Content-Type: multipart/mixed; boundary="-------------0312230733362" This is a multi-part message in MIME format. ---------------0312230733362 Content-Type: text/plain; name="03-549.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-549.comments" 2000 Mathematics Subject Classification 35B65, 35J10, 35B45, 81Q05, 35J15, 81V55 ---------------0312230733362 Content-Type: text/plain; name="03-549.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-549.keywords" Elliptic PDE, Schr\"odinger Operators, Regularity of Solutions ---------------0312230733362 Content-Type: application/x-tex; name="C-1-1.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="C-1-1.tex" %\documentclass[invmat,numbook]{svjour} \documentclass[leqno]{amsart} \NeedsTeXFormat{LaTeX2e}[1994/12/01] \usepackage{amsmath} \usepackage{amsfonts} %\usepackage{hyperref} \usepackage{enumerate} \theoremstyle{plain} \newtheorem{thm}{Theorem}[section]{\bf}{\it} \newtheorem{prop}[thm]{Proposition}{\bf}{\it} \newtheorem{lemma}[thm]{Lemma}{\bf}{\it} \newtheorem{lem}[thm]{Lemma}{\bf}{\it} \newtheorem{cor}[thm]{Corollary}{\bf}{\it} \newtheorem*{lem3.1'}{Lemma 3.1'}{\bf}{\it} \newtheorem*{lem3.4'}{Lemma 3.4'}{\bf}{\it} \newtheorem*{lem3.5'}{Lemma 3.5'}{\bf}{\it} \newtheorem{defn}[thm]{Definition}{\bf}{\rm} \newtheorem{rem}[thm]{Remark}{\it}{\rm} \newtheorem{remark}[thm]{Remark}{\it}{\rm} \newtheorem*{acknowledgement}{Acknowledgement} \newenvironment{pf}{\par\medskip\noindent\textit{Proof}:\,}{\hspace*{\fill}\qed\medskip\par\noindent} \newenvironment{pf*}[1]{\par\medskip\noindent\textit{#1}\,:}{\hspace*{\fill}\qed\medskip\par\noindent} \numberwithin{equation}{section} %\numberwithin{theorem}{section} \newcommand{\sphere}{{\mathbb S}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\Span}{\operatorname{Span}} \newcommand{\dist}{{\operatorname{dist}}} \newcommand{\Vol}{{\operatorname{Vol}}} \newcommand{\Ran}{{\operatorname{Ran}}} \newcommand{\Tr}{{\operatorname{Tr}}} \newcommand{\N}{{\mathbb N}} \newcommand{\R}{{\mathbb R}} \newcommand{\C}{{\mathbb C}} \begin{document} \title[Sharp Regularity for Wave Functions]{Sharp Regularity Results for Many-Electron Wave Functions\footnote{\copyright\ 2003 by the authors. This article may be reproduced in its entirety for non-commercial purposes.}} %\titlerunning{Sharp Regularity for Wave Functions} \author[S. Fournais and M. \& T. Hoffmann-Ostenhof and T. \O. S\o rensen]{S\o ren Fournais, Maria Hoffmann-Ostenhof, Thomas Hoffmann-Ostenhof and Thomas \O stergaard S\o rensen} %\authorrunning{S. Fournais and M. \& T. Hoffmann-Ostenhof and % T. \O. S\o rensen} \address[S. Fournais] {CNRS and Laboratoire de Math\'{e}matiques, UMR CNRS 8628, Universit\'{e} Paris-Sud - B\^{a}t 425, F-91405 Orsay Cedex, France} \address[M. Hoffmann-Ostenhof] {Institut f\"ur Mathematik, Universit\"at Wien, Nordbergstra\ss e 15, A-1090 Vienna, Austria} \address [T. Hoffmann-Ostenhof] {The Erwin Schr\"odinger International Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria} \address[T. Hoffmann-Ostenhof] {Institut f\"ur Theoretische Chemie, Universit\"at Wien, W\"ahringer\-stra\ss e 17, A-1090 Vienna, Austria} \address[T. \O stergaard S\o rensen (on leave from)] {Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg East, Denmark} \address[T. \O stergaard S\o rensen (present address)] {Mathematisches Institut, Universit\"at M\"unchen, Theresienstra\ss e 39, D-80333 Munich, Germany} \address[]{} \email[S. Fournais]{soeren.fournais@math.u-psud.fr} \email[M. Hoffmann-Ostenhof]{maria.hoffmann-ostenhof@univie.ac.at} \email[T. Hoffmann-Ostenhof]{thoffman@esi.ac.at} \email[T. \O. S\o rensen]{sorensen@mathematik.uni-muenchen.de} %\date{\today} \begin{abstract} We show that electronic wave functions $\psi$ of atoms and molecules have a representation $\psi=\mathcal F \phi$, where $\mathcal F$ is an explicit universal factor, locally Lipschitz, and independent of the eigenvalue and the solution $\psi$ itself, and $\phi$ has locally bounded second derivatives. This representation turns out to be optimal as can already be demonstrated with the help of hydrogenic wave functions. The proofs of these results are, in an essential way, based on a new elliptic regularity result which is of independent interest. Some identities that can be interpreted as cusp conditions for second order derivatives of $\psi$ are derived. \end{abstract} \maketitle \section{Introduction} \subsection{Motivation and results} \label{chap:intro} The non-relativistic quantum mechanical Hamiltonian of an \(N\)-elec\-tron molecule with L fixed nuclei is given by \begin{equation*} H_{N,L}(\mathbf X,\mathbf Z)=-\Delta+V(\mathbf X,\mathbf Z)+U(\mathbf X,\mathbf Z), \end{equation*} where $V$, the Coulombic potential, is given by \begin{equation}\label{V} V\equiv V(\mathbf X,\mathbf Z)=-\sum_{j=1}^N\sum_{k=1}^L\frac{Z_k}{|X_k-x_j|} +\sum_{1\le i0$ and if we had an $\mathcal F$ which would allow more regularity of the \(\phi^{(i)}\)'s,\, then \begin{equation}\label{quotient} \frac{\phi^{(2)}}{\phi^{(1)}}= \frac{\psi_2}{\psi_1}=x_1e^{\frac{Z}{4}|x|} \end{equation} would be better behaved than just $C^{1,1}$. But near the origin the right hand side of \eqref{quotient} behaves like $x_1(1+\tfrac{Z}{4}|x|)$ and this is just $C^{1,1}$, i.e., the second derivatives are bounded but not continuous. \end{pf*} The results in Theorem \ref{thm:main:Jastrow} are not well suited for obtaining {\it a~priori} estimates. In particular neither $F_2$ nor $F_3$ stay bounded as $|\mathbf x|$ tends to infinity so that if, say, $\psi\in L^2(\mathbb R^{3N})$ then $\phi_3$ is not necessarily in $L^2(\mathbb R^{3N})$. These shortcomings will be dealt with below in a similar way as in \cite{HHO:2001}. \begin{defn}\label{Cutoff} Let \(\chi\in C_{0}^{\infty}(\R)\), \(0\leq \chi\leq1\), with \begin{align} \label{eq:def_cutoff} \chi(x)= \begin{cases} 1& \text{ for } |x|\leq1 \\ 0& \text{ for } |x|\geq2. \end{cases} \end{align} We define \begin{equation}\label{F23cut} F_{\text{\rm cut}}=F_{2,\text{\rm cut}}+F_{3,\text{\rm cut}}, \end{equation} where \begin{align} \label{eq:def_F2cut} &F_{2,\text{\rm cut}}({\mathbf x})=-\frac{1}{2}\sum_{\ell=1}^{L}\sum_{i=1}^NZ_{\ell}\, \chi(|y_{i,\ell}|)\,|y_{i,\ell}| \\&\qquad\qquad\qquad\qquad\qquad\qquad +\frac{1}{4} \sum_{1\le i0\) around \(x\). We denote by \(Y_{l,m}(\omega)\) the normalised (in \(L^{2}(\sphere^{n-1})\)) real valued spherical harmonics of degree \(l, l\in\N_{0}\), with \(m=1,\ldots,h(l)-1\), where \begin{align} \label{eq:h(l)} h(l)=\frac{(2l+n-2)(l+n-3)!}{(n-2)!\,l\,!}. \end{align} Then \(\{Y_{l,m}\}_{l,m}\) constitutes an orthonormal basis in \(L^{2}(\sphere^{n-1})\). The \(Y_{l,m}\)'s are the eigenfunctions for \(\mathcal{L}^{2}\), the Laplace-Beltrami operator on \(\sphere^{n-1}\): \begin{align*} \mathcal{L}^{2}Y_{l,m}=l(l+n-2)Y_{l,m}, \end{align*} where \({}-\frac{\mathcal{L}^{2}}{r^{2}}\) is the angular part of the Laplacian in \(\R^{n}\), so \begin{align*} {}-\Delta={}-\frac{\partial^{2}}{\partial r^{2}}-\frac{n-1}{r}\frac{\partial}{\partial r}+\frac{\mathcal{L}^{2}}{r^{2}}. \end{align*} We define \(\mathcal{P}_{l,m}^{(n)}\) to be the orthogonal projection in \(L^{2}(\sphere^{n-1})\) on \(Y_{l,m}\): \begin{align*} \big(\mathcal{P}_{l,m}^{(n)}f\big)(\omega) =Y_{l,m}(\omega)\int_{\sphere^{n-1}}Y_{l,m}(\omega)f(\omega)\,d\omega \quad,\quad f\in L^{2}(\sphere^{n-1}), \end{align*} and \begin{align} \label{def:proj} \mathcal{P}_{l}^{(n)}=\sum_{m=0}^{h(l)-1}\mathcal{P}_{l,m}^{(n)}. \end{align} We denote \(\mathfrak{h}_{l}^{(n)}=\Ran(\mathcal{P}_{l}^{(n)})\). By abuse of notation, for a function \(\ f:\R^{n}\to\C\) we write \(f(r\omega)=f(x)\), and, whenever \(f(r_{0}\,\cdot):\sphere^{n-1}\to\C\) is in \(L^{2}(\sphere^{n-1})\) for some \(r_{0}\in(0,\infty)\), we write \begin{align*} \big(\mathcal{P}_{l,m}^{(n)}f\big)(r_{0}\omega) =Y_{l,m}(\omega)\int_{\sphere^{n-1}}Y_{l,m}(\omega)f(r_{0}\omega)\,d\omega \equiv f_{l,m}(r_{0})Y_{l,m}(\omega). \end{align*} \section{Elliptic regularity} \label{chap:GT} In this section we collect results on regularity of solutions to second order elliptic equations needed for the proof of Theorems~\ref{thm:main:Jastrow}. and \ref{thm:main:apriori}. The results fall in two parts, known ones (in subsection~\ref{subsection4.1}) and new ones, developed for our purpose, and of interest in themselves. The latter are in subsection~\ref{subsection4.2}. The result of main interest is Theorem~\ref{thm:abstract}, which is proved in subsection~\ref{subsection4.3}. \subsection{Known results} \label{subsection4.1} $\, $ We start by recalling the definition of H\"older continuity: \begin{defn} \label{def:Holder} Let \(\Omega\) be a domain in \(\R^{n}\), \(k\in\N\), and \(\alpha\in(0,1]\). We say that a function \(u\) belongs to \(C^{k,\alpha}(\Omega)\) whenever \(u\in C^{k}(\Omega)\), and for all \(\beta\in\N^{n}\) with \(|\beta|=k\), and all open balls $B_{n}(x_{0},r)$ with $\overline{B_{n}(x_{0},r)}\subset\Omega $, we have \begin{align*} \sup_{x,y\in B_{n}(x_{0},r),\,x\neq y} \!\!\!\!\!\!\!\!