x_j|:j\in P^{\ast }\text{or}{S}^{\ast },2^{-1/2}|x-x_k|:k\in P^c,\hfill \\ \hfill & 2^{-1/2}|y-x_k|:k\in S^c,2^{-1/2}|x_j-x_k|:j\in P^{\ast },k\in P^c\text{or}j\in S^{\ast },k\in S^c\}.\hfill \end{array}$$ 4.11

Lemma 4.3

For any $x,y\in {\mathbb{R}}^3$ with $|x|,|y|\ge ϵ$,

• (i)
  there exists C, independent of $\delta ,ϵ$ and $x,y,\widehat{\mathbf{x}}$, such that when $\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\ne 0$,

  $$\begin{array}{cc}\hfill \pi (x,y,\widehat{\mathbf{x}})& \ge Cϵ,\hfill \end{array}$$ 4.12

  $$\begin{array}{cc}\hfill \lambda (x,y,\widehat{\mathbf{x}})& \ge C\delta ,\hfill \end{array}$$ 4.13
• (ii)
  for any $b\ge 0$ be such that $b\ne 3$, there exists C, independent of $\delta ,ϵ$ and x, y, such that

  $$\begin{array}{cc}& {\int }_{\text{supp}\mathrm{\Phi }(x,y,·)}\lambda {(x,y,\widehat{\mathbf{x}})}^{-b}{f}_{\infty }(x,\widehat{\mathbf{x}})f_{\infty }(y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & \le C({ϵ}^{-b}+{\delta }^{min\{0,3-b\}}){∥\rho ∥}_{L^1(B(x,1))}^{1/2}{∥\rho ∥}_{L^1(B(y,1))}^{1/2}.\hfill \end{array}$$ 4.14

Proof

We can prove (i) using the expressions (4.10) and (4.11). The bound follows from Lemma 3.2 along with (3.14) of Lemma 3.3, since $P,S\subset Q$. For (ii) we use (4.11) to write

$$\begin{array}{c}\hfill \lambda {(x,y,\widehat{\mathbf{x}})}^{-b}\le C\left({ϵ}^{-b},,+,,\sum _{k\in P^c},|x-,x_k,{|}^{-b},,+,,\sum _{k\in S^c},|y-,x_k,{|}^{-b},,+,,\sum _{\begin{array}{c}j\in P^{\ast },k\in P^c\text{or}\\ j\in S^{\ast },k\in S^c\end{array}},{|x_j-x_k|}^{-b}\right)\end{array}$$

for some C depending on b. The required inequality then follows from (3.29) of Proposition 3.10 for the first term above and from Lemma 3.11 for the remaining terms. $\square$

Corollary 4.4

For any $x,y\in {\mathbb{R}}^3$ with $|x|,|y|\ge ϵ$,

$$\begin{array}{cc}& {\int }_{{\mathbb{R}}^{3N-3}}\lambda {(x,y,\widehat{\mathbf{x}})}^{-2}{f}_{\infty }(x,\widehat{\mathbf{x}})f_{\infty }(y,\widehat{\mathbf{x}})|\mathrm{\nabla }\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})|\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & \le C{ϵ}^{-3}{∥\rho ∥}_{L^1(B(x,1))}^{1/2}{∥\rho ∥}_{L^1(B(y,1))}^{1/2}\hfill \end{array}$$ 4.15

for some C, independent of $\delta ,ϵ$ and x, y.

Proof

Using Lemma 3.9 and (4.13), the integral can be bounded by some constant multiplying

$$\begin{array}{c}\hfill {\int }_{\text{supp}\mathrm{\Phi }(x,y,·)}({ϵ}^{-1}\lambda {(x,y,\widehat{\mathbf{x}})}^{-2}+{\delta }^{-3}{M}_{\delta }(x,y,\widehat{\mathbf{x}}))f_{\infty }(x,\widehat{\mathbf{x}})f_{\infty }(y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}.\end{array}$$

Expanding, the first term is bounded by (4.14) and second is bounded using (3.31) of Proposition 3.10. $\square$

We now collect certain results which will be used throughout this section. We first consider cluster derivatives of the cutoff $\mathrm{\Phi }$ of the form (3.21),(3.22). Such derivatives can be expanded as partial derivatives. Therefore, taking some $\eta \in {\mathbb{N}}_0^3$ and $\mathit{\nu }=({\nu }_1,{\nu }_2)\in {\mathbb{N}}_0^6$, there exists C by Lemma 3.9, depending on $\eta ,\mathit{\nu }$ but independent of $\delta ,ϵ$, such that

$$\begin{array}{c}\hfill |D_{x,y,Q}^{\eta }{D}_{x,P}^1{D}_{y,S}^2{\mathrm{\Phi }}_{\delta ,ϵ}(x,y,\widehat{\mathbf{x}})|\le C{ϵ}^{-|\eta |}({ϵ}^{-|\mathit{\nu }|}+{\delta }^{-|\mathit{\nu }|}{M}_{\delta }(x,y,\widehat{\mathbf{x}}))\end{array}$$ 4.16

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

Now take any $x,y\in {\mathbb{R}}^3$ with $|x|,|y|\ge ϵ$, and suppose $\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\ne 0$. Then, as a consequence of Lemma 4.3 we have that both $\pi (x,y,\widehat{\mathbf{x}})$ and $\lambda (x,y,\widehat{\mathbf{x}})$ are positive. Therefore, by the definitions (4.8), (4.9) and (1.20), (1.21),

$$\begin{array}{c}\hfill (x,\widehat{\mathbf{x}})\in {\mathrm{\Sigma }}_Q^c\cap {\mathrm{\Sigma }}_P^c\cap {\mathrm{\Sigma }}_{S^{\ast }}^c\text{and}(y,\widehat{\mathbf{x}})\in {\mathrm{\Sigma }}_Q^c\cap {\mathrm{\Sigma }}_{P^{\ast }}^c\cap {\mathrm{\Sigma }}_S^c.\end{array}$$

This allows us to apply Theorem 1.3 for cluster derivatives of clusters $Q,P,S^{\ast }$ on $\psi$ at the point $(x,\widehat{\mathbf{x}})$, and of $Q,P^{\ast },S$ on $\psi$ at the point $(y,\widehat{\mathbf{x}})$. Indeed, using the definitions (4.8) and (4.9), we have for every $\eta \in {\mathbb{N}}_0^3$, $\mathit{\nu }=({\nu }_1,{\nu }_2)\in {\mathbb{N}}_0^6$ some C such that

$$\begin{array}{cc}\hfill \sum _{k=0,1}|D_Q^{\eta }{D}_1^{\mathit{\nu }}{\mathrm{\nabla }}^k\psi (x,\widehat{\mathbf{x}})|& \le C\pi {(x,y,\widehat{\mathbf{x}})}^{-|\eta |}\lambda {(x,y,\widehat{\mathbf{x}})}^{-|\mathit{\nu }|}{f}_{\infty }(x,\widehat{\mathbf{x}}),\hfill \end{array}$$ 4.17

$$\begin{array}{cc}\hfill \sum _{k=0,1}|D_Q^{\eta }{D}_2^{\mathit{\nu }}{\mathrm{\nabla }}^k\psi (y,\widehat{\mathbf{x}})|& \le C\pi {(x,y,\widehat{\mathbf{x}})}^{-|\eta |}\lambda {(x,y,\widehat{\mathbf{x}})}^{-|\mathit{\nu }|}{f}_{\infty }(y,\widehat{\mathbf{x}}),\hfill \end{array}$$ 4.18

where, for convenience, we define the cluster sets

$$\begin{array}{c}\hfill {\mathbf{L}}_1=(P,S^{\ast }),{\mathbf{L}}_2=(P^{\ast },S).\end{array}$$ 4.19

Lemma 4.5

Let ${\mu }_1,{\mu }_2,{\mu }_3\in {\mathbb{N}}_0^3$ be arbitrary and $l,m\in {\mathbb{N}}_0^3$ be such that $|l|,|m|\le 1$. The derivative ${\partial }_{x+y}^1{\partial }_x^2{\partial }_y^3{\gamma }_{l,m}(x,y;\mathrm{\Phi })$ is equal to a linear combination of integrals of the form

$$\begin{array}{cc}& {\int }_{{\mathbb{R}}^{3N-3}}(D_Q^1{D}_{{\mathbf{L}}_1}^{({\alpha }_2,{\alpha }_3)}{\partial }_x^l\psi (x,\widehat{\mathbf{x}}))(D_Q^1{D}_{{\mathbf{L}}_2}^{({\beta }_2,{\beta }_3)}{\partial }_y^m\psi (y,\widehat{\mathbf{x}}))(D_{x,y,Q}^1{D}_{x,P}^2{D}_{y,S}^3\mathrm{\Phi }(x,y,\widehat{\mathbf{x}}))\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & \text{where}|{\alpha }_i|+|{\beta }_i|+|{\sigma }_i|=|{\mu }_i|,i=1,2,3\hfill \end{array}$$ 4.20

for all $|x|,|y|\ge ϵ$. In particular, the derivative exists across the diagonal $x=y$ for $|x|,|y|\ge ϵ$.

