# A note on the Compactness of Poincaré-Einstein manifolds

Fang Wang  School of Mathematical Sciences, Shanghai Jiao Tong University, 800
Dongchuan Rd, Shanghai, 200240<EMAIL_ADDRESS>and  Huihuang Zhou
School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan
Rd, Shanghai, 200240<EMAIL_ADDRESS>
###### Abstract.

For a conformally compact Poincaré-Einstein manifold $(X,g_{+})$, we consider
two types of compactifications for it. One is $\bar{g}=\rho^{2}g_{+}$, where
$\rho$ is a fixed smooth defining function; the other is the adapted
(including Fefferman-Graham) compactification $\bar{g}_{s}=\rho^{2}_{s}g_{+}$
with a continuous parameter $s>\frac{n}{2}$. In this paper, we mainly prove
that for a set of conformally compact Poincaré-Einstein manifolds
$\\{(X,g_{+}^{(i)})\\}$ with conformal infinity of positive Yamabe type,
$\\{\bar{g}^{(i)}\\}$ is compact in $C^{k,\alpha}(\overline{X})$ topology if
and only if $\\{\bar{g}_{s}^{(i)}\\}$ is compact in some
$C^{l,\beta}(\overline{X})$ topology, provided that
$\bar{g}^{(i)}|_{TM}=\bar{g}_{s}^{(i)}|_{TM}=\hat{g}^{(i)}$ and
$\hat{g}^{(i)}$ has positive scalar curvature for each $i$. See Theorem 1.1
and Corollary 1.1 for the exact relation of $(k,\alpha)$ and $(l,\beta)$.

Wang and Zhou’s research are supported in part by National Natural Science
Foundation of China No. 11871331 and Shanghai Science and Technology
Innovation Action Plan No. 20JC1413100.

##  1\. Introduction

Suppose $\overline{X}^{n+1}$ is a $(n+1)$-dimensional smooth compact manifold
with boundary. Denote $X$ the interior of $\overline{X}$ and $M=\partial X$
the boundary. Let $\rho$ be a smooth boundary defining function, i.e.,

$0\leq\rho\in C^{\infty}(\overline{X}),\quad\rho>0\ \textrm{in
$X$},\quad\rho=0\ \textrm{on $M$},\quad d\rho|_{M}\neq 0.$

We call $(X,g_{+})$ a $C^{k,\alpha}$ conformally compact Poincaré-Einstein
manifold with conformal infinity $(M,[\hat{g}])$, if $g_{+}$ is a complete
metric in $X$ such that $\bar{g}=\rho^{2}g_{+}$ can be $C^{k,\alpha}$ extended
to $\overline{X}$ and satisfies

$\begin{cases}\mathrm{Ric}_{g_{+}}=-ng_{+}&\ \mathrm{in}\ X,\\\
\bar{g}|_{TM}\in[\hat{g}]&\ \mathrm{on}\ M.\end{cases}$

In this paper, we always require that $n\geq 3$, $k\geq 3,0<\alpha<1$ and
$\rho$ is fixed.

Given $(M,[\hat{g}])$, the existence and uniqueness of $(X,g_{+})$, such that
$(X,g_{+})$ is Poincaré-Einstein with conformal infinity $(M,[\hat{g}])$, are
difficult problems in general. See [An1, An2, An3, GL, ST, LSQ] and many other
works for related results. The compactness problem was developed recently in a
series of papers by Chang-Ge-Qing. See [CGQ1, CGQ2].

The compactness problem is formulated as follows: Given a set of
$C^{k,\alpha}$ conformally compact Poincaré-Einstein metrics
$\\{(X,g_{+}^{(i)})\\}$ with conformal infinity $\\{(M,[\hat{g}^{(i)}])\\}$,
if the boundary representatives $\\{\hat{g}^{(i)}\\}$ is compact in
$C^{k,\alpha}(M)$ topology, whether
$\displaystyle\\{\bar{g}^{(i)}=(\rho^{(i)})^{2}g_{+}^{(i)}\\}$ is also compact
in $C^{k,\alpha}(\overline{X})$ topology by a suitable choice of
$\\{\rho^{(i)}\\}$ such that $\bar{g}^{(i)}|_{TM}=\hat{g}^{(i)}$. In [CGQ1,
CGQ2], the authors proved that it is true under certain geometric assumptions
for the case $M=\mathbb{S}^{3}$. In particular, they chose
$\\{\bar{g}^{(i)}\\}$ to be the Fefferman-Graham compactification of
$\\{g_{+}^{(i)}\\}$ when deriving the compactness, because for $n=3$, this
imposes an extra curvature vanishing condition for $\\{\bar{g}^{(i)}\\}$.

In this paper, we mainly want to show that once we get the compactness for one
type of compactification, then it may also hold for some other type. We
consider a family of compactifications defined by a real parameter $s$ for a
given $C^{k,\alpha}$ conformally compact Poincaré-Einstein manifold
$(X,g_{+})$ with conformal infinity $(M,[\hat{g}])$:

  * (i)

For $s>\frac{n}{2},s\neq n$, if $\mathrm{Spec}(-\Delta_{+})>s(n-s)$, let
$v_{s}$ satisfy the following equation:

$-\Delta_{+}v_{s}-s(n-s)v_{s}=0,\quad
v_{s}=\rho^{n-s}\left(1+O(\rho^{\epsilon})\right)\ \textrm{for some
$\epsilon>0$}.$

Then the adapted compactification of $g_{+}$ is defined by

$\bar{g}_{s}=\rho_{s}^{2}g_{+},\quad\mathrm{where}\quad\rho_{s}=v_{s}^{\frac{1}{n-s}}.$

  * (ii)

For $s=n$, let $w$ satisfy the following equation

$-\Delta_{+}w=n,\quad w=\log\rho+O(\rho^{\epsilon})\ \textrm{for some
$\epsilon>0$}.$

Then the Fefferman-Graham compactification of $g_{+}$ is defined by

$\bar{g}_{F}=\rho_{F}^{2}g_{+},\quad\mathrm{where}\quad\rho_{F}=e^{w}.$

The spectrum condition required in (i) can be satisfied by a nonnegativity
assumption on the Yamabe constant of the conformal infinity. In [Le1], Lee
showed that if the boundary Yamabe constant $\mathcal{Y}(M,[\hat{g}])\geq 0$,
then $\mathrm{Spec}(-\Delta_{+})\geq\frac{n^{2}}{4}$. In this case, we can
view the Fefferman-Graham compactification as a limit of the adapted
compactificaions:

$\lim_{s\rightarrow n}\bar{g}_{s}=\bar{g}_{F}.$

Hence by taking $g_{n}=g_{F}$, $\\{\bar{g}_{s}:s>\frac{n}{2}\\}$ forms a
continuous family of compactifications. See Section 4 for the details.

The adapted and Fefferman-Graham compactifications were first introduced in
[CC] and [FG2] respectively, which satisfy certain geometric curvature
conditions and have certain advantage to study the fractional GJMS operators
on the boundary, as well as the correspondence of interior Einstein metric
with boundary conformal geometry. For example, while
$s=\frac{n}{2}+N-\frac{1}{2}$ for some positive integer $N$, then the $2N$-th
order Q-curvature of $\bar{g}_{s}$ vanishes, i.e. $Q_{2N}^{\bar{g}_{s}}=0$.

The main theorem of this paper is the following.

######  Theorem 1.1.

Suppose $\\{(X,g^{(i)}_{+})\\}$ is a set of conformally compact Poincaré-
Einstein manifolds with conformal infinity $\\{(M,[\hat{g}^{(i)}])\\}$
satisfying $\mathcal{Y}(M,[\hat{g}^{(i)}])\geq 0$. Let $\rho$ be a fixed
smooth defining function and

$\bar{g}^{(i)}=\rho^{2}g^{(i)}_{+},\quad\bar{g}^{(i)}|_{TM}=\hat{g}^{(i)}.$

For $s>\frac{n}{2}$, let $\\{\bar{g}^{(i)}_{s}\\}$ be the adapted (including
Fefferman-Graham) compactification of $g^{(i)}_{+}$ with the fixed boundary
representatives

$\bar{g}_{s}^{(i)}|_{TM}=\bar{g}^{(i)}|_{TM}=\hat{g}^{(i)}.$

  * (a)

If $\\{\bar{g}^{(i)}\\}$ is compact in $C^{k,\alpha}(\overline{X})$ $(k\geq
3,0<\alpha<1)$ topology, then $\\{\bar{g}_{s}^{(i)}\\}$ is compact in
$C^{l,\beta}(\overline{X})$ $(l\geq 0,0<\beta<1)$ topology, where

    * (a.1)

$l=k,0<\beta<\alpha$ if $k+\alpha<2s-n$;

    * (a.2)

$l=k,0<\beta<\alpha$ if $k+\alpha\geq 2s-n$ and $2s-n=2N-1$ for some positive
integer $N$;

    * (a.3)

$l+\beta<2s-n$ if $k+\alpha\geq 2s-n$ and $2s-n$ is not a positive odd
integer.

  * (b)

If $\\{\bar{g}_{s}^{(i)}\\}$ is compact in $C^{k,\alpha}(\overline{X})$
$(k\geq 3,0<\alpha<1)$ topology for some $s>\frac{n}{2}+1$ or
$s=\frac{n+1}{2}$, and the boundary scalar curvature $\hat{R}^{(i)}>0$ for all
$i$, then $\\{\bar{g}^{(i)}\\}$ is compact in $C^{k,\beta}(\overline{X})$
$(0<\beta<\alpha)$ topology.

The proof of statement (a) is mainly based on the uniform Hölder norm
estimates for the adapted (including Fefferman-Graham) boundary defining
function $\rho_{s}$ and the ratio $\rho_{s}/\rho$, where $\rho$ is the fixed
boundary defining function (background coordinate). Here the main difficulty
is that the equations which determine $\rho_{s}$ and $\rho_{s}/\rho$ are
degenerate near the boundary and hence the classical Schauder estimates fail.
We overcome this difficulty by combining asymptotic analysis and delicate
extension theorem (Theorem 7.1) with improved Schauder estimates on weighted
Hölder spaces (Definition 3.1) for uniformly degenerate operators given by Lee
in [Le2]. These improved Schauder estimates are only valid for the weight in a
certain indicial gap determined by the operators. Hence for $k+\alpha\geq
2s-n$, it implies a loss of regularity in general. However, while $2s-n=2N-1$
for some positive integer $N$, we can recover the full regularity by using the
Q-curvature equation $Q_{2N}^{\bar{g}_{s}}=0$. Otherwise, we can view this
loss of regularity as the result of the fractional order term $\rho^{2s-n}$ in
the asymptotical expansion of $\rho_{s}$ at boundary.

The proof of statement (b) is mainly by exploring the geometric positivities
for $\bar{g}_{s}$ under the assumption $\hat{R}>0$, which gives $L^{\infty}$
estimates for $\rho_{s}$ and $|d\rho_{s}|_{\bar{g}_{s}}$ directly. In this
case, the equations that determine $\rho_{s}$ are uniformly elliptic with
background metric $\bar{g}_{s}$, and hence higher order norm estimates follow
from the classical Schauder estimates. Moreover, $\hat{R}>0$ also implies that
for $\gamma\in(0,1)$ the fractional Q-curvature $Q^{\hat{g}}_{2\gamma}$ is
positive on the boundary by [GQ]. Hence if $s\in(\frac{n}{2},\frac{n}{2}+1)$,
$\\{\bar{g}_{s}\\}$ can not have $C^{k,\alpha}(\overline{X})$ regularity
except for $s=\frac{n+1}{2}$. This is why we only consider $s>\frac{n}{2}+1$
or $s=\frac{n+1}{2}$.

Theorem 1.1 implies the following corollary directly.

######  Corollary 1.1.

Suppose $\\{(X,g^{(i)}_{+})\\}$ is a family of conformally compact Poincaré-
Einstein manifold with conformal infinity $\\{(M,[\hat{g}^{(i)}])\\}$
satisifying $\mathcal{Y}(M,[\hat{g}^{(i)}])\geq 0$. Let $\rho$ be a fixed
smooth defining function and $\rho^{2}g^{(i)}_{+}|_{TM}=\hat{g}^{(i)}.$ Let
$N,K$ be two positive integers and
$s=\frac{n}{2}+N-\frac{1}{2},t=\frac{n}{2}+K-\frac{1}{2}$. Assume that

$\bar{g}_{s}^{(i)}|_{TM}=\bar{g}^{(i)}_{t}|_{TM}=\hat{g}^{(i)}$

satisfying $\hat{R}^{(i)}>0$ for all $i$. If $\\{\bar{g}_{s}^{(i)}\\}$ is
compact in $C^{k,\alpha}(\overline{X})$ $(k\geq 3,0<\alpha<1)$ topology, then
$\\{\bar{g}_{t}^{(i)}\\}$ is also compact in $C^{k,\beta}(\overline{X})$
$(0<\beta<\alpha)$ topology.

The outline of the paper is as follows. In Section 2, we review some basic
properties for Poincaré-Einstein manifolds and its conformal compactification.
In Section 3, we introduce the improved Schauder estimates and invertibility
theorem for $\Delta_{+}+s(n-s)$ given by Lee in [Le2]. In Section 4, we prove
the global regularity theorem for $\bar{g}_{s}$. In Section 5, we investigate
the geometric positivity results for $\bar{g}_{s}$ under the assumption
$\hat{R}>0$ for $\hat{g}=\bar{g}_{s}|_{TM}$. In Section 6, we prove Theorem
1.1. In Section 7, we provide an extension theorem, which is used in the proof
of global regularity of $\bar{g}_{s}$.

Achknowlegement: The first author wants to thank Professor Sun-Yung Alice
Chang for proposing the question about the compactness in different type of
compactifications for Poincaré-Einstein manifolds.

##  2\. Poincaré-Einstein Manifold and Its Conformal Compactification

Suppose $(X^{n+1},g_{+})$ ($n\geq 3$) is a $C^{k,\alpha}$ conformally compact
Poincaré-Einstein manifold with conformal infinity $(M,[\hat{g}])$ with $k\geq
3$, $0<\alpha<1$. Let $\rho$ be a fixed smooth boundary defining function.
Denote

$\bar{g}=\rho^{2}g_{+}\quad\textrm{and}\quad\bar{g}|_{M}=\hat{g}.$

For any $l\geq 0$ and $0\leq\beta<1$, denote $C^{l,\beta}(X)$ and
$C^{l,\beta}(\overline{X})$ the classical Hölder spaces consists of functions
with local and uniform $C^{l,\beta}$ Hölder continuity.

###  2.1. Conformal Transformation

Let $\bar{R},\bar{R}_{ij}$ be the scalar curvature and Ricci curvature tensor
of $\bar{g}$. Let $\bar{J}=\frac{1}{2n}\bar{R}$ and $\bar{E}_{ij}$ be the
trace free part of $\bar{R}_{ij}$. Let $H$ be the boundary mean curvature for
$(\overline{X},\bar{g})$. Notice that for $k\geq 3$, a $C^{k,\alpha}$
compactification of Poincaré-Einstein manifold, $(\overline{X},\bar{g})$
always has an umbilical boundary. Hence the second fundamental form of the
boundary is

$\Pi=\frac{1}{n}{H}\hat{g}.$

Let $\Delta_{+}$ and $\bar{\Delta}$ be the Beltrami-Laplace operators for
$g_{+}$ and $\bar{g}$ respectively. The conformal transformation implies the
following relation between the two operators:

(2.1) $\Delta_{+}=\rho^{2}\bar{\Delta}-(n-1)\rho\langle
d\rho,d\cdot\rangle_{\bar{g}};$

and it also gives the scalar curvature and Ricci curvature of $\bar{g}$ in
terms of $\rho$:

(2.2)
$\bar{J}=-\rho^{-1}\bar{\Delta}\rho-\frac{n+1}{2}\rho^{-2}\left(1-|d\rho|^{2}_{\bar{g}}\right),$
(2.3)
$\bar{E}=-(n-1)\rho^{-1}\left(\bar{\nabla}^{2}\rho-\frac{1}{n+1}(\bar{\Delta}\rho)\bar{g}\right).$

Notice that by assuming $\bar{g}\in C^{k,\alpha}(\overline{X})$, we have
$|d\rho|^{2}\in C^{k,\alpha}(\overline{X})$ and (2.2) implies that
$|d\rho|_{\bar{g}}^{2}=1$ on $M$. This means the sectional curvature of
$g_{+}$ approaches to $-1$ at the boundary by [Ma]. We also define the
T-curvature of $\bar{g}$ as follows:

(2.4) $\bar{T}=\rho^{-1}\left(1-|d\rho|^{2}_{\bar{g}}\right).$

Then $\bar{T}\in C^{k,\alpha}(X)\cap C^{k-1,\alpha}(\overline{X}).$ By (2.2)
again,

$\rho\bar{J}=-\bar{\Delta}\rho-\frac{n+1}{2}T.$

Therefore, $\bar{J},\bar{E}\in C^{k-1,\alpha}(X)\cap
C^{k-2,\alpha}(\overline{X})$.