\! \frac{|D^{\beta}u(x)-D^{\beta}u(y)|}{|x-y|^{\alpha}} \leq C(x_{0},r). \end{align*} For any domain \(\Omega'\), with \(\Omega'\subset\subset\Omega\), we define the following norms: \begin{align} \nonumber \|u\|_{C^{k,\alpha}(\Omega')}&=\sum_{|\beta| \leq k}\|D^{\beta} u\|_{L^{\infty}(\Omega')} +[u]_{k,\alpha,\Omega'}, \\\nonumber [u]_{k,\alpha,\Omega'} &=\sum_{|\beta| = k}\sup_{x,y\in\Omega',\,x\neq y} \frac{|D^{\beta}u(x)-D^{\beta} u(y)|}{|x-y|^{\alpha}}. \end{align} For \(k=0\) we use the notation \(C^{\alpha}(\Omega)\equiv C^{0,\alpha}(\Omega)\) and \([u]_{\alpha,\Omega'}\equiv[u]_{0,\alpha,\Omega'}\). Furthermore, for a function \(u\in C^{\alpha}(\R^{n}\setminus\{0\})\) we define \begin{align} \|u\|_{C^{\alpha}(\sphere^{n-1})}&=\sup_{\sphere^{n-1}}|u| + [u]_{\alpha,\sphere^{n-1}}, \\ [u]_{\alpha,\sphere^{n-1}}&= \sup_{x,y\in\sphere^{n-1},\,x\neq y}\frac{|u(x)-u(y)|}{|x-y|^{\alpha}}. \nonumber \end{align} \end{defn} We will need the following result on elliptic regularity in order to conclude that the solutions of elliptic second order equations with bounded coefficients are $C^{1,\alpha}$. The proposition is a reformulation of Corollary~8.36 in Gilbarg and Trudinger~\cite{GandT}, adapted for our purposes: \begin{prop} \label{prop:GTbis} Let \(\Omega_{0}\) be a bounded domain in \({\mathbb R}^{n}\) and suppose \(u\in W^{1,2}(\Omega_{0})\) is a weak solution of \(\Delta u +\sum_{j=1}^{n}b_{j}D_{j}u+Wu=g\) in \(\Omega_{0}\), where \(b_{j},W,g\in L^{\infty}(\Omega_{0})\). Then \(u\in C^{1,\alpha}(\Omega_{0})\) for all \(\alpha\in(0,1)\) and for any domains \(\Omega',\Omega\), \(\overline{\Omega'}\subset\Omega\), \(\overline{\Omega}\subset\Omega_{0}\) we have \begin{align*} \|u\|_{C^{1,\alpha}(\Omega')}\leq C\big(\sup_{\Omega}|u|+\sup_{\Omega}|g|\big) \end{align*} for \(C=C(\alpha,n,M,\dist(\Omega',\partial\Omega))\), with \begin{align} \max \{1,\max_{j=1,\ldots,n}\|b_{j}\|_{L^{\infty}(\Omega)}, \|W\|_{L^{\infty}(\Omega)},\|g\|_{L^{\infty}(\Omega)}\} \leq M. \nonumber \end{align} \end{prop} We further need results concerning the regularity of solutions of the Poisson equation. These regularity properties are based on the regularity properties of the Newton potential of the inhomogeneity. For our further considerations we recall here the properties of this function. Let \(g\in L^{\infty}(\Omega)\) for \(\Omega\) a bounded domain in \(\R^{n}\), \(n\geq2\). The Newton potential of \(g\) is the function \(w\) defined on \(\R^{n}\) by \begin{align} \label{Newton} w(x)&=\int_\Omega\Gamma(x-y)g(y)dy \intertext{with} \Gamma(x)&=\left\{ \begin{array}{ll} \frac{1}{2\pi}\ln(|x|),& n=2, \\ \frac{1}{(2-n)\; |\sphere^{n-1}|}\;|x|^{2-n}, & n\geq 3. \\ \end{array}\right. \nonumber \end{align} From \cite[Theorem 10.2 and 10.3]{Li-loss} we have \begin{prop} \label{prop:newton} Let \(\Omega\subset\R^{n}, n\geq2\), be a bounded domain, then: \begin{enumerate}[\rm (i)] \item If \(g\in L^{\infty}(\Omega)\), then \(w\in C^{1,\alpha}(\Omega)\) for all \(\alpha\in(0,1)\), and \(\Delta w=g\) in \(\Omega\) holds in the distributional sense. \item If \(g\in C^{k,\alpha}(\Omega)\) for some \(k\in\N\) and some \(\alpha\in(0,1)\), then \(w\in C^{k+2,\alpha}(\Omega)\). \end{enumerate} \end{prop} Since every solution to the Poisson equation can be written as a sum of the Newton potential of the inhomogeneity and a harmonic function, the above implies in particular the following well-known result: \begin{prop} \label{prop:GT} Let \(g\in C^{k,\alpha}(\Omega_{0})\) for some \(k\in\N\) and some \(\alpha\in(0,1)\), and assume \(u\) is a weak solution to \(\Delta u=g\) in \(\Omega_{0}\). Then \(u\in C^{k+2,\alpha}(\Omega_{0})\). Furthermore, for any domains \(\Omega',\Omega\), \(\overline{\Omega'}\subset\Omega\), \(\overline{\Omega}\subset\Omega_{0}\), we have \begin{align} \|u\|_{C^{k+2,\alpha}(\Omega')}\leq C\big(\sup_{\Omega}|u|+ \|g\|_{C^{k,\alpha}(\Omega)}\big), \end{align} with \(C=C(n, k, \alpha,\dist(\Omega',\partial\Omega))\). \end{prop} The next lemma, which is taken from Gilbarg and Trudinger~\cite[Lemma 4.2]{GandT}, is essential for the proof of the main regularity result in subsection~\ref{subsection4.2}. \begin{lem} \label{GT42} Let \(\Omega\) be a bounded domain in \(\R^{n}, \: n\ge 2\) and let \(g\in C^\alpha(\Omega)\cap L^{\infty}(\Omega)\) for some \(\alpha\in (0,1]\). Then the Newton potential \(w\) of \(g\) (given in \eqref{Newton}) satifies, for \(x\in\Omega\) and \(i,j=1,2,\dots,n\), \begin{align} \label{Dij} D_{ij}w(x)&=\int_{\Omega_0}D_{ij}\Gamma(x-y)\big(g(y)-g(x)\big)\,dy \nonumber\\ &\quad\quad\quad -g(x)\int_{\partial\Omega_0}D_i\Gamma(x-y)\nu_j(y)\,d\sigma(y). \end{align} Here, \(\Omega_0\) is any bounded domain containing \(\Omega\) for which the divergence theorem holds, and \(g\) is extended to vanish outside \(\Omega\). In the last integral, \(d\sigma\) denotes the surface measure of \(\partial\Omega_{0}\), and \(\nu_{j}\) the \(j\)-th coordinate of its (outwards directed) normal vector. \end{lem} \subsection{New results} \label{subsection4.2} $\, $ We here collect a number of more explicit regularity results needed in the proof of Theorems~\ref{thm:main:Jastrow} and \ref{thm:main:apriori}. The following result shows that one can push the $C^{1,\alpha}, \alpha\in(0,1)$, in Proposition~\ref{prop:GTbis} to $C^{1,1}$ in certain cases. \begin{thm}\label{thm:abstract} Let \(g\in L^\infty(\mathbb R^k)\), \(k\geq2\), be a homogeneous function of degree \(0\) which has the properties $g\in C^\alpha(\R^{k}\setminus\{0\})$ and $g|_{\sphere^{k-1}}$ is orthogonal to \(\mathfrak{h}_{2}^{(k)}\) (the subspace of \(L^2(\sphere^{k-1})\) spanned by the spherical harmonics of degree \(2\)). Let \(f\in C^\alpha(\R^{d})\) for some \(d\ge0\) and let \(u\in C^{1,\alpha}(\R^{k+d})\) be a weak solution of the equation \begin{equation} \label{eq:Poisson} \Delta u(x',x'')=g(x')f(x'') \end{equation} where \(x'\in\R^{k}, \:x''\in\mathbb R^d\), \(\Delta=\Delta_{x'}+\Delta_{x''}\). Then \(u\in W_{\rm loc}^{2,\infty}(\R^{n})\), \(n=k+d\), and the following {\it a~priori} estimate holds:\\ For all balls \(B_{n}(z,R)\) and \(B_{n}(z,R_1)\) in \(\R^{n}\) where $00\); for \(d=0, \pi_{d}(x')=0\). \end{thm} \begin{rem} \label{rem:abstract} \(\, \) \begin{enumerate}[\rm (i)] \item \label{abstract(ii)} The case $d=0$ means that $f$ is a constant and the terms in \eqref{eq:apriori} with $f$ then equal this constant. \item\label{abstract(iii)} The reason for the condition \(k\geq2\) will become clear in the proof of the theorem, when Lemma~\ref{GT42} is applied. \item\label{abstract(iv)} Note that if \(k=0, d\geq2\), one has stronger conclusions: Equation \eqref{eq:Poisson} becomes \(\Delta u(y)=f(y)\) with \(f\in C^{\alpha}(\R^{d})\), so by Proposition~\ref{prop:GT}, \(u\in C^{2,\alpha}(\R^{d})\). The {\it a~priori} estimate analogous to \eqref{eq:apriori} is then a consequence of H\"older-estimates for \(u\) (see e.~g., \cite[Corollary 6.3]{GandT}). \item\label{abstract(v)} Using the standard fact (\cite[Theorem 4 in 5.8]{Evans}) that \(W^{2,\infty}_{\text{\rm loc}}(\R^{n})\) \(=C^{1,1}_{\text{\rm loc}}(\R^{n})\) (with equivalent norms) we may replace the term \(\sup_{B_{n}(z,R)}|D_{ij}u|\) by \([u]_{1,1,B_{n}(z,R)}\) on the left hand side in \eqref{eq:apriori}. \item\label{abstract(vi)} For the special solution to \eqref{eq:Poisson} given by the Newton potential of $g f$, the estimate \eqref{eq:apriori} holds without the term $\sup_{B_n(z,R_1)} |u|$ on the right hand side (see \eqref{eq:apriori1}). \end{enumerate} \end{rem} Since the proof of Theorem~\ref{thm:abstract} is a bit lenghty we present it separately in subsection~\ref{subsection4.3}. The following proposition, on solutions to Poisson's equation, when the inhomogeneity \(f\) in \(\Delta u=f\) is a homogeneous function, is needed often in the paper. \begin{prop} \label{prop:homogen} Assume that the function \(g\) satisfies\par\noindent \(g(r\omega)=r^{k}G(\omega)\) with \(G\in L^{\infty}(\sphere^{n-1})\) and \(\mathcal{P}_{k+2}^{(n)}G=0\). Then there exists a solution \(u\) to \begin{align} \label{eq:homogen} \Delta u=g \quad \text{ on } \quad B_{n}(0,R)\subset\R^{n}, \end{align} satisfying \(u(r\omega)=r^{k+2}U(\omega)\) with \(U\in C^{1,\alpha}(\sphere^{n-1})\) for all \(\alpha\in(0,1)\). \end{prop} \begin{pf} Let \begin{align*} g_{l,m}(r)=\int_{\sphere^{n-1}}g(r\omega)Y_{l,m}(\omega)\,d\omega =r^{k}\int_{\sphere^{n-1}}G(\omega)Y_{l,m}(\omega)\,d\omega =r^{k}g_{l,m}. \end{align*} Then (see \eqref{eq:h(l)} for \(h(l)\)) \begin{align*} g(r\omega)=r^{k}\sum_{l=0,l\neq k+2}^{\infty}\sum_{m=0}^{h(l)-1} g_{l,m}Y_{l,m}(\omega), \end{align*} since \(g_{k+2,m}=0\) for all \(m\). Now define \begin{align} \label{eq:def_U} U(\omega)=\sum_{l=0,l\neq k+2}^{\infty}\sum_{m=0}^{h(l)-1} \frac{g_{l,m}}{b_{l}(n,k)}\,Y_{l,m}(\omega) \end{align} with \(b_{l}(n,k)\equiv(k+2)((k+2)+n-2)-l(l+n-2)\). Note that \(b_{l}(n,k))\neq 0\) for \(l\neq k+2\). Since \(\sum_{l,m}g_{l,m}Y_{l,m}\in L^{2}(\sphere^{n-1})\) (since \(G\in L^{\infty}(\sphere^{n-1})\)) the sum \eqref{eq:def_U} therefore converges in \(L^{2}(\sphere^{n-1})\). Make the `Ansatz' \(u(r\omega)=r^{k+2}U(\omega)\), and denote for \(N\in\N\) \begin{align*} g_{N}(r\omega)&= \sum_{l=0,l\neq k+2}^{N}\sum_{m=0}^{h(l)-1} g_{l,m}r^{k}\,Y_{l,m}(\omega), \\ u_{N}(r\omega)&=r^{k+2} \sum_{l=0,l\neq k+2}^{N}\sum_{m=0}^{h(l)-1} \frac{g_{l,m}}{b_{l}(n,k)}\,Y_{l,m}(\omega). \end{align*} Now let \(\phi\in C_{0}^{\infty}\big(B_{n}(0,R)\big)\), then, using that \(\mathcal{L}^{2}Y_{l,m}=l(l+n-2)Y_{l,m}\), \begin{align} \label{eq:indskud} \int_{B_{n}(0,R)}\!\!\!\!\!\!\phi(\Delta u-g)\, dx=\int_{B_{n}(0,R)}\!\!\!\!\!\!(\Delta\phi)(u-u_{N})\,dx +\int_{B_{n}(0,R)}\!\!\!\!\!\!