Proof

For each choice of x and y, we define a x- and y-dependent change of variables for the integral ${\gamma }_{l,m}(x,y;\mathrm{\Phi })$. To start, we define two vectors $\widehat{\mathbf{a}}=(a_2,\cdots ,a_N),\widehat{\mathbf{b}}=(b_2,\cdots ,b_N)\in {\mathbb{R}}^{3N-3}$ by

$$\begin{array}{cccc}\hfill a_k& =\left\{\begin{array}{cc}x\hfill & \text{if}k\in P^{\ast }\hfill \\ y\hfill & \text{if}k\in S^{\ast }\hfill \\ 0\hfill & \text{if}k\in P^c\cap S^c,\hfill \end{array}\right)\hfill & \hfill b_k& =\left\{\begin{array}{cc}(x+y)/2\hfill & \text{if}k\in Q^{\ast }\hfill \\ 0\hfill & \text{if}k\in Q^c.\hfill \end{array}\right)\hfill \end{array}$$ 4.21

By a translational change of variables using $\widehat{\mathbf{a}}$, we can write

$$\begin{array}{c}\hfill {\gamma }_{l,m}(x,y;\mathrm{\Phi })={\int }_{{\mathbb{R}}^{3N-3}}{\partial }_x^l\psi (x,\widehat{\mathbf{x}}+\widehat{\mathbf{a}})\overline{{\partial }_y^m\psi (y,\widehat{\mathbf{x}}+\widehat{\mathbf{a}})}\mathrm{\Phi }(x,y,\widehat{\mathbf{x}}+\widehat{\mathbf{a}})\text{d}\widehat{\mathbf{x}}.\end{array}$$ 4.22

We will then apply differentiation under the integral. Beforehand, we show how such derivatives will act on each function within the integrand. As an illustration, we take a function f defined on ${\mathbb{R}}^{3N}$ and any $\eta \in {\mathbb{N}}_0^3$ to see by the chain rule that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\partial }_{x+y}^{\eta }[f(x,\widehat{\mathbf{x}}+\widehat{\mathbf{b}})]& =D_Q^{\eta }f(x,\widehat{\mathbf{x}}+\widehat{\mathbf{b}}),\hfill \\ \hfill {\partial }_{x+y}^{\eta }[f(y,\widehat{\mathbf{x}}+\widehat{\mathbf{b}})]& =D_Q^{\eta }f(y,\widehat{\mathbf{x}}+\widehat{\mathbf{b}}),\hfill \\ \hfill {\partial }_x^{\eta }[f(x,\widehat{\mathbf{x}}+\widehat{\mathbf{a}})]& =D_P^{\eta }f(x,\widehat{\mathbf{x}}+\widehat{\mathbf{a}}),\hfill \end{array}\begin{array}{cc}\hfill {\partial }_x^{\eta }[f(y,\widehat{\mathbf{x}}+\widehat{\mathbf{a}})]& =D_{\ast }^{\eta }f(y,\widehat{\mathbf{x}}+\widehat{\mathbf{a}}),\hfill \\ \hfill {\partial }_y^{\eta }[f(x,\widehat{\mathbf{x}}+\widehat{\mathbf{a}})]& =D_{\ast }^{\eta }f(x,\widehat{\mathbf{x}}+\widehat{\mathbf{a}}),\hfill \\ \hfill {\partial }_y^{\eta }[f(y,\widehat{\mathbf{x}}+\widehat{\mathbf{a}})]& =D_S^{\eta }f(y,\widehat{\mathbf{x}}+\widehat{\mathbf{a}}).\hfill \end{array}\end{array}$$

Analogous expressions arise when the cutoff $\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})$ is differentiated which involve cluster derivatives of the form (3.21), (3.22).

We now differentiate (4.22) in x and y and use the product rule to get that the function ${\partial }_x^2{\partial }_y^3{\gamma }_{l,m}(x,y;\mathrm{\Phi })$ is a linear combination of terms of the form

$$\begin{array}{cc}& I_{({\alpha }_2,{\alpha }_3,{\beta }_2,{\beta }_3,{\sigma }_2,{\sigma }_3)}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^{3N-3}}(D_{{\mathbf{L}}_1}^{({\alpha }_2,{\alpha }_3)}{\partial }_x^l\psi (x,\widehat{\mathbf{x}}+\widehat{\mathbf{a}}))(D_{{\mathbf{L}}_2}^{({\beta }_2,{\beta }_3)}{\partial }_y^m\psi (y,\widehat{\mathbf{x}}+\widehat{\mathbf{a}}))D_{x,P}^2{D}_{y,S}^3\mathrm{\Phi }(x,y,\widehat{\mathbf{x}}+\widehat{\mathbf{a}}))\text{d}\widehat{\mathbf{x}},\hfill \end{array}$$

for some ${\alpha }_2+{\beta }_2+{\sigma }_2={\mu }_2$ and ${\alpha }_3+{\beta }_3+{\sigma }_3={\mu }_3$.

We will now apply a $(x+y)$-derivative to an integral of this form. However, first we apply a change of variables to replace $\widehat{\mathbf{a}}$ with $\widehat{\mathbf{b}}$ as the translation of the integration variable $\widehat{\mathbf{x}}$. Then, in the same way as before, we obtain that ${\partial }_{x+y}^1{I}_{({\alpha }_2,{\alpha }_3,{\beta }_2,{\beta }_3,{\sigma }_2,{\sigma }_3)}$ is a linear combination of terms of the form

$$\begin{array}{cc}& {\int }_{{\mathbb{R}}^{3N-3}}(D_Q^1{D}_{{\mathbf{L}}_1}^{({\alpha }_2,{\alpha }_3)}{\partial }_x^l\psi (x,\widehat{\mathbf{x}}+\widehat{\mathbf{b}}))(D_Q^1{D}_{{\mathbf{L}}_2}^{({\beta }_2,{\beta }_3)}{\partial }_y^m\psi (y,\widehat{\mathbf{x}}+\widehat{\mathbf{b}}))·\hfill \\ \hfill & D_{x,y,Q}^1{D}_{x,P}^2{D}_{y,S}^3\mathrm{\Phi }(x,y,\widehat{\mathbf{x}}+\widehat{\mathbf{b}}))\text{d}\widehat{\mathbf{x}}.\hfill \end{array}$$

for some ${\alpha }_1+{\beta }_1+{\sigma }_1={\mu }_1$. Finally, another translational change of variables removes $\widehat{\mathbf{b}}$ to give the expression (4.20). $\square$

We can now prove Lemma 4.2 by bounding the integrals produced in the previous lemma.

Proof of Lemma 4.2

Consider ${\mu }_1,{\mu }_2,{\mu }_3$ satisfying either (i) or (ii) of Propostion 4.1. By Lemma 4.5 it suffices to prove the required bound for integrals of the form (4.20). Define $a=|{\alpha }_1|+|{\beta }_1|+|{\sigma }_1|$, $b=|{\alpha }_2|+|{\alpha }_3|+|{\beta }_2|+|{\beta }_3|$ and $c=|{\sigma }_2|+|{\sigma }_3|$. Notice that $a=|{\mu }_1|$ and $b+c=|{\mu }_2|+|{\mu }_3|$. Then, using (4.16)-(4.18) and (4.12) it is possible to bound (4.20) in absolute value by some constant multiplying

$$\begin{array}{cc}& {ϵ}^{-a-c}{\int }_{\text{supp}\mathrm{\Phi }(x,y,·)}\lambda {(x,y,\widehat{\mathbf{x}})}^{-b}{f}_{\infty }(x,\widehat{\mathbf{x}})f_{\infty }(y,\widehat{\mathbf{x}})P\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & +{ϵ}^{-a}{\delta }^{-c}{\int }_{\text{supp}\mathrm{\Phi }(x,y,·)}{M}_{\delta }(x,y,\widehat{\mathbf{x}})\lambda {(x,y,\widehat{\mathbf{x}})}^{-b}{f}_{\infty }(x,\widehat{\mathbf{x}})f_{\infty }(y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}=:I_1+I_2.\hfill \end{array}$$

For $I_2$, we use (4.13) to bound this by some constant multiplying

$$\begin{array}{c}\hfill {ϵ}^{-a}{\delta }^{-c-b}{\int }_{{\mathbb{R}}^{3N-3}}{M}_{\delta }(x,y,\widehat{\mathbf{x}})f_{\infty }(x,\widehat{\mathbf{x}})f_{\infty }(y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\le C{ϵ}^{-a}{\delta }^{3-b-c}{∥\rho ∥}_{L^1(B(x,1))}^{1/2}{∥\rho ∥}_{L^1(B(y,1))}^{1/2}\end{array}$$

for some C, where we used (3.31). This gives (4.5) for $I_2$.