###  2.2. GJMS Operators

We recall the GJMS operators for $(X,g_{+})$ and $(\overline{X},\bar{g})$
here. Since $g_{+}$ and $\bar{g}$ are (conformal) Einstein metrics, the GJMS
operators of all orders are well defined for them. Let $P^{+}_{2N}$ and
$\bar{P}_{2N}$ be the GJMS operators of order $2N$ w.r.t. $g_{+}$ and
$\bar{g}$ respectively. Here we only consider the case $2\leq 2N\leq k$
because of the finite regularity. Then

$P^{+}_{2N}=\Pi_{j=1}^{N}\left(-\Delta_{+}-\frac{(n+2N-4j+3)(n-2N+4j-3)}{4}\right),$
$\bar{P}_{2N}=\rho^{-\frac{n+1}{2}-N}P^{+}_{2N}\rho^{\frac{n+1}{2}-N}=\left(-\bar{\Delta}\right)^{N}+L.O.T.,$

where $L.O.T.$ denotes a differential operator of order $\leq 2N-2$. The
Q-curvature of the corresponding order is given by

$\bar{Q}_{2N}=\frac{2}{n+1-2N}\bar{P}_{2N}(1).$

If $2N=n+1$, then the formula is understood as a formal ”limit” of
$n+1\rightarrow 2N$. See [GJMS, FG1, Go, CC] for more details. In particular,
for $N=1$,

$\bar{Q}_{2}=\frac{1}{2n}\bar{R}=\bar{J}.$

For $N=2$,

$\bar{Q}_{4}=-\bar{\Delta}\bar{J}+\frac{n+1}{2}\bar{J}^{2}-2|\bar{A}|^{2},$

where $\bar{A}$ is the Schouten tensor of $\bar{g}$, i.e.

$\bar{A}=\frac{1}{n-1}\left(\bar{R}_{ij}-\bar{J}\bar{g}_{ij}\right).$

Recursive formulae given in [Ju] show that for all $N\geq 1$,

$\bar{Q}_{2N}=(-\Delta)^{N-1}\bar{J}+L.O.T.,$

where $L.O.T$ is a polynomial of derivatives of metric $\bar{g}$ of order
$\leq 2N-2$ with coefficients given by $\bar{g}$ and $\bar{g}^{-1}$ in local
coordinates.

##  3\. Analysis on Poincaré-Einstein manifold

In this section, we recall some analysis on a $C^{k,\alpha}$ conformally
compact Poincaré-Einstein manifold $(X^{n+1},g_{+})$. Since our background
metric $g_{+}$ has only finite regularity, we will mainly follow from [Le2].
Let $\rho$ be the fixed smooth boundary defining function and

$\bar{g}=\rho^{2}g_{+},\quad\hat{g}=\bar{g}|_{TM}.$

Denote $X_{\epsilon}=\\{0<\rho<\epsilon\\}$ for $\epsilon>0$ small. Then
$X\backslash X_{\epsilon}$ is a compact subset of $X$.

###  3.1. Möbius Coordinates

First, we can cover $M$ by finitely many smooth coordinate charts
$(\Omega,\Theta)$, called the background coordinates, where
$\Theta=(\rho,\theta)=(\rho,\theta^{1},\cdots,\theta^{n})$. After fixing a
choice of such background coordinates $(\Omega,\Theta)$, they form a finite
covering of a neighbourhood $W$ of $M$ in $\overline{X}$. Then there is a
positive number $c>0$ such that $X_{c}\subset W$ and every point $p\in X_{c}$
is contained in a background coordinate chart containing a set of the
following form

$\\{(\rho,\theta):|\theta-\theta_{p}|<c,0\leq\rho<c\\}.$

Let $\mathbb{H}$ be the half plane model of the hyperbolic space with
coordinates $(x,y)=(x,y^{1},\cdots y^{n})$. Let $g_{\mathbb{H}}$ be the
hyperbolic metric $g_{\mathbb{H}}=x^{-2}(dx^{2}+dy^{2})$. Let $B_{r}$ be the
geodesic ball in $\mathbb{H}$ with radius $r$ and center $(1,0,\cdots,0)$. If
$p_{0}\in X_{c/8}$, then choose the background chart containing $p_{0}$ and
define a Möbius chart centered at $p_{0}$ by

$\Phi_{p_{0}}:B_{2}\longrightarrow
X,\quad(\rho,\theta)=\Phi_{p_{0}}(x,y)=(\theta_{0}+\rho_{0}x,\rho_{0}y),$

where $(\rho_{0},\theta_{0})$ is the background coordinates of $p_{0}$. It is
easy to check that for all $0<r\leq 2$,
$V_{r}(p_{0})=\Phi_{p_{0}}(B_{r})\subset\Phi_{p_{0}}(\overline{B_{2}})\subset
X_{c}$. We can also choose finite smooth coordinate charts
$\Phi_{i}:B_{2}\longrightarrow X$ such that the sets $\Phi_{i}(B_{1})$ cover a
neighborhood of $X\backslash X_{c/8}$. For consistency, we will also call
these ”Möbius Charts”. Without loss of generality, we can require that $\Phi$
extends smoothly to a neighborhood of $B_{2}$. We recall the following two
properties for Möbius charts given in [Le2].

######  Lemma 3.1.

There exists a constant $C>0$ such that if $\Phi_{p_{0}}:B_{2}\rightarrow X$
is any Möbius chart, then

$\|\Phi^{*}_{p_{0}}g_{+}-g_{\mathbb{H}}\|_{C^{k,\alpha}(\overline{B_{2}})}\leq
C,\quad\sup_{B_{2}}\left(|(\Phi^{*}_{p_{0}}g_{+})^{-1}g_{\mathbb{H}}|+|(\Phi^{*}_{p_{0}}g_{+})g_{\mathbb{H}}^{-1}|\right)\leq
C,$

where $\|\cdot\|_{C^{k,\alpha}(\overline{B_{2}})}$ is the classical Euclidean
Hölder norm.

######  Lemma 3.2.

There exists a countable collection of points $p_{i}\subset X$ and
corresponding Möbius chart: $\Phi_{i}=\Phi_{p_{i}}:B_{2}\longrightarrow
V_{2}(p_{i})$ such that $\\{V_{1}(p_{i})\\}$ cover $X$ and
$\\{V_{2}(p_{i})\\}$ are uniformly locally finite.

In this paper, $(\overline{X},\rho)$ are always fixed and hence the choice of
background coordinates and Möbius charts can also be fixed, while dealing with
different metrics.

######  Remark 3.1.

If $\\{g_{+}^{(i)}\\}$ is a set of $C^{k,\alpha}$ conformally compact
Poincaré-Einstein metric such that $\\{\bar{g}^{(i)}=\rho^{2}g_{+}^{(i)}\\}$
is compact in the $C^{k,\alpha}(\overline{X})$ topology, then Lemma 3.1 holds
for all $i$ with a uniform constant $C>0$.

###  3.2. Function Spaces

For $l\geq 0$ and $0\leq\beta<1$, recall that $C^{l,\beta}(X)$ and
$C^{l,\beta}(\overline{X})$ are the classical Hölder spaces consisting of
functions with local and uniform $C^{l,\beta}$ Hölder continuity respectively.
Here we introduce two types of function spaces dealing with the asymptotical
behavior of functions at the boundary.

For $0\leq\delta\leq l+\beta$, define a subspace
$C^{l,\beta}_{(\delta)}(\overline{X})\subset C^{l,\beta}(\overline{X})$ by

$C^{l,\beta}_{(\delta)}(\overline{X})=\\{u\in
C^{l,\beta}(\overline{X}):u=O(\rho^{\delta})\\}.$

It is obviously that
$C^{l,\beta}_{(0)}(\overline{X})=C^{l,\beta}(\overline{X})$. The following
Lemma is proved in [Le2, Lemma 3.1].

######  Lemma 3.3 (Lee).

Let $0\leq\beta<1$ and $0\leq\delta\leq l+\beta$.

  * (1)

$\displaystyle C^{l,\beta}_{(\delta)}(\overline{X})=\\{u\in
C^{l,\beta}(\overline{X}):\partial_{\rho}^{j}u|_{M}=0,\ \forall\ 0\leq
j<\delta\\}.$

  * (2)

$\displaystyle C^{l,\beta}_{(\delta)}(\overline{X})$ is a closed subspace of
$C^{l,\beta}(\overline{X})$.

  * (3)

If $j$ is a positive integer and $j-1<\delta\leq j\leq l$, then $\displaystyle
C^{l,\beta}_{(\delta)}(\overline{X})=C^{l,\beta}_{(j)}(\overline{X})$.

  * (4)

If $l<\delta\leq l+\beta$, then $\displaystyle
C^{l,\beta}_{(\delta)}(\overline{X})=C^{l,\beta}_{(l+\beta)}(\overline{X})$.

######  Definition 3.1.

For $\delta\in\mathbb{R}$, define the weighted Hölder space
$\Lambda^{l,\beta}_{\delta}(X)$ as follows

$\Lambda^{l,\beta}_{\delta}(X)=\\{u\in
C^{l,\beta}(X):\|u\|_{\Lambda^{l,\beta}_{\delta}(X)}<\infty\\};$

the norm $\|\cdot\|_{\Lambda^{l,\beta}_{\delta}(X)}$ is given by

$\|u\|_{\Lambda^{l,\beta}_{\delta}(X)}:=\|\rho^{-\delta}u\|_{\Lambda^{l,\beta}_{0}(X)}\quad\textrm{and}\quad\|u\|_{\Lambda^{l,\beta}_{0}(X)}:=\sup_{\Phi}\|\Phi^{*}u\|_{C^{l,\beta}(\overline{B_{2}})},$

where the supremum is over all Möbius charts.

Notice that $\Lambda^{l,\beta}_{0}(X)\subsetneq C^{l,\beta}(X)$. The main
relation between $C_{(\delta)}^{l,\beta}(\overline{X})$ and
$\Lambda^{l,\beta}_{\delta}(X)$ can be seen from the following lemma, where
the proof is given in [Le2, Lemma 3.7].

######  Lemma 3.4 (Lee).

Let $0<\beta<1$ and $0\leq\delta\leq l+\beta$. The following inclusions are
continuous

  * (1)

$C^{l,\beta}_{(\delta)}(\overline{X})\hookrightarrow\Lambda^{l,\beta}_{\delta}(X)$.

  * (2)

$\Lambda^{l,\beta}_{l+\beta}(X)\hookrightarrow C^{l,\beta}(\overline{X})$.

######  Lemma 3.5.

For $0\leq l+\beta<l^{\prime}+\beta^{\prime}$ and $\delta<\delta^{\prime}$,
the inclusion
$\Lambda^{l^{\prime},\beta^{\prime}}_{\delta^{\prime}}(X)\hookrightarrow\Lambda^{l,\beta}_{\delta}(X)$
is compact.

###  3.3. Schauder Estimates

In this paper, we mainly deal with the following elliptic operator:

(3.1) $P_{0}=\Delta_{+}+s(n-s)$

for some given $s>\frac{n}{2}$. By Lemma 4.8 of [Le2], we have the following.

######  Lemma 3.6 (Schauder Estimates).

Suppose $\delta\in\mathbb{R}$, $0\leq\beta<1$ and $2\leq l+\beta\leq
k+1+\alpha$. If $u\in\Lambda_{\delta}^{0,0}(X)$ such that
$P_{0}u\in\Lambda_{\delta}^{l-2,\beta}(X)$, then there exists a constant $C>0$
such that

$\|u\|_{\Lambda^{l,\beta}_{\delta}(X)}\leq
C\left(\|P_{0}u\|_{\Lambda^{l-2,\beta}_{\delta}(X)}+\|u\|_{\Lambda^{0,0}_{\delta}(X)}\right).$

###### Proof.

Let $\\{\Phi_{i}\\}$ be the uniform locally finite covering of $X$ by Möbius
charts as in Lemma 3.2. Then the estimate we want reduces to the interior
Schauder estimates for $L_{i}=\Phi_{i}^{*}\circ P\circ(\Phi_{i}^{-1})^{*}$ in
$B_{2}$. By assumption, $L_{i}$ has uniform $C^{k-1,\alpha}(\overline{B_{2}})$
coefficients. Hence

$\|u\|_{C^{l,\beta}(\overline{B_{1}})}\leq
C(\|L_{i}u\|_{C^{l-2,\beta}(\overline{B_{2}})}+\|u\|_{C^{0,0}(\overline{B_{2}})}).$

Here $C$ only depends on the $C^{l-2,\beta}(\overline{B_{2}})$ norm of the
coefficients of $L_{i}$. ∎

######  Remark 3.2.

If $\\{g_{+}^{(i)}\\}$ is a set of $C^{k,\alpha}$ conformally compact
Poincaré-Einstein metric such that $\\{\bar{g}^{(i)}=\rho^{2}g_{+}^{(i)}\\}$
is compact in the $C^{k,\alpha}(\overline{X})$ topology, then Lemma 3.6 holds
for all $P_{0}^{(i)}$ with a constant $C>0$ independent of $i$.

###  3.4. Fredholm and Invertibility

First by [Le2, Theorem C] (which was also proved in the smooth setting in [MM,
Ma]), $P_{0}$ defined in (3.1) has the following properties.

######  Lemma 3.7.

For $0\leq\beta<1$, $2\leq l+\beta\leq k+\alpha$ and $n-s<\delta<s$, the map

(3.2)
$P_{0}:\Lambda_{\delta}^{l,\beta}(X)\longrightarrow\Lambda_{\delta}^{l-2,\beta}(X)$

is Fredholm with kernel equal to the $L^{2}$-kernel of $P_{0}$. If further
$\mathrm{Spec}(-\Delta_{+})>s(n-s)$, then (3.2) is invertible.

