81(2009) 225-250

BLOW-UP PHENOMENA FOR THE YAMABE EQUATION II

Simon Brendle & Fernando C. Marques

Abstract

Let *n* be an integer such that 25 *≤* *n* *≤* 51. We construct a
smooth metric*g* on*S** ^{n}* with the property that the set of constant
scalar curvature metrics in the conformal class of

*g*is not compact.

1. Introduction

Let (M, g) be a compact Riemannian manifold of dimension *n* *≥*
3. The Yamabe problem is concerned with finding metrics of constant
scalar curvature in the conformal class of *g. This problem leads to*
a semi-linear elliptic PDE for the conformal factor. More precisely, a
conformal metric of the form *u*^{n−2}^{4} *g* has constant scalar curvature *c* if
and only if

(1) 4(n*−*1)

*n−*2 ∆_{g}*u−R*_{g}*u*+*c u*^{n+2}* ^{n−2}* = 0,

where ∆* _{g}* is the Laplace operator with respect to

*g*and

*R*

*denotes the scalar curvature of*

_{g}*g. Every solution of (1) is a critical point of the*functional

(2) *E** _{g}*(u) =

R

*M*

¡_{4(n−1)}

*n−2* *|du|*^{2}* _{g}*+

*R*

_{g}*u*

^{2}¢

*dvol*

_{g}¡ R

*M**u*^{n−2}^{2n} *dvol** _{g}*¢

^{n−2}*n*

*.*

In this paper, we address the question whether the set of all solutions to
the Yamabe PDE is compact in the*C*^{2}-topology. It has been conjectured
that this should be true unless (M, g) is conformally equivalent to the
round sphere (see [15], [16], [17]). The case of the round sphere *S** ^{n}* is
special in that (1) is invariant under the action of the conformal group
on

*S*

*, which is non-compact. It follows from a theorem of Obata [14]*

^{n}that every solution of the Yamabe PDE on *S** ^{n}* is minimizing, and the
space of all solutions to the Yamabe PDE on

*S*

*can be identified with the unit ball*

^{n}*B*

*. Note that the round sphere is the only compact manifold for which the set of minimizing solutions is non-compact.*

^{n+1}The Compactness Conjecture has been verified in low dimensions and in the locally conformally flat case. If (M, g) is locally conformally

Received 03/14/2007.

225

flat, compactness follows from work of R. Schoen [15], [16]. Moreover, Schoen proposed a strategy, based on the Pohozaev identity, for proving the conjecture in the non-locally conformally flat case. In [12], Y.Y. Li and M. Zhu followed this strategy to prove compactness in dimension 3. O. Druet [6] proved the conjecture in dimensions 4 and 5.

The case *n≥*6 is more subtle, and requires a careful analysis of the
local properties of the background metric *g* near a blow-up point. The
Compactness Conjecture is closely related to the Weyl Vanishing Con-
jecture, which asserts that the Weyl tensor should vanish to an order
greater than ^{n−6}_{2} at a blow-up point (see [17]). The Weyl Vanishing
Conjecture has been verified in dimensions 6 and 7 by F. Marques [13]

and, independently, by Y.Y. Li and L. Zhang [10]. Using these re-
sults and the positive mass theorem, these authors were able to prove
compactness for *n≤*7. Moreover, Li and Zhang showed that compact-
ness holds in all dimensions provided that *|W** _{g}*(p)|+

*|∇W*

*(p)|*

_{g}*>*0 for all

*p*

*∈*

*M. In dimensions 10 and 11, it is sufficient to assume that*

*|W** _{g}*(p)|+

*|∇W*

*(p)|+*

_{g}*|∇*

^{2}

*W*

*(p)|*

_{g}*>*0 for all

*p∈M*(see [11]).

Very recently, M. Khuri, F. Marques, and R. Schoen [9] proved the Weyl Vanishing Conjecture up to dimension 24. This result, combined with the positive mass theorem, implies the Compactness Conjecture for those dimensions. After proving sharp pointwise estimates, they reduce these questions to showing a certain quadratic form is positive definite.

It turns out the quadratic form has negative eigenvalues if*n≥*25.

In a recent paper [4], it was shown that the Compactness Conjecture
fails for *n≥*52. More precisely, given any integer *n≥*52, there exists
a smooth Riemannian metric *g* on *S** ^{n}* such that set of constant scalar
curvature metrics in the conformal class of

*g*is non-compact. Moreover, the blowing-up sequences obtained in [4] form exactly one bubble. The construction relies on a gluing procedure based on some local model metric. These local models are directions in which the quadratic form of [9] is negative definite. We refer to [5] for a survey of this and related results.

In the present paper, we extend these counterexamples to the dimen-
sions 25*≤n≤*51. Our main theorem is:

Theorem. *Assume that* 25*≤n≤*51. Then there exists a Riemann-
*ian metric* *g* *on* *S** ^{n}* (of class

*C*

*)*

^{∞}*and a sequence of positive functions*

*v*

_{ν}*∈C*

*(S*

^{∞}*) (ν*

^{n}*∈*N)

*with the following properties:*

(i) *g* *is not conformally flat,*

(ii) *v*_{ν}*is a solution of the Yamabe PDE (1) for all* *ν* *∈*N,

(iii) *E** _{g}*(v

*)*

_{ν}*< Y*(S

*)*

^{n}*for allν*

*∈*N, and

*E*

*(v*

_{g}*)*

_{ν}*→Y*(S

*)*

^{n}*as*

*ν*

*→ ∞,*(iv) sup

_{S}

^{n}*v*

_{ν}*→ ∞*

*as*

*ν→ ∞.*

(Here, *Y*(S* ^{n}*)

*denotes the Yamabe energy of the round metric on*

*S*

^{n}*.)*

We note that O. Druet and E. Hebey [7] have constructed blow-up examples for perturbations of (1) (see also [8]).

In Section 2, we describe how the problem can be reduced to find-
ing critical points of a certain function *F** _{g}*(ξ, ε), where

*ξ*is a vector in R

*and*

^{n}*ε*is a positive real number. This idea has been used by many authors (see, e.g., [1], [2], [3], [4]). In Section 3, we show that the func- tion

*F*

*(ξ, ε) can be approximated by an auxiliary function*

_{g}*F*(ξ, ε). In Section 4, we prove that the function

*F*(ξ, ε) has a critical point, which is a strict local minimum. Finally, in Section 5, we use a perturbation argument to construct critical points of the function

*F*

*(ξ, ε). From this the non-compactness result follows.*

_{g}The authors would like to thank Professor Richard Schoen for con- stant support and encouragement. The first author was supported by the Alfred P. Sloan foundation and by the National Science Founda- tion under grant DMS-0605223. The second author was supported by CNPq-Brazil, FAPERJ and the Stanford Mathematics Department.

