\section{Introduction and statement of the results}
In recent years, there has been a lot of study about fractional order operators in the context of elliptic theory with nonlocal operators, nonlinear diffusion involving nonlocal operators, nonlocal aggregations, and balance between nonlinear diffusions and nonlocal attractions. The elliptic theory of  fractional order operators is well understood in many directions like semilinear equations, free-boundary value problems, and non-local minimal surfaces (see \cite{cafroq}, \cite{cafsyl}, \cite{cafsoug}, \cite{cafval}, \cite{cafvas}, \cite{fabkenser}). Furthermore a connection to classical conformally covariant operators arising in conformal geometry and their associated conformally invariant geometric variational problems is established (see \cite{cg}, \cite{fg}, \cite{guil}, \cite{gq} \cite{gz}, \cite{qr}, \cite{gms}). In this paper, we are interested in the latter aspect of fractional order operators, precisely in the fractional Yamabe problem introduced by Gonzalez-Qing\cite{gq}. 
\vspace{6pt}

\noindent
To discuss the fractional Yamabe problem, we first recall some definitions in the theory of asymptotically hyperbolic metrics. Given \;$X=X^{n+1}$\; a smooth manifold with boundary $M=M^n$\; and \;$n\geq 2$, we say that \;$\varrho$\; is a defining function of the boundary \;$M$\; in \;
$X$,\; if 
$$
\varrho>0\;\;\text{in}\;\;X,\;\;\varrho=0\;\;\text{on}\;\;M\;\;\text{and}\;\;d\varrho\neq 0\;\;\text{on}\;\; M.
$$
A Riemannian metric \;$g^+$\; on \;$X$\; is said to be conformally compact, if for some defining function \;$\varrho$, the Riemannian metric 
\begin{equation}\label{eq:cmetric}
 g:=\varrho^2g^+
\end{equation}
 extends to \;$\overline{X}:=X\cup M$\; so that \;$(\overline{X}, \, g)$\; is a compact Riemannian manifold with boundary \;$M$ and interior \;$X$. Clearly, this induces a conformal class of Riemannian metrics 
$$ [h]=[ g|_{TM}]$$ on \;$M$, where $TM$ denotes the tangent bundle of \;$M$, when the defining functions \;$\varrho$\; vary and the resulting conformal manifold \;$(M, [h])$\; is called conformal infinity of \;$(X, \;g^+)$. Moreover  a Riemannian metric \;$g^+$\; in \;$X$\; is said to be asymptotically hyperbolic, if it is conformally compact and its sectional curvature tends to \;$-1$\; as one approaches the conformal infinity of \;$(X, \;g^+)$,
 which is equivalent to 
\begin{equation*
|d\varrho|_{\bar g}=1
\end{equation*}
 on\;$M$, see \cite{mazzeo1}, and in such a case \;$(X, \;g^+)$\; is called an asymptotically hyperbolic manifold. Furthermore  a Riemannian metric \;$g^+$\; on \;$X$\; is said to be conformally compact Einstein or Poincar\'e-Einstein, if it is asymptotically hyperbolic and satisfies the Einstein equation 
\begin{equation*
Ric_{g^+}=-ng^+,
\end{equation*}
where $Ric_{g^+}$\; denotes the Ricci tensor of \;$(X, \;g^+)$.
\vspace{6pt}

\noindent
On one hand, for every asymptotically hyperbolic manifold \;$(X, \;g^+)$\; and every choice of the representative \;$h$\; of its conformal infinity \;$(M, [h])$, there exists a  geodesic defining function \;$ y $\; of \;$M$\; in \;$X$\; such that  in a tubular neighborhood of \;$M$\; in \;$X$,  the Riemannian metric \;$g^+$\; takes the following normal form
\begin{equation}\label{eq:uniqdef}
g^+=\frac{d y ^2+h_{ y }}{ y ^2},
\end{equation}
where \;$h_{ y }$\; is a family of Riemannian metrics on \;$M$\; satisfying \;$h_0=h$\; and \;$ y $\; is the unique such a one in a tubular neighborhood of $M$. We remark that in the conformally compact Einstein case, \;$h_{ y }$\; has an asymptotic expansion which contains only even power of \;$ y $, at least up to order \;$n$, see \cite{cg}. Furthermore  we say that the conformal infinity \;$(M, \;[\hat h])$\; of an asymptotically hyperbolic manifold \;$(X, \;g^+)$\; is locally flat, if \;$h$\; is locally conformally flat, and clearly this is independent of the representative \;$h$\; of \;$[h]$. Moreover  we say that \;$(M, [h])$\; is umbilic, if \;$(M, h)$\; is umbilic in \;$(X, \; g)$ \;where \;$g$\; is given by \eqref{eq:cmetric} and \;$ y $\; is the unique geodesic defining function given by \eqref{eq:uniqdef}, and this is clearly independent of the representative \;$h$\; of \;$[h]$, as easily seen from the unicity of the normal form \eqref{eq:uniqdef} or Lemma 2.3 in \cite{gq}. Similarly, we say that \;$(M, [h])$\; is minimal if \;$H_{g}=0$\; with \;$H_{g}$\; denoting the mean curvature of \;$(M,  \;h)$\; in \;$(\overline X, \;g)$ with respect to the inward direction, and this is again clearly independent of the representative of \;$h$\; of \;$[h]$, as easily seen from Lemma 2.3 in \cite{gq}. Finally, we say that \;$(M, [h])$\; is totally geodesic, if \;$(M, [h])$\; is umbilic and minimal.                                                                                                                                         
\begin{rem}\label{eq:minimal}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
We would like to emphasize the fact that the conformal infinity \;$(M, [h])$\; of  any Poincar\'e-Einstein manifold \;$(X, g^+)$\; is totally geodesic.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          \end{rem}
\begin{rem}\label{defpe}
As every \;$2$-dimensional Riemannian manifold is locally conformally flat, we will say locally flat conformal infinity of a Poincar\'e-Einstein manifold  to mean just the conformal infinity of a Poincar\'e-Einstein manifold when the dimension is either \;$2$\; or which is further locally flat if the dimension is bigger than \;$2$. 
\end{rem}

\noindent
On the other hand, for any asymptotically hyperbolic manifold \;$(X, g^+)$\; with conformal  infinity \;$(M, [h])$, Graham-Zworsky\cite{gz}\; have attached a family of scattering operators \;$S(s)$\; which is a meromorphic family of pseudo-differential operators on \;$M$ defined on \;$\mathbb{C}$, by considering Dirichlet-to-Neumann operators for the scattering problem for \;$(X, \;g^+)$\; and a meromorphic continuation argument. Indeed,  it follows from  \cite{gz} and \cite{mazmel} that for every \;$f\in C^{\infty}(M)$, and for every \;$s\in \mathbb{C}$\; such that \;$Re(s)>\frac{n}{2}$\;and\; \;$s(n-s)$ is not an \;$L^2$-eigenvalue of \;$-\D_{g^+}$, the following generalized eigenvalue problem
\begin{equation}\label{eq:geneig}
 -\D_{g^+}u-s(n-s)u=0\,\;\;\text{in}\;\;X
\end{equation}
has a solution of the form
$$
u=F y ^{n-s}+G y ^s, \;\; F,G\in C^{\infty}(\overline X), \;\;F|_{ y =0}=f,
$$
where \;$ y $\; is given by the normal form \eqref{eq:uniqdef} and for those values of \;$s$, the scattering operator \;$S(s)$\; on \;$M$\; is defined as follows
\begin{equation}\label{eq:dtnscat}
S(s)f:=G|_{M}.
\end{equation}
Furthermore  using a meromorphic continuation argument, Graham-Zworsky\cite{gz} extend \;$S(s)$\; defined by \eqref{eq:dtnscat} to a meromorphic family of pseudo-differential operators on \;$M$\; defined on all \;$\mathbb{C}$\; and still denoted by $S(s)$\;with only a discrete set of poles including the trivial ones \;$s=\frac{n}{2}, \frac{n}{2}+1, \cdots, $\; which are simple poles of finite rank, and possibly some others corresponding to the \;$L^2$-eigenvalues of \;$-\D_{g^+}$. Using the regular part of the scattering operators \;$S(s)$, to any \;$\gamma\in (0, 1)$\; such that \;$\left(\frac{n}{2}\right)^2-\gamma^2<\l_1(-\D_{g^+})$ \;with \;$\l_1(-\D_{g^+})$\: denoting the first eigenvalue of \;$-\D_{g_+}$ ,\; Chang-Gonzalez\cite{cg} have attached the following fractional order pseudo-differential operators referred to as fractional conformal Laplacians or fractional Paneitz operators
\begin{equation}\label{eq:fracopd}
P^{\gamma}[g^+, \;h]:=-d_\gamma S\left(\frac{n}{2}+\gamma\right),
\end{equation}
where \;$d_\gamma$\; is a positive constant depending only on \;$\gamma$\; and chosen  such that the principal symbol of \;$P^{\gamma}[g^+,  h]$\; is exactly the same as the one of the fractional Laplacian \;$(-\D_{h})^{\gamma}$. When there is no possible confusion with the  metric \;$g^+$, we will just use the simple notation
\begin{equation*
P^{\gamma}_{h}:=P^{\gamma}[g^+,  \;h].
\end{equation*}
Similarly to the other well studied conformally covariant differential operators, Chang-Gonzalez\cite{cg} associate to each \;$P^{\gamma}_{h}$\; the following curvature quantity
\begin{equation*}
 Q^{\gamma}_{h}:=P^{\gamma}_{h}(1).
\end{equation*}
The \;$Q^\gamma_{h}$\; are referred to as fractional scalar curvatures, or fractional \;$Q$-curvatures, or simply \;$Q^\gamma$-curvatures. Of particular importance to conformal geometry is the following conformal covariance property verified by \;$P^{\gamma}_{h}$
\begin{equation}\label{eq:confinv}
P^{\gamma}_{h_u}(v)=v^{-\frac{n+2\gamma}{n-2\gamma}}P^{\gamma}_{h}(uv)
\;\text{ for }\; h_v=v^{\frac{4}{n-2\gamma}}  \;\text{ and }\; 0<v\in C^{\infty}(M).
\end{equation}
As for the classical scalar curvature, for the \;$Q^\gamma$-curvature, Gonzalez-Qing\cite{gq} have introduced the fractional Yamabe problem (or \;$\gamma$-Yamabe problem) which consists of finding conformal metrics with constant fractional scalar curvature \;$Q^{\gamma}$. Moreover  for an asymptotically hyperbolic manifold \;$(X, g^+)$\; with conformal infinity \;$(M, [h])$, Chang-Gonzalez\cite{cg} showed the equivalence between the Dirichlet-to-Neumann operators of the scattering problem \eqref{eq:geneig} with the ones of some uniformly degenerate elliptic boundary value problems defined on $(X, g)$\; which coincide with the extension problem of Caffarelli-Sylvestre\cite{cafsyl} when 
\begin{equation*
(X, g^{+},  h)=(\mathbb{H}^{n+1}, g_{\mathbb{H}^{n+1}}, g_{\mathbb{R}^n}),
\end{equation*} 
and hence 
\begin{equation*
(-\D_{g_{\mathbb{R}^n}})^{\gamma}=P^{\gamma}[g_{\mathbb{H}^{n+1}}, g_{\mathbb{R}^n}].
\end{equation*}
\vspace{6pt}

\noindent
The latter established relation allows Gonzalez-Qing\cite{gq} to derive a Hopf-type maximum principle, by taking inspiration from the work of Cabre-Sire\cite{cabsir} which deals with the Euclidean half-space, see Theorem 3.5 and Corollary 3.6 in \cite{gq}. Clearly, the latter Hopf-type maximum principle of Gonzalez-Qing\cite{gq} opens the door to variational arguments for existence for the \;$\gamma$-Yamabe problem, as explored by Gonzalez-Qing\cite{gq}, Gonzalez-Wang\cite{gw}, and Kim-Musso-Wei\cite{kmw1} . 
\vspace{6pt}

\noindent
In terms of geometric differential equations, the fractional Yamabe problem is equivalent to finding a smooth and positive solution to the following semilinear pseudo-differential equation with critical Sobolev nonlinearity
\begin{equation}\label{eq:fracyam}
P^{\gamma}_{h}u+Q^{\gamma}_{h}=cu^{\frac{n+2\gamma}{n-2\gamma}}\;\;\text{on}\;\;M,
\end{equation}
for some constant \;$c$. The non-local equation \eqref{eq:fracyam} has a variational structure and  thanks to the regularity theory for uniformly degenerate elliptic boundary value problems (see \cite {cabsir},\cite{fabkenser}, \cite{gq}), the above cited local interpretation of \;$P^\gamma_h$\; of Chang-Gonzalez\cite{cg}, and the Hope-type maximum principle of Gonzalez-Qing\cite{gq}, we have that smooth and positive solutions to \eqref{eq:fracyam} can be found by looking at critical points of the following fractional Yamabe functional 
\begin{equation*
\mathcal{E}_{h}^\gamma(u):=\frac{\langle u, u\rangle_{P^{\gamma}_{ h}}}{\left(\int_M u^{\frac{2n}{n-2\gamma}}dV_{ h}\right)^{\frac{n-2\gamma}{n}}}, \;\; u\in W^{\gamma, 2}_+(M,  h):=\{v\in W^{\gamma, 2}(M, h), \;\;v>0\}, 
\end{equation*}
where \;$W^{\gamma, 2}(M,  h)$\; denotes the usual fractional Sobolev space on \;$M$\;with respect to the Riemannian metric \;$h$
and $\langle u, u\rangle_{P^{\gamma}_{ h}}=\langle P^{\gamma}_{ h}u, u\rangle_{L^{2}_{h}(M)}$, for more informations see  \cite{dinpalval}, \cite{gt}, and \cite{s}. 
\vspace{6pt}

\noindent
However, as for the classical Yamabe problem and for the same reasons, the variational analysis of \;$\mathcal{E}_{h}^\gamma$\; has a local regime-namely the situation where the local geometry can be used  to ensure a solution (even a minimizer) and a global one-which is the case where the local geometry can not be used to find a solution and that just a global one usually called {\em mass} can be used. Furthermore  still as for the classical Yamabe problem, there is a natural invariant called the \,$\gamma$-Yamabe invariant of \;$(M, [h])$\; denoted by $\mathcal{Y}^{\gamma}(M, [h])$\; and defined by the following formula
\begin{equation*
\mathcal{Y}^{\gamma}(M, [h]):=\inf_{u\in W^{\gamma, 2}_+(M)}\frac{\langle u, u\rangle_{P^{\gamma}_{h}}}{(\int_{M}u^{\frac{2n}{n-2\gamma}}dV_{h})^{\frac{n-2\gamma}{2}}}.
\end{equation*}
From the work of Gonzalez-Qing\cite{gq}, it is known that \;$\mathcal{Y}^{\gamma}(M, [h])$\; satisfies the following rigidity estimate
\begin{equation*
\mathcal{Y}^{\gamma}(M, [h])\leq \mathcal{Y}^{\gamma}(S^n, [g_{S^n}]),
\end{equation*}
where \;$(S^n, [g_{S^n}])$\; is the conformal infinity of the Poincaré ball model of the hyperbolic space. Moreover  as mentioned in the abstract, the global situation clearly includes the case of locally flat conformal infinity of a Poincar\'e-Einstein manifold  as observed by Kim-Musso-Wei\cite{kmw1}, while the existence results of Gonzalez-Qing\cite{gq}, Gonzalez-Wang\cite{gw}, and some part of the existence results in Kim-Musso-Wei\cite{kmw1} deal with situations which clearly belong to the local regime. Moreover  in the global situation, as for the classical Yamabe problem, to run the minimizing technique of Aubin\cite{au}-Schoen\cite{sc}, one needs an analogue of the positive mass theorem of Schoen-Yau\cite{sy1} (see the work of Kim-Musso-Wei\cite{kmw}), which is not known to hold, see conjecture 1.6 in Kim-Musso-Wei\cite{kmw1}. 
\vspace{12pt}

\noindent
Our main goal in this work is to show that  with the analytical results provided by the works of Caffarelli-Sylvestre\cite{cafsyl}, Chang-Gonzalez\cite{cg}, and Gonzalez-Qing\cite{gq} cited above at hand, and the classical regularity theory for uniformly degenerate elliptic equations, one can perform a variational argument for existence for the fractional Yamabe problem for a locally flat conformal infinity of a Poincar\'e-Einstein manifold, bypassing the lack of knowledge of the fractional analogue of the positive mass theorem of Schoen-Yau\cite{sc}. Indeed, using a suitable scheme of the algebraic topological argument (called also barycenter technique) of Bahri-Coron\cite{bc} in the form implemented in our previous works \cite{martndia1} and \cite{martndia2}, we show the following existence theorem.
\begin{thm}\label{eq:thm}
Let \;$n\geq 2$\; be a positive integer, \;$(X^{n+1},\; g^+)$\; be a Poincar\'e-Einstein manifold with conformal infinity \;$(M^{n}, \;[h])$, \;$\gamma \in(0, 1)$, and \;$\left(\frac{n}{2}\right)^2-\gamma^2<\l_1(-\D_{g^+})$. Assuming that either\;$n=2$\; or \;$n\ge 3$\; and \;$(M, [h])$\; is locally flat, then \;$(M, [h])$\; carries a Riemannian metric of constant \;$Q^{\gamma}$-curvature. 
\end{thm}
\vspace{6pt}

\noindent
\begin{rem}\label{eq:half}
We point out that as observed by Gonzalez-Qing\cite{gq}, for Poincar\'e-Einstein manifolds, the \;$\frac{1}{2}$-Yamabe problem is equivalent to the Riemann mapping problem of Cherrier\cite{cherrier}-Escobar\cite{es2}, which is completely solved after the series of works of \cite{almaraz1}, \cite{chen}, \cite{es2}, \cite{marques}, \cite{ould1}, \cite{martndia2}.
\end{rem}



\noindent
As already mentioned, to prove Theorem \ref{eq:thm}, we use variational arguments by applying a suitable scheme of the barycenter technique of Bahri-Coron\cite{bc}. In fact, exploiting the fact that  the conformal infinity is locally flat, the ambient space is Poincar\'e-Einstein, the conformal covariance property \eqref{eq:confinv}, the works of Caffarelli-Sylvestre\cite{cafsyl} and Chang-Gonzalez\cite{cg} , the Hopf-type maximum principle of Gonzalez-Qing\cite{gq}, and the standard bubbles attached to the related optimal trace Sobolev inequality, we define some bubbles  and show that they can be used to run a suitable scheme of the barycenter technique of Bahri-Coron\cite{bc} for existence. For a discussion about the barycenter technique of Bahri-Coron\cite{bc}, we refer the reader to the introduction of our paper\cite{martndia2} (see also \cite{bc}, \cite{gam1}, \cite{gam2}, \cite{martndia1}, and \cite{ould1}).
\vspace{6pt}

\noindent
We would like to make some comments about the application of the barycenter technique of Bahri-Coron\cite{bc} in our situation and in the case of the classical Yamabe problem for locally conformally flat closed Riemannian manifolds studied by Bahri\cite{bah1}. Clearly the latter situations are counterpart to each other, however the nonlocal aspect of our situation creates an additional difficulty which is not in its counterpart for the classical Yamabe problem which is of local nature. In fact, even if both problems are conformally invariant and that after a conformal change, we are locally in the corresponding model space of singularity, we have that the lower order term, i.e. the scalar curvature, {\em vanishes} locally for the classical Yamabe problem, because of its  local nature, while for the fractional Yamabe problem, that does not necessarily imply that the \;$Q^{\gamma}$-curvature vanishes, because of the nonlocal aspect of the problem, and this is the source of the difficulty we mentioned before, which is similar to the one  encountered by Bahri-Brezis\cite{BB} (and Brendle\cite{bre2} in a different framework). To overcome the latter issue, we use the works of Caffarelli-Sylvestre\cite{cafsyl}, Chang-Gonzalez\cite{cg}, and Gonzalez-Qing\cite{gq}, to reduce ourself to a local situation. However, after having done that  one encounters the problem of not having an explicit knowledge of the standard bubble corresponding to the reduced local situation, and clearly an explicit knowledge of the standard bubble in the corresponding situation for the classical Yamabe problem plays an important role in the variational theory of the classical Yamabe functional. To deal with the lack of  an explicit knowledge of the standard bubbles corresponding to the reduced local situation on the \;$1$-dimension augmented half space, we observe that its integral representation  given in Caffarelli-Sylvestre\cite{cafsyl} can be interpreted as a suitable interaction of standard bubbles on the boundary of the latter augmented half space with different points and scales of concentration. 
Then one can think that the latter way of interpretation gives the needed estimates to make use of the barycenter technique of Bahri-Coron\cite{bc}, that one gets for free from an explicit knowledge, and this is made rigorous in this work, see lemma \ref{lem_standard interaction on Rn} and corollary \ref{cor_sharp_estimates} . 
\vspace{6pt}

\noindent
\begin{rem}
 We remark that the estimates of  corollary \ref{cor_sharp_estimates} are sharp,  new, of independent interest, and that some weak versions of them are known in the literature, see for example \cite{bah1}, \cite{gq}, \cite{gw}, \cite{kmw}. Furthermore  the method of proof of the estimates of  corollary \ref{cor_sharp_estimates}\ which is based on  lemma \ref{lem_standard interaction on Rn} is different from its weak versions whose strategy of proof is based on Fourier transform.
\end{rem}
\vspace{6pt}

\noindent
\begin{rem}
We would like to emphasize the analogy between our function \;$\mathcal{M}_{\gamma}$\; given by definition \ref{fracmass} and the notion of mass appearing in the context of the classical Yamabe problem. Moreover  we want to point out that the analysis giving rise to the definition of $\mathcal{M}_{\gamma}$\; answers the first part of the conjecture of Kim-Musso-Wei\cite{kmw1} about the structure of the Green's function, see corollary \ref{cor_kernels_for_poincare_einstein_metrics}.  Furthermore  we believe that the work of Schoen-Yau\cite{sy1} can be adapted to this fractional framework and answer the second part of conjecture of Kim-Musso-Wei\cite{kmw1} about sign, and hence as checked by Kim-Musso-Wei\cite{kmw1} 
ensure the existence of a minimizer for \;$\mathcal{E}^\gamma_h$. 
We add that under the validity of the conjecture of Kim-Musso-Wei\cite{kmw1}, our sharp estimate given by lemma \ref{sharpenergy} gives directly the existence of a fractional Yamabe minimizer.
\end{rem}
\vspace{6pt}


\noindent
The structure of the paper is as follows . In Section \ref{eq:notpre}, we fix some notations, define our bubbles and give some preliminaries, like the set of formal barycenters of the underlying manifold of the conformal infinity and present some useful topological properties of them. In Section \ref{nonhom}, we develop a non-homogeneous extension of some aspect of the work of Chang-Gonzalez\cite{cg} and Graham-Zworsky\cite{gz}. It is divided into subsections. In the first one, namely subsection \ref{eq:nhscatdeg},  we develop a non-homogeneous scattering theory, define the associated non-homogeneous fractional operator and its relation to a non-homogeneous uniformly degenerate boundary value problem. In subsection \ref{confnonhom}, we discuss the conformal property of the non-homogeneous fractional operator. We point out that section \ref{nonhom} even being of independent interest contains a regularity result that we use in section \ref{pceinsten}.  Section \ref{fundso} is concerned with the study of some fundamental solutions in the general case of asymptotically hyperbolic manifolds. While section \ref{bubinter} deals with the introduction of some bubbles and the derivation of some related interactions, in section \ref{pceinsten} we sharpen the results obtained so far in the particular case of a locally flat conformal infinity of a Poincaré-Einstein manifold (recalling Remark \ref{defpe} ). Moreover  we recall the Schoen's bubbles associated to our problem and use them to define an other family of bubbles--and call them Projective bubbles-- which "talk" more to the equivalent local formulation. Furthermore  we establish  sharp \;$L^{\infty}$-estimates for the difference between the latter two type of bubbles and derive a sharp selfaction estimate for our Projective bubbles. In section \ref{varagtop}, we present the variational and algebraic topological argument to prove Theorem \ref{eq:thm}. It is divided in two subsections.  In the first one, namely subsection \ref{var}, we present a variational principle which extends the classical variational principle by taking into account the non-compactness phenomena of our Projective bubbles. In the second one, namely subsection \ref{agtop}, we present the barycenter technique or algebraic topological argument for existence.  Finally in section \ref{appendix}, we collect the proof of some technical lemmas and estimates. 

 

\vspace{6pt}

\noindent
\begin{center}
{\bf Acknowledgements}
\end{center}
The authors worked on this project when they were visiting the department of Mathematics of the University of Ulm in Germany. Part of this paper was written when the authors were visiting the Mathematical Institute of Oberwolfach in Germany as "Research in Pairs".  We are very grateful to all these institutions for their kind hospitality.


\section{Notations and preliminaries}\label{eq:notpre} 
In this section, we fix some notations, and give some preliminaries. First of all let \;$X=X^{n+1}$\; be a manifold of dimension \;$n+1$\; with boundary \;$M=M^{n}$\; and closure \;$\overline{X}$\; with\;$n\geq 2$. 
\vspace{10pt}

\noindent
In the following, for any Riemannian metric \;$\bar h$\; defined on \;$M$, $a\in M$\; and \;$r>0$, we will use the notation \;$B^{\bar h}_{r}(a)$\; to denote the geodesic ball with respect to $\bar h$\; of radius \;$r$\;and center \;$a$.  We also denote by \;$d_{\bar h}(x,y)$\; the geodesic distance with respect to \;$\bar h$\; between two points \;$x$\;and \;$y$\; of \;$M$.  $inj_{\bar h}(M)$\;stands for the injectivity radius of \;$(M, \bar h)$. $dV_{\bar h}$\;denotes the Riemannian measure associated to the metric\;$\bar h$\; on \;$M$. For \;$a\in M$, we use the notation \;$\exp^a_{\bar h}$\; to denote the exponential map with respect to \;$\bar h$\; on \;$M$. 
\vspace{10pt}

\noindent
Similarly, for any Riemannian metric \;$\bar g$\; defined on \;$\overline{X}$, $a\in M$\; and \;$r>0$, we will use the notation \;$B^{\bar g, + }_{a}(r)$\; to denote the geodesic half ball with respect to \;$\bar g$\; of radius \;$r$\;and center \;$a$.  We also denote by \;$d_{\bar g}(x,y)$\; the geodesic distance with respect to \;$\bar g$\; between two points \;$x\in M$\;and \;$y\in \overline {X}$.  $inj_{\bar g }(\overline{X})$\;stands for the injectivity radius of \;$(\overline{X}, \bar g)$. $dV_{\bar g}$\;denotes the Riemannian measure associated to the metric\;$\bar g$\; on \;$\overline{X}$. For \;$a\in M^{n}$, we use the notation \;$\exp_a^{\bar g, +}$\; to denote the exponential map with respect to \;$\bar g$\; on \;$\overline{X}$. 
\vspace{10pt}


\noindent
$\mathbb{N}$\;denotes the set of nonnegative integers, $\mathbb{N}^*$\;stands for the set of positive integers, and  for $k\in \mathbb{N}^*$, $\mathbb{R}^k$\;stands for the standard $k$-dimensional Euclidean space,  $\mathbb{R}^k_+$ the open positive half-space of $\mathbb{R}^k$, and $\bar \mathbb{R}^k_+$ its closure in $\mathbb{R}^k$. For simplicity, we will use the notation \;$\mathbb{R}_+:=\mathbb{R}^1_+$, and $\bar \mathbb{R}_+:=\bar \mathbb{R}^1_+$. For $r>0$, \;$B^{\mathbb{R}^k}_r(0)$\; denotes the open ball of \;$\mathbb{R}^k$\; of center \;$0$\; and radius \;$r$. Furthermore   for  \;$r>0$, we set
$B^{\mathbb{R}^k_+}_r(0):=B_r^{\mathbb{R}^{k}}(0)\cap \bar \mathbb{R}^k_+\simeq [0, r[\times B_r^{\mathbb{R}^{k-1}}(0)$. For simplicity,  we set \;$B_r=B_r^{\mathbb{R}^n}$\; and $B_r^+=B_r^{\mathbb{R}^{n+1}_+}$. 
\vspace{10pt}

\noindent
For \;$p\in \mathbb{N}^*$, $\sigma_p$\; denotes the permutation group of \;$p$ elements and \;$M^p$\; the Cartesian product of \;$p$\; copies of \;$M$. We define \;$(M^2)^*:=M^2\setminus Diag(M^2)$, where \;$Diag(M^2)=\{(a, a): \,a\in M\}$\; is the diagonal of \;$M^2$.  
\vspace{10pt}

\noindent
For \;$1\leq p\leq \infty$ and $k\in \mathbb{N}$, $s\in \mathbb{R}_+$, $\beta\in  ]0, 1[$, and \;$\bar h$\; a Riemannian metric defined on \;$M$, $L^p(M, \bar h)$, \;$W^{s, p}(M, \bar h)$, \;$C^k(M, \bar h)$, and \;$C^{k, \beta}(M, \bar h)$\; stand respectively for the standard $p$-Lebesgue space, \;$(s, p)$-Sobolev space, $k$-continuously differentiable space, and \;$k$-continuously differential space of H\"older exponent \;$\beta$, all on \;$M$ and with respect to \;$\bar h$\; (if the definition needs a metric structure) and for precise definitions and properties, see  for example \cite{aubin}, \cite{dinpalval}, \cite{gt}, and \cite{s}. Similarly, for\;$1\leq p\leq \infty$ and $k\in \mathbb{N}$, $s\in \mathbb{R}_+$, $\beta\in  ]0, 1[$, and \;$\bar g$\; a Riemannian metric defined on \;$\overline{X}$, $L^p(\overline{X}, \bar g)$, \;$W^{s, p}(\overline{X}, \bar g )$, \;$C^k(\overline{X}, \bar g)$, and \;$C^{k, \beta}(\overline{X}, \bar g)$\; stand respectively for the standard $p$-Lebesgue space, \;$(s, p)$-Sobolev space, $k$-continuously differentiable space, and \;$k$-continuously differential space of H\"older exponent \;$\beta$, all on \;$\overline{X}$, and as above, with respect to \;$g$\; (if the definition needs a metric structure) and still for precise definitions and properties, see  for example \cite{aubin}, \cite{dinpalval}, \cite{gt}, and \cite{s}. $C_0^{\infty}(X)$ means element in \;$C^{\infty}(X)$\; vanishing on \;$M$\; to infinite order.
\vspace{10pt} 

\noindent
 For \;$\varepsilon>0$\;  and small, \;$o_{\varepsilon}(1)$\; means quantities which tend to \;$0$\; as \;$\varepsilon$\; tends to \;$0$. $O(1)$ stands for quantities which are bounded.  For \;$x\in \mathbb{R}$, we will use the notation \;$O(x)$, and \;$o(x)$,  to mean  respectively\;$|x|O(1)$, and \; $|x|o(1)$.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Large positive constants are usually denoted by \;$C$\; and the value of \;$C$\; is allowed to vary from formula to formula and also within the same line. Similarly small positive constants are denoted by \;$c$\; and their values may vary from formula to formula and also within the same line. 
  \vspace{10pt}

\noindent
For \;$Z$ a topological space, \;$H_{*}(Z)$\; will denote the singular homology of \;$Z$ with \;$\mathbb{Z}_2$\; coefficients, and $H^*(Z)$ for the cohomology. For \;$Y$ a subspace of \;$X$, $H_*(Z, Y)$ will stand for the relative homology. The symbol \;$\frown$\; will denote the cap product between cohomology and homology. For a map $f:Z\rightarrow Y$, with \;$Z$\; and \;$Y$\; topological spaces, \;$f_*$\; stands for the induced map in homology, and \;$f^*$\; for the induced map in cohomology.
\vspace{6pt}

\noindent
We also define the following quantities
\begin{equation*
c_{n, 1}^{\gamma}:=\int_{\mathbb{R}^{n}}\left(\frac{1}{1+|x|^2}\right)^{n}dx  \;\;\text{and}\;\;c_{n, 2}^{\gamma}:=d_\gamma^*\int_{\mathbb{R}^{n+1}_+}y^{1-2\gamma}\left|\n \hat \d_{0, 1}(y,x)\right|^2dxdy,
\end{equation*}
where
\begin{equation}\label{dsgamma}
d_{\gamma}^*:=\frac{d_\gamma}{2\gamma}.
\end{equation}
Furthermore  we set
\begin{equation}\label{defc3}
c_{n, 3 }^{\gamma}:=\int_{\mathbb{R}^{n}}\left(\frac{1}{1+|x|^2}\right)^{\frac{n+2\gamma}{2}}dx
\end{equation}
and  define
\begin{equation}\label{eq:defc4}
c_{n, 4}^{\gamma}=
c_{n, \gamma}c_{n, 3}^{\gamma}
\end{equation}
and 
\begin{equation}\label{dfcngs}
c_{n, \gamma}^*=d_\gamma^*c_{n, 3}^\gamma.
\end{equation}
Furthermore for \;$\gamma\in (0, 1)$ we set
\begin{equation*
\mathcal{Y}^\gamma(S^n):=\mathcal{Y}^\gamma(S^n, [g_{S^n}]),
\end{equation*}
and for\;$a\in \mathbb{R}^n$\; and\;$\l$\; positive, we define the standard bubbles on \;$\mathbb{R}^n$\; as follows
\begin{equation*
\delta_{a, \l}(x):=\left(\frac{\l}{1+\l^2|x-b|^2}\right)^{\frac{n-2\gamma}{2}}, \;\;x\in \mathbb{R}^n.
\end{equation*}
They are solutions of the following pseudo-differential equation
\begin{equation*
(-\Delta_{\mathbb{R}^n})^{\gamma}\delta_{a, \l}=c_{n, \gamma}\delta_{a, \l}^\frac{2n}{n-2\gamma}\;\;\text{on}\;\; \mathbb{R}^n,
\end{equation*}
where \;$c_{n, \gamma}$\; is a positive constant depending only on \;$n$\; and \;$\gamma$. In particular the numbers \;$c_{n,\gamma},c_{n,1}^{\gamma},c_{n,2}^{\gamma}$ and \;$\mathcal{Y}^{\gamma}(S^n)$\; verify the following relations
\begin{equation}\label{eq:relationcy}
 c_{n, 2}^{\gamma}=c_{n, \gamma}c_{n, 1}^{\gamma}\;\;\text{and}\;\;\mathcal{Y}_{\gamma}(S^n)=\frac{c_{n, 2}^{\gamma}}{(c_{n, 1}^{\gamma})^{\frac{n-2\gamma}{n}}}.
\end{equation}
Using $\delta_{a, \l}$, we define $\hat \d_{a, \l}$ via
\begin{equation*}\begin{split}
\begin{cases}
div(y^{1-2\gamma}\nabla \tilde \delta_{a,\lambda})=0\;\; \text{in}\;\; \mathbb{R}^{n+1}_+, \\
\tilde \delta_{a,\lambda}=\delta_{a, \l} \;\; \text{on}\;\;\mathbb{R}^{n}.
\end{cases}
\end{split}\end{equation*}
Equivalently by \cite{cafsyl}
$$
\tilde \delta_{a,\lambda}(\cdot)=\hat \delta_{0,\lambda}(\cdot-a)
$$
and
\begin{equation*}\begin{split}
\hat \delta_{0,\lambda}(y,x)
=K*\delta_{0, 1}(y,x)=p_{n, \gamma} &
\int_{\mathbb{R}^n} \frac{y^{2\gamma}}{(y^{2}+\vert x-\xi\vert^{2})^{\frac{n+2\gamma}{2}}}
\left(\frac{\lambda}{1+\lambda^{2}\vert \xi\vert^{2}}\right)^{\frac{n-2\gamma}{2}}d\xi \\
= & p_{n, \gamma}
y^{-\frac{n-2\gamma}{2}}\int_{\mathbb{R}} \left(\frac{y^{-1}}{1+y^{-2}\vert x-\xi\vert^{2}}\right)^{\frac{n+2\gamma}{2}}
\left(\frac{\lambda}{1+\lambda^{2}\vert \xi\vert^{2}}\right)^{\frac{n-2\gamma}{2}} d\xi \\
= & p_{n, \gamma}
y^{-\frac{n-2\gamma}{2}}
\int_{\mathbb{R}^n} \delta_{x,y^{-1}}^{\frac{n+2\gamma}{n-2\gamma}} \;\delta_{0,\lambda}\; d\xi.
\end{split}\end{equation*}
with \;$K$\; being the Poisson kernel at $0$ given by
\begin{equation}\label{pngamma}
K(y,x):=p_{n, \gamma}\frac{y^{2\gamma}}{(|x|^2+|y|^2)^{\frac{n+2\gamma}{2}}} ,\;
p_{n,\gamma}=\frac{1}{c_{n,3}^{\gamma}}.
\end{equation}
                                                           
\vspace{10pt}

\noindent
Now, we are going to present some topological properties of the space of formal barycenter of \;$M$\; that we will need for our barycenter technique for existence. To do that   for \;$p\in \mathbb{N}^*$, we recall that the set of formal barycenters of \;$M$\; of order \;$p$\; is defined as follows
 \begin{equation*
B_{p}(M):=\{\sum_{i=1}^{p}\alpha_i\d_{a_i}\;:\;a_i\in \partial M, \;\alpha_i\geq 0,\;\; i=1,\cdots, p,\;\,\sum_{i=1}^{p}\alpha_i=1\}, 
\end{equation*}
where $\delta_{a}$ for $a\in M$ denotes the Dirac measure at $a$, and set
\begin{equation*}
B_0(M)=\emptyset. 
\end{equation*}
Furthermore  we have the existence of  \;$\mathbb{Z}_2$\; orientation classes 
\begin{equation}\label{orientation_classes}
w_p\in H_{np-1}(B_{p}(M), B_{p-1}(M))
\end{equation}
and that the cap product acts as follows 
\begin{equation}\label{eq:actioncap}
\begin{CD}
 H^l(M^p/\s_p))\times H_k(B_{p}(M), B_{p-1}(M))@>\frown>> H_{k-l}(B_{p}(M), B_{p-1}(M)).
 \end{CD}
\end{equation}
On the other hand, since \;$M$\; is a closed \;$n$-dimensional manifold, then we have 
\begin{equation*
\text{an orientation class}\,\;0\neq O^{*}_{M}\in H^{n}(M).
\end{equation*}
Furthermore  there is a natural way to see \;$O^{*}_{M}\in H^{n}(M)$\; as a nontrivial element of $H^n(M^p/\s_p)$, namely 
\begin{equation}\label{eq:defom1}
O^*_{M}\simeq O^*_p \;\; \text{with}\;\;0\neq O^{*}_p\in H^{n}((M^p)/\sigma_p).
\end{equation}
Recalling \eqref{eq:actioncap}, and Identifying \;$O^*_{M}$\; and \;$O^*_p$\; via  \eqref{eq:defom1}, we have the following well-know formula.
\begin{lem
There holds 
$$
\begin{CD}
 H^{n}((M^p)/\s_p)\times H_{(n+1)p-1}(B_{p}(M), B_{p-1}(M))@>\frown>> H_{(n+1)p-(n+1)}(B_{p}(M), B_{p-1}(M))\\@>\partial>>H_{(n+1)p-n-2}(B_{p-1}(M), B_{p-2}(M)),
 \end{CD}
$$
and 
\begin{equation*
\omega_{p-1}=\partial(O^*_{M}\frown w_p).
\end{equation*}
\end{lem}