\phi(g_{N}-g)\,dx. \end{align} Since \(u-u_{N}\to0, g-g_{N}\to 0\) (in \(L^{2}\) - sense) for \(N\to0\), the RHS of \ref{eq:indskud} tends to zero for \(N\to0\). Hence \(u=r^{k+2}U(\omega)\) solves \ref{eq:homogen} in the distributional sense. With \(w\) the Newton potential corresponding to \(g\) (see \ref{Newton}), we have \(w\in C^{1,\alpha}(B_{n}(0,R))\) due to Proposition~\ref{prop:newton}, and \(u-w\) is harmonic, so \(u\in C^{1, \alpha}(B_{n}(0,R))\). This implies that \(U\in C^{1,\alpha}(\sphere^{n-1})\). \end{pf} We prove the following useful lemma: \begin{lem} \label{lem:XdotG} Let \(G:U\to\R^{n}\) for \(U\subset\R^{n+m}\) a neighbourhood of a point \((0,y_{0})\in\R^{n}\times\R^{m}\). Assume \(G(0,y)=0\) for all \(y\) such that \((0,y)\in U\). Let \begin{align*} f(x,y)=\left\{\begin{array}{cc} \frac{x}{|x|}\cdot G(x,y)& x\neq 0, \\ 0& x=0. \\ \end{array}\right. \end{align*} Then, for \(\alpha\in(0,1]\), \begin{align} \label{eq:lem_G=0} G\in C^{0,\alpha}(U;\R^{n})\Rightarrow f\in C^{0,\alpha}(U). \end{align} Furthermore, $\| f \|_{C^{\alpha}(U)} \leq 2\| G \|_{C^{\alpha}(U)}$. \end{lem} \begin{pf} Let \(\alpha \in (0,1]\). We need to estimate \(\frac{f(x_1,y_1)-f(x_2,y_2)}{|(x_1,y_1)-(x_2,y_2)|^{\alpha}}\). Suppose first that $x_2=0$. Then $f(x_2,y_2)=0$ and we get \begin{align*} \frac{|f(x_1,y_1)-f(0,y_2)|}{|(x_1,y_1)-(0,y_2)|^{\alpha}} &\leq\frac{\left|\frac{x_1}{|x_1|}\cdot G(x_1,y_1)\right|}{|x_1|^{\alpha}} \leq \left|\frac{x_1}{|x_1|}\right|\cdot\frac{|G(x_1,y_1)|}{|x_1|^{\alpha}}\\ &\leq \| G \|_{C^{\alpha}(U)}, \end{align*} since \(G\in C^{\alpha}(U;\R^{n})\) and \(G(0,y_1)=0\). Next, suppose $0<|x_2| \leq |x_1|$. By the triangle inequality: \begin{align*} \big|f(x_1,y_1)-f(x_2,y_2)\big| &\leq \Big|\frac{x_1}{|x_1|}\cdot\big(G(x_1,y_1)-G(x_2,y_2)\big)\Big| \\&\quad + \Big|( \frac{x_1}{|x_1|} - \frac{x_2}{|x_2|})\cdot G(x_2,y_2)\Big| . \end{align*} Using that $G$ is $C^{\alpha}$ and that $G(0,y_2) = 0$, we get \begin{align*} \big|f&(x_1,y_1)-f(x_2,y_2)\big| \\&\leq \| G \|_{C^{\alpha}(U)}\left(\big|(x_1,y_1)-(x_2,y_2)\big|^{\alpha}+ \Big|\big(\frac{x_1}{|x_1|}-\frac{x_2}{|x_2|}\big)\Big|\ |x_2 |^{\alpha}\right). \end{align*} To control the last term---divided by $\big|(x_1,y_1)-(x_2,y_2)\big|^{\alpha}$---we first derive a lower bound on $\big|(x_1,y_1)-(x_2,y_2)\big|^{\alpha}$: \begin{align*} &\big|(x_1,y_1)-(x_2,y_2)\big|^2 \geq|x_1-x_2|^2 \\&= \big(|x_1|-|x_2|\big)^2 +|x_1|\,|x_2|\Big(\frac{x_1}{|x_1|}- \frac{x_2}{|x_2|}\Big)^2 \geq |x_1|\,|x_2|\Big(\frac{x_1}{|x_1|}-\frac{x_2}{|x_2|}\Big)^2. \end{align*} Therefore, using the assumption $0<|x_2| \leq|x_1|$, \begin{align*} \big|(x_1,y_1) - (x_2,y_2)\big|^{\alpha} \geq |x_2|^{\alpha}\, \Big|\frac{x_1}{|x_1|} - \frac{x_2}{|x_2|}\Big| ^{\alpha}. \end{align*} This finishes the proof of the lemma. \end{pf} The following obvious lemma is used repeatedly throughout the paper: \begin{lem} \label{lem:r2G} Assume \(f(r\omega)=r^2 G(\omega)\) with \(G\in C^{1,1}(\R^{n}\setminus\{0\})\cap L^{\infty}(\R^{n})\). Then \(f\in C^{1,1}(\R^{n})\). \end{lem} \begin{pf} The first derivatives of \(f\) trivially exist and are continuous. Therefore it suffices to show that all derivatives of \(f\) of second order belong to \(L_{\rm loc}^{\infty}(\R^{n})\); the result then follows from Remark~\ref{rem:abstract} \eqref{abstract(v)}. \begin{align*} \frac{\partial^2 f}{\partial x_j\partial x_k}=2\delta_{j,k} G(\omega)+2\Big(x_j \frac{\partial G}{\partial x_k}+x_k\frac{\partial G}{\partial x_j}\Big) + r^2\frac{\partial^2 G}{\partial x_j\partial x_k}\in L_{\rm loc}^{\infty}(\R^{n}), \end{align*} since \(G\in C^{1,1}(\R^{n}\setminus\{0\})\). This proves the lemma. \end{pf} Note that better regularity cannot be expected without assuming continuity of \(G\) at \(x=0\). On the other hand, if \(G\) only depends on \(\omega\in\sphere^{n-1}\), and \(G\) is continuous at \(x=0\), then \(G\) is a constant. \subsection{Proof of Theorem~\ref{thm:abstract}} \label{subsection4.3} \(\, \) We first investigate, for \(x_0\in B_{n}(z,R_1)\), the behaviour of the Newton potential $w$ as given in \eqref{Newton}, namely \begin{align} \label{y'} w(x_0)=\underset{B_{n}(z,R_1)}\int\Gamma(x_0-y)g(y')f(y'')\,dy \end{align} with \(y=(y',y'')\in\R^{k+d}=\R^{n}\). Since $u$ and $w$ are $C^{1,\alpha}$ - solutions of \eqref{eq:Poisson} in $B_{n}(z,R_1)$ (see Proposition~\ref{prop:GTbis}), $h=u-w$ is harmonic. Any harmonic function $h$ in a bounded domain $\Omega$ satisfies the following {\it a~priori} estimate (see \cite[Theorem 2.10]{GandT}): \begin{equation}\label{hap} \sup_K |D_{ij}h|\le \frac{C(n)}{\delta^2}\sup_\Omega |h| \quad,\quad i,j\in\{1,\ldots,n\}, \end{equation} with \(K\) compact, \(K\subset\Omega\subset\R^{n}\), and \(\delta=\dist(K,\,\partial\Omega)\). So, by \eqref{y'} and \eqref{hap}, for \(x_{0}\in B_{n}(z,R)\) (recall that \(h=u-w\)) \begin{align} \label{eq:enough_w} |D_{ij}u(x_{0})|&\leq\frac{C(n)}{(R_{1}-R)^{2}} \Big(\sup_{B_{n}(z,R_{1})}|u| \nonumber\\ &\qquad\qquad\qquad\quad+ C(n,R_{1})\big(\sup_{\sphere^{k-1}}|g|\big) \big(\sup_{\pi_{d}B_{n}(z,R_{1})}|f|\big)\Big) \nonumber\\ &\quad+ |D_{ij}w(x_{0})|. \end{align} Therefore to prove that $u\in W^{2,\infty}_{\rm loc}(\mathbb R^n)$ and that $u$ satisfies \eqref{eq:apriori} it obviously remains to show that $w$ satisfies the {\it a~priori} estimate \eqref{eq:apriori}. This will be done via Lemma \ref{GT42} and will finish the proof of Theorem~\ref{thm:abstract}. We proceed as follows: Define $N=\{(x',x'')\in \mathbb R^n\:|\:x'=0\}$ and note that $|N|=0$ ($|N|$ denotes \(n\)-dimensional Lebesgue measure of $N$) and that for every ball $B_{n}\subset\R^{n}$, $B_{n}\setminus N$ is still a domain. For this the assumption \(k\geq2\) is vital (see also Remark~\ref{rem:abstract} \eqref{abstract(iii)}). Note also that (still with \(x_0\in B_{n}(z,R_1)\)) $w$ can be written as \begin{align}\label{(.)} w(x_0)=\int_{B_{n}(z,R_1)\setminus N} \!\!\!\!\! \Gamma(x_0-y)g(y')f(y'')\,dy. \end{align} Taking into account the H\"older continuity assumptions on $g$ and $f$ it is easily seen that for every domain $\Omega\subset\R^{n}$, $g f\in C^{\alpha}(\Omega\setminus N)$. Hence \eqref{(.)} and Proposition~\ref{prop:GT} implies that $w\in C^{2,\alpha}\big(B_{n}(z,R_1)\setminus N\big)$. Now we are ready to apply Lemma \ref{GT42}: Pick $\Omega=B_{n}(z,R_1)\setminus N$ and $\Omega_0=B_{n}(z,R_2)$ with $R_11\)) and write \(K(s_1,s_2)=A_1+A_2+A_3\) where (when \(d=0, A_{2}=0\) for \(s_{1}>1\)) \begin{align} A_1&=\int_{s_1}^{s_2}s^{-1}\int_{\Sigma(s)^{c}}P_2(\omega)\: (g(\eta+s\omega')-g(s\omega'))\,d\omega\,ds,\nonumber\\ A_2&=\int_{s_1}^{s_2}s^{-1}\int_{\Sigma(s)}P_2(\omega)\;(g(\eta+s\omega') -g(s\omega'))\,d\omega\,ds,\nonumber\\ A_3&=\int_{s_1}^{s_2}s^{-1}\int_{\sphere^{n-1}}P_2(\omega)\;g(s\omega')\, d\omega\,ds.\nonumber \end{align} The estimate \eqref{eq:K(iii)} is a direct consequence of the following lemma. Proving it will finish the proof of Lemma~\ref{lem:K}. \end{pf*} \begin{lem} \label{lem:As} We have \begin{align} \label{A1<} |A_1|&\le C(n,\alpha) \big[g\big]_{\alpha,\; \sphere^{k-1}},\\ \label{A2<} |A_2|&\le C(n)\sup_{\sphere^{k-1}}|g|,\\ A_3&=0. \end{align} \end{lem} \begin{pf} $\mathbf{A_1:}$ Note first that since $s|\omega'|\ge 2$ and $|\eta+s\omega'|\ge 1$ in $\Sigma(s)^{c}$ we obtain, using the homogeneity of degree zero of $g$ and the H\"older continuity of $g$ on $\sphere^{k-1}$, that \begin{equation*} |A_1|\le C(n) \big[g\big]_{\alpha,\sphere^{k-1}} \int_{s_1}^{s_2}s^{-1} \int_{\Sigma^c(s)}\Big|\frac{\eta+s\omega'}{|\eta+s\omega'|}- \frac{s\omega'}{|s\omega'|}\Big|^\alpha\;d\omega \,ds. \end{equation*} Then by using the triangle inequality and that $s|\omega'|\ge \sqrt s\ge 2$, we get \begin{align*} \Big|\frac{\eta+s\omega'}{|\eta+s\omega'|}-\frac{s\omega'}{|s\omega'|}\Big| &\le \frac{1+\big|\,|s\omega'|-|\eta+s\omega'|\big|}{|\eta+s\omega'|} \\& \le \frac{2}{|\eta+s\omega'|}\le\frac{2}{\sqrt s-1}\le \frac{4}{\sqrt s} \end{align*} which leads to \begin{equation*}%\label{A1<} |A_1|\le C(n,\alpha)\big[g\big]_{\alpha,\;\sphere^{k-1}}s_1^{-\frac{\alpha}{2}} \le C(n,\alpha) \big[g\big]_{\alpha,\; \sphere^{k-1}}, \end{equation*} verifying \eqref{A1<}. \\ \noindent $\mathbf{A_2:}$ For \(d=0\), \(A_{2}=0\). For \(d>0\), the estimate \eqref{A2<} is a consequence of the following lemma, which is not hard to prove using polar coordinates in \(\R^{n}\) (we omit the proof): \begin{lem}\label{Soeren} Let $|\Sigma(s)|$ denote the $n-1$-dimensional surface measure of $\Sigma(s)$. Then \begin{equation}\label{Sbs} \big|\Sigma(s)\big|\le C(n)s^{-1/2}. \end{equation} \end{lem} From \eqref{Sbs} we immediately get \eqref{A2<}: \begin{equation*} |A_2|\le\Big(\int_4^\infty s^{-1}\big|\Sigma(s)\big|\,ds\Big)\:C(n)\sup_{\sphere^{k-1}}|g| \le C(n)\sup_{\sphere^{k-1}}|g|. \end{equation*} \noindent $\mathbf{A_3:}$ We have \begin{equation}\label{A3=0} A_3=0 \end{equation} as a consequence of the lemma below (when \(d=0\), \eqref{A3=0} is trivially true, due to the assumptions on \(g\)), since, by assumption, $g|_{\sphere^{k-1}}$ is orthogonal to \(\mathfrak{h}_{2}^{(k)}\) (the subspace of \(L^2(\sphere^{k-1})\) spanned by the sphe\-ri\-cal harmonics of degree \(2\)). \begin{lem} Let $00\) the following estimate holds: \begin{align} \label{eq:G_3-r-est} \| G_{{3,\rm cut}}\|_{C^{1,1}(B_{3N}({\mathbf x_0},\rho))}+ \| r_{{\rm cut}}\|_{C^{\alpha}(B_{3N}({\mathbf x_0},\rho))} \leq C, \end{align} for some constant $C=C(\rho) > 0$ independent of ${\mathbf x_0} \in {\mathbb R}^{3N}$. \end{lem3.1'} \begin{pf} The proof of Lemma~3.1' is analogous to that of Lemma~\ref{lem:F3}. Instead of $\mu, \kappa, \nu$ we will use functions $\mu_{{\rm cut}}, \kappa_{{\rm cut}}$ and $\nu_{{\rm cut}}$ to be defined presently. With $\chi$ being the function defined in \eqref{eq:def_cutoff} we define \begin{align} \mu_{{\rm cut}}(x) &=\chi(|x|) \mu(x)=\chi(|x|)|x|^{2}, \\ \label{def:kappa_cut} \kappa_{{\rm cut}}(x,y) &= \chi(|x|)\chi(|y|)\kappa(x,y)\\ &\quad-\frac{1}{4}\chi(3|y|)\big(1-\chi(|x|)\big)\Big(|y|^{2}\frac{x\cdot y}{|x||y|}\Big)\nonumber\\ &\quad-\frac{1}{4}\chi(3|x|)\big(1-\chi(|y|)\big)\Big(|x|^{2}\frac{x\cdot y}{|x||y|}\Big)\nonumber\\ & \equiv\chi(|x|)\chi(|y|)\frac{2-\pi}{3\pi}(x\cdot