2. Lyapunov-Schmidt reduction

In this section, we collect some basic results established in [4]. Let
*E* =

½

*w∈L*^{n−2}^{2n} (R* ^{n}*)

*∩W*

_{loc}^{1,2}(R

*) : Z*

^{n}R^{n}

*|dw|*^{2} *<∞*

¾
*.*

By Sobolev’s inequality, there exists a constant *K, depending only on*
*n, such that*

µ Z

R^{n}

*|w|*^{n−2}^{2n}

¶^{n−2}

*n* *≤K*
Z

R^{n}

*|dw|*^{2}
for all *w* *∈ E. We define a norm on* *E* by *kwk*^{2}* _{E}* =R

R^{n}*|dw|*^{2}. It is easy
to see that *E, equipped with this norm, is complete.*

Given any pair (ξ, ε)*∈*R^{n}*×*(0,*∞), we define a functionu*_{(ξ,ε)} :R^{n}*→*
Rby

*u*_{(ξ,ε)}(x) =

³ *ε*

*ε*^{2}+*|x−ξ|*^{2}

´^{n−2}

2 *.*

The function *u*_{(ξ,ε)} satisfies the elliptic PDE

∆u_{(ξ,ε)}+*n(n−*2)*u*

*n+2*
*n−2*

(ξ,ε)= 0.

It is well known that Z

R^{n}

*u*

2n
*n−2*

(ξ,ε)=

³ *Y*(S* ^{n}*)
4n(n

*−*1)

´^{n}

2

for all (ξ, ε)*∈*R^{n}*×*(0,*∞). We next define*
*ϕ*_{(ξ,ε,0)}(x) =

³ *ε*

*ε*^{2}+*|x−ξ|*^{2}

´^{n+2}

2 *ε*^{2}*− |x−ξ|*^{2}
*ε*^{2}+*|x−ξ|*^{2}

and

*ϕ*_{(ξ,ε,k)}(x) =

³ *ε*

*ε*^{2}+*|x−ξ|*^{2}

´^{n+2}

2 2ε(x_{k}*−ξ** _{k}*)

*ε*

^{2}+

*|x−ξ|*

^{2}

for*k*= 1, . . . , n. Finally, we define a closed subspace *E*_{(ξ,ε)}*⊂ E* by
*E*_{(ξ,ε)}=

½

*w∈ E* :
Z

R^{n}

*ϕ*_{(ξ,ε,k)}*w*= 0 for *k*= 0,1, . . . , n

¾
*.*
Clearly,*u*_{(ξ,ε)}*∈ E*_{(ξ,ε)}.

Proposition 1. *Consider a Riemannian metric on* R^{n}*of the form*
*g(x) = exp(h(x)), where* *h(x)* *is a trace-free symmetric two-tensor on*
R^{n}*satisfying* *h(x) = 0* *for* *|x| ≥* 1. There exists a positive constant
*α*_{0}*≤*1, depending only on*n, with the following significance: if|h(x)|*+

*|∂h(x)|*+*|∂*^{2}*h(x)| ≤* *α*_{0} *for all* *x* *∈* R^{n}*, then, given any pair* (ξ, ε) *∈*
R^{n}*×*(0,*∞)* *and any function* *f* *∈* *L*^{n+2}^{2n} (R* ^{n}*), there exists a unique

*function*

*w*=

*G*

_{(ξ,ε)}(f)

*∈ E*

_{(ξ,ε)}

*such that*

Z

R^{n}

³

*hdw, dψi** _{g}*+

*n−*2

4(n*−*1)*R*_{g}*wψ−n(n*+ 2)u

4
*n−2*

(ξ,ε)*wψ*

´

= Z

R^{n}

*f ψ*
*for all test functions* *ψ∈ E*_{(ξ,ε)}*.*

*Moreover, we have* *kwk*_{E}*≤* *Ckfk*

*L*^{n+2}^{2n} (R* ^{n}*)

*, where*

*C*

*is a constant*

*that depends only on*

*n.*

Proposition 2. *Consider a Riemannian metric on* R^{n}*of the form*
*g(x) = exp(h(x)), where* *h(x)* *is a trace-free symmetric two-tensor on*
R^{n}*satisfying* *h(x) = 0* *for* *|x| ≥*1. Moreover, let (ξ, ε)*∈*R^{n}*×*(0,*∞).*

*There exists a positive constant* *α*_{1} *≤α*_{0}*, depending only on* *n, with the*
*following significance: if* *|h(x)|+|∂h(x)|*+*|∂*^{2}*h(x)| ≤α*_{1} *for allx∈*R^{n}*,*
*then there exists a functionv*_{(ξ,ε)}*∈ E* *such that* *v*_{(ξ,ε)}*−u*_{(ξ,ε)} *∈ E*_{(ξ,ε)} *and*
Z

R^{n}

³

*hdv*_{(ξ,ε)}*, dψi** _{g}*+

*n−*2

4(n*−*1)*R*_{g}*v*_{(ξ,ε)}*ψ−n(n−2)|v*_{(ξ,ε)}*|*^{n−2}^{4} *v*_{(ξ,ε)}*ψ*

´

= 0
*for all test functions* *ψ∈ E*_{(ξ,ε)}*. Moreover, we have the estimate*

*kv*_{(ξ,ε)}*−u*_{(ξ,ε)}*k*_{E}

*≤C*

°°

°∆_{g}*u*_{(ξ,ε)}*−* *n−*2

4(n*−*1)*R*_{g}*u*_{(ξ,ε)}+*n(n−*2)*u*

*n+2*
*n−2*

(ξ,ε)

°°

°*L*^{n+2}^{2n} (R* ^{n}*)

*,*

*where*

*C*

*is a constant that depends only on*

*n.*

We next define a function*F** _{g}* :R

^{n}*×*(0,

*∞)→*Rby

*F*

*(ξ, ε) =*

_{g}Z

R^{n}

³

*|dv*_{(ξ,ε)}*|*^{2}* _{g}*+

*n−*2

4(n*−*1)*R*_{g}*v*_{(ξ,ε)}^{2} *−*(n*−*2)^{2}*|v*_{(ξ,ε)}*|*^{n−2}^{2n}

´

*−*2(n*−*2)