\section{Non-homogeneous  theory}\label{nonhom}
In this section, we discuss some aspect of the work of Chang-Gonzalez\cite{cg} and Graham-Zworsky\cite{gz}  in the non-homogeneous setting and in the general framework of asymptotically  hyperbolic manifolds. We remark that this section is of independent interest, but in it we derive a regularity result used in section \ref{pceinsten}.  We divide this section in two subsections.
\subsection{Scattering operators and uniformly degenerate equations}\label{eq:nhscatdeg}
In this subsection, we extend some parts of the work of Chang-Gonzalez\cite{cg} to a non-homogeneous setting in the context of asymptotically hyperbolic manifolds. First of all, let \;$(X, g^{+})$\; be an asymptotically hyperbolic manifold with conformal infinity \;$(M, [h])$\; and \;$y$\; the unique geodesic defining function associated to \;$h$\; given by \eqref{eq:uniqdef}. Then we have the normal form 
\begin{equation*}\begin{split}
\;y^{2}g^{+}=g=dy^{2}+h_{y}, \;\;\text{near}\;\;M
\end{split}\end{equation*}
with
\begin{equation*}
\begin{split}
y>0\;\; \text{ in }  \;\;X,\;\;y=0\;\;\text{on}\;\; M\;\; \text{and}\;\;\vert dy \vert_{g}=1 \;\;\text{ near }\;\; M
\end{split}
\end{equation*} 
 Furthermore  let
\begin{equation*}
\square_{g^{+}}=-\Delta_{g^{+}}-s(n-s),
\end{equation*}
where by definition
\begin{equation*}
s=\frac{n}{2}+\gamma,\;\;\gamma\in (0,1), \;\,\gamma\neq \frac{1}{2}\;\; \text{and}\; \;\;s(n-s)\in (0,\frac{n^{2}}{4}).
\end{equation*} 
According to Mazzeo and Melrose \cite{mazzeo1}, \cite{mazzeo2}, \cite{mazmel}
\begin{equation*}
\sigma(-\Delta_{g^{+}})=\sigma_{pp}(-\Delta_{g^{+}})\cup [\frac{n^{2}}{4},\infty),
\; \sigma_{pp}(-\Delta_{g^{+}})\subset (0,\frac{n^{2}}{4}),
\end{equation*}
where \;$\sigma(-\Delta_{g^{+}})$\; and \;$\sigma_{pp}(-\Delta_{g^{+}})$ are respectively the spectrum and  the pure point spectrum of \;$L^{2}$-eigenvalues of \;$-\Delta_{g^{+}}$.  
Using the work of  Graham-Zworski\cite{gz} (see equation (3.9)), we may solve
\begin{equation*}\begin{split}
\begin{cases}
\square_{g^{+}}u=f\;\;\text{ in }\;\;X, \\
y^{s-n}u=\underline{v} \;\;\text{ on } \;\;M,
\end{cases}
\end{split}\end{equation*}
for \;$s(n-s)\not \in \sigma_{pp}(-\Delta_{g^{+}})$\; and \;$f\in y^{n-s+1}C^{\infty}(\overline X)+y^{s+1}C^{\infty}(\overline X)$\; in the form
\begin{equation*}\begin{split}
\begin{cases}
u=y^{n-s}A+y^{s}B \;\;\text{in}\;\;X,\\
A,\;\; B\in C^{\infty}(\overline X),\;\;\,A=\underline{v} \;\;\text{on} \;\;M,
\end{cases}.
\end{split}\end{equation*}
As in the case \;$f=0$, which corresponds to the generalized eigenvalue problem of Graham-Zworsky\cite{gz}, this gives rise to a Dirichlet-to-Neumann map \;$S_{f}(s)$\; via
\begin{equation*}\begin{split}
\underline{v}=A\lfloor _{M}\longrightarrow\-B\lfloor _{M}=\overline{v},
\end{split}\end{equation*}
which we refer to as non-homogeneous scattering operator and denote it by \;$S_{f}(s)$. Clearly $S_0(s)=S(s)$ and $S_{f}(s)$ is invertible, since the standard scattering operator \;$S_0(s)$\; 
is invertible (see equation (1.2) in \cite{jb}). We define the non-homogeneous fractional operators by
$$
P_{\gamma, f}^{h}=-d_\gamma S_{f}(s),
$$
where \;$d_\gamma$\; is as in \eqref{eq:fracopd}. Following \cite{gq} we find by conformal covariance of the conformal Laplacian that 
\begin{equation}\begin{split}\label{transformation_scattering_extension}
\square_{g^{+}}u=f
\xLeftrightarrow{U=y^{s-n}u} D_{g}U=y^{-s-1}f,
\end{split}\end{equation}
where
\begin{equation}\label{eqdg}
\begin{split}
D_{g}U=-div_{g}(y^{1-2\gamma}\nabla_{g}U)+E_{g}U
\end{split}
\end{equation}
and with \;$L_g=-\Delta_{g}+\frac{R_{g}}{c_{n}}$\; denoting the conformal Laplacian on \;$(X, g)$
\begin{equation}\begin{split}\label{Eg_general}
E_{g} =y^{\frac{1-2\gamma}{2}}L_{g}y^{\frac{1-2\gamma}{2}} -(\frac{R_{g^{+}}}{c_{n}}+s(n-s))y^{(1-2\gamma)-2}
,\;
c_{n}=\frac{4n}{n-1}.
\end{split}
\end{equation}
Thus we find for \;$\phi,\psi \in C^{\infty}(\overline X)$, that 
\begin{equation*}\begin{split
\begin{cases}
\square_{g^{+}}u=y^{n-s+1}\phi +y^{s+1}\psi\;\;\text{ in }\;\;X \\
y^{s-n}u=\underline{v} \;\; \text{ on } \;\;M
\end{cases}
\xLeftrightarrow{U=y^{s-n}u}
\begin{cases}
D_{g}U=y^{-2\gamma}\phi + \psi\text{ in }X \\
U=\underline{v} \text{ on } M
\end{cases}.
\end{split}\end{equation*}
Note, that such a solution \;$U$\; is of the form
\begin{equation*}
\begin{split}
U=A+By^{2\gamma}=\sum A_{i}y^{i}+\sum B_{i}y^{i+2\gamma}+U_0 
\end{split}
\end{equation*} 
for some \;$U_0\in C^{\infty}_0(X)$\; and has  principal terms
\begin{equation*}
\begin{cases} 
\underline{v}+\overline{v}  y^{2\gamma}\; \text{ for }\; \gamma<\frac{1}{2}\\
\underline{v}+A_{1}y+\overline{v} y^{2\gamma}\; \text{ for }\; \gamma>\frac{1}{2}
\end{cases}.
\end{equation*}
As for the case $\gamma>\frac{1}{2}$, expanding the boundary metric  
\;$h_y$,\;  we find 
\begin{equation*}
\begin{split}
h_{y}=h_{0}+h_{1}y+O(y^{2}) \;\;\text{ with }\;\; h_{1}=2\Pi_g
\end{split}
\end{equation*} 
and \;$\Pi_g$\; denoting the second fundamental form of \;$(M, h)$\; in $(\overline{X}, g)$. Still according to \cite{gz} we may solve
\begin{equation*}\begin{split}
\begin{cases}
\square_{g^{+}}u=\square_{g^{+}}u=y^{n-s+2}\phi +y^{s+1}\psi\;\;\text{ in }\;\;X \\
y^{s-n}u=\underline{v} \;\;\text{on} \;\;M
\end{cases}
\end{split}\end{equation*}
for \;$\phi,\psi \in C^{\infty}(\overline X)$ in the form
\begin{equation*}\begin{split}
\begin{cases}
u=y^{n-s}A+y^{s}B \;\;\text{ in }\;\;X\\
A, \;B\in C^{\infty}(\overline X),\;\;A=\underline{v} \;\;\text{ on } \;\;M
\end{cases}
\end{split}\end{equation*}
with asymptotic 
\begin{equation*}
\begin{split}
A=\sum A_{i}y^{i},\;\; A_{0}=\underline{v},\;\;A_{1}=0
\end{split}
\end{equation*} 
at a point, where \;$H_g=0$, i.e. the mean curvature vanishes. Thus for \;$\gamma>\frac{1}{2}$
\begin{equation*}\begin{split}
\begin{cases}
\square_{g^{+}}u=y^{n-s+2}\phi +y^{s+1}\psi\text{ in }X \\
y^{s-n}u=\underline{v}  \text{ on } M
\end{cases}
\xLeftrightarrow{U=y^{s-n}u}
\begin{cases}
D_{g}U=y^{1-2\gamma}\phi+\psi\text{ in }X \\
U= \underline{v} \text{ on } M
\end{cases}
\end{split}\end{equation*}
with principal terms
\begin{equation*}
U=\underline{v}+\overline{v} y^{2\gamma}+o(y^{2\gamma})
\end{equation*}
at a point with \;$H_{g}=0$ - just like in the case \;$\gamma<\frac{1}{2}$ - and there holds
\begin{equation*}
\overline{v}=\frac{1}{2\gamma}\lim_{y\to 0}y^{1-2\gamma}\partial_{y}U.
\end{equation*}
We summarize the latter discussion in the following proposition.
 \begin{pro}\label{prop_scattering}
 Let \;$(X, g^{+})$\; be a $(n+1)$-dimensional asymptotically hyperbolic manifold with minimal conformal infinity \;$(M, [h])$ and \;$y$ the unique geodesic defining function associated to \;$h$\; given by \eqref{eq:uniqdef}. Assuming that $$s=\frac{n}{2}+\gamma,\;\;\gamma\in (0,1), \;\,\gamma\neq \frac{1}{2},\; \;\;s(n-s)\not \in \sigma_{pp}(\Delta_{g^{+}})$$
and \;$f\in y^{n-s+2}C^{\infty}(\overline {X})+y^{s+1}C^{\infty}(\overline {X})$, then for every \;$\underline{v}\in C^{\infty}(M)$
$$
P_{\gamma, f}^h(\underline{v}):=-d_\gamma^*\lim_{y\to 0}y^{1-2\gamma}\partial_{y}U^{f}
$$
where \;$U^{f}$\; is the unique solution to
\begin{equation*}
\begin{cases}
D_{g}U=y^{-s-1}f\;\;\text{in}\;\;X
\\
U=\underline{v}\;\;\text{on} \;\;M
\end{cases}
\end{equation*} 
and \;$d_\gamma^*$ is as in \eqref{dsgamma}.  Moreover  $U^{f}$\; satisfies
 $$U^{f}=A+y^{2\gamma}B,\; \;\;A, \;B\in C^{\infty}(\overline{X})$$ and \;$A$\; and \,$B$\; satisfy the asymptotics
\begin{equation*}
\begin{cases}
A=\sum A_{i}y^{i},\;\;\,A_{i}\in C^{\infty}(M),\;\; A_{0}=\underline{v}\; \text{ and }\;\; A_{1}=0 \\
B=\sum B_{i}y^{i},\;\;B_{i}\in C^{\infty}(M)\; \;\;\text{ and }\;\; -d_\gamma B_{0}=-d_\gamma\overline{v}:=P_{\gamma, f}^h(\underline{v}),
\end{cases}
\end{equation*}
where \;$d_\gamma$\; is as in \eqref{eq:fracopd}, hence
$$U^{f}=\underline{v}+\overline{v} y^{2\gamma}+o(y^{2\gamma}).$$
\end{pro}
\subsection{Conformal property of the non-homogeneous scattering operator}\label{confnonhom}
In this subsection, we study the conformal property of the non-homogeneous scattering operator \;$P_{h, f}^{\gamma}$\; of the previous subsection.  To this end, we first 
consider as background data \;$(X, g^{+})$\; with conformal  infinity \;$(M, [h])$\; with \;$n\geq 2$\; and \;$y$\; the associated unique geodesic definition function  such that 
\begin{equation*}
g=y^{2}g^{+},\;\; g=dy^{2}+h_{y}\;\; \text{ close to \;$M$\; and }\;\;h=g\lfloor_{M}
\end{equation*}
as in \eqref{eq:uniqdef}. From \eqref{Eg_general} it is easy to see, that in \;$g$-normal Fermi coordinates \;$(y, x)$
\begin{equation}\label{Eg_local}
E_{g}=\frac{n-2\gamma}{2}\frac{\partial_{y}\sqrt{g}}{\sqrt{g}}y^{-2\gamma}\;\;
\text{ close to }\;\; M.
\end{equation}
We assume further that \;$(M, [h])$\; is minimal and \;$\square_{g^+}$\; is positive, i.e. 
\begin{equation*}
H_{g}=0 \;\; \text{ and }\;\; \lambda_{1}(-\Delta_{g^{+}})>s(n-s).
\end{equation*}
Then \;$\partial_{y}\sqrt{g}=0$\; on \;$M^{n+1}$\; and we may assume \;
\begin{equation}\label{sqrt_g_sim_yC_infty}
\partial_{y}\sqrt{g}\in 
yC^{\infty}(\overline {X}) 
\end{equation} whence 
$D_{g}$\; is well defined on 
$$
W^{1,2}_{y^{1-2\gamma}}=W^{1,2}_{y^{1-2\gamma}}(X,g)
=
\overline{C^{\infty}(X)}^{\Vert \cdot \Vert_{W^{1,2}_{y^{1-2\gamma}}(X,g)}},
\;
\Vert u \Vert_{W^{1,2}_{y^{1-2\gamma}}(X,g)}^{2}
=
\int_{X} y^{1-2\gamma}(\vert du \vert^{2}_{g}+u^{2})dV_{g}
$$
and becomes positive under Dirichlet condition, cf. \eqref{transformation_scattering_extension}, so
\begin{equation*}
\partial_{y}\sqrt{g}\in yC^{\infty}(\overline{X})\;\;\text{ and }\;\;\langle\cdot,\cdot \rangle_{D_{g}} \simeq \langle \cdot,\cdot \rangle_{W^{1,2}_{y^{1-2\gamma}}}.
\end{equation*} 
Let us consider now a conformal metric \;$\tilde h=\varphi^{\frac{4}{n-2\gamma}}h$\; on \;$M$. We then find a unique geodesic defining function \;$\tilde y>0$, precisely unique in a tubular neighborhood  of \;$M$, such that
\begin{equation*}
\tilde g=d\tilde y^{2}+\tilde h_{y}\;\;\text{ close to }\;\;M, \;\;\tilde{y}^{-2}\tilde g=g^{+}=y^{-2}g 
\;\;\text{ and }\;\;
\tilde h=\varphi^{\frac{4}{n-2}}h=(\frac{\tilde y}{y})^{2}h\;\;\text{ on }\;\;M.
\end{equation*} 
So we may naturally extend \;$\varphi=(\frac{\tilde y}{y})^{\frac{n-2\gamma}{2}}$\; onto \;$X$\; and
by the conformal  relation 
\begin{equation*}
\tilde g=(\frac{\tilde y}{y})^{2}g=\varphi^{\frac{4}{n-2\gamma}}g,
\end{equation*}
we still have \;$\langle \cdot,\cdot \rangle_{D_{\tilde g}}\simeq \langle \cdot,\cdot \rangle_{W^{1,2}_{\tilde y^{1-2\gamma}}}$. Putting \;$\tilde y=\alpha y$, the equation
\begin{equation*}
\vert d y\vert^{2}_{g}
=
1
=
\vert d \tilde y\vert^{2}_{\tilde g}
=
1
+
2\frac{y}{\alpha}\langle d\alpha,dy\rangle_{g}+ (\frac{y}{\alpha})^{2}\vert d\alpha \vert_{g}^{2}
\end{equation*} 
for the geodesic defining functions implies 
$\partial_{y}\alpha=-\frac{1}{2}\frac{y}{\alpha}\vert d\alpha\vert^{2}_{g}$. Since $\tilde g =\alpha^{2}g$ by definition, we firstly find
$H_{g}=0 \Longrightarrow H_{\tilde g}=0$, i.e. minimality is preserved  as already observed by Gonzalez-Qing\cite{gq}, and secondly  \;$\tilde y=\alpha_{0}y+O(y^{3})$.
Thus on the one hand side  the properties
\begin{equation*}
\partial_{\tilde y}\sqrt{\tilde g}\in \tilde y C^{\infty}
\text{ and }
\langle \cdot,\cdot \rangle_{D_{\tilde g}}\simeq \langle \cdot,\cdot \rangle_{W^{1,2}_{\tilde y^{1-2\gamma}}}\;\;˛
\end{equation*}
are preserved under a conformal change of the metric on the boundary. Moreover we obtain a conformal transformation for the extension operators \;$D_{\tilde g}$\; and $\;D_{g}$\; subjected to Dirichlet and weighted Neumann boundary conditions. Put \;$\tilde u=(\frac{y}{\tilde y})^{n-s}u$. 
As for the  Dirichlet case, \eqref{transformation_scattering_extension} directly shows
\begin{equation*}
\begin{cases}
D_{g}u=f \;\;\text{ in } \;\;X, \\
u=  v\;\;\text{ on } \;\;M
\end{cases}
\Longleftrightarrow
\begin{cases}
D_{\tilde g}\tilde u=(\frac{y}{\tilde y})^{s+1}f \;\;\text{ in} \;\;X, \\
\tilde u=(\frac{y}{\tilde y})^{n-s}v \;\;\text{on} \;\;M
\end{cases}
\end{equation*} 
Moreover there holds
$$
\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_{y}u=v
\Longleftrightarrow 
\lim_{\tilde y\rightarrow 0}\tilde y^{1-2\gamma}\partial_{\tilde y}\tilde u=(\frac{y}{\tilde y})^{n-s+2\gamma}v,
$$
since \;$\tilde y=\alpha_{0}y+O(y^{3})$, whence for the weighted Neumann case we obtain
\begin{equation*}
\begin{cases}
D_{g}u=f \;\;\text{ in } \;\;X \\
\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_{y}u=v \;\;\text{ on } \;\;M
\end{cases}
\Longleftrightarrow
\begin{cases}
D_{\tilde g}\tilde u=(\frac{y}{\tilde y})^{s+1}f \;\;\text{ in } \;\;X \\
\lim_{\tilde y\rightarrow 0}\tilde y^{1-2\gamma}\partial_{\tilde y}\tilde u=(\frac{y}{\tilde y})^{n-s+2\gamma}v \;\;\text{ on } \;\;M
\end{cases}.
\end{equation*}
We may rephrase this via \;$\varphi=(\frac{\tilde y}{y})^{\frac{n-2\gamma}{2}}=(\frac{\tilde y}{y})^{n-s}$\; as
\begin{equation*}
\begin{cases}
D_{g}(\varphi u)=\varphi^{\frac{s+1}{n-s}} f\;\; \text{ in } \;\;X \\
\varphi u=\varphi v \;\,\;\text{ on } \;\,\;M
\end{cases}
\Longleftrightarrow
\begin{cases}
D_{\tilde g} u=f \;\,\;\text{in} \;\;X \\
u=v\;\;\text{on} \;\;M
\end{cases}
\end{equation*} 
and
\begin{equation*}
\begin{cases}
D_{g}(\varphi u)=\varphi^{\frac{s+1}{n-s}}f \;\;\text{in}\;\; X \\
\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_{y}(\varphi u)=\varphi^{\frac{n+2\gamma}{n-2\gamma}}v \;\;\text{ on } \;\;M^{n}
\end{cases}
\Longleftrightarrow
\begin{cases}
D_{\tilde g}\tilde u=f \;\;\text{ in } \;\;X \\
\lim_{\tilde y\rightarrow 0}\tilde y^{1-2\gamma}\partial_{\tilde y} u=v \;\;\text{ on } \;\;M
\end{cases}.
\end{equation*}
Noticing \;$\frac{s+1}{n-s}=\frac{n+2+2\gamma}{n-2\gamma}$\; we thus have 
shown

\begin{equation*}
\xymatrix@R=6ex@C=-11ex
{
*+[l]
{
P^{\tilde h}_{\gamma, f}(\underline v)=\overline{v}
}
\ar@{<=>}[d] \ar@{<=>}[r]
&
*+[r]
{
{
\begin{cases}
D_{\tilde g}u=f \;\;\text{ in } \;\;X \\
u=\underline v \;\;\text{ on } \;\;M \\
-d_\gamma^*\lim_{\tilde y\rightarrow 0}\tilde y^{1-2\gamma}\partial_{\tilde y} u=\overline v \;\;\text{ on } \;\;M
\end{cases}
}
}
\ar@{<=>}[d]
\\
*+[l]
{
P^{h}_{\gamma, \varphi^{\frac{s+1}{n-s}}f }(\varphi \underline v)=\varphi^{\frac{n+2\gamma}{n-2\gamma}}\overline v
}
\ar@{<=>}[r]
&
*+[r]
{
{
\begin{cases}
D_{ g}(\varphi u)=\varphi^{\frac{n+2+2\gamma}{n-2\gamma}}f \text{ in } X \\
\varphi u=\varphi \underline v \text{ on } M \\
-d_\gamma^*\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_{ y} (\varphi 
u)=\varphi^{\frac{n+2\gamma}{n-2\gamma}}\overline v \text{ on } M
\end{cases}
}
}
}
\end{equation*}
Therefore the non-homogeneous fractional operator verifies the conformal  property
 $$P^{\tilde h}_{\gamma, f}(\underline v)=\varphi^{-\frac{n+2\gamma}{n-2\gamma}}P^{h}_{\gamma, \varphi^{\frac{s+1}{n-s}}f }(\varphi \underline v)$$
for \;$\tilde h=\varphi^{\frac{4}{n-2\gamma}}h$\;
or equivalently
$$P^{\tilde h}_{\gamma, \tilde f}(\underline v)=\varphi^{-\frac{n+2\gamma}{n-2\gamma}}P^{h}_{\gamma, f }(\varphi \underline v)$$
for \;$\tilde h=\varphi^{\frac{4}{n-2\gamma}}h$\; and \;$\tilde f=\varphi^{\frac{-s-1}{n-s}}f$, hence extending the conformal property of  the homogeneous fractional operator to the non-homogeneous setting.  We remark that $$P_\gamma^h=P_{\gamma, 0}^h.$$

\section{Fundamental solutions in the asymptotically hyperbolic case}\label{fundso}
In this section, keeping the notations of the previous section, for an asymptotically hyperbolic manifold\;$(X, g^+)$ with conformal infinity 
\;$(M, [h])$, we study the existence and asymptotic behavior of the Poisson kernel \;$K_g$ \; of $D_g$, the Green's function \;$\Gamma_g$\; of \;$D_g$\; under weighted normal boundary condition and \;$G^\gamma_h$\; of the fractional conformal  Laplacian \;$P^{\gamma}_{h}$\; satisfying  the boundary value problems
\begin{equation*}\begin{split}
\begin{cases}
D_{g}K_g(\cdot, \xi)=0\; \text{ in } X \;\;\text{and for all}\;\; \xi\in M\\
\lim_{y\rightarrow 0}K_g(y, x, \xi)=\d_{x}(\xi)\;\;\text{and for all}\;\; x,\xi\in M,
\end{cases}
\end{split}\end{equation*}
and
\begin{equation*}\begin{split}
\begin{cases}
D_{g}\Gamma_g(\cdot, \xi)=0\; \text{ in } X \;\;\text{and for all}\;\; \xi\in M\\
-d_\gamma^*\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_y\Gamma_g(y, x, \xi)=\d_{x}(\xi)\;\;\text{and for all}\;\; x,\xi\in M,
\end{cases}
\end{split}\end{equation*}
where \;$d_\gamma^*$ is given by \eqref{dsgamma}, and 
\begin{equation*}
P^\gamma_h G^\gamma_h(x, \xi)=\d_{x}(\xi),\;\;x\in M.
\end{equation*}
Thus by definition, we have
\begin{equation*}\begin{split}
K_{g}:(\overline X \times M)\setminus Diag(M)\longrightarrow\mathbb{R}_{+}
\end{split}\end{equation*}
is the Green's  function to the extension problem
\begin{equation*}\begin{split}
\begin{cases}
D_{g}U=0\; \text{ in } \;\;X \\
U=\underline{v}\;\; \text{on} \;\;M
\end{cases},
\end{split}\end{equation*}
while
\begin{equation*}\begin{split}
\Gamma_{g}:(\overline X\times M)\setminus Diag(M)\longrightarrow\mathbb{R}_{+}
\end{split}\end{equation*}
is the Green's function to the dual problem
\begin{equation*}\begin{split}
\begin{cases}
D_{g}U=0\;\;\text{ in }\;\;X\\
-d_\gamma^*\lim_{y\to 0}y^{1-2\gamma}\partial_{y}U=\overline{v}  \;\;\text{ on } \;\;M
\end{cases}
\end{split}\end{equation*}
and 
\begin{equation*}\begin{split}
G_\gamma^{h}:(M\times M)\setminus Diag(M)\longrightarrow\mathbb{R}_{+}.
\end{split}\end{equation*}
is the Green's function of the nonlocal problem
\begin{equation*}
P_\gamma^h \underline{v}=\overline{v}  \;\;\text{ on } \;\;M.
\end{equation*}
They are linked via
\begin{equation}\label{regreen}
\Gamma_{g}=K_{g}*G_\gamma^{h}.
\end{equation}


\subsection{Poisson kernel for \;$D_g$}
In this subsection, we study the Poisson kernel \;$K_g$ focusing on the existence issue and its asymptotics. We will follow the method of Lee-Parker\cite{lp}, however due to the rigidity property enjoyed by  the problem, we loose symmetry when looking for good coordinates and has to overcome this issue. To do that  we  work in the space of homogeneous functions rather than in the one of polynomials which is the case in \cite{lp}. We start with recalling  some related facts in the  case of the standard Euclidean space \;$\mathbb{R}^{n+1}_+$. Indeed,   according to \cite{cafsyl} on $\mathbb{R}^{n+1}_+$
\begin{equation*}
\begin{split}
K(y,x,\xi)=K_\gamma(y, x, \xi)= p_{n, \gamma}\frac{y^{2\gamma}}{(y^{2}+\vert x-\xi\vert^{2})^{\frac{n+2\gamma}{2}}},
\end{split}
\end{equation*} 
where \;$p_{n, \gamma}$ is as in \eqref{pngamma},  is the Poisson kernel of the operator
\begin{equation*
D=-div(y^{1-2\gamma}\nabla (\,\cdot\,))
\end{equation*}
 namely the Green's function of the extension problem 
\begin{equation*}
\begin{cases}
Du=0 \;\;\text{in}\;\; \mathbb{R}^{n+1}_+ \\
u=f \;\;\text{on}\;\;\mathbb{R}^{n}
\end{cases}
\end{equation*} 
 i.e.
\begin{equation}\label{poisson_defining_equation}
\begin{cases}
DK(y,x,\xi)=0 \;\;\text{ in}\;\;\mathbb{R}^{n+1}_+\;\; \\
K(y,x,\xi)\rightarrow\delta_{x}(\xi)\; \;\;\text{ for }\;\; y\rightarrow  0
\end{cases}.
\end{equation} 
We will construct the Poisson kernel for \;$D_{g}$ as in \eqref{eqdg}, namely the Green's function of the analogous extension problem
\begin{equation*}
\begin{cases}
D_{g}u=0 \;\;\text{in} \;\;X \\
u=f \;\;\text{ on } \;\;M
\end{cases}
\end{equation*} 
 i.e. \;$K_g$\; solves for \;$z\in X$\; and \;$\xi \in M$
\begin{equation*}
\begin{cases}
D_{g}K_{g}(z,\xi)=0\;\;\text{ in } \;\;X\\
K(z,\xi)\rightarrow\delta_{x}(\xi)\;\;\text{for}\; \;\;y\rightarrow 0
\end{cases},
\end{equation*}
where \;$z=(y,x)\in X$\; for \;$z$\; close to \;$M$. To that end, we identify
\begin{equation*}
\xi \in M\cap U\subset U\cap X \text{ with }0\in B^{\mathbb{R}^{n+1}}_{\epsilon}(0)\cap 
\mathbb{R}^{n}\subset  B_{\epsilon}^{\mathbb{R}^{n+1}}(0)\cap \mathbb{R}^{n+1}_{+}
\end{equation*}
for some small \;$\epsilon>0$, and write \;$K(z)=K(z,0)$. On $\;B_{\epsilon}^{\mathbb{R}^{n+1}}(0)\cap \mathbb{R}^{n+1}_{+}$\; we then have 
\begin{equation}\label{poisson_local_equation_for_the_poisson_kernel}
D_{g}K=-\frac{\partial_{p}}{\sqrt{g}}(\sqrt{g}g^{p,q}y^{1-2\gamma}\partial_{q}
K)+E_{g}K=f\in
y H_{-n-2\gamma-1}C^{\infty}
\end{equation}
due \eqref{sqrt_g_sim_yC_infty}, which relies on minimality \;$H_{g}=0$,
where by definition 
\begin{equation}\label{definition_H_l}
H_{l}=\{\varphi\in C^{\infty}(\overline{\mathbb{R}^{n+1}_{+}}\setminus \{0\})\mid 
\varphi \text{ is homogeneous of degree } l\}.
\end{equation}
The next lemma allows us to solve homogeneous deficits 
homogeneously.
\begin{lem}[Homogeneous solvability, Dirichlet case 
]\label{poisson_homogeneous_solvability_dirichlet}$_{}$\\
For \;$f_{l}\in yH_{l-1}, \;l\in \mathbb{N}-n-2\gamma$\; there exists \;$K_{1+2\gamma+l}\in 
y^{2\gamma}H_{l+1}$\; such, that 
$$DK_{1+2\gamma+l}=f_{l},$$
provided \;$\frac{1}{2}\neq \gamma \in (0, 1)$.
\end{lem}

\begin{pf
First of all the Stone-Weierstra{\ss} Theorem implies
\begin{equation*}
\langle Q^{k}_{l}(y,x)=y^{2\gamma+2k}P_{l}(x)\mid k,l\in \mathbb{N}\text{ and }P_{l}\in 
\Pi_{l}\rangle \underset{\text{dense}}{\subset} y^{2\gamma}C^{0}(\overline 
B_{1}(0)\cap\mathbb{R}^{n+1}_{+})
\end{equation*}
and an easy induction argument shows, that we have a unique representation
\begin{equation*}
Q^{k}_{l}=\sum \vert z \vert^{2i}A_{2k+l-2i}
\end{equation*}
with \;$D$-harmonics of the form \;$A_{m}(y,x)=\sum y^{2\gamma+2l}P_{m-2l}(x),\;
DA_{m}=0$. Since 
\begin{equation*}
y^{2\gamma}C^{0}(S^{n}_{+})\underset{\text{dense}}{\subset} 
L^{2}_{y^{1-2\gamma}}(S^{n}_{+}),
\end{equation*}
we thus obtain a \;$D$-harmonic basis \;$E=\{e^{i}_{k}\}$\; for 
\;$L^{2}_{y^{1-2\gamma}}(S^{n}_{+})$\; with
\begin{equation*}
De^{i}_{k}=0, \;\;k=deg(e^{i}_{k}), \;\; k\in \mathbb{N} +2\gamma\text{ and } \;\;i\in 
\{1,\dots,d_{k}\},
\end{equation*}
where \;$d_{k}$\; denotes the dimension of the space of \;$D$-harmonics of degree 
$k$. We may assume, that \;$e^{i}_{k},e^{j}_{k}$ for $i\neq j$\; are orthogonal 
with respect to the scalar product on \;$L^{2}_{y^{1-2\gamma}}(S^{n}_{+})$. 
Moreover on \;$S^{n}_{+}$\; we have
\begin{equation*}
\begin{split}
0
= &
-De^{i}_{k}
=
\partial_{y}(y^{1-2\gamma}\partial_{y}e^{i}_{k})+y^{1-2\gamma}\Delta_{x}e^{i}_{k
}
=
\nabla y^{1-2\gamma}\nabla e^{i}_{k}+y^{1-2\gamma}\Delta e^{i}_{k} \\
= &
\nabla y^{1-2\gamma} \nabla 
e^{i}_{k}+y^{1-2\gamma}\frac{\Delta_{S^{n}}}{r^{2}}e^{i}_{k}+y^{1-2\gamma}[
\partial_{r}^{2}+\frac{n\partial_{r}}{r}]e^{i}_{k} \\
= &
\nabla_{S^{n}_{+}}^{\perp} y^{1-2\gamma}\nabla_{S^{n}_{+}}^{\perp} e^{i}_{k}
+
div_{S^{n}_{+}}(y^{1-2\gamma}\nabla_{S^{n}_{+}}e^{i}_{k})
+
k(k+n-1)y^{1-2\gamma}e^{i}_{k},
\end{split}
\end{equation*}
whence due to 
\begin{equation*}
\begin{split}
\nabla_{S^{n}_{+}}^{\perp} y^{1-2\gamma}\nabla_{S^{n}_{+}}^{\perp} e^{i}_{k}
= &
\langle \nabla y^{1-2\gamma},\nu_{S^{n}_{+}}\rangle \langle 
\nu_{S^{n}_{+}},\nabla e^{i}_{k}\rangle
=
(1-2\gamma)y^{-2\gamma}\langle e_{n+1},\nu_{S^{n}_{+}}\rangle 
r\partial_{r}e^{i}_{k} \\
= &
(1-2\gamma)ky^{1-2\gamma}e^{i}_{k}
\end{split}
\end{equation*}
there holds for\; 
$D_{S^{n}_{+}}=-div_{S^{n}_{+}}(y^{1-2\gamma}\nabla_{S^{n}_{+}}\,\cdot\,)$
\begin{equation*}
D_{S^{n}_{+}}e^{i}_{k}=k(k+n-2\gamma)y^{1-2\gamma}e^{i}_{k}.
\end{equation*}
Therefore \;$E=\{e^{i}_{k}\}$\; is an orthogonal basis of 
\;$y^{2\gamma-1}D_{S^{n}_{+}}$-eigenfunctions with eigenvalues 
$$\lambda_{k}=k(k+n-2\gamma).$$ 
By the same argument solving
\begin{equation}\label{poisson_homogeneous_to_solve}
\begin{cases}
Du=f\in L^{2}_{y^{2\gamma-1}}(\mathbb{R}^{n+1}_{+})\;\; \text{ in } \;\,\;\mathbb{R}^{n+1}_{+}\\
u=0  \;\;\text{ on } \;\; \mathbb{R}^{n}
\end{cases}
\end{equation}
with homogeneous \;$f, \;u$\; of degree \;$\lambda, \;\lambda+1+2\gamma$\; is equivalent to 
solving
\begin{equation*}
\begin{cases}
D_{S^{n}_{+}}u=f+(\lambda+1+2\gamma)(\lambda+n+1)y^{1-2\gamma}u\;\;\text{ in 
}\;\;S^{n}_{+} \\
u=0  \;\;\text{ on }\;\; \partial S^{n}_{+}=S^{n-1}
\end{cases},
\end{equation*}
and thus, writing 
\begin{equation*}
u=\sum a_{i,k}e^{i}_{k},\,y^{2\gamma-1}f=\sum 
b_{j,l}e^{j}_{l}
\end{equation*}
also equivalent to solving
\begin{equation*}
\sum a_{i,k}(k(k+n-2\gamma)-(\lambda+1+2\gamma)(\lambda+n+1))e^{i}_{k}=\sum 
b_{j,l}e^{j}_{l}
\end{equation*}
and the latter system is always solvable in case
\begin{equation}\label{solvability_homogeneous_poisson}
k(k+n-2\gamma)-(\lambda+1+2\gamma)(\lambda+n+1)\neq 0 \;\;\text{ for all } \;\;
k, \;n, \;\lambda\in \mathbb{N}.
\end{equation}
This observation allows us to prove the lemma, by whose assumptions  
\begin{equation*}
deg(f_{l})=\lambda=m-n-2\gamma, \;\; m\in \mathbb{N}.
\end{equation*}
And we know 
\begin{equation*}
deg(e^{i}_{k})=k=m^{\prime}+2\gamma, \;\; m^{\prime}\in \mathbb{N}. 
\end{equation*}
Plugging these values into \eqref{solvability_homogeneous_poisson}, solvability 
of \eqref{poisson_homogeneous_to_solve} is a consequence of
\begin{equation*}
(m^{\prime}+2\gamma)(m^{\prime}+n)-(m-n+1)(m+1-2\gamma)\neq 0\;\;\text{ for all } \;\;
m^{\prime}, \; \;n,m\in \mathbb{N}
\end{equation*}
and this holds true for \;$\frac{1}{2}\neq\gamma \in (0,1)$. Thus we have 
proven solvability of
\begin{equation*}
\begin{cases}
DK_{1+2\gamma+l}=f_{l} \;\;\text{ in }\;\;\mathbb{R}^{n+1}_{+} \\
K_{1+2\gamma+l}=0  \;\;\text{ on } \;\;\mathbb{R}^{n}\setminus \{0\}
\end{cases}
\end{equation*}
with \;$K_{1+2\gamma+l}$\; being homogeneous of degree \;$1+2\gamma+l$. We are left 
with showing \;$K_{1+2\gamma+l}\in y^{2\gamma}H_{l+1}$. But this follows easily from 
proposition \ref{poisson_regularity} below.
\end{pf}
\vspace{6pt}

\noindent
Now, recalling \eqref{poisson_local_equation_for_the_poisson_kernel}, we may use 
lemma \ref{poisson_homogeneous_solvability_dirichlet} to solve 
\eqref{poisson_defining_equation} successively, since
\begin{equation*}
D_{g}K_{1+2\gamma+l}=f_{l}+(D_{g}-D)K_{1+2\gamma+l}\in f_{l}+ yH_{l}C^{\infty}
\end{equation*}
due to \eqref{sqrt_g_sim_yC_infty} and  \;$K_{1+2\gamma+l}\in y^{2\gamma}H_{l+1}$. Let
\begin{equation}\label{etaxi}
\eta_{\xi}:\overline X\longrightarrow \mathbb{R}^{+}, \;\;supp(\eta_{\xi})=B_{\epsilon}^{+}(\xi)=B_{\epsilon}^{g, +}(\xi)\;\;\text{ 
for } \;\;M\ni \xi \sim 0 \in \mathbb{R}^{n} \;\;\text{and}\;\;\epsilon>0\;\;\text{ small}
\end{equation}
denote a suitable cut-off function. For \;$m\in \mathbb{N}$\; we then find 
\begin{equation*}
K_{g}=\eta_{\xi}(K+\sum_{l=-n-2\gamma}^{m+2-2\gamma} K_{1+2\gamma+l})+\kappa_{m}
\end{equation*} 
for a weak solution
\begin{equation*}
\begin{cases}
D_{g}\kappa_{m}=-D_{g}\left(\eta_{\xi}(K+\sum_{l=-n-2\gamma}^{m+2-2\gamma} 
K_{1+2\gamma+l})\right)=h_{m} \;\;\text{ in }\;\,\;X \\
\kappa_{m}=0\;\;\text{ on } \;\;M 
\end{cases}
\end{equation*}
with \;$h_{m} \in yC^{m,\alpha}$. 
The following weak regularity statement will be sufficient for our purpose.
\begin{pro}\label{poisson_regularity}[Higher regularity for the 
Dirichlet problem]$_{}$\\
Let \;$h\in yC^{2k+3,\alpha}(X)$\; and \;$u\in W^{1,2}_{y^{1-2\gamma}}(X)$\;  be a 
weak solution of
\begin{equation*}
\begin{cases}
D_{g}u=h\;\;\text{ in }\;\;X \\
u=0\;\; \text{ on } \;\;M 
\end{cases}.
\end{equation*}
Then \;$u$\; is of class \;$y^{2\gamma}C^{2k,\beta}(X)$, provided \;$H_{g}=0$.
\end{pro}
\vspace{6pt}