y)\ln(x^{2}+y^{2})+\kappa_{1,\text{\rm cut}}(x,y). \nonumber \end{align} (Note that \(\kappa_{1,\text{\rm cut}}(x,y)\neq\chi(|x|)\chi(|y|)\kappa_{1}(x,y)\)). Let \(\nu_{{\rm cut}}\) be as in Lem\-ma~\ref{lem:construct_nu_cut}, we then have \begin{align} \label{eq:nu_cut} &\Delta\nu_{{\rm cut}} = \gamma_3 + h_{\nu},\\ &\| \nu_{{\rm cut}} \|_{C^{1,1}(B_{9}((x_0,y_0,z_0),\rho))}+ \| h_{\nu} \|_{C^{\alpha}(B_{9}((x_0,y_0,z_0),\rho))} \leq C, \nonumber \end{align} with $\gamma_3$ as in \eqref{def:gamma_2&gamma_3} and with \(C\) independent of \((x_0,y_0,z_0)\in\R^{9}\) and \(\rho>0\). For \(\mu_{\text{\rm cut}}\), note that \begin{align} \label{eq:mu_cut} \Delta\mu_{\text{\rm cut}}&=\Delta|x|^{2}+\Delta(\mu_{\text{\rm cut}}-|x|^{2})\\ &=6-\Delta\big((1-\chi(|x|))|x|^{2}\big) \equiv6-h_{\mu},\nonumber \end{align} where obviously, \begin{align} \label{est:h_mu} \| \mu_{{\rm cut}} \|_{C^{1,1}(B_{3}(x_0,\rho))}+ \big\|h_{\mu}\big\|_{C^{\alpha}(B_{3}(x_{0},\rho))}\leq C, \end{align} with \(C\) independent of \(x_{0}\in\R^{3}\) and \(\rho>0\). For \(\kappa_{\text{\rm cut}}\), using \(\Delta\kappa=\gamma_{2}\) (see \eqref{def:gamma_2&gamma_3} and \eqref{eq:eq_for_kappa}), that \(\Delta_{y}(|y|^{2}\frac{x\cdot y}{|x||y|})=4\frac{x\cdot y}{|x||y|}\), and the support properties of \(\chi\), we have that \begin{align} \label{eq:kappa_cut1} \Delta\kappa_{\text{\rm cut}} &=\gamma_{2} -\big\{1-\chi(|x|)\chi(|y|)\big\}\big(1-\chi(3|x|)-\chi(3|y|)\big)\gamma_{2}\nonumber\\ &-\Big\{\chi(3|y|)\big(1-\chi(|x|)\big) + \chi(3|x|)\big(1-\chi(|y|)\big)\Big\} \big(\gamma_{2}+\frac{x\cdot y}{|x||y|}\big)\nonumber\\ &+R_1+R_2+R_3,\nonumber\\ &\equiv \gamma_{2}- \mathcal{H}\gamma_{2}-\mathcal{G}\big(\gamma_{2}+\frac{x\cdot y}{|x||y|}\big) +R_1+R_2+R_3, \end{align} where \begin{align*} R_1&=\chi(|y|)\kappa\Delta_{x}\chi(|x|) +\chi(|y|)2\nabla_{x}\chi(|x|)\cdot\nabla_{x}\kappa\nonumber\\ &\,+\chi(|x|)\kappa\Delta_{y}\chi(|y|) +\chi(|x|)2\nabla_{y}\chi(|y|)\cdot\nabla_{y}\kappa,\\ R_2&=-\frac{1}{4} \chi(3|y|)|y|^{2}\frac{y}{|y|}\cdot\Delta_{x}\big((1-\chi(|x|))\frac{x}{|x|}\big)\nonumber\\ &-\frac{1}{4}\big(\Delta_{y}\chi(3|y|)\big)|y|^{2}\frac{y}{|y|} \cdot\big((1-\chi(|x|)\frac{x}{|x|}\big)\nonumber\\ &-\frac{1}{2}\big(\nabla_{y}\chi(3|y|)\big)\cdot\nabla_{y}\Big(\frac{x\cdot y}{|x||y|}|y|^{2}\big(1-\chi(|x|)\big)\Big), \end{align*} and where \(R_{3}\) is \(R_{2}\) with \(x\) and \(y\) interchanged. Using that \(\kappa\in C^{1,\alpha}(\R^{6})\) for all \(\alpha\in(0,1)\), and the support properties of \(\chi\), it is easily seen that \begin{align} \label{est:R_j} \|R_{j}\|_{C^{\alpha}(B_{6}((x_{0},y_{0}),\rho))}\leq C, \end{align} with a constant \(C\) independent of \((x_{0},y_{0}))\in\R^{6}\) and \(\rho>0\). Since for all \((x,y)\in\R^{6}\), \begin{align*} \big|\nabla\gamma_{2}\big|\leq 6\sqrt{2}\Big(\frac{1}{|x|}+\frac{1}{|y|}\Big) \quad,\quad \Big|\nabla\big(\gamma_{2}+\frac{x\cdot y}{|x||y|}\big)\Big|\leq\frac{8\sqrt{2}}{|x-y|} \end{align*} we get, using the support properties of \(\mathcal{H}\) and \(\mathcal{G}\), that \begin{align*} \|\mathcal H\,\nabla\gamma_{2}\|_{L^{\infty}(\R^{6})}\leq C\quad,\quad \big\|\mathcal G\nabla\big(\gamma_{2}+\tfrac{x\cdot y}{|x||y|}\big)\big\|_{L^{\infty}(\R^{6})}\leq C. \end{align*} Again using the support properties of \(\mathcal{H}\) and \(\mathcal{G}\), this implies that \begin{align} \label{est:F&G} &\|\mathcal H\,\gamma_{2}\|_{C^{0,1}(B_{6}((x_{0},y_{0}),\rho))}\leq C,\\ &\big\|\mathcal G\big(\gamma_{2}+\tfrac{x\cdot y}{|x||y|}\big)\big\|_{C^{0,1}(B_{6}((x_{0},y_{0}),\rho))}\leq C, \nonumber \end{align} with a constant \(C\) independent of \((x_{0},y_{0})\in\R^{6}\) and \(\rho>0\). From \eqref{eq:kappa_cut1}, \eqref{est:R_j}, and \eqref{est:F&G} we get \begin{align} \label{eq:kappa_cut} \Delta\kappa_{\text{\rm cut}}=\gamma_{2}+h_{\kappa}\quad,\quad \|h_{\kappa}\|_{C^{\alpha}(B_{6}((x_{0},y_{0}),\rho))}\leq C, \end{align} with a constant \(C\) independent of \((x_{0},y_{0})\in\R^{6}\) and \(\rho>0\). Note that (see \eqref{def:kappa_cut} and \eqref{eq:formula_kappa}) \begin{align*} \kappa_{1,\text{\rm cut}}(x,y)=\chi(|x|)\chi(|y|)\Big((x^{2}+y^{2}) G_{\kappa_{1}}\big(\frac{(x,y)}{|(x,y)|}\big)\Big)\ ,\ G_{\kappa_{1}}\in C^{1,1}(\sphere^{5}). \end{align*} Therefore, due to the compact support of \(\chi\), \begin{align} \label{eq:est_kappa_1_cut} \|\kappa_{1,\text{\rm cut}}\|_{C^{1,1}(B_{6}((x_{0},y_{0}),\rho))}\leq C \end{align} with \(C\) independent of \((x_{0},y_{0})\in\R^{6}\) and \(\rho>0\). Observe that \begin{align} \label{eq:square_prod} |\nabla F_{2}|^2= |\nabla F_{2,{\rm cut}} |^2 +\nabla(F_{2}-F_{2,{\rm cut}})\cdot\nabla(F_{2}+ F_{2,{\rm cut}}) \end{align} and that \begin{align*} \nabla(F_{2}- F_{2,{\rm cut}})&\cdot\nabla(F_{2}+ F_{2,{\rm cut}})\\ &= \sum_{j=1}^{N}\nabla_{j}(F_{2}- F_{2,{\rm cut}})\cdot\nabla_{j}(F_{2}+ F_{2,{\rm cut}})\\ &=\sum_{j=1}^{N}\vec{b}_{j}\cdot\frac{x_{j}}{|x_{j}|} +\sum_{1\leq j 0$ independent of ${\mathbf x_0} \in {\mathbb R}^{3N}$. Using \(\Delta(|x_{j}|^{2}\frac{x_{j}}{|x_{j}|})=4\frac{x_{j}}{|x_{j}|}\) and \(\Delta(|\frac{x_{j}-x_{k}}{\sqrt{2}}|^{2}\frac{x_{j}-x_{k}}{|x_{j}-x_{k}|}) =4\frac{x_{j}-x_{k}}{|x_{j}-x_{k}|}\), we see that \begin{align} \label{eq:G_1_cut} \Delta G_{1,{\rm cut}}=\nabla(F_{2}-F_{2,{\rm cut}})\cdot\nabla(F_{2}+ F_{2,{\rm cut}})+R, \end{align} with \begin{align*} R&= \frac{1}{4}\sum_{j=1}^{N}\Delta\big(\chi(|x_{j}|) \vec{b}_{j}\big)\cdot\Big(|x_{j}|^{2}\frac{x_{j}}{|x_{j}|}\Big)\\ &+\frac{1}{2}\sum_{j=1}^{N}\sum_{i=1}^{3}\nabla_{j}\big(\chi(|x_{j}|) \vec{b}_{j,i}\big)\cdot\nabla_{j}\Big(|x_{j}|^{2}\frac{x_{j,i}}{|x_{j}|}\Big)\\ &+\frac{1}{4}\sum_{1\leq j0\). Define \begin{align} \label{def:G_3_cut} G_{{3,\rm cut}}=G_{{1,\rm cut}}+G_{{2,\rm cut}} \end{align} with \begin{align*} G_{{2,\rm cut}}&=\hat\mu_{\rm cut}+\hat \kappa_{1,\rm cut}+\hat\nu_{\rm cut},\\ \hat\mu_{\rm cut}({\bf x})& =-\frac{1}{6}\Big( \sum_{j=1}^{N}\frac{Z^2}{4}\mu_{\rm cut}(x_{j}) +\sum_{1\le j 0$ independent of ${\mathbf x_0} \in {\mathbb R}^{3N}$. Also, using \eqref{def:G_3_cut}, \eqref{eq:nu_cut}, \eqref{est:h_mu}, \eqref{est:c-1-1_G_1} and \eqref{eq:est_kappa_1_cut}, \begin{align} \label{est:G_3_cut} \| G_{{3,\rm cut}}\|_{C^{1,1}(B_{3N}({\mathbf x_0},\rho))}\leq C, \end{align} for some constant \(C\) independent of ${\mathbf x_0} \in {\mathbb R}^{3N}$ and \(\rho>0\). Now, \eqref{eq:G_3-r-est} follows from \eqref{est:r_cut} and \eqref{est:G_3_cut}. This finishes the proof of Lemma~3.1'. \end{pf} Let $K_{3,{\rm cut}}$ be the function constructed in Lemma~3.1' above. Define (see \eqref{def:K_3_cut}, \eqref{F23cut}, and \eqref{phcut}) \begin{align} \zeta_{3,{\rm cut}} = e^{-F_{2,{\rm cut}} - K_{3,{\rm cut}}} \psi=e^{-G_{3,{\rm cut}}}\phi_{3,{\rm cut}}. \end{align} Since for all \(\rho>0\) (using Lemma~3.1') \begin{align*} \| F_{3,{\rm cut}}- K_{3,{\rm cut}} \|_{C^{1,1}(B_{3N}({\mathbf x_0}, \rho))}=\|G_{3,\text{\rm cut}} \|_{C^{1,1}(B_{3N}({\mathbf x_0}, \rho))} \end{align*} is bounded independently of ${\mathbf x_0}$, to prove \eqref{eq:a_priori} is equivalent to proving \begin{align} \label{eq:a_priori_zeta} \| \zeta_{3,{\rm cut}} \|_{C^{1,1}(B_{3N}({\mathbf x_0}, R))} \leq C(R) \| \zeta_{3,{\rm cut}} \|_{L^{\infty}(B_{3N}({\mathbf x_0}, 2R))}. \end{align} Using that \(\zeta_{3,\text{\rm cut}}=e^{-G_{3,\text{\rm cut}}}\phi_{3,\text{\rm cut}}\), the estimate \eqref{eq:G_3-r-est} (twice), and the bound \eqref{eq:a_priori_1der}, we get, for all \(0<\rho<\rho'\), \begin{align} \label{eq:whatever2} \|\zeta_{3,\text{\rm cut}}\|_{C^{1,\alpha}(B_{3N}({\mathbf x_0},\rho))} \leq C \|\zeta_{3,{\rm cut}}\|_{L^{\infty}(B_{3N}({\mathbf x_0}, \rho'))}, \end{align} with \(C=C(\rho,\rho')\). Proving \eqref{eq:a_priori_zeta} is improving \eqref{eq:whatever2} to \(\alpha=1\). The function $\zeta_{3,{\rm cut}}$ satisfies the equation \begin{align*} &\Delta \zeta_{3,{\rm cut}} + 2 \big( \nabla F_{2,{\rm cut}} + \nabla K_{3,{\rm cut}}\big) \cdot \nabla \zeta_{3,{\rm cut}} \\ &+ \big( \Delta F_{2,{\rm cut}} + \Delta K_{3,{\rm cut}} + |\nabla F_{2,{\rm cut}} + \nabla K_{3,{\rm cut}}|^2 + (E-V) \big)\zeta_{3,{\rm cut}}=0. \nonumber \end{align*} We can rewrite this as \begin{align} \label{eq:2nd_eq_for_zetaBIS} \Delta \zeta_{3,{\rm cut}} + 2 \nabla F_{2,{\rm cut}} \cdot \big(\nabla \zeta_{3,{\rm cut}} &+ \zeta_{3,{\rm cut}} \nabla K_{3,{\rm cut}}\big) \\ &+ r_{1,{\rm cut}} \cdot \nabla \zeta_{3,{\rm cut}} + r_{2,{\rm cut}}\zeta_{3,{\rm cut}}=0, \nonumber \end{align} with (since \(\Delta F_2=V\) and \(\Delta K_{3,\text{\rm cut}}=-|\nabla F_{2,\text{\rm cut}}|^{2}+r_{\text{\rm cut}}\)) \begin{align*} r_{1,\text{\rm cut}}&=2\nabla K_{3,\text{\rm cut}},\\ r_{2,\text{\rm cut}}&=\Delta F_{2,\text{\rm cut}}+r_{\text{\rm cut}} +|\nabla K_{3,\text{\rm cut}}|^{2}+(E-V)\\ &=\Delta(F_{2,\text{\rm cut}}-F_{2})+r_{\text{\rm cut}}+|\nabla K_{3,\text{\rm cut}}|^{2}+E. \end{align*} By the construction of \(F_{2}\) and \(F_{2,\text{\rm cut}}\) (see \eqref{F2}, \eqref{eq:def_cutoff}, and \eqref{eq:def_F2cut}) it is clear that for all \(\rho>0\) \begin{align*} \|\Delta(F_{2,\text{\rm cut}}-F_{2})\|_{C^{\alpha}(B_{3N}({\mathbf x_0}, \rho))} \leq C, \end{align*} with \(C=C(\rho)\) independent of \(\mathbf x_0\in\R^{3N}\). Due to Lemma~3.1' (see also \eqref{eq:def_cutoff}), \(\nabla K_{3,\text{\rm cut}}\) is \(C^{\alpha}\), and we have for all \(\rho>0\) \begin{align} \label{eq:est_nabla_K_3} \|\nabla K_{3,\text{\rm cut}}\|_{C^{\alpha}(B_{3N}({\mathbf x_0}, \rho))}\leq C, \end{align} with \(C=C(\rho)\) independent of \({\mathbf x_0}\in\R^{3N}\). This, together with \eqref{eq:G_3-r-est}, means that \begin{align} \label{eq:est_r_j} \| r_{j,{\rm