³ *Y*(S* ^{n}*)
4n(n

*−*1)

´^{n}

2*.*

If we choose *α*_{1} small enough, then we obtain the following result:

Proposition 3. *The function* *F*_{g}*is continuously differentiable.*

*Moreover, if*(¯*ξ,ε)*¯ *is a critical point of the functionF*_{g}*, then the function*
*v*_{(¯}_{ξ,¯}_{ε)}*is a non-negative weak solution of the equation*

∆_{g}*v*_{(¯}_{ξ,¯}_{ε)}*−* *n−*2

4(n*−*1)*R*_{g}*v*_{( ¯}_{ξ,¯}* _{ε)}*+

*n(n−*2)

*v*

*n+2*
*n−2*

(¯*ξ,¯**ε)*= 0.

3. An estimate for the energy of a “bubble”

Throughout this paper, we fix a real number *τ* and a multi-linear
form *W* :R^{n}*×*R^{n}*×*R^{n}*×*R^{n}*→*R. The number *τ* depends only on the
dimension *n. The exact choice of* *τ* will be postponed until Section 4.

We assume that *W** _{ijkl}* satisfies all the algebraic properties of the Weyl
tensor. Moreover, we assume that some components of

*W*are non-zero, so that

X*n*
*i,j,k,l=1*

(W* _{ijkl}*+

*W*

*)*

_{ilkj}^{2}

*>*0.

For abbreviation, we put
*H** _{ik}*(x) =

X*n*
*p,q=1*

*W*_{ipkq}*x*_{p}*x** _{q}*
and

*H** _{ik}*(x) =

*f*(|x|

^{2})

*H*

*(x),*

_{ik}where*f*(s) =*τ*+5s−s^{2}+_{20}^{1} *s*^{3}. It is easy to see that*H** _{ik}*(x) is trace-free,
P

_{n}*i=1**x*_{i}*H** _{ik}*(x) = 0, andP

_{n}*i=1**∂*_{i}*H** _{ik}*(x) = 0 for all

*x∈*R

*.*

^{n}We consider a Riemannian metric of the form *g(x) = exp(h(x)),*
where*h(x) is a trace-free symmetric two-tensor on*R* ^{n}*satisfying

*h(x) =*0 for

*|x| ≥*1,

*|h(x)|*+*|∂h(x)|*+*|∂*^{2}*h(x)| ≤α*_{1}
for all *x∈*R* ^{n}*, and

*h** _{ik}*(x) =

*µ λ*

^{6}

*f*(λ

^{−2}*|x|*

^{2})

*H*

*(x)*

_{ik}for *|x| ≤* *ρ. We assume that the parameters* *λ,* *µ, and* *ρ* are chosen
such that *µ* *≤* 1 and *λ* *≤* *ρ* *≤* 1. Note that P_{n}

*i=1**x*_{i}*h** _{ik}*(x) = 0 and
P

_{n}*i=1**∂*_{i}*h** _{ik}*(x) = 0 for

*|x| ≤ρ.*

Given any pair (ξ, ε) *∈* R^{n}*×*(0,*∞), there exists a unique function*
*v*_{(ξ,ε)} such that*v*_{(ξ,ε)}*−u*_{(ξ,ε)}*∈ E*_{(ξ,ε)} and

Z

R^{n}

³

*hdv*_{(ξ,ε)}*, dψi** _{g}*+

*n−*2

4(n*−*1)*R*_{g}*v*_{(ξ,ε)}*ψ−n(n−2)|v*_{(ξ,ε)}*|*^{n−2}^{4} *v*_{(ξ,ε)}*ψ*

´

= 0
for all test functions*ψ∈ E*_{(ξ,ε)} (see Proposition 2).

For abbreviation, let Ω =

½

(ξ, ε)*∈*R^{n}*×*R:*|ξ|<*1, 1

2 *< ε <*2

¾
*.*

The following result is proved in the Appendix A of [4]. A similar formula is derived in [2].

Proposition 4. *Consider a Riemannian metric on* R^{n}*of the form*
*g(x) = exp(h(x)), where* *h(x)* *is a trace-free symmetric two-tensor on*
R^{n}*satisfying* *|h(x)| ≤*1 *for all* *x* *∈*R^{n}*. Let* *R*_{g}*be the scalar curvature*
*of* *g. There exists a constant* *C, depending only on* *n, such that*

¯¯

¯R_{g}*−∂*_{i}*∂*_{k}*h** _{ik}*+

*∂*

*(h*

_{i}

_{il}*∂*

_{k}*h*

*)*

_{kl}*−*1

2*∂*_{i}*h*_{il}*∂*_{k}*h** _{kl}*+1

4*∂*_{l}*h*_{ik}*∂*_{l}*h*_{ik}

¯¯

¯

*≤C|h|*^{2}*|∂*^{2}*h|*+*C|h| |∂h|*^{2}*.*

Proposition 5. *Assume that* (ξ, ε)*∈λ*Ω. Then we have

°°

°°∆_{g}*u*_{(ξ,ε)}*−* *n−*2

4(n*−*1)*R*_{g}*u*_{(ξ,ε)}+*n(n−*2)*u*

*n+2*
*n−2*

(ξ,ε)

°°

°°

*L*^{n+2}^{2n} (R* ^{n}*)

*≤C λ*^{8}*µ*+*C*
µ*λ*

*ρ*

¶^{n−2}

2

*and*

°°

°°∆_{g}*u*_{(ξ,ε)}*−* *n−*2

4(n*−*1)*R*_{g}*u*_{(ξ,ε)}+*n(n−*2)*u*

*n+2*
*n−2*

(ξ,ε)

+
X*n*
*i,k=1*

*µ λ*^{6}*f(λ*^{−2}*|x|*^{2})*H** _{ik}*(x)

*∂*

_{i}*∂*

_{k}*u*

_{(ξ,ε)}

°°

°°

*L*^{n+2}^{2n} (R* ^{n}*)