For $I_1$, we first consider when $b\ne 3$. By (4.14), this is bounded by some constant multiplying

$$\begin{array}{c}\hfill ({ϵ}^{-a-c-b}+{ϵ}^{-a-c}{\delta }^{min\{0,3-b\}}){∥\rho ∥}_{L^1(B(x,1))}^{1/2}{∥\rho ∥}_{L^1(B(y,1))}^{1/2}\end{array}$$

which is then bounded by (4.5).

Now we consider $I_1$ for $b=3$. If we are in case (i) then we must have $c\ge 1$ since $b+c=|{\mu }_1|+|{\mu }_2|\ne 3$. Here, we use (4.13) followed by (4.14) to bound $I_1$ by some constant multiplying

$$\begin{array}{cc}& {ϵ}^{-a-c}{\delta }^{-1}{\int }_{\text{supp}\mathrm{\Phi }(x,y,·)}\lambda {(x,y,\widehat{\mathbf{x}})}^{-2}{f}_{\infty }(x,\widehat{\mathbf{x}})f_{\infty }(y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & \le C{ϵ}^{-a-c-2}{\delta }^{-1}{∥\rho ∥}_{L^1(B(x,1))}^{1/2}{∥\rho ∥}_{L^1(B(y,1))}^{1/2}\hfill \end{array}$$

for some C, which is then bounded by (4.5). Finally, if we are in case (ii) for $b=3$ then $I_1$ cannot be bounded appropriately. Instead, we use that the integral (4.20) that we are attempting to bound must have ${\alpha }_1={\beta }_1=0$ and ${\sigma }_i=0$, $i=1,2,3$. This allows us to use Lemma 4.6, which is stated and proven below, to bound (4.20) directly. $\square$

Proof of Lemma 4.6

As before, $\mathrm{\Phi }={\mathrm{\Phi }}_{\delta ,ϵ}$ is an arbitrary cutoff of the form (3.3) with corresponding clusters $P=P(\mathrm{\Phi })$, $S=S(\mathrm{\Phi })$ and $Q=Q(\mathrm{\Phi })$. Also as before, we use the shorthand (4.19) in writing ${\mathbf{L}}_1=(P,S^{\ast })$ and ${\mathbf{L}}_2=(P^{\ast },S)$. We assume throughout that $P^{\ast }\cap S^{\ast }=\mathrm{\varnothing }$ and $0<2\delta \le ϵ\le 1$.

Lemma 4.6

For any $\mathit{\alpha }=({\alpha }_1,{\alpha }_2),\mathit{\beta }=({\beta }_1,{\beta }_2)\in {\mathbb{N}}_0^6$ with $|\mathit{\alpha }|+|\mathit{\beta }|=3$, and any $l,m\in {\mathbb{N}}_0^3$ with $|l|,|m|\le 1$ there exists C, independent of $\delta ,ϵ$, such that

$$\begin{array}{cc}& |{\int }_{{\mathbb{R}}^{3N-3}}(D_{{\mathbf{L}}_1}^{\mathit{\alpha }}{\partial }_x^l\psi (x,\widehat{\mathbf{x}}))(D_{{\mathbf{L}}_2}^{\mathit{\beta }}{\partial }_y^m\psi (y,\widehat{\mathbf{x}})){\mathrm{\Phi }}_{\delta ,ϵ}(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}|\hfill \\ \hfill & \le C{ϵ}^{-3}{∥\rho ∥}_{L^1(B(x,1))}^{1/2}{∥\rho ∥}_{L^1(B(y,1))}^{1/2}\hfill \end{array}$$ 4.23

for all $|x|,|y|\ge ϵ$ with $|x-y|\le 2\delta$.

First, we recall from Sect. 2.1 that

$$\begin{array}{c}\hfill F=F_c-F_s=F_c^{(en)}+F_c^{(ee)}-F_s\end{array}$$ 4.24

where $F_s$ is smooth and where

$$\begin{array}{c}\hfill F_c^{(en)}(\mathbf{x})=-\frac{Z}{2}\sum _{1\le j\le N}|x_j|,F_c^{(ee)}(\mathbf{x})=\frac{1}{4}\sum _{1\le j<k\le N}|x_j-x_k|.\end{array}$$ 4.25

The proof of Lemma 4.6 involves examining the cluster derivatives of $\psi$ using the refined estimates in (1.36) of Theorem 1.3. These allow us to write such derivatives in terms of a “bad” term, $F_c$, and a “good” term, $G_{\mathbf{P}}^{\mathit{\alpha }}$, which is of higher regularity near certain singularities. The contributions involving $F_c$ are handled explicitly using integration by parts.

Let $\mathit{\alpha }=({\alpha }_1,{\alpha }_2)\in {\mathbb{N}}_0^6$, $\mathit{\beta }=({\beta }_1,{\beta }_2)\in {\mathbb{N}}_0^6$ and let $l,m\in {\mathbb{N}}_0^3$ obey $|l|=|m|=1$. In the following, the cluster derivatives in (4.26) are understood to act with respect to the ordered variables $(x,x_2,\cdots ,x_N)$ and the cluster derivatives in (4.27) are understood to act with respect to the ordered variables $(y,x_2,\cdots ,x_N)$. Now assume $|\mathit{\alpha }|,|\mathit{\beta }|\ge 1$, then

$$\begin{array}{cc}\hfill D_{{\mathbf{L}}_1}^{\mathit{\alpha }}{\partial }_x^l|x-x_j|& =\left\{\begin{array}{cc}{(-1)}^{|{\alpha }_1|+1}{\partial }_{x_j}^{{\alpha }_1+{\alpha }_2+l}|x-x_j|\hfill & \text{if}|{\alpha }_2|\ge 1\text{and}j\in S^{\ast }\hfill \\ 0\hfill & \text{if}|{\alpha }_2|\ge 1\text{and}j\in S^c\hfill \\ {(-1)}^{|{\alpha }_1|+1}{\partial }_{x_j}^{{\alpha }_1+l}|x-x_j|\hfill & \text{if}{\alpha }_2=0\text{and}j\in P^c\hfill \\ 0\hfill & \text{if}{\alpha }_2=0\text{and}j\in P^{\ast }\hfill \end{array}\right)\hfill \end{array}$$ 4.26

$$\begin{array}{cc}\hfill D_{{\mathbf{L}}_2}^{\mathit{\beta }}{\partial }_y^m|y-x_j|& =\left\{\begin{array}{cc}{(-1)}^{|{\beta }_2|+1}{\partial }_{x_j}^{{\beta }_1+{\beta }_2+m}|y-x_j|\hfill & \text{if}|{\beta }_1|\ge 1\text{and}j\in P^{\ast }\hfill \\ 0\hfill & \text{if}|{\beta }_1|\ge 1\text{and}j\in P^c\hfill \\ {(-1)}^{|{\beta }_2|+1}{\partial }_{x_j}^{{\beta }_2+m}|y-x_j|\hfill & \text{if}{\beta }_1=0\text{and}j\in S^c\hfill \\ 0\hfill & \text{if}{\beta }_1=0\text{and}j\in S^{\ast }\hfill \end{array}\right)\hfill \end{array}$$ 4.27

where we made use of $P^{\ast }\cap S^{\ast }=\mathrm{\varnothing }$ and the identity $({\mathrm{\nabla }}_t+{\mathrm{\nabla }}_s)|t-s|\equiv 0$ for $t,s\in {\mathbb{R}}^n$. By the definition of $F_c^{(ee)}$, which is rewritten above, we have therefore proven the following lemma.