######  Remark 3.3.

If $\\{g_{+}^{(i)}\\}$ is a set of $C^{k,\alpha}$ conformally compact
Poincaré-Einstein metric such that $\\{\bar{g}^{(i)}=\rho^{2}g_{+}^{(i)}\\}$
is compact in the $C^{k,\alpha}(\overline{X})$ topology and
$\mathcal{Y}(M,[\hat{g}^{(i)}])\geq 0$ for all $i$, then for $0\leq\beta<1$,
$2<l+\beta\leq k+\alpha$ and $n-s<\delta<s$, the map

$P^{(i)}_{0}:\Lambda_{\delta}^{l,\beta}(X)\longrightarrow\Lambda_{\delta}^{l-2,\beta}(X)$

is invertible for each $i$ and there is a constant $C>0$ independent of $i$
such that

$\|u\|_{\Lambda_{\delta}^{l,\beta}(X)}\leq
C\|P_{0}^{(i)}u\|_{\Lambda_{\delta}^{l-2,\beta}(X)}.$

###### Proof.

By Lemma 3.6, 3.7 and Remark 3.2, we only need to show that if
$\|P_{0}^{(i)}u\|_{\Lambda_{\delta}^{l-2,\beta}(X)}=1$ then there is a uniform
constant $C$ such that $\|u\|_{\Lambda_{\delta}^{0,0}(X)}\leq C$. Otherwise,
there exists a sequence $\\{u_{k}\\}_{k\geq 1}$ such that

$\|P_{0}^{(k)}u_{k}\|_{\Lambda_{\delta}^{l-2,\beta}(X)}=1,\quad\|u_{k}\|_{\Lambda_{\delta}^{0,0}(X)}\geq
k.$

Take $v_{k}=\frac{u_{k}}{\|u_{k}\|_{\Lambda_{\delta}^{0,0}(X)}}$. Then

$\displaystyle\|P_{0}^{(k)}v_{k}\|_{\Lambda_{\delta}^{l-2,\beta}(X)}\leq\frac{1}{k},\quad\|v_{k}\|_{\Lambda_{\delta}^{0,0}(X)}=1,\quad\Longrightarrow\quad\|v_{k}\|_{\Lambda_{\delta}^{l,\beta}(X)}\leq
C^{\prime}$

for some $C^{\prime}>0$ independent of $i$. Take $2\leq
l^{\prime}+\beta^{\prime}<l+\beta$ and $n-s<\delta^{\prime}<\delta<s$. Then by
passing to a subsequence, we can assume

$v_{k}\longrightarrow v_{\infty}\quad\textrm{in
$\Lambda_{\delta^{\prime}}^{l^{\prime},\beta^{\prime}}(X)$}\quad\textrm{and}\quad\bar{g}^{k}\longrightarrow\bar{g}^{(\infty)}\quad\textrm{in
$C^{k,\alpha}(\overline{X})$}.$

Hence $P_{0}^{(k)}v_{k}$ converges to $P_{0}^{(\infty)}v_{\infty}$ in
$\Lambda_{\delta^{\prime}}^{l^{\prime}-2,\beta^{\prime}}(X)$ and by assumption
$P_{0}^{(\infty)}v_{\infty}=0$. However,
$P_{0}^{(\infty)}:\Lambda_{\delta^{\prime}}^{l^{\prime},\beta^{\prime}}(X)\longrightarrow\Lambda_{\delta^{\prime}}^{l^{\prime}-2,\beta^{\prime}}(X)$
is invertible. Hence $v_{\infty}=0$, which contradicts with
$\|v_{k}\|_{\Lambda_{\delta}^{0,0}(X)}=1$ for all $k$. We finish the proof. ∎

For $\lambda\in\mathbb{R}$, we can also define

(3.3) $P_{\lambda}=\rho^{-\lambda}(\Delta_{+}+s(n-s))\rho^{\lambda}.$

Then a shifting of the weight shows the following.

######  Lemma 3.8.

For $0\leq\beta<1$, $2\leq l+\beta\leq k+1+\alpha$ and
$n-s-\lambda<\delta<s-\lambda$, the map

(3.4)
$P_{\lambda}:\Lambda_{\delta}^{l,\beta}(X)\longrightarrow\Lambda_{\delta}^{l-2,\beta}(X)$

is Fredholm with kernel consisting of $\rho^{-\lambda}f$ where $f$ is in the
$L^{2}$-kernel of $\Delta_{+}+s(n-s)$. If further
$\mathrm{Spec}(-\Delta_{+})>s(n-s)$, then (3.4) is invertible.

######  Remark 3.4.

If $\\{g_{+}^{(i)}\\}$ is a set of $C^{k,\alpha}$ conformally compact
Poincaré-Einstein metric such that $\\{\bar{g}^{(i)}=\rho^{2}g_{+}^{(i)}\\}$
is compact in the $C^{k,\alpha}(\overline{X})$ topology and
$\mathcal{Y}(M,[\hat{g}^{(i)}])\geq 0$ for all $i$, then for $0\leq\beta<1$,
$2<l+\beta\leq k+\alpha$ and $n-s-\lambda<\delta<s-\lambda$, the map

$P_{\lambda}^{(i)}:\Lambda_{\delta}^{l,\beta}(X)\longrightarrow\Lambda_{\delta}^{l-2,\beta}(X)$

is invertible for each $i$ and there is a constant $C>0$ independent of $i$
such that

$\|u\|_{\Lambda_{\delta}^{l,\beta}(X)}\leq
C\|P_{\lambda}^{(i)}u\|_{\Lambda_{\delta}^{l-2,\beta}(X)}.$

###  3.5. Geodesic Normal Defining Function

While working with the boundary asymptotics, we usually choose the geodesic
normal defining function. However, in our setting, it is no longer smooth. A
general regularity version of [Le1, Lemma 5.1] gives the following.

######  Lemma 3.9 (Lee).

There exists a boundary defining function $x\in C^{k-1,\alpha}(\overline{X})$
such that

  * (1)

$x=\rho+O(\rho^{2})$;

  * (2)

$\tilde{g}=x^{2}g_{+}$ has a $C^{k-1,\alpha}(\overline{X})$ extension to
$\overline{M}$ with $\tilde{g}|_{TM}=\bar{g}|_{TM}=\hat{g}$;

  * (3)

$\|dx\|^{2}_{\tilde{g}}\equiv 1$ in a collar neighbourhood of $M$.

######  Lemma 3.10.

Let $x$ and $\tilde{g}$ be given in Lemma 3.9. Then

$\tilde{g}=dx^{2}+\hat{g}+x^{2}g_{2}+O(x^{2+\epsilon}),$

where $g_{2}=-\hat{A}$ and
$\hat{A}_{ij}=\frac{1}{n-2}\left(\hat{R}_{ij}-\hat{J}\hat{g}_{ij}\right)$ is
the Schouten tensor of $\hat{g}$.

###### Proof.

By a similar proof as [Le1, Lemma 5.2], we have

$\displaystyle
x\tilde{R}_{ij}+(n-1)\tilde{x}_{ij}+(\tilde{\Delta}x)\tilde{g}_{ij}=0,$
$\displaystyle\tilde{\Delta}x=-\frac{1}{2n}x\tilde{R}=-x\tilde{R}_{ij}\tilde{x}^{i}\tilde{x}^{j},$

where $\tilde{\Delta}$ is the Laplace operator of $\tilde{g}$, and
$\tilde{x}^{i},\tilde{x}_{ij}$ denote the covariant derivatives w.r.t.
$\tilde{g}$, and $\tilde{R},\tilde{R}_{ij}$ are the scalar and Ricci
curvatures of $\tilde{g}$. It is easy to see that $\tilde{x}_{ij}|_{M}=0$,
which implies that $(M,\hat{g})$ is totally geodesic in
$(\overline{X},\tilde{g})$. Moreover,

$\hat{R}=\frac{n-1}{n}\tilde{R}|_{M}\quad\Leftrightarrow\quad\hat{J}=\tilde{J}|_{M}.$

Hence by an identification of a collar neighborhood of $M$ with $[0,c)\times
M$, we can write

$\tilde{g}=dx^{2}+\hat{g}+x^{2}g_{2}+O(x^{2+\epsilon}).$

Since $g_{+}=x^{-2}\tilde{g}$ satisfies $Ric_{g_{+}}=-ng_{+}$, the same
calculation as in [CDLS] gives that $g_{2}=-\hat{A}.$ ∎

##  4\. The Adapted and Fefferman-Graham compactifications

In this section, we study some basic properties for the adapted and Fefferman-
Graham compactifications for a Poincaré-Einstein manifold $(X,g_{+})$, which
were introduced in [CC] and [FG2] respectively. Suppose $(X^{n+1},g_{+})$
($n\geq 3$) is a $C^{k,\alpha}$ conformally compact Poincaré-Einstein manifold
with $k\geq 3,0<\alpha<1$. Let $\rho$ be a fixed smooth boundary defining
function. Denote

$\bar{g}=\rho^{2}g_{+}\quad\textrm{and}\quad\bar{g}|_{TM}=\hat{g}.$

###  4.1. Definition of $\rho_{s}$ and $\bar{g}_{s}$

####  4.1.1. Case 1: $s>\frac{n}{2},s\neq n$

In this case, if $\mathrm{Spec}(-\Delta_{+})>s(n-s)$, let $v_{s}$ satisfy the
following equation:

(4.1) $-\Delta_{+}v_{s}-s(n-s)v_{s}=0,\quad
v_{s}=\rho^{n-s}\left(1+O(\rho^{\epsilon})\right)\ \textrm{for some
$\epsilon>0$}.$

Then the adapted compactification of $g_{+}$ is defined by

$\bar{g}_{s}=\rho_{s}^{2}g_{+},\quad\mathrm{where}\quad\rho_{s}=v_{s}^{\frac{1}{n-s}}.$

Notice that the adapted defining function $\rho_{s}\sim\rho$, which may only
have finite regularity.

######  Lemma 4.1.

For $s>\frac{n}{2},s\neq n$, if $\mathrm{Spec}(-\Delta_{+})>s(n-s)$, then
$\rho_{s}$ is uniquely determined by $\hat{g}$.

###### Proof.

First, with fixed $\rho$, (4.1) has a solution. Notice that

$-\Delta_{+}\rho^{n-s}-s(n-s)\rho^{n-s}=-\rho^{n-s+1}\left[(n-s)\bar{\Delta}\rho+s(n-s)T\right].$

where $T$ is given in (2.4). Moreover, $\bar{\Delta}\rho,T\in
C^{k-1,\alpha}(\overline{X}).$ Hence

$-\rho^{n-s+1}\left[(n-s)\bar{\Delta}\rho+s(n-s)T\right]\in\rho^{n-s+1}C^{k-1,\alpha}(\overline{X})\subset\Lambda^{k-1,\alpha}_{n-s+1}(X).$

If $s>\frac{n+1}{2}$, then $n-s+1\in(n-s,s)$. By Lemma 3.7, there exists a
unique $v_{s}^{\prime}\in\Lambda^{k+1,\alpha}_{n-s+1}(X)$ such that

$(-\Delta_{+}-s(n-s))(\rho^{n-s}+v_{s}^{\prime})=0.$

If $\frac{n+1}{2}\geq s>\frac{n}{2}$, then $n-s+1\geq s$. Take
$\delta\in(n-s,s)$. Then by Lemma 3.7 again, there exists a unique
$v_{s}^{\prime\prime}\in\Lambda^{k+1,\alpha}_{\delta}(X)$ such that

$(-\Delta_{+}-s(n-s))(\rho^{n-s}+v_{s}^{\prime\prime})=0.$

Second, $v_{s}>0$ in $X$. It is obvious that $v_{s}$ is positive near $M$. If
$v_{s}$ has negative part in the interior, then by denoting
$\Omega=\\{v_{s}<0\\}$, we have $\Omega\Subset X$ and

$\begin{cases}-\Delta_{+}v_{s}-s(n-s)v_{s}=0,&\textrm{in $\Omega$,}\\\
v_{s}=0,&\textrm{on $\partial\Omega$.}\end{cases}$

This contradicts to
$\mathrm{Spec}(-\Delta_{+}|_{\Omega})\geq\mathrm{Spec}(-\Delta_{+})>s(n-s)$.
So $v_{s}\geq 0$ and by strong maximum principle, $v_{s}>0$ in $X$.

Third, the positivity also implies the uniqueness with given $\rho$. If there
is a second solution $\tilde{v}_{s}$ satisfies (4.1), then $\tilde{v}_{s}>0$
in $X$ and

$\Delta_{+}\left(\frac{v_{s}}{\tilde{v}_{s}}\right)+2\left\langle
d\log\tilde{v_{s}},d\left(\frac{v_{s}}{\tilde{v}_{s}}\right)\right\rangle_{g_{+}}=0,\quad\left.\frac{v_{s}}{\tilde{v}_{s}}\right|_{M}=1.$

By the strong maximum principle, this implies that $v_{s}\equiv\tilde{v}_{s}$
in $X$.

Finally, if there is another smooth defining function $\tilde{\rho}$ such that
$\tilde{\rho}^{2}g_{+}|_{TM}=\hat{g}$ then we have $\tilde{\rho}\sim\rho$,
which does not change the definition of $v_{s}$. Hence by above uniqueness,
$\rho_{s}$ is only determined by the boundary representative $\hat{g}$. ∎

####  4.1.2. Case 2: $s=n$

In this case, the equation(4.1) is replaced by

(4.2) $-\Delta_{+}w=n,\quad w=\log\rho+O(\rho^{\epsilon})\ \textrm{for some
$\epsilon>0$}.$

Then the Fefferman-Graham compactification of $g_{+}$ is defined by

$\bar{g}_{F}=\rho_{F}^{2}g_{+},\quad\mathrm{where}\quad\rho_{F}=e^{w}.$

Similarly the Fefferman-Graham defining function $\rho_{F}\sim\rho$, which may
only have finite regularity.

######  Lemma 4.2.

The Fefferman-Graham defining function $\rho_{F}$ is uniquely determined by
$\hat{g}$.

###### Proof.

First, (4.2) has a solution. Notice that

$-\Delta_{+}(\log\rho)=n-\rho[\bar{\Delta}\rho+nT],$

where

$\rho[\bar{\Delta}\rho+nT]\in\rho
C^{k-1,\alpha}(\overline{X})\subset\Lambda_{1}^{k-1,\alpha}(X).$

By Lemma 3.7, there exists a unique
$w^{\prime}\in\Lambda_{1}^{k+1,\alpha}(X)$, such that

$-\Delta_{+}(\log\rho+w^{\prime})=n.$

Since $\mathrm{Spec}(-\Delta_{+})>0$, then $w>0$ and is uniquely determined by
$\hat{g}$ by a similar proof of Lemma 4.1. ∎

####  4.1.3. Continuous Family

In lots of discussions, we can view the Fefferman-Graham compactification as a
special case of the adapted compactification in the following sense.

######  Lemma 4.3.

If the Yamabe constant $\mathcal{Y}(M,[\hat{g}])\geq 0$, then the adapted
defining function $\rho_{s}$ is a continuous family of boundary defining
functions for $s>\frac{n}{2}$ such that

$\lim_{s\rightarrow n}\rho_{s}=\rho_{F}.$

###### Proof.

For $\mathcal{Y}(M,[\hat{g}])\geq 0$, $v_{s}$ is a continuous family of
solutions to (4.1). While $s=n$, $v_{s}=1$. Take a derivative in $s$ at $s=n$
and denote

$\left.\frac{dv_{s}}{ds}\right|_{s=n}=-w.$

Then $w$ satisfies (4.2). By L’Hospital rule,

$\lim_{s\rightarrow n}\rho_{s}=\lim_{s\rightarrow
n}v_{s}^{\frac{1}{n-s}}=\lim_{s\rightarrow n}e^{\frac{1}{n-s}\ln
v_{s}}=e^{w}=\rho_{F}.$

∎

In the following of the paper, we will always assume

$\mathcal{Y}(M,[\hat{g}])\geq 0.$

So that for all $s>\frac{n}{2}$, $\rho_{s}$ and $\bar{g}_{s}$ are well defined
with given boundary representative $\hat{g}$. We will denote
$\rho_{n}=\rho_{F}$ for consistency.

###  4.2. Interior Regularity and Curvature Vanishing

Notice that $g_{+}=\rho^{-2}\bar{g}$ has $C^{k,\alpha}$ regularity in the
interior $X$. Therefore by the interior Schauder estimates for equations (4.1)
and (4.2), for all $s>\frac{n}{2}$

$\rho_{s}\in C^{k+1,\alpha}(X),\quad\bar{g}_{s}\in C^{k,\alpha}(X).$

This also can be seen from the proof of Lemma 4.1 and 4.2.

