Ann. I. H. Poincaré – AN 31 (2014) 591–614
www.elsevier.com/locate/anihpc
Constant Q-curvature metrics near the hyperbolic metric
Gang Li
a,b,∗aDepartment of Mathematics, University of Notre Dame, 295 Hurley Hall, Notre Dame, IN 46556, USA bDepartment of Mathematics, Nanjing University, Nanjing 210093, China
Received 22 May 2012; received in revised form 25 April 2013; accepted 30 April 2013 Available online 11 June 2013
Abstract
Let(M, g)be a Poincaré–Einstein manifold with a smooth defining function. In this note, we prove that there are infinitely many asymptotically hyperbolic metrics with constantQ-curvature in the conformal class of an asymptotically hyperbolic metric close enough tog. These metrics are parametrized by the elements in the kernel of the linearized operator of the prescribed constant Q-curvature equation. A similar analysis is applied to a class of fourth order equations arising in spectral theory.
©2013 Elsevier Masson SAS. All rights reserved.
1. Introduction
In this note we will discuss the prescribed constantQ-curvature problem for asymptotically hyperbolic manifolds.
We obtain the existence of a family of constantQ-curvature metrics in a small neighborhood of any Poincaré–Einstein metric, parametrized by elements in the null space of the linearized operatorLin(1.3). Much of the analysis follows from Mazzeo’s microlocal analysis method for elliptic edge operators. Results in this setting have been proved for the scalar curvature equation, see[1].
Forn4, a natural conformal invariant and the corresponding conformal covariant operator are theQ-curvature and the fourth order Paneitz operator. Let Ricg andRg be the Ricci curvature and the scalar curvature of(M, g).
TheQ-curvatureand thePaneitz operatorare defined as follows, Qg=
⎧⎨
⎩
−121(gRg−Rg2+3|Ricg|2), n=4,
−(n−22)2|Ricg|2+n8(n3−4n−1)2+2(n16n−−2)162 R2g−2(n1−1)gRg, n5, Pg(ϕ)=
2gϕ−div(23Rgg−2 Ricg) dϕ, n=4, 2gϕ−divg(anRgg−bnRicg)∇gϕ+n−24Qgϕ, n5,
wherean=2(n(n−−1)(n2)2+−42),bn=n−42, divgX= ∇iXifor any smooth vector fieldX, andϕis any smooth function onM.
* Correspondence to: Department of Mathematics, University of Notre Dame, 295 Hurley Hall, Notre Dame, IN 46556, USA & Department of Mathematics, Nanjing University, 210093, Nanjing, China.
E-mail address:gli3@nd.edu.
0294-1449/$ – see front matter ©2013 Elsevier Masson SAS. All rights reserved.
http://dx.doi.org/10.1016/j.anihpc.2013.04.008
Letg˜=ρg, withρa positive function onM, so that ρ=
e2u, n=4, un−44 , n5.
TheQ-curvature has the following transformation, Pgu+2Qg=2Qg˜e4u, n=4,
Pgu=n−4
2 Qg˜unn−4+4, n >4.
Note that Paneitz operator satisfies the following conformal covariance property forϕ∈C∞(M), Pg˜ϕ=e−4uPgϕ, n=4,
Pg˜(ϕ)=u−nn+−44Pg(uϕ), n >4.
We want to find a functionu so that the metricg˜ satisfiesQg˜= ˜Qfor a given function Q. For the prescribed˜ Q-curvature problem on closed manifold M of dimension four there are many results, see[3,7,12,13]. In [24]a boundary value problem for this problem is solved. A flow approach is performed in[2], see also[4]. Forn5, see [6,23,27].
There are some interesting results for complete non-compact manifolds. For Euclidean spaceRn,n4, see[17]
and[26]. In [11], using shooting method, the authors proved that there are infinitely many complete metrics with constantQ-curvature in the conformal class of the Poincaré disk with dimensionn5, which are radially symmetric ODE solutions to the initial value problem parametrized by distinct given initial data at the origin. It is not difficult to prove that similar results hold for n=4. Mazzeo pointed out that there should be a more general result of this type. In this paper, we solve a perturbation problem in the setting of asymptotically hyperbolic metrics close to a Poincaré–Einstein metric. To give a precise statement we first need some definitions.