Lemma 4.7

For any $|l|=|m|=1$ and $|\mathit{\alpha }|,|\mathit{\beta }|\ge 1$ we have

$$\begin{array}{cc}& D_{{\mathbf{L}}_1}^{\mathit{\alpha }}{\partial }_x^l{F}_c^{(ee)}(x,\widehat{\mathbf{x}})=\left\{\begin{array}{cc}\frac{{(-1)}^{|{\alpha }_1|+1}}{4}{\sum }_{j\in S^{\ast }}{\partial }_{x_j}^{{\alpha }_1+{\alpha }_2+l}|x-x_j|\hfill & \text{if}|{\alpha }_2|\ge 1\hfill \\ \frac{{(-1)}^{|{\alpha }_1|+1}}{4}{\sum }_{j\in P^c}{\partial }_{x_j}^{{\alpha }_1+l}|x-x_j|\hfill & \text{if}{\alpha }_2=0,\hfill \end{array}\right)\hfill \end{array}$$ 4.28

$$\begin{array}{cc}& D_{{\mathbf{L}}_2}^{\mathit{\beta }}{\partial }_y^m{F}_c^{(ee)}(y,\widehat{\mathbf{x}})=\left\{\begin{array}{cc}\frac{{(-1)}^{|{\beta }_2|+1}}{4}{\sum }_{j\in P^{\ast }}{\partial }_{x_j}^{{\beta }_1+{\beta }_2+m}|y-x_j|\hfill & \text{if}|{\beta }_1|\ge 1\hfill \\ \frac{{(-1)}^{|{\beta }_2|+1}}{4}{\sum }_{j\in S^c}{\partial }_{x_j}^{{\beta }_2+m}|y-x_j|\hfill & \text{if}{\beta }_1=0.\hfill \end{array}\right)\hfill \end{array}$$ 4.29

We now consider bounds to these derivatives. We will use $\lambda (x,y,\widehat{\mathbf{x}})$ as the quantity defined in (4.8) for the clusters P and S. By (4.11) and see, for example, the comment preceeding Lemma 3.6, we have for each $|\alpha |\ge 1$ some C such that

$$\begin{array}{cc}\hfill |{\partial }_{x_j}^{\alpha }|x-x_j||+|{\partial }_{x_k}^{\alpha }|y-x_k||& \le C\lambda {(x,y,\widehat{\mathbf{x}})}^{1-|\alpha |}\hfill \end{array}$$ 4.30

for all $2\le j,k\le N$ if $|\alpha |=1$, and all $j\in P^c$, $k\in S^c$ if $|\alpha |\ge 2$. Therefore it is clear from Lemma 4.7 and $P^{\ast }\cap S^{\ast }=\mathrm{\varnothing }$, that for each $\mathit{\nu }\in {\mathbb{N}}_0^6$ there exists C such that

$$\begin{array}{c}\hfill |D_1^{\mathit{\nu }}{\mathrm{\nabla }}_x{F}_c^{(ee)}(x,\widehat{\mathbf{x}})|+|D_2^{\mathit{\nu }}{\mathrm{\nabla }}_y{F}_c^{(ee)}(y,\widehat{\mathbf{x}})|\le C\lambda {(x,y,\widehat{\mathbf{x}})}^{-|\mathit{\nu }|}.\end{array}$$ 4.31

In addition, direct differentiation shows that for each $\mathit{\nu }\in {\mathbb{N}}_0^6$ there is a C such that

$$\begin{array}{c}\hfill |D_{{\mathbf{L}}_1}^{\mathit{\nu }}{\mathrm{\nabla }}_x{F}_c^{(en)}(x,\widehat{\mathbf{x}})|+|D_{{\mathbf{L}}_2}^{\mathit{\nu }}{\mathrm{\nabla }}_y{F}_c^{(en)}(y,\widehat{\mathbf{x}})|\le C{ϵ}^{-|\mathit{\nu }|}\end{array}$$ 4.32

whenever $|x|,|y|\ge ϵ$.

We now proceed with the proof of the main result of this subsection.

Proof of Lemma 4.6

We first consider the more difficult case of $|l|=|m|=1$. By Theorem 1.3 with $b=1/2$, we can write for $j=1,2$ and any $\mathit{\nu }\in {\mathbb{N}}_0^6$ with $|\mathit{\nu }|\ge 1$,

$$\begin{array}{cc}\hfill D_{{\mathbf{L}}_j}^{\mathit{\nu }}\mathrm{\nabla }\psi & =G_{{\mathbf{L}}_j}^{\mathit{\nu }}+\psi (D_{{\mathbf{L}}_j}^{\mathit{\nu }}\mathrm{\nabla }{F}_c),\hfill \end{array}$$ 4.33

and in these cases we have the estimates

$$\begin{array}{cc}\hfill |G_1^{\mathit{\nu }}(x,\widehat{\mathbf{x}})|& \le C\lambda {(x,y,\widehat{\mathbf{x}})}^{1/2-|\mathit{\nu }|}{f}_{\infty }(x,\widehat{\mathbf{x}}),\hfill \end{array}$$ 4.34

$$\begin{array}{cc}\hfill |G_2^{\mathit{\nu }}(y,\widehat{\mathbf{x}})|& \le C\lambda {(x,y,\widehat{\mathbf{x}})}^{1/2-|\mathit{\nu }|}{f}_{\infty }(y,\widehat{\mathbf{x}}).\hfill \end{array}$$ 4.35

With these preliminary estimates, we can prove the result in the case where $|\mathit{\alpha }|,|\mathit{\beta }|\ge 1$. By (4.33) and $F_c=F_c^{(en)}+F_c^{(ee)}$ we bound the integral (4.23) by

$$\begin{array}{cc}& {\int }_{{\mathbb{R}}^{3N-3}}|G_{{\mathbf{L}}_1}^{\mathit{\alpha }}(x,\widehat{\mathbf{x}})||D_{{\mathbf{L}}_2}^{\mathit{\beta }}{\mathrm{\nabla }}_y\psi (y,\widehat{\mathbf{x}})|\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^{3N-3}}|D_{{\mathbf{L}}_1}^{\mathit{\alpha }}{\mathrm{\nabla }}_x{F}_c(x,\widehat{\mathbf{x}})||\psi (x,\widehat{\mathbf{x}})||G_{{\mathbf{L}}_2}^{\mathit{\beta }}(y,\widehat{\mathbf{x}})|\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^{3N-3}}|D_{{\mathbf{L}}_1}^{\mathit{\alpha }}{\mathrm{\nabla }}_x{F}_c^{(en)}(x,\widehat{\mathbf{x}})||\psi (x,\widehat{\mathbf{x}})||D_{{\mathbf{L}}_2}^{\mathit{\beta }}{\mathrm{\nabla }}_y{F}_c(y,\widehat{\mathbf{x}})||\psi (y,\widehat{\mathbf{x}})|\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^{3N-3}}|D_{{\mathbf{L}}_1}^{\mathit{\alpha }}{\mathrm{\nabla }}_x{F}_c^{(ee)}(x,\widehat{\mathbf{x}})||\psi (x,\widehat{\mathbf{x}})||D_{{\mathbf{L}}_2}^{\mathit{\beta }}{\mathrm{\nabla }}_y{F}_c^{(en)}(y,\widehat{\mathbf{x}})||\psi (y,\widehat{\mathbf{x}})|\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & +|{\int }_{{\mathbb{R}}^{3N-3}}(D_{{\mathbf{L}}_1}^{\mathit{\alpha }}{\partial }_x^l{F}_c^{(ee)}(x,\widehat{\mathbf{x}}))(D_{{\mathbf{L}}_2}^{\mathit{\beta }}{\partial }_y^m{F}_c^{(ee)}(y,\widehat{\mathbf{x}}))\psi (x,\widehat{\mathbf{x}})\overline{\psi (y,\widehat{\mathbf{x}})}\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}|.\hfill \end{array}$$

By Lemma 4.7, the final term above can be expanded by terms of the form (4.37), since $S^{\ast }\subset P^c$ and $P^{\ast }\subset S^c$. Lemma 4.8 then gives the appropriate bound. We will now show that each of the remaining integrals above can be bounded by a constant multiplying an integral of the form

$$\begin{array}{c}\hfill {ϵ}^{-a}{\int }_{{\mathbb{R}}^{3N-3}}\lambda {(x,y,\widehat{\mathbf{x}})}^{-b}{f}_{\infty }(x,\widehat{\mathbf{x}})f_{\infty }(y,\widehat{\mathbf{x}})\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}.\end{array}$$ 4.36

for some $a,b\ge 0$, $b\le 5/2$ and $a+b\le 3$. This follows from using the estimates (4.31), (4.32), (4.34) and (4.35). We also use (4.18) to bound the cluster derivatives of $\psi (y,\widehat{\mathbf{x}})$ in the first term. The required bound then follows by Lemma 4.3.

Next we consider the case where $|\mathit{\alpha }|=3$ and $\mathit{\beta }=0$. The other case where $|\mathit{\beta }|=3$ and $\mathit{\alpha }=0$ is similar with obvious modifications. By (4.33) and $F_c=F_c^{(en)}+F_c^{(ee)}$, we bound the integral (4.23) by

$$\begin{array}{cc}& {\int }_{{\mathbb{R}}^{3N-3}}|G_{{\mathbf{L}}_1}^{\mathit{\alpha }}(x,\widehat{\mathbf{x}})||\mathrm{\nabla }\psi (y,\widehat{\mathbf{x}})|\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^{3N-3}}|D_{{\mathbf{L}}_1}^{\mathit{\alpha }}{\partial }_x^l{F}_c^{(en)}(x,\widehat{\mathbf{x}})||\psi (x,\widehat{\mathbf{x}})||\mathrm{\nabla }\psi (y,\widehat{\mathbf{x}})|\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & +|{\int }_{{\mathbb{R}}^{3N-3}}(D_{{\mathbf{L}}_1}^{\mathit{\alpha }}{\partial }_x^l{F}_c^{(ee)}(x,\widehat{\mathbf{x}}))\psi (x,\widehat{\mathbf{x}})\overline{{\partial }_y^m\psi (y,\widehat{\mathbf{x}})}\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}|=:I_1+I_2+I_3.\hfill \end{array}$$