The adapted and Fefferman-Graham compactifications are introduced so that the
compactified metric $\bar{g}_{s}$ possesses some certain geometric features.
Let $\bar{R}_{s}$ be the scalar curvature of $\bar{g}_{s}$, with
$\bar{J}_{s}=\frac{1}{2n}\bar{R}_{s}$, and $\bar{Q}_{2N}^{\bar{g}_{s}}$ its
$2N$-th order Q-curvautre. Let $\bar{\Delta}_{s}$ be the Laplace-Beltrami
operator for $\bar{g}_{s}$. Direct calculation shows that in $X$,

(4.3) $\bar{\Delta}_{s}\rho_{s}=-s\bar{T}_{s},$

where

(4.4) $\bar{T}_{s}=\rho_{s}^{-1}\left(1-|d\rho_{s}|^{2}_{\bar{g}_{s}}\right),$

and

(4.5)
$\bar{J}_{s}=\frac{2s-n-1}{2}\rho_{s}^{-2}\left(1-|d\rho_{s}|^{2}_{\bar{g}_{s}}\right).$

Therefore $\bar{T}_{s},\bar{J}_{s}\in C^{k,\alpha}(X)$ and
$\bar{Q}_{2N}^{\bar{g}_{s}}\in C^{k-2N+2,\alpha}(X)$.

######  Lemma 4.4.

If $s=\frac{n}{2}+N-\frac{1}{2}$ for some integer $1\leq
N\leq[\frac{k+1}{2}]$, then $(\overline{X},\bar{g}_{s})$ has vanishing
$\bar{Q}^{\bar{g}_{s}}_{2N}$ curvature in $X$.

###### Proof.

If $s=\frac{n}{2}+N-\frac{1}{2}\neq n$, then (4.1) implies that

$\displaystyle
P_{2N}^{+}(v_{s})=P_{2N}^{+}\left(\rho^{\frac{n+1}{2}-N}_{s}\right)=0,$
$\displaystyle\Longrightarrow$
$\displaystyle\bar{P}^{\bar{g}_{s}}_{2N}(1)=\rho^{-\frac{n+1}{2}-N}P_{2N}^{+}\left(\rho^{\frac{n+1}{2}-N}_{s}\right)=0,$
$\displaystyle\Longrightarrow$
$\displaystyle\bar{Q}^{\bar{g}_{s}}_{2N}=0,\quad\textrm{in $X$}.$

If $s=\frac{n}{2}+N-\frac{1}{2}=n$, i.e. $2N=n+1$, by [Go, Theorem 1.1-1,2],
the $(n+1)$-order GJMS operator and Q-curvature of $(X,g_{+})$ is

$\begin{gathered}P^{+}_{n+1}=\left(-\Delta_{+}-(n-1)\right)\left(-\Delta_{+}-3(n-3)\right)\cdots\left(-\Delta_{+}-(n-2)2\right)\left(-\Delta_{+}\right),\\\
Q^{+}_{n+1}=(-1)^{\frac{n+1}{2}}n!.\end{gathered}$

Then (4.2) implies that

$0=P^{+}_{n+1}w+Q^{+}_{n+1}=\bar{Q}^{\bar{g}_{s}}_{n+1}e^{(n+1)w}\quad\Longrightarrow\quad\bar{Q}^{\bar{g}_{s}}_{n+1}=0,\
\textrm{in $X$}.$

∎

We want to point out that in [CC] the authors considered the metric measure
space and defined the more general $\bar{Q}_{2N,\phi}^{m}$ curvatures, and
proved that if $s=\frac{n+2N-1-m}{2}$, then $\bar{Q}_{2N,\phi}^{m}=0$ for
$\bar{g}_{s}$. In later applications, the equation
$\bar{Q}^{\bar{g}_{s}}_{2N}=0$ will help us to lift the regularity of
$\bar{J}_{s}$. However, $\bar{Q}_{2N,\phi}^{m}=0$ can not. So we avoid going
to the details of $\bar{Q}_{2N,\phi}^{m}$ here and ask the readers to refer to
[CC] if interested.

###  4.3. Global Regularity of $\rho_{s}$ and $\bar{g}_{s}$

It is obvious that the regularity of $\bar{g}_{s}$ depends on both the initial
regularity of $\bar{g}$ and the parameter $s$. Even if $g_{+}$ is $C^{\infty}$
conformally compact and the geodesic normal defining function $x$ is smooth in
Lemma 3.9, we know that $v_{s}$ takes the form $v_{s}=x^{s}F+x^{n-s}G$ for
some $F,G\in C^{\infty}(\overline{X})$ if $2s-n$ is fractional. And the
regularity of $\rho_{s}$ is stoped by this fractional order. Therefore it is
nature to conjecture that the regularity of $\rho_{s}$ and $\bar{g}_{s}$ is
given by the minimum of the background regularity of $\bar{g}$ and the
fractional order $2s-n$. We give a strict proof here.

####  4.3.1. Case 1: $s>\frac{n}{2},s\neq n$

In this case, denote

$\bar{g}_{s}=\rho_{s}^{2}g_{+}=\phi_{s}^{\frac{2}{n-s}}\bar{g},\quad\phi_{s}=\left(\frac{\rho_{s}}{\rho}\right)^{n-s}.$

Then $v_{s}=\phi_{s}\rho^{n-s}$ and (4.1) implies that

$\Delta_{+}v_{s}+s(n-s)v_{s}=\rho^{n-s}P(\phi_{s}),$

where $P=\rho^{s-n}(\Delta_{+}+s(n-s))\rho^{n-s}$. Let us write $P$ in
background metric $\bar{g}$:

$P\phi_{s}=\rho^{2}\bar{\Delta}\phi_{s}-(2s-n-1)\rho\langle
d\rho,d\phi_{s}\rangle_{\bar{g}}+(n-s)\rho\left(\bar{\Delta}\rho+sT\right)\phi_{s},$

where

$T=\frac{1-|d\rho|^{2}_{\bar{g}}}{\rho}\in C^{k-1,\alpha}(\overline{X}).$

Therefore, $\phi_{s}$ satisfies

(4.6) $\begin{cases}P\phi_{s}=0,&\textrm{in $X$},\\\ \phi_{s}=1,&\textrm{on
$M$}.\end{cases}$

We begin by constructing an asymptotical solution for (4.6).

######  Lemma 4.5.

For $s>\frac{n}{2},s\neq n$, there exists an approximate solution
$\tilde{\phi}_{s}\in C^{k,\alpha}(\overline{X})$ such that

$P(\tilde{\phi}_{s})=\tilde{F},\quad\tilde{F}\in\Lambda^{k-1,\alpha}_{\delta}(X).$

where $\delta=k+\alpha$ if $k+\alpha\leq 2s-n$ or $2s-n\notin\mathbb{N}$;
$\delta=2s-n$ if $k+\alpha>2s-n$ and $2s-n\in\mathbb{N}$.

Moreover, there exists a constant $C$ only depending on $k,\alpha$ and the
$C^{k,\alpha}(\overline{X})$ norm of $\bar{g}$ such that

$\|\tilde{\phi}_{s}\|_{C^{k,\alpha}(\overline{X})}\leq
C,\quad\|\tilde{F}\|_{\Lambda^{k-1,\alpha}_{\delta}(X)}\leq C;$

and as $\rho\rightarrow 0$, $\tilde{\phi}_{s}$ has asymptotic expansion

$\tilde{\phi}_{s}=1+\rho\phi_{s,1}+\cdots+\rho^{l}\phi_{s,l},$

where $l=k$ if $k+\alpha\leq 2s-n$ or $2s-n\notin\mathbb{N}$; $l=2s-n-1$ if
$k+\alpha>2s-n$ and $2s-n\in\mathbb{N}$. Here $\phi_{s,j}\in
C^{k-j,\alpha}(M)$ is determined by the derivatives of $\bar{g}$ up to $j$-th
order and

$\|\phi_{s,j}\|_{C^{k-j,\alpha}(M)}\leq C.$

###### Proof.

Let $s=\frac{n}{2}+\gamma$. First $\bar{g}$ is $C^{k,\alpha}(\overline{X})$
and $\rho\in C^{\infty}(\overline{X})$, which together implies that the
coefficients in $P$ are in $C^{k-1,\alpha}(\overline{X})$. By Theorem 7.1, for
any $\phi_{s,j}\in C^{k-j,\alpha}(M)$, we can take $\tilde{\phi}_{s,j}\in
C^{k,\alpha}(\overline{X})$ as an extension of $\rho^{j}\phi_{s,j}$ such that
$\tilde{\phi}_{s,j}\sim\rho^{j}\phi_{s,j}$ as $\rho\rightarrow 0$. Moreover,
there exists some constant $C^{\prime}$ only depending on $k,j,\alpha$ such
that

$\|\tilde{\phi}_{s,j}\|_{C^{k,\alpha}(\overline{X})}\leq
C^{\prime}\|\phi_{s,j}\|_{C^{k-j,\alpha}(M)}.$

Notice that under the background coordinates $(\rho,\theta)$, $P$ is in the
form

$P=\rho\left(a^{\mu\nu}\rho\partial_{\mu}\partial_{\nu}+b^{\mu}\rho\partial_{\mu}+b^{\prime\mu}\partial_{\mu}+c\right),$

where $a^{\mu\nu},b^{\prime\mu}\in C^{k,\alpha}(\overline{X})$ and
$b^{\mu},c\in C^{k-1,\alpha}(\overline{X})$. By (3) in Theorem 7.1,
$\rho^{-1}P\tilde{\phi}_{s,j}\in C^{k-1,\alpha}(\overline{X})$, and there
exists some constant $C^{\prime\prime}$ only depending on the
$C^{k,\alpha}(\overline{X})$ norm of $\bar{g}$, such that

$\|\rho^{-1}P\tilde{\phi}_{s,j}\|_{C^{k-1,\alpha}(\overline{X})}\leq
C^{\prime\prime}\|\phi_{s,j}\|_{C^{k-j,\alpha}(M)}.$

As $\rho\rightarrow 0$,

$P(\tilde{\phi}_{s,j})=-j(2\gamma-j)\rho^{j}\phi_{s,j}(\theta)+O(\rho^{j+\epsilon}),$

where $\epsilon=1$ if $j<k$ and $\epsilon=\alpha$ if $j=k$.

Now we are ready to construct the asymptotical solution $\tilde{\phi}_{s}$.
Take $\tilde{\phi}_{s,0}=\phi_{s,0}=1$. Then

$P(1)=(n-s)\rho\left(\bar{\Delta}\rho+sT\right)=\tilde{F_{0}}\sim\rho F_{0},$

where $\tilde{F}_{0}\in\rho C^{k-1,\alpha}(\overline{X})$ and
$F_{0}=(n-s)\left(\bar{\Delta}\rho+sT\right)|_{M}\in C^{k-1,\alpha}(M)$.
Moreover, for some constant $C_{1}$ only depending on the
$C^{k,\alpha}(\overline{X})$ norm of $\bar{g}$,

$\|\rho^{-1}\tilde{F}_{0}\|_{C^{k-1,\alpha}(\overline{X})}\leq
C_{1},\quad\|F_{0}\|_{C^{k-1,\alpha}(M)}\leq C_{1}.$

Choose $\phi_{s,1}=(2\gamma-1)^{-1}F_{0}$. Let $\tilde{\phi}_{s,1}\in
C^{k,\alpha}(\overline{X})$ be an extension of $\rho\phi_{s,1}$ by Theorem
7.1. Then

$P(1+\tilde{\phi}_{s,1})=\tilde{F_{1}}\sim\rho^{2}F_{1},$

where $\tilde{F}_{1}\in\rho C^{k-1,\alpha}(\overline{X})$ and $F_{1}\in
C^{k-2,\alpha}(M)$. And similar as above for some constant $C_{2}$ only
depending on the $C^{k,\alpha}(\overline{X})$ norm of $\bar{g}$,

$\|\rho^{-1}\tilde{F}_{1}\|_{C^{k-1,\alpha}(\overline{X})}\leq
C^{\prime}_{2},\quad\|F_{1}\|_{C^{k-2,\alpha}(M)}\leq C_{2}.$

Choose $\phi_{s,2}=[2(2\gamma-2)]^{-1}F_{1}$.

If $k+\alpha\leq 2\gamma$ or $2\gamma\notin\mathbb{N}$, we can repeat this
progress until order $k$ such that $\phi_{s,k}\in C^{0,\alpha}(M)$. Let
$\tilde{\phi}_{s,k}\in C^{k,\alpha}(\overline{X})$ be an extension of
$\rho^{k}\phi_{s,k}$ by Theorem 7.1.

$P(1+\tilde{\phi}_{s,1}+\cdots+\tilde{\phi}_{s,k})=\tilde{F_{k}}=O(\rho^{k+\alpha}),$

where $\tilde{F}_{k}\in\rho C^{k-1,\alpha}(\overline{X})$. And similar as
above for some constant $C_{k}$ only depending on the
$C^{k,\alpha}(\overline{X})$ norm of $\bar{g}$,

$\|\rho^{-1}\tilde{F}_{k}\|_{C^{k-1,\alpha}(\overline{X})}\leq C_{k}.$

Finally, we take
$\tilde{\phi}_{s}=1+\tilde{\phi}_{s,1}+\cdots+\tilde{\phi}_{s,k}$ and
$\tilde{F}=\tilde{F}_{k}$. By Lemma 3.4, $\rho^{-1}\tilde{F}_{k}\in
C^{k-1,\alpha}(\overline{X})$ and
$\rho^{-1}\tilde{F}_{k}=O(\rho^{k-1+\alpha})$ imply that

$\rho^{-1}\tilde{F}\in
C^{k-1,\alpha}_{(k-1+\alpha)}(\overline{X})\subset\Lambda^{k-1,\alpha}_{k-1+\alpha}(X)\quad\Longrightarrow\quad\tilde{F}\in\Lambda^{k-1,\alpha}_{k+\alpha}(X).$

By Lemma 3.3, 3.4, the inclusion
$C^{k-1,\alpha}_{(k-1+\alpha)}(\overline{X})\hookrightarrow\Lambda^{k-1,\alpha}(\overline{X})$
is continuous and $C^{k-1,\alpha}_{(k-1+\alpha)}$ is a closed subspace of
$C^{k-1,\alpha}(\overline{X})$. Hence there exists a constant $\tilde{C}$ only
depending on $X$ such that

$\|\tilde{F}\|_{\Lambda^{k-1,\alpha}_{k+\alpha}(X)}=\|\rho^{-1}\tilde{F}\|_{\Lambda^{k-1,\alpha}_{k-1+\alpha}(X)}\leq\tilde{C}\|\rho^{-1}\tilde{F}\|_{C^{k-1,\alpha}(\overline{X})}\leq\tilde{C}C_{k}.$

If $k+\alpha>2\gamma$ and $2\gamma\in\mathbb{N}$, then we stop at the order
$2\gamma-1$. And

$P(1+\tilde{\phi}_{s,1}+\cdots+\tilde{\phi}_{s,2\gamma-1})=\tilde{F}_{2\gamma-1}=O(\rho^{2\gamma}),$

where $\tilde{F}_{2\gamma-1}\in\rho C^{k-1,\alpha}(\overline{X})$ and

$\|\rho^{-1}\tilde{F}_{2\gamma-1}\|_{C^{k-1,\alpha}(\overline{X})}\leq
C_{2\gamma-1}.$

Finally, we take
$\tilde{\phi}_{s}=1+\tilde{\phi}_{s,1}+\cdots+\tilde{\phi}_{s,2\gamma-1}$ and
$\tilde{F}=\tilde{F}_{2\gamma-1}$. Then

$\rho^{-1}\tilde{F}\in
C^{k-1,\alpha}_{(2\gamma-1)}(\overline{X})\subset\Lambda^{k-1,\alpha}_{2\gamma-1}(X)\quad\Longrightarrow\quad\tilde{F}\in\Lambda^{k-1,\alpha}_{2\gamma}(X).$

Moreover,

$\|\tilde{F}\|_{\Lambda^{k-1,\alpha}_{2\gamma}(X)}=\|\rho^{-1}\tilde{F}\|_{\Lambda^{k-1,\alpha}_{2\gamma-1}(X)}\leq\tilde{C}\|\rho^{-1}\tilde{F}\|_{C^{k-1,\alpha}(\overline{X})}\leq\tilde{C}C_{2\gamma-1}.$

We finish the proof. ∎

######  Remark 4.1.

While $k+\alpha>2s-n$ and $2s-n=2\gamma\in\mathbb{N}$, actually we can
construct a full asymptotical solution such that
$P(\tilde{\phi})=O(\rho^{k+\alpha-\epsilon})$ if we allow logrithmic terms in.
However, here it does not help to discuss the global Hölder regularity, so we
omit it.