cut}} \|_{C^{\alpha}(B_{3N}({\mathbf x_0}, \rho))} \leq C,\qquad j=1,2, \end{align} where $C=C(\rho)$ is independent of ${\mathbf x_0}\in\R^{3N}$. In order to finish the proof, we introduce a localisation. Let \(f:\R\to\R, 0\leq f\leq1\), be decreasing and such that \(f(t)=1\) for \(t<0\) and \(f(t)=0\) for \(t>1\), and define, for \(\rho>0, \lambda>1\), \begin{align} \theta(x)\equiv\theta_{\rho,\lambda}(x) =f\big(\tfrac{1}{\lambda-1}(\tfrac{|x-x_{0}|}{\rho}-1)\big). \end{align} (So \(\theta(x)=1\) on \(B_{3N}(x_{0},\rho)\) and \(\theta(x)=0\) outside \(B_{3N}(x_{0},\lambda\rho)\)). Clearly the derivatives of \(\theta\) are bounded independently of ${\mathbf x_0}$. Below, all constants \(C=C(\rho)\) also depend on \(\lambda>1\); we omit this dependence in the notation. On the set $B_{3N}({\mathbf x_0}, \rho))$, $\theta \zeta_{3,{\rm cut}}$ satisfies the following equation: \begin{align} \label{eq:3rd_eq_for_zeta} \Delta (\theta \zeta_{3,{\rm cut}}) + 2 \nabla F_{2,{\rm cut}} &\cdot \big(\nabla (\theta \zeta_{3,{\rm cut}}) + (\theta \zeta_{3,{\rm cut}}) \nabla K_{3,{\rm cut}}\big) \\ &+ r_{1,{\rm cut}} \cdot \nabla (\theta \zeta_{3,{\rm cut}}) + r_{2,{\rm cut}}( \theta \zeta_{3,{\rm cut}})=0. \nonumber \end{align} Using \eqref{eq:3rd_eq_for_zeta} we will deduce that \begin{align} \label{eq:a_priori_chizeta} \|\theta_{R,\sqrt{2}}\, \zeta_{3,{\rm cut}}\|_{C^{1,1}(B_{3N}({\mathbf x_0}, R))} \leq C(R) \|\zeta_{3,{\rm cut}}\|_{L^{\infty}(B_{3N}({\mathbf x_0}, 2R))}, \end{align} from which \eqref{eq:a_priori_zeta} clearly follows (since \(\theta\equiv 1\) on \(B_{3N}(\mathbf x_{0},R)\)). To prove Theorem~\ref{thm:main:apriori}, it therefore remains to prove \eqref{eq:a_priori_chizeta}. \begin{pf*}{Proof of \eqref{eq:a_priori_chizeta}} Let $\Psi_{j,i,\text{\rm cut}}$ be defined as $\Psi_{j,i,}$ was in \eqref{def:PsiJI} but with $\zeta_3, K_3$ replaced by $\theta\zeta_{3,{\rm cut}}$, $K_{3,{\rm cut}}$, that is (\(j\in\{1,\ldots,N\}, i\in\{1,2,3\}\)), \begin{align} \label{eq:3.11bis} \Psi_{j,i,\text{\rm cut}} = 2 \frac{\partial (\theta\zeta_{3,{\rm cut}})}{\partial x_{j,i}} + 2 (\theta\zeta_{3,{\rm cut}}) \frac{\partial K_{3,{\rm cut}}}{\partial x_{j,i}}. \end{align} (Here, \(\theta\equiv\theta_{R,\sqrt{2}}\)). We define \(\hat{\Psi}_{j,i,\text{\rm cut}}\), \(\Phi_{(j,k),i,\text{\rm cut}}\) analogously to \(\hat{\Psi}_{j,i}\), \(\Phi_{(j,k),i}\) defined in \eqref{def:PsiHats} and \eqref{def:PhiJK}. Using \eqref{eq:est_nabla_K_3} and \eqref{eq:whatever2} we get that for all \(0<\rho<\rho'\), \begin{align} \label{eq:alpha-bound_PsiCut} \big\|\Psi_{j,i,\text{\rm cut}}\big\|_{C^{\alpha}(B_{3N}({\mathbf x_0}, \rho))}&\leq C(\rho)\|\theta\zeta_{3,\text{\rm cut}}\|_{C^{1,\alpha}(B_{3N}({\mathbf x_0}, \rho))} \\ &\leq C(\rho,\rho',R)\|\zeta_{3,\text{\rm cut}}\|_{L^{\infty}(B_{3N}({\mathbf x_0}, \rho'))}. \nonumber \end{align} We then have the following result, similar to Lemma~\ref{lem:ExplPoisson}: \begin{lem3.4'} Let $u_{j,i,\text{\rm cut}}, v_{(j,k),i,\text{\rm cut}}$ be the solutions to the equations \eqref{eq:u_s}, \eqref{eq:v_s} (with \(\hat{\Psi}_{j,i}\), \(\Phi_{(j,k),i}\) replaced by \(\hat{\Psi}_{j,i,\text{\rm cut}}\), \(\Phi_{(j,k),i,\text{\rm cut}}\)) given by the Newton potential on \(B_{3N}({\mathbf x_0}, \sqrt{2}R)\). Then, for all \(\rho<\sqrt{2}R<\rho'\), there exists a constant $C=C(\rho,\rho',R)$ (independent of ${\mathbf x_0}\in\R^{3N}$) such that \begin{align}\label{eq:est_uijCut} \| u_{j,i} \|_{C^{1,1}(B_{3N}({\mathbf x_0}, \rho))} & \leq C \|\zeta_{3,{\rm cut}}\|_{L^{\infty}(B_{3N}({\mathbf x_0}, \rho'))}, \\ \label{eq:est_vjkCut} \| v_{(j,k),i} \|_{C^{1,1}(B_{3N}({\mathbf x_0}, \rho))} & \leq C \|\zeta_{3,{\rm cut}}\|_{L^{\infty}(B_{3N}({\mathbf x_0}, \rho'))}. \end{align} \end{lem3.4'} \begin{pf} Using Theorem~\ref{thm:abstract} and Remark~\ref{rem:abstract} \eqref{abstract(v)} and \eqref{abstract(vi)}, we get the {\it a~priori} estimate \begin{align} \label{eq:whatever3} \| u_{j,i,\text{\rm cut}} \|_{C^{1,1}(B_{3N}({\mathbf x_0}, \rho))} &\leq C \Big( \sup \left| \frac{x_{j,i}}{|x_j|} \right| \| \hat{\Psi}_{j,i,\text{\rm cut}} \|_{C^{\alpha}(\pi_{3N-3} B_{3N}({\mathbf x_0}, \sqrt{2}R))}\nonumber \\ &+\Big( \sup_{\pi_{3N-3} B_{3N}({\mathbf x_0}, \sqrt{2}R))} |\hat{\Psi}_{j,i,\text{\rm cut}} |\Big) \Big\| \frac{x_{j,i}}{|x_j|} \Big\|_{C^{\alpha}({\mathbb S}^2)} \Big).\nonumber\\ \end{align} Using \eqref{eq:3.11bis} and \eqref{eq:est_nabla_K_3} we have \begin{align*} \| \hat{\Psi}_{j,i,\text{\rm cut}} \|_{C^{\alpha}(\pi_{3N-3} B_{3N}({\mathbf x_0}, \sqrt{2}R))}&\leq \| \Psi_{j,i,\text{\rm cut}} \|_{C^{\alpha}((\pi_{3N-3} B_{3N}({\mathbf x_0}, \sqrt{2}R))\times\R^{3}})\\ &\!\!\!\!\!\!\leq C \|\theta\zeta_{3,{\rm cut}} \|_{C^{1,\alpha}((\pi_{3N-3}B_{3N}({\mathbf x_0}, \sqrt{2}R))\times\R^{3})}. \end{align*} This, the compact support of \(\theta\), and \eqref{eq:whatever3} implies the estimate \begin{align} \label{eq:whatever1} \| u_{j,i,\text{\rm cut}} \|_{C^{1,1}(B_{3N}({\mathbf x_0}, \rho))} \leq C \,\|\zeta_{3,{\rm cut}} \|_{C^{1,\alpha}(B_{3N}({\mathbf x_0}, \sqrt{2}R))}. \end{align} Combining \eqref{eq:whatever1} and \eqref{eq:whatever2}, we arrive at \eqref{eq:est_uijCut}. This finishes the proof of the estimate \eqref{eq:est_uijCut} for $u_{j,i,\text{\rm cut}}$. The analogous estimate \eqref{eq:est_vjkCut} for $v_{(j,k),i,\text{\rm cut}}$ is proved in the same manner using the same coordinate transformation as in the proof of Lemma~\ref{lem:ExplPoisson} (see also the proof of Lemma~3.5' below). We omit the details. \end{pf} \begin{lem3.5'} Let $\Psi_{j,i,\text{\rm cut}}$ be defined by \eqref{eq:3.11bis} and let $\hat{\Psi}_{j,i,\text{\rm cut}}$ and $\Phi_{(j,k),i,\text{\rm cut}}$ be defined by \eqref{def:PsiHats} and \eqref{def:PhiJK} (with $\Psi_{j,i}$ replaced by $\Psi_{j,i,\text{\rm cut}}$). Then the functions defined by \eqref{eq:regRest1} and \eqref{eq:regRest2} (again, with an extra index `{\rm cut}') belong to $C^{\alpha}({\mathbb R}^{3N})$ for all $\alpha \in (0,1)$. Furthermore, for any \(\rho<\sqrt{2}R<\rho'\), their $C^{\alpha}$-norms on the ball $B_{3N}({\mathbf x_0}, \rho)$ are bounded by \begin{align} \label{bound:Lemma3.5bis} C \|\zeta_{3,\text{\rm cut}}\|_{L^{\infty}(B_{3N}({\mathbf x_0}, \rho'))} \end{align} with \(C=C(\rho,\rho',R)\) independent of \({\mathbf x_0}\in\R^{3N}\). \end{lem3.5'} \begin{pf} That the functions belong to $C^{\alpha}({\mathbb R}^{3N})$ for all $\alpha \in (0,1)$ follows like in the proof of Lemma~\ref{lem:regRest}. To prove the bounds on the norms it suffices, by Lemma~\ref{lem:XdotG} and the triangle inequality, to prove them for \begin{align*} \big\|\Psi_{k,i,\text{\rm cut}}\big\|_{C^{\alpha}(B_{3N}({\mathbf x_0}, \rho))} \quad\text{and}\quad \big\|\Phi_{(j,k),i,\text{\rm cut}})\big\|_{C^{\alpha}(B_{3N}({\mathbf x_0}, \rho))}. \end{align*} For \( \Psi_{k,i,\text{\rm cut}}\), the estimate follows from \eqref{eq:alpha-bound_PsiCut}. To bound \(\Phi_{(j,k),i,\text{\rm cut}}\), denote by \(t_{j,k}:\R^{3N}\to\R^{3N}\) the linear transformation (see also \eqref{def:PhiJK}), \begin{align*} &t_{j,k}(\mathbf x)=\\&(x_{1},\ldots,x_{j-1},\tfrac{1}{2}(x_{j}+x_{k}), x_{j+1},\ldots,x_{k-1},\tfrac{1}{2}(x_{j}+x_{k}),x_{k+1}, \ldots,x_{N}), \end{align*} so that \begin{align*} \Phi_{(j,k),i,\text{\rm cut}}(\mathbf x)=\Psi_{j,i,\text{\rm cut}}(t_{j,k}(\mathbf x))-\Psi_{k,i,\text{\rm cut}}(t_{j,k}(\mathbf x)). \end{align*} Then, since \(|t_{j,k}(\mathbf z)|\leq|\mathbf z|\), \begin{align} \label{eq:cut-alpha-norms} &\frac{\big|\Phi_{(j,k),i,\text{\rm cut}}(\mathbf x) -\Phi_{(j,k),i,\text{\rm cut}}(\mathbf y)\big|}{|\mathbf x-\mathbf y|^{\alpha}} \leq \frac{\big|\Psi_{j,i,\text{\rm cut}}(t_{j,k}(\mathbf x))-\Psi_{j,i,\text{\rm cut}}(t_{j,k}(\mathbf y))\big|}{|t_{j,k}(\mathbf x)-t_{j,k}(\mathbf y)|^{\alpha}} \nonumber \\ &\qquad\qquad\qquad\qquad+\frac{\big|\Psi_{k,i,\text{\rm cut}}(t_{j,k}(\mathbf x))-\Psi_{k,i,\text{\rm cut}}(t_{j,k}(\mathbf y))\big|}{|t_{j,k}(\mathbf x)-t_{j,k}(\mathbf y)|^{\alpha}} \end{align} Due to the localisation \(\theta\) in the definition of \(\Psi_{k,i,\text{\rm cut}}\) (see \eqref{eq:3.11bis}), both of the terms on the RHS of \eqref{eq:cut-alpha-norms} are bounded by \begin{align*} C(\rho)\|\zeta_{3,\text{\rm cut}}\|_{C^{1,\alpha}(B_{3N}({\mathbf x_0}, \sqrt{2}R))}. \end{align*} The bound \eqref{bound:Lemma3.5bis} for \(\Phi_{(j,k),i,\text{\rm cut}}\) now follows using \eqref{eq:whatever2}. This finishes the proof of the bound \eqref{bound:Lemma3.5bis} for the functions \((\Psi_{j,i,\text{\rm cut}}-\Psi_{k,i,\text{\rm cut}})-\Phi_{(j,k),i,\text{\rm cut}}\). The proof for the functions \(\Psi_{j,i,\text{\rm cut}}-\hat\Psi_{j,i,\text{\rm cut}}\) is similar (see also the proof of Lemma~3.4' above), so we omit the details. \end{pf} To finish the proof of Theorem~\ref{thm:main:apriori}, define $U_{{\rm cut}}$ analogously to \eqref{def:U}, using the functions $u_{j,i,\text{\rm cut}}$, $v_{(j,k),i,\text{\rm cut}}$ from Lemma~3.4'. Then, by Lemma~3.4', for any \(\rho<\sqrt{2}R<\rho'\), \begin{align} \label{eq:Ucut} \Delta &U_{{\rm cut}}= \sum_{j=1}^{N}\frac{Z}{2}\frac{x_{j}}{|x_{j}|}\cdot \hat\Psi_{j,{{\rm cut}}}\ -\sum_{1\leq j0\), \(\hat\gamma_2\) is \(C^{\infty}\) away from points of type (a) and (b), see \eqref{def2:gamma2} and \eqref{eq:def_gamma2_tilde}). (a): Let \(U_{a}\subset\R^{6}\) be a neighbourhood of a point \((x_{0},x_{0})\in\sphere^{5}\) (i.e. \(2|x_{0}|^{2}=1\)) such that for some \(c>0\), \(|x|\geq c, |y|\geq c\) for \((x,y)\in U_{a}\). Choose new coordinates: Let \begin{align*} (x_{1}, x_{2})=t(x,y)=(x-y, x+y). \end{align*} Then \begin{align*} \big(\gamma_{2}\circ t^{-1}\big)(x_{1},x_{2})=\frac{x_{1}}{|x_{1}|}\cdot \Big(\frac{x_{1}-x_{2}}{|x_{1}-x_{2}|} +\frac{x_{1}+x_{2}}{|x_{1}+x_{2}|}\Big) \equiv\frac{x_{1}}{|x_{1}|}\cdot G_{a}(x_{1},x_{2}) \end{align*} with \(G_{a}\in C^{\infty}\big(t(U_{a})\big)\). Since \(G_{a}(0,x_{2})=0\) for \(x_{2}\neq0\) (that is, for \(x=y\neq0\) in the original coordinates), we have, by Lemma~\ref{lem:XdotG}, that \(\gamma_{2}\circ t^{-1}\in C^{0,1}\big(t(U_{a})\big)\), and therefore \(\gamma_{2}\in C^{0,1}(U_{a})\subset C^{\alpha}(U_{a})\) for all \(\alpha\in(0,1)\). Since \((x\cdot y)/(x^{2}+y^{2})\in C^{\infty}(U_{a})\), we have (see \eqref{eq:def_gamma2_tilde}) \(\hat\gamma_{2}\in C^{\alpha}(U_{a})\) for all \(\alpha\in(0,1)\). By Proposition~\ref{prop:GT} we get from \eqref{eq:tilde_f_3} that \(\kappa_{1}\in C^{2,\alpha}(U_{a})\). (b): Let \(U_{b}\subset\R^{6}\) be a neighbourhood of a point \((0,y_{0})\in\sphere^{5}\) (i.e.