The integral $I_1$ is bounded using (4.34), from which we can then apply Lemma 4.3. Bounding $I_2$ is a simple application of (4.32) and Proposition 3.10. For $I_3$, we use the expression $\psi =e^F\varphi$, where $\varphi$ was introduced in (2.7). Since $F=F_c^{(ee)}+F_c^{(en)}-F_s$, we can write

$$\begin{array}{cc}& {\partial }_y^m\psi (y,\widehat{\mathbf{x}})=e^{F(y,\widehat{\mathbf{x}})}{\partial }_y^m\varphi (y,\widehat{\mathbf{x}})+\psi (y,\widehat{\mathbf{x}}){\partial }_y^m{F}_c^{(ee)}(y,\widehat{\mathbf{x}})\hfill \\ \hfill & +\psi (y,\widehat{\mathbf{x}})({\partial }_y^m{F}_c^{(en)}(y,\widehat{\mathbf{x}})-{\partial }_y^m{F}_s(y,\widehat{\mathbf{x}})).\hfill \end{array}$$

We expand $I_3$ according to this expression. By Lemma 4.7, the first term can be expanded in terms of the form (4.45) and therefore can bounded by Lemma 4.9. Also by Lemma 4.7, the second term can be expanded in terms of the form (4.37) for $\beta =m$ and $|\alpha |=4$ and therefore can bounded by Lemma 4.8. It therefore suffices to bound

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^{3N-3}}(D_{{\mathbf{L}}_1}^{\mathit{\alpha }}{\partial }_x^l{F}_c^{(ee)}(x,\widehat{\mathbf{x}}))({\partial }_y^m{F}_c^{(en)}(y,\mathbf{x})-{\partial }_y^m{F}_s(y,\mathbf{x}))\psi (x,\widehat{\mathbf{x}})\overline{\psi (y,\mathbf{x})}\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}.\end{array}$$

We expand the derivative of $F_c^{(ee)}$ using Lemma 4.7. A general term in this expansion will have the form

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^{3N-3}}{\partial }_{x_j}^{\alpha +l}|x-x_j|({\partial }_y^m{F}_c^{(en)}(y,\mathbf{x})-{\partial }_y^m{F}_s(y,\mathbf{x}))\psi (x,\widehat{\mathbf{x}})\overline{\psi (y,\mathbf{x})}\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\end{array}$$

for some $j\in P^c$ and $|\alpha |=3$. We write this integral as follows, using integration by parts to remove one derivative from $|x-x_j|$,

$$\begin{array}{cc}& -{\int }_{{\mathbb{R}}^{3N-3}}{\partial }_{x_j}^{\alpha }|x-x_j|({\partial }_y^m{F}_c^{(en)}(y,\mathbf{x})-{\partial }_y^m{F}_s(y,\mathbf{x})){\partial }_{x_j}^l(\psi (x,\widehat{\mathbf{x}})\overline{\psi (y,\mathbf{x})})\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & -{\int }_{{\mathbb{R}}^{3N-3}}{\partial }_{x_j}^{\alpha }|x-x_j|({\partial }_y^m{F}_c^{(en)}(y,\mathbf{x})-{\partial }_y^m{F}_s(y,\mathbf{x}))\psi (x,\widehat{\mathbf{x}})\overline{\psi (y,\mathbf{x})}{\partial }_{x_j}^l\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^{3N-3}}{\partial }_{x_j}^{\alpha }|x-x_j|{\partial }_{x_j}^l{\partial }_y^m{F}_s(y,\mathbf{x})\psi (x,\widehat{\mathbf{x}})\overline{\psi (y,\mathbf{x})}\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\hfill \end{array}$$

since ${\partial }_{x_j}^l{\partial }_y^m{F}_c^{(en)}(y,\mathbf{x})\equiv 0$. We now use (4.30) that $\mathrm{\nabla }{F}_c^{(en)}\in L^{\infty }({\mathbb{R}}^{3N})$ and that ${\mathrm{\nabla }}^k{F}_s\in L^{\infty }({\mathbb{R}}^{3N})$ for any integer $k\ge 1$, see (2.5). Therefore, these terms can be bounded by some constant multiplying

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^{3N-3}}\lambda {(x,y,\widehat{\mathbf{x}})}^{-2}{f}_{\infty }(x,\widehat{\mathbf{x}})f_{\infty }(y,\widehat{\mathbf{x}})(\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})+|\mathrm{\nabla }\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})|)\text{d}\widehat{\mathbf{x}}\end{array}$$

which is bounded appropriately by Lemma 4.3 and Corollary 4.4.

Finally, it remains to consider the case where one or both of l, m are zero. The strategy is analogous but simpler. We remark that we can use the following improved version of (4.17) and (4.18) in the case where $\eta =0$ and $k=0$. For any $\mathit{\nu }=({\nu }_1,{\nu }_2)\in {\mathbb{N}}_0^6$ with $|\mathit{\nu }|\ge 1$ there exists some C such that

$$\begin{array}{cc}\hfill |D_1^{\mathit{\nu }}\psi (x,\widehat{\mathbf{x}})|& \le C\lambda {(x,y,\widehat{\mathbf{x}})}^{1-|\mathit{\nu }|}{f}_{\infty }(x,\widehat{\mathbf{x}}),\hfill \\ \hfill |D_2^{\mathit{\nu }}\psi (y,\widehat{\mathbf{x}})|& \le C\lambda {(x,y,\widehat{\mathbf{x}})}^{1-|\mathit{\nu }|}{f}_{\infty }(y,\widehat{\mathbf{x}}).\hfill \end{array}$$

These inequalities follow immediately from (1.34) of Theorem 1.3. $\square$

We now state and prove two the two auxiliary lemmas which were used in the preceeding proof.

Lemma 4.8

Let $|\alpha |,|\beta |\ge 1$ be such that $|\alpha |+|\beta |\le 5$. Then for any pair $2\le j,k\le N$ such that $j\in P^c$ if $|\alpha |\ge 2$ and $k\in S^c$ if $|\beta |\ge 2$ we have for

$$\begin{array}{c}\hfill I_{j,k}:={\int }_{{\mathbb{R}}^{3N-3}}{\partial }_{x_j}^{\alpha }|x-x_j|{\partial }_{x_k}^{\beta }|y-x_k|\psi (x,\widehat{\mathbf{x}})\overline{\psi (y,\widehat{\mathbf{x}})}\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}},\end{array}$$ 4.37

that there exists C, independent of $\delta$ and $ϵ$, such that

$$\begin{array}{c}\hfill |I_{j,k}|\le C{ϵ}^{2-|\alpha |-|\beta |}{∥\rho ∥}_{L^1(B(x,1))}^{1/2}{∥\rho ∥}_{L^1(B(y,1))}^{1/2}\end{array}$$

for all $|x|,|y|\ge ϵ$ and $|x-y|\le 2\delta$.

Proof

We suppose that $|\alpha |+|\beta |=5$ otherwise the result follows immediately from (4.30) and Lemma 4.3.

We first prove the case where $j\ne k$, where the integration by parts is particularly simple. Suppose without loss that $|\alpha |\ge |\beta |$. Then take any $l\le \alpha$ with $|l|=1$ and use integration by parts to obtain

$$\begin{array}{cc}\hfill I_{j,k}& =-{\int }_{{\mathbb{R}}^{3N-3}}{\partial }_{x_j}^{\alpha -l}|x-x_j|{\partial }_{x_k}^{\beta }|y-x_k|{\partial }_{x_j}^l(\psi (x,\widehat{\mathbf{x}})\psi (y,\widehat{\mathbf{x}})\mathrm{\Phi }(x,y,\widehat{\mathbf{x}}))\text{d}\widehat{\mathbf{x}}.\hfill \end{array}$$ 4.38

This can then be bounded in absolute value as required using (4.30) followed by either Lemma 4.3 or Corollary 4.4. This completes the proof where $j\ne k$.

For the remainder of the proof, we consider the case where $j=k$. First, suppose $j\in P^c\cap S^c$. In the procedure that follows, we will use integration by parts to remove all derivatives from $|y-x_j|$ in $I_{j,j}$.