*≤C λ*^{8(n+2)}^{n−2}*µ*^{2}+*C*
µ*λ*

*ρ*

¶^{n−2}

2 *.*

*Proof.* Note that P_{n}

*i=1**∂*_{i}*h** _{ik}*(x) = 0 for

*|x| ≤*

*ρ. Hence, it follows*from Proposition 4 that

*|R** _{g}*(x)| ≤

*C|h(x)|*

^{2}

*|∂*

^{2}

*h(x)|*+

*C|∂h(x)|*

^{2}

*≤C µ*

^{2}(λ+

*|x|)*

^{14}for

*|x| ≤ρ. This implies*

¯¯

¯¯∆_{g}*u*_{(ξ,ε)}*−* *n−*2

4(n*−*1)*R*_{g}*u*_{(ξ,ε)}+*n(n−*2)*u*

*n+2*
*n−2*

(ξ,ε)

¯¯

¯¯

=

¯¯

¯¯

¯¯
X*n*
*i,k=1*

*∂** _{i}*
h

(g^{ik}*−δ** _{ik}*)

*∂*

_{k}*u*

_{(ξ,ε)}i

*−* *n−*2

4(n*−*1)*R*_{g}*u*_{(ξ,ε)}

¯¯

¯¯

¯¯

*≤C λ*^{n−2}^{2} *µ*(λ+*|x|)*^{8−n}

and

¯¯

¯∆_{g}*u*_{(ξ,ε)}*−* *n−*2

4(n*−*1)*R*_{g}*u*_{(ξ,ε)}+*n(n−*2)*u*

*n+2*
*n−2*

(ξ,ε)+
X*n*
*i,k=1*

*h*_{ik}*∂*_{i}*∂*_{k}*u*_{(ξ,ε)}

¯¯

¯

=

¯¯

¯
X*n*
*i,k=1*

*∂** _{i}*£

(g^{ik}*−δ** _{ik}*+

*h*

*)*

_{ik}*∂*

_{k}*u*

_{(ξ,ε)}¤

*−* *n−*2

4(n*−*1)*R*_{g}*u*_{(ξ,ε)}

¯¯

¯

*≤C λ*^{n−2}^{2} *µ*^{2}(λ+*|x|)*^{16−n}

*≤C λ*^{n−2}^{2} *µ*^{2}(λ+*|x|)*^{8(n+2)}^{n−2}^{−n}

for*|x| ≤ρ. From this the assertion follows.*

Corollary 6. *The function* *v*_{(ξ,ε)}*−u*_{(ξ,ε)} *satisfies the estimate*
*kv*_{(ξ,ε)}*−u*_{(ξ,ε)}*k*

*L*^{n−2}^{2n} (R* ^{n}*)

*≤C λ*

^{8}

*µ*+

*C*

³*λ*
*ρ*

´^{n−2}

2

*whenever* (ξ, ε)*∈λ*Ω.

*Proof.* It follows from Proposition 2 that
*kv*_{(ξ,ε)}*−u*_{(ξ,ε)}*k*

*L*^{n−2}^{2n} (R* ^{n}*)

*≤C*

°°

°∆_{g}*u*_{(ξ,ε)}*−* *n−*2

4(n*−*1)*R*_{g}*u*_{(ξ,ε)}+*n(n−*2)*u*

*n+2*
*n−2*

(ξ,ε)

°°

°*L*^{n+2}^{2n} (R* ^{n}*)

*,*where

*C*is a constant that depends only on

*n. Hence, the assertion*follows from Proposition 5.

We next establish a more precise estimate for the function *v*_{(ξ,ε)}*−*
*u*_{(ξ,ε)}. Applying Proposition 1 with*h*= 0, we conclude that there exists
a unique function *w*_{(ξ,ε)}*∈ E*_{(ξ,ε)} such that

Z

R^{n}

³

*hdw*_{(ξ,ε)}*, dψi −n(n*+ 2)*u*

4
*n−2*

(ξ,ε)*w*_{(ξ,ε)}*ψ*

´

=*−*
Z

R^{n}

X*n*
*i,k=1*

*µ λ*^{6}*f*(λ^{−2}*|x|*^{2})*H** _{ik}*(x)

*∂*

_{i}*∂*

_{k}*u*

_{(ξ,ε)}

*ψ*for all test functions

*ψ∈ E*

_{(ξ,ε)}.

Proposition 7. *The function* *w*_{(ξ,ε)} *is smooth. Moreover, if* (ξ, ε)*∈*
*λ*Ω, then the function *w*_{(ξ,ε)} *satisfies the estimates*

*|w*_{(ξ,ε)}(x)| ≤*C λ*^{n−2}^{2} *µ*(λ+*|x|)*^{10−n}

*|∂w*_{(ξ,ε)}(x)| ≤*C λ*^{n−2}^{2} *µ*(λ+*|x|)*^{9−n}

*|∂*^{2}*w*_{(ξ,ε)}(x)| ≤*C λ*^{n−2}^{2} *µ*(λ+*|x|)*^{8−n}
*for all* *x∈*R^{n}*.*

*Proof.* There exist real numbers*b** _{k}*(ξ, ε) such that
Z

R^{n}

³

*hdw*_{(ξ,ε)}*, dψi −n(n*+ 2)*u*

4
*n−2*

(ξ,ε)*w*_{(ξ,ε)}*ψ*

´

=*−*
Z

R^{n}

X*n*
*i,k=1*

*µ λ*^{6}*f*(λ^{−2}*|x|*^{2})*H** _{ik}*(x)

*∂*

_{i}*∂*

_{k}*u*

_{(ξ,ε)}

*ψ*

+
X*n*
*k=0*

*b** _{k}*(ξ, ε)
Z

R^{n}

*ϕ*_{(ξ,ε,k)}*ψ*

for all test functions *ψ* *∈ E. Hence, standard elliptic regularity theory*
implies that*w*_{(ξ,ε)} is smooth.