\noindent
Putting these facts together before giving the proof of proposition \ref{poisson_regularity},  we have the existence of \;$K_g$\; and can describe its asymptotic as follows.
\begin{cor}[Asymptotics of the Poisson 
kernel \;$K_g$]\label{Poisson_kernel_asymptotics}$_{}$\\
Let \;$\frac{1}{2}\neq \gamma \in (0,1)$. We then have the existence of \;$K_g$\; and may expand in \;$g$-normal 
Fermi-coordinates
\begin{equation*}
K_{g}(z,\xi)
\;\in
\eta_{\xi}(z)
\left( p_{n, \gamma}
\frac{y^{2\gamma}}{\vert z 
\vert^{n+2\gamma}}+\sum^{2m+5-2\gamma}_{l=-n-2\gamma}y^{2\gamma}H_{1+l}(z)
\right)
+
y^{2\gamma}C^{2m,\alpha}(X)
\end{equation*}
with \;$H_{l}\in C^{\infty}(\mathbb{R}^{n+1}_+\setminus \{0\})$\; being homogeneous of order 
$l$\; and \;$p_{n, \gamma}$\; is as in \eqref{pngamma}, provided \;$H_{g}=0$.
\end{cor}


\begin{pfn}{ of proposition \ref{poisson_regularity}}\\
We use the Moser iteration argument. For this, we first let 
$
p,q=1,\ldots,n+1\text{ and } i,j=1,\ldots,n
$
such, that $g_{n+1,i}=g_{y,i}=0$. The statement clearly holds by standard local 
regularity away from the boundary, since $D_{g}$ is strongly elliptic there. 
Now fixing a point $\xi\in M$ and for $0<r_{1}<r_{2}\ll 1$ a cut-off function
\begin{equation*}
\eta\in C^{\infty}_{0}(B^{+}_{r_{2}}(0),\mathbb{R}_{+}),\,\; \eta\equiv 1\text{ on } 
B^{+}_{r_{1}}(0)\;\;\text{ where } \;\;M\ni \xi \sim 0\in \mathbb{R}^{n},
\end{equation*}
we pass to $g$-normal Fermi-coordinates around $\xi$ and estimate for some \;$\lambda\geq 2$\; and \;$\alpha\in \mathbb{N}^{n}$ 
\begin{equation}\label{poisson_kernel_moser_iteration_1}
\begin{split}
\int y^{1-2\gamma}\vert \nabla_{z}(\vert 
\partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\eta)\vert^{2} 
\leq &
2\int y^{1-2\gamma}\vert \nabla_{z}\vert 
\partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert^{2}\eta^{2}
+
2\int y^{1-2\gamma}\vert \partial^{\alpha}_{x}u\vert^{\lambda}\vert 
\nabla_{z}\eta\vert^{2}
\end{split}
\end{equation}
and
\begin{equation*}
\begin{split}
\int y^{1-2\gamma}&\vert \nabla_{z}\vert 
\partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert^{2}\eta^{2}
=
\frac{\lambda^{2}}{4}\int y^{1-2\gamma} \nabla_{z} \partial^{\alpha}_{x}u
\nabla_{z} \partial^{\alpha}_{x}u \vert  
\partial^{\alpha}_{x}u\vert^{\lambda-2}\eta^{2}\\
= &
\frac{\lambda^{2}}{4(\lambda-1)}\int y^{1-2\gamma} \nabla_{z} 
\partial^{\alpha}_{x}u
\nabla_{z} (\partial^{\alpha}_{x}u \vert  
\partial^{\alpha}_{x}u\vert^{\lambda-2}\eta^{2}) 
-
\frac{\lambda^{2}}{2(\lambda-1)}\int y^{1-2\gamma} \nabla_{z} 
\partial^{\alpha}_{x}u
\partial^{\alpha}_{x}u \vert  
\partial^{\alpha}_{x}u\vert^{\lambda-2}\nabla_{z}\eta \eta \\
\leq &
\frac{\lambda^{2}}{4(\lambda-1)}\int y^{1-2\gamma} \nabla_{z} 
\partial^{\alpha}_{x}u
\nabla_{z} (\partial^{\alpha}_{x}u \vert  
\partial^{\alpha}_{x}u\vert^{\lambda-2}\eta^{2}) \\
& +
\frac{\lambda^{2}}{8}\int y^{1-2\gamma} \vert \nabla_{z} 
\partial^{\alpha}_{x}u\vert^{2}
\vert  \partial^{\alpha}_{x}u\vert^{\lambda-2}\eta^{2} 
+
\frac{\lambda^{2}}{2(\lambda-1)^{2}}\int y^{1-2\gamma} 
 \vert  \partial^{\alpha}_{x}u\vert^{\lambda}\vert \nabla_{z}\eta \vert^{2}.
\end{split}
\end{equation*}
Absorbing the second summand above this implies
\begin{equation}\label{poisson_kernel_moser_iteration_3}
\begin{split}
\int y^{1-2\gamma}\vert \nabla_{z}(\vert 
\partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}})\vert^{2}\eta^{2}
\leq &
\frac{\lambda^{2}}{2(\lambda-1)}\int D(\partial^{\alpha}_{x}u)
\partial^{\alpha}_{x}u \vert  \partial^{\alpha}_{x}u\vert^{\lambda-2}\eta^{2} +
\frac{\lambda^{2}}{(\lambda-1)^{2}}\int y^{1-2\gamma}\vert 
\partial^{\alpha}_{x}u\vert^{\lambda}\vert \nabla_{z}\eta\vert^{2}.
\end{split}
\end{equation}
Due to
$
D(\partial^{\alpha}_{x}u)=\partial^{\alpha}_{x}Du,
$
and the structure of the metric 
\begin{equation}\label{poisson_kernel_moser_iteration_4}
\begin{split}
\int \partial_{x}^{\alpha}(D  u)\partial^{\alpha}_{x}u\vert 
\partial^{\alpha}_{x}u\vert^{\lambda-2}\eta^{2} 
= &
\int \partial_{x}^{\alpha}(D_{g}u)\partial^{\alpha}_{x}u\vert 
\partial^{\alpha}_{x}u\vert^{\lambda-2}\eta^{2} 
-
\int \partial_{x}^{\alpha}((D_{g}-D)u)\partial^{\alpha}_{x}u\vert 
\partial^{\alpha}_{x}u\vert^{\lambda-2} \eta^{2} \\
= &
\int 
\partial_{x}^{\alpha}
[
h
+
\frac{\partial_{p}\sqrt{g}}{\sqrt{g}}y^{1-2\gamma}g^{p,q}\partial_{q}u
+
y^{1-2\gamma}\partial_{i}((g^{i,j}-\delta^{i,j})\partial_{j}u) \\
& \quad \quad\quad\quad\quad\quad\quad\quad\quad -
\frac{n-2\gamma}{2}\frac{\partial_{y}\sqrt{g}}{\sqrt{g}}y^{-2\gamma}u
]
\partial^{\alpha}_{x}u\vert \partial^{\alpha}_{x}u\vert^{\lambda-2}\eta^{2}\\
= &
I_{1}+\ldots+I_{4}.
\end{split}
\end{equation}
We may assume $\nabla^{k}_{z}\eta\leq \frac{C}{\epsilon^{k}}$ for $k=0,1,2$, 
where $\epsilon=r_{2}-r_{1}$. Then
\begin{enumerate}[label=(\roman*)]
\item \quad 
$
\vert I_{1} \vert
\leq  
C\int \vert \nabla ^{\vert \alpha \vert}_{x}h\vert  
\vert \nabla^{\vert \alpha \vert}_{x}u\vert^{\lambda-1} \eta^{2}
$


\item using integrations by parts and \eqref{sqrt_g_sim_yC_infty}
\begin{equation*}
\begin{split}
\vert I_{2}\vert
\leq &
\vert \int 
y^{1-2\gamma}\partial_{q}\partial^{\alpha}_{x}(\frac{\partial_{p}\sqrt{g}}{\sqrt
{g}}g^{p,q}u)\partial^{\alpha}_{x}u\vert \partial^{\alpha}_{x}u\vert 
^{\lambda-2} \eta^{2}\vert 
+
\vert \int 
y^{1-2\gamma}\partial^{\alpha}_{x}(\partial_{q}(\frac{\partial_{p}\sqrt{g}}{
\sqrt{g}}g^{p,q})u)\partial^{\alpha}_{x}u\vert \partial^{\alpha}_{x}u\vert 
^{\lambda-2} \eta^{2}\vert \\
\leq &
\vert \int 
y^{1-2\gamma}\partial_{y}\partial^{\alpha}_{x}(\frac{\partial_{y}\sqrt{g}}{\sqrt
{g}}u)\partial^{\alpha}_{x}u\vert \partial^{\alpha}_{x}u\vert ^{\lambda-2} 
\eta^{2} \vert   +
\frac{C_{\vert \alpha \vert}}{\lambda}\sum_{m\leq \vert \alpha \vert} \int 
y^{1-2\gamma}\vert \nabla_{x}^{m}u\vert \vert 
\partial^{\alpha}_{x}u\vert^{\frac{\lambda-2}{2}}
\vert \nabla_{x} \vert \partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert 
\eta^{2} \\
& +
C_{\vert \alpha \vert}\sum_{m\leq \vert \alpha \vert} \int y^{1-2\gamma}\vert 
\nabla_{x}^{m}u\vert \vert \partial^{\alpha}_{x}u\vert^{\lambda-1} [\vert 
\nabla_{x}\eta\vert \eta+\eta^{2}] \\
\leq &
\frac{C_{\vert \alpha \vert}}{\lambda}\sum_{m\leq \vert \alpha \vert} 
\int y^{1-2\gamma}\vert \nabla_{x}^{m}u\vert^{\frac{\lambda}{2}} 
\vert \nabla_{z} \vert \partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert 
\eta^{2} 
+
C_{\vert \alpha \vert}\sum_{m\leq \vert \alpha \vert} \int y^{1-2\gamma}\vert 
\nabla_{x}^{m}u \vert^{\lambda} [\vert \nabla_{z}\eta\vert \eta+\eta^{2}] \\
\end{split}
\end{equation*}

\item using integration by parts and recalling $i,j=1,\ldots,n$
\begin{equation*}
\begin{split}
\vert I_{3} \vert
\leq &
\frac{C}{\lambda} \int y^{1-2\gamma}\vert 
\partial^{\alpha}_{x}((g^{i,j}-\delta^{i,j})\partial_{j}u)\vert \vert 
\partial^{\alpha}_{x}u\vert^{\frac{\lambda-2}{2}}\vert \partial_{i}\vert 
\partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert \eta^{2} \\
& +
C\int y^{1-2\gamma}\vert 
\partial^{\alpha}_{x}((g^{i,j}-\delta^{i,j})\partial_{j}u)\vert \vert 
\partial^{\alpha}_{x}u\vert^{\lambda-1}\vert \partial_{i}\eta \vert \eta \\
\leq &
\frac{C}{\lambda^{2}}\sup_{B^{+}_{r_{2}}}\vert g^{i,j}-\delta^{i,j}\vert \int 
y^{1-2\gamma}
\vert \partial_{i}\vert \partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert
\vert \partial_{j}\vert \partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert
\eta^{2} \\
& +
\frac{C_{\vert \alpha \vert}}{\lambda}\sum_{m\leq \vert \alpha }
 \int y^{1-2\gamma}\vert \nabla_{x}^{m}u\vert \vert 
\partial^{\alpha}_{x}u\vert^{\frac{\lambda-2}{2}}\vert \nabla_{x}\vert 
\partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert \eta^{2}\\
& +
\frac{C}{\lambda}\sup_{B^{+}_{r_{2}}}\vert g^{i,j}-\delta^{i,j}\vert \int 
y^{1-2\gamma} \vert \partial_{j}\vert 
\partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert \vert 
\partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert\partial_{i} \eta \vert 
\eta 
\\
& +
C_{\vert \alpha \vert}\sum_{m\leq \vert \alpha }
 \int y^{1-2\gamma}\vert \nabla_{x}^{m}u\vert^{\lambda} \vert 
\nabla_{x}\eta\vert \eta 
\end{split}
\end{equation*}

\item \quad 
$
\vert I_{4} \vert 
\leq 
C_{\vert \alpha \vert}\sum_{m\leq \vert\alpha\vert}\int y^{1-2\gamma} \vert 
\nabla^{m}_{x}u\vert^{\lambda}
 \eta^{2}
$
using \eqref{sqrt_g_sim_yC_infty}.
\end{enumerate}
Applying H\"older's and Young's inequality to (i)-(vi)  we obtain
\begin{equation*}
\begin{split}
\sum^{4}_{i=1}\vert I_{i}\vert
\leq &
\frac{C\sup_{B^{+}_{r_{2}}}\vert g-\delta\vert}{\lambda^{2}}
\int y^{1-2\gamma}\vert \nabla_{z}\vert \partial^{\alpha}_{x}u 
\vert^{\frac{\lambda}{2}}\vert^{2}\eta^{2} 
+
\frac{C_{\vert \alpha\vert}}{\epsilon^{2}}\sum_{k\leq \vert \alpha \vert}
\Vert 
\nabla^{k}_{x}u\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}^{\lambda} \\
& +
C\Vert y^{2\gamma-1}\nabla^{\vert \alpha 
\vert}_{x}h\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}
\Vert \nabla^{\vert \alpha 
\vert}_{x}u\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}^{\lambda-1}
.
\end{split}
\end{equation*}
We may assume \;$C\sup_{B^{+}_{r_{2}}}\vert g-\delta\vert<\frac{1}{2}$, whence in 
view of \eqref{poisson_kernel_moser_iteration_3} and 
\eqref{poisson_kernel_moser_iteration_4} 
\begin{equation*}
\begin{split}
\int y^{1-2\gamma}\vert \nabla_{z}\vert 
\partial^{\alpha}_{x}u\vert^{\frac{\lambda}{2}}\vert^{2}\eta^{2}
\leq &
\frac{C_{\vert \alpha \vert}\lambda}{\epsilon^{2}}
\sum_{k\leq \vert \alpha \vert}
\Vert 
\nabla^{k}_{x}u\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}^{\lambda}
\\
& +
C\lambda\Vert y^{2\gamma-1}\nabla^{\vert \alpha 
\vert}_{x}h\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}
\Vert \nabla^{\vert \alpha 
\vert}_{x}u\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}^{\lambda-1},
\end{split}
\end{equation*} 
so \eqref{poisson_kernel_moser_iteration_1} implies
\begin{equation}\label{poisson_kernel_moser_iteration_2}
\begin{split}
\int y^{1-2\gamma}\vert \nabla_{z}(\vert \partial^{\alpha}_{x} 
u\vert^{\frac{\lambda}{2}}\eta)\vert^{2}
\leq &
\frac{C_{\vert \alpha \vert}\lambda}{\epsilon^{2}}
\sum_{k\leq \vert \alpha \vert}
\Vert 
\nabla^{k}_{x}u\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}^{\lambda}
\\
& +
C\lambda\Vert y^{2\gamma-1}\nabla^{\vert \alpha 
\vert}_{x}h\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}
\Vert \nabla^{\vert \alpha 
\vert}_{x}u\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}^{\lambda-1}.
\end{split}
\end{equation} 
The weighted Sobolev inequality  of Fabes-Kenig-Seraponi \cite{fabkenser}  Theorem 1.2 with \;$\kappa=\frac{n+1}{n}$\; 
then shows
\begin{equation*}
\begin{split}
r_{2}^{-\frac{n+2\gamma}{n+1}}\Vert \partial^{ \alpha }_{x}u \Vert_{L^{\kappa 
\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{1}})}^{\lambda}
\leq &
\frac{C_{\vert \alpha \vert}\lambda}{\epsilon^{2}}
\sum_{k\leq \vert \alpha \vert}
\Vert 
\nabla^{k}_{x}u\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}^{\lambda}
\\
& +
C\lambda\Vert y^{2\gamma-1}\nabla^{\vert \alpha 
\vert}_{x}h\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}
\Vert \nabla^{\vert \alpha 
\vert}_{x}u\Vert_{L^{\lambda}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}^{\lambda-1}
.
\end{split}
\end{equation*}
By rescaling we may assume for some \;$0<\epsilon_{0}\ll 1$, that
\begin{equation}\label{poisson_rescaling_deficit}
\Vert u \Vert_{L^{2}_{y^{1-2\gamma}}}+\sum^{\vert \alpha \vert}_{k=0}\Vert  
y^{2\gamma-1}\nabla^{k}_{x}h\Vert_{L^{\infty}_{y^{1-2\gamma}}(B^{+}_{(2+\vert 
\alpha \vert)\epsilon_{0}})}= 1,
\end{equation}
and putting \;$\lambda_{i}=2(\frac{n+1}{n})^{i}$\; and 
\;$\rho_{i}=\epsilon_{0}(1+\frac{1}{2^{i}})$\; we obtain
\begin{equation*}
\begin{split}
\Vert \nabla^{ \vert \alpha \vert }_{x}u 
\Vert_{L^{\lambda_{i+1}}_{y^{1-2\gamma}}(B^{+}_{\rho_{i+1}})}
\leq 
\sqrt[\lambda_{i}]{C_{\vert\alpha\vert,\epsilon_{0}}\lambda_{i}2^{2i}}
\cdot
\sup_{m\leq \vert \alpha \vert}
[
 & \Vert \nabla^{m}_{x}u\Vert_{L^{\lambda_{i}}_{y^{1-2\gamma}}(B^{+}_{r_{2}})} 
+
\Vert 
\nabla^{m}_{x}u\Vert_{L^{\lambda_{i}}_{y^{1-2\gamma}}(B^{+}_{r_{2}})}^{\frac{1}{
2}}
],
\end{split}
\end{equation*}
where we have used \;$\frac{1}{2}\leq \frac{\lambda_{i}-1}{\lambda_{i}}<1$. 
Iterating this inequality then shows
\begin{equation*}
\begin{split}
\Vert \nabla^{ \vert \alpha  \vert}_{x}u 
\Vert_{L^{\infty}_{y^{1-2\gamma}}(B^{+}_{\epsilon_{0}})}
\leq 
C_{\alpha,\epsilon_{0}}(1+\sup_{m\leq \vert \alpha \vert}
\Vert \nabla^{m}_{x}u \Vert_{L^{2}_{y^{1-2\gamma}}(B^{+}_{2\epsilon_{0}})})
\leq 
C_{\alpha,\epsilon_{0}},
\end{split}
\end{equation*}
where the last inequality follows from iterating 
\eqref{poisson_kernel_moser_iteration_2} with \;$\lambda=2$
and \eqref{poisson_rescaling_deficit}. Rescaling back we conclude
\begin{equation}\label{poisson_kernel_moser_iteration_5}
\begin{split}
\sum^{m}_{k=0}\Vert \nabla^{ k }_{x}u 
\Vert_{L^{\infty}_{y^{1-2\gamma}}(B^{+}_{\epsilon_{0}})}
\leq 
C_{m,\epsilon_{0}}
[
\Vert u \Vert_{L^{2}_{y^{1-2\gamma}}}
+
\sum^{m}_{k=0}\Vert y^{2\gamma-1}\nabla^{ k }_{x}h 
\Vert_{L^{\infty}_{y^{1-2\gamma}}}
].
\end{split}
\end{equation}
Note, that
$
D(\partial^{\alpha}_{x}u)
= 
\partial_{x}^{\alpha}h
+
\partial^{\alpha}_{x}((D-D_{g})u),
$
where
\begin{equation}\label{poisson_x_deriviative_versus_difference_of_operators}
\begin{split}
\partial^{\alpha}_{x}((D& -D_{g})u)
= 
\partial^{\alpha}_{x}
[
\frac{\partial_{p}\sqrt{g}}{\sqrt{g}}y^{1-2\gamma}g^{p,q}\partial_{q}u 
+
\partial_{i}(y^{1-2\gamma}(g^{i,j}-\delta^{i,j})\partial_{j}u)
-
\frac{n-2\gamma}{2}\frac{\partial_{y}\sqrt{g}}{\sqrt{g}}y^{-2\gamma}u
]
\\
= &
\partial_{q}\partial_{x}^{\alpha}(\frac{\partial_{p}\sqrt{g}}{\sqrt{g}}y^{
1-2\gamma}g^{p,q}u)
-
\partial^{\alpha}_{x}(\partial_{q}(\frac{\partial_{p}\sqrt{g}}{\sqrt{g}}y^{
1-2\gamma}g^{p,q})u) \\
& +
\partial_{i}\partial^{\alpha}_{x}(y^{1-2\gamma}(g^{i,j}-\delta^{i,j})\partial_{j
}u)
-
\frac{n-2\gamma}{2}\partial^{\alpha}_{x}(\frac{\partial_{y}\sqrt{g}}{\sqrt{g}}y^
{-2\gamma}u).
\end{split}
\end{equation}
In particular, since 

-\partial_{p}(y^{1-2\gamma}g^{p,q}\partial_{q}v) 
=
Dv
-
\partial_{i}(y^{1-2\gamma}(g^{i,j}-\delta^{i,j})v)
$
we may write
\begin{equation*}
\partial_{p}(y^{1-2\gamma}g^{p,q}\partial_{q}\partial^{\alpha}_{x}u)
=
\partial^{\alpha}_{x}h
+
h^{\alpha}
+
\sum \partial_{p}h^{\alpha}_{p},
\end{equation*}
where \;$h^{\alpha}, \;h^{\alpha}_{p}$\; depend only on \;$x-$derivatives of \;$u$\; of 
order up to \;$\vert \alpha \vert$, and
due to \eqref{sqrt_g_sim_yC_infty},\;\eqref{poisson_kernel_moser_iteration_5},  
there holds 
\begin{equation*}
\sum^{m}_{\vert \alpha \vert=0}\Vert 
\frac{h^{\alpha}}{y^{1-2\gamma}},\frac{h^{\alpha}_{p}}{y^{1-2\gamma}}\Vert_{L^{
\infty}_{y^{1-2\gamma}}(B^{+}_{\epsilon_{0}})}
\leq
C_{m,\epsilon_{0}}
[
\Vert u \Vert_{L^{2}_{y^{1-2\gamma}}}
+
\sum^{m}_{k=0}\Vert y^{2\gamma-1}\nabla^{ k }_{x}h 
\Vert_{L^{\infty}_{y^{1-2\gamma}}}
]
\;\;\text{for all}\;\; m\in \mathbb{N}.
\end{equation*} 
Then Zamboni\cite{Zamboni} Theorem 5.2 shows H\"older regularity, i.e. for all \;$m\in \mathbb{N}$
\begin{equation}\label{poisson_zamboni}
\sum^{m}_{k=0}\Vert 
\nabla^{k}_{x}u\Vert_{C^{0,\alpha}(B^{+}_{\frac{\epsilon_{0}}{2}})}
\leq 
C_{m,\epsilon_{0}}
[
\Vert u \Vert_{L^{2}_{y^{1-2\gamma}}}
+
\sum^{m}_{k=0}\Vert y^{2\gamma-1}\nabla^{ k }_{x}h 
\Vert_{L^{\infty}_{y^{1-2\gamma}}}
].
\end{equation} 
This allows us to integrate the equation directly. Indeed from 
\eqref{poisson_x_deriviative_versus_difference_of_operators} we have
\begin{equation*}
D(\partial^{\alpha}_{x}u)
=
\partial^{\alpha}_{x}h
+
\partial^{\alpha}_{x}((D-D_{g})u)
=
\partial^{\alpha}_{x}h
+
\partial_{y}(y^{2-2\gamma}f^{\alpha}_{1})+y^{1-2\gamma}f^{\alpha}_{2},
\end{equation*}
where by definition 
\begin{equation*
f^{\alpha}_{1}=\partial^{\alpha}_{x}(\frac{\partial_{y}\sqrt{g}}{y\sqrt{g}}u)
\end{equation*}
and 
\begin{equation*
\begin{split}
f_{2}^{\alpha}
= &
\partial_{i}\partial_{x}^{\alpha}(\frac{\partial_{j}\sqrt{g}}{\sqrt{g}}g^{i,j}u)
-
\partial^{\alpha}_{x}(y^{2\gamma-1}\partial_{q}(\frac{\partial_{p}\sqrt{g}}{
\sqrt{g}}y^{1-2\gamma}g^{p,q})u) \\
& +
\partial_{i}\partial^{\alpha}_{x}((g^{i,j}-\delta^{i,j})\partial_{j}u)
-
\frac{n-2\gamma}{2}\partial^{\alpha}_{x}(\frac{\partial_{y}\sqrt{g}}{y\sqrt{g}}
u)
.
\end{split}
\end{equation*}
This implies 
\begin{equation*}
-\partial_{y}(y^{1-2\gamma}\partial_{y}\partial^{\alpha}_{x}u)
=
\partial_{y}(y^{2-2\gamma}f^{\alpha}_{1})
+
y^{1-2\gamma}(f^{\alpha}_{2}+\Delta_{x}\partial^{\alpha}_{x}u+y^{2\gamma-1}
\partial^{x}_{\alpha}h)
\end{equation*}
and we obtain
\begin{equation}\label{poisson_equation_integrated_1}
\begin{split}
\partial^{\alpha}_{x}u(y,x)
= 
y^{2\gamma}\bar u_{0}^{\alpha}(x) 
- 
\int^{y}_{0}\sigma \tilde f^{\alpha}_{1}(\sigma,x)dV 
- 
\int^{y}_{0}\sigma^{2\gamma-1}\int^{\sigma}_{0}\tau^{1-2\gamma} \tilde 
f^{\alpha}_{2}(\tau,x)d\tau dV,
\end{split}
\end{equation}
where by definition we may write with smooth coefficients \;$f_{i,\beta}$
\begin{equation}\label{poisson_tilde_f_structure}
\tilde f^{\alpha}_{1}=\sum _{\vert \beta\vert \leq \vert \alpha 
\vert}f_{1,\beta}\partial^{\beta}_{x}u
\;\text{ and }\;
\tilde f^{\alpha}_{2}=\frac{\partial^{\alpha}_{x}h}{y^{1-2\gamma}}+\sum _{\vert 
\beta\vert \leq \vert \alpha \vert+2}f_{2,\beta} \partial_{x}^{\beta}u.
\end{equation}
Let \;$h\in yC^{l,\lambda^{\prime}}$. Then \eqref{poisson_zamboni} shows
\begin{equation*}
\forall \vert \alpha \vert\leq l : \;\;\nabla^{\vert \alpha\vert}_{x}u \in 
C^{0,\lambda},
\end{equation*}
whence \;$\forall\; \vert \alpha\vert \leq l-2 : \;\tilde  
f_{i}^{\alpha}\in C^{0,\lambda}$\; due to \eqref{poisson_tilde_f_structure}. In 
particular  \eqref{poisson_equation_integrated_1} implies
\begin{equation*}
\partial^{\alpha}_{x}u(y,x)
=
y^{2\gamma}\bar u_{0}^{\alpha}(x) +o(y^{2\gamma}),
\end{equation*}
so \;$\bar u_{0}^{\alpha}\in C^{l+2,\lambda}$\; anyway by interior regularity. We define
\begin{equation}\label{poisson_y_minus_2_gamma_redefinition}
\bar u^{\alpha}=y^{-2\gamma}\partial^{\alpha}_{x}u
,\;
\bar f_{1}^{\alpha}=y^{-2\gamma}f_{1}^{\alpha}
,\;
\bar f_{2}^{\alpha}=y^{-2\gamma}\tilde f^{\alpha}_{2}.
\end{equation} 
We then find from  \eqref{poisson_equation_integrated_1}, that 
\begin{equation}\label{poisson_equation_integrated}
\begin{split}
\bar u^{\alpha}(y,x)
= &
\bar u_{0}^{\alpha}(x) 
-
y^{-2\gamma}\int^{y}_{0}\sigma^{1+2\gamma} \bar f^{\alpha}_{1}(\sigma,x)dV 
-
y^{-2\gamma}\int^{y}_{0}\sigma^{2\gamma-1}\int^{\sigma}_{0}\tau\bar 
f^{\alpha}_{2}(\tau,x)d\tau dV\\
= &
\bar u_{0}^{\alpha}(x) + \bar u_{1}^{\alpha}(y,x) +\bar u_{2}^{\alpha}(y,x),
\end{split}
\end{equation}
where according to 
\eqref{poisson_tilde_f_structure},\eqref{poisson_y_minus_2_gamma_redefinition} 
we may write with smooth coefficients \;$f_{i,\beta}$
\begin{equation}\label{poisson_bar_f_structure}
\bar f^{\alpha}_{1}=\sum _{\vert \beta\vert \leq \vert \alpha 
\vert}f_{1,\beta}\bar u^{\beta}
\;\;\text{ and }\;\;
\bar f^{\alpha}_{2}=\frac{\partial^{\alpha}_{x}h}{y}+\sum _{\vert \beta\vert 
\leq \vert \alpha \vert+2}f_{2,\beta}\bar u^{\beta}.
\end{equation}
\eqref{poisson_equation_integrated_1} and \;$\forall \;\vert 
\alpha\vert \leq l-2 : \;\tilde  f_{i}^{\alpha}\in C^{0,\lambda}$\; then already show
\begin{equation*}
\forall \vert \alpha\vert \leq l-2 :\;\;\bar u^{\alpha}\in C^{0,\lambda},
\end{equation*}
and we may assume inductively
\begin{equation*}
\forall \vert \alpha\vert\leq  l-2-2m: \;\;\partial^{2m}_{y}\bar u^{\alpha}\in 
C^{0,\lambda}.
\end{equation*}
whence according to \eqref{poisson_bar_f_structure} 
\begin{equation*}
\forall \vert \alpha \vert\leq l-2-2(m+1) : \;\;\partial^{2m}_{y}\bar f^{\alpha}_{i}\in 
C^{0,\lambda}. 
\end{equation*}
Then \eqref{poisson_equation_integrated} implies via Taylor expansion
\begin{equation*}
\forall \vert \alpha \vert\leq l-2-2(m+1):\;\;\partial_{y}^{2m+2}\bar 
u^{\alpha}_{i}, \;\partial_{y}^{2m+2}\bar u^{\alpha}\in C^{0,\alpha}.
\end{equation*}
Thus we have proven
$\forall\; \vert \alpha \vert\leq l-2-2m : \;\partial^{2m}_{y}\partial^{\alpha}_{x} 
u\in C^{0,\lambda}$\; for some \;$\lambda>0$. However, since there are only even 
powers in the \;$y$-derivative, we only find 
\;$u\in C^{l-3,\lambda}$\; for $\;l\in 2\mathbb{N}$.
The proof is thereby complete.\end{pfn}
\subsection{ Green's function for \;$D_g$\; under weighted Neumann boundary condition}\label{subsec:Greens_function}
In this subsection we study the Green's function \;$\Gamma_g$. As in the previous one, we consider the existence issue and its asymptotics. Furthermore  still as in the case of the Poisson kernel, to do that  we use the method of  Lee-Parker\cite{lp} and have  the same difficulties to overcome as in the previous subsection. We first note that on \;$\mathbb{R}^{n+1}_+$,
\begin{equation}\label{gngamma}
\Gamma(y,x,\xi)=\Gamma^\gamma(y,x,\xi)=g_{n, \gamma}\frac{1}{(y^{2}+\vert x-\xi\vert^{2})^{\frac{n-2\gamma}{2}}}, \;\; (y, x)\in \mathbb{R}^{n+1}_+, \;\;\xi\in \mathbb{R}^n
\end{equation}
for some \;$g_{n, \gamma}>0$\; is the Green's function to the dual problem 
\begin{equation*}
\begin{cases}
Du=0\;\;\text{ in }\;\;\mathbb{R}^{n+1}_+ \\
-d_\gamma^*\lim_{y\to 0}y^{1-2\gamma}\partial_{y}u(y,\cdot)=f\;\; \text{ on }\;\; \mathbb{R}^{n}
\end{cases}
\end{equation*}
i.e.
\begin{equation*}
\begin{cases}
D\Gamma(, \xi)=0\;\;\text{ in }\;\;\mathbb{R}^{n+1}_+, \;\;\xi\in \mathbb{R}^n \\
-d_\gamma^*\lim_{y\to 0}y^{1-2\gamma}\Gamma(y,x,\xi)=\delta_{x}(\xi), \;\;x, \;\xi\in \mathbb{R}^n
\end{cases}.
\end{equation*}
We will construct the Green's function \;$\Gamma_g$\; for the analogous problem
\begin{equation*}
\begin{cases}
D_{g}u=0\;\; \text{ in }\;\;X \\
-d_\gamma^*\lim_{y\to 0}y^{1-2\gamma}\partial_{y}u(y,\cdot)=f \;\;\text{ on }\;\;M
\end{cases}
\end{equation*} 
for \;$D_{g}=-div_{g}(y^{1-2\gamma}\nabla_{g} (\,\cdot\,))+E_{g}$, i.e. for \;$z\in X$\; and \;$\xi \in M$
\begin{equation}\label{green_defining_equation} 
\begin{cases}
D_{g}\Gamma_{g}(\cdot,\xi)=0 \;\;\text{ in }\;\; X\\
-d_\gamma^*\lim_{y\to 0}y^{1-2\gamma}\Gamma_{g}(z,\xi)=\delta_{x}(\xi)
\end{cases},
\end{equation}
where \;$z=(y,x)\in X$\;in $g$-normal Fermi-coordinates close to \;$M$. To that end  we identify
\begin{equation*}
\xi \in M\cap U\subset U\cap X \text{ with }0\in B_{\epsilon}^{\mathbb{R}^{n+1}}(0)\cap 
\mathbb{R}^{n}\subset  B_{\epsilon}^{\mathbb{R}^{n+1}}(0)\cap \mathbb{R}^{n+1}_{+}
\end{equation*}
as in the previous subsection, and write \;$\Gamma(z)=\Gamma(z,0)$. On \;$B_{\epsilon}^{\mathbb{R}^{n+1}}(0)\cap \mathbb{R}^{n+1}_{+}$\; we 
then have 
\begin{equation}\label{green_local_equation_for_the_greens_function}
D_{g}\Gamma=-\frac{\partial_{p}}{\sqrt{g}}(\sqrt{g}g^{p,q}y^{1-2\gamma}\partial_
{ q}\Gamma)+E_{g}\Gamma
=
f
\in 
y^{1-2\gamma}H_{-n+2\gamma-1}C^{\infty}.
\end{equation}
Again we may solve homogeneous deficits homogeneously.
\begin{lem}[Homogeneous solvability, Neumann case 
]\label{green_homogeneous_solvability_neumann}$_{}$\\
For \;$f_{l}\in y^{1-2\gamma}H_{l+2\gamma-1},\; l\in \mathbb{N}-n$\; there exists 
\;$\Gamma_{1+2\gamma+l}\in H_{1+2\gamma+l}$\; such, that 
$$D\Gamma_{1+2\gamma+l}=f_{l}\;\;\text{ in }\;\mathbb{R}^{n+1}\;\;\text{ and } \;\;\lim_{y\rightarrow 0}
y^{1-2\gamma}\partial_{y}\Gamma_{1+2\gamma+l}=0\text{ on } \mathbb{R}^{n}\setminus 
\{0\},$$
provided \;$\frac{1}{2}\neq \gamma \in (0,1)$.
\end{lem}

\begin{pf
This time we use
\begin{equation*}
\langle Q^{k}_{l}(y,x)=y^{2k}P_{l}(x)\mid \;\,k, \;l\in \mathbb{N}\;\;\text{ and }\;\;P_{l}\in 
\Pi_{l}\rangle \underset{\text{dense}}{\subset} C^{0}(\overline 
B_{1}^{\mathbb{R}^{n+1}}(0)\cap\mathbb{R}^{n+1}_{+}),
\end{equation*}
to obtain a orthogonal basis \;$E=\{e^{i}_{k}\}$\; for\; 
$L^{2}_{y^{1-2\gamma}}(S^{n}_{+})$
consisting of $D$-harmonics of the form 
\begin{equation*}
e^{i}_{k}=A_{m}\lfloor_{S^{n}_{+}},\;\;\,A_{m}(y,x)=\sum 
y^{2l}P_{k-2l}(x), \;\;DA_{m}=0
\end{equation*}
and we have \;$
D_{S^{n}_{+}}e^{i}_{k}=k(k+n-2\gamma)y^{1-2\gamma}e^{i}_{k}.
$
As before solving
\begin{equation*}
\begin{cases}
Du=f\in L^{2}_{y^{2\gamma-1}}(\mathbb{R}^{n+1}_{+})\;\;\text{ in }\;\;\mathbb{R}^{n+1}_{+} \\
\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_{y}u=0  \;\;\text{ on } \;\; \mathbb{R}^{n}
\end{cases}
\end{equation*}
with homogeneous \;$f, \;u$\; of degree \;$\lambda,\; \lambda+1+2\gamma$ is, when writing 
\begin{equation*}
u=\sum a_{i,k}e^{i}_{k}\,,y^{2\gamma-1}f=\sum b_{j,l}e^{j}_{l},
\end{equation*}
equivalent to solving 
\begin{equation*}
\sum a_{i,k}(k(k+n-2\gamma)-(\lambda+1+2\gamma)(\lambda+n+1))e^{i}_{k}=\sum 
b_{j,l}e^{j}_{l}
\end{equation*}
and the latter system is always solvable in case
\begin{equation}\label{solvability_homogeneous_neumann}
k(k+n-2\gamma)-(\lambda+1+2\gamma)(\lambda+n+1)\neq 0 \,\,\text{ for all 
}\;\;k, \;n, \;\lambda \in \mathbb{N}. 
\end{equation}
As for proving the lemma there holds 
\begin{equation*}
deg(f_{l})=\lambda=m-n\;\;
\text{ and }\;\;
deg(e^{i}_{k})=k=m^{\prime}
\;\;\text{ for some }\,\;
m, \;m^{\prime}\in \mathbb{N}
\end{equation*}
and plugging this into 
\eqref{solvability_homogeneous_neumann} we verify
\begin{equation*}
m^{\prime}(m^{\prime}+n-2\gamma)-(m-n+1+2\gamma)(m+1)\neq 0\;\;\text{ for all } \;\;
n \;,m, \;m^{\prime}\in \mathbb{N},
\end{equation*}
provided \;$\frac{1}{2}\neq \gamma \in (0,1)$. This shows homogeneous 
solvability, whereas regularity of the solution follows from proposition 
\ref{green_regularity} below.
\end{pf}
\vspace{6pt}

\noindent
Analogously to the case of the Poisson kernel  we may solve 
\eqref{green_defining_equation} successively using lemma 
\ref{green_homogeneous_solvability_neumann}  and obtain
\begin{equation*}
\Gamma_{g}=\eta_{\xi}(\Gamma+\sum_{l=-n}^{m} \Gamma_{1+2\gamma+l})+\gamma_{m}
\end{equation*} 
for \;$m\geq 0$\,where \;$\eta_{\xi}$\; is as in \eqref{etaxi} and a weak solution
\begin{equation*}
\begin{cases}
D_{g}\gamma_{m}=-D_{g}\left(\eta_{\xi}(\Gamma+\sum_{l=-n}^{m} 
\Gamma_{1+2\gamma+l})\right)=y^{1-2\gamma}h_{m}\;\; \text{ in }\;\;X \\
\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_{y}\gamma_{m}=0 \,\;\text{ on } \;\;M 
\end{cases}
\end{equation*}
with \;$h_{m} \in C^{m,\alpha}$. As in the previous subsection, the following weak regularity statement will be 
sufficient for our purpose.
\begin{pro}\label{green_regularity}[Higher regularity for the Neumann 
problem]$_{}$\\
Let \;$h\in y^{1-2\gamma}C^{2k+3,\alpha}(X)$\; and \;$u\in 
W^{1,2}_{y^{1-2\gamma}}(X)$\; be a weak solution of
\begin{equation*}
\begin{cases}
D_{g}u=h\;\;\text{ in }\;\;X \\
\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_{y}u=0 \,\;\text{ on } \;\;M 
\end{cases}.
\end{equation*}
Then \;$u$\; is of class \;$C^{2k,\beta}(X)$, provided \;$H_{g}=0$.
\end{pro}
\vspace{6pt}