Since $|\beta |\ge 1$, we can find some multiindex ${\beta }_1\le \beta$ with $|{\beta }_1|=1$. Integration by parts then gives

$$\begin{array}{cc}\hfill I_{j,j}=& -{\int }_{{\mathbb{R}}^{3N-3}}{\partial }_{x_j}^{\alpha +{\beta }_1}|x-x_j|{\partial }_{x_j}^{\beta -{\beta }_1}|y-x_j|\psi (x,\widehat{\mathbf{x}})\psi (y,\widehat{\mathbf{x}})\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & -{\int }_{{\mathbb{R}}^{3N-3}}{\partial }_{x_j}^{\alpha }|x-x_j|{\partial }_{x_j}^{\beta -{\beta }_1}|y-x_j|{\partial }_{x_j}^1(\psi (x,\widehat{\mathbf{x}})\psi (y,\widehat{\mathbf{x}})\mathrm{\Phi }(x,y,\widehat{\mathbf{x}}))\text{d}\widehat{\mathbf{x}}.\hfill \end{array}$$ 4.39

We leave untouched the second integral above. For the first, if $|\beta |-|{\beta }_1|\ge 1$, we can remove another first-order derivative from $|y-x_j|$ by the same procedure, that is, using integration by parts to give two new terms as in (4.39). We retain the term where the derivative falls on $\psi (x,\widehat{\mathbf{x}})\psi (y,\widehat{\mathbf{x}})\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})$, whereas on the term where the derivative falls on $|x-x_j|$ we repeat the procedure, so long as there remains at least one derivative on $|y-x_j|$. Through this process, we obtain a formula for $I_{j,j}$. To express this formula, we first set $T=|\beta |$ and write $\beta ={\sum }_{s=1}^T{\beta }_s$ for some collection $|{\beta }_s|=1$ where $1\le s\le T$. Furthermore, we define

$$\begin{array}{c}\hfill {\beta }_{<i}=\left\{\begin{array}{cc}0\hfill & \text{if}i=1\hfill \\ {\beta }_1\hfill & \text{if}i=2\hfill \\ {\beta }_1+\cdots +{\beta }_{i-1}\hfill & \text{if}i\ge 3\hfill \end{array}\right){\beta }_{>i}=\left\{\begin{array}{cc}0\hfill & \text{if}i=T\hfill \\ {\beta }_T\hfill & \text{if}i=T-1\hfill \\ {\beta }_{i+1}+\cdots +{\beta }_T\hfill & \text{if}i\le T-2.\hfill \end{array}\right)\end{array}$$

Then,

$$\begin{array}{c}\hfill I_{j,j}={(-1)}^T{\int }_{{\mathbb{R}}^{3N-3}}{\partial }_{x_j}^{\alpha +\beta }|x-x_j||y-x_j|\psi (x,\widehat{\mathbf{x}})\psi (y,\widehat{\mathbf{x}})\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}+\sum _{i=1}^T{(-1)}^i{I}_{j,j}^{(i)}\end{array}$$ 4.40

where

$$\begin{array}{c}\hfill I_{j,j}^{(i)}={\int }_{{\mathbb{R}}^{3N-3}}{\partial }_{x_j}^{\alpha +{\beta }_{<i}}|x-x_j|{\partial }_{x_j}^{>i}|y-x_j|{\partial }_{x_j}^i(\psi (x,\widehat{\mathbf{x}})\psi (y,\widehat{\mathbf{x}})\mathrm{\Phi }(x,y,\widehat{\mathbf{x}}))\text{d}\widehat{\mathbf{x}}.\end{array}$$

We can bound $|I_{j,j}^{(i)}|$ for $1\le i\le T-1$ in the same way as (4.38) since $j\in P^c\cap S^c$. It remains to bound $I_{j,j}^{(T)}$, along with the first integral in formula (4.40).

We begin by expanding $|y-x_j|=|x-x_j|+(|y-x_j|-|x-x_j|)$ and noticing that

$$\begin{array}{c}\hfill ||y-x_j|-|x-x_j||\le |x-y|\le 2\delta .\end{array}$$ 4.41

It is then straightforward to show (for example see the comment preceeding Lemma 3.6) that for some $C,C^{\prime }$,

$$\begin{array}{cc}\hfill |I_{j,j}^{(T)}|& \le C{\int }_{{\mathbb{R}}^{3N-3}}|x-x_j{|}^{-3}(2\delta +|x-x_j|)|{\partial }_{x_j}^T(\psi (x,\widehat{\mathbf{x}})\psi (y,\widehat{\mathbf{x}})\mathrm{\Phi }(x,y,\widehat{\mathbf{x}}))|\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & \le C^{\prime }{\int }_{{\mathbb{R}}^{3N-3}}\lambda {(x,y,\widehat{\mathbf{x}})}^{-2}{f}_{\infty }(x,\widehat{\mathbf{x}})f_{\infty }(y,\widehat{\mathbf{x}})(\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})+|\mathrm{\nabla }\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})|)\text{d}\widehat{\mathbf{x}}\hfill \end{array}$$ 4.42

where in the second step we used formula (4.11), since $j\in P^c$, along with (4.13). This is then bounded as required by Lemma 4.3 and Corollary 4.4.

We now bound the first integral in (4.40). Once again, we use $|y-x_j|=|x-x_j|+(|y-x_j|-|x-x_j|)$, (4.41) and (4.11) to show it suffices to bound the two expressions

$$\begin{array}{cc}& \delta {\int }_{{\mathbb{R}}^{3N-3}}\lambda {(x,y,\widehat{\mathbf{x}})}^{-4}{f}_{\infty }(x,\widehat{\mathbf{x}})f_{\infty }(y,\widehat{\mathbf{x}})\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}},\hfill \end{array}$$ 4.43

$$\begin{array}{cc}& {\int }_{{\mathbb{R}}^{3N-3}}{\partial }_{x_j}^{\alpha +\beta }|x-x_j||x-x_j|\psi (x,\widehat{\mathbf{x}})\psi (y,\widehat{\mathbf{x}})\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}.\hfill \end{array}$$ 4.44

The first is readily bounded as required using (4.14) of Lemma 4.3 with $b=4$.

It remains to bound (4.44). We simplify the calculation by denoting $\sigma =\alpha +\beta$ and writing $\sigma ={\sigma }_1+\cdots +{\sigma }_5$ for some $|{\sigma }_s|=1$, $s=1,\cdots ,5$. Define (in the same way as ${\beta }_{<i}$ and ${\beta }_{>i}$ above)

$$\begin{array}{c}\hfill {\sigma }_{<i}=\left\{\begin{array}{cc}0\hfill & \text{if}i=1\hfill \\ {\sigma }_1\hfill & \text{if}i=2\hfill \\ {\sigma }_1+\cdots +{\sigma }_{i-1}\hfill & \text{if}3\le i\le 5,\hfill \end{array}\right){\sigma }_{>i}=\left\{\begin{array}{cc}0\hfill & \text{if}i=5\hfill \\ {\sigma }_5\hfill & \text{if}i=4\hfill \\ {\sigma }_{i+1}+\cdots +{\sigma }_5\hfill & \text{if}1\le i\le 3.\hfill \end{array}\right)\end{array}$$

We now apply the same method used above, that is, we transfer successive first-order derivatives via integration by parts. At each step, we leave as a remainder the term where the derivative falls on $\psi (x,\widehat{\mathbf{x}})\psi (y,\widehat{\mathbf{x}})\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})$. Since $|\sigma |=5$ is odd, the result after this procedure has occured five times is that (4.44) is precisely equal to minus the same integral plus remainder terms. This explains the $\frac{1}{2}$-factor in the following formula

$$\begin{array}{cc}& {\int }_{{\mathbb{R}}^{3N-3}}{\partial }_{x_j}^{\sigma }|x-x_j||x-x_j|\psi (x,\widehat{\mathbf{x}})\psi (y,\widehat{\mathbf{x}})\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & =\frac{1}{2}\sum _{i=1}^5{(-1)}^i{\int }_{{\mathbb{R}}^{3N-3}}{\partial }_{x_j}^{>i}|x-x_j|{\partial }_{x_j}^{<i}|x-x_j|{\partial }_{x_j}^i(\psi (x,\widehat{\mathbf{x}})\psi (y,\widehat{\mathbf{x}})\mathrm{\Phi }(x,y,\widehat{\mathbf{x}}))\text{d}\widehat{\mathbf{x}}.\hfill \end{array}$$

By (4.30), each of these integrals can be bounded by the right-hand side of (4.42) for a new constant $C^{\prime }$. This completes the proof in the case where $j\in P^c\cap S^c$.