It remains to prove quantitative estimates for*w*_{(ξ,ε)}. To that end, we
consider a pair (ξ, ε)*∈λ*Ω. One readily verifies that

°°

°
X*n*
*i,k=1*

*µ λ*^{6}*f*(λ^{−2}*|x|*^{2})*H** _{ik}*(x)

*∂*

_{i}*∂*

_{k}*u*

_{(ξ,ε)}

°°

°*L*^{n+2}^{2n} (R* ^{n}*)

*≤C λ*

^{8}

*µ.*

As a consequence, the function*w*_{(ξ,ε)}satisfies*kw*_{(ξ,ε)}*k*

*L*^{n−2}^{2n} (R* ^{n}*)

*≤C λ*

^{8}

*µ.*

Moreover, we haveP_{n}

*k=0**|b** _{k}*(ξ, ε)| ≤

*C λ*

^{8}

*µ. This implies*

¯¯∆w_{(ξ,ε)}+*n(n*+ 2)*u*

4
*n−2*

(ξ,ε)*w*_{(ξ,ε)}¯

¯

=

¯¯

¯¯
X*n*
*i,k=1*

*µ λ*^{6}*f*(λ^{−2}*|x|*^{2})*H** _{ik}*(x)

*∂*

_{i}*∂*

_{k}*u*

_{(ξ,ε)}

*−*X

*n*

*k=0*

*b** _{k}*(ξ, ε)
Z

R^{n}

*ϕ*_{(ξ,ε,k)}

¯¯

¯¯

*≤C λ*^{n−2}^{2} *µ*(λ+*|x|)*^{8−n}
for all *x∈*R* ^{n}*. We claim that

sup

*x∈R*^{n}

(λ+*|x|)*^{n−2}^{2} *|w*_{(ξ,ε)}(x)| ≤*C λ*^{8}*µ.*

To show this, we fix a point *x*_{0} *∈*R* ^{n}*. Let

*r*=

^{1}

_{2}(λ+

*|x*

_{0}

*|). Then*

*u*

_{(ξ,ε)}(x)

^{n−2}^{4}

*≤C r*

^{−2}and ¯¯∆w_{(ξ,ε)}+*n(n*+ 2)*u*

4
*n−2*

(ξ,ε)*w*_{(ξ,ε)}¯¯*≤C λ*^{n−2}^{2} *µ r*^{8−n}

for all *x* *∈* *B** _{r}*(x

_{0}). Hence, it follows from standard interior estimates that

*r*^{n−2}^{2} *|w*_{(ξ,ε)}(x_{0})| ≤*Ckw*_{(ξ,ε)}*k*

*L*^{n−2}^{2n} (B*r*(x0))

+*C r*^{n+2}^{2} °

°∆w_{(ξ,ε)}+*n(n*+ 2)*u*

4
*n−2*

(ξ,ε)*w*_{(ξ,ε)}°

°*L** ^{∞}*(B

*r*(x0))

*≤C λ*^{8}*µ*+*C λ*^{n−2}^{2} *µ r*^{−}^{n−18}^{2}

*≤C λ*^{8}*µ.*

Therefore, we have sup

*x∈R** ^{n}*(λ+

*|x|)*

^{n−2}^{2}

*|w*

_{(ξ,ε)}(x)| ≤

*C λ*

^{8}

*µ,*

as claimed. Since sup_{x∈R}^{n}*|x|*^{n−2}^{2} *|w*_{(ξ,ε)}(x)| *<* *∞, we can express the*
function*w*_{(ξ,ε)} in the form

(3) *w*_{(ξ,ε)}(x) =*−* 1

(n*−*2)*|S*^{n−1}*|*
Z

R^{n}

*|x−y|*^{2−n}∆w_{(ξ,ε)}(y)*dy*
for all *x∈*R* ^{n}*.

We are now able to use a bootstrap argument to prove the desired
estimate for *w*_{(ξ,ε)}. It follows from (3) that

sup

*x∈R*^{n}

(λ+*|x|)*^{β}*|w*_{(ξ,ε)}(x)| ≤*C* sup

*x∈R*^{n}

(λ+*|x|)*^{β+2}*|∆w*_{(ξ,ε)}(x)|

for all 0*< β < n−*2. Since

*|∆w*_{(ξ,ε)}(x)| ≤*n(n*+ 2)*u*_{(ξ,ε)}(x)^{n−2}^{4} *|w*_{(ξ,ε)}(x)|

+*C λ*^{n−2}^{2} *µ*(λ+*|x|)*^{8−n}
for all *x∈*R* ^{n}*, we conclude that

sup

*x∈R*^{n}

(λ+*|x|)*^{β}*|w*_{(ξ,ε)}(x)| ≤*C λ*^{2} sup

*x∈R*^{n}

(λ+*|x|)*^{β−2}*|w*_{(ξ,ε)}(x)|

+*C λ*^{β−}^{n−18}^{2} *µ*

for all 0*< β≤n−*10. Iterating this inequality, we obtain
sup

*x∈R*^{n}

(λ+*|x|)*^{n−10}*|w*_{(ξ,ε)}(x)| ≤*C λ*^{n−2}^{2} *µ.*

The estimates for the first and second derivatives of *w*_{(ξ,ε)} follow now
from standard interior estimates.

Corollary 8. *The functionv*_{(ξ,ε)}*−u*_{(ξ,ε)}*−w*_{(ξ,ε)} *satisfies the estimate*
*kv*_{(ξ,ε)}*−u*_{(ξ,ε)}*−w*_{(ξ,ε)}*k*

*L*^{n−2}^{2n} (R* ^{n}*)

*≤C λ*

^{8(n+2)}

^{n−2}*µ*

^{n+2}*+*

^{n−2}*C*

³*λ*
*ρ*

´^{n−2}

2

*whenever* (ξ, ε)*∈λ*Ω.

*Proof.* Consider the functions
*B*_{1}=

X*n*
*i,k=1*

*∂** _{i}*£

(g^{ik}*−δ** _{ik}*)

*∂*

_{k}*w*

_{(ξ,ε)}¤

*−* *n−*2

4(n*−*1)*R*_{g}*w*_{(ξ,ε)}
and

*B*_{2}=
X*n*
*i,k=1*

*µ λ*^{6}*f*(λ^{−2}*|x|*^{2})*H** _{ik}*(x)

*∂*

_{i}*∂*

_{k}*u*

_{(ξ,ε)}

*.*

By definition of*w*_{(ξ,ε)}, we have
Z

R^{n}

³

*hdw*_{(ξ,ε)}*, dψi** _{g}*+

*n−*2

4(n*−*1)*R*_{g}*w*_{(ξ,ε)}*ψ−n(n*+ 2)*u*

4
*n−2*

(ξ,ε)*w*_{(ξ,ε)}*ψ*

´

=*−*
Z

R^{n}

(B_{1}+*B*_{2})*ψ*

for all functions *ψ∈ E*_{(ξ,ε)}. Since *w*_{(ξ,ε)}*∈ E*_{(ξ,ε)}, it follows that
*w*_{(ξ,ε)} =*−G*_{(ξ,ε)}(B_{1}+*B*_{2}).