######  Lemma 4.6.

For $s>\frac{n}{2},s\neq n$, the equation (4.6) has a unique solution
$\phi_{s}\in C^{l,\beta}(\overline{X})$, where $l+\beta=k+\alpha$ if
$k+\alpha<2s-n$; $l+\beta<2s-n$ if $k+\alpha\geq 2s-n$. Moreover, there exists
some constant $C>0$ only depending on the $C^{k,\alpha}(\overline{X})$ norm of
$\bar{g}$ such that

$\|\phi_{s}\|_{C^{l,\beta}(\overline{X})}\leq C.$

###### Proof.

Denote $s=\frac{n}{2}+\gamma$. Take $\tilde{\phi}_{s}$ the approximate
solution in Lemma 4.5. Now we modify it to be a true solution. First, consider
$k+\alpha<2\gamma$. In this case,

$P\tilde{\phi}_{s}=\tilde{F}\in\Lambda_{k+\alpha}^{k-1,\alpha}(X).$

By Lemma 3.8,

$P:\Lambda_{k+\alpha}^{k+1,\alpha}(X)\longrightarrow\Lambda_{k+\alpha}^{k-1,\alpha}(X),$

is invertible. Hence we can solve a unique
$\tilde{\phi}^{\prime}_{s}\in\Lambda_{k+\alpha}^{k+1,\alpha}(X)$ such that

$P(\tilde{\phi}_{s}+\tilde{\phi}_{s}^{\prime})=0.$

Moreover, there exists a constant $C$ only depending on the
$C^{k,\alpha}(\overline{X})$ norm of $\bar{g}$ such that

$\|\tilde{\phi}_{s}^{\prime}\|_{\Lambda_{k+\alpha}^{k+1,\alpha}(X)}\leq
C\|\tilde{F}\|_{\Lambda_{k+\alpha}^{k-1,\alpha}(X)}.$

By Lemma 3.4, $\Lambda_{k+\alpha}^{k+1,\alpha}(X)\hookrightarrow
C^{k,\alpha}(\overline{X})$ is continuous. Hence for some constant $\tilde{C}$
only depending on $X$, we have

$\|\tilde{\phi}_{s}^{\prime}\|_{C^{k,\alpha}(\overline{X})}\leq\tilde{C}C\|\tilde{F}\|_{\Lambda_{k+\alpha}^{k-1,\alpha}(X)}.$

Second, if $k+2\alpha\geq 2\gamma$, then

$P\tilde{\phi}_{s}=\tilde{F}\in\Lambda_{2\gamma}^{k-1,\alpha}(X)\subset\Lambda_{l+\beta}^{k-1,\alpha}(X),$

for $l\geq 0$, $0<\beta<1$ such that $l+\beta<2\gamma$. By Lemma 3.8 again

$P:\Lambda_{l+\beta}^{k+1,\alpha}(X)\longrightarrow\Lambda_{l+\beta}^{k-1,\alpha}(X),$

is invertible. Hence we can solve a unique
$\tilde{\phi}^{\prime}_{s}\in\Lambda_{l+\beta}^{k+1,\alpha}(X)$ such that

$P(\tilde{\phi}_{s}+\tilde{\phi}_{s}^{\prime})=0.$

Moreover, there exists a constant $C$ only depending on the
$C^{k,\alpha}(\overline{X})$ norm of $\bar{g}$ and $C^{\prime}$ only depending
on the maximum value of $\rho$ such that

$\|\tilde{\phi}_{s}^{\prime}\|_{\Lambda_{l+\beta}^{k+1,\alpha}(X)}\leq
C\|\tilde{F}\|_{\Lambda_{l+\beta}^{k-1,\alpha}(X)}\leq
CC^{\prime}\|\tilde{F}\|_{\Lambda_{2\gamma}^{k-1,\alpha}(X)}.$

By Lemma 3.4, $\Lambda_{l+\beta}^{k+1,\alpha}(X)\hookrightarrow
C^{l,\beta}(\overline{X})$ is continuous. Hence for some constant $\tilde{C}$
only depending on $X$, we have

$\|\tilde{\phi}_{s}^{\prime}\|_{C^{l,\beta}(\overline{X})}\leq\tilde{C}CC^{\prime}\|\tilde{F}\|_{\Lambda_{2\gamma}^{k-1,\alpha}(X)}.$

Combing above discussion with Lemma 4.5, we finish the proof. ∎

######  Lemma 4.7.

If $s=\frac{n}{2}+N-\frac{1}{2}$ for some integer $N\geq 1$, then $\phi_{s}\in
C^{k,\alpha}(\overline{X})$ and there exists some constant $C>0$ only
depending on the $C^{k,\alpha}(\overline{X})$ norm of $\bar{g}$ such that

$\|\phi_{s}\|_{C^{k,\alpha}(\overline{X})}\leq C.$

###### Proof.

First, if $N=1$, then $s=\frac{n+1}{2}$ and (2.2) implies that

$\rho\bar{\Delta}\rho+s(1-|d\rho|^{2}_{\bar{g}})=-\bar{J}\rho^{2},$

where $\bar{J}=\frac{1}{2n}\bar{R}\in C^{k-2,\alpha}(\overline{X})$. Hence
equation (4.6) becomes

$\begin{cases}\bar{\Delta}\phi_{s}-\frac{n-1}{2}\bar{J}\phi_{s}=0,&\textrm{in
$X$}\\\ \phi_{s}=1,&\textrm{on $M$}.\end{cases}$

By classical Schauder estimates, we have $\phi_{s}\in
C^{k,\alpha}(\overline{X})$.

Next, consider $N\geq 2$. Notice that if $k+\alpha<2N-1$, it is already proved
in Lemma 4.6. So we only need to consider the case $k+\alpha\geq 2N-1$. First
choose $l=2N-2$. Then $l+\alpha<2N-1$ and by Lemma 4.6, we have

$\phi_{s}\in C^{k,\alpha}(X)\cap C^{l,\alpha}(\overline{X}).$

Recall that by Lemma 4.4, in $X$,

(4.7) $Q^{\bar{g}_{s}}_{2N}=\Delta_{\bar{g}_{s}}^{N-1}\bar{J}_{s}+L.O.T.=0,$

where $L.O.T$ consists of derivatives of $\bar{g}_{s}$ of order $\leq 2N-2$.
Hence

$L.O.T.\in C^{k-2N+2,\alpha}(X)\cap C^{0,\alpha}(\overline{X}).$

By Lemma 5.1, on the boundary $\bar{J}_{s}|M\in C^{k-2,\alpha}(M)$. Hence by
global Schauder estimate,

$\bar{J}_{s}\in C^{l,\alpha}(\overline{X}).$

Consider the conformal transformation from $\bar{g}$ to $\bar{g}_{s}$, where
$\bar{g}_{s}=\phi_{s}^{\frac{2}{n-s}}\bar{g}$. Then

(4.8)
$-\frac{2n}{n-s}\frac{\bar{\Delta}\phi_{s}}{\phi_{s}}-\frac{n(n-3)}{n-s}\frac{|d\phi_{s}|^{2}_{\bar{g}}}{\phi_{s}^{2}}+\bar{R}=\phi_{s}^{\frac{2}{n-s}}\bar{R}_{s}.$

Notice that by the definition of $\phi_{s}$ and Lemma 4.6, $c<\phi_{s}<C$ for
some $0<c<C$ only depending on the $C^{k,\alpha}(\overline{X})$ norm of
$\bar{g}$. Now $k-2\geq l-1$, $0<c<\phi_{s}<C$, $\phi_{s},\bar{R}_{s}\in
C^{l,\alpha}(\overline{X})$, $|d\phi_{s}|^{2}_{\bar{g}}\in
C^{l-1,\alpha}(\overline{X})$ and $\bar{R}\in C^{k-2,\alpha}(\overline{X})$.
These imply that $\phi_{s}\in C^{l+1,\alpha}(\overline{X})$ by the global
Schauder estimates. If $k-2\geq l$, using $|d\phi_{s}|^{2}_{\bar{g}}\in
C^{l,\alpha}(\overline{X})$ again in equation (4.8), we have $\phi_{s}\in
C^{l+2,\alpha}(\overline{X})$. Repeat this lifting progress and apply the
global Schauder estimates to (4.7) and (4.8), until the regularity lifting is
stoped by $C^{k,\alpha}(\overline{X})$ assumption on $\bar{g}$. Finally we get
$\phi_{s}\in C^{k,\alpha}(\overline{X})$. And the estimates of
$\|\phi_{s}\|_{C^{k,\alpha}(\overline{X})}$ is directly from Lemma 4.6 and the
global Schauder estimates for (4.7) and (4.8). ∎

######  Remark 4.2.

If $\\{g_{+}^{(i)}\\}$ is a set of $C^{k,\alpha}$ conformally compact
Poincaré-Einstein metric such that $\\{\bar{g}^{(i)}=\rho^{2}g_{+}^{(i)}\\}$
is compact in the $C^{k,\alpha}(\overline{X})$ topology and
$\mathcal{Y}(M,[\hat{g}^{(i)}])\geq 0$ for all $i$, then Lemma 4.5, 4.6, 4.7
hold with uniform estimates, i.e.

$\|\phi_{s}^{(i)}\|_{C^{l,\beta}(\overline{X})}\leq C,$

where $C>0$ is independent of $i$. Here $l+\beta=k+\alpha$ if $k+\alpha<2s-n$
or $2s-n=2N-1$ for some positive integer $N$; $l+\beta<2s-n$ if $k+\alpha\geq
2s-n$ and $2s-n$ is not an odd positive integer.

####  4.3.2. Case 2: $s=n$

For $s=n$, denote

$\bar{g}_{n}=\rho_{n}^{2}g_{+}=e^{2\varphi}\bar{g},\quad\varphi=w-\log\rho.$

Then $\varphi$ satisfies

(4.9)
$\begin{cases}\Delta_{+}\varphi+\rho\left(\bar{\Delta}\rho+nT\right)=0,&\textrm{in
$X$},\\\ \varphi=0,&\textrm{on $M$},\end{cases}$

where

$\Delta_{+}\varphi=\rho^{2}\bar{\Delta}\varphi-(n-1)\rho\langle
d\rho,d\varphi\rangle_{\bar{g}}.$

A similar proof of Lemma 4.5, 4.6, 4.7 gives the following.

######  Lemma 4.8.

There exists an approximate solution $\tilde{\varphi}\in
C^{k,\alpha}(\overline{X})$ such that

$\Delta_{+}\tilde{\varphi}=\tilde{F},\quad\tilde{F}\in\Lambda^{k-1,\alpha}_{\delta}(X).$

where $\delta=k+\alpha$ if $k+\alpha\leq n$; $\delta=n$ if $k+\alpha>n$.
Moreover, there exists a constant $C$ only depending on $k,\alpha$ and the
$C^{k,\alpha}(\overline{X})$ norm of $\bar{g}$ such that

$\|\tilde{\varphi}\|_{C^{k,\alpha}(\overline{X})}\leq
C,\quad\|\tilde{F}\|_{\Lambda^{k-1,\alpha}_{\delta}(X)}\leq C;$

and as $\rho\rightarrow 0$,

$\tilde{\varphi}=1+\rho\varphi_{1}+\cdots+\rho^{l}\varphi_{l},$

where $l=k$ if $k+\alpha\leq n$; $l=n-1$ if $k+\alpha>n$. Here $\varphi_{j}\in
C^{k-j,\alpha}(M)$ is determined by the derivatives of $\bar{g}$ up to $j$-th
order and

$\|\varphi_{j}\|_{C^{k-j,\alpha}(M)}\leq C.$

######  Lemma 4.9.

Let $l+\beta=k+\alpha$ if $k+\alpha<n$ and $l+\beta<n$ if $k+\alpha\geq n$.
Then $\varphi\in C^{l,\beta}(\overline{X})$, and there exists some constant
$C>0$ only depending on the $C^{k,\alpha}(\overline{X})$ norm of $\bar{g}$
such that

$\|\varphi\|_{C^{l,\beta}(\overline{X})}\leq C.$

######  Lemma 4.10.

If $n$ is odd, then $\varphi\in C^{k,\alpha}(\overline{X})$ and there exists
some constant $C>0$ only depending on the $C^{k,\alpha}(\overline{X})$ norm of
$\bar{g}$ such that

$\|\varphi\|_{C^{k,\alpha}(\overline{X})}\leq C.$

######  Remark 4.3.

If $\\{g_{+}^{(i)}\\}$ is a set of $C^{k,\alpha}$ conformally compact
Poincaré-Einstein metric such that $\\{\bar{g}^{(i)}=\rho^{2}g_{+}^{(i)}\\}$
is compact in the $C^{k,\alpha}(\overline{X})$ topology and
$\mathcal{Y}(M,[\hat{g}^{(i)}])\geq 0$ for all $i$, then Lemma 4.8, 4.9, 4.10
hold with uniform estimates, i.e.

$\|\varphi^{(i)}\|_{C^{l,\beta}(\overline{X})}\leq C,$

where $C>0$ is independent of $i$. Here $l+\beta=k+\alpha$ if $k+\alpha<n$ or
$n$ is odd; $l+\beta<n$ if $k+\alpha\geq n$ and $n$ is even.

##  5\. Geometric Positivity Results for $\bar{g}_{s}$.

In this section we investigate some more geometric features for the adapted
and Fefferman-Graham compactifications. Suppose $(X^{n+1},g_{+})$ ($n\geq 3$)
is a $C^{k,\alpha}$ conformally compact Poincaré-Einstein manifold with $k\geq
3$, $0<\alpha<1$. For $s>\frac{n}{2}$, $\bar{g}_{s}$ is the adapted (including
Fefferman-Graham) compactification defined in last section. Denote

$\bar{g}_{s}=\rho_{s}^{2}g_{+}\quad\textrm{and}\quad\bar{g}_{s}|_{TM}=\hat{g}.$

Let $H_{s}$ be the boundary mean curvature of $\bar{g}_{s}$ if it is well
defined. Let $\hat{R}$ be the scalar curvature of $\hat{g}$ and
$\hat{J}=\frac{1}{2n}\hat{R}$.