Definition 1.1.LetM be a smooth manifold of dimensionaln, with smooth boundary∂Mof dimensionn−1. Let g be a complete metric onM=Int(M). We say thatgis conformally compact if there exists a smooth functionx onM, with the property thatx >0 inM, andx=0 on ∂M, so that the metrich=x2g extends continuously to a Riemannian metric onM. Herex is called a defining function ofg. Moreover, ifh∈Ck orCk,α, for some positive integerk, we say thatgis conformally compact of orderCkorCk,α.
Note that a direct calculation shows that the sectional curvatures of a conformally compact metric g approach
−|dx|gnear∂M. We call(M, g)asymptotically hyperbolic, if|dx|h|∂M=1; and heregis called anasymptotically hyperbolic metric, for which the sectional curvatures approach−1 near∂M.
Definition 1.2. Let(M, g)be an asymptotically hyperbolic manifold. If g is also Einstein, we callg a Poincaré–
Einstein metric, and(M, g)a Poincaré–Einstein manifold.
Let (Mn, g)be an asymptotically hyperbolic manifold of dimension n, withx as its smooth defining function.
Actually, we can choose x so that|dx|h=1 in a neighborhood of∂M, see[8], and to simplify notation we always choose a defining function in this sense except in Section4. We will mainly focus on the asymptotic behavior of the metric near∂M, which is a local matter. Lety be local coordinates on∂M. In a neighborhood of∂MinM, we introduce the local coordinates in the following way:(x, y)∈ [0, ε)×∂Mrepresents the point moving from the point on∂Mwith local coordinatey, along the geodesic which is the integral curve of∇hx for a lengthxin the metrich.
In local coordinates(x, y), h=x2g=dx2+
n−1
i,j=1
hijdyidyj.
For convenience, letg˜=ρg, withρa positive function onM, so that ρ=
e2u, n=4, (1+u)n−44, n5.
For a given functionQ, let the operator˜ Ebe defined by E(u)=
Pgu+2Qg−2Qe˜ 4u, forn=4, Pg(1+u)−n−24Q(1˜ +u)n+4n−4, forn5.
(1.1) To solve the prescribedQ-curvature problem amounts to finding a solution to
E(u)=0. (1.2)
ForQ˜=Qg, we define the linear operatorL=LgofEas follows, L(u)=
Pgu−8Qgu, n=4,
Pgu−n+24Qgu, n5. (1.3)
Let(x, y)be the local coordinates ofMnear the boundary described above. LetVebe the collection of the smooth vector fields onM, which restricted to a neighborhood of∂Mare generated by{x∂x, x∂y1, . . . , x∂yn−1}with smooth coefficients onM.
Next we introduce the weighted spaces that we will be using. First, the weighted Sobolev spaces, xδHem
M, Ω12
= u=xδv: V1. . . Vjv∈L2 M, Ω12
, ∀j m, Vi∈Ve
,
wherem∈N,δ∈R, andΩ12 =√
dx dyis the half-density. We also introduce the weighted Hölder space, xδΛm,α=xδΛm,α
M, Ω12
= u=xδv
dx dy: V1. . . Vjv∈Λ0,α, ∀jm, Vi∈Ve
,
withm∈N,δ∈R, and 0< α <1, whereΛ0,α(M)is the space of half-densitiesu=v√
dx dy such that vΛ0,α(M)=sup|v| +sup(x+ ˜x)α|v(x, y)−v(x,˜ y)˜ |
|x− ˜x|α+ |y− ˜y|α <∞. We will use the norm
uxδΛk,α(M)= k m=0
|γ|=m
∂eγv
0,α, with∂e∈Veandu=xδv.
In this paper, we always assumen4 to be the dimension ofM. With these definitions, we can now state our main result:
Theorem 1.3.Let(Mn, g),n4, be a Poincaré–Einstein manifold with defining functionx, and assume the metric h=x2gis smooth up to the boundary. LetL:xνΛ4,α(M)→xνΛ0,α(M), where0< ν <n−21 and0< α <1be the linearized operator defined in(1.3). Then,
(i) The kernel of√ L⊂xνΛ4,α(M) is infinite dimensional, and L is surjective. Furthermore, givenQ˜ ∈Λ0,α(M, dx dy )with ˜Q−QgxνΛ0,αsufficiently small, for eachvin the kernel ofLthere exists a unique solutionuto the problem(1.2)withΠ1(u)=v, whereΠ1:xνΛ4,α(M)→ker(L)is the projection map(seeTheorem1.5).