Moreover, we have

*v*_{(ξ,ε)}*−u*_{(ξ,ε)}=*G*_{(ξ,ε)}¡

*B*_{3}+*n(n−*2)*B*_{4}¢
*,*
where

*B*_{3}= ∆_{g}*u*_{(ξ,ε)}*−* *n−*2

4(n*−*1)*R*_{g}*u*_{(ξ,ε)}+*n(n−*2)*u*

*n+2*
*n−2*

(ξ,ε)

and

*B*_{4}=*|v*_{(ξ,ε)}*|*^{n−2}^{4} *v*_{(ξ,ε)}*−u*

*n+2*
*n−2*

(ξ,ε)*−n*+ 2
*n−*2*u*

4
*n−2*

(ξ,ε)(v_{(ξ,ε)}*−u*_{(ξ,ε)}).

Thus, we conclude that

*v*_{(ξ,ε)}*−u*_{(ξ,ε)}*−w*_{(ξ,ε)}=*G*_{(ξ,ε)}¡

*B*_{1}+*B*_{2}+*B*_{3}+*n(n−*2)*B*_{4}¢
*,*
where*G*_{(ξ,ε)}denotes the solution operator constructed in Proposition 1.

As a consequence, we obtain
*kv*_{(ξ,ε)}*−u*_{(ξ,ε)}*−w*_{(ξ,ε)}*k*

*L*^{n−2}^{2n} (R* ^{n}*)

*≤C*°

°*B*_{1}+B_{2}+B_{3}+n(n−2)*B*_{4}°

°

*L*^{n+2}^{2n} (R* ^{n}*)

*.*It follows from Proposition 7 that

*|B*_{1}(x)| ≤*C λ*^{n−2}^{2} *µ*^{2}(λ+*|x|)*^{16−n}*≤C λ*^{n−2}^{2} *µ*^{2}(λ+*|x|)*^{8(n+2)}^{n−2}* ^{−n}*
for

*|x| ≤ρ*and

*|B*_{1}(x)| ≤*C λ*^{n−2}^{2} *µ|x|*^{8−n}
for*|x| ≥ρ. This implies*

*kB*_{1}*k*

*L*^{n+2}^{2n} (R* ^{n}*)

*≤C λ*

^{8(n+2)}

^{n−2}*µ*

^{2}+

*C ρ*

^{8}

*µ*

³*λ*
*ρ*

´^{n−2}

2 *.*

Moreover, observe that
*kB*_{2}+*B*_{3}*k*

*L*^{n+2}^{2n} (R* ^{n}*)

*≤C λ*

^{8(n+2)}

^{n−2}*µ*

^{2}+

*C*

³*λ*
*ρ*

´^{n−2}

2

by Proposition 5. Finally, Corollary 6 implies that
*kB*_{4}*k*

*L*^{n+2}^{2n} (R* ^{n}*)

*≤Ckv*

_{(ξ,ε)}

*−u*

_{(ξ,ε)}

*k*

*n+2*
*n−2*

*L*^{n−2}^{2n} (R* ^{n}*)

*≤C λ*^{8(n+2)}^{n−2}*µ*^{n+2}* ^{n−2}* +

*C*

³*λ*
*ρ*

´^{n+2}

2 *.*

Putting these facts together, we obtain
*kv*_{(ξ,ε)}*−u*_{(ξ,ε)}*−w*_{(ξ,ε)}*k*

*L*^{n−2}^{2n} (R* ^{n}*)

*≤C λ*

^{8(n+2)}

^{n−2}*µ*

^{n+2}*+*

^{n−2}*C*

³*λ*
*ρ*

´^{n−2}

2 *.*
This completes the proof.

Proposition 9. *We have*

¯¯

¯¯ Z

R^{n}

³

*|dv*_{(ξ,ε)}*|*^{2}_{g}*− |du*_{(ξ,ε)}*|*^{2}* _{g}*+

*n−*2

4(n*−*1)*R** _{g}*(v

_{(ξ,ε)}

^{2}

*−u*

^{2}

_{(ξ,ε)})

´

+ Z

R^{n}

*n(n−*2) (|v_{(ξ,ε)}*|*^{n−2}^{4} *−u*

4
*n−2*

(ξ,ε))*u*_{(ξ,ε)}*v*_{(ξ,ε)}

*−*
Z

R^{n}

*n(n−*2) (|v_{(ξ,ε)}*|*^{n−2}^{2n} *−u*

2n
*n−2*

(ξ,ε))

*−*
Z

R^{n}

X*n*
*i,k=1*

*µ λ*^{6}*f*(λ^{−2}*|x|*^{2})*H** _{ik}*(x)

*∂*

_{i}*∂*

_{k}*u*

_{(ξ,ε)}

*w*

_{(ξ,ε)}

¯¯

¯¯

*≤C λ*^{n−2}^{16n} *µ*^{n−2}^{2n} +*C λ*^{8}*µ*

³*λ*
*ρ*

´^{n−2}

2 +*C*

³*λ*
*ρ*

´_{n−2}

*whenever* (ξ, ε)*∈λ*Ω.

*Proof.* By definition of*v*_{(ξ,ε)}, we have
Z

R^{n}

³

*|dv*_{(ξ,ε)}*|*^{2}_{g}*− hdu*_{(ξ,ε)}*, dv*_{(ξ,ε)}*i** _{g}*+

*n−*2

4(n*−*1)*R*_{g}*v*_{(ξ,ε)}(v_{(ξ,ε)}*−u*_{(ξ,ε)})

´

*−*
Z

R^{n}

*n(n−*2)*|v*_{(ξ,ε)}*|*^{n−2}^{4} *v*_{(ξ,ε)}(v_{(ξ,ε)}*−u*_{(ξ,ε)}) = 0.