\ \(|y_{0}|=1\)) such that for some \(c>0\), \(|y|\geq c, |x-y|\geq c\) for \((x,y)\in U_{b}\). Then \begin{align} \gamma_{2}(x,y) &=\Big(\frac{x}{|x|}-\frac{y}{|y|}\Big) \cdot\frac{x-y}{|x-y|} \nonumber\\& ={}-\frac{x}{|x|}\cdot\frac{y}{|y|}+\frac{x}{|x|} \cdot\Big(\frac{y}{|y|}-\frac{y-x}{|y-x|}\Big) -\frac{y}{|y|}\cdot\frac{x-y}{|x-y|}. \nonumber \end{align} Note that \begin{align*} {}-\frac{y}{|y|}\cdot\frac{x-y}{|x-y|} \in C^{\infty}(U_{b}) \end{align*} and that \begin{align*} \frac{x}{|x|} \cdot\Big(\frac{y}{|y|}- \frac{y-x}{|y-x|}\Big) \equiv\frac{x}{|x|}\cdot G_{b}(x,y), \end{align*} with \(G_{b}\in C^{\infty}(U_{b})\), \(G_{b}(0,y)=0\) for \(y\neq0\). Therefore, by Lemma~\ref{lem:XdotG} and \eqref{eq:def_gamma2_tilde}, \begin{align*} \hat\gamma_{2}(x,y)-\Big({}-\frac{x}{|x|}\cdot\frac{y}{|y|}\Big) \in C^{0,1}(U_{b})\subset C^{\alpha}(U_{b}) \text{ for all }\alpha\in(0,1). \end{align*} Let \(\kappa_{2}\) be such that \begin{align} \big(\Delta_{x}+\Delta_{y}\big)\kappa_{2} ={}-\frac{x}{|x|}\cdot\frac{y}{|y|}\quad,\quad \kappa_{2}\in C^{1,1}(U_{b}). \nonumber \end{align} The existence of such a function is ensured by Theorem~\ref{thm:abstract}, since \(y\neq0\) for \((x,y)\in U_{b}\), and \(\mathcal{P}_{2}^{(3)}\big(\frac{x}{|x|}\big)=0\) due to the anti-symmetry of \(\frac{x}{|x|}\). Then (see \eqref{eq:tilde_f_3}) \(\kappa_{3}=\kappa_{1}-\kappa_{2}\) solves \begin{align} \big(\Delta_{x}+\Delta_{y}\big)\kappa_{3}= \hat\gamma_{2}(x,y)-\Big({}-\frac{x}{|x|}\cdot\frac{y}{|y|}\Big) \in C^{\alpha}(U_{b}) \text{ for all }\alpha\in(0,1), \nonumber \end{align} so by elliptic regularity \(\kappa_{3}\in C^{2,\alpha}(U_{b})\subset C^{1,1}(U_{b})\). Since \(\kappa_{2}\in C^{1,1}(U_{b})\), this proves \(\kappa_{1}=\kappa_{2}+\kappa_{3}\in C^{1,1}(U_{b})\). Together with \(\kappa_{1}\in C^{2,\alpha}(U_{a})\) from above, this implies \(G_{\kappa_{1}}\!=\kappa_{1}/r^{2}\in C^{1,1}(\sphere^{5})\), and so \(\kappa_{1}=r^{2}\,G_{\kappa_{1}}\in C^{1,1}(\R^{6})\). This finishes the proof of the existence of \(\kappa\) solving \eqref{eq:eq_for_kappa}, and having the form \eqref{eq:formula_kappa}, with \(G=G_{\kappa_{1}}\). \end{proof} \section{Construction of the function \(\nu\)} \label{chap:nu} In this appendix we construct a function \(\nu\) solving \eqref{eq:eq_for_nu}. \begin{lem} \label{lem:construct_nu} There exists a solution $\nu = \nu(x,y,z)$ to the equation \eqref{eq:eq_for_nu} satisfying \begin{enumerate}[\rm (i)] \item \label{invariance} $\nu$ is invariant under cyclic permutation, i.e., $\nu(x,y,z) = (\nu \circ \sigma)(x,y,z)$ for all $x,y,z \in {\mathbb R}^3$, where $\sigma (x,y,z) = (z,x,y)$. \item $\nu \in C^{1,1}({\mathbb R}^9)$. \end{enumerate} \end{lem} The idea is to change coordinates, to the centre-of-mass frame for \((x,y,z)\). In these new coordinates, the problem of solving \eqref{eq:eq_for_nu} turns out to reduce to a problem in \(6\) variables only. By an extra symmetry of the function \(\gamma_{3}\) (see \eqref{def:gamma_2&gamma_3}), namely permutation of the three electron-coordinates \(x, y\), and \(z\), the logarithmic term that occured in the function \(\kappa\) (see \eqref{eq:formula_kappa}) does not occur here. This is because the projection on \(\mathfrak{h}_{2}^{(6)}\) of \(\tilde\gamma_{3}\) (the function that \(\gamma_{3}\) transforms into in the new coordinates, see \eqref{eq:inv_gamma3} below) vanishes, due to this extra symmetry. \begin{pf} Make the following change of coordinates (each entry below is a diagonal \(3\times3\)-matrix with the listed number in the diagonal; we will use this notation repeatedly; here, \(x,y,z\in\R^{3}\)) \begin{align} \label{eq:change_coord} \left(\begin{array}{c} x \\ y \\ z \end{array}\right) = \mathcal{T}\, \left(\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array}\right) =\left( \begin{array}{ccc} \frac{1}{\sqrt{3}} & 0 & \frac{2}{\sqrt{6}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{6}} \\ \frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{6}} \\ \end{array}\right) \left(\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array}\right). \end{align} Then \begin{align} \label{eq:inv_gamma3} &\big(\gamma_{3}\circ \mathcal{T}\big)(x_{1},x_{2},x_{3}) = \\ \nonumber &\frac{x_{2}}{|x_{2}|}\cdot\frac{x_{2}+\sqrt{3}x_{3}}{|x_{2}+\sqrt{3}x_{3}|} +\frac{x_{2}}{|x_{2}|}\cdot\frac{x_{2}-\sqrt{3}x_{3}}{|x_{2}-\sqrt{3}x_{3}|} -\frac{x_{2}+\sqrt{3}x_{3}}{|x_{2}+\sqrt{3}x_{3}|}\cdot \frac{x_{2}-\sqrt{3}x_{3}}{|x_{2}-\sqrt{3}x_{3}|} \\ &\equiv\tilde\gamma_{3}(x_{1},x_{2},x_{3}). \nonumber \end{align} That \(\tilde\gamma_{3}\) is independent of \(x_{1}\) is the fact that \(\gamma_{3}\) only depends on the inter-electron coordinates (\(x-y\), \(y-z\), \(z-x\) respectively), and not on the centre-of-mass coordinate (\(x_{CM}=\frac{1}{\sqrt{3}}(x+y+z)=x_{1}\)). The function \(\gamma_{3}\) is invariant under cyclic permutation of the elec\-tron-coordinates \(x, y\) and \(z\), that is, \(\big(\gamma_{3}\circ\sigma\big)(x,y,z)=\gamma_{3}(x,y,z)\) for all \(x, y, z\in \R^{3}\) with \(\sigma(x,y,z)=(z,x,y)\). This gives that \begin{align} \label{eq:inv_R} \big(\tilde\gamma_{3}\circ\mathcal{R}\big)(x_{1},x_{2},x_{3}) =\tilde\gamma_{3}(x_{1},x_{2},x_{3}) \text{ for all } x_{1}, x_{2}, x_{3}\in\R^{3}, \end{align} with \(\mathcal{R}\) the orthogonal transformation given by \(\mathcal{R}=\mathcal{T}^{-1}\circ\sigma\circ \mathcal{T}\), that is by the \(9\times9\)-matrix (again, each entry is a diagonal \(3\times3\)-matrix) \begin{align} \mathcal{R}= \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos(\frac{2\pi}{3}) & \sin(\frac{2\pi}{3}) \\ 0 & -\sin(\frac{2\pi}{3}) & \cos(\frac{2\pi}{3}) \\ \end{array}\right). \nonumber \end{align} Note that \(\mathcal{R}\) is a rotation of \((x_{2},x_{3})\) by \(\frac{2\pi}{3}\) around \(x_{1}\) (all in \(\R^{9}\)), that is, \(\mathcal{R}^{3}= I_{9}\), where \(I_{9}\) is the identity on \(\R^{9}\). Define the function \(\bar\gamma_{3}\) by \begin{align} \label{eq:def_gamma3Bar} \bar\gamma_{3}(x_{2},x_{3}) =\tilde\gamma_{3}(x_{1},x_{2},x_{3})\quad,\quad (x_{2},x_{3})\in\R^{6} \end{align} (since \(\tilde\gamma_{3}\) is independent of \(x_{1}\), this is well defined). Then, due to \eqref{eq:inv_R}, \begin{align} \label{eq:R_invBis} \big(\bar\gamma_{3}\circ\bar{\mathcal{R}}\big)(x_{2},x_{3}) =\bar\gamma_{3}(x_{2},x_{3}) \text{ for all } x_{2}, x_{3}\in\R^{3}, \end{align} with (each entry still being a diagonal \(3\times3\)-matrix) \begin{align} \label{def:barR} \bar{\mathcal{R}}= \left(\begin{array}{cc} -\frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \end{array}\right) = \left(\begin{array}{cc} \cos(\frac{2\pi}{3}) & \sin(\frac{2\pi}{3}) \\ -\sin(\frac{2\pi}{3}) & \cos(\frac{2\pi}{3}) \\ \end{array}\right). \end{align} Observe that if \(\bar\nu=\bar\nu(x_{2},x_{3})\) solves (for \(\bar\gamma_{3}\), see \eqref{eq:inv_gamma3} and \eqref{eq:def_gamma3Bar}) \begin{align} \label{eq:gamma4} \big(\Delta_{x_{2}}+\Delta_{x_{3}}\big)\bar\nu&=\bar\gamma_{3}, \end{align} then trivially the function \(\tilde\nu\) defined by \(\tilde\nu(x_{1},x_{2},x_{3})=\bar\nu(x_{2},x_{3})\) solves \begin{align*} \big(\Delta_{x_{1}}+\Delta_{x_{2}}+\Delta_{x_{3}}\big)\tilde\nu =\tilde\gamma_{3}. \end{align*} Since \(\mathcal{T}\) is orthogonal, the function \(\nu=\tilde\nu\circ \mathcal{T}^{-1}\) will then solve (recall that \(\tilde\gamma_{3}=\gamma_{3}\circ \mathcal{T}\)) \(\big(\Delta_{x}+\Delta_{y}+\Delta_{z}\big)\nu=\gamma_{3}\), that is, \eqref{eq:eq_for_nu}. The problem of solving \eqref{eq:eq_for_nu} therefore reduces to solving \eqref{eq:gamma4}. Observe next that (see \eqref{eq:inv_gamma3} and \eqref{eq:def_gamma3Bar}) \begin{align} \bar\gamma_{3}(\mathcal{O}x_{2},\mathcal{O}x_{3}) =\bar\gamma_{3}(x_{2},x_{3}) \text{ for all } \mathcal{O}\in SO(3), x_{2}, x_{3}\in\R^{3}. \nonumber \end{align} This and \eqref{eq:R_invBis} gives, by \eqref{lem:NO_l=2} of Lemma~\ref{lemma:first_invariance}, that \(\mathcal{P}^{(6)}_{2}\bar\gamma_{3}=0\) . Therefore, by Proposition~\ref{prop:homogen}, there exists a solution \(\bar\nu\) to \eqref{eq:gamma4} with \begin{align} \bar\nu(x_{2},x_{3})=(x_{2}^{2}+x_{3}^{2})\, G_{\bar\nu}\left(\frac{(x_{2},x_{3})}{|(x_{2},x_{3})|}\right)&, \nonumber\\ G_{\bar\nu}\in &C^{1,\alpha}(\sphere^{5}) \text{ for all } \alpha\in(0,1). \nonumber \end{align} We proceed to prove that in fact \(G_{\bar\nu}\in C^{2,\alpha}(\sphere^{5})\) for all \(\alpha\in(0,1)\). We do this by showing that \(\bar\nu\in C^{2,\alpha}(\R^{6}\setminus\{0\})\), using \eqref{eq:gamma4} and elliptic regularity (Proposition \ref{prop:GT}). Note that there are two kinds of singular points of \(\bar\gamma_{3}\) on \(\sphere^{5}\): (a) \(x_{2}=0\) (and so \(x_{3}\neq0\)), (b) \(x_{2}=\sqrt{3}x_{3}\) (and so \(x_{2}\neq0\neq x_{3}\)) (resp. \(x_{2}=-\sqrt{3}x_{3}\)). The function \(\bar\nu\) (and therefore, \(G_{\bar\nu}\)) is \(C^{\infty}\) in a neighbourhood of all other points on \(\sphere^{5}\) due to elliptic regularity (Proposition \ref{prop:GT}). (a): Let \(U_{a}\subset\R^{6}\) be a neighbourhood of a point \((0,x_{3}^{0})\in\sphere^{5}\) (i.e., \(x_{3}^{0}\neq0\)), such that for some \(c>0\), \(|x_{2}+\sqrt{3}x_{3}|\geq c\), \(|x_{2}-\sqrt{3}x_{3}|\geq c\) for \((x_{2},x_{3})\in U_{a}\). Note that \begin{align} \label{eq:gamma_3_bar} \bar\gamma_{3}(x_{2},x_{3})&= \frac{x_{2}}{|x_{2}|}\cdot\left(\frac{x_{2} +\sqrt{3}x_{3}}{|x_{2}+\sqrt{3}x_{3}|} +\frac{x_{2}-\sqrt{3}x_{3}}{|x_{2}-\sqrt{3}x_{3}|}\right) \nonumber\\ &\quad-\frac{x_{2}+\sqrt{3}x_{3}}{|x_{2}+\sqrt{3}x_{3}|}\cdot \frac{x_{2}-\sqrt{3}x_{3}}{|x_{2}-\sqrt{3}x_{3}|}. \end{align} Write \begin{align*} \frac{x_{2}}{|x_{2}|}\cdot\left(\frac{x_{2} +\sqrt{3}x_{3}}{|x_{2}+\sqrt{3}x_{3}|} +\frac{x_{2}-\sqrt{3}x_{3}}{|x_{2}-\sqrt{3}x_{3}|}\right) \equiv\frac{x_{2}}{|x_{2}|}\cdot G_{a}(x_{2},x_{3}) \end{align*} where \(G_{a}\in C^{\infty}(U_{a})\), \(G_{a}(0,x_{3})=0\). Furthermore, \begin{align*} \frac{x_{2}+\sqrt{3}x_{3}}{|x_{2}+\sqrt{3}x_{3}|}\cdot \frac{x_{2}-\sqrt{3}x_{3}}{|x_{2}-\sqrt{3}x_{3}|} \in C^{\infty}(U_{a}). \end{align*} Therefore, due to Lemma~\ref{lem:XdotG}, \(\bar\gamma_{3}\in C^{0,1}(U_{a})\subset C^{\alpha}(U_{a})\) for all \(\alpha\in (0,1)\), and so, by \eqref{eq:gamma4} and elliptic regularity (Proposition \ref{prop:GT}), \(\bar\nu\in C^{2,\alpha}(U_{a})\). (b): Let \(U_{b}\) be a neighbourhood of a point \((x_{2}^{0},x_{3}^{0})\in\sphere^{5}\) with \(x_{2}^{0}=\sqrt{3}x_{3}^{0}\) (i.e., \(x_{2}^{0}\neq0\neq x_{3}^{0}\)), such that for some \(c>0\), \(|x_{2}|\geq c\), \(|x_{2}+\sqrt{3}x_{3}|\geq c\) for \((x_{2},x_{3})\in U_{b}\). Choose new coordinates: Let \begin{align*} (u,v)=\tau(x_{2},x_{3})=(x_{2}-\sqrt{3}x_{3},x_{2}+\sqrt{3}x_{3}). \end{align*} Then \begin{align*} \big(\bar\gamma_{3}\circ \tau^{-1}\big)(u,v)= \frac{u}{|u|}\cdot\left(\frac{u+v}{|u+v|}-\frac{v}{|v|}\right) +\frac{u+v}{|u+v|}\cdot\frac{v}{|v|}. \end{align*} We proceed as above. Write \begin{align*} \frac{u}{|u|}\cdot\left(\frac{u+v}{|u+v|}-\frac{v}{|v|}\right) \equiv\frac{u}{|u|}\cdot G_{b}(u,v) \end{align*} where \(G_{b}\in C^{\infty}\big(\tau(U_{b})\big)\) (since \(v\neq0, u+v\neq0\) in \(\tau(U_{b})\)), \(G_{b}(0,v)=0\) for \(v\neq0\). Furthermore, \begin{align*} \frac{u+v}{|u+v|}\cdot\frac{v}{|v|}\in C^{\infty}(U_{b}). \end{align*} Lemma~\ref{lem:XdotG} implies that \(\bar\gamma_{3}\circ \tau^{-1}\in C^{0,1}\big(\tau(U_{b})\big)\), and so \(\bar\gamma_{3}\in C^{0,1}(U_{b})\) \(\subset C^{\alpha}(U_{b})\) for all \(\alpha\in(0,1)\). By \eqref{eq:gamma4} and elliptic regularity (Proposition \ref{prop:GT}) follows that \(\bar\nu\in C^{2,\alpha}(U_{b})\). Singular points of the form \(x_{2}^{0}=-\sqrt{3}x_{3}^{0}\) are treated analogously. From the above follows that \(\bar\nu\in C^{2,\alpha}(\R^{6}\setminus\{0\})\), and therefore \(G_{\bar\nu}\in C^{2,\alpha}(\sphere^{5})\), for all \(\alpha\in(0,1)\). This finishes the construction of a function \(\bar\nu\in C^{1,1}(\R^{6})\) that solves \eqref{eq:gamma4}, and has the form \begin{align} \label{eq:form_nu_bar} \bar\nu(x_{2},x_{3})=(x_{2}^{2}+x_{3}^{2})\, G_{\bar\nu}\left(\frac{(x_{2},x_{3})}{|(x_{2},x_{3})|}\right)&, \\ G_{\bar\nu}\in &C^{2,\alpha}(\sphere^{5}) \text{ for all } \alpha\in(0,1).\nonumber \end{align} As discussed above $\overline{\nu}$ defines a function $\nu$ solving the equation \eqref{eq:eq_for_nu}. Clearly, since $\overline{\nu} \in C^{1,1}({\mathbb R}^6)$, we get $\nu \in C^{1,1}({\mathbb R}^9)$. The solution $\nu$ constructed in this manner does not necessarily satisfy the invariance property (\ref{invariance}). In order to force this invariance, we consider $$ \nu_{\text{sym}} = \frac{1}{3} \sum_{j=1}^3 (\nu \circ \sigma^j)(x,y,z). $$ Since the Laplace operator commutes with $\sigma$, and $\gamma_3$ is invariant under $\sigma$, $\nu_{\text{sym}}$ satisfies the conclusion of Lemma~\ref{lem:construct_nu}. \end{pf} With the notation from the proof of Lemma~\ref{lem:construct_nu}, we define $$ \overline{\nu}_{{\rm cut}}(x_2,x_3) = \chi(x_2^2 + x_3^2)\, \overline{\nu}(x_2,x_3), $$ with $\chi$ as in \eqref{eq:def_cutoff}, and \(\tilde\nu_{{\rm cut}}(x_1,x_2,x_3)\equiv\overline{\nu}_{{\rm cut}}(x_2,x_3)\) (as already defined). As discussed above (for $\nu$) the function $\tilde\nu_{{\rm cut}}$ defines a function $\nu_{{\rm cut}}=\tilde\nu_{{\rm cut}}\circ\mathcal T^{-1} : {\mathbb R}^9 \rightarrow {\mathbb R}$ (by the linear transformation \(\mathcal T\) in \eqref{eq:change_coord}). We then get: \begin{lem} \label{lem:construct_nu_cut} The function $\nu_{{\rm cut}}$ satisfies $$ \Delta\nu_{{\rm cut}} = \gamma_3+h, $$ with $\gamma_3$ as in \eqref{def:gamma_2&gamma_3} and $h\in C^{\alpha}({\mathbb R}^9)$ for all $\alpha \in (0,1)$. Furthermore, we have the estimate \begin{align} \label{eq:construct_nu_cut} \| \nu_{{\rm cut}} \|_{C^{1,1}(B_{9}((x_0,y_0,z_0),R))}+ \| h \|_{C^{\alpha}(B_{9}((x_0,y_0,z_0),R))} \leq C, \end{align} with \(C\) independent of \((x_0,y_0,z_0)\in\R^{9}\) and \(R>0\). \end{lem} \begin{pf} We calculate, using \eqref{eq:gamma4}, \begin{align*} \big(\Delta_{x_1}+\Delta_{x_2}+\Delta_{x_3}\big)\tilde\nu_{{\rm cut}}&= \big(\Delta_{x_2}+\Delta_{x_3}\big) \overline{\nu}_{{\rm cut}} \equiv\Delta \overline{\nu}_{{\rm cut}}\\ &=\overline{\gamma}_3+ \big\{(\Delta \chi) \overline{\nu} + 2 \nabla \chi \cdot \nabla \overline{\nu}\big\} - (1-\chi) \overline\gamma_3\\ &\equiv \tilde\gamma_3+\tilde h. \end{align*} Using \eqref{eq:gamma_3_bar} and \eqref{eq:form_nu_bar} we see that the term in $\{\cdot \}$ is $C^{\alpha}$ and has compact support. The function $(1-\chi) \overline{\gamma}_3$ is $C^{\alpha}$ (this was proved in the proof of Lemma~\ref{lem:construct_nu}) and homogeneous of degree zero outside \(B_{6}(0,2)\). Therefore, \begin{align*} \| \tilde h \|_{C^{\alpha}(B_{9}((x_1^0,x_2^0, x_3^0),R))} \leq C, \end{align*} with \(C\) independent of \((x_1^0,x_2^0,x_3^0)\in\R^{9}\) and \(R>0\). Since \(\chi\) has compact support, and \(\overline{\nu}\in C^{1,1}(\R^{6})\), we have \begin{align*} \| \tilde \nu_{{\rm cut}} \|_{C^{1,1}(B_{9}((x_1^0,x_2^0, x_3^0),R))} \leq C, \end{align*} with \(C\) independent of \((x_1^0,x_2^0,x_3^0)\in\R^{9}\) and \(R>0\). Since \(\mathcal T\) is an orthogonal transformation, \eqref{eq:construct_nu_cut} follows. This finishes the proof of the lemma. \end{pf} \section{Computation of \(\mathcal{P}_{2}^{(6)}\gamma_2\)} \label{chap:P_2_two_elec} In this appendix we compute \(\mathcal{P}_{2}^{(6)}\gamma_{2}\), the singular part of the two-particle terms in \(|\nabla F_{2}|^{2}\), see \eqref{eq:grad_F2_squared} and \eqref{def:gamma_2&gamma_3}. This is Lemma~\ref{prop:proj_gamma2} below. It follows from general results on \(\mathcal{P}_{2}^{(6)}\eta\) when \(\eta\) has certain symmetry-properties (Lemma~\ref{lemma:first_invariance}). The latter is also responsable for the non-occurence of terms of order \(r^{2}\ln(r)\) (of regularity \(C^{1,\alpha}\) only) in the function \(\nu\) constructed in the previous appendix; see Lemma~\ref{lem:construct_nu}. \begin{lem} \label{prop:proj_gamma2} Let \begin{align} \label{eq:gamma3_again} \gamma_{2}(x,y)=\Big(\frac{x}{|x|}-\frac{y}{|y|}\Big) \cdot\frac{x-y}{|x-y|},\quad(x,y)\in\R^{3}\times\R^{3}. \end{align} Then \begin{align*} \big(\mathcal{P}_{2}^{(6)}\gamma_{2}\big)(x,y) =\frac{16(2-\pi)}{3\pi}\frac{x\cdot y}{x^{2}+y^{2}}, \quad(x,y)\in\R^{3}\times\R^{3}. \end{align*} \end{lem} \begin{pf}This will follow from Lemma~\ref{lemma:first_invariance} and Lemma~\ref{lem:value_c0} below. Na\-me\-ly, by (\ref{lem:invarianceI}) and (\ref{cor:x*y}) in Lemma~\ref{lemma:first_invariance} we get that, due to symmetry, \begin{align*} \big(\mathcal{P}_{2}^{(6)}\gamma_{2}\big)(x,y) =c_{1}\,\frac{x\cdot y}{x^{2}+y^{2}}\ \text{ for some }\ c_{1}\in\R, \end{align*} that is, only the function \(x\cdot y\) (restricted to \(\sphere^{5}\)) contributes to the projection onto \(\mathfrak h_{2}^{(6)}\) of the function \(\gamma_{2}\) in \eqref{eq:gamma3_again}. That \(c_{1}=\frac{16(2-\pi)}{3\pi}\) is the result of Lemma~\ref{lem:value_c0} (which is merely two computations). \end{pf} \begin{lem} \label{lemma:first_invariance} Assume \(\eta\in L^{2}({\mathbb S}^5)\) satisfies \begin{align} \label{eq:inv_SO3} \eta({\mathcal O}x, {\mathcal O}y)&=\eta(x,y) \end{align} for all \(\mathcal{O} \in SO(3)\) and almost all \((x, y)\in\sphere^{5}\subset\R^{3}\times\R^{3}\). Let \(\mathcal{Q}_{1}\) be the orthogonal projection (in \(L^{2}(\sphere^{5})\)) onto \begin{align*} \Span\left\{\left.P_{1}\right|_{\sphere^{5}}, \left.P_{2}\right|_{\sphere^{5}}\right\}, \end{align*} and \(\mathcal{Q}_{2}\) the orthogonal projection onto \begin{align*} \Span\left\{\left.P_{1}\right|_{\sphere{^5}}\right\}, \end{align*} where \(P_{1}(x,y)=x\cdot y\), \(P_{2}(x,y)=x^{2}-y^{2}\), \((x,y)\in\R^{3}\times\R^{3}\). Then \begin{enumerate}[\rm (i)] \item\label{lem:invarianceI} \(\mathcal{P}_{2}^{(6)}\eta = \mathcal{Q}_{1}\eta\). \item \label{cor:x*y} Let \(\eta\) satisfy \begin{align} \label{eq:symmetry1} \eta(x, y) = \eta(y, x) \text{ for almost all } (x, y)\in\sphere^{5}\subset\R^{3}\times\R^{3}. \end{align} Then \(\mathcal{P}_{2}^{(6)}\eta = \mathcal{Q}_{2}\eta\). \item \label{lem:NO_l=2} Let \(\bar{\mathcal{R}}\) be as in \eqref{def:barR}. Assume \(\eta\) satisfies \begin{align} \label{eq:symmetry2} \!\!\!\!\!\!\!