\noindent
As in the previous subsection, putting these facts together before presenting the proof of proposition \ref{green_regularity}, we  have the existence of \;$\Gamma_g$\; and can describe its asymptotics as follows.
\begin{cor}[Asymptotics of the Green's 
function \;$\Gamma_g$]\label{Greens_function_asymptotics}$_{}$\\
Let \;$\frac{1}{2}\neq \gamma\in (0,1)$. We then have the existence of \;$\Gamma_g$\; and may expand in \;$g$-normal 
Fermi-coordinates
\begin{equation*}
\Gamma_{g}(z,\xi)
\;\in
\eta_{\xi}(z)
\left( 
\frac{g_{n, \gamma}}{\vert z \vert^{n-2\gamma}}+\sum^{2m+3}_{l=-n}H_{1+2\gamma+l}(z)
\right)
+
C^{2m,\alpha}(X)
\end{equation*}
with \;$H_{l}\in C^{\infty}(\mathbb{R}^{n+1}_+\setminus \{0\})$\; being homogeneous of order 
\;$l$\; and \;$g_{n, \gamma}$\; is as in \eqref{gngamma}, provided \;$H_{g}=0$.
\end{cor}
\begin{pfn}{ of proposition \ref{green_regularity}}\\
As in the previous subsection, we use the Moser iteration argument. Indeed, by exactly the same arguments as the ones used when proving proposition 
\ref{poisson_regularity} we recover H\"older regularity \eqref{poisson_zamboni} 
and integrating the equation directly as in the proof of 
proposition \ref{poisson_regularity} we find the analogue of
\eqref{poisson_equation_integrated_1}, namely
\begin{equation}\label{green_equation_integrated}
\begin{split}
\partial^{\alpha}_{x}u(y,x)
= &
u_{0}^{\alpha}(x) 
-
\int^{y}_{0}\sigma \tilde f^{\alpha}_{1}(\sigma,x)dV 
-
\int^{y}_{0}\sigma^{2\gamma-1}\int^{\sigma}_{0}\tau^{1-2\gamma} \tilde 
f^{\alpha}_{2}(\tau,x)d\tau dV \\
= &
u^{\alpha}_{0}(x)+u^{\alpha}_{1}(y,x)+u^{\alpha}_{2}(y,x),
\end{split}
\end{equation}
where \;$\tilde f_{1}, \;\tilde f_{2}$\; are given by 
\eqref{poisson_tilde_f_structure}.
Let \;$h\in y^{1-2\gamma}C^{l,\lambda^{\prime}}$. Then 
\eqref{poisson_zamboni} and \eqref{poisson_tilde_f_structure} show
\begin{equation*}
\forall \vert \alpha\vert \leq l-2 :\,\;\tilde  
f_{i}^{\alpha}\in C^{0,\lambda}.
\end{equation*}
In particular \eqref{green_equation_integrated} implies
\begin{equation*}
\partial^{\alpha}_{x}u(y,x)
=
u_{0}^{\alpha}(x) +O(y),
\end{equation*} 
so \;$u_{0}^{\alpha}\in C^{l+2,\lambda}$\; anyway by interior regularity
and we may assume inductively
\begin{equation*}
\forall \vert \alpha\vert\leq  l-2-2m: \;\;\partial^{2m}_{y}\partial^{\alpha}_{x}u\in 
C^{0,\lambda}, 
\end{equation*}
whence according to \eqref{poisson_tilde_f_structure} 
\begin{equation*}
\forall \vert \alpha \vert\leq l-2-2(m+1) : \;\;\partial^{2m}_{y}\tilde 
f^{\alpha}_{i}\in C^{0,\lambda}. 
\end{equation*}
Then \eqref{green_equation_integrated} implies via Taylor expansion 
\begin{equation*}
\forall \vert \alpha \vert\leq 
l-2-2(m+1):\;\;\partial_{y}^{2m+2}u_{i}^{\alpha}, \;\partial_{y}^{2m+2}\partial^{\alpha
}_{x}u \in C^{0,\lambda} 
\end{equation*}
Thus we have proven  \;$
\forall \vert \;\alpha \vert\leq 
l-2-2m: \;\;\partial_{y}^{2m}\partial^{\alpha
}_{x}u \in C^{0,\lambda} 
$
for some \;$\lambda>0$. However, since there are only even powers in 
the \;$y$-derivative, we only find \;$u\in C^{l-3,\lambda}$\; for \;$l\in 2\mathbb{N}$.
The proof is thereby complete.
\end{pfn}

\subsection{The  Green's function \;$G_h^\gamma$\; of \;$P_h^\gamma$
In this short subsection, we study the  Green's function \;$G_h^\gamma$\; of \;$P_h^\gamma$. As in the previous one, we derive its existence and asymptotics. Indeed as a consequence of the results of the previous subsections and formula \eqref{regreen}, we have the following corollary 

\begin{cor}[Asymptotics of the Green's 
function \;$G_h^\gamma$]\label{Greens_function_asymptoticsgamma}$_{}$\\
Let \;$\frac{1}{2}\neq \gamma\in (0,1)$. We then have the existence of \;$G_h^\gamma$\; and may expand in \;$h$-normal-coordinates
\begin{equation*}
G_{h}^\gamma(\xi)
\;\in
\eta_{\xi}(x)
\left( 
\frac{g_{n, \gamma}}{\vert x \vert^{n-2\gamma}}+\sum^{2m+3}_{l=-n}H_{1+2\gamma+l}(x)
\right)
+
C^{2m,\alpha}(M)
\end{equation*}
with \;$H_{l}\in C^{\infty}(\mathbb{R}^{n}\setminus \{0\})$\; being homogeneous of order 
\;$l$, provided \;$H_{g}=0$.
\end{cor}
\section{Bubbles and related interaction estimates}\label{bubinter}
In this section, we introduce some bubbles and establish some related interaction estimates. Indeed, we recall the definition of the standard bubbles on  \;$\mathbb{R}^{n+1}_+$\; and their interpretation as a suitable interaction of standard bubbles on \;$\mathbb{R}^n$. Furthermore  we derive new sharp estimates of independent interest for the interaction of the standard bubbles on \;$\mathbb{R}^n$ and use the latter to drive sharp estimates for the standard bubbles on \;$\mathbb{R}^{n+1}_+$.  Moreover  we recall the Schoen's  bubbles associated to the standard bubbles on $\mathbb{R}^{n+1}_+$, and use them to define other bubbles which talk to the local formulation of the problem and call them Projective bubbles. Finally, we derive some sharp interaction estimates for the projective bubbles.
\subsection{The standard bubbles on \;$\mathbb{R}^{n+1}_{+}$
In this subsection, we deal with the standard bubbles on \;$\mathbb{R}^{n+1}_+$. We recall that they are the natural extension \;$\hat \delta_{a, \l} $\; on \;$\mathbb{R}^{n+1}_+$\; of the fractional 
bubble $\delta_{a, \l}$\; on \;$\mathbb{R}^{n}$\;with respect to $D$ ($a\in \mathbb{R}^n$),  and are  given by the 
convolution of the Poisson kernel $K$\; of \;$D$, i.e. 
\begin{equation}\label{bubint}
\begin{split}
\hat \delta_{a,\lambda}(y,x)
= &
p_{n, \gamma}\int_{\mathbb{R}^{n}} \frac{y^{2\gamma}}{(y^{2}+\vert 
x-\xi\vert^{2})^{\frac{n+2\gamma}{2}}}
(\frac{\lambda}{1+\lambda^{2}\vert \xi-a\vert^{2}})^{\frac{n-2\gamma}{2}} \\
= &p_{n, \gamma}
y^{-\frac{n-2\gamma}{2}}\int_{\mathbb{R}^{n}} (\frac{y^{-1}}{1+y^{-2}\vert 
x-\xi\vert^{2}})^{\frac{n+2\gamma}{2}}
(\frac{\lambda}{1+\lambda^{2}\vert \xi-a\vert^{2}})^{\frac{n-2\gamma}{2}}d\xi.
\end{split}
\end{equation}
The last integral can be treated as a standard bubble interaction on \;$\mathbb{R}^n$. This interpretation will be used to derive sharp estimate for \;$\hat \d_{a, \l}$.  In order to do that  we will need the following interaction estimate for the standard bubbles on \;$\mathbb{R}^{n}$, which are new, sharp and of independent interest. We anticipate that its proof is postponed to the appendix given by section \ref{appendix}.
\begin{lem}[The standard interaction estimates on \;$\mathbb{R}^{n}$]\label{lem_standard 
interaction on Rn}$_{}$\\
For \;$c_{n, 3}^{\gamma}$ given by \eqref{defc3}\; and \;$\varepsilon_{i,j}=(\frac{\lambda_{i}}{\lambda_{j}}+\frac{\lambda_{j}}{\lambda_{
i
}}+\lambda_{i}\lambda_{j}\vert a_{i}-a_{j}\vert^{2})^{\frac{2\gamma-n}{2}}
$,  there holds
\begin{enumerate}[label=(\roman*)]
 \item \quad 
$\int_{\mathbb{R}^{n}}\delta_{a_{i},\lambda_{i}}^{\frac{n+2\gamma}{n-2\gamma}}
\delta_{a_{j},\lambda_{j}}
=
c_{n, 3}^{\gamma}\varepsilon_{i,j}
+
O(\min(\frac{\lambda_{i}}{\lambda_{j}},\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}
\varepsilon_{i,j}^{\frac{n}{n-2\gamma}})$
 \item \quad 
$
\lambda_{k}\partial_{\lambda_{k}}\int_{\mathbb{R}^{n}}\delta_{a_{i},\lambda_{i}}^{\frac{
n+2\gamma}{n-2\gamma}}
\delta_{a_{j},\lambda_{j}}
=
c_{n, 3}^{\gamma}\lambda_{k}\partial_{\lambda_{k}}\varepsilon_{i,j}
+
O(\min(\frac{\lambda_{i}}{\lambda_{j}},\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}
\varepsilon_{i,j}^{\frac{n}{n-2\gamma}})
$

\item \quad 
$
\nabla_{a_{i}}\int_{\mathbb{R}^{n}}\delta_{a_i, \l_i}^{\frac{n+2\gamma}{n-2\gamma}}\delta_{a_j, \l_j}
= 
c_{n, 3}^{\gamma}\lambda_{i}\lambda_{j}(a_{i}-a_{j})
\varepsilon_{i,j}^{\frac{n+2-2\gamma}{n-2\gamma}}
+
O(\min(\frac{\lambda_{i}}{\lambda_{j}},\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}
\sqrt{\lambda_{i}\lambda_{j}}\varepsilon_{i,j}^{\frac{n+1}{n-2\gamma}})
$

\item \quad 
$
\nabla^{2}_{a_{k},a_{l}}\int_{\mathbb{R}^{n}}\delta_{a_i, \l_i}^{\frac{n+2\gamma}{n-2\gamma}}
\delta_{a_j, \l_j}
= 
c_{n, 3}^{\gamma}\lambda_{i}\lambda_{j}
(
id-\frac{(n+2-2\gamma)\lambda_{i}\lambda_{j}(a_{i}-a_{j}).(a_{i}-a_{j})}{\frac{
\lambda_{i}}{\lambda_{j}}+\frac{\lambda_{j}}{\lambda_{i}}+\lambda_{i}\lambda_{j}
\vert a_{i}-a_{j} \vert^{2}}
)
\varepsilon_{i,j}^{\frac{n+2-2\gamma}{2}}
\\ _{}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\quad \,\,  +
O(\min(\frac{\lambda_{i}}{\lambda_{j}},\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}
\lambda_{i}\lambda_{j}\varepsilon_{i,j}^{\frac{n+2}{n-2\gamma}}).
$
\end{enumerate}
\end{lem}

\vspace{6pt}

\noindent
Lemma \ref{lem_standard interaction on Rn} has the following impacts on the standard bubbles \;$\hat \d_{a, \lambda}$\; on \;$\mathbb{R}^{n+1}_+$.
\begin{cor}[Rough estimates for the standard bubbles on \;$\mathbb{R}^{n+1}_+$]\label{cor_rough_estimates}$_{}$\\
On \;$\mathbb{R}^{n+1}_{+}$\; we have for \;$r_{a}=\vert z-a \vert=\vert (y,x-a)\vert$
\begin{enumerate}[label=(\roman*)]
 \item \quad
$
\hat \delta_{a,\lambda}(y,x) 
=
O((\frac{\lambda}{1+\lambda^{2}r_{a}^{2}})^{\frac{n-2\gamma}{2}})
$
 \item \quad
$
\partial_{y}\hat 
\delta_{a,\lambda}(y,x)
=
O(\lambda^{\gamma}y^{2\gamma-1}(\frac{\lambda}{1+\lambda^{2}r_{a}^
{ 2 } }
)^{\frac{n}{2}})
$
 \item \quad
$
\nabla_{x}\hat\delta_{a,\lambda}(y,x)
=
O(\sqrt{\lambda}(\frac{\lambda}{1+\lambda^{2}
r_{a}^{2}})^{\frac{n+1-2\gamma}{2}})
$
 \item \quad
$
\nabla^{2}_{x}\hat\delta_{a,\lambda}(y,x)
=
O(\lambda(\frac{\lambda}{1+\lambda^{2}r_{a}^{2}})^{\frac{n+2-2\gamma}{2}})
$
\end{enumerate}
\end{cor}
\begin{pf}
Let \;$a_{i}=x,\lambda_{i}=\frac{1}{y},a_{j}=0$ and $\lambda_{j}=\lambda$. Then \eqref{bubint} and 
lemma \ref{lem_standard interaction on Rn} (i) show
\begin{equation*}
\begin{split}
\hat \delta_{a,\lambda}(y,x)
\leq
Cy^{-\frac{n-2\gamma}{2}}\varepsilon_{i,j}
=
O(\frac{y^{-\frac{n-2\gamma}{2}}}{(\frac{1}{\lambda y}+\lambda 
y+\frac{\lambda}{y}\vert x-a\vert^{2})^{\frac{n-2\gamma}{2}}}),
\end{split}
\end{equation*}
whence (i) follows. From \eqref{pngamma},\eqref{bubint} and lemma \ref{lem_standard interaction on Rn} 
(ii) we infer using \;$y\partial_{y}=-y^{-1}\partial_{y^{-1}}$\; that
\begin{equation*}
\begin{split}
y\partial_{y}&\hat \delta_{a,\lambda}(y,x)
= 
p_{n, \gamma}y\partial_{y}(y^{-\frac{n-2\gamma}{2}}\int 
\delta^{\frac{n+2\gamma}{n-2\gamma}}_{x,y^{-1}}\delta_{a,\lambda})
\\ = &
y\partial_{y}y^{-\frac{n-2\gamma}{2}}\frac{1}{(\frac{1}{\lambda y}+\lambda 
y+\frac{\lambda}{y}\vert x-a \vert^{2})^{\frac{n-2\gamma}{2}}} \\
& +
y^{-\frac{n-2\gamma}{2}}y\partial_{y}\frac{1}{(\frac{1}{\lambda y}+\lambda 
y+\frac{\lambda}{y}\vert x-a \vert^{2})^{\frac{n-2\gamma}{2}}}
+
O(\frac{\min(\frac{1}{\lambda y},\lambda 
y)^{\gamma}y^{-\frac{n-2\gamma}{2}}}{(\frac{1}{\lambda y}+\lambda 
y+\frac{\lambda}{y}\vert x-a \vert^{2})^{\frac{n}{2}}}) \\
& = 
-(n-2\gamma)(\frac{\lambda}{1+\lambda^{2}r_{a}^{2}})^{\frac{n-2\gamma}{2}}
[
\frac{\lambda^{2}y^{2}}{1+\lambda^{2}r_{a}^{2}}
+
O((\frac{\min(1,\lambda^{2} y^{2})}{1+\lambda^{2}r_{a}^{2}})^{\gamma})
],
\end{split}
\end{equation*}  
whence (ii) follows. Likewise (iii), (iv) follow from \eqref{bubint} and lemma \ref{lem_standard interaction on Rn} (iii), (iv).
\end{pf}
\vspace{6pt}

\noindent
\begin{cor}[Sharp estimates for the standard bubbles on \;$\mathbb{R}^{n+1}_+$]\label{cor_sharp_estimates}$_{}$\\
On \;$B_{\varepsilon}(a)^{c}\cap \mathbb{R}^{n+1}_{+}$\; we have for \;$r_{a}=\vert z-a 
\vert=\vert 
(y,x-a)\vert$
\begin{enumerate}[label=(\roman*)]
 \item \quad 
$\hat \delta_{a,\lambda}(y,x)
= 
\frac{1}{\lambda^{\frac{n-2\gamma}{2}}r_{a}^{n-2\gamma}}
+
o_{\frac{1}{1+\lambda^{2\gamma}\varepsilon^{n}}}(\lambda^{\frac{2\gamma-n}{2}})$
 \item \quad
$y\partial_{y}\hat \delta_{a,\lambda}(y,x)
= 
y\partial_{y}\frac{1}{\lambda^{\frac{n-2\gamma}{2}}r_{a}^{n-2\gamma}}
+
o_{\frac{1}{1+\lambda^{2\gamma}\varepsilon^{n}}}(\lambda^{\frac{2\gamma-n}{2}})$
 \item \quad
$x\nabla_{x}\hat \delta_{a,\lambda}(y,x)
= 
x\nabla_{x}\frac{p1}{\lambda^{\frac{n-2\gamma}{2}}r_{a}^{n-2\gamma}}
+
o_{\frac{1}{1+\lambda^{2\gamma}\varepsilon^{n}}}(\lambda^{\frac{2\gamma-n}{2}})$
\end{enumerate}
\end{cor}
\begin{pf
Let $a_{i}=x,\lambda_{i}=\frac{1}{y},a_{j}=0$ and $\lambda_{j}=\lambda$. \eqref{pngamma},\eqref{bubint} and 
Lemma \ref{lem_standard interaction on Rn} (i) give
\begin{equation*}
\begin{split}
\hat \delta_{a,\lambda}(y,x)
= &
y^{-\frac{n-2\gamma}{2}}(\varepsilon_{i,j}
+
O(\min(\frac{\lambda_{i}}{\lambda_{j}},\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}
\varepsilon_{i,j}^{\frac{n}{n-2\gamma}})) \\
= &
\frac{y^{-\frac{n-2\gamma}{2}}}{(\frac{1}{\lambda y}+\lambda 
y+\frac{\lambda}{y}\vert x-a\vert^{2})^{\frac{n-2\gamma}{2}}}
+
O(\frac{1}{(\lambda 
y)^{\gamma}}\frac{y^{-\frac{n-2\gamma}{2}}}{(\frac{1}{\lambda y}+\lambda 
y+\frac{\lambda}{y}\vert x-a\vert^{2})^{\frac{n}{2}}}) \\
= & 
(\frac{\lambda}{1+\lambda^{2}r_{a}^{2})})^{\frac{n-2\gamma}{2}}
+
O(\frac{1}{\lambda^{\frac{n+2\gamma}{2}}r_{a}^{n}})
,
\end{split}
\end{equation*} 
whence (i) follows.
Moreover we find from \eqref{bubint} and lemma \ref{lem_standard interaction on Rn} (ii) as before
\begin{equation*}
\begin{split}
y\partial_{y}\hat \delta_{a,\lambda}(y,x)
& = 
-(n-2\gamma)(\frac{\lambda}{1+\lambda^{2}r_{a}^{2}})^{\frac{n-2\gamma}{2}}
[
\frac{\lambda^{2}y^{2}}{1+\lambda^{2}r_{a}^{2}}
+
O((\frac{\min(1,\lambda^{2} y^{2})}{1+\lambda^{2} r_{a}^{2} 
})^{\gamma})
],
\end{split}
\end{equation*}  
whence 
\begin{equation*}
y\partial_{y}\hat \delta_{a,\lambda}(y,x)
=
y\partial_{y}\frac{1}{\lambda^{\frac{n-2\gamma}{2}}r_{a}^{n-2\gamma}}
+
\rfrac
{
O
(
\frac
{1}
{\lambda^{2}r_{a}^{n-2\gamma+2}}
+
\frac
{1}
{\lambda^{2\gamma}r_{a}^{n}
} 
)
}
{\lambda^ {\frac {n-2\gamma}{2}} 
}
\end{equation*}
and (ii) follows. (iii) follows analogously from \eqref{bubint} and lemma \ref{lem_standard 
interaction on Rn} (iii).
\end{pf}

\subsection{Schoen's bubbles and Projective bubbles}
In this subsection we recall the Schoen's  bubbles associated to the standard bubbles on \;$\mathbb{R}^{n+1}_+$\; for an asymptotically hyperbolic manifold \;$(X, g^+)$ with minimal conformal infinity\;$(M, [h])$\; with $n\geq 2$. Moreover  we will use them to define an other type of bubbles whose construction is motived by the local interpretation of the problem under study, and call them {\em Projective bubbles}. Furthermore  we will establish some interaction estimates for our Projective bubbles, which by the way coincide with the  Schoen's  bubbles on the conformal infinity. Thus, because of \eqref{eq:uniqdef}, and minimality of the conformal infinity, we can consider a geodesic defining function \;$y$\; splitting the metric 
\begin{equation*}
g=y^{2}g^{+},\;\; g=dy^{2}+h_{y}\;\;\text{near}\;\;M\;\;\text{ and } \;\;h=h_{y}\lfloor_{M}
\end{equation*} 
in such a way, that \;$H_{g}=0$.
Moreover  using the existence of conformal normal coordinates (see \cite{gun}), there exists for every \;$a\in M$\; a conformal factor
\begin{equation*}
0<u_{a}\in C^{\infty}(M)\;\;\text{ satisfying }\;\;\frac{1}{C}\leq u_{a}\leq 
C, \;\;u_{a}(a)=1\;\;\text{ and }\;\;\nabla u_{a}(a)=0,
\end{equation*}
inducing a conformal normal coordinate system close to \;$a$\; on \;$M$, in 
particular in normal coordinates with respect to  
$$h_{a}=u_{a}^{\frac{4}{n-2\gamma}}h$$  we have for some small \;$\epsilon>0$
\begin{equation*}
h_{a}=\delta+O(\vert x \vert^{2}), \;\;\det h_{a}\equiv 1\;\;\text{ on } \;\;
B_{\epsilon}^{h_a}(a).
\end{equation*}
As clarified in 
subsection \ref{confnonhom} the conformal factor \;$u_{a}\;$ then 
naturally extends onto \;$X$ via 
$$u_{a}=(\frac{y_{a}}{y})^{\frac{n-2\gamma}{2}},$$
where \;$y_{a}$\; close to the 
boundary \;$M$ is the unique geodesic defining function, for which
\begin{equation*}
g_{a}=y_{a}^{2}g^{+}, \;\; g_{a}=dy_{a}^{2}+h_{a, y_a}\;\;\text{near}\;\;M\;\;\text{ with } 
\;\;h_{a}=h_{a, y_a}\lfloor_{M}
\end{equation*} 
and there still holds \;$H_{g_{a}}=0$. Consequently
\begin{equation*}
g_{a}=\delta+O(y+\vert x \vert^{2})\;\;\text{ and }\;\; \det g_{a}=1+O(y^{2}) \;\;\text{in 
} \;\;B_{\epsilon}^{g_a, +}(a).
\end{equation*}
With respect to \;$h_{a}$-normal coordinates\; $x=x_{a}$\; centered at \;$a$, we define the standard bubble 
\begin{equation}\begin{split}\label{delta_boundary}
\delta_{a,\lambda}
=
(\frac{\lambda}{1+\lambda^{2}\vert x\vert^{2}})^{\frac{n-2\gamma}{2}}\,\;\;
\text{ on } \;\;B_{4\rho_{0}}^{h_a}(a)
\end{split}\end{equation}
for some small \;$\rho_0>0$\; and identifying \;$M\ni a \sim 0\in \mathbb{R}^{n}$\; and the Schoen's  bubble associated to \;$\d_{a, \l}$\; by 
\begin{equation}\begin{split}\label{bubble_boundary}
\varphi_{a,\lambda,\varepsilon}
=
\eta_{a,\varepsilon}\delta_{a,\lambda}
+
(1-\eta_{a,\varepsilon})\lambda^{\frac{2\gamma-n}{2}}G_{h_a}^{\gamma}
 \;\;\text{ on } \;\;M
\end{split}\end{equation}
with a cut-off function in \;$g_{a}$-normal Fermi-coordinates ($\varepsilon>0$\; and small)
\begin{equation}\label{cut_off}
\begin{split}
\eta_{a,\varepsilon}=\eta(\frac{y^{2}+\vert x \vert^{2}}{\varepsilon^{2}}), \;\;
\eta\equiv 1 \,\,\;\text{ on } [0,1]\;\;
\text{ and }\;\;
\eta\equiv 0\;\; \text{ on } \;\;\mathbb{R}_{\geq 2}.
\end{split}
\end{equation} 
In what follows we will always choose 
$$
\varepsilon \sim \lambda^{-\frac{1}{k}} \text{ with }1\ll k\in \mathbb{N},
$$
hence 
$\varphi_{a,\lambda,\varepsilon}=\varphi_{a,\lambda}$.  We define our projective bubbles for \;$X$\; by
\;$\overline{\varphi_{a,\lambda}}^{a}=K_{g_{a}}*\varphi_{a,\lambda}$,  i.e.
\begin{equation}\begin{split}\label{canonical_bubble}
\begin{cases}
D_{g_{a}}\overline{\varphi_{a,\lambda}}^{a}=0 \;\;\text{ in }\;\; X \\
\overline{\varphi_{a,\lambda}}^{a}=\varphi_{a,\lambda} \;\;\text{ on }\;\;M
\end{cases}
\end{split}\end{equation}
namely  the canonical extension of \;$\varphi_{a,\lambda}$\; to \;$X$\; with respect to \;$D_{g_a}$. Using $g_{a}$-normal Fermi-coordinates 
\begin{equation*}
z=z_{a}=(y ,x_{a})=(y,x) 
\end{equation*}
we consider as a 
second extension \begin{equation}\begin{split}\label{constructed_bubble}
\widetilde{\varphi_{a,\lambda}}^{a}
=
\eta_{a,\varepsilon}\hat 
\delta_{a,\lambda}+(1-\eta_{a,\varepsilon})\lambda^{\frac{2\gamma-n}{2}}\Gamma_{
a},
\end{split}\end{equation}
namely the Schoen's bubbles associated to the standard bubbles $\hat\delta_{a,\lambda}$ on $\mathbb{R}^{n+1}_+$ solving
\begin{equation*}\begin{split
\begin{cases}
D\hat \delta_{a,\lambda}=0\;\;\text{ in }\;\;\mathbb{R}^{n+1}_{+} \\
\hat \delta_{a,\lambda}=\delta_{a,\lambda} \;\; \text{ on } \;\;\mathbb{R}^{n}\\
-d_\gamma^*\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_{y}\hat \delta_{a,\lambda}= 
c_{n, \gamma}\delta_{a,\lambda}^{\frac{n+2\gamma}{n-2\gamma}} \text{ on } \;\;\mathbb{R}^{n}
\end{cases}.
\end{split}\end{equation*}
Readily \;$\widetilde{\varphi_{a,\lambda}}^{a}$\; enjoys analogous, but weaker 
identities, namely
\begin{equation}\label{properties_of_tilde_phi}
\begin{cases}
D_{g_{a}}\widetilde{\varphi_{a,\lambda}}^{a}=0 \;\;\text{ in }\;\; 
B_{2\varepsilon}^{g_a, +}(a)\setminus B_{\varepsilon}^{g_a, +}(a)\subset X \\
\widetilde{\varphi_{a,\lambda}}^{a}=\varphi_{a,\lambda} \;\;\text{ on }\;\;M \\
-d_\gamma^*\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_{y}\widetilde{\varphi_{a,\lambda}}^{a}=c_{n, \gamma}\eta_{a,
\varepsilon}\delta_{a,\lambda}^{\frac{n+2\gamma}{n-2\gamma}}
\end{cases}.
\end{equation}
Finally, we set
\begin{equation*
v_{a, \l}=u_a\varphi_{a, \l},
\end{equation*}
\begin{equation*
G_a=G_{h_a}^\gamma,
\end{equation*}
and
\begin{equation}\label{dfbua}
\overline u^a=K_{g_a}*u.
\end{equation}
\subsection{The interaction estimates for \;$v_{a, \l}$
In this subsection, we derive several interaction estimates for \;$v_{a, \l}$.
\begin{lem}[Higher interaction estimates]\label{lem_higher_interaction_estimates
For $ i \neq j$ there holds
\begin{enumerate}[label=(\roman*)]
 \item \quad 
$
\int_{M} v_{a_{i},\lambda_{i}}^{\alpha}v_{a_{j},\lambda_{j}}^{\beta}dV_{h}
=
O(\varepsilon_{i,j}^{\beta})
$
for all $\alpha +\beta=\frac{2n}{n-2\gamma}, \;\alpha>\frac{n}{n-2\gamma}>\beta>0 $
\item \quad
$
\int v_{a_{i},\lambda_{i}}^{\frac{n}{n-2\gamma}}v_{a_{j},\lambda_{j}}^{\frac{n}{n-2\gamma}} 
=
O(\varepsilon^{\frac{n}{n-2\gamma}}_{i,j}\ln \varepsilon_{i,j}).
$ 
\end{enumerate}
\end{lem}
\noindent
Since the type of estimates stated in this lemma are standard, we delay their proof to the appendix. 
\begin{lem}[The interaction estimates for \;$v_{a, \l}$]${}$\label{interactest
For
\begin{enumerate}[label=(\roman*)]
 \item \quad 
$
e_{i,j}
=
\langle 
v_{a_{i},\lambda_{i}},v_{a_{j},\lambda_{j}}
\rangle_{P_{h}^\gamma}
$
 \item \quad
$
\epsilon_{i,j}
=
\int_{M}(v_{a_{i},\lambda_{i}})^{\frac{n+2\gamma}{
n-2\gamma } }v_{a_{j}}dV_{h}
$
 \item \quad 
$
\varepsilon_{i,j}
=
\left(
\frac
{1}
{
\frac{\lambda_{i}}{\lambda_{j}}
+
\frac{\lambda_{j}}{\lambda_{i}}
+
\lambda_{i}\lambda_{j}(G_{h}^\gamma)^{\frac{2}{2\gamma-n}}(a_{i},a_{j})
}
\right)^{\frac{n-2\gamma}{2}}
$
\end{enumerate}
there holds
\begin{enumerate}[label=(\roman*)]
 \item \quad
$
c \leq 
\frac{e_{i,j}}{\varepsilon_{i,j}},\frac{\epsilon_{i,j}}{\varepsilon_{i,j}}
\leq C
$
 \item \quad 
$
e_{i,j}=c_{n,\gamma}\epsilon_{i,j}(1+o_{\max(\frac{1}{\lambda_{i}},\frac{1}{
\lambda_{j}} ) }(1))
$
 \item \quad 
$e_{i,j}=c_{n, 4}^\gamma\varepsilon_{i,j}(1+o_{\varepsilon_{i,j}}(1))$\;\; and  \;$\epsilon_{i,j}=c_{n, 3}^\gamma\varepsilon_{i,j}(1+o_{\varepsilon_{i,j}
} (1))$
\end{enumerate}

\end{lem}
\begin{pf}
Writing abbreviatively \;$\varphi_{k}=\varphi_{a_{k},\lambda_{k}}$\; we first remark
\begin{equation}\label{epsilon_i_j_symmetry}
\epsilon_{i,j}=\epsilon_{j,i}+o_{\max(\frac{1}{\lambda_{i}},\frac{1}{\lambda_{j}
} )} (\varepsilon_{i,j} ).
\end{equation} 
We will prove \eqref{epsilon_i_j_symmetry} in the appendix.
Then, since \;$e_{i,j}$\; and \;$\varepsilon_{i,j}$\; are symmetric in \;$i\;$ and \;$j$\;, we may assume \;$\frac{1}{\lambda_{i}}\leq 
\frac{1}{\lambda_{j}}$\; for the rest of the proof. Consider
\begin{equation*}\begin{split}
e_{i,j}
= &
\int_{M}P_{h}^\gamma(u_{a_{i}}  \varphi_{i}) (u_{a_{j}}\varphi_{j}) dV_{h} 
= 
\int_{M}P_{h_{a_{i}}}^\gamma\varphi_{i}\frac{u_{a_{j}}}{u_{a_{i}}}\varphi_{j}dV_{h_{a_
{i}}}
= 
\langle \varphi_{i}  , 
\frac{u_{a_{j}}}{u_{a_{i}}}\varphi_{j}\rangle_{P^\gamma_{h_{a_{i}}}}.
\end{split}\end{equation*}
Denoting by \;$y=y_{a_{i}}$\; the geodesic defining function attached to \;$h_{a_{i}}=u_{a_{i}}^{\frac{4}{n-2\gamma}}h$\; and choosing close to \;$a_{i}$\; a 
corresponding \;$g_{a_{i}}$-normal Fermi-coordinate system
$
z=z_{a_{i}}=(y_{a_{i}},x_{a_{i}})=(y,x),\,\;\; r=\vert z \vert
$
we find
\begin{equation}\begin{split}\label{quadratic_interaction}
\frac{1}{d_\gamma^*}e_{i,j}
= &
-\lim_{y\rightarrow 0}\int_{M}y^{1-2\gamma}\partial_{y}
\widetilde{\varphi_{i}}^{a_{i } }
\frac{u_{a_{j}}}{u_{a_{i}}}\varphi_{j}dV_{h_{a_{i}}} \\
& +
\int_{X}div_{g_{a_{i}}}(y^{1-2\gamma}\nabla_{g_{a_{i}}}(\overline{
\varphi_{i}}^{a_{i}}-\widetilde{\varphi_{i}}^{a_{i}})
\overline{\frac{u_{a_{j}}}{u_{a_{i}}}\varphi_{j}}^{a_{i}})
dV_{g_{a_{i}}}
= 
I_{1}+I_{2}.
\end{split}\end{equation}
Since  \;$D_{g_{a_{i}}}\overline{\frac{u_{a_{j}}}{u_{a_{i}}}\varphi_{j}}^{a_{i}}=0$\;( for the meaning of \;$\overline{\frac{u_{a_{j}}}{u_{a_{i}}}\varphi_{j}}^{a_{i}}$, see \eqref{dfbua}) by 
definition, we have
\begin{equation*}\begin{split
I_{2}
= &
-\int_{X}D_{g_{a_{i}}}(\overline{\varphi_{i}}^{a_{i}}-\widetilde{\varphi_{i}}^{
a_{i}})
\overline{\frac{u_{a_{j}}}{u_{a_{i}}}\varphi_{j}}^{a_{i}}
dV_{g_{a_{i}}}
\end{split}\end{equation*}
and there holds, as we will prove in the appendix,
\begin{equation}\label{varphi_in_a_j_coordinates}
\begin{split}
\vert \overline{\frac{u_{a_{j}}}{u_{a_{i}}}\varphi_{j}}^{a_{i}} \vert
\leq 
C\widetilde{\varphi_{j}}^{a_{j}} 
 \;\;\text{ on }\;\,\; B_{\rho_{0}}^{g_{a_i}, +}(a_{i}).
\end{split}
\end{equation} 
Let us estimate
\begin{equation*}
D_{g_{a_{i}}}(\overline{\varphi_{i}}^{a_{i}}-\widetilde{\varphi_{i}}^{a_{i}})
=
D_{g_{a_{i}}}\widetilde{\varphi_{i}}^{a_{i}}.
\end{equation*}
Clearly  \;$D_{g_{a_{i}}}\widetilde{\varphi_{i}}^{a_{i}}=0$\; on
$B^{g_{a_i}, +}_{2\varepsilon_{i}}(a_{i})^{c}$,
since \;$D_{g_{a_{i}}}\Gamma_{a_{i}}=0$, whereas on \;$B^{g_{a_i}, +}_{2\varepsilon}(a_{i})$  
\begin{equation*}
\begin{split}
D_{g_{a_{i}}}\widetilde{\varphi_{i}}^{a}
= &
D(\eta_{a_{i},\varepsilon_{i}}(\hat 
\delta_{a_{i},\lambda_{i}}-\frac{\lambda_{i}^{\frac{2\gamma-n}{2}}}{r 
^{n-2\gamma}})) +
(D_{g_{a_{i}}}-D)
(\eta_{a_{i},\varepsilon_{i}}\hat 
\delta_{a_{i},\lambda_{i}} 
-
(1-\eta_{a_{i},\varepsilon_{i}})
\lambda_{i}^{\frac{2\gamma-n}{2}}\Gamma_{a_{i}})
\\
& - 
D_{g_{a_{i}}}((1-\eta_{a_{i},\varepsilon_{i}})
\frac{H_{a_{i}}}{\lambda_{i}^{\frac{n-2\gamma}{2}}})
=
J_{1}+J_{2}+J_{3}
\end{split}
\end{equation*}
writing \;$\Gamma_{a_{i}}=r ^{2\gamma-n}+H_{a_{i}}$\; in \;$g_{a_{i}}$-normal 
Fermi-coordinates thanks to subsection \ref{subsec:Greens_function}. Then
\begin{equation*}\begin{split
J_{1}
= &
-div (y^{1-2\gamma}\nabla
(
\eta_{a_{i},\varepsilon_{i}}(\hat \delta_{a_{i},\lambda_{i}}
-
\frac{\lambda_{i}^{\frac{2\gamma-n}{2}}}{r ^{n-2\gamma}})))
\\
= &
-
\partial_{y}y^{1-2\gamma}\partial_{y}\eta_{a_{i},\varepsilon_{i}}(\hat 
\delta_{a_{i},\lambda_{i}}-\frac{\lambda_{i}^{\frac{2\gamma-n}{2}}}{r 
^{n-2\gamma}}) 
+
2y^{1-2\gamma}\nabla \eta_{a_{i},\varepsilon_{i}}\nabla (\hat 
\delta_{a_{i},\lambda_{i}}-\frac{\lambda_{i}^{\frac{2\gamma-n}{2}}}{r 
^{n-2\gamma}}) \\
& +
y^{1-2\gamma}\Delta \eta_{a_{i},\varepsilon_{i}}(\hat 
\delta_{a_{i},\lambda_{i}}-\frac{\lambda_{i}^{\frac{2\gamma-n}{2}}}{r 
^{n-2\gamma}}),
\end{split}\end{equation*}
due to \;$D\hat \delta_{a_{i},\lambda_{i}}=Dr^{2\gamma-n}=0$.
\eqref{cut_off} and  corollary \ref{cor_sharp_estimates} then show 
\begin{equation*}
J_{1}
=
o_{\frac{1}{1+\lambda_{i}^{2\gamma}\varepsilon_{i}^{n} 
}}(\frac{y^{1-2\gamma}}{\varepsilon_{i}^{2}  
\lambda_{i}^{\frac{n-2\gamma}{2}}})
=
O(y^{1-2\gamma}\widetilde{\varphi_{i}}^{a_{i}}),
\end{equation*}
whenever \;$\varepsilon_{i}\sim \lambda_{i}^{-\frac{1}{k}}$\; and \;$k\gg 1$\; is 
chosen sufficiently large. Secondly letting  
\begin{equation*}
p, \;q=1,\ldots, n+1 \;\;\text{ and } \;\;k, \;l=1,\ldots, n
\end{equation*}
we evaluate 
\begin{equation}\label{OperatorsDifference_interaction}
\begin{split}
D - D_{g_{a_{i}}}
= &
\frac{\partial_{p}}{\sqrt{g_{a_{i}}}}(\sqrt{g_{a_{i}}}g_{a_{i}}^{p,q}y^{
1-2\gamma } \partial_{
q}\cdot)
-
E_{g_{a_{i}}}
-
\partial_{p}(\delta^{p,q}y^{1-2\gamma}\partial_{q}\cdot) \\
= &
-
E_{g_{a_{i}}}
+
\frac{\partial_{y}\sqrt{g_{a_{i}}}}{\sqrt{g_{a_{i}}}}y^{1-2\gamma}\partial_{y}
+
\frac{\partial_{k}\sqrt{g_{a_{i}}}}{\sqrt{g_{a_{i}}}}g_{a_{i}}^{k,l}y^{1-2\gamma
} \partial_{l
}
+
\partial_{k}((g_{a_{i}}^{k,l}-\delta^{k,l})y^{1-2\gamma}\partial_{l}\cdot)
 \\
= &
O(y^{1-2\gamma})
+
O(y^{2-2\gamma})\partial_{y}
+
O(y^{1-2\gamma}r)\nabla_{x}
+
O(y^{1-2\gamma}r^{2}+y^{2-2\gamma})\nabla^{2}_{x},
\end{split}
\end{equation}
where we made use of \eqref{Eg_local}, and
\begin{equation*}
g_{a_{i}}=\delta+O(y+\vert x \vert^{2})\;\;\text{ and } \;\;\det g_{a_{i}}=1+O(y^{2}) 
\;\;\text{ 
on } \;\;
B_{2\varepsilon_{i}}^{g_{a_i}, +}(a_{i}).
\end{equation*}
From \eqref{OperatorsDifference_interaction} and corollary 
\ref{cor_rough_estimates} we 
then find 
\begin{equation*}
J_{2}
=
O(y^{1-2\gamma}\widetilde{\varphi_{i}}^{a_{i}})
+
O(y^{1-2\gamma}\lambda_{i}y(\widetilde{\varphi_{i}}^{a_{i}})^{\frac{n+2-2\gamma}
{ n-2\gamma } } ).
\end{equation*}
Moreover from \eqref{OperatorsDifference_interaction} we find 
\begin{equation*}
D_{g_{a_{i}}}H_{a_{i}}=-D_{g_{a_{i}}}r^{2\gamma-n}=(D-D_{g_{a_{i}}})r^{2\gamma-n
}
=
O
( 
y^ { 1-2\gamma } r^ { 2\gamma-n-1 } )
, 
\end{equation*}
whence \;$H_{a_{i}}=O(r^{1+2\gamma-n})$, cf. the subsection 
\ref{subsec:Greens_function}.
Thus \eqref{OperatorsDifference_interaction} yields
\begin{equation*}
\begin{split}
\vert J_{3} \vert
\leq 
\frac{Cy^{1-2\gamma}\chi_{B_{\varepsilon_{i}}(a_{i})^{c}}}
{\lambda_{i}^{\frac{n-2\gamma}{2}}r^{n-2\gamma}}
=
O(y^{1-2\gamma}\widetilde{\varphi_{i}}^{a_{i}})
\text{ on }
B_{2\varepsilon_{i}}^{+}(a_{i})
\end{split}
\end{equation*}
for any choice \;$\varepsilon_{i}\sim \lambda_{i}^{-\frac{1}{k}},\, k\gg 1$.
Collecting terms we conclude
\begin{equation*
\begin{split}
D_{g_{a_{i}}}\widetilde{\varphi_{i}}^{a_{i}}
= &
O(y^{1-2\gamma}\widetilde{\varphi_{i}}^{a_{i}})
+
O(y^{1-2\gamma}\lambda_{i}y(\widetilde{\varphi_{i}}^{a_{i}})^{\frac{n+2-2\gamma}
{ n-2\gamma } } )\;\;
\text{ on }\;\;
B_{2\varepsilon_{i}}^{g_{a_i}, +}(a_{i}),
\end{split}
\end{equation*}
which in conjunction with \eqref{varphi_in_a_j_coordinates} implies
\begin{equation*}
I_{2}
=
O
(
\int_{B^{g_{a_i}, +}_{2\varepsilon_{i}}(a_{i})}y^{1-2\gamma}\widetilde{\varphi_{i}}^{
a_{i }}\widetilde{\varphi_{j}}^{a_{j}}
)
+
O(\int_{B^{g_{a_i}, +}_{2\varepsilon_{i}}(a_{i})}y^{1-2\gamma}\lambda_{i}y(\widetilde{
\varphi_{i}}^{a_{i}})^{\frac{n+2-2\gamma}
{ n-2\gamma } }\widetilde{\varphi_{j}}^{a_{j}})
. 
\end{equation*}
As we will prove in the appendix, this gives
\begin{equation}\label{I2_interaction_integral_estimate}
I_{2}
=
o_{\frac{1}{\lambda_{i}}}(\varepsilon_{i,j}). 
\end{equation} 
Recalling \eqref{quadratic_interaction} we thus arrive at
\begin{equation*}\begin{split}
\frac{1}{d_\gamma^*}e_{i,j}
= &
-\lim_{y\rightarrow 0}\int_{M}y^{1-2\gamma}\partial_{y}\widetilde{\varphi_{i}}^{a_{i}
}
\frac{u_{a_{j}}}{u_{a_{i}}}\varphi_{j}dV_{h_{a_{i}}} 
= 
\int_{M}
\frac{c_{n, \gamma}}{d_\gamma^*}\eta_{a_{i},\varepsilon_{i}}\delta_{i}^{\frac{n+2\gamma}{n-2\gamma}}
\frac{u_{a_{j}}}{u_{a_{i}}}\varphi_{j}dV_{h_{a_{i}}}
\end{split}\end{equation*}
up to some \;$o_{\frac{1}{\lambda_{i}}}(\varepsilon_{i,j})$\; using 
\eqref{properties_of_tilde_phi} for the last equality. 
Moreover setting \;$\d_{i}=\d_{a_{i}, \l_{i}}$, we obtain
\begin{equation}\label{interaction_expansion_intermediate_form}
\begin{split}
\int_{M} 
\eta_{a_{i},\varepsilon_{i}}\delta_{i}^{\frac{n+2\gamma}{n-2\gamma}}\frac{u_{a_
{j}}}{u_{a_{i}}}\varphi_{j}dV_{h_{a_{i}}} 
= &
\int_{M} 
\eta_{a_{i},\varepsilon_{i}}(u_{a_{i}}\delta_{i})^{\frac{n+2\gamma}{n-2\gamma}}
(u_{a_{j}}\varphi_{j})dV_{h}  \\
= &
\underset{B^{h_{a_i}}_{\varepsilon_{i}}(a_{i})}\int 
(u_{a_{i}}\delta_{i})^{\frac{n+2\gamma}{n-2\gamma}}
(u_{a_{j}}\varphi_{j})dV_{h} 
+
O(\frac{\varepsilon_{i}^{-n-2\gamma}}{\lambda_{i}^{\frac{n+2\gamma}{2}}\lambda_{
j}^{\frac{n-2\gamma}{2}}})\\
= &
\int_{M} (u_{a_{i}}\varphi_{i})^{\frac{n+2\gamma}{n-2\gamma}}
(u_{a_{j}}\varphi_{j})dV_{h} 
+
o_{\frac{1}{\lambda_{i}}}(\varepsilon_{i,j})
=
\epsilon_{i,j}+o_{\frac{1}{\lambda_{i}}}(\varepsilon_{i,j})
\end{split}
\end{equation}
for any choice \;$\varepsilon_{i}\sim \lambda_{i}^{-\frac{1}{k}},\, k$\; large. 
We conclude
\begin{equation}\begin{split}\label{e_ij_epsilon_ij_relation}
e_{i,j}
= &
c_{n,\gamma}\epsilon_{i,j}
+
o_{\frac{1}{\lambda_{i}}}(\varepsilon_{i,j}).
\end{split}\end{equation}
We turn to analyse \;$\epsilon_{i,j}$.
From \eqref{interaction_expansion_intermediate_form} we have up to some \;$o_{\frac{1}{\lambda_{i}}}(\varepsilon_{i,j})$
\begin{equation*}\begin{split}
\epsilon_{i,j}
= &
\underset{B^{h_{a_i}}_{\varepsilon_{i}}(a_{i})}\int 
(u_{a_{i}}\delta_{i})^{\frac{n+2\gamma}{n-2\gamma}}
(u_{a_{j}}\varphi_{j})dV_{h}
= 
\underset{B^{h_{a_i}}_{\varepsilon_{i}}(a_{i})}\int 
\delta_{i}^{\frac{n+2\gamma}{n-2\gamma}}
\frac{u_{a_{j}}}{u_{a_{i}}}\varphi_{j}dV_{h_{a_{i}}}.
\end{split}\end{equation*}
Changing coordinates and rescaling we obtain up to some 
$o_{\frac{1}{\lambda_{i}}}(\varepsilon_{i,j})$
\begin{equation*}\begin{split}
\epsilon_{i,j}
= &
\underset{B^{h_{a_i}}_{ \varepsilon_{i}\lambda_{i} }(0)}{\int}
\frac{\frac{u_{ a _{j}}}{u_{a_{i}}}( 
\exp^{h_{a_{i}}}_{a_i}\frac{x}{\lambda_{i}})}{(1+r^{2})^{\frac{n+2\gamma}{2}}} 
\left(\frac{1}{\frac{\lambda_{i} }{ \lambda_{j} }+\lambda_{i}\lambda_{j}G_{ a 
_{j}}^{\frac{2}{2\gamma-n}}(\exp^{h_{ a _{i}}}_{a_i} 
\frac{x}{\lambda_{i}})}\right)^{\frac{n-2\gamma}{2}}
dV_{h_{a_{i}}}.
\end{split}\end{equation*}
In particular
\begin{equation}\label{smallness_if_and_only_if}
\epsilon_{i,j}\ll 1 \Longleftrightarrow \varepsilon_{i,j}\ll 1
\end{equation}
and, as we will prove in the appendix, 
\begin{equation}\label{epsilon_varepsilon_relation}
\begin{split}
\epsilon_{i,j}
= &
c_{n, 3}^{\gamma}\varepsilon_{i,j}(1+o_{\varepsilon_{i,j}}(1)),
\end{split}\end{equation}
Now
\eqref{smallness_if_and_only_if} and \eqref{epsilon_varepsilon_relation} show
\begin{equation*}
c\leq \rfrac{\epsilon_{i,j}}{\varepsilon_{i,j}}\leq C,
\end{equation*}
whence by virtue 
of \eqref{e_ij_epsilon_ij_relation} for \;$\frac{1}{\lambda_{i}}\ll 1$\; sufficiently small
\begin{equation*}
c\leq \rfrac{e_{i,j}}{\varepsilon_{i,j}}\leq C,
\end{equation*}
Therefore (i) is proven, i.e. \;$e_{i,j},\epsilon_{i,j}$\; and  \;$\varepsilon_{i,j}$\; are pairwise comparable. In particular (ii) follows from 
\eqref{e_ij_epsilon_ij_relation}. In view of 
\eqref{epsilon_varepsilon_relation} we are left with proving 
\;$e_{i,j}=c_{n, 4}^\gamma\varepsilon_{i,j}(1+o_{\varepsilon_{i,j}}(1))$, but 
this follows easily from (ii) and \eqref{epsilon_varepsilon_relation} combined with \eqref{eq:defc4}.
Thus the lemma is proven.
\end{pf}