(ii) Letube a solution of(1.2)withQ˜ =Qg. Thenuhas an expansion near the boundary u(x, y)∼
u00(y)xn−12 +iβ+u10(y)xn−12 −iβ +o
xn−12
, (1.4)
withβ=n2+22n−9andi=√
−1, whereu00andu10are generally distributions of negative order.
Moreover, supposev=Π1uhas an expansion of the form
v(x, y)∼
+∞
j=0
v0j(y)xn−12 +iβ+j+v1j(y)xn−12 −iβ+j+v2j(y)xn+j
, (1.5)
in the sense that v(x, y)−
k j=0
v0j(y)xn−21+iβ+j+v1j(y)xn−21−iβ+j
=o xn−21+k
,
for eachk0, whereβ=n2+22n−9 and the coefficient functions are smooth. Thenuhas an expansion of the same form with different smooth coefficients.
For kernel elements having an expansion with smooth coefficients as in(1.5), one can prescribe the leading terms;
seeRemark 2.2.
Remark 1.1.Our proof uses in a crucial way that the Laplacian operatorhas no embedded eigenvalues in its essen- tial spectrum. In[21], Mazzeo showed that this holds for any asymptotically hyperbolic manifold(M, g)with smooth defining functionxsuch that the compactified metrich=x2gis smooth up to the boundary. The smoothness assump- tion comes up when he uses boundary regularity results of kernel elements and the unique continuation property on the boundary.
Remark 1.2.For examples of smooth Poincaré–Einstein manifolds(Mn, g), we have the Poincaré ball(B1n(0), gH), geometrically finite quotients of hyperbolic spaceHn/Γ with infinite volume. For nontrivial examples, Graham and Lee [9]and Lee[16]proved that there are infinitely many Poincaré–Einstein metricsg near the hyperbolic metric and a class of known Poincaré–Einstein metricsgwith prescribed data ofx2g|∂M. Moreover, by the regularity result in[5], forgasymptotically hyperbolic of orderC2, with smooth defining functionx andx2g|∂M is a smooth metric on∂M, forneven, orn=3, up to aC1,αdiffeomorphism near the boundary,h=x2gextends to boundary smoothly;
while fornodd andn >3,hhas expansion with possible log(x)-terms appearing atx=0, which does not satisfy the regularity condition in[21].
Remark 1.3.The ODE result in[11]only gives existence of radially symmetric constantQ-curvature metrics in the conformal class of the hyperbolic metric, but allows the metric to be far away from the hyperbolic metric. As a pertur- bation result, our theorem gives the existence of solutions in the conformal class of metrics in a small neighborhood of the hyperbolic metric, more precisely, seeTheorem 4.1.
Since this is a perturbation result, we first discuss the linear problem. We say that a bounded linear operatorLis essentially injective, if the null space ofLis at most finitely dimensional; andLisessentially surjectiveifLhas closed range and with at most finitely dimensional cokernel. Using Mazzeo’s approach in[18], we obtain the semi-Fredholm property for the linear operator(1.3):
Theorem 1.4.Let(Mn, g)be an asymptotically hyperbolic manifold with defining functionx and the metrich=x2g smooth up to the boundary, then the linear operatorL:xδHe4(M)→xδL2(M,√
dx dy )as in(1.3), is essentially injective ifδ > n2 and δ =n+12, with infinite dimensional cokernel, andL is essentially surjective ifδ < n2 and δ = −12, with infinite dimensional kernel. Moreover, in both cases, L has closed range, and admits a generalized inverse Gand orthogonal projectorsΠ1onto the nullspace andΠ2onto orthogonal complement of the range ofL which are edge operators, such that,
GL=I−Π1, LG=I−Π2.
The corresponding theorem for the weighted Hölder space is as follows.
Theorem 1.5.Let(Mn, g)be an asymptotically hyperbolic manifold with defining functionxand the metrich=x2g smooth up to the boundary. Let0< α <1. The linear operatorL:xνΛ4,α(M)→xνΛ0,α(M)as in(1.3), is essentially injective if ν >n−21 and ν=n, with infinite dimensional cokernel;and Lis essentially surjective if ν <n−21 and ν= −1, with infinite dimensional kernel. Moreover, in both cases, L has closed range. Also,xνΛ4,α(M)has the topological splitting of the following direct sumxνΛ4,α(M)=Π1(xνΛ4,α(M))⊕(I−Π1)(xνΛ4,α(M)), which is the projection to the null space ofLand its topological complement for the second case. Similarly as the theorem with weighted Sobolev spaces, there is a corresponding splitting ofxνΛ0,α(M)forν >n−21.