\! \eta(\bar{\mathcal{R}}(x, y)) = \eta(x, y) \text{ for almost all } (x, y)\in\sphere^{5}\subset\R^{3}\times\R^{3}. \end{align} Then \(\mathcal{P}_{2}^{(6)}\eta = 0\). \end{enumerate} \end{lem} \begin{pf*}{Proof of Lemma~\ref{lemma:first_invariance}} Suppose (\ref{lem:invarianceI}) is proven then the proofs of (\ref{cor:x*y}) and (\ref{lem:NO_l=2}) are simple: \begin{pf*}{Proof of {\rm\eqref{cor:x*y}}} Due to (\ref{lem:invarianceI}) we only need to prove that \begin{align*} \int_{{\mathbb S}^5} \eta(x,y) (x^2-y^2)\,d\omega = 0. \end{align*} This follows using the symmetry \eqref{eq:symmetry1} of \(\eta\) (which preserves the measure \(d\omega\) of \(\sphere^{5}\)): \begin{align*} \int_{{\mathbb S}^5} \eta(x,y) P(x,y)\,d\omega = \frac{1}{2} \int_{{\mathbb S}^5} \eta(x,y) \big(P(y,x)+P(x,y)\big)\,d\omega, \end{align*} and when \(P(x,y)=P_{2}(x,y)=x^2-y^2\), then \(P(y,x)+P(x,y)=0.\) This proves (\ref{cor:x*y}). \end{pf*} \begin{pf*}{Proof of {\rm\eqref{lem:NO_l=2}}} Using (\ref{lem:invarianceI}) and \eqref{eq:symmetry2} it is enough to show that \begin{align*} P(x,y) + P(\bar{\mathcal R}(x,y)) + P(\bar{\mathcal R}^2(x,y)) = 0, \end{align*} when \(P(x,y) = x \cdot y\) or \( x^2 - y^2\) (since \(\bar{\mathcal R}\) preserves the measure \(d\omega\) of \(\sphere^{5}\)). This follows by direct calculation. \end{pf*} It remains to prove (\ref{lem:invarianceI}): \begin{pf*}{Proof of {\rm\eqref{lem:invarianceI}}} Recall that \({\mathfrak h}_{2}^{(6)}=\Ran(\mathcal{P}_{2}^{(6)})\). Define \({\mathfrak h}_{2,inv}\) by \begin{align*} {\mathfrak h}_{2,inv}=\Span\left\{ f\in{\mathfrak h}_{2}^{(6)} \,\big|\, f({\mathcal O} x, {\mathcal O} y) = f(x,y) \, \text{ for all } {\mathcal O} \in SO(3) \right\}. \end{align*} Note that \(\mathcal{P}_{2}^{(6)}\eta\in {\mathfrak h}_{2,inv}\) because of \eqref{eq:inv_SO3}. We need to prove that \begin{align*} {\mathfrak h}_{2,inv}=\Span\left\{\left.P_{1}\right|_{\sphere^{5}}, \left.P_{2}\right|_{\sphere^{5}}\right\}. \end{align*} Since every function in \(\mathfrak{h}_{2,inv}\) can be written as a finite sum of spherical harmonics of degree \(2\) it suffices to consider a real, harmonic polynomial \(P\) which is homogeneous of degree \(2\), and which is invariant under the action of \(SO(3)\): \begin{align} \label{eq:inv_P} P({\mathcal O} x, {\mathcal O} y)=P(x,y) \text{ for all } {\mathcal O} \in SO(3). \end{align} Identifying \(P\) with a quadratic form on \(\R^{6}\), there exist real symmetric matrices \(A, B\), and \(C\), such that \begin{align} \label{eq:A_B_C} P(x,y)= x\cdot Ax + y\cdot B y + x\cdot C y. \end{align} The condition of harmonicity of \(P\) becomes \(\Tr[ A + B ] = 0\). We prove that \(A,B\), and \(C\) have to be multiples of the identity matrix \(I_{3}\) on \(\R^{3}\). To do so, let us first restrict to \(x=0\). Using \eqref{eq:inv_P} and \eqref{eq:A_B_C} we get \begin{align*} y\cdot B y = P(0,y) = P({\mathcal O}0,{\mathcal O}y) = \mathcal{O}y\cdot B{\mathcal O} y, \end{align*} for all \({\mathcal O} \in SO(3)\). Let \(\lambda\) be a (real) eigenvalue of \(B\), with corresponding eigenvector \(v\): \(Bv=\lambda v\). Let \(y\) be any vector in \({\mathbb R}^3\). Then there exists an \({\mathcal{O}_{y}} \in SO(3)\) such that \(\mathcal{O}_{y} y = \mu_{y} v\) for some \(\mu_{y}\in\R\), and therefore \(y\cdot B y=\mathcal{O}_{y}y\cdot B\mathcal{O}_{y}y=\lambda\|y\|^2\). Since this is true for all \(y \in {\mathbb R}^3\), we get \(B=\lambda I_{3}\). A similar argument (with \(y=0\), and letting \(x\) vary) shows that also \(A\) is a multiple of the identity. Finally, the condition of harmonicity, \(\Tr[ A + B] = 0\), implies that \(A = -B = -\lambda I_{3}\). Finally the term \(x\cdot C y\). This will be treated similarly. Due to the above (see \eqref{eq:A_B_C}), \(x\cdot C y=P(x,y)-\lambda(y^{2}-x^{2})\). Therefore, \eqref{eq:inv_P} implies \begin{align*} x\cdot C y = \mathcal{O}x\cdot C {\mathcal O} y \quad \text{ for all } {\mathcal O} \in SO(3). \end{align*} By arguments similar to the above, we find that \(C\) is also a multiple of the identity \(I_{3}\). Since \(P(x,y)=\lambda(x^{2}-y^{2})+x\cdot C y\), this finishes the proof of (\ref{lem:invarianceI}). \end{pf*} This finishes the proof of Lemma~\ref{lemma:first_invariance}. \end{pf*} \begin{lem} \label{lem:value_c0} Let \(\mathcal{Q}_{2}\) be the orthogonal projection (in \(L^{2}(\sphere^{5})\)) onto \begin{align*} \Span\left\{\left.P_{1}\right|_{\sphere{^5}}\right\}\quad,\quad P_{1}(x,y)=x\cdot y\quad,\quad (x,y)\in\R^{3}\times\R^{3}, \end{align*} and let \begin{align*} \gamma_{2}(x,y)=\left(\frac{x}{|x|} -\frac{y}{|y|}\right)\cdot\frac{x-y}{|x-y|} \quad,\quad (x,y)\in\R^{3}\times\R^{3}. \end{align*} Then \begin{align} \label{eq:value_c0} \mathcal{Q}_{2}\gamma_{2} = c_{1}\frac{x\cdot y}{x^{2}+y^{2}}\quad,\quad c_{1}=\frac{16(2-\pi)}{3\pi}. \end{align} \end{lem} \begin{pf} Note that, with \begin{align*} Y(\omega)=\frac{\left. P_{1} \right|_{\sphere^{5}}\!(\omega) }{ \big\| \left. P_{1} \right|_{\sphere^{5}} \big\|_{L^{2}(\sphere^{5})} } \quad,\quad \omega=\frac{(x,y)}{\sqrt{x^{2}+y^{2}}}, \end{align*} we have \(\|Y\|_{L^{2}(\sphere^{5})}=1\), and so \begin{align} \label{eq:proj_Y} \mathcal{Q}_{2}\gamma_{2}(\omega) &=Y(\omega)\int_{\sphere^5}Y(\omega)\gamma_{2}(\omega)\,d\omega \\& =\left\{\frac{1} { \big\| \left. P_{1} \right|_{\sphere^{5}} \big\|_{L^{2}(\sphere^{5})}^{2} } \cdot \int_{\sphere^5} \left. P_{1} \right|_{\sphere^{5}}\!(\omega)\, \gamma_{2}(\omega)\,d\omega \right\}\cdot \frac{x\cdot y}{x^{2}+y^{2}}. \nonumber \end{align} We need to compute the two integrals in the brackets. Since \(P_{1}\) is homogeneous of order \(2\) and \(\gamma_{2}\) of order \(0\) (as functions on \(\R^{6}\)), we have \begin{align*} \int_{B_{6}(0,R)} \!\!\!\!\!\!\!\!&P_{1}(x,y)\gamma_{2}(x,y)\,dx\,dy =\frac{R^{8}}{8}\, \int_{\sphere^5} \left. P_{1} \right|_{\sphere^{5}}\!(\omega)\, \gamma_{2}(\omega)\,d\omega. \end{align*} Therefore, \begin{align} \label{eq:scalingP2} \int_{\sphere^5} \left. P_{1} \right|_{\sphere^{5}}\!(\omega)\, \gamma_{2}(\omega)\,d\omega =8\int_{B_{6}(0,1)} \!\!\!\!\!\!\!\!\! P_{1}(x,y)\gamma_{2}(x,y)\,dx\,dy. \end{align} Choose coordinates \((|x|, |y|, |x-y|, \Omega)\) for \(\R^{6}\) (with \(\Omega\) three necessary angles). Note that \begin{align*} P_{1}(x,y)&=x\cdot y = \frac{1}{2}\big(|x|^{2}+|y|^{2}-|x-y|^{2}\big) \quad,\quad (x,y)\in\R^{3}\times\R^{3}, \intertext{and } \gamma_{2}(x,y) &=\frac{|x|+|y|}{|x-y|} \Big(1-\frac{|x|^{2}+|y|^{2}-|x-y|^{2}}{2|x||y|}\Big) \ ,\ (x,y)\in\R^{3}\times\R^{3}. \end{align*} Then (see Hylleraas \cite[(45d)]{Hylleraas}; let \(s=|x|, t=|y|, r=|x-y|\)) \begin{align} \label{eq1:projection} &\int_{B_{6}(0,1)} P_{1}(x,y)\gamma_{2}(x,y)\,dx\,dy \nonumber =\frac{1}{4}\Big(\int\,d\Omega\Big) \times \nonumber \\&\times \int_{0}^{1}\!\int_{0}^{\sqrt{1-s^{2}}}\!\!\!\int_{|s-t|}^{s+t} (s^{2}+t^{2}-r^{2})(s+t) \big(2st-(s^{2}+t^{2}-r^{2})\big)\,dr\,dt\,ds\nonumber\\ &= \frac{1}{4}\frac{(2-\pi)}{48}\int\,d\Omega. \end{align} Using \eqref{eq:scalingP2} and \eqref{eq1:projection} this means that \begin{align} \label{eq:firstIntegral} \int_{\sphere^5} \left. P_{1} \right|_{\sphere^{5}}\!(\omega)\, \gamma_{2}(\omega)\,d\omega =\frac{2-\pi}{24}\int\,d\Omega. \end{align} Next, observe that, again due to homogeneity, we have \begin{align*} \int_{B_{6}(0,R)}(x\cdot y)^{2}\,dx\,dy =\frac{R^{10}}{10}\left.\big\|P_{1}\right|_{\sphere^{5}} \big\|_{L^{2}(\sphere^{5})}^{2} \end{align*} and so \begin{align} \label{eq:homgen2} \left.\big\|P_{1}\right|_{\sphere^{5}}\big\|_{L^{2}(\sphere^{5})}^{2} =10\int_{B_{6}(0,1)}(x\cdot y)^{2}\,dx\,dy. \end{align} Since \(x\cdot y=\frac12\big(|x|^{2}+|y|^{2}-|x-y|^{2}\big)\) we get (using coordinates as above) \begin{align*} &\int_{B_{6}(0,1)}(x\cdot y)^{2}\,dx\,dy \nonumber \\& =\frac14\Big(\int d\Omega\Big)\int_{0}^{1}\int_{0}^{\sqrt{1-s^{2}}}\int_{|s-t|}^{s+t} \big(s^2+t^2-r^2\big)^{2}srt\,dr\,dt\,ds \nonumber \\& =\frac{\pi}{1280} \int\,d\Omega. \end{align*} This means (see \eqref{eq:homgen2}) that \begin{align} \label{eq:normL2} \left.\big\|P_{1}\right|_{\sphere^{5}}\big\|_{L^{2}(\sphere^{5})}^{2} =\frac{\pi}{128}\int\,d\Omega. \end{align} Now \eqref{eq:value_c0} follows from \eqref{eq:proj_Y}, \eqref{eq:firstIntegral}, and \eqref{eq:normL2}. This finishes the proof of Lemma~\ref{lem:value_c0}. \end{pf} \begin{acknowledgement} All four authors thank the organizers of the program {\it Partial Differential Equations and Spectral Theory} for invitations to the Mittag-Leffler Institute in 2003 where part of the work was done. Furthermore, parts of this work have been carried out at various institutions, whose hospitality is gratefully acknowledged: Aalborg University (SF, MHO, THO), The Erwin Schr\"{o}dinger Institute (T\O S), Universit\'{e} Paris-Sud (T\O S), and the IH\'ES (T\O S). Financial support from the Danish Natural Science Research Council, European Science Foundation Programme {\it Spectral Theory and Partial Differential Equations} (SPECT), and EU IHP network {\it Postdoctoral Training Program in Mathematical Analysis of Large Quantum Systems}, contract no.\ HPRN-CT-2002-00277 is gratefully acknowledged.\\ SF has been supported by a Marie Curie Fellowship of the European Community Programme `Improving the Human Research Potential and the Socio-Economic Knowledge Base' under contract number HPMF-CT-2002-01822, and by a grant from the Carlsberg Foundation. \\ Finally, SF and T\O S wish to thank I. Herbst for useful discussions at the Mittag-Leffler Institute. \end{acknowledgement} %\bibliographystyle{hamsplain} \bibliographystyle{amsplain} \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \begin{thebibliography}{10} \bibitem{CFKS:1987} Hans~L. Cycon, Richard~G. 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