###  5.1. Boundary Asymptotics

First recall the geodesic normal defining function $x$ from Lemma 3.9. Notice
that $x\in C^{k-1,\alpha}(\overline{X})$ is determined by a given boundary
representative $\hat{g}$. Then

$\tilde{g}=x^{2}g_{+}\quad\textrm{and}\quad\tilde{g}|_{TM}=\bar{g}|_{TM}=\bar{g}_{s}|_{TM}=\hat{g}.$

Recall that

$\bar{J}_{s}=\frac{2s-n-1}{2}\left(\frac{1-|d\rho_{s}|^{2}_{\bar{g}_{s}}}{\rho_{s}^{2}}\right),\quad\bar{T}_{s}=\frac{1-|d\rho_{s}|^{2}_{\bar{g}_{s}}}{\rho_{s}}.$

######  Lemma 5.1.

For $s>\frac{n}{2}+1$, we have

$\rho_{s}=x\left(1-\frac{1}{2(2s-n-2)}\hat{J}x^{2}+O(x^{2+\epsilon})\right),$

and

$\bar{J}_{s}|_{M}=\frac{2s-n-1}{2s-n-2}\hat{J}.$

###### Proof.

For $s>\frac{n}{2}+1,s\neq n$, we work on the equation (4.1). By Lemma 3.10
and the calculation in its proof, we have

$-\Delta_{+}x^{n-s}-s(n-s)x^{n-s}=-x^{n-s+1}[(n-s)\tilde{\Delta}x+s(n-s)\tilde{T}],$

where

$\tilde{T}=\frac{1-|dx|^{2}_{\tilde{g}}}{x}.$

By the definition of $x$, $\tilde{T}\equiv 0$ in a neighbourhood of $M$ and
hence for some $c>0$ small enough,

$-x^{n-s+1}[(n-s)\tilde{\Delta}x+s(n-s)\tilde{T}]=x^{n-s+2}\left(\frac{n-s}{2n}\tilde{R}\right),\quad\textrm{in
$X_{c}$}.$

Here $\tilde{R}\in C^{k-3,\alpha}(\overline{X})$. Notice that

$\displaystyle\ \left[-\Delta_{+}-s(n-s)\right](x^{r}f)$ $\displaystyle=$
$\displaystyle\
x^{r}\left[(r-s)(r+s-n)f-r(r-n)(1-|dx|^{2}_{\tilde{g}})f+rx\tilde{\Delta}xf+(2r-n+1)x\langle
dx,df\rangle_{\tilde{g}}+x^{2}\tilde{\Delta}f\right].$

Take $r=n-s+2$ and $f\in C^{k-2,\alpha}(\overline{X})$ be an extension of

$f_{0}=-\frac{n-s}{2(2s-n-2)}\hat{J}\in C^{k-2,\alpha}(M)$

according to Theorem 7.1. Then

$\left[-\Delta_{+}-s(n-s)\right](x^{n-s}+x^{n-s+2}f)=x^{n-s+3}F,\quad F\in
C^{k-3,\alpha}(\overline{X}).$

By Lemma 3.7, there exists $v^{\prime}\in\Lambda_{0}^{k-1,\alpha}(X)$ such
that

$\left[-\Delta_{+}-s(n-s)\right](x^{n-s}+x^{n-s+2}f+x^{n-s+2+\epsilon}v^{\prime})=0,$

So $v_{s}$ has asymptotics

$v_{s}=x^{n-s}\left(1-\frac{n-s}{2(2s-n-2)}\hat{J}x^{2}+O(x^{2+\epsilon})\right)$

which implies the asymptotics of $\rho_{s}$.

If $s=n$, we work on the equation (4.2). Then similar proof as above gives the
asymptotical expansion of $\rho_{n}$, which also can be read from the
continuity of $\rho_{s}$ in parameter $s$. ∎

######  Lemma 5.2.

For $s=\frac{n+1}{2}$, we have

$\rho_{s}=x\left(1-\frac{1}{n}H_{s}x+\frac{1}{2}\hat{J}x^{2}+O(x^{2+\epsilon})\right),$

and

$\bar{T}_{s}|_{M}=\frac{2}{n}H_{s}.$

###### Proof.

By Lemma 4.7, we know for $s=\frac{n+1}{2}$, $\rho_{s}\in
C^{k,\alpha}(\overline{X})$ and $\bar{g}_{s}\in C^{k,\alpha}(\overline{X})$.
Assume

$\rho_{s}=x\psi,\quad\psi=1+x\rho_{s,1}+x^{2}\rho_{s,2}+O(x^{2+\epsilon}).$

By Lemma 3.10, $(M,\hat{g})$ is totally geodesic in $(\overline{X},\tilde{g})$
and

$\bar{g}_{s}=\psi^{2}\tilde{g}=\left(1+x\rho_{s,1}+x^{2}\rho_{s,2}+O(x^{2+\epsilon})\right)^{2}(dx^{2}+\hat{g}-x^{2}\hat{A}+O(x^{2+\epsilon})),$

which implies that

$H_{s}=-n\rho_{s,1}.$

Notice that $v_{s}=\rho_{s}^{\frac{n-1}{2}}$ satisfies
$\Delta_{+}v_{s}+\frac{n^{2}-1}{4}v_{s}=0$. Then a similar calculation as the
proof of Lemma 5.1 gives $\rho_{s,2}$.

∎

###  5.2. Positivity

First, we see [CC, Proposition 6.2] still holds if the background metric has
only finite regularity.

######  Lemma 5.3 (Case-Chang).

If $\hat{R}>0$ on $M$, then for all $s>\frac{n}{2}+1$, $\bar{J}_{s}>0$ all
over $\overline{X}$.

###### Proof.

Let $s=\frac{n}{2}+\gamma$. If $s=n+1$, $\bar{J}_{n+1}>0$ in $X$ is proved in
[Le1]. Let $\mathcal{E}\subset(\frac{n}{2}+1,+\infty)$ be the subset such that
for each $s\in\mathcal{E}$ such that $\bar{J}_{s}>0$ in $X$. It is obvious
that $\mathcal{E}$ is nonempty and open. If $s_{0}\in(\frac{n}{2}+1,+\infty)$
is a limit point of $\mathcal{E}$, then $\bar{J}_{s_{0}}\geq 0$. Recall
$\bar{J}_{s}\in C^{k-1,\alpha}(X)$ and satisfies

(5.1)
$\frac{1}{2s-n-1}\left(\bar{\Delta}_{s}\bar{J}_{s}+(3+n-2s)\rho_{s}^{-1}\langle
d\rho_{s},d\bar{J}_{s}\rangle_{\bar{g}_{s}}\right)=-|\bar{A}_{s}|^{2}_{\bar{g}_{s}}+\frac{n+1}{(2s-n-1)^{2}}\bar{J_{s}}^{2}.$

By Lemma 5.1, $\bar{J}_{s_{0}}|_{M}>0$. If $s_{0}\notin\mathcal{E}$, then
there is an interior point $p\in X$ such that $\bar{J}_{s_{0}}(p)=0$, which
contradicts with the strong maximum principle. Hence $s_{0}\in\mathcal{E}$ and
$\mathcal{E}$ is closed. ∎

######  Lemma 5.4.

If $\hat{R}>0$ on $M$, then for $s=\frac{n+1}{2}$, $\bar{T}_{s}>0$ and
$\bar{J}_{s}=0$ all over $\overline{X}$.

###### Proof.

First, we can apply the same comparison argument as the proof of [GQ] show
that if $\hat{R}\geq 0$ on $M$, then $H_{s}>0$ on $M$. Let
$v^{\prime}=(\rho_{n+1})^{\frac{n-1}{2}}=(v_{n+1})^{-\frac{n-1}{2}}$. Then by
Lemma 5.1,

$v^{\prime}=x^{\frac{n-1}{2}}\left(x-\frac{n-1}{4n}\hat{J}x^{3}+O(x^{3+\epsilon})\right).$

Moreover, by Lemma 5.1,

$\frac{\Delta_{+}v^{\prime}}{v^{\prime}}+\frac{n^{2}-1}{4}=-\frac{n-1}{2}\rho_{n+1}^{2}\bar{J}_{n+1}<0.$

Apply the maximum principle to the following equation,

$\Delta_{+}\left(\frac{v_{s}}{v^{\prime}}\right)=-\left(\frac{\Delta_{+}v^{\prime}}{v^{\prime}}+\frac{n^{2}-1}{4}\right)+2\left\langle
d\left(\frac{v_{s}}{v^{\prime}}\right),d\log
v^{\prime}\right\rangle_{g_{+}},\quad\left.\frac{v_{s}}{v^{\prime}}\right|_{M}=1$

we get $v_{s}<v^{\prime}$ and hence $\rho_{s}<\rho_{n+1}$ in $X$. Therefore,
$\bar{H}_{s}>0$ on M.

Second, $\bar{T}_{s}\in C^{k-1,\alpha}(\overline{X})$ and direct calculation
shows that $\bar{T}_{s}$ satisfies the equation for $x=\frac{n+1}{2}$,

$\bar{\Delta}_{s}\bar{T}_{s}=-\frac{2}{(n-1)^{2}}\rho_{s}|\bar{E}_{s}|^{2}_{\bar{g}_{s}}.$

By the maximum principle again, we have $\bar{T}_{s}>0$ in $X$. ∎

##  6\. Proof of Theorem 1.1

In this section, we prove Theorem 1.1. Suppose $\\{(X,g^{(i)}_{+})\\}$ is a
family of $C^{k,\alpha}$ ($k\geq 3,0<\alpha<1$) conformally compact Poincaré-
Einstein manifold with conformal infinity $\\{(M,[\hat{g}^{(i)}])\\}$
satisifying $\mathcal{Y}(M,[\hat{g}^{(i)}])\geq 0$. Let $\rho$ be a fixed
smooth defining function and

$\bar{g}^{(i)}=\rho^{2}g^{(i)}_{+},\quad\hat{g}^{(i)}=\bar{g}^{(i)}|_{TM}.$

For $s>\frac{n}{2}$, let $\\{\bar{g}^{(i)}_{s}\\}$ be the adapted (including
Fefferman-Graham) compactification of $g^{(i)}_{+}$ with fixed boundary
representatives:

$\bar{g}_{s}^{(i)}|_{TM}=\bar{g}^{(i)}|_{TM}=\hat{g}^{(i)}.$

###  6.1. Proof of Theorem 1.1-(a)

Assume $\\{\bar{g}^{(i)}\\}$ is compact in $C^{k,\alpha}(\overline{X})$
topology.

####  6.1.1. Case 1: $s>\frac{n}{2},s\neq n$

In this case

$\bar{g}_{s}^{(i)}=\left(\rho_{s}^{(i)}\right)^{2}g_{+}^{(i)}=\left(\phi_{s}^{(i)}\right)^{\frac{2}{n-s}}\bar{g}^{(i)},\quad\phi_{s}^{(i)}=\left(\frac{\rho^{(i)}_{s}}{\rho}\right)^{n-s},\quad\phi_{s}^{(i)}|_{M}=1,$

and $\phi_{s}^{(i)}>0$ satisfies the equation (4.6) with background metric
$\bar{g}^{(i)}$. By the global estimates given in Lemma 4.6, 4.7 and Remark
4.2, there is a uniform constant $C>0$ independent of $i$ such that

(6.1) $\|\phi_{s}^{(i)}\|_{C^{l,\beta}(\overline{X})}\leq C,\quad\forall\ i,$

where

  * •

$l+\beta=k+\alpha$ if $k+\alpha<2s-n$ or $2s-n=2N-1$ for some positive integer
$N$;

  * •

$l+\beta<2s-n$ if $k+\alpha\geq 2s-n$ and $2s-n$ is not a positive odd
integer.

Therefore, choosing $0<\beta^{\prime}<\beta$, we have $\\{\phi_{s}^{(i)}\\}$
is compact in $C^{l,\beta^{\prime}}(\overline{X})$.

Moreover, there exist constants $0<C_{1}<C_{2}$ independent of $i$, such that

$C_{1}<\phi_{s}^{(i)}<C_{2},\quad\forall\ i.$

The uniform upper bound is obvious from (6.1). If there is no uniform lower
bound, then by passing to a subsequence, we can assume

$\displaystyle\bar{g}^{(i)}\longrightarrow\bar{g}^{(\infty)}\quad\textrm{in
$C^{k,\alpha}(\overline{X})$},$
$\displaystyle\phi_{s}^{(i)}\longrightarrow\phi_{s}^{(\infty)}\quad\textrm{in
$C^{l,\beta^{\prime}}(\overline{X})$},$

where $\phi_{s}^{(\infty)}|_{M}=1$, $\phi_{s}^{(\infty)}\geq 0$ and
$\phi_{s}^{(\infty)}(p)=0$ for some $p\in X$. However, $\phi_{s}^{(\infty)}$
satisfies the equation (4.6) with background metric $\bar{g}^{(\infty)}$.
Hence $v_{s}^{(\infty)}=\phi_{s}^{(\infty)}\rho^{n-s}$ satisfies the equation
(4.1) with background metric $g_{+}^{(\infty)}=\rho^{-2}\bar{g}^{(\infty)}$.
Since $Y(M,[\hat{g}^{(\infty)}])\geq 0$ and
$\mathrm{Spec(-\Delta_{+}^{(\infty)})}>\frac{n^{2}}{4}$, as a solution
$v^{(\infty)}_{s}(p)=0$ can not happen.

Finally, the compactness of $\\{\phi_{s}^{(i)}\\}$ in
$C^{l,\beta^{\prime}}(\overline{X})$ and uniform boundness
$0<C_{1}<\phi_{s}^{(i)}<C_{2}$ imply that $\\{\bar{g}_{s}^{(i)}\\}$ is compact
in $C^{l,\beta^{\prime}}(\overline{X})$ topology.

####  6.1.2. Case 2: $s=n$

. In this case,

$\bar{g}^{(i)}_{n}=\left(\rho^{(i)}_{n}\right)^{2}g_{+}=e^{2\varphi^{(i)}}\bar{g}^{(i)},\quad\varphi^{(i)}=w^{(i)}-\log\rho,\quad\varphi^{(i)}|_{M}=0,$

and $w^{(i)}$ satisfies the equation (4.2), $\varphi^{(i)}$ satisfies the
equation (4.9) with background metric $\bar{g}^{(i)}$. By the global estimates
given in Lemma 4.9, 4.10 and Remark 4.3, there is a uniform constant $C>0$
independent of $i$ such that

(6.2) $\|\varphi^{(i)}\|_{C^{l,\beta}(\overline{X})}\leq C,\quad\forall\ i,$

where

  * •

$l+\beta=k+\alpha$ if $k+\alpha<n$ or $n$ is odd;

  * •

$l+\beta<n$ if $k+\alpha\geq n$ and $n$ is even.

Therefore, choosing $0<\beta^{\prime}<\beta$, we have $\\{\phi_{s}^{(i)}\\}$
is compact in $C^{l,\beta^{\prime}}(\overline{X})$. And (6.2) implies that
there is a uniform constant $C_{1}>0$ such that

$-C_{1}<\varphi^{(i)}<C_{1},\quad\forall\ i.$

These imply that $\\{\bar{g}_{n}^{(i)}\\}$ is compact in
$C^{l,\beta^{\prime}}(\overline{X})$ topology.

###  6.2. Proof of Theorem 1.1-(b)

Assume $\\{\bar{g}_{s}^{(i)}\\}$ is compact in $C^{k,\alpha}(\overline{X})$
topology for some $s>\frac{n}{2}+1$ or $s=\frac{n+1}{2}$ and the boundary
scalar curvature $\hat{R}^{(i)}>0$ for all $i$. We first derive some uniform
estimates for $\rho_{s}$.

####  6.2.1. Case 1: $s>\frac{n}{2}+1$

In this case, first notice that $\rho^{(i)}_{s}$ satisfies the following
equation for each $i$ with background metric $\bar{g}^{(i)}$:

(6.3)
$\begin{cases}\displaystyle\bar{\Delta}_{s}\rho_{s}+\frac{2s}{2s-n-1}\bar{J}_{s}\rho_{s}=0,&\textrm{in
$X$},\\\ \displaystyle\rho_{s}=0,&\textrm{on $M$}.\end{cases}$

where

$\bar{J}_{s}=\frac{2s-n-1}{2}\left(\frac{1-|d\rho_{s}|^{2}_{\bar{g}_{s}}}{\rho_{s}^{2}}\right)\in
C^{k-2,\alpha}(\overline{X}).$

Recall that

$T_{s}=\frac{1-|d\rho_{s}|^{2}_{\bar{g}_{s}}}{\rho_{s}}.$

Notice that (6.3) is also equivalent to

(6.4)
$\begin{cases}\displaystyle\bar{\Delta}_{s}\rho_{s}=-s\bar{T}_{s},&\textrm{in
$X$},\\\ \displaystyle\rho_{s}=0,&\textrm{on $M$}.\end{cases}$

And $\bar{T}_{s}$ satisfies

(6.5)
$\begin{cases}\displaystyle\bar{\Delta}_{s}\bar{T}_{s}+\bar{J}_{s}\bar{T}_{s}=-2|\bar{A}_{s}|_{\bar{g}_{s}}^{2}\rho_{s}+2\langle
d\rho_{s},d\bar{J}_{s}\rangle_{\bar{g}_{s}},&\textrm{in $X$},\\\
\displaystyle\bar{T}_{s}=0,&\textrm{on $M$}.\end{cases}$

where $\bar{A}_{s}$ is the Schouten tensor of $\bar{g}_{s}$.