Finally, we must bound (4.37) for when $j=k$ but where $j\in P^{\ast }$ or $j\in S^{\ast }$. We assume that $j\in P^{\ast }$, the case of $j\in S^{\ast }$ is similar. Hence, we need only consider $|\alpha |=1$. To bound the integral, we first apply integration by parts to obtain

$$\begin{array}{cc}\hfill I_{j,j}& =-{\int }_{{\mathbb{R}}^{3N-3}}|x-x_j|{\partial }_{x_j}^{\beta +\alpha }|y-x_j|\psi (x,\widehat{\mathbf{x}})\psi (y,\widehat{\mathbf{x}})\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & -{\int }_{{\mathbb{R}}^{3N-3}}|x-x_j|{\partial }_{x_j}^{\beta }|y-x_j|{\partial }_{x_j}^{\alpha }(\psi (x,\widehat{\mathbf{x}})\psi (y,\widehat{\mathbf{x}})\mathrm{\Phi }(x,y,\widehat{\mathbf{x}}))\text{d}\widehat{\mathbf{x}}.\hfill \end{array}$$

By Lemma 3.3, we have $|x-x_j|<\delta /2$ when $\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\ne 0$. Along with (4.30) and (4.13), we then get

$$\begin{array}{cc}\hfill |I_{j,j}|& \le C\delta {\int }_{{\mathbb{R}}^{3N-3}}\lambda {(x,y,\widehat{\mathbf{x}})}^{-4}{f}_{\infty }(x,\widehat{\mathbf{x}})f_{\infty }(y,\widehat{\mathbf{x}})\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & +C{\int }_{{\mathbb{R}}^{3N-3}}\lambda {(x,y,\widehat{\mathbf{x}})}^{-2}{f}_{\infty }(x,\widehat{\mathbf{x}})f_{\infty }(y,\widehat{\mathbf{x}})(\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})+|\mathrm{\nabla }\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})|)\text{d}\widehat{\mathbf{x}}\hfill \end{array}$$

for some C. The two terms are just (4.43) and the right-hand side of (4.42), respectively, and hence can be suitably bounded. This completes all required cases. $\square$

Lemma 4.9

Let $|\alpha |=4$, $j\in P^c$ and $k\in S^c$. Then the integrals

$$\begin{array}{cc}& {\int }_{{\mathbb{R}}^{3N-3}}{\partial }_{x_j}^{\alpha }|x-x_j|\psi (x,\widehat{\mathbf{x}})e^{F(y,\widehat{\mathbf{x}})}\overline{{\mathrm{\nabla }}_y\varphi (y,\widehat{\mathbf{x}})}\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}},\hfill \end{array}$$ 4.45

$$\begin{array}{cc}& {\int }_{{\mathbb{R}}^{3N-3}}{e}^{F(x,\widehat{\mathbf{x}})}{\mathrm{\nabla }}_x\varphi (x,\widehat{\mathbf{x}}){\partial }_{x_k}^{\alpha }|y-x_k|\overline{\psi (y,\widehat{\mathbf{x}})}\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\hfill \end{array}$$ 4.46

can be bounded as in (4.23).

Proof

We prove the bound for (4.45); the case of (4.46) is similar. We use the cutoff function $\xi \in C_c^{\infty }(\mathbb{R})$, introduced in (3.1), to define

$$\begin{array}{c}\hfill {\chi }_1(z):=\xi (8|z|),{\chi }_2(z):=1-{\chi }_1(z)\end{array}$$

$z\in {\mathbb{R}}^3$. Then ${\chi }_1(z)\ne 0$ implies $|z|<1/4$ and ${\chi }_2(z)\ne 0$ implies $|z|>1/8$.

We write the integral (4.45) as

$$\begin{array}{cc}& {\int }_{{\mathbb{R}}^{3N-3}}({\chi }_1(x-x_j)+{\chi }_2(x-x_j)){\partial }_{x_k}^{\alpha }|x-x_j|\psi (x,\widehat{\mathbf{x}})e^{F(y,\widehat{\mathbf{x}})}\overline{{\partial }_y^m\varphi (y,\widehat{\mathbf{x}})}\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}.\hfill \end{array}$$ 4.47

Expanding this and using, for example, the comment before Lemma 3.6, the integral involving ${\chi }_2$ can be bounded in absolute value by some constant multiplying

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^{3N-6}}{\int }_{\{x_j:|x-x_j|>1/8\}}|x-x_j{|}^{-3}|\psi (x,\widehat{\mathbf{x}})e^{F(y,\widehat{\mathbf{x}})}\mathrm{\nabla }\varphi (y,\widehat{\mathbf{x}})|\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}{x}_j\text{d}{\widehat{\mathbf{x}}}_{1,j}.\end{array}$$

We then use (2.9), (2.10) and that F is uniformly bounded on ${\mathbb{R}}^{3N}$ to bound this integral by some constant multiplying (3.29) of Proposition 3.10 with, say, $R=1/2$.

We now consider the ${\chi }_1$ term of (4.47). Firstly, we recall the notation introduced in (1.10)-(1.13); namely, we can write

$$\begin{array}{c}\hfill (y,x,{\widehat{\mathbf{x}}}_{1,j})=(y,x_2,\cdots ,x_{j-1},x,x_{j+1},\cdots ,x_N).\end{array}$$

Take any $\theta \in (0,1)$. We know that $\varphi \in C^{1,\theta }({\mathbb{R}}^{3N})$. In particular, using (2.9) and (2.10) there exists a constant C such that when $|x-x_j|<1/4$,

$$\begin{array}{cc}& |{\partial }_y^m\varphi (y,x,{\widehat{\mathbf{x}}}_{1,j})|\le {∥\mathrm{\nabla },\varphi ∥}_{L^{\infty }(B((y,\widehat{\mathbf{x}}),1/4))}\le C{f}_{\infty }(y,\widehat{\mathbf{x}}),\hfill \end{array}$$ 4.48

$$\begin{array}{cc}& |{\partial }_y^m\varphi (y,\widehat{\mathbf{x}})-{\partial }_y^m\varphi (y,x,{\widehat{\mathbf{x}}}_{1,j})|\le |x-x_j{|}^{\theta }{[\mathrm{\nabla }\varphi ]}_{\theta ,B((y,\widehat{\mathbf{x}}),1/4)}\le C{|x-x_j|}^{\theta }{f}_{\infty }(y,\widehat{\mathbf{x}}).\hfill \end{array}$$ 4.49

The constant C depends on $\theta$ but is independent of x, y and $\widehat{\mathbf{x}}$.

We now write the ${\chi }_1$ term in (4.47) as the sum of the following two integrals

$$\begin{array}{cc}& {\int }_{{\mathbb{R}}^{3N-3}}{\chi }_1(x-x_j){\partial }_{x_j}^{\alpha }|x-x_j|\psi (x,\widehat{\mathbf{x}})e^{F(y,\widehat{\mathbf{x}})}\overline{{\partial }_y^m\varphi (y,x,{\widehat{\mathbf{x}}}_{1,j})}\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}},\hfill \\ \hfill & {\int }_{{\mathbb{R}}^{3N-3}}{\chi }_1(x-x_j){\partial }_{x_j}^{\alpha }|x-x_j|\psi (x,\widehat{\mathbf{x}})e^{F(y,\widehat{\mathbf{x}})}\overline{({\partial }_y^m\varphi (y,\widehat{\mathbf{x}})-{\partial }_y^m\varphi (y,x,{\widehat{\mathbf{x}}}_{1,j}))}\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}.\hfill \end{array}$$ 4.50

The second integral can be bounded by some constant multiplying

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^{3N-3}}\lambda {(x,y,\widehat{\mathbf{x}})}^{-3+\theta }{f}_{\infty }(x,\widehat{\mathbf{x}})f_{\infty }(y,\widehat{\mathbf{x}})\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\end{array}$$

using (4.49) and that F is uniformly bounded. Lemma 4.3 can then be used to obtain the required bound.

It suffices to bound (4.50). Integration by parts in the variable $x_j$ is used in (4.50) to remove a single derivative from ${\partial }_{x_j}^{\alpha }|x-x_j|$. Take any $l\le \alpha$ with $|l|=1$. Integral (4.50) can therefore be rewritten as

$$\begin{array}{cc}& {\int }_{{\mathbb{R}}^{3N-3}}{\partial }^l{\chi }_1(x-x_j){\partial }_{x_j}^{\alpha -l}|x-x_j|\psi (x,\widehat{\mathbf{x}})e^{F(y,\widehat{\mathbf{x}})}\overline{{\partial }_y^m\varphi (y,x,{\widehat{\mathbf{x}}}_{1,j})}\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})\text{d}\widehat{\mathbf{x}}\hfill \\ \hfill & -{\int }_{{\mathbb{R}}^{3N-3}}{\chi }_1(x-x_j){\partial }_{x_j}^{\alpha -l}|x-x_j|\overline{{\partial }_y^m\varphi (y,x,{\widehat{\mathbf{x}}}_{1,j})}{\partial }_{x_j}^l(e^{F(y,\widehat{\mathbf{x}})}\psi (x,\widehat{\mathbf{x}})\mathrm{\Phi }(x,y,\widehat{\mathbf{x}}))\text{d}\widehat{\mathbf{x}}.\hfill \end{array}$$

We can use (4.30), (4.48) and the fact that both F and $\mathrm{\nabla }F$ are uniformly bounded, to bound this in absolute value by some constant multiplying

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^{3N-3}}\lambda {(x,y,\widehat{\mathbf{x}})}^{-2}{f}_{\infty }(x,\widehat{\mathbf{x}})f_{\infty }(y,\widehat{\mathbf{x}})(\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})+|\mathrm{\nabla }\mathrm{\Phi }(x,y,\widehat{\mathbf{x}})|)\text{d}\widehat{\mathbf{x}}.\end{array}$$

The relevant bound then follows by Lemma 4.3 and Corollary 4.4. $\square$

Appendix A. Second Derivatives of $\varphi$

In [22], it was shown that $\varphi$, as defined in (2.7), has improved smoothness upon multiplication by a certain exponential factor, depending only on N and Z. It is the aim of this section to use these results to give pointwise bounds to the second derivatives of $\varphi$ itself, as required in the proof of Theorem 1.3.