The paper is organized as follows. In Section2, we study the linear elliptic edge operatorLdefined in(1.3), and obtain the semi-Fredholm property of the linear operatorL. In Section3, we obtain that if the linear operatorLwith respect to the initial asymptotically hyperbolic metricgis surjective in a suitable weighted Hölder space, there are infinitely many solutions to the prescribedQ-curvature problem withQ˜ a small perturbation ofQg, and the solutions are parametrized by the elements in the kernel ofL. Then we give the proof ofTheorem 1.3. Using a special weighted Hölder space, in Section 4, we prove a perturbation result for the prescribed constant Q-curvature problem for a Poincaré–Einstein metric. In Section5, we give a similar discussion to the prescribedU-curvature equations.
2. Semi-Fredholm properties of the linearized operator
In this section, we will discuss the local parametrix forLand the Fredholm property ofL. An important feature is that the elliptic operatorLunder consideration here is degenerate near infinity. Here we review some of the material developed by Mazzeo and others in the theory of elliptic edge operators, see[19].
As in the introduction, let(Mn, g)be an asymptotically hyperbolic manifold of dimensionn, with defining func- tionxand the metrich=x2gsmooth up to the boundary. Let(x, y)be the local coordinates ofMnear the boundary, andVeas defined in the introduction. The one forms dual to the vector fields which are elements inVeare smooth one forms inM, restricted on the neighborhood of∂Mgenerated linearly by{dxx,dyx1, . . . ,dyxn−1}with coefficients smooth up to∂M. Generally, a left or right parametrixEof an elliptic operatorLonMis a pseudo-differential operator with the property that
EL=Id+R1, or LE=Id+R2, withR1, R2compact operators.
The Schwartz kernel of an interior parametrix of the linear operatorLis a distribution onM×M, and for “interior”
we mean that the parametrix has singularity near the boundary which will be explained in the following. Let(x, y) and(x,˜ y)˜ be local coordinates on each copy ofMnear the boundary. We know that the parametrix is smooth, except for the singularity along the diagonal= {x= ˜x, y= ˜y}, as in the case of compact manifolds. Moreover, due to the degeneration of the edge operatorL, asx,x˜→0, we also have the important additional singularity at the intersection ofand the corner, which isS= {x= ˜x=0, y= ˜y}. To deal with the boundary singularity, we introduce a new manifoldM02=M×0M, by blowing-upM×MalongS. Actually, if we use polar coordinates forM×Mnear the corner,
r=
x2+ |y− ˜y|2+ ˜x21/2
∈R+,
Θ=(x, y− ˜y,x)/r˜ ∈S++n = Θ∈Sn, Θ0, Θn0 ,
we know that the level set ofr=Ris a submanifold of dimensional 2n−1 forR >0, whileS= {r=0}is singular.
More precisely, letM02be the lift ofM×Msuch that it is the same asM×Maway fromS, but near the corner, it is represented by the lift of the polar coordinates, smoothly. Hence,S11= {r=0}is a(2n−1)-dimensional submanifold ofM02. Letb be the natural projection map fromM02toM×M. For the convenience of calculation, as in[18], we introduce two systems of local coordinates onMe2,(s, v,x,˜ y)˜ and(x, y, t, w), where
s=x/x,˜ v=y− ˜y
˜
x ; t= ˜x/x, w=y˜−y x . Changing variables in these two coordinates,
x∂x=s∂s=x∂x−w∂w−t ∂t, and x∂y=s∂v=x∂y−∂w.
In the following without loss of generality we only need to consider(s, v,x,˜ y). Viewing elements in˜ Veas first order differential operators, we denote Diff∗e(M)the algebra generated byVewith coefficients in the ringC∞(M), and with the product given by composition of operators. We call an element in Diff∗e(M)anedge operator. Let Diffme(M)be the linear subspace of differential operators which are ofm-th order. ThenL∈Diffme(M)has the form
L=
j+|α|m
aj,α(x, y)(x∂x)j(x∂y)α, (2.1)
withaj,α∈C∞(M), in the coordinate chart(x, y). The symbol ofLis σe(L)(x, y;ξ, η)=
j+|α|=m
aj,α(x, y)ξjηα.