######  Lemma 6.1.

For $s>\frac{n}{2}+1$, if $\\{\bar{g}_{s}^{(i)}\\}$ is compact in
$C^{k,\alpha}(\overline{X})$ topology and $\hat{R}^{(i)}>0$ for all $i$, then
there exist constants $C>c>0$ such that for all $i$,

  * (i)

$\displaystyle\|\rho_{s}^{(i)}\|_{L^{\infty}(X)}\leq C$;

  * (ii)

$\displaystyle\|\rho_{s}^{(i)}\|_{C^{k+1,\alpha}(\overline{X})}\leq C$;

  * (iii)

$\displaystyle c\rho<\rho_{s}^{(i)}<C\rho$;

  * (iv)

$\displaystyle c<|d\rho^{(i)}_{s}|_{\bar{g}^{(i)}_{s}}\leq 1$ if $\rho<c$.

###### Proof.

For (i), notice that $\hat{R}^{(i)}>0$ implies that $\bar{J}^{(i)}_{s}>0$ by
Lemma 5.3. Therefore,

$|d\rho^{(i)}_{s}|_{\bar{g}^{(i)}_{s}}\leq
1\quad\Longrightarrow\quad\|\rho_{s}^{(i)}\|_{L^{\infty}(X)}\leq\mathrm{diam}(\overline{X},\bar{g}_{s}^{(i)}),\quad\forall\
i.$

For (ii), since $\\{\bar{g}_{s}^{(i)}\\}$ is $C^{k,\alpha}(\overline{X})$
compact, there is a uniform constant $C_{1}>0$ such that

$\|\bar{A}^{(i)}_{s}\|_{C^{k-2,\alpha}(\overline{X})}\leq
C_{1},\quad\|\bar{J}^{(i)}_{s}\|_{C^{k-2,\alpha}(\overline{X})}\leq
C_{1},\quad\forall\ i.$

By the classical Schauder estimates for equation (6.3), we first get

$\|\rho_{s}^{(i)}\|_{C^{k,\alpha}(\overline{X})}\leq C_{2},\quad\forall\ i.$

for some constant $C_{2}>0$. Then (6.5) implies that

$\|\bar{T}_{s}^{(i)}\|_{C^{k-1,\alpha}(\overline{X})}\leq C_{3},\quad\forall\
i.$

for some constant $C_{3}>0$. And hence (6.4) implies that

$\|\rho_{s}^{(i)}\|_{C^{k+1,\alpha}(\overline{X})}\leq C_{4},\quad\forall\ i.$

for some constant $C_{4}>0$.

For (iii), since $\rho$ is a fixed coordinates in a collar neighborhood
$X_{c}$, the uniform upper bound of $\rho_{s}^{(i)}/\rho$ is directly from
(ii). If there is no uniform lower bound, then by passing to a subsequence, we
can assume

$\displaystyle\rho_{s}^{(i)}\longrightarrow\rho_{s}^{(\infty)},\quad\textrm{in
$C^{k,\alpha^{\prime}}(\overline{X})$},$
$\displaystyle\bar{g}_{s}^{(i)}\longrightarrow\bar{g}_{s}^{(\infty)},\quad\textrm{in
$C^{k,\alpha}(\overline{X})$}.$

and $(\rho_{s}^{(\infty)}/\rho)|_{M}=1$, $(\rho_{s}^{(\infty)}/\rho)|_{p}=0$
for some $p\in X$. However, $\rho_{s}^{(\infty)}$ satisfies the equation (6.3)
with background metric $\bar{g}_{s}^{(\infty)}$ and $\bar{J}_{s}^{(\infty)}>0$
in $X$. By the strong maximum principle, $\rho_{s}^{(\infty)}>0$ in $X$, which
contradicts with $\rho_{s}^{(\infty)}(p)=0$. Hence there must be some
constants $0<c<C$ such that

$c<\frac{\rho_{s}^{(i)}}{\rho}<C,\quad\forall\ i.$

For (iv), notice that
$|d\rho^{(i)}_{s}|^{2}_{\bar{g}^{(i)}_{s}}=1-\frac{2}{2s-n-1}\bar{J}^{(i)}_{s}(\rho^{(i)}_{s})^{2}$.
So it follows immediately from (iii). ∎

####  6.2.2. Case 2: $s=\frac{n+1}{2}$

In this case, notice that $\rho_{s}^{(i)}$ satisfies the following equation

(6.6)
$\begin{cases}\displaystyle\bar{\Delta}_{s}\rho_{s}=-s\bar{T}_{s},&\textrm{in
$X$},\\\ \displaystyle\rho_{s}=0,&\textrm{on $M$},\end{cases}$

where

$\bar{T}_{s}=\frac{1-|d\rho_{s}|^{2}_{\bar{g}_{s}}}{\rho_{s}}$

satisfies

(6.7)
$\begin{cases}\displaystyle\bar{\Delta}_{s}\bar{T}_{s}=-\frac{2}{(n-1)^{2}}|\bar{E}_{s}|^{2}_{\bar{g}_{s}}\rho_{s},&\textrm{in
$X$},\\\ \displaystyle\bar{T}_{s}=\frac{2}{n}H_{s},&\textrm{on
$M$}.\end{cases}$

Here $\bar{E}_{s}$ is the trace free Ricci tensor for $\bar{g}_{s}$.

######  Lemma 6.2.

For $s=\frac{n+1}{2}$, if $\\{\bar{g}_{s}^{(i)}\\}$ is compact in
$C^{k,\alpha}(\overline{X})$ topology and $\hat{R}^{(i)}>0$ for all $i$, then
there exist constants $C>c>0$ such that for all $i$,

  * (i)

$\displaystyle\|\rho_{s}^{(i)}\|_{L^{\infty}(X)}\leq C$ and
$\displaystyle\|\bar{T}_{s}^{(i)}\|_{L^{\infty}(X)}\leq C$;

  * (ii)

$\displaystyle\|\rho_{s}^{(i)}\|_{C^{k+1,\alpha}(\overline{X})}\leq C$ and
$\displaystyle\|\bar{T}_{s}^{(i)}\|_{C^{k-1,\alpha}(\overline{X})}\leq C$;

  * (iii)

$\displaystyle c\rho<\rho_{s}^{(i)}<C\rho$;

  * (iv)

$\displaystyle c<|d\rho^{(i)}_{s}|_{\bar{g}^{(i)}_{s}}\leq 1$ if $\rho<c$.

###### Proof.

For (i), first by Lemma 5.4, $\bar{T}_{s}>0$ implies that

$|d\rho^{(i)}_{s}|_{\bar{g}^{(i)}_{s}}\leq
1\quad\Longrightarrow\quad\|\rho_{s}^{(i)}\|_{L^{\infty}(X)}\leq\mathrm{diam}(\overline{X},\bar{g}_{s}^{(i)}),\quad\forall\
i.$

If there is no uniform constant $C$ such that
$\displaystyle\|\bar{T}_{s}^{(i)}\|_{L^{\infty}(X)}\leq C$, then by passing to
a subsequence, we can assume

$\displaystyle\|\bar{T}_{s}^{(i)}\|_{L^{\infty}(X)}\longrightarrow+\infty.$

Take $\tilde{\rho}^{(i)}=\rho_{s}^{(i)}/\|\bar{T}_{s}^{(i)}\|_{L^{\infty}(X)}$
and $\tilde{T}^{(i)}=\bar{T}_{s}^{(i)}/\|\bar{T}_{s}^{(i)}\|_{L^{\infty}(X)}$.
Then $(\tilde{\rho}^{(i)},\tilde{T}^{(i)})$ satisfies

$\begin{cases}\displaystyle\bar{\Delta}^{(i)}_{s}\tilde{\rho}^{(i)}+s\tilde{T}^{(i)}=0,&\textrm{in
$X$},\\\ \displaystyle\tilde{\rho}^{(i)}=0,&\textrm{on
$M$},\end{cases}\quad\mathrm{and}\quad\begin{cases}\displaystyle\bar{\Delta}^{(i)}_{s}\tilde{T}^{(i)}+\frac{2}{(n-1)^{2}}\tilde{\rho}^{(i)}|\bar{E}^{(i)}_{s}|^{2}_{\bar{g}_{s}}=0,&\textrm{in
$X$},\\\
\displaystyle\tilde{T}^{(i)}=\frac{2}{n}\frac{H_{s}}{\|\bar{T}_{s}^{(i)}\|_{L^{\infty}(X)}},&\textrm{on
$M$}.\end{cases}$

Notice that $\\{\bar{g}_{s}^{(i)}\\}$ is compact in
$C^{k,\alpha}(\overline{X})$ topology implies that

$\|H_{s}^{(i)}\|_{C^{k-1,\alpha}(M)}\leq
C_{1},\quad\|\bar{E}^{(i)}_{s}\|_{C^{k-2,\alpha}(\overline{X})}\leq C_{1},$

for some constant $C_{1}>0$ independent of $i$. Hence applying the classical
Schauder estimate to above coupled system, we have

$\|(\tilde{\rho}^{(i)},\tilde{T}^{(i)})\|_{C^{k-1,\alpha}(\overline{X})}\leq
C^{\prime},$

for some constant $C^{\prime}>0$ independent of $i$. By passing to a
subsequence again, we assume

$\displaystyle\tilde{T}^{(i)}\longrightarrow\tilde{T}^{(\infty)},\quad\textrm{in
$C^{2}(\overline{X})$},$
$\displaystyle\bar{g}_{s}^{(i)}\longrightarrow\bar{g}_{s}^{(\infty)},\quad\textrm{in
$C^{k,\alpha}(\overline{X})$}.$

Then $\tilde{T}^{(\infty)}$ satisfies

$\begin{cases}\bar{\Delta}_{s}^{(\infty)}\tilde{T}^{(\infty)}=0,&\quad\textrm{in
$X$},\\\ \tilde{T}^{(\infty)}=0,&\quad\textrm{on $M$}.\end{cases}$

This implies that $\tilde{T}^{(\infty)}\equiv 0$ which contradicts with
$\|\tilde{T}^{(\infty)}\|_{L^{\infty}(X)}=1$.

For (ii), it follows from (i) together with the classical Schauder estimates
for the coupled equations (6.6) and (6.7).

For (iii), the uniform upper bound of $\rho_{s}^{(i)}/\rho$ is directly from
(ii). If there is no uniform lower bound, then by passing to a subsequence, we
can assume

$\displaystyle\rho_{s}^{(i)}\longrightarrow\rho_{s}^{(\infty)},\quad\textrm{in
$C^{k,\alpha^{\prime}}(\overline{X})$},$
$\displaystyle\bar{g}_{s}^{(i)}\longrightarrow\bar{g}_{s}^{(\infty)},\quad\textrm{in
$C^{k,\alpha}(\overline{X})$},$

and $(\rho_{s}^{(\infty)}/\rho)|_{M}=1$, $(\rho_{s}^{(\infty)}/\rho)|_{p}=0$
for some $p\in X$. Here $\rho_{s}^{(\infty)}$ and $\bar{T}_{s}^{(\infty)}$
satisfies the equation (6.6) and (6.7) with background metric
$\bar{g}^{(\infty)}_{s}$. And $H_{s}^{(\infty)}\geq 0$ on $M$ implies that
$\bar{T}_{s}^{\infty}>0$ in $X$. By the strong maximum principle,
$\rho_{s}^{(\infty)}>0$ in $X$, which contradicts with
$\rho_{s}^{(\infty)}(p)=0$. Hence there must be some uniform constants $0<c<C$
such that

$c<\frac{\rho_{s}^{(i)}}{\rho}<C,\quad\forall\ i.$

For (iv), notice that
$|d\rho^{(i)}_{s}|^{2}_{\bar{g}^{(i)}_{s}}=1-\rho_{s}^{(i)}\bar{T}_{s}^{(i)}$.
Hence (iv) follows immediately from (iii). ∎

Now we continue the proof of Theorem 1.1-(b). Lemma 6.1 and 6.2 imply that for
$s>\frac{n}{2}+1$ or $s=\frac{n+1}{2}$, there is a constant $C^{\prime}>0$
such that

$\|\rho_{s}^{(i)}\|_{C^{k+1,\alpha}(\overline{X})}\leq C,\quad\forall\ i.$

Since $\rho_{s}|_{M}=0$, by Corollary 7.1 in the Appendix, there is a constant
$C^{\prime}>0$ such that

$\left\|\frac{\rho_{s}^{(i)}}{\rho}\right\|_{C^{k,\alpha}(\overline{X})}\leq
C^{\prime},\quad\forall\ i.$

This together with Lemma 6.1-(iii) and Lemma 6.2-(iii) implies that for some
constant $C^{\prime\prime}>0$,

$\left\|\frac{\rho}{\rho_{s}^{(i)}}\right\|_{C^{k,\alpha}(\overline{X})}\leq
C^{\prime\prime},\quad\forall\ i.$

Recall that

$\bar{g}^{(i)}=\left(\frac{\rho}{\rho_{s}^{(i)}}\right)^{2}\bar{g}_{s}^{(i)}.$

Therefore $\\{\bar{g}^{(i)}\\}$ is compact in
$C^{k,\alpha^{\prime}}(\overline{X})$ topology for $0<\alpha^{\prime}<\alpha$.
We finish the proof.

##  7\. Appendix

In this section we derive an extension theorem.