Using Proposition 5 and Corollary 6, we obtain

¯¯

¯¯ Z

R^{n}

³

*hdu*_{(ξ,ε)}*, dv*_{(ξ,ε)}*i*_{g}*− |du*_{(ξ,ε)}*|*^{2}* _{g}*+

*n−*2

4(n*−*1)*R*_{g}*u*_{(ξ,ε)}(v_{(ξ,ε)}*−u*_{(ξ,ε)})

´

*−*
Z

R^{n}

*n(n−*2)*u*

*n+2*
*n−2*

(ξ,ε)(v_{(ξ,ε)}*−u*_{(ξ,ε)})

*−*
Z

R^{n}

X*n*
*i,k=1*

*µ λ*^{6}*f*(λ^{−2}*|x|*^{2})*H** _{ik}*(x)

*∂*

_{i}*∂*

_{k}*u*

_{(ξ,ε)}(v

_{(ξ,ε)}

*−u*

_{(ξ,ε)})

¯¯

¯¯

*≤*

°°

°∆_{g}*u*_{(ξ,ε)}*−* *n−*2

4(n*−*1)*R*_{g}*u*_{(ξ,ε)}+*n(n−*2)*u*

*n+2*
*n−2*

(ξ,ε)

+
X*n*
*i,k=1*

*µ λ*^{6}*f*(λ^{−2}*|x|*^{2})*H** _{ik}*(x)

*∂*

_{i}*∂*

_{k}*u*

_{(ξ,ε)}

°°

°*L*^{n+2}^{2n} (R* ^{n}*)

*· kv*_{(ξ,ε)}*−u*_{(ξ,ε)}*k*

*L*^{n−2}^{2n} (R* ^{n}*)

*≤C λ*^{n−2}^{16n} *µ*^{3}+*C λ*^{8}*µ*

³*λ*
*ρ*

´^{n−2}

2 +*C*

³*λ*
*ρ*

´_{n−2}*.*

Moreover, we have

¯¯

¯¯ Z

R^{n}

X*n*
*i,k=1*

*µ λ*^{6}*f*(λ^{−2}*|x|*^{2})*H** _{ik}*(x)

*∂*

_{i}*∂*

_{k}*u*

_{(ξ,ε)}(v

_{(ξ,ε)}

*−u*

_{(ξ,ε)}

*−w*

_{(ξ,ε)})

¯¯

¯¯

*≤C λ*^{8}*µkv*_{(ξ,ε)}*−u*_{(ξ,ε)}*−w*_{(ξ,ε)}*k*

*L*^{n−2}^{2n} (R* ^{n}*)

*≤C λ*^{n−2}^{16n} *µ*^{n−2}^{2n} +*C λ*^{8}*µ*

³*λ*
*ρ*

´^{n−2}

2

by Corollary 8. Putting these facts together, the assertion follows.

Proposition 10. *We have*

¯¯

¯¯ Z

R^{n}

(|v_{(ξ,ε)}*|*^{n−2}^{4} *−u*

4
*n−2*

(ξ,ε))*u*_{(ξ,ε)}*v*_{(ξ,ε)}*−* 2
*n*

Z

R^{n}

(|v_{(ξ,ε)}*|*^{n−2}^{2n} *−u*

2n
*n−2*

(ξ,ε))

¯¯

¯¯

*≤C λ*^{n−2}^{16n} *µ*^{n−2}^{2n} +*C*

³*λ*
*ρ*

´_{n}

*whenever* (ξ, ε)*∈λ*Ω.

*Proof.* We have the pointwise estimate

¯¯

¯(|v_{(ξ,ε)}*|*^{n−2}^{4} *−u*

4
*n−2*

(ξ,ε))*u*_{(ξ,ε)}*v*_{(ξ,ε)}*−* 2

*n*(|v_{(ξ,ε)}*|*^{n−2}^{2n} *−u*

2n
*n−2*

(ξ,ε))

¯¯

¯

*≤C|v*_{(ξ,ε)}*−u*_{(ξ,ε)}*|*^{n−2}^{2n} *,*

where *C* is a constant that depends only on*n. This implies*

¯¯

¯¯ Z

R^{n}

(|v_{(ξ,ε)}*|*^{n−2}^{4} *−u*

4
*n−2*

(ξ,ε))*u*_{(ξ,ε)}*v*_{(ξ,ε)}*−* 2
*n*

Z

R^{n}

(|v_{(ξ,ε)}*|*^{n−2}^{2n} *−u*

2n
*n−2*

(ξ,ε))

¯¯

¯¯

*≤Ckv*_{(ξ,ε)}*−u*_{(ξ,ε)}*k*

2n
*n−2*

*L*^{n−2}^{2n} (R* ^{n}*)

*≤C λ*^{n−2}^{16n} *µ*^{n−2}^{2n} +*C*

³*λ*
*ρ*

´_{n}

by Corollary 6.

Proposition 11. *We have*

¯¯

¯¯ Z

R^{n}

³

*|du*_{(ξ,ε)}*|*^{2}* _{g}*+

*n−*2

4(n*−*1)*R*_{g}*u*^{2}_{(ξ,ε)}*−n(n−*2)*u*

2n
*n−2*

(ξ,ε)

´

*−*
Z

*B**ρ*(0)

1 2

X*n*
*i,k,l=1*

*h*_{il}*h*_{kl}*∂*_{i}*u*_{(ξ,ε)}*∂*_{k}*u*_{(ξ,ε)}

+ Z

*B**ρ*(0)

*n−*2
16(n*−*1)

X*n*
*i,k,l=1*

(∂_{l}*h** _{ik}*)

^{2}

*u*

^{2}

_{(ξ,ε)}

¯¯

¯¯

*≤C λ*^{n−2}^{16n} *µ*^{3}+*C*

³*λ*
*ρ*

´_{n−2}*whenever* (ξ, ε)*∈λ*Ω.

*Proof.* Note that

¯¯

¯g* ^{ik}*(x)

*−δ*

*+*

_{ik}*h*

*(x)*

_{ik}*−*1 2

X*n*
*l=1*

*h** _{il}*(x)

*h*

*(x)*

_{kl}¯¯

¯

*≤C|h(x)|*^{3} *≤C µ*^{3}(λ+*|x|)*^{24}*≤C µ*^{3}(λ+*|x|)*^{n−2}^{16n}
for*|x| ≤ρ. This implies*

¯¯

¯¯ Z

R^{n}

¡*|du*_{(ξ,ε)}*|*^{2}_{g}*− |du*_{(ξ,ε)}*|*^{2}¢
+

Z

R^{n}

X*n*
*i,k=1*

*h*_{ik}*∂*_{i}*u*_{(ξ,ε)}*∂*_{k}*u*_{(ξ,ε)}

*−*
Z

*B**ρ*(0)

1 2

X*n*
*i,k,l=1*

*h*_{il}*h*_{kl}*∂*_{i}*u*_{(ξ,ε)}*∂*_{k}*u*_{(ξ,ε)}

¯¯

¯¯

*≤C λ*^{n−2}*µ*^{3}
Z

*B**ρ*(0)