\section{Locally flat conformal infinities of Poincaré-Einstein manifolds}\label{pceinsten}
In this section, we sharpen the results of section \ref{fundso} in the case  of Poincaré-Einstein manifold \;$(X, g^+)$\; with  locally conformally flat infinity \;$(M, [h])$. Furthermore in this particular case, we  establish a sharp \;$L^{\infty}$-estimate for \;$\overline{\varphi_{a, \l}}^a-\wtilde{\varphi_{a, \l}}^a$, and derive selfaction estimates for \;$v_{a, \l}$. By our assumptions, we have  
\begin{enumerate}[label=(\roman*)]
 \item for a geodesic defining function \;$y$\; splitting the metric
\begin{equation*}
g=y^{2}g^{+},\; g=dy^{2}+h_{y}\;\;\text{near}\;\;M\;\;\text{ and }\;\; h=h_{y}\lfloor_{M}
\end{equation*} 
there exists for every \;$a\in M$\;  a conformal factor 
\begin{equation*}
\frac{1}{C}\leq u_{a}\leq C,\;\;\,u_{a}(a)=1\;\;\text{ and }\;\;\nabla u_{a}(a)=0,
\end{equation*}
whose conformal metric \;$h_{a}=u_{a}^{\frac{4}{n-2\gamma}}h$\; close to \;$a$\; admits an Euclidean 
coordinate system, \;$h_{a}=\delta$\; on $B_{\epsilon}^{h_a}(a)$. As clarified in 
subsection \ref{confnonhom} and recalling Remark \ref{eq:minimal}, this gives rise to a geodesic defining 
function \;$y_{a}$, for which
\begin{equation}\label{dfmga}
g_{a}=y_{a}^{2}g^{+}, \;\; g_{a}=dy_{a}^{2}+h_{a,y}\;\;\text{near}\;\;M\;\;\text{ with }\;\; 
h_{a}=h_{a, y_a}\lfloor_{M}\;\;\text{ and }\;\;\delta=h_{a}\lfloor_{ 
B_{\epsilon}^{h_a}(a)},
\end{equation} 
the boundary \;$(M, [h_a])$\; is totally geodesic and the extension operator \;$D_{g_{a}}$\; is positive. 



 \item
 as observed by Kim-Musso-Wei \cite{kmw1} in the case \;$n\geq 3$\; (see lemma 43 in \cite{kmw1}) and for \;$n=2$\; due to Remark \ref{eq:minimal} and the existence of isothermal coordinates we have
\begin{equation}\label{flatness}
g_{a}=\delta+O(y^{n}) \;\;\text{ on } \;\;B_{\epsilon}^{g_a, +}(a)
\end{equation} 
in \;$g_{a}$-normal 
Fermi-coordinates around $a$for some small \;$\epsilon >0$. 
Therefore the previous results on the fundamental 
solutions in the case of asymptotically hyperbolic manifold of minimal conformal infinity of section \ref{fundso} are applicable. We collect them in the following subsection.
\end{enumerate}
\subsection{Fundamental solutions in this case}
In this subsection we sharpen the results of section \ref{fundso} in the case of a Poincar\'e-Einstein manifold \;$(X, g^+)$\; with  locally conformally flat infinity \;$(M, [h])$ . 
\vspace{6pt}

\noindent
To do that  
let us first write  \;$K_{a}=K_{g_{a}}(\cdot, a)$, \;$\Gamma_{a}=\Gamma_{g_{a}}(\cdot, 
a)$ and \;$G_a^\gamma=G_{h_a}^\gamma(\cdot,a)$.
From \eqref{flatness} we then find  
\begin{equation*}
D_{g_{a}}K\in yH_{-2\gamma-2}C^{\infty},
\;\,\;\;
D_{g_{a}}\Gamma\in y^{1-2\gamma}H_{2\gamma-2}C^{\infty}
\end{equation*}
for the lowest order 
deficits in \eqref{poisson_local_equation_for_the_poisson_kernel} and 
\eqref{green_local_equation_for_the_greens_function}.
Then in view of lemmas \ref{poisson_homogeneous_solvability_dirichlet}, 
\ref{green_homogeneous_solvability_neumann} the corresponding expansions given 
by corollaries
\ref{Poisson_kernel_asymptotics}, \ref{Greens_function_asymptotics},  \ref{Greens_function_asymptoticsgamma}
are 
\begin{enumerate}[label=(\roman*)]
 \item \quad 
$
K_{a}(z)
\in
\eta_{\xi}(z)
\left(p_{n, \gamma} 
\frac{y^{2\gamma}}{\vert z 
\vert^{n+2\gamma}}+\sum^{2m+6}_{l=0}y^{2\gamma}H_{l-2\gamma}(z)
\right)
+
y^{2\gamma}C^{2m,\alpha}(X)
$ 

 \item \quad 
$
\Gamma_{a}(z)
\;\in
\eta_{\xi}(z)
\left( 
\frac{g_{n, \gamma}}{\vert z \vert^{n-2\gamma}}+\sum^{2m+4}_{l=0}H_{l+2\gamma}(z)
\right)
+
C^{2m,\alpha}(X)
$ 

 \item \quad 
$
G_{a}^\gamma(x)
\;\in
\eta_{\xi}(x)
\left( 
\frac{g_{n, \gamma}}{\vert x \vert^{n-2\gamma}}+\sum^{2m+4}_{l=0}H_{l+2\gamma}(x)
\right)
+
C^{2m,\alpha}(M).
$ 

\end{enumerate}
Recalling \eqref{definition_H_l} there holds \;$y^{2\gamma}H_{l-2\gamma}\subset 
C^{m,\alpha}$\; for \;$l>m $\; and
$H_{l+2\gamma}\subset C^{m,\alpha}$\; for \;$l\geq m$. We have therefore proven the following result.
\begin{cor}[The kernels in the case locally flat conformal infinities of Poincar\'e-Einstein  manifolds
]\label{cor_kernels_for_poincare_einstein_metrics}
\noindent
Let \;$(X,  \;g^{+})$\; be a Poincar\'e-Einstein manifold with 
conformally infinity  \;$(M, [h])$\; of dimension \;$n= 2$\; or \;$n\geq 3$\; and \;$(M, [h])$\; is locally flat. If
\begin{equation*}
\frac{1}{2}\neq \gamma \in (0,1) \;\;\text{ and } \;\,\;
\lambda_{1}(-\Delta_{g^{+}})>s(n-s),\;\;
\text{ for } s=\frac{n}{2}+\gamma
\end{equation*}
then the Poison kernel \;$K_{g}$\; and the Green's functions \;$\Gamma_{g}$\;and \;$G_h^{\gamma}$ 
respectively for
\begin{equation*}
\begin{cases}
D_{g}U=0 \;\;\text{ in }\;\; X \\
U=f \,\,\;\text{ on } \;\;M
\end{cases} 
\quad\quad
\begin{cases}
D_{g}U=0\;\; \text{ in } \;\;X \\
-d_{\gamma}^*\lim_{y\rightarrow 0}y^{1-2\gamma}\partial_{y}U=f \;\;\text{ on }\;\; M
\end{cases} 
\quad \text{and}\quad
\begin{cases}
P_h^\gamma u=f\;\;\text{on}\;\;M
\end{cases}
\end{equation*}
respectively are of class \;$y^{2\gamma}C^{2,\alpha}$\; and \;$C^{2,\alpha}$\; away from the 
singularity and admit for every \;$a\in M$ locally in \;$g_{a}$-normal 
Fermi-coordinates an expansion  (where \;$g_a$\;is as in  \eqref{dfmga})
\begin{enumerate}[label=(\roman*)]
 \item \quad 
$
K_{a}(z)\;
\in
p_{n, \gamma}\frac{y^{2\gamma}}{\vert z \vert^{n+2\gamma}}
+
y^{2\gamma}H_{-2\gamma}(z)
+
y^{2\gamma}H_{1-2\gamma}(z)
+
y^{2\gamma}H_{2-2\gamma}(z)
+
y^{2\gamma}C^{2,\alpha}(X)
$

 \item \quad 
$
\Gamma_{a}(z)\;
\in
\frac{g_{n, \gamma}}{\vert z \vert^{n-2\gamma}}+H_{2\gamma}(z)+H_{1+2\gamma}(z)
+
C^{2,\alpha}(X)
$

 \item \quad 
$
G_{a}^{\gamma}(x)\;
\in
\frac{g_{n, \gamma}}{\vert x \vert^{n-2\gamma}}+H_{2\gamma}(z)+H_{1+2\gamma}(x)
+
C^{2,\alpha}(X)
$

\end{enumerate}
with \;$H_{k}\in C^{\infty}(\overline{\mathbb{R}^{n}_{+}}\setminus \{0\})$\; being 
homogeneous of degree \;$k$. 
\end{cor}
\subsection{Comparing  Schoen's bubbles and the Projective ones}
In this subsection, we compare the  Schoen's bubbles \;$\wtilde{\varphi_{a, \l}}^a$ and our Projective bubbles \;$\overline{\varphi_{a, \l}}^a$. 
Indeed, we are going to establish a sharp $L^{\infty}$-estimates for \;$D_{g_a}\left(\overline{\varphi_{a, \l}}^a-\wtilde{\varphi_{a, \l}}^a\right)$.  Furthermore  using the Maximum principle for \;$D_{g_a}$\; under Dirichlet boundary condition, we are going to derive a  sharp $\; L^{\infty}$-estimate\; for $\overline{\varphi_{a, \l}}^a-\wtilde{\varphi_{a, \l}}^a$.  This is the content of the following lemma. 

\begin{lem}[Comparing the  Schoen's bubbles and the Projective ones]\label{lem_bubble comparsion} $_{}$\\
Writing in \;$g_{a}$-normal Fermi-coordinates 
\begin{equation*}
\Gamma_{a}=r ^{2\gamma-n}+H_{a}=r^{2\gamma-n}+M_{a}+O(\max(r ,r ^{
2\gamma})),
\end{equation*}
cf. corollary \ref{cor_kernels_for_poincare_einstein_metrics}, there holds
\begin{enumerate}[label=(\roman*)]
 \item \quad
$
D_{g_{a}}\left(\overline{\varphi_{a,\lambda}}^{a}-\widetilde{\varphi_{a,\lambda}}^{a}\right)
= -D_{g_{a}}(\widetilde{\varphi_{a,\lambda}}^{a})=
D_{g_{a}}(\eta_{a,\varepsilon}\frac{H_{a}}{\lambda^{\frac{n-2\gamma}{2}}})
+
O(\frac{\lambda^{\frac{2\gamma-n}{2}}}{y^{\gamma}r ^{1-\gamma}}) \\
_{}\quad\quad\quad\quad\;\,\, = 
\frac{M_{a}}{\lambda^{\frac{n-2\gamma}{2}}}div_{g_{a}}(y^{1-2\gamma}\nabla_{g_{
a}}\eta_{a,\varepsilon})
+
O(\frac{\max(1,\varepsilon^{1-2\gamma})}{\lambda^{\frac{n-2\gamma}{2}}y^{\gamma}
r ^{1-\gamma}}) 
$
 \item \quad
$
\overline{\varphi_{a,\lambda}}^{a}-\widetilde{\varphi_{a,\lambda}}^{a}
=
O(\frac{1}{\lambda^{\frac{n-2\gamma}{2}}}),
$
\end{enumerate}
provided \;$\varepsilon\sim \lambda^{-\frac{1}{k}}$\; for some \;$k=k(n,\gamma)$ 
sufficiently large.
\end{lem}
\begin{pf
Since \;$D_{g_{a}}\Gamma_{a}=0$, we have 
\begin{equation*}
\begin{split}
D_{g_{a}}\widetilde{\varphi_{a,\lambda}}^{a}
= &
D_{g_{a}}(\eta_{a,\varepsilon}(\hat 
\delta_{a,\lambda}-\lambda^{\frac{2\gamma-n}{2}}\Gamma_{a}))
\\
= &
D(\eta_{a,\varepsilon}(\hat 
\delta_{a,\lambda}-\frac{\lambda^{\frac{2\gamma-n}{2}}}{r ^{n-2\gamma}}))
+
(D_{g_{a}}-D)(\eta_{a,\varepsilon}(\hat 
\delta_{a,\lambda}-\frac{\lambda^{\frac{2\gamma-n}{2}}}{r ^{n-2\gamma}})) \\
& - 
D_{g_{a}}(\eta_{a,\varepsilon} \frac{H_{a}}{\lambda^{\frac{n-2\gamma}{2}}})
=
I_{1}+I_{2}+I_{3}
\end{split}
\end{equation*}
writing
$\Gamma_{a}=r ^{2\gamma-n}+H_{a}$ in $g_{a}$-normal Fermi-coordinates
and start evaluating
\begin{equation*}\begin{split
I_{1}
= &
-div (y^{1-2\gamma}\nabla
(
\eta_{a,\varepsilon}(\hat \delta_{a,\lambda}
-
\frac{\lambda^{\frac{2\gamma-n}{2}}}{r ^{n-2\gamma}})))
\\
= &
-
\partial_{y}y^{1-2\gamma}\partial_{y}\eta_{a,\varepsilon}(\hat 
\delta_{a,\lambda}-\frac{\lambda^{\frac{2\gamma-n}{2}}}{r ^{n-2\gamma}}) 
+
2y^{1-2\gamma}\nabla \eta_{a,\varepsilon}\nabla (\hat 
\delta_{a,\lambda}-\frac{\lambda^{\frac{2\gamma-n}{2}}}{r ^{n-2\gamma}})  +
y^{1-2\gamma}\Delta \eta_{a,\varepsilon}(\hat 
\delta_{a,\lambda}-\frac{\lambda^{\frac{2\gamma-n}{2}}}{r ^{n-2\gamma}}),
\end{split}\end{equation*}
where we made use of \;$D\hat \delta_{a,\lambda}=Dr^{2\gamma-n}=0$.
\eqref{cut_off} and  corollary \ref{cor_sharp_estimates} then show 
\begin{equation*}
I_{1}
=
o_{\frac{1}{1+\lambda^{2\gamma}\varepsilon^{n} 
}}(\frac{y^{1-2\gamma}}{\varepsilon^{2}  
\lambda^{\frac{n-2\gamma}{2}}})
=
O(\frac{\lambda^{\frac{2\gamma-n}{2}}}{y^{\gamma}r ^{1-\gamma}}),
\end{equation*}
whenever \;$\varepsilon\sim \lambda^{-\frac{1}{k}}$\; and \;$k\gg 1$\; is 
chosen sufficiently large. Secondly letting 
\begin{equation*}
p, \;q=1,\ldots,n+1 \;\;\text{ and } \;\;i, \;j=1,\ldots, n
\end{equation*}
we evaluate 
\begin{equation}\label{OperatorsDifference}
\begin{split}
D - D_{g_{a}}
= &
\frac{\partial_{p}}{\sqrt{g_{a}}}(\sqrt{g_{a}}g_{a}^{p,q}y^{1-2\gamma}\partial_{
q}\cdot)
-
E_{g_{a}}
-
\partial_{p}(\delta^{p,q}y^{1-2\gamma}\partial_{q}\cdot) \\
= &
\frac{\partial_{y}\sqrt{g_{a}}}{\sqrt{g_{a}}}y^{1-2\gamma}\partial_{y}
+
\frac{\partial_{i}\sqrt{g_{a}}}{\sqrt{g_{a}}}g_{a}^{i,j}y^{1-2\gamma}\partial_{j
} +
\partial_{i}((g_{a}^{i,j}-\delta^{i,j})y^{1-2\gamma}\partial_{j}\cdot)
-
E_{g_{a}} \\
= &
O(y^{n-1-2\gamma})
+
O(y^{n-2\gamma})\partial_{y}
+
O(y^{n+1-2\gamma})\nabla_{x}
+
O(y^{n+1-2\gamma})\nabla^{2}_{x},
\end{split}
\end{equation}
where we made use of \eqref{Eg_local},\eqref{flatness} and the splitting of the 
metric.
From \eqref{OperatorsDifference} and corollary \ref{cor_rough_estimates} we 
then find 
\begin{equation*}
I_{2}=O(\frac{\lambda^{\frac{2\gamma-n}{2}}}{y^{\gamma}r ^{1-\gamma}}).
\end{equation*}
Moreover \eqref{OperatorsDifference} and corollary 
\ref{cor_kernels_for_poincare_einstein_metrics} imply 

D_{g_{a}}H_{a}=(D-D_{g_{a}})r ^{2\gamma-n}=O(\frac{1}{y^{\gamma}r ^{
1-\gamma}}),
$
whence
\begin{equation*}
\begin{split}
I_{3}
=
-
\frac{H_{a}}{\lambda^{\frac{n-2\gamma}{2}}}div_{g_{a}}(y^{1-2\gamma}\nabla_{g_{a
}}\eta_{a,\varepsilon})
-
2y^{1-2\gamma}
\langle \nabla_{g_{a}} \eta_{a,\varepsilon},\frac{\nabla_{g_{a}} 
H_{a}}{\lambda^{\frac{n-2\gamma}{2}}}\rangle_{g_{a}}
+
O(\frac{\lambda^{\frac{2\gamma-n}{2}}}{y^{\gamma}r ^{1-\gamma}}).
\end{split}
\end{equation*}
Due to  the structure of \;$H_{a}$, cf. corollary 
\ref{cor_kernels_for_poincare_einstein_metrics} we have 
\begin{equation*}
H_{a}=M_{a}+O(\max( r ,r ^{2\gamma})) \text{ and }\nabla_{g_{a}}H_{a}=
O(\max(1,r ^{2\gamma-1})
\end{equation*}
with \;$M_{a}$\; denoting the constant in the expansion of the Green's 
function \;$\Gamma_a$.
Recalling \eqref{cut_off} we then get
\begin{equation*}
\begin{split}
I_{3}
=
-\frac{M_{a}}{\lambda^{\frac{n-2\gamma}{2}}}div_{g_{a}}(y^{1-2\gamma}\nabla_{g_{
a}}\eta_{a,\varepsilon})
+
O(\frac{\max(1,\varepsilon^{1-2\gamma})}{\lambda^{\frac{n-2\gamma}{2}}y^{\gamma}
r ^{1-\gamma}}).
\end{split}
\end{equation*}
Collecting terms we conclude
\begin{equation}\label{Dga_Tilde_Bubble_Estimate}
\begin{split}
D_{g_{a}}\widetilde{\varphi_{a,\lambda}}^{a}
= &
-D_{g_{a}}(\eta_{a,\varepsilon}\frac{H_{a}}{\lambda^{\frac{n-2\gamma}{2}}})
+
O(\frac{\lambda^{\frac{2\gamma-n}{2}}}{y^{\gamma}r ^{1-\gamma}}) \\
= &
-\frac{M_{a}}{\lambda^{\frac{n-2\gamma}{2}}}div_{g_{a}}(y^{1-2\gamma}\nabla_{g_{
a}}\eta_{a,\varepsilon})
+
O(\frac{\max(1,\varepsilon^{1-2\gamma})}{\lambda^{\frac{n-2\gamma}{2}}y^{\gamma}
r ^{1-\gamma}}) 
.
\end{split}
\end{equation}
This with the definition of \;$\overline{\varphi_{a,\lambda}}^{a}$ (which implies $D_{g_a}\overline{\varphi_{a,\lambda}}^{a}=0$) shows (i). (ii) follows from the maximum principle for $D_{g_a}$\; under Dirichlet boundary condition. Indeed consider

\begin{equation*}
\chi_{a,\rho}=\chi(\frac{y^{2}}{\rho^{2}}), \,\;\;\chi=1 \;\;\text{ on }\;\; [0,1] \;\;\text{ 
and \;\;
} \;\;\chi=0 \text{ on } \;\;(2,+\infty),
\end{equation*}
i.e. a cut-off function for the boundary and 
\begin{equation*}
\psi
=
\psi_{a,\lambda,\varepsilon,\rho}
=
\overline{\varphi_{a,\lambda}}^{a}
-
\widetilde{\varphi_{a,\lambda}}^{a}
-
\eta_{a,\varepsilon}\frac{H_{a}}{\lambda^{\frac{n-2\gamma}{2}}}
\pm
C\eta_{a,\rho}\frac{y^{2\gamma}\ln y}{\lambda^{\frac{n-2\gamma}{2}}}
\end{equation*}
with constants \;$\rho, \;C>0$.
We then find from \eqref{canonical_bubble},\eqref{constructed_bubble},
\eqref{Dga_Tilde_Bubble_Estimate} and
\begin{equation*}
D(y^{2\gamma}\ln y)=-\frac{2\gamma}{y},
\end{equation*}
that for suitable \;$0<\rho, \;C^{-1}\ll 1$
\begin{equation*}
\begin{cases}
D_{g_{a}}\psi
=
O(\lambda^{\frac{2\gamma-n}{2}}) \;\;\text{in} \;\;X
\\
D_{g_{a}}
\psi
\;\substack{\leq\\\geq}\;
0\;\;
\text{ in } \;\;[d_{g_{a}}(\cdot,M)\leq \rho]
\\
\psi
=
O(\lambda^{\frac{2\gamma-n}{2}})\;\;
\text{ on } \;\;M
\end{cases}
\end{equation*}
and according to proposition \ref{prop_scattering} we 
may solve
\begin{equation*}
\begin{cases}
D_{g_{a}}u=c_{1}\;\; \text{ in } \;\;X \\
u=c_{2} \;\,\;\text{ on }\,\;\; M
\end{cases}
\end{equation*}
for constants \;$c_{1}, c_{2}\in \mathbb{R}$\; with \;$u\in 
C^{\infty}(\overline X)+y^{2\gamma}C^{\infty}(\overline X)$.
\end{pf}
\subsection{Selfaction estimates for \;$v_{a, \l}$}
In this subsection we derive two selfaction estimate for \;$v_{a, \l}$. Indeed we show

\begin{lem}[The 
selfaction estimates for \;$v_{a, \l}$]$_{}$\label{selfactestv}\\
Writing in \;$g_{a}$-normal Fermi-coordinates \;$z=z_{a}=(y_{a},x_{a})=(y,x)$ 
\begin{equation*}
\Gamma_{a}=r ^{2\gamma-n}+H_{a}=r^{2\gamma-n}+M_{a}+O(\max(r ,r ^{
2\gamma})),
\end{equation*}
cf. corollary \ref{cor_kernels_for_poincare_einstein_metrics}, there holds
\begin{enumerate}[label=(\roman*)]
 \item \quad 
$
\langle v_{a,\lambda}  , v_{a,\lambda}\rangle_{P_{h}^\gamma}
= 
\langle \varphi_{a,\lambda}  , \varphi_{a,\lambda}\rangle_{P_{h_{a}}^\gamma} 
=  
c_{n, 2}^\gamma-c_{n, \gamma}^*\rfrac{M_{a}}{\lambda^{n-2\gamma}}+o_{\frac{1}{\lambda}}(\frac{
1}{\lambda^{n-2\gamma}}),
$
 \item \quad 
$
\int_{M}v_{a,\lambda}^{\frac{2n}{n-2\gamma}}dV_{h}
=
\int_{M}\varphi_{a,\lambda}^{\frac{2n}{n-2\gamma}}dV_{h_{a}}
=
c_{n, 1}^\gamma+o_{\frac{1}{\lambda}}(\lambda^{2\gamma-n})
$
\end{enumerate}
provided \;$\varepsilon\sim \lambda^{-\frac{1}{k}}$\; for \;$k\in \mathbb{N}$\; sufficiently 
large.
\end{lem}
\begin{pf}
Using \eqref{properties_of_tilde_phi} we calculate the quadratic form
\begin{equation*}\begin{split}
\frac{1}{d_\gamma^*}\langle \varphi_{a,\lambda}  ,\varphi_{a,\lambda}\rangle_{P_{h_{a}}^\gamma}
= &
\frac{1}{d_\gamma^*}
\int_{M}P_{g_{a}}\varphi_{a,\lambda}  \varphi_{a,\lambda}dV_{h_{a}} \\
= & 
-\lim_{y\rightarrow 0}\int_{M}y^{1-2\gamma}\partial_{y}\widetilde{\varphi_{a,\lambda}}^{a}\widetilde
{\varphi_{a,\lambda}}^{a} dV_{h_{a}} 
-
\lim_{y\rightarrow 0}\int_{M}y^{1-2\gamma}\partial_{y}(\overline{\varphi_{a,\lambda}}^{a}
-\widetilde{\varphi_{a,\lambda}}^{a})\widetilde{\varphi_{a,\lambda}}^{a} 
dV_{h_{a}}  \\
= &\frac{c_{n, \gamma}}{d_\gamma^*}
\int_{M} 
\eta_{a,\varepsilon}\delta_{a,\lambda}^{\frac{n+2\gamma}{n-2\gamma}}\varphi_{a,
\lambda}dV_{h_{a}} 
+
\int_{X}div_{g_{a}}(y^{1-2\gamma}\nabla_{g_{a}}(\overline{\varphi_{a,\lambda}}^{
a}-\widetilde{\varphi_{a,\lambda}}^{a})\widetilde{\varphi_{a,\lambda}}^{a})dV_
{g_{a}}
\end{split}\end{equation*}
and from \eqref{delta_boundary},\eqref{bubble_boundary} and 
$h_{a}=g_{a}\lfloor_{M}=\delta$\; close to \;$a$ \;we easily find
\begin{equation}\label{super_quadratic_bubble_energy_expansion}
\begin{split}
\int_{M}\eta_{a,\varepsilon}\delta_{a,\lambda}^{\frac{n+2\gamma}{n-2\gamma}}
\varphi_{a,\lambda}dV_{h_{a}}
= &
\int_{\mathbb{R}^{n}}\delta_{0,\lambda}^{\frac{2n}{n-2\gamma}}
+
O(\int_{\mathbb{R}^{n}\setminus 
B_{\varepsilon}(0)}\delta_{0,\lambda}^{\frac{2n}{n-2\gamma}})
=
c_{n,1}^{\gamma}+O(\frac{1}{(\varepsilon \lambda)^{n}}),
\end{split}
\end{equation}
whence by integration by parts and any choice \;$\varepsilon \sim 
\lambda^{-\frac{1}{k}}, \,k$\; sufficiently large
\begin{equation*}\begin{split}
\frac{1}{d_\gamma^*}\langle \varphi_{a,\lambda}  , \varphi_{a,\lambda}\rangle_{P_{h_{a}}^\gamma}
= &
\frac{c_{n, 2}^\gamma}{d_\gamma^*} 
+
\int_{X} y^{1-2\gamma}
\langle 
\nabla_{g_{a}}(\overline{\varphi_{a,\lambda}}^{a}-\widetilde{\varphi_{a,\lambda}
}^{a}),
\nabla_{g_{a}}\widetilde{\varphi_{a,\lambda}}^{a}\rangle_{g_{a}} 
+
E_{g_{a}}(\overline{\varphi_{a,\lambda}}^{a}-\widetilde{\varphi_{a,\lambda}}^{a}
)\widetilde{\varphi_{a,\lambda}}^{a} dV_{g_{a}}\\
&  -
\int_{X}D_{g_{a}}(\overline{\varphi_{a,\lambda}}^{a}-\widetilde{\varphi_{a,
\lambda}}^{a})\widetilde{\varphi_{a,\lambda}}^{a}dV_{g_{a}}
+
o_{\frac{1}{\lambda}}(\lambda^{2\gamma-n})
\\
= &
\frac{c_{n, 2}^\gamma}{d_\gamma^*}
+
\int_{X}div_{g_{a}}(y^{1-2\gamma}\nabla_{g_{a}}\widetilde{\varphi_{a,\lambda}}^{
a}(\overline{\varphi_{a,\lambda}}^{a}-\widetilde{\varphi_{a,\lambda}}^{a})) 
dV_{g_{a}} \\
& -
\int_{X}D_{g_{a}}(\overline{\varphi_{a,\lambda}}^{a}-\widetilde{\varphi_{a,
\lambda}}^{a})\widetilde{\varphi_{a,\lambda}}^{a} dV_{g_{a}}   
+
\int_{X} 
D_{g_{a}}\widetilde{\varphi_{a,\lambda}}^{a}(\overline{\varphi_{a,\lambda}}^{a}
-\widetilde{\varphi_{a,\lambda}}^{a})dV_{g_{a}}
+
o_{\frac{1}{\lambda}}(\lambda^{2\gamma-n})\\
= &
I_{1}+\ldots+I_{5}.
\end{split}\end{equation*}
Clearly \;$I_{2}=0$\;  and from lemma \ref{lem_bubble comparsion} (i) we obtain
\begin{equation*}\begin{split}
I_{3}
= &
-\int_{X}D_{g_{a}}(\eta_{a,\varepsilon}\frac{H_{a}}{\lambda^{\frac{n-2\gamma}{2}
}})\widetilde{\varphi_{a,\lambda}}^{a}dV_{g_{a}}
+
o_{\frac{1}{\lambda}}(\frac{1}{\lambda^{n-2\gamma}})
,
\end{split}\end{equation*}
since, whenever we choose \;$\varepsilon\sim \lambda^{-\frac{1}{k}}$, then
\begin{equation*}
\begin{split}
\int_{B_{2\varepsilon}^{+}}
\frac{\widetilde{\varphi_{a,\lambda}}^{a}}{y^{\gamma} r^{1-\gamma}}
\leq &
\frac{C}{\lambda^{\frac{n+2\gamma}{2}}}\int_{B_{2\varepsilon \lambda}^{+}}
\frac{1}{y^{\gamma}(y^{2}+\vert x 
\vert)^{\frac{1-\gamma}{2}}}(\frac{1}{1+y^{2}+\vert x 
\vert^{2}})^{\frac{n-2\gamma}{2}} \\
\leq & 
\frac{C}{\lambda^{\frac{n+2\gamma}{2}}}\int^{\varepsilon\lambda}_{0}\frac{dy}{y^
{ \gamma } }
\int_{Q^{n}_{\varepsilon\lambda}}\frac{1}{\vert x 
\vert^{1-\gamma}}(\frac{1}{1+\vert x \vert^{2}})^{\frac{n-2\gamma}{2}}
=
\frac{\varepsilon^{2\gamma}}{\lambda^{\frac{n-2\gamma}{2}}}.
\end{split}
\end{equation*}
Moreover from lemma \ref{lem_bubble comparsion} we easily find 
\begin{equation*}
D_{g_{a}}\widetilde{\varphi_{a,\lambda}}^{a}=O(\frac{\varepsilon^{-2\gamma}}{
\lambda^{\frac{n-2\gamma}{2}}y^{\gamma}r^{1-\gamma}} )
\text{ and }
\overline{\varphi_{a,\lambda}}^{a}-\widetilde{\varphi_{a,\lambda}}^{a}=O(\frac{1
} {\lambda^{\frac{n-2\gamma}{2}}}),
\end{equation*}
whence \;$I_{4}=O(\frac{\varepsilon^{n-2\gamma}}{\lambda^{n-2\gamma}})$. 
Collecting terms we obtain
\begin{equation*}\begin{split}
\frac{1}{d_\gamma^*}\langle \varphi_{a,\lambda}  , &\varphi_{a,\lambda}\rangle_{P_{h_{a}}^\gamma}
= 
\frac{c_{n, 2}^\gamma}{d_\gamma^*}
-
\int_{X}D_{g_{a}}(\eta_{a,\varepsilon}\frac{H_{a}}{\lambda^{\frac{n-2\gamma}{2}}
})\widetilde{\varphi_{a,\lambda}}^{a}dV_{g_{a}}
+
o_{\frac{1}{\lambda}}(\frac{1}{\lambda^{n-2\gamma}}).
\end{split}
\end{equation*}
From lemma \ref{lem_bubble comparsion} (i) we then find
\begin{equation*}
\begin{split}
\int_{X}D_{g_{a}}(\eta_{a,\varepsilon}\frac{H_{a}}{\lambda^{\frac{n-2\gamma}{2}}
})\widetilde{\varphi_{a,\lambda}}^{a}dV_{g_{a}}
= &
\frac{M_{a}}{\lambda^{\frac{n-2\gamma}{2}}}
\int_{X}div_{g_{a}}(y^{1-2\gamma}\nabla_{g_{a}}\eta_{a,\varepsilon})\widetilde{
\varphi_{a,\lambda}}^{a}dV_{g_{a}}  
+
O(\frac{\max(\varepsilon,\varepsilon^{2\gamma})}{\lambda^{n-2\gamma}})
,
\end{split}
\end{equation*}
where integrating by parts using \eqref{cut_off} 
\begin{equation*}
\begin{split}
\int_{X}div_{g_{a}}(y^{1-2\gamma}\nabla_{g_{a}}\eta_{a,\varepsilon})\widetilde{
\varphi_{a,\lambda}}^{a}dV_{g_{a}} 
= &
-\int_{X}div_{g_{a}}(y^{1-2\gamma}\eta_{a,\varepsilon}\nabla_{g_{a}}\widetilde{
\varphi_{a,\lambda}}^{a})dV_{g_{a}} \\
& +
\int_{X}\eta_{a,\varepsilon}div_{g_{a}}(y^{1-2\gamma}\nabla_{g_{a}}\widetilde{
\varphi_{a,\lambda}}^{a})dV_{g_{a}}
\end{split}
\end{equation*}
with the latter integral being of order  \;$O(\frac{\varepsilon^{n-2\gamma}}{\lambda^{n-2\gamma}})$, since
\begin{equation*}
\begin{split}
div_{g_{a}}(y^{1-2\gamma}\nabla_{g_{a}}\widetilde{\varphi_{a,\lambda}}^{a})
= &
-D_{g_{a}}\widetilde{\varphi_{a,\lambda}}^{a}-E_{g_{a}}\widetilde{\varphi_{a,
\lambda}}^{a} \\
= &
-
\frac{M_{a}}{\lambda^{\frac{n-2\gamma}{2}}}div_{g_{a}}(y^{1-2\gamma}\nabla_{g_{a
}}\eta_{a,\varepsilon})
+
O(\frac{\max(1,\varepsilon^{1-2\gamma})}{\lambda^{\frac{n-2\gamma}{2}}y^{\gamma}
r_{a}^{1-\gamma}}) 
= 
O(\frac{\varepsilon^{-2\gamma}\lambda^{\frac{2\gamma-n}{2}}}{y^{\gamma}r_{a}^{
1-\gamma}})
\end{split}
\end{equation*}
due to \eqref{Eg_local},\eqref{flatness},\eqref{Dga_Tilde_Bubble_Estimate}.
Collecting terms we derive
\begin{equation*}\begin{split}
\frac{1}{d_\gamma^*}\langle \varphi_{a,\lambda}  , &\varphi_{a,\lambda}\rangle_{P_{h_{a}}^\gamma}
= 
\frac{c_{n, 2}^\gamma}{d_\gamma^*}
-
\frac{M_{a}}{\lambda^{\frac{n-2\gamma}{2}}}\int_{X}div_{g_{a}}(y^{1-2\gamma}
\eta_{a,\varepsilon}\nabla_{g_{a}}\widetilde{\varphi_{a,\lambda}}^{a})dV_{g_{a
}}
+
o_{\frac{1}{\lambda}}(\frac{1}{\lambda^{n-2\gamma}}),
\end{split}
\end{equation*}
provided we choose \;$\varepsilon\sim \lambda^{-\frac{1}{k}}$\; and \;$k\gg1$. Recalling \eqref{eq:relationcy}, from 
\eqref{properties_of_tilde_phi} we then find 
\begin{equation*}\begin{split}
\langle \varphi_{a,\lambda}  , &\varphi_{a,\lambda}\rangle_{P_{h_{a}}^\gamma}
= 
c_{n, 2}^\gamma
-
d_\gamma^*\frac{M_{a}}{\lambda^{\frac{n-2\gamma}{2}}}\int_{M}\eta^{2}_{a,\varepsilon}
\delta_{a,\lambda}^{\frac{n+2\gamma}{n-2\gamma}}dV_{h_{a}}
+
o_{\frac{1}{\lambda}}(\frac{1}{\lambda^{n-2\gamma}}).
\end{split}
\end{equation*}
and arguing as for \eqref{super_quadratic_bubble_energy_expansion} we conclude (recalling \eqref{dfcngs})
\begin{equation*}\begin{split}
\langle \varphi_{a,\lambda}  , &\varphi_{a,\lambda}\rangle_{P_{h_{a}}^\gamma}
= 
c_{n, 2}^\gamma
-c_{n, \gamma}^*
\frac{M_{a}}{\lambda^{n-2\gamma}}
+
o_{\frac{1}{\lambda}}(\frac{1}{\lambda^{n-2\gamma}}).
\end{split}
\end{equation*}
Likewise
\begin{equation*}
\begin{split}
\int_{M}\varphi_{a,\lambda}^{\frac{2n}{n-2\gamma}}dV_{h_{a}}
= &
\int_{B_{\varepsilon}(a)}\delta_{a,\lambda}^{\frac{2n}{n-2\gamma}}dV_{h_{a}
}
+
\int_{M\setminus 
B_{\varepsilon}(a)}O((\frac{1}{\lambda^{\frac{n-2\gamma}{2}}\varepsilon^{
n-2\gamma}})^{\frac{2n}{n-2\gamma}})dV_{h_{a}} \\
= &
\int_{\mathbb{R}^{n}}\delta_{a,\lambda}^{\frac{2n}{n-2\gamma}}dV_{h_{a}}
+
\int_{\mathbb{R}^{n}\setminus 
B_{\varepsilon}(a)}\delta_{a,\lambda}^{\frac{2n}{n-2\gamma}}
+
O(\frac{\varepsilon^{-2n}}{\lambda^{n}}) \\
= &
c_{n, 1}^\gamma
+
O(\frac{\varepsilon^{-n}}{\lambda^{n}})
+
O(\frac{\varepsilon^{-2n}}{\lambda^{n}})
=
c_{n, 1}^\gamma+o_{\frac{1}{\lambda}}(\lambda^{2\gamma-n})
\end{split}
\end{equation*}
provided \;$\varepsilon\sim \lambda^{-\frac{1}{k}}$\; for $\;k\gg 1$ sufficiently 
large. The assertions now follow from 
conformal covariance of \;$P_{h}^\gamma$, see \cite{cg} or  cf. subsection \eqref{confnonhom}, and the one of volume forms.\end{pf}
\begin{df}\label{fracmass}
In analogy to the classical Yamabe problem, we define the fractional mass  map\;$\mathcal{M}_\gamma$\; as 
$$
\mathcal{M}_{\gamma}(a)=M_a,
$$
where \;$M_a$\; is given by lemma \ref{lem_bubble comparsion},and the fractional minimal mass as follows
$$
\mathcal{M}^{\min}_\gamma=\min_{a\in M}\mathcal{M}_{\gamma}(a).
$$
\end{df}
\vspace{10pt}