Lis elliptic ifσe(L)(x, y;ξ, η)=0, for(ξ, η)=0. It is easy to check thatgand the linear operatorLin(1.3)are elliptic.Lin(2.1)can be considered as a lift toMe2as follows,
L=
j+|α|m
aj,α(x, y)(x∂x)j(x∂y)α=
j+|α|m
aj,α(sx,˜ y˜+ ˜xv)(s∂s)j(s∂v)α.
LetN (L)be the normal operator ofL, so that N (L)=
j+|α|m
aj,α(0,y)(s∂˜ s)j(s∂v)α,
is the restriction to S11 of the lift of L toMe2. The normal operator is an important approximation of Lnear the boundary. For the linear operatorLin(2.1),
Lφ=
j+|α|m
aj,α(0, y)(x∂x)j(x∂y)αφ+Eφ,
for any smooth functionφ, with the error term
Eφ=x
j+|α|m
bj,α(x, y)(x∂x)j(x∂y)αφ,
forx >0 small, with the coefficientsbj,αsmooth up to the boundary.
Definition 2.1.The indicial familyIζ(L)ofL∈Diffke(M)is defined to be the family of operators L
xζ
log(x)p
f (x, y)
=xζ
log(x)p
Iζ(L)f (0, y)+O xζ
log(x)p−1 , forf∈C∞(M),ζ ∈C,p∈N0.
There exists a unique dilation-invariant operatorI (L), which is called the indicial operator, such that I (L)(y, s∂s)sζf (y)=sζIζ(L)f (y).
In local coordinates near the boundary,I (L)=
jkaj,0(0, y)(s∂s)j.
Definition 2.2.IfL∈Diff∗e(M)is elliptic, we denotespecb(L)as the boundary spectrum ofL, which is the set of ζ ∈C, for whichIζ(L)=0.
Let(M, g),x, andhbe defined as above. DenoteSxas the level set ofx(xis also denoted asy0for convenience), and the coordinates(y1, . . . , yn−1)=y. We now use this point of view to analyze our linearized operator(1.3).
In a neighborhood of∂M, we have the following, Ricg=Rich+x−1
(n−2)Hesshx+hxh
−(n−1)x−2|dx|2hh, (2.2)
and
Rg= −n(n−1)|dx|2h+(2n−2)x(hx)+x2Rh, (2.3) where|dx|h=1, and
(Hessh)ij(x)= ∇ih∇jh(x)=∂i∂j(x)−Γijs∂s(x)= −Γij0=1
2∂xhij=Bij,
withBij the second fundamental form ofSx, fori, j >0; while(Hessh)ij(x)=0 fori=0 orj =0. Also hx= trh(Hessh)=H (h), withH (h)the mean curvature of the level set ofx in the metrich. AlsoΓijk is the Christoffel symbol with respect toh. Note thatgin our paper is the trace of Hessg, with negative eigenvalues:
gu=gij
∂i∂j−Γijk∂k
u (2.4)
=x2hu+(2−n)x(∇hx, du) (2.5)
=(2−n)x∂xu+x2
∂x2u+yu+H (h)∂xu
, (2.6)
whereyis the Laplacian on the level setSxofx, in the induced metrich|Sx. Near the boundary, theQ-curvature is
Qg= − 2
(n−2)2(n−1)2n+n3−4n2+16n−16
8(n−1)2(n−2)2 n2(n−1)2+O(x)
=n(n2−4)
8 +O(x),
forn5, andQg=3+O(x), forn=4.
In the rest of this section we will consider the linear operatorLin(1.3). Note that Lφ=2gφ−divg(anRgg−bnRicg)∇gφ−4f φ
=2gφ−anRggφ+bnRicgij∇gi∇gjφ−an(∇gRg,∇gφ)+bn∇giRicij∇gjφ−4f φ
=2gφ−anRggφ+bnRicgij∇gi∇gjφ+
−an+bn
2
(∇gRg,∇gφ)−4f φ
=2gφ−anRggφ+bnRicgij∇gi∇gjφ+ 6−n
2(n−1)(∇gRg,∇gφ)−4f φ,
withf =Qgforn5, andf =2Qgforn=4. For the third equality, we use the second Bianchi identity. Also, gφ=x2hφ−(n−2)x(∇hx, dφ)h=x2hφ−(n−2)x∂xφ,
and
Rij(g)∇gi∇gjφ∼
−(n−1)x2hij+O x3
x−4∇i∇gjφ
= −(n−1)
gφ+O(x)p(x, y, x∂x, x∂y)φ , for some smooth functionp(·). As a consequence,
Lφ=2gφ−anRggφ+bnRicgij∇gi∇gjφ+ 6−n
2(n−1)(∇gRg,∇gφ)−4f φ
=2gφ−an
−n(n−1)+O(x)
gφ+bn
−(n−1)gφ+O(x)p(x, y, x∂x, x∂y)φ + 6−n
2(n−1)
−(2n−2)x2H (h|Sx)∂xφ+O x3
|∇yφ|
− 1
2n n2−4
+O(x)
φ.