(λ+*|x|)*^{n−2}^{16n}^{+2−2n}+*C λ** ^{n−2}*
Z

R^{n}*\B**ρ*(0)

(λ+*|x|)*^{2−2n}

*≤C λ*^{n−2}^{16n} *µ*^{3}+*C*

³*λ*
*ρ*

´_{n−2}*.*

By Proposition 4, the scalar curvature of *g* satisfies the estimate

¯¯

¯R* _{g}*(x) + 1
4

X*n*
*i,k,l=1*

(∂_{l}*h** _{ik}*(x))

^{2}

¯¯

¯

*≤C|h(x)|*^{2}*|∂*^{2}*h(x)|*+*C|h(x)| |∂h(x)|*^{2}

*≤C µ*^{3}(λ+*|x|)*^{22}*≤C µ*^{3}(λ+*|x|)*^{n−2}^{16n}* ^{−2}*
for

*|x| ≤ρ. This implies*

¯¯

¯¯ Z

R^{n}

*R*_{g}*u*^{2}_{(ξ,ε)}+
Z

*B**ρ*(0)

1 4

X*n*
*i,k,l=1*

(∂_{l}*h** _{ik}*)

^{2}

*u*

^{2}

_{(ξ,ε)}

¯¯

¯¯

*≤C λ*^{n−2}*µ*^{3}
Z

*B**ρ*(0)

(λ+*|x|)*^{n−2}^{16n}^{+2−2n}+*C λ** ^{n−2}*
Z

R^{n}*\B**ρ*(0)

(λ+*|x|)*^{4−2n}

*≤C λ*^{n−2}^{16n} *µ*^{3}+*C ρ*^{2}

³*λ*
*ρ*

´_{n−2}*.*
At this point, we use the formula

*∂*_{i}*u*_{(ξ,ε)}*∂*_{k}*u*_{(ξ,ε)}*−* *n−*2

4(n*−*1)*∂*_{i}*∂** _{k}*(u

^{2}

_{(ξ,ε)})

= 1
*n*

³

*|du*_{(ξ,ε)}*|*^{2}*−* *n−*2

4(n*−*1)∆(u^{2}_{(ξ,ε)})

´
*δ*_{ik}*.*
Since *h** _{ik}* is trace-free, we obtain

X*n*
*i,k=1*

*h*_{ik}*∂*_{i}*u*_{(ξ,ε)}*∂*_{k}*u*_{(ξ,ε)}= *n−*2
4(n*−*1)

X*n*
*i,k=1*

*h*_{ik}*∂*_{i}*∂** _{k}*(u

^{2}

_{(ξ,ε)}),

hence Z

R^{n}

X*n*
*i,k=1*

*h*_{ik}*∂*_{i}*u*_{(ξ,ε)}*∂*_{k}*u*_{(ξ,ε)}=
Z

R^{n}

*n−*2
4(n*−*1)

X*n*
*i,k=1*

*∂*_{i}*∂*_{k}*h*_{ik}*u*^{2}_{(ξ,ε)}*.*
Since P_{n}

*i=1**∂*_{i}*h** _{ik}*(x) = 0 for

*|x| ≤ρ, it follows that*

¯¯

¯¯ Z

R^{n}

X*n*
*i,k=1*

*h*_{ik}*∂*_{i}*u*_{(ξ,ε)}*∂*_{k}*u*_{(ξ,ε)}

¯¯

¯¯*≤C*
Z

R^{n}*\B**ρ*(0)

*u*^{2}_{(ξ,ε)}*≤C ρ*^{2}

³*λ*
*ρ*

´_{n−2}*.*
Putting these facts together, the assertion follows.

Corollary 12. *The functionF** _{g}*(ξ, ε)

*satisfies the estimate*

¯¯

¯¯*F** _{g}*(ξ, ε)

*−*Z

*B**ρ*(0)

1 2

X*n*
*i,k,l=1*

*h*_{il}*h*_{kl}*∂*_{i}*u*_{(ξ,ε)}*∂*_{k}*u*_{(ξ,ε)}

+ Z

*B**ρ*(0)

*n−*2
16(n*−*1)

X*n*
*i,k,l=1*

(∂_{l}*h** _{ik}*)

^{2}

*u*

^{2}

_{(ξ,ε)}

*−*
Z

R^{n}

X*n*
*i,k=1*

*µ λ*^{6}*f(λ*^{−2}*|x|*^{2})*H** _{ik}*(x)

*∂*

_{i}*∂*

_{k}*u*

_{(ξ,ε)}

*w*

_{(ξ,ε)}

¯¯

¯¯

*≤C λ*^{n−2}^{16n} *µ*^{n−2}^{2n} +*C λ*^{8}*µ*

³*λ*
*ρ*

´^{n−2}

2 +*C*

³*λ*
*ρ*

´_{n−2}

*whenever* (ξ, ε)*∈λ*Ω.

4. Finding a critical point of an auxiliary function
We define a function*F* :R^{n}*×*(0,*∞)→*R as follows: given any pair
(ξ, ε)*∈*R^{n}*×*(0,*∞), we define*

*F(ξ, ε) =*
Z

R^{n}

1 2

X*n*
*i,k,l=1*

*H** _{il}*(x)

*H*

*(x)*

_{kl}*∂*

_{i}*u*

_{(ξ,ε)}(x)

*∂*

_{k}*u*

_{(ξ,ε)}(x)

*−*
Z

R^{n}

*n−*2
16(n*−*1)

X*n*
*i,k,l=1*

(∂_{l}*H** _{ik}*(x))

^{2}

*u*

_{(ξ,ε)}(x)

^{2}

+ Z

R^{n}

X*n*
*i,k=1*

*H** _{ik}*(x)

*∂*

_{i}*∂*

_{k}*u*

_{(ξ,ε)}(x)

*z*

_{(ξ,ε)}(x), where

*z*

_{(ξ,ε)}

*∈ E*

_{(ξ,ε)}satisfies the relation

Z

R^{n}

³

*hdz*_{(ξ,ε)}*, dψi −n(n*+ 2)*u*_{(ξ,ε)}(x)^{n−2}^{4} *z*_{(ξ,ε)}*ψ*

´

=*−*
Z

R^{n}

X*n*
*i,k=1*

*H*_{ik}*∂*_{i}*∂*_{k}*u*_{(ξ,ε)}*ψ*