\noindent
\section{Variational and algebraic topological argument}\label{varagtop}
In this section, we present the proof of Theorem \ref{eq:thm}. Therefore we will assume in this section that we are under the assumptions of Theorem \ref{eq:thm}. Furthermore  because of Remark \ref{eq:half} and the work of Gonzalez-Qing\cite{gq}, we will assume also that \;$\gamma\neq \frac{1}{2}$\; and \; $\mathcal{Y}^\gamma(M, [h])>0$, and hence the Green's function  \;$G_h^\gamma>0$.   With the above agreement, we are going to carry out the variational and algebraic topological argument for existence.  
\subsection{Variational principle in a non-compact setting}\label{var}
In this subsection, we extend the classical variational principle to this non-compact setting.  Clearly our bubbles \;$v_{a, \l}$\; can be used to replace the standard bubbles in the analysis of diverging Palais-Smale (PS) sequences of \;$\mathcal{E}^\gamma_{h}$\; in \cite{fanggonz}. Relying on this fact we derive a deformation lemma that takes into account the bubbling phenomena via our bubbles \;$v_{a, \l}$. To do so,  first for \;$p\in \mathbb{N}$, we define
\begin{equation*
W_p=W_{p, h}^\gamma:=\{u\in W^{\gamma, 2}_+(M, h):\;\;\mathcal{E}^{\gamma}_{h}(u)\leq (p+1)^{\frac{2\gamma}{n}}\mathcal{Y}_{\gamma}(S^n)\}.
\end{equation*}
Next  we introduce the {\em neighborhood of potential critical points at infinity} of  \;$\mathcal{E}^\gamma_h$. Precisely, for \;$p\in \mathbb{N}^*$, $0<\varepsilon\leq \varepsilon_0$ (for some small \;$\varepsilon_{0}>0$),  we define \;$V(p, \varepsilon)$\; the \;$(p, \varepsilon)$-neighborhood of potential critical points at infinity of \;$\mathcal{E}^\gamma_{h}$, namely
\begin{equation*
\begin{split}
V(p, \varepsilon):=\{u\in W^{
\gamma, 2}(M,  h): &\;\;\exists a_1, \cdots, a_{p}\in M,\;\;\alpha_1, \cdots, \alpha_{p}>0, \;\l_1, \cdots,\l_{p}\geq \frac{1}{\varepsilon},\\ &\;\; \Vert u-\sum_{i=1}^{p}\alpha_i\varphi_{a_i, \l_i}\Vert\leq \varepsilon, \;\;\frac{\alpha_i}{\alpha_j}\leq \nu_0\;\;\text{and}\;\;\varepsilon_{i, j}\leq \varepsilon, \;\;i\neq j=1, \cdots, p\},
\end{split}
\end{equation*}
where $\Vert\cdot\Vert$\; denotes the standard \;$W^{\gamma, 2}$-norm, $\varepsilon_{i, j}:=\varepsilon_{i, j}(A_{i, j}, \bar \l)$\; with \;$A_{i, j}:=(a_i, a_{j})$, $\bar \l:=(\l_i,\l_{j})$. 
\vspace{6pt}

\noindent
Concerning the sets \;$V(p, \varepsilon)$, for every \;$p\in \mathbb{N}^*$, we have that there exists \;$0<\varepsilon_p\leq\varepsilon_0$\; such that for every \;$0<\varepsilon\leq \varepsilon_p$, we have that
\begin{equation}\label{eq:mini}
\begin{cases}
\forall u\in V(p, \varepsilon)\;\; \text{the minimization problem}\;\;\min_{B_{C_1\varepsilon}^{p}}\Vert u-\sum_{i=1}^{p}\alpha_i\varphi_{a_i, \l_i}\Vert \\
\text{has a solution, which is unique up to permutations,}
\end{cases}
\end{equation}
where \;$B^{p}_{C_1\varepsilon}$\; is defined as follows
\begin{equation*}
\begin{split}
B_{C_1\varepsilon}^{p}:=\{(\bar\alpha, A, \bar \l)\in \mathbb{R}^{p}_+\times M^p\times (0, +\infty)^{p}: \;\;&\l_i\geq \frac{1}{\varepsilon}, i=1, \cdots, p,\\ &\frac{\alpha_i}{\alpha_j}\leq \nu_0\;\; \text{and}\;\ \varepsilon_{i, j}\leq C_1\varepsilon, i\neq j=1, \cdots, p\}
\end{split}
\end{equation*}
and \;$C_1>1$\; and \;$\nu_0$\; is as in proposition \ref{eq:baryest}. Furthermore  we define the selection map  as follows
\begin{equation*
s_{p}: V(p, \varepsilon)\longrightarrow ( M)^p/\sigma_p
:
u\longrightarrow s_{p}(u)=A
\;\,\text{and} \,\;A\;\;\text{is given by}\;\,\eqref{eq:mini}.
\end{equation*}
\vspace{6pt}

\noindent
Now having introduced the neighborhoods of potential critical points at infinity of  \;$\mathcal{E}^\gamma_{\hat h}$, we are ready to present our deformation lemma which follows from the same arguments as for its counterparts in classical application of the barycenter technique for existence of Bahri-Coron\cite{bc} and the fact that the \;$v_{a, \l}$'s can replace the standard bubbles in the analysis of diverging (PS) sequences of \; $\mathcal{E}^\gamma_{\hat h}$. Indeed, we have the result which was the aim of this subsection.
\begin{lem}\label{eq:classicdeform}
Assuming that \;$\mathcal{E}^\gamma_{h}$ \;has no critical points, then for every \;$p\in \mathbb{N}^*$, up to taking \;$\varepsilon_p$\; smaller (where \;$\varepsilon_p$\; is given by \eqref{eq:mini}), we have that for every\; $0<\varepsilon\leq \varepsilon_p$, there holds \;$(W_p,\; W_{p-1})$\; retracts by deformation onto \;$(W_{p-1}\cup A_p, \;W_{p-1})$\; with \;$V(p, \;\tilde \varepsilon)\subset A_p\subset V(p, \;\varepsilon)$\; where \;$0<\tilde \varepsilon<\frac{\varepsilon}{4}$\; is a very small positive real number and depends on \;$\varepsilon$.
\end{lem}
\subsection{Algebraic topological argument}\label{agtop}
In this subsection, we present the algebraic topological argument or barycenter technique of Bahri-Coron\cite{bc}  for existence to prove  Theorem \ref{eq:thm}. In order to do that  we start by establishing the sharp \;$\mathcal{E}_h^\gamma$-energy estimate for \;$v_{a, \l}$.
\begin{lem}[Sharp \;$\mathcal{E}_h^\gamma$-energy estimate for \;$v_{a, \l}$]${}$\label{sharpenergy}\\
There holds
$$
\mathcal{E}_h^\gamma(v_{a, \l})=\mathcal{Y}^\gamma(S^n)\left(1- c_{n, \gamma}^*\frac{\mathcal{M}_{\gamma}(a)}{\l^{n-2\gamma}}\right)\left(1+o_{\frac{1}{\l}}(1)\right),
$$
where \;$c_{n, \gamma}^*$ is as in \eqref{dfcngs}.
\end{lem}
\vspace{6pt}

\noindent
\begin{pf}
The proof is a direct application of lemma \ref{selfactestv}.
\end{pf}
\vspace{6pt}

\noindent
Next  we  define for \;$p\in \mathbb{N}^*$ and $\lambda >0$
\begin{equation*
f_p(\l): \;B_p(M)\longrightarrow W^{\gamma, 2}_+(M)
:
\sigma=\sum_{i=1}^p\alpha_i\d_{a_i}\in B_p(M)
\longrightarrow
f_p(\l)(\sigma)
=
\sum_{i=1}^p\alpha_i v_{a_i, \l}
\end{equation*}
Using lemma \ref{interactest} and lemma \ref{sharpenergy}, we are going to prove that at \textit{true infinity}, the following quantity 
\begin{equation*
\D^{\l}_p\mathcal{E}^{h}_{\gamma}(\sigma):=\l^{n-2\gamma}\left(\frac{\mathcal{E}_\gamma^{\hat h}(f_p(\sigma))}{p^{\frac{2\gamma}{n}}{}\mathcal{Y}_\gamma(S^n)}-1\right), \;\;\sigma:=\sum_{i=1}^p\alpha_i\d_{a_i}\in B_p(M)
\end{equation*}
has at most a linear growth in \;$p$\; with a negative slope and this uniformly in \;$\sigma$ and $\l$, namely at infinity
\begin{equation}\label{eq:ling1}
\sup_{\l>1, \sigma\in B_p(M)}\D^{\l}_p\mathcal{E}^{\hat h}_{\gamma}(\sigma)\leq -cp+C.
\end{equation}
 About estimate \eqref{eq:ling1}, we precisely show the following proposition.
\begin{pro}\label{eq:baryest}
There exists  \;$\nu_0>1$\; such that for every \;$p\in \mathbb{N}^*$, $p\geq 2$ and every \;$0<\varepsilon\leq \varepsilon_0$, there exists \;$\l_p:=\l_p(\varepsilon)$ such that for every \;$\l\geq \l_p$ and for every $\sigma=\sum_{i=1}^p\alpha_i\delta_{a_i}\in B_p(M)$, we have
\begin{enumerate}[label=(\roman*)]
 \item
If \;$\sum_{i\neq j}\varepsilon_{i, j}> \varepsilon$\; or there exist \;$i_0\neq j_0$\; such that \;$\frac{\alpha_{i_0}}{\alpha_{j_0}}>\nu_0$, then
 $$
\mathcal{E}^\gamma_{h}(f_p(\l))(\sigma))\leq p^{\frac{2\gamma}{n}}\mathcal{Y}(S^n).
$$ 
 \item
If \;$\sum_{i\neq j}\varepsilon_{i, j}\leq \varepsilon$\; and for every \;$i\neq j$\; we have \;$\frac{\alpha_{i}}{\alpha_j}\leq\nu_0$, then
$$
\mathcal{E}^\gamma_{h}(f_p(\l))(\sigma))\leq p^{\frac{2\gamma}{n}}\mathcal{Y}^{\gamma}(S^n)\left(1- c_{n, \gamma}^*\frac{\mathcal{M}^{\min}_{\gamma}}{\l^{n-2}}-c_{h}^\gamma\frac{(p-1)}{\l^{n-2}}\right),
$$
where \;$c_{h}^{\gamma}=\frac{c^\gamma_{n, 3}}{4c^1_{n, 1}}\min_{(M^2)^*}G_h^{\gamma}$.
\end{enumerate}

\end{pro}
\begin{pf}
The proof is the same as the one of proposition 3.1 in \cite{martndia2} using lemma \ref{interactest}, lemma \ref{selfactestv}, lemma \ref{sharpenergy}, lemma \ref{lem_higher_interaction_estimates}.
\end{pf}
\vspace{6pt}

\noindent
Now we start transporting the topology of the manifold \;$M$\; into the sublevels of the Euler-Lagrange functional \;$\mathcal{E}_h^\gamma$\; by bubbling via \;$v_{a, \l}$. Recalling \eqref{orientation_classes} we have
\begin{lem}\label{eq:nontrivialf1}
Assuming that \;$\mathcal{E}_{\hat h}^{\gamma}$\; has no critical points and \:$0<\varepsilon\leq  \varepsilon_1$, then up to taking \;$\varepsilon_1$\; smaller and \;$\l_1$\; larger, we have that for every \;$\l\geq \l_1$, there holds
$$
f_1(\l): \;(B_1(M),\; B_0(M))\longrightarrow (W_1, \;W_0)
$$
is well defined and satisfies
$$
(f_1(\l))_*(w_1)\neq 0\;\;\text{in}\;\;H_{n}(W_1, \;W_0).
$$
\end{lem}
\begin{pf}
The proof follows from the same arguments as the ones used in the proof of  lemma 4.2  in \cite{martndia1} by using the selection map \;$s_1$\; lemma \ref{eq:classicdeform} and lemma \ref{sharpenergy}.
\end{pf}
\vspace{6pt}

\noindent
Next  we use the previous lemma and {\em pile up masses}  by bubbling  via \;$v_{a, \l}$\, in a recursive way. 
\begin{lem}\label{eq:nontrivialrecursive}
Assuming that \;$\mathcal{E}^{\gamma}_{\hat h}$\; has no critical points and \;$0<\varepsilon\leq  \varepsilon_{p+1}$, then up to taking \;$\varepsilon_{p+1}$\; smaller, and \;$\l_p$\; and \;$\l_{p+1}$\; larger, we have that for every \;$\l\geq \max\{\l_p, \l_{p+1}\}$, there holds
$$
f_{p+1}(\l): (B_{p+1}(M),\; B_{p}(M))\longrightarrow (W_{p+1}, \;W_{p})
$$
and 
$$
f_p(\l): (B_p(M), \;B_{p-1}(M))\longrightarrow (W_p, \; W_{p-1})
$$
are well defined and satisfy
$$(f_p(\l))_*(w_p)\neq 0\;\; \text{in}\;\; \;\;H_{np-1}(W_p, \;W_{p-1})$$ implies
$$(f_{p+1}(\l))_*(w_{p+1})\neq 0\;\; \text{in} \;\;H_{n(p+1)-1}(W_{p+1}, \;W_{p}).$$
\end{lem}
\begin{pf}
The proof follows from the same arguments as the ones used in the proof of lemma 4.3  in \cite{martndia1}, by using the selection map \;$s_p$\; lemma \ref{eq:classicdeform} and proposition \ref{eq:baryest}.
\end{pf}
\vspace{6pt}

\noindent
Finally, we use the strength of proposition \ref{eq:baryest} - namely point (ii) - to give a criterion ensuring that the recursive process of {{\em pilling up} mass given by lemma \ref{eq:nontrivialrecursive} must stop after a very large number of steps.
\begin{lem}\label{eq:largep}
Setting \;$$p^{\gamma}_{h}:=[1-c_{n, \gamma}^*\frac{\mathcal{M}^{\min}_{\gamma}}{c^{\gamma}_{h}} ]+2,$$ we have that \;$\forall\;\;0<\varepsilon\leq\varepsilon_0$\; and \;$\forall\l\geq\l_{p_{\gamma}^{h}}$ there holds 
$$
f_{p^{\gamma}_{h}}(\l)(B_{p^{\gamma}_{h}}(M))\subset W_{p_h^\gamma-1}.
$$
\end{lem}
\begin{pf}
The proof is a direct application of proposition \ref{eq:baryest} .
\end{pf}
\vspace{6pt}

\noindent
\begin{pfn} {of Theorem \ref{eq:thm}}\\
It follows by a contradiction argument from lemma \ref{eq:nontrivialf1}-lemma\ref{eq:largep}.
\end{pfn}

\section{Appendix}\label{appendix}
In this section we collect  the proves of lemmas \ref{lem_standard interaction on Rn},\ref{lem_higher_interaction_estimates} and of  some technical estimates used in this paper. We start with providing the proof of lemma \ref{lem_standard interaction on Rn}.
\vspace{6pt}

\noindent


\begin{pfn}{ of lemma \ref{lem_standard interaction on Rn}}
\begin{enumerate}[label=(\roman*)]
 \item 
Let us write \;$\delta_{i}=\delta_{a_{i},\lambda_{i}},\delta_{j}=\delta_{a_{j},\lambda_{j}}$ \;for simplicity and agree that whenever we omit the domain of integration, it is understood to be \;$\mathbb{R}^n$. Due to
\begin{equation*}\begin{split}
\int \delta_{i}^{\frac{n+2\gamma}{n-2\gamma}}\delta_{j}
= &
\frac{1}{c_{n, \gamma}}\int (-\Delta)^{\gamma}\delta_{i}\delta_{j}=\frac{1}{c_{n, \gamma}}\int 
\delta_{i}(-\Delta)^{\gamma}\delta_{j}
\int \delta_{i}\delta_{j}^{\frac{n+2}{n-2}}
\end{split}
\end{equation*} 
we may assume \;$ \frac{1}{\lambda_{i}} \leq  \frac{1}{\lambda_{j}}$. We then have
\begin{equation*}\begin{split}
\int \delta_{i}^{\frac{n+2\gamma}{n-2\gamma}} \delta_{j}
= &
\int
(\frac{1}{1+r^{2}})^{\frac{n+2\gamma}{2}}
(\frac{1}{\frac{\lambda_{i} }{ \lambda_{j} }+\lambda_{i}\lambda_{j}\vert 
\frac{x}{\lambda_{i}}+a_{i}-a_{j}\vert^{2}})^{\frac{n-2\gamma}{2}}.
\end{split}\end{equation*}
Since \;$\varepsilon_{i,j}^{\frac{2}{2\gamma-n}}
\sim 
\lambda_{i}\lambda_{j}\vert a_{i}-a_{j}\vert^{2}
\;\text{ or }\;
\varepsilon_{i,j}^{\frac{2}{2\gamma-n}}
\sim
\frac{\lambda_{i}}{\lambda_{j}},
$\; and  may expand on
\begin{equation*}\begin{split}
\mathcal{A}
=
\left[
\vert  \frac{x}{\lambda_{i}}\vert \leq \epsilon\vert a_{i}-a_{j}\vert
\right]
\cup
\left[
\vert  \frac{x}{\lambda_{i}}\vert \leq \epsilon  \frac{1}{\lambda_{j}} 
\right]
\end{split}\end{equation*}
for \;$\epsilon>0$\; sufficiently small
\begin{equation*}\begin{split}
(
\frac{\lambda_{i} }{ \lambda_{j} }
& +
\lambda_{i}\lambda_{j}\vert \frac{x}{\lambda_{i}} 
+a_{i}-a_{j}\vert^{2})
)^{\frac{2\gamma-n}{2}} \\
= & 
(
\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}
\vert a_{i}-a_{j}\vert^{2}
)^{\frac{2\gamma-n}{2}} 
+
(2\gamma-n)
\frac
{
\langle a_{i}-a_{j},\lambda_{j}x\rangle
+
O(\frac{ \lambda_{j} }{ \lambda_{i} }\vert x\vert^{2})
}
{
(
\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}\vert a_{i}-a_{j}\vert^{2}
)^{\frac{n+2-2\gamma}{2}}
}.
\end{split}\end{equation*}
Thus  \;$
\int \delta_{i}^{\frac{n+2}{n-2}}\delta_{j}
= 
\sum_{k=1}^{4}I_{k}
$\; with 
\begin{enumerate}[label=(\roman*)]
 \item \quad
$
I_{1}
= 
\frac
{1}
{(\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}
\vert a_{i}-a_{j}\vert^{2}
)^{\frac{n-2\gamma}{2}}
}
\int_{\mathcal{A}}\frac{1}{(1+r^{2})^{\frac{n+2\gamma}{2}}}
$
 \item \quad
$
I_{2}
=
\frac
{2\gamma-n}
{(\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}
\vert a_{i}-a_{j}\vert^{2}
)^{\frac{n+2-2\gamma}{2}}
}
\int_{\mathcal{A}} \frac{\langle a_{i}-a_{j}, 
\lambda_{j}x\rangle}{(1+r^{2})^{\frac{n+2\gamma}{2}}}
$
 \item \quad
$
I_{3}
= 
\frac
{1}
{(\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}
\vert a_{i}-a_{j}\vert^{2}
)^{\frac{n+2-2\gamma}{2}}
}
\int_{\mathcal{A}}
\frac{O(\frac{ \lambda_{j} }{ \lambda_{i} }\vert x 
\vert^{2})}{(1+r^{2})^{\frac{n+2\gamma}{2}}}
$
 \item \quad
$
I_{4}
= 
\underset{\mathcal{A}^{c}}{\int} (\frac{ 1}{1+r^{2}})^{\frac{n+2\gamma}{2}}
(\frac{1}{\frac{\lambda_{i} }{ \lambda_{j} }+\lambda_{i}\lambda_{j}\vert 
\frac{x}{\lambda_{i}}+a_{i}-a_{j}\vert^{2}})^{\frac{n-2\gamma}{2}}.
$
\end{enumerate}
We get
\begin{equation*}\begin{split}
I_{1}
= &
\frac
{b_{1}}
{(\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}
\vert a_{i}-a_{j}\vert^{2}
)^{\frac{n-2\gamma}{2}}
}
+
O((\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}\varepsilon_{i,j}^{\frac{n}{n-2\gamma}})
\end{split}\end{equation*}
with \;$b_{1}=\int (\frac{1}{1+r^{2}})^{\frac{n+2\gamma}{2}}=c_{n,3}^{\gamma}$,  $I_{2}=0$\;  and \;$
I_{3}
= 
O((\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}\varepsilon_{i,j}^{\frac{n}{n-2\gamma}})
$. Moreover  
\begin{equation*}
\begin{split}
I_{4}
\leq 
\frac{C}{(\frac{\lambda_{i}}{\lambda_{j}})^{\frac{n+2\gamma}{2}}}
=
O((\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}\varepsilon_{i,j}^{\frac{n}{n-2\gamma}})\;\;
\text{ in case }\;\;\varepsilon_{i,j}^{\frac{2}{2\gamma-n}}\sim 
\frac{\lambda_{i}}{\lambda_{j}}.
\end{split}
\end{equation*} 
Otherwise we may assume \;$\frac{\lambda_{j}}{\lambda_{i}}\leq 
\frac{\lambda_{i}}{\lambda_{j}}\ll \lambda_{i}\lambda_{j}\vert 
a_{i}-a_{j}\vert^{2}$\; and decompose 
\begin{equation*}
\begin{split}
\mathcal{A}^{c}
\subseteq &
\mathcal{B}_{1}\cup \mathcal{B}_{2},
\end{split}
\end{equation*}
where for a sufficiently large constant $E>0$
\begin{equation*}
\begin{split}
\mathcal{B}_{1}
= &
\left[\epsilon \vert a_{i}-a_{j}\vert
\leq 
\vert \frac{x}{\lambda_{i}}\vert 
\leq 
E\vert a_{i}-a_{j}\vert\right
] \\
\end{split}
\end{equation*} 
and
\begin{equation*}
\begin{split}
\mathcal{B}_{2}
= &
\left[
E\vert a_{i}-a_{j}\vert
\leq 
\vert \frac{x}{\lambda_{i}}\vert 
<\infty
\right].
\end{split}
\end{equation*} 
We then may estimate
\begin{equation*}
\begin{split}
I_{4}^{1}
= &
\underset{\mathcal{B}_{1}}{\int} (\frac{1}{1+r^{2}})^{\frac{n+2\gamma}{2}}
(\frac{1}{\frac{\lambda_{i} }{ \lambda_{j} }+\lambda_{i}\lambda_{j}
\vert \frac{x}{\lambda_{i}}+a_{i}-a_{j}\vert^{2}})^{\frac{n-2\gamma}{2}} \\
\leq &
\frac{C(\frac{\lambda_{i}}{\lambda_{j}})^{\frac{n+2\gamma}{2}}}{(1+\lambda_{i}^{
2}\vert a_{i}-a_{j}\vert^{2})^{\frac{n+2\gamma}{2}}} 
\underset{[\vert \frac{x}{\lambda_{j}}\vert \leq E\vert 
a_{i}-a_{j}\vert]}{\int} 
(\frac{1}{1+\lambda_{j}^{2}\vert 
\frac{x}{\lambda_{j}}+a_{i}-a_{j}\vert^{2}})^{\frac{n-2\gamma}{2}}
\\
\leq &
\frac{C}{(\frac{\lambda_{j}}{\lambda_{i}}+\lambda_{i}\lambda_{j}\vert 
a_{i}-a_{j}\vert^{2})^{\frac{n+2\gamma}{2}}}
\underset{B_{C\lambda_{j}\vert a_{i}-a_{j}\vert}(0)}{\int} 
(\frac{1}{1+r^{2}})^{\frac{n-2\gamma}{2}}\\
\leq &
\frac{C(\lambda_{j}^{2}\vert 
a_{i}-a_{j}\vert^{2})^{\gamma}}{(\frac{\lambda_{j}}{\lambda_{i}}+\lambda_{i}
\lambda_{j}\vert a_{i}-a_{j}\vert^{2})^{\frac{n+2\gamma}{2}}}
=
O((\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}\varepsilon_{i,j}^{\frac{n}{n-2\gamma}})
\end{split}
\end{equation*} 
and 
\begin{equation*}
\begin{split}
I_{4,2}
= &
\underset{\mathcal{B}_{2}}{\int} (\frac{ 1}{1+r^{2}})^{\frac{n+2\gamma}{2}}
(\frac{1}{\frac{\lambda_{i} }{ \lambda_{j} }+\lambda_{i}\lambda_{j}\vert  
\frac{x}{\lambda_{i}}+a_{i}-a_{j}\vert^{2}})^{\frac{n-2\gamma}{2}} \\
\leq &
\frac{C}{(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
a_{i}-a_{j}\vert^{2})^{\frac{n-2\gamma}{2}}} 
\underset{[\vert x \vert \geq \lambda_{i}\vert a_{i}-a_{j}\vert]}{\int}
(\frac{1}{1+r^{2}})^{\frac{n+2\gamma}{2}} 
= 
O((\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}\varepsilon_{i,j}^{\frac{n}{n-2\gamma}}),
\end{split}
\end{equation*} 
so  \;$
I_{4}\leq 
I_{4}^{1}+I_{4}^{2}=O((\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}\varepsilon_{i,j}^{
\frac{n}{n-2\gamma}}).
$
Collecting terms the claim follows.

\item It is sufficient to consider the case \;$k=j$, since 
\begin{equation*}
\begin{split}
\lambda_{j}\partial_{\lambda_{j}}\int 
\delta_{i}\delta_{j}^{\frac{n+2\gamma}{n-2\gamma}}
=
\lambda_{j}\partial_{\lambda_{j}}\int 
\delta_{i}\delta_{j}^{\frac{n+2\gamma}{n-2\gamma}}
\end{split}
\end{equation*} 
First we deal with the case \;$\frac{1}{\lambda_{i}}\leq \frac{1}{\lambda_{j}}$. 
We have
\begin{equation*}\begin{split}
-\lambda_{j}\int 
\delta_{i}^{\frac{n+2\gamma}{n-2\gamma}}\partial_{\lambda_{j}}\delta_{j}
= &
\frac{n-2\gamma}{2}
\int
(\frac{ \lambda_{i} }{ 1+\lambda_{i}^{2}\vert 
x-a_{i}\vert^{2}})^{\frac{n+2\gamma}{2}} 
(\frac{ \lambda_{j} }{ 1+\lambda_{j}^{2}\vert 
x-a_{j}\vert^{2}})^{\frac{n-2\gamma}{2}}
\frac{\lambda_{j}^{2}\vert x-a_{j}\vert^{2}-1}{\lambda_{j}^{2}\vert 
x-a_{j}\vert^{2}+1},
\end{split}\end{equation*}
whence
\begin{equation*}\begin{split}
-\lambda_{j}\int  
\delta_{i}^{\frac{n+2\gamma}{n-2\gamma}}\partial_{\lambda_{j}}\delta_{j} 
= &
\frac{n-2\gamma}{2}
\int
(\frac{1}{1+r^{2}})^{\frac{n+2\gamma}{2}} 
\frac{\lambda_{j}^{2}\vert\frac{x}{\lambda_{i}}+a_{i}-a_{j}\vert^{2}-1}
{\lambda_{j}^{2}\vert\frac{x}{\lambda_{i}}+a_{i}-a_{j}\vert^{2}+1}
(\frac{1}{\frac{\lambda_{i} }{ \lambda_{j} 
}+\lambda_{i}\lambda_{j}\vert\frac{x}{\lambda_{i}}+a_{i}-a_{j}\vert^{2}})^{\frac
{n-2\gamma}{2}} 
.
\end{split}\end{equation*}
Since \;$
\varepsilon_{i,j}^{\frac{2}{2-n}}
\sim 
\lambda_{i}\lambda_{j}\vert a_{i}-a_{j}\vert^{2}
\;\;\text{ or }\;\;
\varepsilon_{i,j}^{\frac{2}{2-n}}
\sim
\frac{\lambda_{i}}{\lambda_{j}},
$
\; we may expand on
\begin{equation*}\begin{split}
\mathcal{A}
= &
\left[
\vert  \frac{x}{\lambda_{i}}\vert \leq \epsilon\vert a_{i}-a_{j}\vert
\right]
\cup
\left[\vert  \frac{x}{\lambda_{i}}\vert \leq \epsilon  \frac{1}{\lambda_{j}} 
\right]
\end{split}\end{equation*}
for \;$\epsilon>0$\; sufficiently small
\begin{equation*}\begin{split}
& 
\frac{1}
{
(\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}\vert\frac{x}{\lambda_{i}}+a_{i}-a_{j}\vert^{2})^{\frac{
n-2\gamma}{2}} }
\frac{\lambda_{j}^{2}\vert\frac{x}{\lambda_{i}}+a_{i}-a_{j}\vert^{2}-1}
{\lambda_{j}^{2}\vert\frac{x}{\lambda_{i}}+a_{i}-a_{j}\vert^{2}+1}
\\
& \quad = 
(
\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}
\vert a_{i}-a_{j}\vert^{2}
)^{\frac{2\gamma-n}{2}} 
\frac
{\lambda_{j}^{2}\vert a_{i}-a_{j}\vert^{2}-1}
{\lambda_{j}^{2}\vert a_{i}-a_{j}\vert^{2}+1}
\\
& \quad \;\; 
+
\frac{2\gamma-n}{2}
\frac
{
\langle a_{i}-a_{j},\lambda_{j}x\rangle
}
{
(
\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}\vert a_{i}-a_{j}\vert^{2}
)^{\frac{n+2-2\gamma}{2}}
}
\frac
{\lambda_{j}^{2}\vert a_{i}-a_{j}\vert^{2}-1}
{\lambda_{j}^{2}\vert a_{i}-a_{j}\vert^{2}+1}
\\
& \quad \;\;+
\frac
{
2\frac{\lambda_{j}}{\lambda_{i}}
}
{
(
\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}\vert a_{i}-a_{j}\vert^{2}
)^{\frac{n+2-2\gamma}{2}}
}
\frac
{\langle a_{i}-a_{j},\lambda_{j}x\rangle}
{1+\lambda_{j}^{2}\vert a_{i}-a_{j}\vert^{2}}
+
\frac
{
O(\frac{ \lambda_{j} }{ \lambda_{i} }\vert x\vert^{2})
}
{
(
\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}\vert a_{i}-a_{j}\vert^{2}
)^{\frac{n+2-2\gamma}{2}}
}.
\end{split}\end{equation*}
From this we derive as before with \;$b_{2}=\frac{n-2\gamma}{2}c_{n,3}^{\gamma}$
\begin{equation*}\begin{split}
-\lambda_{j}\int  
\delta_{i}^{\frac{n+2\gamma}{n-2\gamma}}\partial_{\lambda_{j}}\delta_{j} 
= &
b_{2}
\frac
{
\rfrac
{\lambda_{j}^{2}\vert a_{i}-a_{j}\vert^{2}-1}
{\lambda_{j}^{2}\vert a_{i}-a_{j}\vert^{2}+1}
}
{(\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}
\vert a_{i}-a_{j}\vert^{2}
)^{\frac{n-2\gamma}{2}}
} 
+
O((\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}\varepsilon_{i,j}^{\frac{n}{
n-2\gamma}}),
\end{split}\end{equation*}
whence
\begin{equation*}\begin{split}
-\lambda_{j}\int 
\delta_{i}^{\frac{n+2\gamma}{n-2\gamma}}\partial_{\lambda_{j}}\delta_{j}
= &
b_{2}
\frac
{
\lambda_{i}\lambda_{j}\vert a_{i}-a_{j}\vert^{2}-\frac{\lambda_{i}}{\lambda_{j}}
}
{(\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}
\vert a_{i}-a_{j}\vert^{2}
)^{\frac{n+2-2\gamma}{2}}
} 
+
O((\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}\varepsilon_{i,j}^{\frac{n}{
n-2\gamma}}).
\end{split}\end{equation*}
We turn to the case  \;$ 
\frac{1}{\lambda_{i}}\geq \frac{1}{\lambda_{j}}.
$\; We then have
\begin{equation*}\begin{split}
-\lambda_{j}\int \delta_{i}^{\frac{n+2\gamma}{n-2\gamma}}\partial_{ 
\lambda_{j}}\delta_{j}
= &
-\lambda_{j}\int \delta_{i}\partial_{\lambda_{j}} 
\delta_{j}^{\frac{n+2\gamma}{n-2\gamma}}
\end{split}\end{equation*}
and
\begin{equation*}\begin{split}
-\lambda_{j}\int \delta_{i} 
\partial_{\lambda_{j}}\delta_{j}^{\frac{n+2\gamma}{n-2\gamma}}
= &
\frac{n+2\gamma}{2}\int
(\frac{ \lambda_{i} }{ 1+\lambda_{i}^{2}\vert 
x-a_{i}\vert^{2}})^{\frac{n-2\gamma}{2}} \\
& \quad\quad\quad\quad\;\,
(\frac{ \lambda_{j} }{ 1+\lambda_{j}^{2}\vert 
x-a_{j}\vert^{2}})^{\frac{n+2\gamma}{2}}
\frac{\lambda_{j}^{2}\vert x-a_{j}\vert^{2}-1}{\lambda_{j}^{2}\vert 
x-a_{j}\vert^{2}+1},
\end{split}\end{equation*}
whence
\begin{equation*}\begin{split}
-\lambda_{j}\int  
\delta_{i}\partial_{\lambda_{j}}\delta_{j}^{\frac{n+2\gamma}{n-2\gamma}} 
= &
\frac{n+2\gamma}{2}
\int
\frac
{r^{2}-1}
{r^{2}+1}
(\frac{1}{1+r^{2}})^{\frac{n+2\gamma}{2}}
\frac
{
1
}
{
(\frac{\lambda_{j}}{\lambda_{i}}+\lambda_{i}\lambda_{j}\vert\frac{x}{\lambda_{j}
}+a_{j}-a_{i}\vert^{2})^{\frac{n-2\gamma}{2}}
} .
\end{split}\end{equation*}
We may expand on

\mathcal{A}
=
\left[
\vert  \frac{x}{\lambda_{j}}\leq \epsilon \vert a_{j}-a_{i}\vert
\right]
\cup
\left[\vert  \frac{x}{\lambda_{j}}\vert \leq \epsilon  \frac{1}{\lambda_{i}} 
\right]
$
for \;$\epsilon>0$\; sufficiently small
\begin{equation*}\begin{split}
(
\frac{\lambda_{j} }{ \lambda_{i} }
& +
\lambda_{i}\lambda_{j}\vert 
\frac{x}{\lambda_{j}}+a_{j}-a_{i}\vert^{2})^{\frac{2\gamma-n}{2}} 
\\
= & 
(
\frac{\lambda_{j} }{ \lambda_{i} }
+
\lambda_{i}\lambda_{j}
\vert a_{j}-a_{i}\vert^{2}
)^{\frac{2\gamma-n}{2}} 
+
(2\gamma-n)
\frac
{
\langle a_{j}-a_{i},\lambda_{i}x\rangle
+
O(\frac{ \lambda_{i} }{ \lambda_{j} }\vert x\vert^{2})
}
{
(
\frac{\lambda_{j} }{ \lambda_{i} }
+
\lambda_{i}\lambda_{j}\vert a_{j}-a_{i}\vert^{2}
)^{\frac{n+2-2\gamma}{2}}
}.
\end{split}\end{equation*}
As before we find with \;$
b_{2}
=
\frac{n-2\gamma}{2}\int (\frac{1}{1+r^{2}})^{\frac{n+2\gamma}{2}}
=
\frac{n+2\gamma}{2}\int 
\frac{r^{2}-1}{r^{2}+1}(\frac{1}{1+r^{2}})^{\frac{n+2\gamma}{2}}
$
\begin{equation*}\begin{split}
-\lambda_{j}\int 
\delta_{i}\partial_{\lambda_{j}}\delta_{j}^{\frac{n+2\gamma}{n-2\gamma}}
= &
b_{2}\frac
{1}
{(\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}
\vert a_{j}-a_{i}\vert^{2}
)^{\frac{n-2\gamma}{2}}
} 
+
O((\frac{\lambda_{i}}{\lambda_{j}})^{\gamma}\varepsilon_{i,j}^{\frac{n}{n-2\gamma}}).
\end{split}\end{equation*}
Comparing the results in cases  \;$\frac{1}{\lambda_{i}}\leq \frac{1}{\lambda_{j}}$\;
and \;$\frac{1}{\lambda_{j}}\leq \frac{1}{\lambda_{i}}$\; the claim follows.