By definition, N (L)=
(s∂s)2−(n−1)s∂s+s2v−n
(s∂s)2−(n−1)s∂s+s2v+n2−4 2
.
In addition, I (L)=
(s∂s)2−(n−1)s∂s−n
(s∂s)2−(n−1)s∂s+n2−4 2
.
SupposeI (L)φ=0, and writeφ=sζ. Solving this equation, we get the indicial rootsζ, given by specb(L)=
n,−1,n−1 2 −i
√n2+2n−9
2 ,n−1
2 +i
√n2+2n−9 2
. LetΛbe the index set
Λ= 1
2+Re(δ); δ∈specb(L)
. (2.7)
The operatorN (L)acts on functions defined onR+s ×Rnv−1for each fixedy˜, with coordinates(s, v). For the linear operatorL,N (L)does not depend ony. We now take the Fourier transformation of˜ N (L)in thevdirection,
N (L) =
j+|α|m
ai,α(s∂s)j(isη)α.
We have the symmetry of dilation:
aj α(s∂s)j(s∂y)α=aj α(ks∂ks)j(ks∂ky)α, for anyk∈R− {0}. Lett=s|η|, then
N (L)(s, η) =
j+|α|m
ai,α(0,y)(t ∂˜ t)j(itη)ˆ α,
which is denoted asL0(t,η), whereˆ ηˆ=|ηη|. This is a family of totally characteristic operators onRn+and generally its coefficients depend ony˜. Note that we have fixedηˆin the formula, and there is no scaling freedom in this direction.
LetHm,δ,lbe the weighted Sobolev space Hm,δ,l= f: φ(t)f∈tδHem
R+ ,
1−φ(t)
f ∈t−lHm R+
, withφ∈C0∞(R+), andφ(t)=1 in a neighborhood oft=0. Note that
L0:tδHm,δ,l→tδHm−4,δ,l+4 is bounded.
For the linear operatorL, N (L) =
(s∂s)2−(n−1)s∂s+s2
−|η|2
−n
(s∂s)2−(n−1)s∂s+s2
−|η|2
+n2−4 2
, hence
L0(t,η)ˆ =
(t ∂t)2−(n−1)t ∂t−t2−n
(t ∂t)2−(n−1)t ∂t−t2+n2−4 2
=L1◦L2,
withs∂s=s|η|∂s|η|=t ∂t. Note thatL0does not depend ony˜. We have made the formula into the simplest form.
Let us consider the relationship of Fredholm property amongN (L),N (L) andL0, intδL2, forδ > n2. We know that the first two operators have the same properties of injectivity and surjectivity. Let
L0ϕ(t )=0,
by definition, it holds if and only if N (L)ϕ
s|η|
=0.
But then N (L)
a(η)ϕ s|η|
=a(η)N (L)ϕ s|η|
=0,
for alla(η)smooth, since the derivative is only insdirection, with fixedη. Then, using the inverse Fourier transfor- mation,
N (L)
Rn−1
e2π iy,ηa(η)ϕ s|η|
dη=0.
This means one-dimensional kernel ofL0corresponds to the infinite dimensional kernel ofN (L), and this construction also gives the fact that the kernel ofN (L)is either trivial or of infinite dimension. But ifN (L) is injective, thenL0
is injective. Conversely, ifL0 is injective, thenN (L) is injective, and so is N (L). We have a dual argument of the surjectivity forδ < n2. As in[18],L0is Fredholm whenδ /∈Λ, with the setΛin(2.7), andN (L)is semi-Fredholm with either infinite dimensional kernel or cokernel. Roughly speaking,Lis a small perturbation ofN (L)near∂M.
WhenN (L)is injective or surjective,Lis essentially injective or essentially surjective, which will beTheorem 1.4 andTheorem 1.5.