The above authors introduce the following functions

$$\begin{array}{cc}\hfill \tilde{F}& =-\frac{Z}{2}\sum _{j=1}^N\xi (|x_j|)|x_j|+\frac{1}{4}\sum _{1\le j<k\le N}\xi (|x_j-x_k|)|x_j-x_k|,\hfill \\ \hfill \tilde{G}& =C_{0}Z\sum _{1\le j<k\le N}\xi (|x_j|)\xi (|x_k|)(x_j·x_k)log(|x_j{|}^2+{|x_k|}^2),\hfill \end{array}$$

where $\xi \in C_c^{\infty }(\mathbb{R})$ is defined in (3.1) and $C_0=(2-\pi )/12\pi$. The authors obtain $e^{-\tilde{F}-\tilde{G}}\psi \in W_{\mathit{loc}}^{2,\infty }({\mathbb{R}}^{3N})$, [22, Theorem 1.5]. We note that this is an improvement upon the well-known fact that $\varphi \in C^{1,\theta }({\mathbb{R}}^{3N})$ for all $\theta \in (0,1)$, see Sect. 2.

For our purposes, it will be convenient to restate their results using a slightly different form of the exponential factor. We define

$$\begin{array}{c}\hfill {\varphi }^{\prime }=e^{-F-G}\psi \end{array}$$

where $F=F_c-F_s$ was defined in (2.2) and

$$\begin{array}{cc}\hfill G& =C_{0}Z\sum _{j<k}(x_j·x_k)log(|x_j{|}^2+|x_k{|}^2)-C_{0}Z\sum _{j<k}(x_j·x_k)log(|x_j{|}^2+{|x_k|}^2+1)\hfill \\ \hfill & =:G_c-G_s.\hfill \end{array}$$

We can then write ${\varphi }^{\prime }=e^H{e}^{-\tilde{F}-\tilde{G}}\psi$ for

$$\begin{array}{c}\hfill H=-F_c+F_s+\tilde{F}-G_c+G_s+\tilde{G}.\end{array}$$

It can be verified, with the help of (2.5) and (2.6) and direct calculation, that H is smooth and ${\partial }^{\alpha }H\in L^{\infty }({\mathbb{R}}^{3N})$ for all $|\alpha |\le 2$, $\alpha \in {\mathbb{N}}_0^{3N}$. Therefore, we can restate [22, Theorem 1.5] as follows.

Theorem A.1

(S. Fournais, M. and T. Hoffmann-Ostenhof, T. Ø. Sørensen). For all $0<r<R$, we have a constant C, depending on r and R, such that

$$\begin{array}{c}\hfill {∥{\varphi }^{\prime }∥}_{W^{2,\infty }(B(\mathbf{x},r))}\le C{∥{\varphi }^{\prime }∥}_{L^{\infty }(B(\mathbf{x},R))}\end{array}$$ Appendix A.1

for all $\mathbf{x}\in {\mathbb{R}}^{3N}$. The constant does not depend on $\mathbf{x}$.

Remark

We use the equivalence of the spaces $C^{1,1}(\overline{B})$ and $W^{2,\infty }(B)$ in the case of an open ball B, see, for example, [23].

The following lemma can be verified by straightforward calculations. Here, $e_i$ represents the i-th unit basis vector in ${\mathbb{R}}^3$, $i=1,2,3$. We understand these as multiindicies in ${\mathbb{N}}_0^3$.

Lemma A.2

$G,\mathrm{\nabla }G\in L^{\infty }({\mathbb{R}}^{3N})$. Also, for all $j,k=1,\cdots N$ and $r,s\in \{1,2,3\}$ there exists some C such that

$$\begin{array}{c}\hfill |{\partial }_j^r{\partial }_k^{s}G(\mathbf{x})|\le \left\{\begin{array}{cc}C(1-log(min\{1,|x_j{|}^2+|x_k{|}^2\}))\hfill & \text{if}j\ne k\text{and}r=s\hfill \\ C\hfill & \text{otherwise}.\hfill \end{array}\right)\end{array}$$

Since $\varphi$ does not contain the additional exponential factor involving G, we do not expect it’s second-order derivatives to be bounded. However, we obtain the following corollary of Theorem A.1 to obtain pointwise bounds showing the second-order derivatives of $\varphi$ have only logarithmic singularities. We recall that ${\nu }_P$ was defined in (1.22).

Corollary A.3

For all $0<r<R<1$, there exists C, depending on r and R, such that for any non-empty cluster P and any $\eta \in {\mathbb{N}}_0^3$ with $|\eta |=1$,

$$\begin{array}{c}\hfill {∥D_P^{\eta },\mathrm{\nabla },\varphi ∥}_{L^{\infty }(B(\mathbf{x},r{\nu }_P(\mathbf{x})))}\le C(1-log{\nu }_P(\mathbf{x})){∥\varphi ∥}_{L^{\infty }(B(\mathbf{x},R))}\end{array}$$

for all $\mathbf{x}$ with ${\nu }_P(\mathbf{x})>0$.

Remark A.4

By the definitions, it is immediate that ${\nu }_P\ge {\lambda }_P$ and hence, in particular, the above inequality holds on ${\mathrm{\Sigma }}_P^c$.

Proof

By the definition of cluster derivatives, (1.15), it follows from Lemma A.2 that there is some C such that

$$\begin{array}{c}\hfill |D_P^{\eta }\mathrm{\nabla }G(\mathbf{x})|\le C(1-log({\nu }_P(\mathbf{x})))\end{array}$$ Appendix A.2

for all $\mathbf{x}$ with ${\nu }_P(\mathbf{x})>0$.

By (1.23), we get that $(1-r){\nu }_P(\mathbf{x})\le {\nu }_P(\mathbf{y})$ for all $\mathbf{y}\in B(\mathbf{x},r{\nu }_P(\mathbf{x}))$. Therefore, for C as in (Appendix A.2), we have for all $\mathbf{x}$ with ${\nu }_P(\mathbf{x})>0$,

$$\begin{array}{cc}\hfill {∥D_P^{\eta },\mathrm{\nabla },G∥}_{L^{\infty }(B(\mathbf{x},r{\nu }_P(\mathbf{x})))}& \le C(1-log(1-r)-log({\nu }_P(\mathbf{x})))\hfill \end{array}$$ Appendix A.3

$$\begin{array}{cc}& \le C^{\prime }(1-log({\nu }_P(\mathbf{x})))\hfill \end{array}$$ Appendix A.4

for some $C^{\prime }$ depending on r.

We recall from the definition, (2.7), that $\varphi =e^{-F}\psi$. Therefore, $\varphi =e^G{\varphi }^{\prime }$. Then, we have $\mathrm{\nabla }\varphi =e^G{\varphi }^{\prime }\mathrm{\nabla }G+e^G\mathrm{\nabla }{\varphi }^{\prime }$. And therefore the following formula holds

$$\begin{array}{c}\hfill D_P^{\eta }\mathrm{\nabla }\varphi =(D_P^{\eta }\mathrm{\nabla }G+D_P^{\eta }G\mathrm{\nabla }G)e^G{\varphi }^{\prime }+e^G{D}_P^{\eta }{\varphi }^{\prime }\mathrm{\nabla }G+e^G{D}_P^{\eta }G\mathrm{\nabla }{\varphi }^{\prime }+e^G{D}_P^{\eta }\mathrm{\nabla }{\varphi }^{\prime }.\end{array}$$

Taking the norm and using that $G,\mathrm{\nabla }G\in L^{\infty }$ by Lemma A.2, we can then obtain C such that

$$\begin{array}{cc}\hfill {∥D_P^{\eta },\mathrm{\nabla },\varphi ∥}_{L^{\infty }(B(\mathbf{x},r{\nu }_P(\mathbf{x})))}& \le C(1-log({\nu }_P(\mathbf{x}))){∥{\varphi }^{\prime }∥}_{W^{2,\infty }(B(\mathbf{x},r{\nu }_P(\mathbf{x})))}.\hfill \end{array}$$ Appendix A.5

Using that ${\nu }_P\le 1$, we then apply Theorem A.1 followed by a use of the equality $\varphi =e^G{\varphi }^{\prime }$ to obtain the required result. $\square$

Funding

Open access funding provided by Copenhagen University. This work was partially supported by the Villum Centre of Excellence for the Mathematics of Quantum Theory (QMATH) with Grant No.10059, along with previous support by the EPSRC grant EP/W522636/1.

Declarations

Conflict of interest

The author has no conflict of interest to declare that are relevant to the content of this article.