\item 
By symmetry we may assume 

a_{i}=0, \;\;a_{j}=a \;\;\text{ and }\;\;\frac{1}{\lambda_{i}}\leq \frac{1}{\lambda_{j}}
$
and calculate
\begin{equation*}
\begin{split}
\nabla_{a}  \int \delta_{i}^{\frac{n+2\gamma}{n-2\gamma}}\delta_{j}
= &
\nabla_{a}\int(\frac{\lambda_{i}}{1+\lambda_{i}^{2}r^{2}})^{\frac{n+2\gamma}{2}}
(\frac{\lambda_{j}}{1+\lambda_{j}^{2}\vert x+a\vert^{2}})^{\frac{n-2\gamma}{2}}
\\ = &
(2\gamma-n)
\int(\frac{ \lambda_{i} }{ 1+\lambda_{i}^{2}r^{2}})^{\frac{n+2\gamma}{2}} 
(\frac{ \lambda_{j} }{ 1+\lambda_{j}^{2}\vert 
x+a\vert^{2}})^{\frac{n-2\gamma}{2}}
\frac
{\lambda_{j}^{2}( x+a)}
{1+\lambda_{j}^{2}\vert x+a\vert^{2}} \\
= &
(2\gamma-n)
\int(\frac{ 1 }{ 1+r^{2}})^{\frac{n+2\gamma}{2}} 
\frac{ \lambda_{i}\lambda_{j}( \frac{x}{\lambda_{i}}+a) }{ 
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
\frac{x}{\lambda_{i}}+a\vert^{2}))^{\frac{n+2-2\gamma}{2}}}
\end{split}\end{equation*}
Since \;$
\varepsilon_{i,j}^{\frac{2}{2\gamma-n}}
\sim 
\lambda_{i}\lambda_{j}\vert a\vert^{2}
\;\;\text{ or }\;\;
\varepsilon_{i,j}^{\frac{2}{2\gamma-n}}
\sim
\frac{\lambda_{i}}{\lambda_{j}}
$\; we may expand on 
\begin{equation*}\begin{split}
\mathcal{A}
= &
[\vert  \frac{x}{\lambda_{i}}\vert \leq \epsilon\vert a\vert]
\cup
[\vert  \frac{x}{\lambda_{i}}\vert \leq \epsilon  \frac{1}{\lambda_{j}} ]
\end{split}\end{equation*}
for \;$\epsilon>0$\; sufficiently small 
\begin{equation*}
\begin{split}
& \hspace{-20pt}\frac{ \lambda_{i}\lambda_{j}(\frac{x}{\lambda_{i}}+a) }{ 
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
\frac{x}{\lambda_{i}}+a\vert^{2}))^{\frac{n+2-2\gamma}{2}}} \\
= &
\frac{ \lambda_{i}\lambda_{j}a }{ 
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
a\vert^{2}))^{\frac{n+2-2\gamma}{2}}} 
-(n+2-2\gamma)
\frac
{
\lambda_{i}\lambda_{j}a\lambda_{j}\langle x,a\rangle
}
{
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
a\vert^{2})^{\frac{n+4-2\gamma}{2}}
}
\\
& +
\frac{\lambda_{j}x }{ 
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
a\vert^{2}))^{\frac{n+2-2\gamma}{2}}} 
+
\frac{ \sqrt{\lambda_{i}\lambda_{j}} 
O(\frac{\lambda_{j}}{\lambda_{i}}\vert x 
\vert^{2}) }{ (\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
a\vert^{2}))^{\frac{n+3-2\gamma}{2}}}.
\end{split}
\end{equation*} 
Using radial symmetry we obtain with \;$b_{3}=(2\gamma-n)c_{n,3}^{\gamma}$
\begin{equation*}
\begin{split}
\nabla_{a}  \int \delta_{i}^{\frac{n+2\gamma}{n-2\gamma}}\delta_{j} 
= &
b_{3}\frac{ \lambda_{i}\lambda_{j}a }{ 
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
a\vert^{2}))^{\frac{n+2-2\gamma}{2}}}
+
O((\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}\sqrt{\lambda_{i}\lambda_{j}}
\varepsilon_{i,j}^{\frac{n+1}{n-2\gamma}})
\\
& +
(2\gamma-n)
\underset{\mathcal{A}^{C}}\int(\frac{ 1 }{ 1+r^{2}})^{\frac{n+2\gamma}{2}} 
\frac{ \lambda_{i}\lambda_{j}( \frac{x}{\lambda_{i}}+a) }{ 
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
\frac{x}{\lambda_{i}}+a\vert^{2})^{\frac{n+2-2\gamma}{2}}}.
\end{split}
\end{equation*} 
In case \;$\varepsilon_{i,j}^{\frac{2}{2\gamma-n}}\sim 
\frac{\lambda_{i}}{\lambda_{j}}$\; the last summand above is of order
\begin{equation*}
\begin{split}
O((\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}\sqrt{\lambda_{i}\lambda_{j}}
\varepsilon_{i,j}^{\frac{n+1}{n-2\gamma}}).
\end{split}
\end{equation*} 
Thus we may assume \;$\frac{\lambda_{i}}{\lambda_{j}}\ll 
\lambda_{i}\lambda_{j}\vert a \vert^{2}$.  Letting \;$0<\theta<\frac{1}{4}$\; and
\begin{equation*}
\begin{split}
\mathcal{B}_{1}=\left[\vert \frac{x}{\lambda_{i}}+a\vert \leq \theta \vert 
a\vert\right]\subset \subset B^{c}_{\epsilon \lambda_{i}\vert a 
\vert}(0)=\mathcal{A}^{c}
\end{split}
\end{equation*} 
we may expand on \;$\mathcal{B}_{1}$
\begin{equation*}
\begin{split}
(\frac{1}{1+r^{2}})^{\frac{n+2\gamma}{2}}
= &
(\frac{1}{1+\lambda_{i}^{2}\vert \frac{x}{\lambda_{i}} +a -a 
\vert^{2}})^{\frac{n+2\gamma}{2}} 
=
(\frac{1}{1+\lambda_{i}^{2}\vert a \vert^{2}})^{\frac{n+2\gamma}{2}}
+
\frac{O(\lambda_{i}^{2}\vert \frac{x}{\lambda_{i}}+a\vert\vert a 
\vert)}{(1+\lambda_{i}^{2}\vert a \vert^{2})^{\frac{n+2+2\gamma}{2}}}
\end{split}
\end{equation*} 
and find using radial symmetry  
\begin{equation*}
\begin{split}
\vert \underset{\mathcal{B}_{1}}\int(\frac{ 1 }{ 
1+r^{2}})^{\frac{n+2\gamma}{2}} 
 \frac{ \lambda_{i}\lambda_{j}( \frac{x}{\lambda_{i}}+a) }{ 
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
\frac{x}{\lambda_{i}}+a\vert^{2}))^{\frac{n+2-2\gamma}{2}}} 
\vert 
\leq &
\frac{C}{(1+\lambda_{i}^{2}\vert a \vert^{2})^{\frac{n+2+2\gamma}{2}}}
\underset{\mathcal{B}_{1}}\int
\frac{\lambda_{i}^{3}\lambda_{j}\vert \frac{x}{\lambda_{i}}+a\vert^{2}\vert a 
\vert}
{(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert\frac{x}{\lambda_{i
}}+a\vert^{2})^{\frac{n+2-2\gamma}{2}}} \\
\leq &
\frac{C\lambda_{i}}{(1+\lambda_{i}^{2}\vert a 
\vert^{2})^{\frac{n+1+2\gamma}{2}}}
\underset{\mathcal{B}_{1}}\int
\frac{1}
{(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert\frac{x}{\lambda_{i
}}+a\vert^{2})^{\frac{n-2\gamma}{2}}}.
\end{split}
\end{equation*} 
Rescaling this gives 
\begin{equation*} 
\begin{split}
\vert \underset{\mathcal{B}_{1}}\int(\frac{ 1 }{ 
1+r^{2}})^{\frac{n+2\gamma}{2}} 
& \frac{ \lambda_{i}\lambda_{j}( \frac{x}{\lambda_{i}}+a) }{ 
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
\frac{x}{\lambda_{i}}+a\vert^{2}))^{\frac{n+2-2\gamma}{2}}} 
\vert \\
\leq  &
\frac{C\lambda_{i}(\frac{\lambda_{i}}{\lambda_{j}})^{\frac{n+2\gamma}{2}}}{
(1+\lambda_{i}^{2}\vert a \vert^{2})^{\frac{n+1+2\gamma}{2}}}
\underset{[\vert \frac{x}{\lambda_{j}}+a\vert<\theta \vert a\vert]}\int
\frac{1}
{(1+\lambda_{j}^{2}\vert\frac{x}{\lambda_{j}}+a\vert^{2})^{\frac{n-2\gamma}{2}}}
 \\
\leq &
\frac{C\sqrt{\lambda_{i}\lambda_{j}}(\lambda_{j}\vert a 
\vert)^{2\gamma}}{(\frac{\lambda_{j}}{\lambda_{i}}+\lambda_{i}\lambda_{j}\vert 
a \vert^{2})^{\frac{n+1+2\gamma}{2}}}
=
O((\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}\sqrt{\lambda_{i}\lambda_{j}}
\varepsilon_{i,j}^{\frac{n+1}{n-2\gamma}}).
\end{split}
\end{equation*} 
Moreover letting \;$\mathcal{B}_{2}=\mathcal{A}^{c}\setminus \mathcal{B}_{1}$\; we 
may estimate
\begin{equation*}
\begin{split}
\vert \underset{\mathcal{B}_{2}}\int & (\frac{ 1 }{ 
1+r^{2}})^{\frac{n+2\gamma}{2}} 
\frac{ \lambda_{i}\lambda_{j}( \frac{x}{\lambda_{i}}+a) }{ 
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
\frac{x}{\lambda_{i}}+a\vert^{2}))^{\frac{n+2-2\gamma}{2}}} 
\vert \\
& \leq 
\frac{C\sqrt{\lambda_{i}\lambda_{j}}}{(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{
i}\lambda_{j}\vert a \vert^{2})^{\frac{n+1-2\gamma}{2}}}
\int_{\mathcal{A}^{c}}(\frac{1}{1+r^{2}})^{\frac{n+2\gamma}{2}}
=
O((\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}\sqrt{\lambda_{i}\lambda_{j}}
\varepsilon_{i,j}^{\frac{n+1}{n-2\gamma}}).
\end{split}
\end{equation*} 
Collecting terms the claim follows.

\item 
By translation invariance and symmetry we may assume 

a_{i}=0, \;\;a_{j}=a \;\;\text{and}\;\;\frac{1}{\lambda_{i}}\leq \frac{1}{\lambda_{j}}
$.
Then
\begin{equation*}
\begin{split}
\nabla^{2}_{a}  \int \delta_{i}^{\frac{n+2\gamma}{n-2\gamma}}\delta_{j}
= &
(2\gamma-n)\int(\frac{\lambda_{i}}{1+\lambda_{i}^{2}r^{2}})^{\frac{n+2\gamma}{2}
}
(\frac{\lambda_{j}}{1+\lambda_{j}^{2}\vert x+a\vert^{2}})^{\frac{n-2\gamma}{2}} 
\\
& \quad \quad\ \quad\quad\;\;\, 
[
\frac{\lambda_{j}^{2}\,id}{1+\lambda_{j}^{2}\vert x+a\vert^{2}}
-
(n+2-2\gamma)\frac{\lambda_{j}^{4} (x+a).(x+a)}{(1+\lambda_{j}^{2}\vert 
x+a\vert^{2})^{2}}
]
\\
= &
(2\gamma-n)\lambda_{i}\lambda_{j}
\int\frac
{
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
\frac{x}{\lambda_{i}}+a\vert^{2})\,id
-
(n+2-2\gamma)\lambda_{i}\lambda_{j} 
(\frac{x}{\lambda_{i}}+a).(\frac{x}{\lambda_{i}}+a)
}
{
(1+r^{2})^{\frac{n+2\gamma}{2}}(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}
\lambda_{j}\vert \frac{x}{\lambda_{i}}+a\vert^{2})^{\frac{n+4-2\gamma}{2}}
} .
\end{split}\end{equation*}
Since \;$
\varepsilon_{i,j}^{\frac{2}{2\gamma-n}}
\sim 
\lambda_{i}\lambda_{j}\vert a\vert^{2}
\;\;\text{ or }\;\;
\varepsilon_{i,j}^{\frac{2}{2\gamma-n}}
\sim
\frac{\lambda_{i}}{\lambda_{j}}
$\; we may expand on 
\begin{equation*}\begin{split}
\mathcal{A}
= &
[\vert  \frac{x}{\lambda_{i}}\vert \leq \epsilon\vert a\vert]
\cup
[\vert  \frac{x}{\lambda_{i}}\vert \leq \epsilon  \frac{1}{\lambda_{j}} ]
\end{split}\end{equation*}
for \;$\epsilon>0$\; sufficiently small 
\begin{equation*}
\begin{split}
& \hspace{-20pt}\frac
{
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
\frac{x}{\lambda_{i}}+a\vert^{2})\,id
-
(n+2-2\gamma)\lambda_{i}\lambda_{j} 
(\frac{x}{\lambda_{i}}+a).(\frac{x}{\lambda_{i}}+a)
}
{
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
\frac{x}{\lambda_{i}}+a\vert^{2})^{\frac{n+4-2\gamma}{2}}
} 
\\
= &
\frac
{
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert a\vert^{2})\,id
-
(n+2-2\gamma)\lambda_{i}\lambda_{j} a.a
+
O(\frac{\lambda_{j}}{\lambda_{i}}\vert x 
\vert^{2})
}
{
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
a\vert^{2})^{\frac{n+4-2\gamma}{2}}
}
\end{split}
\end{equation*} 
up to some integrable anti-symmetric terms and thus obtain
\begin{equation*}
\begin{split}
\nabla_{a}^{2} \int \delta_{i}^{\frac{n+2\gamma}{n-2\gamma}}\delta_{j} 
= &
b_{4}\lambda _{i}\lambda_{j}\frac
{
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert a\vert^{2})\,id
-
(n+2-2\gamma)\lambda_{i}\lambda_{j} a.a
}
{
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
a\vert^{2})^{\frac{n+4-2\gamma}{2}}
} \\
& +
O((\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}\lambda_{i}\lambda_{j}\varepsilon_{i
,j}^{\frac{n+2}{n-2\gamma}}) 
+
(2\gamma-n)\lambda_{i}\lambda_{j}
\int_{\mathcal{A}^{c}}...
,
\end{split}
\end{equation*} 
where \;$b_{4}=(2\gamma-n)c_{n,3}^{\gamma}$\; 
and
\begin{equation*}
\int_{\mathcal{A}^{c}}\hspace{-5pt}...
=
\int_{\mathcal{A}^{c}}
\hspace{-1pt}
\frac
{
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
\frac{x}{\lambda_{i}}+a\vert^{2})\,id
-
(n+2-2\gamma)\lambda_{i}\lambda_{j} 
(\frac{x}{\lambda_{i}}+a).(\frac{x}{\lambda_{i}}+a)
}
{
(1+r^{2})^{\frac{n+2\gamma}{2}}(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}
\lambda_{j}\vert \frac{x}{\lambda_{i}}+a\vert^{2})^{\frac{n+4-2\gamma}{2}}
}
.
\end{equation*}
In case \;$\varepsilon_{i,j}^{\frac{2}{2\gamma-n}}\sim 
\frac{\lambda_{i}}{\lambda_{j}}$\; the integral above is of order
\begin{equation*}
\begin{split}
O((\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}\varepsilon_{i,j}^{\frac{n+2}{
n-2\gamma}}).
\end{split}
\end{equation*} 
Thus we may assume \;$\frac{\lambda_{j}}{\lambda_{i}}\leq \frac{\lambda_{i}}{\lambda_{j}}\ll 
\lambda_{i}\lambda_{j}\vert a \vert^{2}$, i.e. for \;$0<\theta<\frac{1}{4}$
\begin{equation*}
\begin{split}
\mathcal{B}_{1}=\left[\vert \frac{x}{\lambda_{i}}+a\vert \leq \theta \vert 
a\vert\right]\subset \subset B^{c}_{\epsilon \lambda_{i}\vert a 
\vert}(0)=\mathcal{A}^{c}.
\end{split}
\end{equation*} 
On \;$\mathcal{B}_{1}$\; we then may expand  
\begin{equation*}
\begin{split}
(\frac{1}{1+r^{2}}  )^{\frac{n+2\gamma}{2}}
= &
(\frac{1}{1+\lambda_{i}^{2}\vert\frac{x}{\lambda_{i}}+a-a 
\vert^{2}})^{\frac{n+2\gamma}{2}} \\
= &
(\frac{1}{1+\lambda_{i}^{2}\vert a \vert^{2}})^{\frac{n+2\gamma}{2}}
+
(n+2\gamma)\frac{\lambda_{i}^{2}\langle 
\frac{x}{\lambda_{i}}+a,a\rangle + O(\lambda_{i}^{2}\vert 
\frac{x}{\lambda_{i}}+a\vert^{2})}{(1+\lambda_{i}^{2}\vert a 
\vert^{2})^{\frac{n+2+2\gamma}{2}}} .
\end{split}
\end{equation*}
Using radial symmetry and 
\begin{equation*}
\int_{\mathbb{R}^{n}}\frac{1+r^{2}-(n+2-2\gamma)x_{k}^{2}}{(1+r^{2})^{\frac{n+4-2\gamma}
{2}}}=0\;\; \text{ for }\;\; k=1,\ldots, n
\end{equation*}
as an easy computation shows, we obtain
\begin{equation*}
\begin{split}
\vert 
\int_{\mathcal{B}_{1}} & \frac
{
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
\frac{x}{\lambda_{i}}+a\vert^{2})\,id
-
(n+2-2\gamma)\lambda_{i}\lambda_{j} 
(\frac{x}{\lambda_{i}}+a).(\frac{x}{\lambda_{i}}+a)
}
{
(1+r^{2})^{\frac{n+2\gamma}{2}}(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}
\lambda_{j}\vert \frac{x}{\lambda_{i}}+a\vert^{2})^{\frac{n+4-2\gamma}{2}}
}
\vert 
\\
\leq &
\frac{C}{(1+\lambda_{i}^{2}\vert a \vert^{2})^{\frac{n+2\gamma}{2}}}
\int_{\mathcal{B}_{1}^{c}}  \frac
{
1
}
{
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
\frac{x}{\lambda_{i}}+a\vert^{2})^{\frac{n+2-2\gamma}{2}}
}
\\
& +
\frac{C\lambda_{i}^{2}}{(1+\lambda_{i}^{2}\vert a 
\vert^{2})^{\frac{n+2+2\gamma}{2}}}
\int_{\mathcal{B}_{1}}  \frac
{
\vert \frac{x}{\lambda_{i}}+a\vert^{2}
}
{
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
\frac{x}{\lambda_{i}}+a\vert^{2})^{\frac{n+2-2\gamma}{2}}
}
\\
\leq &
\frac{C(\frac{\lambda_{j}}{\lambda_{i}})^{\frac{-n+2-2\gamma}{2}}}{(1+\lambda_{i
}^{2}\vert a \vert^{2})^{\frac{n+2\gamma}{2}}}
\int_{[\vert x \vert >\theta \lambda_{j}\vert a \vert]}  \frac
{
1
}
{
(1+r^{2})^{\frac{n+2-2\gamma}{2}}
}
\\
& +
\frac{C(\frac{\lambda_{j}}{\lambda_{i}})^{-\frac{n+2+2\gamma}{2}}}{(1+\lambda_{i
}^{2}\vert a \vert^{2})^{\frac{n+2+2\gamma}{2}}}
\int_{[\vert x \vert \leq \theta\lambda_{j}\vert a \vert]}  \frac
{
r^{2}
}
{
(1+r^{2})^{\frac{n+2-2\gamma}{2}}
}
\\
\leq &
\frac{C\frac{\lambda_{j}}{\lambda_{i}}}{(\frac{\lambda_{j}}{\lambda_{i}}
+\lambda_{i}\lambda_{j}\vert a \vert^{2})^{\frac{n+2\gamma}{2}}}
O(\frac{(\lambda_{j}\vert a \vert)^{n}}{(1+\lambda_{j}\vert a 
\vert)^{n+2-2\gamma}})
\\
& +
\frac{C}{(\frac{\lambda_{j}}{\lambda_{i}}+\lambda_{i}\lambda_{j}\vert a 
\vert^{2})^{\frac{n+2+2\gamma}{2}}}
O(\frac{(\lambda_{j}\vert a \vert^{n+2})}{(1+\lambda_{j}\vert a 
\vert)^{n+2-2\gamma}})
=
O((\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}\varepsilon_{i,j}^{\frac{n+2}{
n-2\gamma
}}).
\end{split}
\end{equation*} 
Moreover letting \;$\mathcal{B}_{2}=\mathcal{A}^{c}\setminus \mathcal{B}_{1}$\; we 
may estimate
\begin{equation*}
\begin{split}
\vert \int_{\mathcal{B}_{2}} &\frac
{
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
\frac{x}{\lambda_{i}}+a\vert^{2})\,id
-
(n+2-2\gamma)\lambda_{i}\lambda_{j} 
(\frac{x}{\lambda_{i}}+a).(\frac{x}{\lambda_{i}}+a)
}
{
(1+r^{2})^{\frac{n+2\gamma}{2}}(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}
\lambda_{j}\vert \frac{x}{\lambda_{i}}+a\vert^{2})^{\frac{n+4-2\gamma}{2}}
}
\vert
\\
\leq &
\frac{C}{(\frac{\lambda_{i}}{\lambda_{j}})^{\frac{n+2-2\gamma}{2}}}
\int_{\mathcal{A}^{c}}
\frac{1}{(1+r^{2})^{\frac{n+2\gamma}{2}}}
\leq 
\frac{C(\frac{\lambda_{j}}{\lambda_{i}}+\lambda_{i}\vert a 
\vert)^{-2\gamma}}{(\frac{\lambda_{i}}{\lambda_{j}})^{\frac{n+2-2\gamma}{2}}}
= 
O((\frac{\lambda_{j}}{\lambda_{i}})^{\gamma}\varepsilon_{i,j}^{\frac{n+2}{
n-2\gamma}}).
\end{split}
\end{equation*} 
Collecting terms the claim follows.
\end{enumerate}
\end{pfn}
\vspace{6pt}


\begin{pfn}{ of lemma \ref{lem_higher_interaction_estimates}}
\begin{enumerate}[label=(\roman*)]
 \item 
Let $\alpha'=\frac{n-2\gamma}{2}\alpha, \, \beta'=\frac{n-2\gamma}{2}\beta$, so $\alpha'+\beta'=n$.
We distinguish the cases
\begin{enumerate}
 \item [($\alpha$)] \quad 
"$\varepsilon_{i,j}^{\frac{2}{2\gamma-n}}\sim \frac{\lambda_{i} }{ \lambda_{j} }
\; \vee \;
\varepsilon_{i,j}^{\frac{2}{2\gamma-n}} \sim \lambda_{i}\lambda_{j}\gamma_{n}G^{\frac{2}{2\gamma-n}}( a _{i}, a _{j})$"
\smallskip \\
We estimate for $c>0$ small
\begin{equation*}\begin{split}
\int v_{i}^{\alpha} v_{j}^{\beta}
\leq &
C \underset{B_{c}(0)}{\int}(\frac{\lambda_{i} }{1+\lambda_{i}^2r^{2}})^{\alpha'}(\frac{ \lambda_{j} }{1 +\lambda_{j}^{2}\gamma_{n}G_{ a _{j}}^{\frac{2}{2\gamma-n}}(\exp_{ a _{i}}x)})^{\beta'}
+
C  \frac{1}{\lambda_{i}^{\alpha'}}\underset{B_{c}(0)}{\int}(\frac{ \lambda_{j} }{1+\lambda_{j}^2r^{2}})^{\beta'}
 \\
\leq &
C \underset{B_{ c\lambda_{i} }(0)}{\int}(\frac{1}{1+r^{2}})^{\alpha'}
(\frac{1}{\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}\gamma_{n}G_{ a _{j}}^{\frac{2}{2\gamma-n}}(\exp_{ a _{i}}\frac{x}{\lambda_{i}})})^{\beta'}
+
C  \frac{1}{\lambda_{i}^{\alpha'}} \frac{1}{\lambda_{j}^{n-\beta'}} 
\underset{B_{ c\lambda_{j} }(0)}{\int}(\frac{1}{1+r^{2}})^{\beta'}
\end{split}\end{equation*}
up to some $O(  \frac{1}{\lambda_{i}^{\alpha'}} \frac{1}{\lambda_{j}^{\beta'}} )$. Thus by 
$
\int_{B_{ c\lambda_{j} }(0)}(\frac{1}{1+r^{2}})^{\beta'}
\leq 
C \frac{1}{\lambda_{j} ^{2\beta'-n}}
$ 
we get
\begin{equation*}\begin{split}
\int   v_{i}^{\alpha}  v_{j}^{\beta}  
\leq &
C\underset{B_{ c\lambda_{i} }(0)}{\int}(\frac{1}{1+r^{2}})^{\alpha'}(\frac{1}{\frac{\lambda_{i} }{ \lambda_{j} }+\lambda_{i}\lambda_{j}\gamma_{n}G_{ a _{j}}^{\frac{2}{2\gamma-n}}(\exp_{ a _{i}} \frac{x}{\lambda_{i}})})^{\beta'} 
\end{split}\end{equation*}
up to some $O(  \frac{1}{\lambda_{i}^{\alpha'}} \frac{1}{\lambda_{j}^{\beta'}} )$, whence the claim follows in cases
\begin{equation*}\begin{split}
\frac{ \lambda_{j} }{ \lambda_{i} }+\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}\gamma_{n}G^{\frac{2}{2\gamma-n}}( a _{i}, a _{j})
\sim \frac{\lambda_{i}}{\lambda_{j}}
\; \text{ or }\;
d( a _{i}, a _{j})>3c.
\end{split}\end{equation*}
Else we may assume $d( a _{i}, a _{j})<3c$ and 
\begin{equation*}\begin{split}
\frac{ \lambda_{j} }{ \lambda_{i} }+\frac{\lambda_{i} }{ \lambda_{j} }+\lambda_{i}\lambda_{j}\gamma_{n}G^{\frac{2}{2\gamma-n}}( a _{i}, a _{j}) 
& \sim 
\lambda_{i}\lambda_{j}d^{2}( a _{i}, a _{j}).
\end{split}\end{equation*}
We then get with $\mathcal{B}=[\frac{1}{2}d( a _{i}, a _{j})\leq \vert  \frac{x}{\lambda_{i}}\vert \leq 2 d( a _{i}, a _{j})]$
\begin{equation*}\begin{split}
\int v_{i}^{\alpha}  v_{j}^{\beta} 
\leq &
C \underset{\mathcal{B}}{\int}
(\frac{1}{1+r^{2}})^{\alpha'}(\frac{1}{\frac{\lambda_{i} }{ \lambda_{j} }+\lambda_{i}\lambda_{j}d^{2}( a _{j}, \exp_{ a _{i}}( \frac{x}{\lambda_{i}}))})^{\beta'} +
O(\varepsilon_{i,j}^{\beta})\\
\leq &
C
(\frac{1}{1+\vert\lambda_{i}d( a _{i}, a _{j})\vert^{2}})^{\alpha'}
\underset{ [\vert  \frac{x}{\lambda_{i}}\vert \leq 4 d( a _{i}, a _{j})]}{\int}
(\frac{1}{\frac{\lambda_{i} }{ \lambda_{j} }+\frac{ \lambda_{j} }{ \lambda_{i} }r^{2}})^{\beta'}  +
O(\varepsilon_{i,j}^{\beta})\\
\leq &
C
\frac{(\frac{ \lambda_{j} }{ \lambda_{i} })^{\beta'-n}}{(1+\vert\lambda_{i}d( a _{i}, a _{j})\vert^{2})^{\alpha'}}
 \underset{ [r \leq 4 \lambda_{j}d( a _{i}, a _{j})]}{\int}
(\frac{1}{1+r ^{2}})^{\beta'}
+
O(\varepsilon_{i,j}^{\beta}).
\end{split}\end{equation*}
Note, that in case 
$\lambda_{j}d( a _{i}, a _{j})$ remains bounded, we are done. Else
\begin{equation*}\begin{split}
\int v_{i}^{\alpha}v_{j}^{\beta}
\leq &
C
\frac{(\frac{ \lambda_{j} }{ \lambda_{i} })^{\beta'-n}(\lambda_{j}d( a _{i}, a _{j}))^{n-2\beta'}}{(1+\vert\lambda_{i}d( a _{i}, a _{j})\vert^{2})^{\alpha'}}
+
O(\varepsilon_{i,j}^{\beta}) \\
\leq &
C
(\frac{1}{1+\vert\lambda_{i}d( a _{i}, a _{j})\vert^{2}})^{\alpha'-\frac{n}{2}+\beta'}
(\frac{\lambda_{i} }{ \lambda_{j} })^{\beta'}+
O(\varepsilon_{i,j}^{\beta}),
\end{split}\end{equation*}
whence due to $\alpha'>\frac{n}{2}$ the claim follows.

 \item [$(\beta)$]\quad "$\varepsilon_{i,j}^{\frac{2}{2\gamma-n}}\sim \frac{ \lambda_{j} }{ \lambda_{i} }$"
\smallskip \\
We estimate for $c>0$ small
\begin{equation*}\begin{split}
\int v_{i}^{\alpha}v_{j}^{\beta} 
\leq &
C
\int_{B_{ c\lambda_{j} }(0)}
(\frac{1}{\frac{ \lambda_{j} }{ \lambda_{i} }+\lambda_{i}\lambda_{j}\gamma_{n}G_{ a _{i}}^{\frac{2}{2\gamma-n}}( \exp_{ a _{j}}( \frac{x}{\lambda_{j}} ))})^{\alpha'} 
(\frac{1}{1+r^{2}})^{\beta'} \\
& +
C \frac{1}{\lambda_{j}^{\beta'}} \int_{B_{c}(0)}(\frac{\lambda_{i} }{1+\lambda_{i}^2r^{2}})^{\alpha'}
+
O(  \frac{1}{\lambda_{i}^{\alpha'}} \frac{1}{\lambda_{j}^{\beta'}} ),
\end{split}\end{equation*}
which by
$
\int_{B_{c}(0)}(\frac{\lambda_{i} }{1+\lambda_{i}^2r^{2}})^{\alpha'}
\leq
C\frac{1}{\lambda_{i}^{n-\alpha'}} =  C\frac{1}{\lambda_{i} ^{\beta'}}
$
gives

\begin{equation*}\begin{split}
\int  v_{i}^{\alpha}v_{j}^{\beta} 
\leq &
C
\int_{B_{ c\lambda_{j} }(0)}
(\frac{1}{\frac{ \lambda_{j} }{ \lambda_{i} }+\lambda_{i}\lambda_{j}\gamma_{n}G_{a_{i}}^{\frac{2}{2\gamma-n}}(\exp_{ a _{j}}( \frac{x}{\lambda_{j}}))})^{\alpha'} 
(\frac{1}{1+r^{2}})^{\beta'} 
\end{split}\end{equation*}
up to some $o(\varepsilon_{i,j}^{\beta})$. Since by assumption $d( a _{i}, a _{j})\leq  \frac{2}{\lambda_{i}} $, there holds
\begin{equation*}\begin{split}
\gamma_{n}G_{ a _{i}}^{\frac{2}{2\gamma-n}}(\exp_{ a _{j}}( \frac{x}{\lambda_{j}}))\sim d^{2}( a _{i}, \exp_{ a _{j}}( \frac{x}{\lambda_{j}}))
\;\text{ on }\;
B_{ c\lambda_{j} }(0).
\end{split}\end{equation*}
Thus for $\gamma>3$
\begin{equation*}\begin{split}
\int  v_{i}^{\alpha}v_{j}^{\beta} 
\leq &
C
\underset{[\gamma\frac{ \lambda_{j} }{ \lambda_{i} }\leq \vert x \vert\leq  c\lambda_{j} ]}{\int}
(\frac{1}{\frac{ \lambda_{j} }{ \lambda_{i} }+\lambda_{i}\lambda_{j}d^{2}( a _{i}, \exp_{ a _{j}}( \frac{x}{\lambda_{j}}))})^{\alpha'} 
(\frac{1}{1+r^{2}})^{\beta'} \\
& +C
(\frac{\lambda_{i} }{ \lambda_{j} })^{\alpha'}
\int_{[\vert x \vert<\gamma\frac{ \lambda_{j} }{ \lambda_{i} }]}
(\frac{1}{1+r^{2}})^{\beta'} 
+
o(\varepsilon_{i,j}^{\beta}) \\
\leq &
C
\int_{ [\gamma\frac{ \lambda_{j} }{ \lambda_{i} }\leq \vert x \vert\leq  c\lambda_{j} ]}
(\frac{1}{\frac{ \lambda_{j} }{ \lambda_{i} }+\frac{\lambda_{i} }{ \lambda_{j} }r^{2}})^{\alpha'} 
(\frac{1}{1+r^{2}})^{\beta'} 
+
C
(\frac{\lambda_{i} }{ \lambda_{j} })^{\alpha'}
(\frac{ \lambda_{j} }{ \lambda_{i} })^{n-2\beta} 
+
o(\varepsilon_{i,j}^{\beta}),
\end{split}\end{equation*}
since for $\vert x \vert \geq \gamma\frac{ \lambda_{j} }{ \lambda_{i} }$ we may due to $d( a _{i}, a _{j})\leq \frac{2}{\lambda_{i}} $ assume 

d( a _{i}, \exp_{ a _{j}}( \frac{x}{\lambda_{j}}))\geq  \frac{\vert x \vert}{\lambda_{j}}.