To see the semi-Fredholm property ofL, the strategy is to first study the Fredholm property ofL0 andN (L), and finally obtain the semi-Fredholm property ofLusing Mazzeo’s theorems which we list here asTheorem 2.4and Corollary 2.5.
We consider the Fredholm property of L0,L1 and L2 on weighted spaces. To this end, we introduce Bessel functions, which are solutions to the Bessel equation
x2d2y dx2 +xdy
dx −
x2+α2 y=0, whereαis a complex number.
The Bessel functionsIα andI−α form a basis of linear space of solutions to the Bessel equation above, while {Iα, Kα}is another basis. For Re(α) >−12, and−π2 <arg(x) <π2, the integral representations of these solutions are as follows,
Iα(x)= (x2)α Γ (α+12)Γ (12)
1
−1
e−xt
1−t2α−12dt,
Kα(x)=π 2
Iα(x)−I−α(x)
sin(απ ) =Γ (12)(x2)α Γ (α+12)
∞ 1
e−xt
t2−1α−12dt,
withxa complex number (see pp. 172 and 77 in[25]). Note thatIα is bounded nearx=0, and it increases exponen- tially near+∞, and
Kα(x)∼C(ε)xRe(α)e−x+ε,
for anyε >0, asx→ +∞. AlsoKα(x)is bounded for Re(α)0, nearx=0. The formKα(x)is more useful near x= ∞, since it decays exponentially.
We want to solve the following ODE, by transforming it into the Bessel type equations as above:
L1u=
(t∂t)2−(n−1)t∂t−t2−n u=0.
Letu=tβu, then we obtain that˜ tβ
(t∂t)2u˜+(2β+1−n)t ∂tu˜+
−n−t2+β2−β(n−1)
˜ u
=0. (2.8)
Then, letting 2β+1−n=0, Eq.(2.8)is just the form of the Bessel equation. In this case,β= n−21, and then the indexα=n+21.
Therefore, u(t )=tn−21
C1In+1
2 (t)+C2Kn+1 2 (t)
.
In fact, tn−21In+1
2
t|η|
∼tn|η|n+21, tn−21Kn+1 2
t|η|
∼t−1|η|−n+21, neart=0. Moreover,
tn−21In+1 2
t|η|
∼tn2−1et|η|/
2π|η|, tn−21Kn+1 2
t|η|
∼tn2−1e−t|η| π
2|η|, ast→ ∞.
Similarly, L2u=
(t∂t)2−(n−1)t∂t−t2+n2−4 2
u=0.
Letu(t )=tβu(t ), then˜ tβ
(t∂t)2u˜+(2β+1−n)t ∂tu˜+
n2−4
2 −t2+β2−β(n−1)
˜ u
=0.
Set 2β+1−n=0, so thatβ=n−21, and thenu˜is a solution to the Bessel equation withα=in2+22n−9, u(t )=tn−21
C1Iin2+2n−9 2
(t)+C2Kin2+2n−9 2
(t) .
By the expansion of the series form of the Bessel functions, as in[15, p. 108], we have tn−21Iα
t|η|
∼tn−21+α|η|α/
2αΓ (1+α) , and
tn−21I−α t|η|
∼tn−21−α|η|α/
2αΓ (1−α) ,
withα=in2+22n−9, neart=0. Now it is easy to see that the linear combination xn−21
C1xi
n2+2n−9
2 +C2x−i
n2+2n−9
2
can never vanish to infinite order att=0 if eitherC1=0 orC2=0. Also, tn−21Kα
t|η|
∼tn−21π 2
Iα(t|η|)−I−α(t|η|) sin(απ ) , withα=in2+22n−9, and|η| =0, neart=0.
Using the integral form as above, we have that Iα(t) grows exponentially, while Kα decays exponentially as t→ +∞, forα=in2+22n−9.
DenoteLt0to be theL2adjoint ofL0in the measuredt, and L∗0=t2δL0t−2δ,
to be the adjoint ofL0intδL2in the measuret−2δdt. These are all elliptic operators, with boundary spectra:
specb Lt0
= −ζ−1:ζ ∈specb(L0) , specb
L∗0
= −ζ+2δ−1:ζ ∈specb(L0) . For example, forL1=(t∂t)2−(n−1)(t∂t)−t2−n,
L1uv dt=
uLt1v dt.
Then