######  Lemma 7.1.

For any $k,l\geq 0$ and $0\leq\alpha<1$, suppose $f=f(y)$ is a $C^{k,\alpha}$
function which has compact support in $\mathbb{R}^{n}_{y}$. Then there exists
an extension $F=F(x,y)$ satisfying the following properties:

  * (1)

$F\in C^{k+l,\alpha}([0,1)_{x}\times\mathbb{R}^{n}_{y})$ and there exists some
constant $C_{1}>0$ only depending on $(k,l,\alpha)$ such that

$\|F\|_{C^{k+l,\alpha}([0,1)_{x}\times\mathbb{R}^{n}_{y})}\leq
C_{1}\|f\|_{C^{k,\alpha}(\mathbb{R}^{n}_{y})};$

  * (2)

As $x\rightarrow 0$, $F(x,y)$ has an asymptotical expansion

$F(x,y)=x^{l}f(y)+x^{l+1}f_{1}(y)+\cdots x^{l+k}f_{k}(y)+O(x^{l+k+\alpha})$

where $f_{i}(y)\in C^{k-i,\alpha}(\mathbb{R}_{y})$ determined derivatives of
$f$ up to $i$-th order and there exists a constant $C_{2}>0$ only depending on
$(k,l,\alpha)$,

$\|f_{i}\|_{C^{k-i,\alpha}(\mathbb{R}_{y})}\leq
C_{2}\|f\|_{C^{k,\alpha}(\mathbb{R}^{n}_{y})},\quad\forall\ 0\leq i\leq k;$

  * (3)

For any $j\geq 0$ and multi-index $J\geq 0$, there exists a constant $C_{3}>0$
only depending on $(k,l,\alpha,j,J)$ such that

$\|x^{j+|J|}\partial_{x}^{j}\partial_{y}^{J}F\|_{C^{k+l,\alpha}([0,1)_{x}\times\mathbb{R}^{n}_{y})}\leq
C_{3}\|f\|_{C^{k,\alpha}(\mathbb{R}^{n}_{y})}.$

###### Proof.

Let $\phi\in C_{c}^{\infty}(\mathbb{R}^{n}_{y})$ satisfy

$\begin{gathered}0\leq\phi\leq 1,\quad\phi(0)=1,\quad\phi(y)=0\textrm{ if
$|y|\geq
1$}\quad\mathrm{and}\quad\int_{\mathbb{R}^{n}_{y}}\phi(y)dy=1.\end{gathered}$

For any $0<x<1$, denote
$\displaystyle\phi_{x}(y)=\frac{1}{x^{n}}\phi\left(\frac{y}{x}\right)$ and

$f_{x}(y)=\phi_{x}*f(y)=\int_{\mathbb{R}^{n}}\phi_{x}(y^{\prime})f(y-y^{\prime})dy^{\prime}=\int_{\mathbb{R}^{n}}\phi(z)f(y-xz)dz.$

Then it is easy to check that $f_{x}\in C^{\infty}_{c}(\mathbb{R}^{n}_{y})$
and

$\lim_{x\rightarrow 0}\|f_{x}-f\|_{C^{k,\alpha}(\mathbb{R}^{n}_{y})}=0.$

Hence there is some constant $C_{0}>0$ such that

$\|f_{x}\|_{C^{k,\alpha}(\mathbb{R}^{n}_{y})}\leq
C_{0}+\|f\|_{C^{k,\alpha}(\mathbb{R}^{n}_{y})},\quad\forall\ 0<x<1.$

Denote $h(x,y)=f_{x}(y)$. Then $h$ satisfies the following properties:

  * (i)

If $I$ is a multi index such that $|I|\leq k$, then

$\partial_{y}^{I}h(x,y)=\phi_{x}*\left(\partial_{y}^{I}f\right)=\int_{\mathbb{R}^{n}}\phi(z)f^{(I)}(y-xz)dz,$

and since $\mathrm{supp}(\phi)\subset\\{|z|<1\\}$,

$\displaystyle\
\frac{|\partial_{y}^{I}h(x,y)-\partial_{y}^{I}h(x^{\prime},y^{\prime})|}{|(x,y)-(x^{\prime},y^{\prime})|^{\alpha}}$
$\displaystyle\leq$ $\displaystyle\
\int_{\mathbb{R}^{n}}\phi(z)\frac{|f^{(I)}(y-xz)-f^{(I)}(y^{\prime}-x^{\prime}z)|}{|(x,y)-(x^{\prime},y^{\prime})|^{\alpha}}dz$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}^{n}}\phi(z)\left(\frac{|f^{(I)}(y-xz)-f^{(I)}(y^{\prime}-xz)|}{|y-y^{\prime}|^{\alpha}}+\frac{|f^{(I)}(y^{\prime}-xz)-f^{(I)}(y^{\prime}-x^{\prime}z)|}{|x-x^{\prime}|^{\alpha}}\right)dz.$

These imply that for some constant $C>0$ only depending on $\phi$,

$\|\partial_{y}^{I}h\|_{C^{0,\alpha}((0,1)\times\mathbb{R}^{n}_{y})}\leq
C\|f\|_{C^{k,\alpha}(\mathbb{R}^{n}_{y})}.$

  * (ii)

If $I$ is a multi index such that $|I|>k$, then choosing a multi index $J$
with $J\leq I$ and $|J|=k$, we have

$\partial_{y}^{I}h=\left(\partial_{y}^{I-J}\phi_{x}\right)*\left(\partial_{y}^{J}f\right)=\frac{1}{x^{|I|-k}}\int_{\mathbb{R}^{n}}\phi^{(I-J)}(z)f^{(J)}(y-xz)dz.$

Hence similar as (i), for some constant $C>0$ only depending on $k,|I|$ and
$\phi$, we have

$\|x^{|I|-k}\partial_{y}^{I}h\|_{C^{0,\alpha}((0,1)\times\mathbb{R}^{n}_{y})}\leq
C\|f\|_{C^{k,\alpha}(\mathbb{R}^{n}_{y})}.$

  * (iii)

If $i\leq k$, then

$\partial_{x}^{i}h=\sum_{|I|=i}\int_{\mathbb{R}^{n}}\phi(z)(-z)^{I}\partial_{y}^{I}f(y-xz)dz$

Hence similar as (i), for some constant constant $C>0$ only depending on
$\phi$ we have

$\|\partial_{x}^{i}f_{x}(y)\|_{C^{0,\alpha}((0,1)\times\mathbb{R}^{n}_{y})}\leq
C\|f\|_{C^{k,\alpha}(\mathbb{R}^{n}_{y})}.$

  * (iv)

If $i>k$, then

$\displaystyle\partial_{x}^{i}h=$ $\displaystyle\
\partial_{x}^{i-k}\left(\sum_{|J|=k}\int_{\mathbb{R}^{n}}\phi(z)(-z)^{J}f^{(J)}(y-xz)dz\right)$
$\displaystyle=$ $\displaystyle\
\partial_{x}^{i-k}\left(\sum_{|J|=k}\int_{\mathbb{R}^{n}}\frac{1}{x^{n+k}}\phi\left(\frac{y^{\prime}}{x}\right)(-y^{\prime})^{J}f^{(J)}(y-y^{\prime})dy^{\prime}\right)$
$\displaystyle=$ $\displaystyle\ \sum_{|I|\leq
i-k}\sum_{|J|=k}\int_{\mathbb{R}^{n}}\frac{c_{I}}{x^{n+i+|J|}}\phi^{(I)}\left(\frac{y^{\prime}}{x}\right)(y^{\prime})^{I+J}f^{(J)}(y-y^{\prime})dy^{\prime}$
$\displaystyle=$ $\displaystyle\ \sum_{|I|\leq
i-k}\sum_{|J|=k}\frac{c_{I}}{x^{i-k}}\int_{\mathbb{R}^{n}}\phi^{(I)}(z)z^{I+J}f^{(J)}(y-xz)dz.$

Here $c_{I}$ are constants. For some constant $C>0$ only depending on $i,k$
and $\phi$, we have

$\|x^{i-k}\partial_{x}^{i}f_{x}(y)\|_{C^{0,\alpha}((0,1)\times\mathbb{R}^{n}_{y})}\leq
C\|f\|_{C^{k,\alpha}(\mathbb{R}^{n}_{y})}.$

Now we define

$F(x,y)=\chi(x)x^{l}h(x,y),\quad 0<x<1.$

where $\chi(x)$ is a smooth cutoff function satisfying

$\chi\in C_{c}^{\infty}([0,1))$, $0\leq\chi\leq 1$, $\chi(x)=1$ if $0\leq
x<\frac{1}{2}$ and $\chi(x)=0$ if $x>\frac{3}{4}$.

And $F(0,y)=f(y)$ if $l=0$; $F(0,y)=0$ if $l>0$. By (i)-(iv), there exists
some constant $C>0$ only depending on $k,l$ and $\phi$ such that

$\|F\|_{C^{k+l,\alpha}([0,1)_{x}\times\mathbb{R}^{n}_{y}}\leq
C\|f\|_{C^{k,\alpha}(\mathbb{R}^{n}_{y})}.$

Hence $F$ satisfies (1).

For the asymptotical expansion of $F$, it is simply by taking $x\rightarrow 0$
in (iii). For $i\leq k$,

$\lim_{x\rightarrow
0}\partial_{x}^{i}h=\sum_{|I|=i}\left(\int_{\mathbb{R}^{n}}\phi(z)(-z)^{I}dz\right)f^{(I)}(y)$

In particular, there is some constant $C>0$ depending on $k,\alpha$ such that

$\displaystyle\left|\frac{\partial_{x}^{k}h(x,y)-\partial_{x}^{k}h(0,y)}{x^{\alpha}}\right|=$
$\displaystyle\
\left|\sum_{|I|=k}\int_{\mathbb{R}^{n}}\phi(z)(-z)^{I}\left(\frac{f^{(I)}(y-xz)-f^{(I)}(y)}{x^{\alpha}}\right)dz\right|$
$\displaystyle=$ $\displaystyle\
\sum_{|I|=k}\int_{\mathbb{R}^{n}}\phi(z)|z|^{|I|+\alpha}\left|\frac{f^{(I)}(y-xz)-f^{(I)}(y)}{|xz|^{\alpha}}\right|dz$
$\displaystyle\leq$ $\displaystyle\
C\|f\|_{C^{k,\alpha}(\mathbb{R}^{n}_{y})}.$

Hence $F$ satisfies (2).

To prove (3), we just notice that

$\displaystyle x^{j+J}\partial_{x}^{j}\partial_{y}^{J}h=$ $\displaystyle\
x^{j}\partial_{x}^{j}\int_{\mathbb{R^{n}}}\frac{1}{x^{n}}\phi^{(J)}\left(\frac{y^{\prime}}{x}\right)f(y-y^{\prime})dy^{\prime}$
$\displaystyle=$ $\displaystyle\ \sum_{|I|\leq
j}\int_{\mathbb{R}^{n}}\frac{c_{I}}{x^{n+|I|}}\phi^{(I+J)}\left(\frac{y^{\prime}}{x}\right)(y^{\prime})^{I}f(y-y^{\prime})dy^{\prime}$
$\displaystyle=$ $\displaystyle\ \sum_{|I|\leq
j}\int_{\mathbb{R}^{n}}c_{I}\phi^{(I+J)}(z)z^{I}f(y-xz)dz,$

where $c_{I}$ are some constants. Replace $\phi(z)$ by
$\tilde{\phi}(z)=\sum_{|I|\leq i}c_{I}\phi^{(I+J)}(z)z^{I}$ and $h$ by
$x^{j+J}\partial_{x}^{j}\partial_{y}^{J}h$, then (i)-(iv) are also valid with
an adjustment of the constants. Then (3) follows from a similar proof of (1).
∎

######  Theorem 7.1.

Suppose $\overline{X}$ is a smooth compact manifold with boundary $M$. Let
$\rho$ be a smooth boundary defining function. For any integers $k,l\geq 0$
and $0\leq\alpha<1$, if $f\in C^{k,\alpha}(M)$, then there exists $F\in
C^{k+l,\alpha}(\overline{X})\cap\Lambda_{l}^{\infty}(X)$ satisfies the
following properties:

  * (1)

There exists some constant $C_{1}>0$ only depending on $k,l,\alpha$, such that

$\|F\|_{C^{k+l,\alpha}(\overline{X})}\leq C_{1}\|f\|_{C^{k,\alpha}(M)};$

  * (2)

As $\rho\rightarrow 0$, $F$ has an asymptotical expansion as follows

$F=\rho^{l}f_{0}+\rho^{l+1}f_{1}+\cdots+\rho^{l+k}f_{k}+O(x^{k+l+\alpha})$

where $f_{i}\in C^{k-i,\alpha}(M)$ with $f_{0}=f$ and there exists some
constant $C_{2}>0$ only depending on $k,l,\alpha$,

$\|f_{i}\|_{C^{k-i,\alpha}(M)}\leq C_{2}\|f\|_{C^{k,\alpha}(M)},\quad\forall\
i=1,...,k.$

  * (3)

If $P$ is a differential operators of order $m\leq k+l$ with coefficients in
$C^{k,\alpha}(\overline{X})$, then there exists some constant $C_{3}>0$ only
depending on the $C^{k,\alpha}(\overline{X})$ norm of the coefficients such
that

$\|PF\|_{C^{k+l-m,\alpha}(\overline{X})}\leq
C_{3}\|f\|_{C^{k,\alpha}(M)},\quad\|\rho^{m}PF\|_{C^{k+l,\alpha}(\overline{X})}\leq
C_{3}\|f\|_{C^{k,\alpha}(M)}.$

Theorem 7.1 directly implies the following.

######  Corollary 7.1.

Suppose $\overline{X}$ is a smooth compact manifold with boundary $M$. Let
$\rho$ be a smooth boundary defining function. For any integer $k\geq 0$ and
$0\leq\alpha<1$, if $f\in C^{k+1,\alpha}(\overline{X})$ satisfying $f|_{M}=0$,
then $f/\rho\in C^{k,\alpha}(\overline{X})$. Moreover, there exists a constant
$C>0$ independent of $f$ such that

$\left\|\frac{f}{\rho}\right\|_{C^{k,\alpha}(\overline{X})}\leq
C\|f\|_{C^{k+1,\alpha}(\overline{X})}.$

###### Proof.

First, $f\in C^{k+1,\alpha}(\overline{X})$ and $f|_{M}=0$ implies that $f$ has
Taylor expansion as the following:

$f=\rho f_{1}+O(\rho^{2}),$

where $f_{1}\in C^{k,\alpha}(M)$ and for some constant $C_{1}>0$,

$\|f_{1}\|_{C^{k,\alpha}(M)}\leq C_{1}\|f\|_{C^{k+1,\alpha}(\overline{X})}.$

Take $\tilde{f}_{1}$ be the extension of $\rho f_{1}$ as in Theorem 7.1. Then
for some constant $C_{1}^{\prime}>0$,

$\|\tilde{f}_{1}\|_{C^{k+1,\alpha}(\overline{X})}\leq
C_{1}^{\prime}\|f_{1}\|_{C^{k,\alpha}(M)}.$

Moreover,

$f-\tilde{f}_{1}=\rho^{2}f_{2}+O(\rho^{3})$

where $f_{2}\in C^{k-1,\alpha}(M)$ and for some constant $C_{2}>0$,

$\|f_{2}\|_{C^{k-1,\alpha}(M)}\leq C_{2}\|f\|_{C^{k+1,\alpha}(\overline{X})}.$

Repeat above progress until we get

$f=\tilde{f}_{1}+\tilde{f}_{2}+\cdots+\tilde{f}_{k}+E,$

where $\tilde{f}_{i}\sim\rho^{i}f_{i}$ and for some constants
$C_{i},C_{i}^{\prime}>0$

$\|f_{i}\|_{C^{k+1-i,\alpha}(M)}\leq
C_{i}\|f\|_{C^{k+1,\alpha}(\overline{X})},\quad\|\tilde{f}_{i}\|_{C^{k+1,\alpha}(\overline{X})}\leq
C_{1}^{\prime}\|f_{i}\|_{C^{k+1-i,\alpha}(M)},\quad i=1,...,k.$

Hence for some constant $C>0$,

$\|E\|_{C^{k+1,\alpha}(\overline{X})}\leq
C\|f\|_{C^{k+1,\alpha}(\overline{X})},\quad E=O(\rho^{k+1+\alpha}).$

Therefore ,

$E\in
C^{k+1,\alpha}_{(k+1+\alpha)}(\overline{X})\hookrightarrow\Lambda^{k+1,\alpha}_{k+1+\alpha}(X),\quad\Longrightarrow\quad\frac{E}{\rho}\in\Lambda^{k+1,\alpha}_{k+\alpha}(X)\hookrightarrow
C^{k,\alpha}(\overline{X}).$

By Lemma 3.3-3.4, $C^{k+1,\alpha}_{(k+1+\alpha)}(\overline{X})$ is closed in
$C^{k+1,\alpha}(\overline{X})$ and above two inclusion maps are continuous.
Hence for some constant $C^{\prime}>0$,

$\left\|\frac{E}{\rho}\right\|_{C^{k,\alpha}(\overline{X})}\leq
C^{\prime}\|E\|_{C^{k+1,\alpha}(\overline{X})}.$

From the extension method given in Lemma 7.1 and Theorem 7.1, it is easy to
see for some constant $C^{\prime\prime}>0$,

$\left\|\frac{\tilde{f}_{i}}{\rho}\right\|_{C^{k,\alpha}(\overline{X})}\leq
C^{\prime\prime}\|\tilde{f}_{i}\|_{C^{k+1,\alpha}(\overline{X})},\quad
i=1,2,...,k.$

We finish the proof. ∎

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