Therefore
\begin{equation*}\begin{split}
\int v_{i}^{\alpha}v_{j}^{\beta}
\leq &
C
(\frac{ \lambda_{j} }{ \lambda_{i} })^{\alpha'}\int_{ [\vert x \vert \geq \gamma\frac{ \lambda_{j} }{ \lambda_{i} }]}
r^{-2n}
+
O(\varepsilon_{i,j}^{\beta}) 
= 
O(\varepsilon_{i,j}^{\beta}).
\end{split}\end{equation*}

\end{enumerate}
\item 
By symmetry we may assume $\frac{1}{\lambda_{i}}\leq \frac{1}{\lambda_{j}}$ and thus
\begin{equation*}\begin{split}
\varepsilon_{i,j}^{\frac{2}{2\gamma-n}}\sim \frac{\lambda_{i} }{ \lambda_{j} }
\; \vee \;
\varepsilon_{i,j}^{\frac{2}{2\gamma-n}} \sim \lambda_{i}\lambda_{j}\gamma_{n}G^{\frac{2}{2\gamma-n}}( a _{i}, a _{j})
\end{split}\end{equation*}
In case $d(a_{i},a_{j})>3\epsilon$ for some $\epsilon>0$  we estimate
\begin{equation*}\begin{split}
\int v_{i}^{\frac{n}{n-2\gamma}}v_{j}^{\frac{n}{n-2\gamma}} 
\leq &
C \int_{B_{\epsilon}(0)}(\frac{\lambda_{i} }{1+\lambda_{i}^2r^{2}})^{\frac{n}{2}}
(\frac{ \lambda_{j} }{1 +\lambda_{j}^{2}\gamma_{n}G_{ a _{j}}^{\frac{2}{2\gamma-n}}(\exp_{ a _{i}}x)})^{\frac{n}{2}}\\
& +
\frac{C_{\epsilon}}{\lambda_{i}^{\frac{n}{2}}}
\int_{B_{\epsilon}(0)}(\frac{ \lambda_{j} }{1+\lambda_{j}^2r^{2}})^{\frac{n}{2}}
+
C_{\epsilon} \frac{1}{\lambda_{i}^{\frac{n}{2}}\lambda_{j}^{\frac{n}{2}}}   \\
\leq &
C \int_{B_{ \epsilon\lambda_{i} }(0)}(\frac{1}{1+r^{2}})^{\frac{n}{2}}
(\frac{1}{\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}\gamma_{n}G_{ a _{j}}^{\frac{2}{2\gamma-n}}(\exp_{ a _{i}}( \frac{x}{\lambda_{i}}))})^{\frac{n}{2}}+
C_{\epsilon}\ln \lambda_{j}\varepsilon_{i,j}^{\frac{n}{n-2\gamma}} \\
\leq &
C_{\epsilon}\ln(\lambda_{i}\lambda_{j})\epsilon_{i,j}^{\frac{n}{n-2\gamma}}
\leq C_{\epsilon}\ln \epsilon_{i,j}\epsilon_{i,j}^{\frac{n}{n-2\gamma}}.
\end{split}\end{equation*}
Thus we assume, that $d(a_{i},a_{j})$ is arbitrarily small. Then for
$d(a_{i},a_{j})\ll c \ll 1 $
we estimate passing to normal coordinates around $a_{i}$
\begin{equation*}
\begin{split}
\int v_{i}^{\frac{n}{n-2\gamma}}v_{j}^{\frac{n}{n-2\gamma}} 
\leq &
C\int_{B_{c\lambda_{i}}(0)}
(\frac{1}{1+r^{2}})^{\frac{n}{2}}
(
\frac{1}
{
\frac{\lambda_{i}}{\lambda_{j}}
+
\lambda_{i}\lambda_{j}\gamma_{n}G_{a_{j}}^{\frac{2}{2\gamma-n}}(\exp_{a_{i}}(\frac{x}{\lambda_{i}}))
}
)^{\frac{n}{2}}  
=
C\int_{B_{c\lambda_{i}}(0)} I
\end{split}
\end{equation*}
up to some terms of order $\frac{1}{\lambda_{i}^{\frac{n}{2}}\lambda_{j}^{\frac{n}{2}}}=O(\epsilon_{i,j}^{\frac{n}{n-2\gamma}})$. Decompose $B_{c\lambda_{i}}(0)$ into
\begin{enumerate}
 \item[$(\alpha)$] \quad 
$
\mathcal{A}
=
[
\vert \frac{x}{\lambda_{i}}\vert
\leq 
\epsilon \sqrt{G_{a_{j}}^{\frac{2}{2\gamma-n}}(a_{i})+\frac{1}{\lambda_{j}^{2}}}
]
$
 \item[$(\beta)$] \quad 
$
\mathcal{B}
=
[
\epsilon \sqrt{G_{a_{j}}^{\frac{2}{2\gamma-n}}(a_{i})+\frac{1}{\lambda_{j}^{2}}}
<
\vert \frac{x}{\lambda_{i}}\vert
<
E \sqrt{G_{a_{j}}^{\frac{2}{2\gamma-n}}(a_{i})+\frac{1}{\lambda_{j}^{2}}}
]
$
 \item[$(\gamma)$] \quad 
$
\mathcal{C}
=
[
E \sqrt{G_{a_{j}}^{\frac{2}{2\gamma-n}}(a_{i})+\frac{1}{\lambda_{j}^{2}}}
\leq
\vert \frac{x}{\lambda_{i}}\vert
<
c
]
$
\end{enumerate}
for some fixed $0<\epsilon \ll 1 \ll E <\infty$. We then find
\begin{equation*}
\begin{split}
\int_{\mathcal{A}}I
\leq 
C\varepsilon_{i,j}^{\frac{n}{n-2\gamma}}
\int_{\mathcal{A}}(\frac{1}{1+r^{2}})^{\frac{n}{2}} 
=
O(\ln \epsilon_{i,j}\epsilon_{i,j}^{\frac{n}{n-2\gamma}})
\end{split}
\end{equation*}
and 
\begin{equation*}
\begin{split}
\int_{\mathcal{B}} I
\leq &
\frac{C}
{
(\frac{\lambda_{i}^{2}}{\lambda_{j}^{2}}
+
\lambda_{i}^{2}G_{a_{j}}^{\frac{2}{2\gamma-n}}(a_{i})
)^{\frac{n}{2}}}
\int_{\mathcal{B}}
(
\frac{1}
{
\frac{\lambda_{i}}{\lambda_{j}}
+
\lambda_{i}\lambda_{j}\gamma_{n}G_{a_{j}}^{\frac{2}{2\gamma-n}}(\exp_{a_{i}}(\frac{x}{\lambda_{i}}))
}
)^{\frac{n}{2}}  \\
\leq &
C\varepsilon_{i,j}^{\frac{n}{n-2\gamma}}(\frac{\lambda_{j}}{\lambda_{i}})^{\frac{n}{2}}
\underset
{
[
\vert \frac{x}{\lambda_{i}}\vert <
\tilde E \sqrt{G_{a_{j}}^{\frac{2}{2\gamma-n}}(a_{i})+\frac{1}{\lambda_{j}^{2}}}
]
}
{\int}
(
\frac{1}
{
\frac{\lambda_{i}}{\lambda_{j}}
+
\frac{\lambda_{j}}{\lambda_{i}}\vert x \vert^{2}
}
)^{\frac{n}{2}} 
\\
= &
C\varepsilon_{i,j}^{\frac{n}{n-2\gamma}}
\underset
{
[
\vert x\vert < \lambda_{j}
\tilde E \sqrt{G_{a_{j}}^{\frac{2}{2\gamma-n}}(a_{i})+\frac{1}{\lambda_{j}^{2}}}
]
}
{\int}
(
\frac{1}
{
1
+
\vert x \vert^{2}
}
)^{\frac{n}{2}} 
=
O(\ln \epsilon_{i,j}\epsilon_{i,j}^{\frac{n}{n-2\gamma}})
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
\int_{\mathcal{C}} I
\leq &
C\int_{\mathcal{C}}
(\frac{1}{1+r^{2}})^{\frac{n}{2}}
(
\frac{1}
{
\frac{\lambda_{i}}{\lambda_{j}}
+
\frac{\lambda_{j}}{\lambda_{i}}\vert x \vert^{2}
}
)^{\frac{n}{2}}  
\leq(\frac{\lambda_{i}}{\lambda_{j}})^{\frac{n}{2}}
\int_{\mathcal{C}}
(\frac{1}{1+r^{2}})^{\frac{n}{2}}r^{-n}
\\
\\
\leq &
C(\frac{\lambda_{i}}{\lambda_{j}})^{\frac{n}{2}}
\underset
{
E\lambda^{i} \sqrt{G_{a_{j}}^{\frac{2}{2\gamma-n}}(a_{i})+\frac{1}{\lambda_{j}^{2}}}
}
{\int^{\infty}}
r^{-1+n}
=
O(\epsilon_{i,j}^{\frac{n}{n-2\gamma}}).
\end{split}
\end{equation*}
\end{enumerate}
Collecting terms the assertion follows.
\end{pfn}

We turn now to the proof of the estimates 
\eqref{epsilon_i_j_symmetry},\eqref{varphi_in_a_j_coordinates}, \eqref{I2_interaction_integral_estimate} and
\eqref{epsilon_varepsilon_relation}
\vspace{6pt}

\noindent
\begin{pfn}{ of the estimate \eqref{epsilon_i_j_symmetry}}\\
First we notice, that due to corollary \ref{Greens_function_asymptotics} and 
\eqref{bubble_boundary} we have
\begin{equation*}
\varphi_{k}=(1+o_{\varepsilon_{k}}(1))(\frac{\lambda_{k}}{1+\lambda_{k}^{2}G_{a_
{ k } } ^{ \frac { 2 } { 2\gamma-n } } } )^{\frac{n-2\gamma}{2}}
\end{equation*}
for any choice \;$\varepsilon_{k}\sim \lambda_{k}^{-\frac{1}{l}}$, whence 
\begin{equation}\label{epsilon_i,j_expansion}
\begin{split}
\epsilon_{i,j}
= &
\int_{B_{\varepsilon_{i}}(a_{i})}(u_{a_{i}}\varphi_{a_{i},\lambda_{i}})^{\frac{
n+2\gamma }{
n-2\gamma } }u_{a_{j}}\varphi_{a_{j}}dV_{h} \\
= &
(1+o_{\frac{1}{\lambda_{j}}}(1))
\underset{B_{\varepsilon_{i}\lambda_{i}}(0)}{\int}
(\frac{1}{1+r^{2}})^{\frac{n+2\gamma}{2}}
\frac
{\frac{u_{a_{j}}}{u_{a_{i}}}(\exp_{g_{a_{i}}}\frac{x}{\lambda_{i}})}
{
(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}G_{a_{j}}^{\frac{2}{
2\gamma-n} }(\exp_{g_{a_{i}}}\frac{x}{\lambda_{i}}) )^{\frac{n-2\gamma}{2}}
}
\end{split}
\end{equation}
up to some  \;$o_{\frac{1}{\lambda_{i}}}((\lambda_{i}\lambda_{j})^{\frac{2\gamma-n}{2}})$, where $G_a=G^\gamma_{h_a}$. In case \;$d_{g_{a_{j}}}(a_{i},a_{j})>\sqrt{\varepsilon_{i}}$\; we 
thus get
\begin{equation*}
\epsilon_{i,j}
=
(1+o_{\max(\frac{1}{\lambda_{i}},\frac{1}{\lambda_{j}})}(1))
(\frac{1}{\lambda_{i
}\lambda_{j}G_{h}^{\frac{2}{2\gamma-n}}(a_{i},a_{j}) })^{\frac{n-2\gamma}{2}}
\end{equation*}
by conformal covariance of the Green's function, i.e.
\begin{equation*}\begin{split}
G_{a_{j}}(a_{j},a_{i})=u_{a_{j}}^{-1}(a_{i})u_{a_{j}}^{-1}(a_{j})G_{h}(a_{i}
,a_{j})\;\;\text{ recalling }\;\;
h_{a_{j}}=u_{a_{j}}^{\frac{4}{n-2\gamma}}h,
\end{split}\end{equation*}
where \;$G_h=G^\gamma_h$. Therefore \;\eqref{epsilon_i_j_symmetry}\; follows by symmetry in this case. 
Otherwise we may assume  \;$d_{g_{a_{j}}}(a_{i},a_{j})\leq 
\sqrt{\varepsilon_{i}}$ 
and passing to \;$h_{a_{i}}$-normal coordinates  \eqref{epsilon_i,j_expansion} 
becomes
\begin{equation*}
\begin{split}
\epsilon_{i,j}
= &
(1+o_{\max(\frac{1}{\lambda_{i}},\frac{1}{\lambda_{j}})}(1))
\int_{B_{\varepsilon_{i}\lambda_{i}}(0)}
(\frac{1}{1+r^{2}})^{\frac{n+2\gamma}{2}}
(\frac
{1}
{
\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}\vert 
\exp^{-}_{g_{a_{i}}}a_{j} -\frac{x}{\lambda_{i} 
}\vert^{2}
}
)^{\frac{n-2\gamma}{2}} \\
= &
(1+o_{\max(\frac{1}{\lambda_{i}},\frac{1}{\lambda_{j}})}(1))
\int_{\mathbb{R}^{n}}
(\frac{\lambda_{i}}{1+\lambda_{i}^{2}r^{2}})^{\frac{n+2\gamma}{2}}
(\frac
{\lambda_{j}}
{
1+\lambda_{j}^{2}\vert 
\exp^{-}_{a_{i}}a_{j} -x\vert^{2}
}
)^{\frac{n-2\gamma}{2}}
\end{split}
\end{equation*}
up to some 
$o_{\frac{1}{\lambda_{i}}}((\lambda_{i}\lambda_{j})^{\frac{2\gamma-n}{2}})$.
Using \;$\exp^{-}_{g_{a_{i}}}a_{j}=(1+o_{\frac{1}{\lambda_{i}}}(1))\exp^{-}_{g_{a_{j}}}
a_ {i} $\; and 
\begin{equation*}
\int_{\mathbb{R}^{n}} \delta_{i}^{\frac{n+2\gamma}{n-2\gamma}}\delta_{j}
=\frac{1}{c_{n, \gamma}}
\int_{\mathbb{R}^{n}} (-\Delta)^\gamma\delta_{i}\delta_{j}
=\frac{1}{c_{n, \gamma}}
\int_{\mathbb{R}^{n}} \delta_{i}(-\Delta)^\gamma\delta_{j}
=
\int_{\mathbb{R}^{n}} \delta_{i}\delta_{j}^{\frac{n+2\gamma}{n-2\gamma}}
\end{equation*}
we thus find up to some 
$o_{\frac{1}{\lambda_{i}}}
((\lambda_{i}\lambda_{j})^{\frac{2\gamma-n}{2}})$
\begin{equation*}
\begin{split}
\epsilon_{i,j}
= &
(1+o_{\max(\frac{1}{\lambda_{i}},\frac{1}{\lambda_{j}})}(1))
\int_{\mathbb{R}^{n}}
(\frac{\lambda_{j}}{1+\lambda_{j}^{2}r^{2}})^{\frac{n+2\gamma}{2}}
(\frac
{\lambda_{i}}
{
1+\lambda_{i}^{2}\vert 
\exp^{-}_{g_{a_{j}}}a_{i} -x\vert^{2}
}
)^{\frac{n-2\gamma}{2}}.
\end{split}
\end{equation*}
This shows \;$\epsilon_{i,j}=O(\varepsilon_{i,j})$, cf. lemma 
\ref{lem_standard interaction on Rn} (i), and calculating back we find  
\begin{equation*}
\epsilon_{i,j}
=
(1+o_{\max(\frac{1}{\lambda_{i}},\frac{1}{\lambda_{j}})} 
(1))\epsilon_{j,i}
+
o_{\max(\frac{1}{\lambda_{i}},\frac{1}{\lambda_{j}})} 
((\lambda_{i}\lambda_{j})^{\frac{2\gamma-n}{2}}).
\end{equation*}
Thereby \;\eqref{epsilon_i_j_symmetry} follows.
\end{pfn}
\vspace{6pt}

\noindent

\begin{pfn}{ of the estimate \eqref{varphi_in_a_j_coordinates}}\\
Note, that due to corollary 
\ref{Poisson_kernel_asymptotics}
\begin{equation*}
\begin{split}
K_{g_{a_{i}}}(y,x,\xi)
\leq &
Cy^{2\gamma}
(
(\frac{1}{y^{2}+d^{2}_{g_{a_{i}}}(x,\xi)})^{\frac{n+2\gamma}{2}}
+
1
)
\end{split}
\end{equation*} 
for $(y,x)\in B_{\rho_{0}}^{a_i, +}(a_{i})=B_{\rho_{0}}^{g_{a_i}, +}(a_{i})$, whence
\begin{equation*}
\begin{split}
\vert \overline{\frac{u_{a_{j}}}{u_{a_{i}}}\varphi_{j}}^{a_{i}} \vert(y,x)
\leq & 
C
\underset{M}{\int} 
(
\frac{y^{2\gamma}}{(y^{2}+d^{2}_{g_{a_{i}}}(x,\xi))^{\frac{n+2\gamma}{2}}}
+
y^{2\gamma}
) 
(\frac{\lambda_{j}}{1+\lambda_{j}^{2}d_{g_{a_{j}}}^{2}(a_{j},\xi)})^{\frac{
n-2\gamma } {2}
}dV_{g_{a_{i}}}
\end{split}
\end{equation*} 
and thus 
\begin{equation*}\begin{split}
\vert \overline{\frac{u_{a_{j}}}{u_{a_{i}}}\varphi_{j}}^{a_{i}} & \vert(y,x)
\leq 
\frac{C}{y^{\frac{n-2\gamma}{2}}}
\int_{M} 
(\frac{y^{-1}}{1+y^{-2}d^{2}_{g_{a_{i}}}(x,\xi)})^{\frac{n+2\gamma}{2}}
(\frac{\lambda_{j}}{1+\lambda_{j}^{2}d_{g_{a_{j}}}^{2}(a_{j},\xi)})^{\frac{
n-2\gamma}{2}}dV_{g_{a_{i}}}
+
\frac{Cy^{2\gamma}}{\lambda_{j}^{\frac{n-2\gamma}{2}}}
\\
\leq &
\frac{C}{y^{\frac{n-2\gamma}{2}}} 
\underset{[d_{g_{a_{i}}}(x,\xi)\leq \rho_{0}]\cap[d_{g_{a_{j}}}(a_{j},\xi)\leq 
\rho_{0}]} {\int}
(\frac{y^{-1}}{1+y^{-2}d^{2}_{g_{a_{i}}}(x,\xi)})^{\frac{n+2\gamma}{2}}(\frac{\lambda_{j}}{1+\lambda_{j}^{2}d_{g_{a_{j}}}^{2}(a_{j},\xi)})^{\frac{
n-2\gamma}{2}}dV_{g_{a_{i}}}
+
\frac{Cy^{2\gamma}}{\lambda_{j}^{\frac{n-2\gamma}{2}}}.
\end{split}
\end{equation*}
Since all the distances involved are comparable to $d_{g}$, the first summand 
above vanishes in case  $d_{g}(x,a_{j})\geq C\rho_{0}$ and the claim follows.
Else we may assume \;$d_{g}(x,a_{j})\leq C\rho_{0}$\; and find passing to 
integration over \;$\mathbb{R}^{n}$\; by standard interaction estimates as given in lemma 
\ref{lem_standard interaction on Rn} 
\begin{equation*}\begin{split}
\vert \overline{\frac{u_{a_{j}}}{u_{a_{i}}}\varphi_{j}}^{a_{i}} \vert(y,x)
\leq &
\frac{C}{y^{\frac{n-2\gamma}{2}}}
(\frac{1}
{
\lambda_{j} y+\frac{1}{\lambda_{j} y}+\frac{\lambda_{j}}{y}d^{2}_{g}(a_{j},x)
})^{\frac{n-2\gamma}{2}}
+
\frac{Cy^{2\gamma}}{\lambda_{j}^{\frac{n-2\gamma}{2}}} \\
\leq &
C(\frac{\lambda_{j}}{1+\lambda_{j}^{2}\vert 
y^{2}+d_{g}^{2}(a_{j},x)\vert})^{\frac{n-2\gamma}{2}}
+
\frac{C}{\lambda_{j}^{\frac{n-2\gamma}{2}}}.
\end{split}
\end{equation*} 
\end{pfn}
\vspace{6pt}

\noindent
\begin{pfn}{ of the estimate \eqref{I2_interaction_integral_estimate}}\\
Let us start showing ($B_r^{a, +}(a)=B_r^{g_a, +}(a)$)
\begin{equation}\label{I2_interaction_integral_estimate_proof}
\begin{split}
\int_{B^{a_i, +}_{2\varepsilon_{i}}(a_{i})}y^{1-2\gamma}\widetilde{\varphi_{i}}^{a_{
i }} \widetilde{\varphi_{j}}^{a_{j}} 
=
o_{\frac{1}{\lambda_{i}}}(\varepsilon_{i,j}).
\end{split}
\end{equation}  
In case \;$d_{g}(a_{i},a_{j})>c\gg \varepsilon_{i}$ we have
\begin{equation*}
\begin{split}
\int_{B^a_i, {+}_{2\varepsilon_{i}}(a_{i})}y^{1-2\gamma}\widetilde{\varphi_{i}}^{a_{
i }} \widetilde{\varphi_{j}}^{a_{j}}
\leq \frac{C}{\lambda_{i}^{\frac{n-2\gamma}{2}}}
\int_{B^{a_i, +}_{2\varepsilon_{i}}(a_{i})}y^{1-2\gamma}\widetilde{\varphi_{i}}^{a_{
i }}
=
O(\frac{\varepsilon_{i}^{2}}{(\lambda_{i}\lambda_{j})^{\frac{n-2\gamma}{2}}}),
\end{split}
\end{equation*}
so \;\eqref{I2_interaction_integral_estimate_proof}\; holds true in this case for 
any choice \;$\varepsilon_{i}\sim \lambda_{i}^{-\frac{1}{k}}$. Thus we may 
assume \;$d_{g}(a_{i},a_{j})\ll 1$\; for the rest of the proof and moreover, that
\begin{equation*}
\varepsilon_{i,j}^{\frac{2\gamma-n}{2}}\sim \frac{\lambda_{i}}{\lambda_{j}}\;\;
\text{ or }\,\;\;
\varepsilon_{i,j}^{\frac{2\gamma-n}{2}}\sim 
\lambda_{i}\lambda_{j}G_{h}^{\frac{2}{2\gamma-n}}(a_{i},a_{j}).
\end{equation*}
Passing to \;$g_{a_{i}}$- normal Fermi-coordinates and rescaling we then have
\begin{equation*}
\begin{split}
\int_{B^{a_i, +}_{2\varepsilon_{i}}(a_{i})} 
y^{1-2\gamma}\widetilde{\varphi_{i}}^{a_{
i }} \widetilde{\varphi_{j}}^{a_{j}}  
\leq 
\frac{C}{\lambda_{i}^{2}}
\int_{B^{a_i, +}_{2\varepsilon_{i}\lambda_{i}}(0)}
\frac{y^{1-2\gamma}}{(1+r^{2})^{\frac{n-2\gamma}{2}}}
(\frac{1}{\frac{\lambda_{i}}{\lambda_{j}}
+
\lambda_{i}\lambda_{j}\vert \frac{y}{\lambda_{i}},
a_{j}-a_{i}+\frac{x}{\lambda_{i}})\vert^{2}})^{\frac{n-2\gamma}{2}}.
\end{split}
\end{equation*}
In particular \;\eqref{I2_interaction_integral_estimate_proof} holds
in case \;$\varepsilon_{i,j}^{\frac{2\gamma-n}{2}}\sim 
\frac{\lambda_{i}}{\lambda_{j}}$, so we may assume 
\begin{equation*}
\varepsilon_{i,j}^{\frac{2\gamma-n}{2}}\sim 
\lambda_{i}\lambda_{j}G_{h}^{\frac{2}{2\gamma-n}}(a_{i},a_{j})
\sim 
\lambda_{i}\lambda_{j}\vert a_{i}-a_{j}\vert^{2}.
\end{equation*}
We subdivide the region of integration, 
i.e.  \;$B_{2\varepsilon_{i}\lambda_{i}}^{+}=B_{2\varepsilon_{i}\lambda_{i}}^{+}(0)$ in
\begin{enumerate}[label=(\roman*)]
 \item \quad
$
\mathcal{B}_{1}
=
[\vert \frac{z}{\lambda_{i}}\vert \leq \epsilon \vert a_{i}-a_{j}\vert]
\cap B_{2\varepsilon_{i}\lambda_{i}}^{+}
$
 \item \quad 
$
\mathcal{B}_{2}
=
[\epsilon\vert a_{i}-a_{j}\vert <\vert \frac{z}{\lambda_{i}}\vert \leq E 
\vert a_{i}-a_{j}\vert]
\cap B_{2\varepsilon_{i}\lambda_{i}}^{+}
$
 \item \quad
 $
\mathcal{B}_{1}
=
[\vert \frac{z}{\lambda_{i}}\vert > E\vert a_{i}-a_{j}\vert]
\cap B_{2\varepsilon_{i}\lambda_{i}}^{+}
$
\end{enumerate}
for \;$z=(y,x)$\; and suitable \;$0<\epsilon,E^{-1}\ll 1$\ and obtain easily
\begin{equation*}
\begin{split}
\int_{B^{a_i, +}_{2\varepsilon_{i}}(a_{i})} 
y^{1-2\gamma}\widetilde{\varphi_{i}}^{a_{
i }} \widetilde{\varphi_{j}}^{a_{j}}  
\leq 
\frac{C}{\lambda_{i}^{2}}
\int_{\mathcal{B}_{2}}
\frac{y^{1-2\gamma}}{(1+r^{2})^{\frac{n-2\gamma}{2}}}
(\frac{1}{\frac{\lambda_{i}}{\lambda_{j}}
+
\lambda_{i}\lambda_{j}\vert (\frac{y}{\lambda_{i}},
a_{j}-a_{i}+\frac{x}{\lambda_{i}})\vert^{2}})^{\frac{n-2\gamma}{2}}
\end{split}
\end{equation*}
up to some \;$o_{\frac{1}{\lambda_{i}}}(\varepsilon_{i,j})$\; for any choice \;
$\varepsilon_{i}\sim \lambda_{i}^{-\frac{1}{k}}$. Note, that on 
\;$\mathcal{B}_{2}$\; we have
\begin{equation*}
\epsilon \vert a_{i}-a_{j}\vert <\vert \frac{z}{\lambda_{i}}\vert \leq 
2\varepsilon_{i},
\end{equation*}
so  \;$\vert a_{i}-a_{j}\vert =O(\varepsilon_{i})$\; for 
\;$\mathcal{B}_{2}\neq \emptyset$.
We then find up to some $o_{\frac{1}{\lambda_{i}}}(\varepsilon_{i,j})$
\begin{equation*}
\begin{split}
\int_{B^{a_i, +}_{2\varepsilon_{i}}(a_{i})} &
y^{1-2\gamma}\widetilde{\varphi_{i}}^{a_{
i }} \widetilde{\varphi_{j}}^{a_{j}}  
\leq 
\frac{C}{\lambda_{i}^{2}(1+\lambda_{i}^{2}\vert
a_{i}-a_{j}\vert^{2})^{\frac{n-2\gamma}{2}}}
\int_{[\vert z \vert \leq C\lambda_{i}\vert a_{i}-a_{j}\vert]}
\frac{y^{1-2\gamma}}{(\frac{\lambda_{i}}{\lambda_{j}}+\frac{\lambda_{j}}{\lambda
_{i}} \vert
z\vert^{2})^{\frac{n-2\gamma}{2}}} \\
\leq &
\frac{C(\frac{\lambda_{i}}{\lambda_{j}})^{\frac{n-2\gamma}{2}}}{\lambda_{i}^{2}
(1+\lambda_{i}^{2}\vert
a_{i}-a_{j}\vert^{2})^{\frac{n-2\gamma}{2}}}
\int_{[\vert z \vert \leq C\lambda_{i}\vert a_{i}-a_{j}\vert]}
y^{1-2\gamma}\vert z \vert^{2\gamma-n} 
\leq 
C\vert a_{i}-a_{j}\vert^{2}\varepsilon_{i,j}
=
o_{\frac{1}{\lambda_{i}}}(\varepsilon_{i,j}),
\end{split}
\end{equation*}
whence \;\eqref{I2_interaction_integral_estimate_proof}\; holds again and thus in 
any case. We are left with proving 
\begin{equation*}
\int_{B^{a_i, +}_{2\varepsilon_{i}}(a_{i})}y^{1-2\gamma}\lambda_{i}y(\widetilde{
\varphi_{i}}^{a_{i}})^{\frac{n+2-2\gamma}
{ n-2\gamma } }\widetilde{\varphi_{j}}^{a_{j}}
=
o_{\frac{1}{\lambda_{i}}}(\varepsilon_{i,j}).
\end{equation*}
But this follows line by line as when showing  \;\eqref{I2_interaction_integral_estimate_proof}.
\end{pfn}
\vspace{6pt}

\noindent

\begin{pfn}{ of the estimate \eqref{epsilon_varepsilon_relation}}\\
We know
\begin{equation*}\begin{split}
\epsilon_{i,j}
= &
\underset{B_{ \varepsilon_{i}\lambda_{i} }(0)}{\int}
\frac{\frac{u_{ a _{j}}}{u_{a_{i}}}( 
\exp_{g_{a_{i}}}\frac{x}{\lambda_{i}})}{(1+r^{2})^{\frac{n+2\gamma}{2}}} 
(\frac{1}{\frac{\lambda_{i} }{ \lambda_{j} }+\lambda_{i}\lambda_{j}G_{ a 
_{j}}^{\frac{2}{2\gamma-n}}(\exp_{g_{ a _{i}}} 
\frac{x}{\lambda_{i}})})^{\frac{n-2\gamma}{2}}
dV_{h_{a_{i}}}.
\end{split}\end{equation*}
and have to show 
\begin{equation*
\begin{split}
\epsilon_{i,j}
= &
c_{n,3}^{\gamma}\varepsilon_{i,j}(1+o_{\varepsilon_{i,j}}(1)).
\end{split}\end{equation*}
Since \;$
\varepsilon_{i,j}^{\frac{2}{2\gamma-n}}
\sim 
\lambda_{i}\lambda_{j}G_{h}^{\frac{2}{2\gamma-n}}( a _{i}, a _{j})
\;\;\text{ or }\;\;
\varepsilon_{i,j}^{\frac{2}{2\gamma-n}}
\sim
\frac{\lambda_{i}}{\lambda_{j}}
$, we may expand on 
\begin{equation*}\begin{split}
\mathcal{A}
=
\left(
\left[
\vert  \frac{x}{\lambda_{i}}\vert \leq \epsilon\sqrt{G^{\frac{2}{2-n}}_{ a 
_{j}}( a _{i})}
\right]
\cup
\left[
\vert  \frac{x}{\lambda_{i}}\vert \leq \epsilon  \frac{1}{\lambda_{j}} 
\right]
\right)
\cap
B_{\varepsilon_{i} \lambda_{i}}(0)
\subseteq
B_{\varepsilon_{i} \lambda_{i}}(0)
\end{split}\end{equation*}
for \;$\epsilon>0$\; small  \;$\frac{u_{a_{j}}}{u_{a_{i}}}(\exp_{g_{a_{i}}}\frac{x}{\lambda_{i}})=u_{a_{j}}(a_
{ i} )+\nabla 
u_{a_{j}}(a_{i})\frac{x}{\lambda_{i}}+O(\vert 
\frac{x}{\lambda_{i}} 
\vert^{2})$\; and 
\begin{equation*}\begin{split}
(
\frac{\lambda_{i} }{ \lambda_{j} }
& +
\lambda_{i}\lambda_{j}G_{a_{j}}^{\frac{2}{2\gamma-n}}(\exp_{g_{ a _{i}}} 
\frac{x}{\lambda_{i}}))^{\frac{2\gamma-n}{2}} \\
= &
(
\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}
G_{a _{j}}^{\frac{2}{2\gamma-n}}( a _{i})
)^{\frac{2\gamma-n}{2}}
+
\frac{2\gamma-n}{2}
\frac
{
\nabla G^{\frac{2}{2\gamma-n}}_{ a _{j}}( a _{i})\lambda_{j}x
+
O(\frac{ \lambda_{j} }{ \lambda_{i} }\vert x\vert^{2})
}
{
(
\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}G_{ a _{j}}^{\frac{2}{2\gamma-n}}( a _{i})
)^{\frac{n+2-2\gamma}{2}}
}.
\end{split}\end{equation*}
Thus  \;$\epsilon_{i,j}=\sum_{k=1}^{5}I_{k}+o_{\frac{1}{\lambda_{i}}}(\varepsilon_{i,j}
)$ \;for
\begin{enumerate}[label=(\roman*)]
 \item \quad 
$
I_{1}
= 
\frac
{u_{ a _{j}}( a _{i})}
{(\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}
G_{ a _{j}}^{\frac{2}{2\gamma -n}}( a _{i})
)^{\frac{n-2\gamma}{2}}
}
\int_{\mathcal{A}}
\frac{1}{(1+r^{2})^{\frac{n+2\gamma}{2}}}
$
 \item \quad
$
I_{2}
= 
\frac
{1}
{(\frac{\lambda_{i}}{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}
G_{ a _{j}}^{\frac{2}{2\gamma-n}}( a _{i})
)^{\frac{n-2\gamma}{2}}
}
\int_{\mathcal{A}} \frac{\nabla 
u_{a_{j}}(a_{i})\frac{x}{\lambda_{i}}}{(1+r^{2})^{\frac{n+2\gamma}{2}}}
$
 \item \quad
$
I_{3}
=
\frac
{\frac{2\gamma-n}{2}u_{ a _{j}}( a _{i})}
{(\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}
G_{a _{j}}^{\frac{2}{2\gamma-n}}( a _{i})
)^{\frac{n+2-2\gamma}{2}}
}
\int_{\mathcal{A}} \frac{\nabla G^{\frac{2}{2\gamma-n}}_{a _{j}}(a 
_{i})\lambda_{j}x}{(1+r^{2})^{\frac{n+2\gamma}{2}}}
$
 \item \quad
$
I_{4}
= 
\frac
{1}
{(\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}
G_{ a _{j}}^{\frac{2}{2\gamma-n}}( a _{i})
)^{\frac{n+2-2\gamma}{2}}
}
\int_{\mathcal{A}}
\frac{O(\frac{ \lambda_{j} }{ \lambda_{i} }\vert x 
\vert^{2})}{(1+r^{2})^{\frac{n+2\gamma}{2}}}
$
 \item \quad
$
I_{5}
 = 
\underset{\mathcal{A}^{c}}{\int} \frac{ \frac{u_{ a _{j}}}{u_{a_{i}}}( 
\exp_{g_{a_{i}}}\frac{x}{\lambda_{i}})}{(1+r^{2})^{\frac{n+2\gamma}{2}}}
 (\frac{1}{\frac{\lambda_{i} }{ \lambda_{j} }+\lambda_{i}\lambda_{j}G_{ a 
_{j}}^{\frac{2}{2\gamma-n}}(\exp_{g_{ a _{i}}} 
\frac{x}{\lambda_{i}})})^{\frac{n-2\gamma}{2}}dV_{h_{a_{i}}}.
$
\end{enumerate}
We then find with \;$c_{n,3}^{\gamma}=\int_{\mathbb{R}^{n}}(\frac{1}{1+r^{2}})^{\frac{n+2\gamma}{2}}
$
\begin{equation*}\begin{split}
I_{1}
= &
c_{n,3}^{\gamma}\frac
{u_{ a _{j}}( a _{i})}
{(\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}
G_{ a _{j}}^{\frac{2}{2\gamma-n}}(a _{i})
)^{\frac{n-2\gamma}{2}}
}
(1+o_{\varepsilon_{i,j}}(1))
\end{split}\end{equation*}
whereas  \;$I_{2}=I_{3}=0$\; by radial symmetry and \;
$
I_{4}
= 
o_{\varepsilon_{i,j}}(\varepsilon_{i,j})
$. Moreover  
\begin{equation*}
\begin{split}
I_{5}=o_{\varepsilon_{i,j}}(\varepsilon_{i,j})
\end{split}
\end{equation*} 
in case \;$\varepsilon_{i,j}^{\frac{2}{2\gamma-n}}\sim \frac{\lambda_{i}}{\lambda_{j}}$.  Else we have \;$\frac{\lambda_{j}}{\lambda_{i}}\leq 
\frac{\lambda_{i}}{\lambda_{j}}\leq 
\lambda_{i}\lambda_{j}G_{h}^{\frac{2}{2\gamma-n}}(a_{i},a_{j})$\; and decompose 
\begin{equation*}
\begin{split}
\mathcal{A}^{c}
=
B_{\varepsilon_{i}\lambda_{i}}\setminus \mathcal{A}
\subseteq &
\mathcal{B}_{1}\cup \mathcal{B}_{2},
\end{split}
\end{equation*}
where for a sufficiently large constant \;$E>0$
\begin{equation*}
\begin{split}
\mathcal{B}_{1}
= &
[\epsilon \sqrt{G^{\frac{2}{2\gamma-n}}_{a_{j}}(a_{i})}
\leq 
\vert \frac{x}{\lambda_{i}}\vert 
\leq 
E\sqrt{G^{\frac{2}{2\gamma-n}}_{a_{j}}(a_{i})}] 
\;\; \text{and} \;\;
\mathcal{B}_{2}
= 
[
E\sqrt{G^{\frac{2}{2\gamma-n}}_{a_{j}}(a_{i})}
\leq 
\vert \frac{x}{\lambda_{i}}\vert 
\leq \epsilon_{i}
].
\end{split}
\end{equation*} 
We may assume 

G_{a_{j}}(a_{i})^{\frac{2}{2\gamma-n}}\sim d^{2}_{g_{a_{j}}(a_{j},a_{i})}\sim 
\ll 1,
$
since otherwise \;$I_{5}=o_{\varepsilon_{i,j}}(\varepsilon_{i,j})$, and estimate
\begin{equation*}
\begin{split}
I_{5}^{1}
= &
\underset{\mathcal{B}_{1}}{\int} \frac{\frac{ u_{a _{j}}}{u_{a_{i}}}(\exp_{g_{ 
a _{i}}} 
\frac{x}{\lambda_{i}})}{(1+r^{2})^{\frac{n+2\gamma}{2}}}dV_{h_{a_{i}}}
 (\frac{1}{\frac{\lambda_{i} }{ \lambda_{j} }+\lambda_{i}\lambda_{j}G_{ a 
_{j}}^{\frac{2}{2\gamma-n}}(\exp_{g_{ a _{i}}} 
\frac{x}{\lambda_{i}})})^{\frac{n-2\gamma}{2}} \\
\leq &
c\frac{(\frac{\lambda_{i}}{\lambda_{j}})^{\frac{n+2\gamma}{2}}}{(1+\lambda_{i}^{
2}G^{\frac{2}{2\gamma-n}}_{a_{j}}(a_{i}))^{\frac{n+2\gamma}{2}}} 
\underset{[\vert \frac{x}{\lambda_{j}}\vert \leq 
E\sqrt{G^{\frac{2}{2\gamma-n}}_{a_{j}}(a_{i})}]}{\int} 
 (\frac{1}{1+\lambda_{j}^{2}G_{ a _{j}}^{\frac{2}{2\gamma-n}}(\exp_{g_{ a 
_{i}}} \frac{x}{\lambda_{j}})})^{\frac{n-2\gamma}{2}}.
\end{split}
\end{equation*} 
Changing coordinates via \;$d_{i,j}=\exp_{g_{a_{i}}}^{-1}\exp_{g_{a_{j}}}$\; we get
\begin{equation*}
\begin{split}
I_{5}^{1}
\leq &
\frac{C}{(\frac{\lambda_{j}}{\lambda_{i}}+\lambda_{i}\lambda_{j}G^{\frac{2}{
2\gamma-n}}_{a_{j}}(a_{i}))^{\frac{n+2\gamma}{2}}}
\underset{[\vert \frac{x}{\lambda_{j}}\vert \leq  
C\sqrt{G_{a_{j}}^{\frac{2}{2\gamma-n}}}(a_{i})]}{\int} 
(\frac{1}{1+r^{2}})^{\frac{n-2\gamma}{2}}
\end{split}
\end{equation*} 
and thus \;$I_{4}^{1}=o_{\varepsilon_{i,j}}(\varepsilon_{i,j})$. Moreover
\begin{equation*}
\begin{split}
I_{5,2}
= &
\underset{\mathcal{B}_{2}}{\int} \frac{ \frac{u_{ a _{j}}}{u_{a_{i}}}(\exp_{g_{ 
a _{i}}} 
\frac{x}{\lambda_{i}})}{(1+r^{2})^{\frac{n+2\gamma}{2}}}
 (\frac{1}{\frac{\lambda_{i} }{ \lambda_{j} }+\lambda_{i}\lambda_{j}G_{ a 
_{j}}^{\frac{2}{2\gamma-n}}(\exp_{g_{ a _{i}}} 
\frac{x}{\lambda_{i}})})^{\frac{n-2\gamma}{2}} \\
 \leq &
\frac{C}{(\frac{\lambda_{i} }{ \lambda_{j} }+\lambda_{i}\lambda_{j}G_{a 
_{j}}^{\frac{2}{2\gamma-n}}(a_{i}))^{\frac{n-2\gamma}{2}}} 
\underset{[\vert x \vert \geq 
c\sqrt{\lambda_{i}^{2}G^{\frac{2}{2\gamma-n}}_{a_{j}}(a_{i})}]}{\int}
\frac{1}{(1+r^{2})^{\frac{n+2\gamma}{2}}}
=
o_{\varepsilon_{i,j}}(\varepsilon_{i,j}).
\end{split}
\end{equation*} 
Therefore \;$I_{5}=I_{5}^{1}+I_{5}^{2}=o_{\varepsilon_{i,j}}(\varepsilon_{i,j})$. 
Collecting terms we get 
\begin{equation*}
\begin{split}
\epsilon_{i,j}
= &
c_{n,3}^{\gamma}\frac
{u_{ a _{j}}( a _{i})}
{(\frac{\lambda_{i} }{ \lambda_{j} }
+
\lambda_{i}\lambda_{j}
G_{ a _{j}}^{\frac{2}{2\gamma-n}}(a _{i})
)^{\frac{n-2\gamma}{2}}
}
+
o_{\varepsilon_{i,j}}(\varepsilon_{i,j}).
\end{split}
\end{equation*}
Using conformal covariance of the Green's function, i.e.
\begin{equation*}\begin{split}
G_{a_{j}}(a_{j},a_{i})=u_{a_{j}}^{-1}(a_{i})u_{a_{j}}^{-1}(a_{j})G_{h}(a_{i}
,a_{j})\;\;\text{ recalling}\;\;
G_{a_i}=G_{h_{a_i}}^\gamma, \;\;h_{a_{j}}=u_{a_{j}}^{\frac{4}{n-2\gamma}}h,
\end{split}\end{equation*}
we conclude
$
\epsilon_{i,j}
= 
c_{n,3}^{\gamma}\varepsilon_{i,j}(1+o_{\varepsilon_{i,j}}(1)),
$
what was to be shown.\end{pfn}
\vspace{6pt}





