applications to anti-de Sitter spacetimes Elliptic boundary value problems for Bessel operators, with C.R. Acad. Sci. Paris, Ser.I

Contents lists available atScienceDirect

C. R. Acad. Sci. Paris, Ser. I

www.sciencedirect.com

Partial differential equations

Elliptic boundary value problems for Bessel operators, with applications to anti-de Sitter spacetimes

Problèmes aux limites elliptiques pour les opérateurs de Bessel, avec applications aux espaces-temps anti-de Sitter

Oran Gannot

DepartmentofMathematics,LuntHall,NorthwesternUniversity,Evanston,IL60208,USA

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received21August2018 Accepted30August2018 Availableonline2October2018 PresentedbytheEditorialBoard

Thispaper considers boundary value problems for aclassof singular elliptic operators that appear naturally in the study of asymptotically anti-de Sitter (aAdS) spacetimes.

These problems involve a singular Bessel operator acting in the normal direction.

AfterformulatingaLopatinskiˇıcondition,elliptic estimatesare establishedfor functions supportedneartheboundary.TheFredholm propertyfollowsfromadditionalhypotheses in the interior. This paper provides a rigorous framework for mode analysis on aAdS spacetimesforawiderangeofboundaryconditionsconsideredinthephysicsliterature.

CompletenessofeigenfunctionsforsomeBesseloperatorpencilsisshown.

©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

ré s u m é

CetteNote considèredesproblèmes auxlimitespour uneclasse d’opérateurselliptiques singuliers,quiapparaissentnaturellementdansl’étudedesespaces-tempsasymptotiquement anti-deSitter(aAdS).CesproblèmesimpliquentunopérateurdeBesselsingulieragissant dans la direction normale. Après avoir formulé une condition de Lopatinskiˇı, nous établissons des estimations elliptiques pour les fonctions dont le support est voisin du bord. La propriété de Fredholm suit d’hypothèses additionnelles à l’intérieur. Nous fournissonsiciuncadrerigoureuxpour l’analysedesmodessurlesespaces-tempsaAdS d’unelarge classe de conditions au bord,considérées en physique. Nous montrons que certainspinceauxd’opérateursdeBesselontunensemblecompletdefonctionspropres.

©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

E-mailaddress:[email protected].

https://doi.org/10.1016/j.crma.2018.08.003

1631-073X/©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

1. Introduction

Thestudyoflinearfieldsonasymptoticallyanti-deSitter(aAdS)spaceshasstimulatednewinterestinboundaryvalue problemsfor aclass ofsingular ellipticequationswhereinthe operator D²_x

+ ( ν

²

₋

1

/

4

)

x⁻² actsonone of thevariables [20,27,30,28,46,49,50].Toformulatethisclassofoperatorsmoreprecisely, consideraproductmanifold

[

⁰

, ε ) × ∂

X,where

∂

X iscompact.ThemodelforwhatwecallaBesseloperatorhastheform P

(

x

,

y

,

Dx

,

Dy

) =

^D2x

+ ( ν

²

₋

1

/

4

)

x⁻²

+

^A

(

x

,

y

,

Dy

),

where

(

x

,

y

) ∈ (

0

, ε ) × ∂

X andA isafamilyofsecond-orderdifferentialoperatorson

∂

X dependingsmoothlyonx

∈ [

⁰

, ε )

. Theparameter

ν

isrequiredtoberealandstrictlypositive.InthestudyoflinearwavesonaAdSspacetimes,

ν

isrelatedto themassofascalarfield—seeSection3formoredetails.Thecondition

ν >

0 correspondstotheBreitenlohner–Freedman bound[10,11].

Boundarydataforthisproblemareformallydefinedbythefollowingweightedrestrictions:

γ

₋u

=

^xν−1/2u

|

∂X

, γ

₊u

=

^x1−2ν

∂

x

(

x^ν−1/2u

) |

∂X

.

Somecareisneededtogiveprecisemeaningtotheserestrictions—seeSection4.4,alongwithanearlierdiscussionin[49].

TheboundaryoperatorsinthispaperareoftheformT

=

^T−

γ

₋

₊

T⁺

γ

₊,whereT⁻

,

T⁺aredifferentialoperatorson

∂

X of orderatmostoneandzero,respectively.Thispaperisconcernedwithsolvabilityoftheboundaryvalueproblem

P u

=

^fonX

T u

=

^gon

∂

X (1.1)

when0

< ν <

1,andthesimplerequation

P u

=

^{f onX (1.2)}

when

ν _≥

1.No boundaryconditionsare imposed when

ν ≥

^{1.The differencebetweenthecases 0}

< ν <

1 and

ν _≥

1 is explainedinmoredetailintheintroductiontoSection5.

EllipticityoftheBesseloperator P isdefinedinSection2.4.Asinthestudyofsmoothboundaryvalueproblems,thereis alsoanotionofellipticityfor(1.1) givenbyanaturalLopatinskiˇıconditiononthepair

(

P

,

T

)

.Thisconditionisintroduced inSection 5.4.Ellipticestimatesare proved inTheorem 5.1.When theoperators P

,

T dependpolynomially ona spectral parameter

λ

, there is a notion of parameter-ellipticity for both P and the boundary value problem (1.1). Theorem 5.2 providesellipticestimatesintermsofparameter-dependentnormswhichareuniformas

| λ | → ∞

^intheconeofellipticity.

Fortheglobalproblem, consider acompact manifold X where

[

⁰

, ε ) × ∂

X isidentified witha collarneighborhoodof

∂

X.Supposethattherestrictionof P tothiscollarisaBesseloperator—seeSection2fordetails.Asinthecaseofsmooth boundaryvalueproblems,estimatesforP near

∂

X mayoftenbecombinedwithestimatesintheinteriorX toestablishthe Fredholmproperty(includingsomecaseswhere P failstobeeverywhereellipticon X).InSection6,asufficientcondition ofthistype isdiscussed.Furthermore,inthepresenceofaspectral parameter

λ

,uniquesolvabilityisestablishedfor

λ

in theconeofellipticityprovided

|λ|

^{issufficientlylarge.}

Section 3 recalls thenotion of an aAdS metric,which is the primary motivationfor this paper.It isshown that the FouriertransformedKlein–GordonoperatorisindeedaBesseloperatorwhoseorder

ν

dependsontheKlein–Gordonmass.

OnegoalistostudytheKlein–Gordonequationby Fouriersynthesis onceitsspectral familyisunderstood,corresponding tothestudyofnormalorquasinormalmodes.Thispaperprovidesarigorousframeworkforfutureworkinthatdirection;

see[22,23] forsomerecentprogress.

ForstationaryaAdSspacetimeswithcompacttimeslicesandaneverywheretimelikeKillingfield

∂

t,Section7describes a class of boundary conditions which yield a complete set ofnormal modes associated witha discrete set of eigenval- ues.Ofparticularinterest are boundaryconditionswhich dependon

∂

t (hence depend onthespectral parameter aftera Fouriertransform).Thisisimportantforthestudyofmodeswithtransparentordissipativeboundaryconditions,alongwith superradiancephenomena[3,30,51].

The approach ofthis paperis inspired by thetexts ofRoitberg [44] andKozlov–Maz’ya–Rossman [35] in the smooth setting.ThisapproachisparticularlysuitedtothesingularnatureofBesseloperators,andallowsforthestudyofboundary value problems inlow regularity spacesasneededin applicationsto generalrelativity —see Section 6.All themethods are classical, using only homogeneity properties of differential operators. The key is exploiting the theory of “twisted”

derivativesasfirstemphasizedin[49].Thisisbasedontheclassicalobservationthattheone-dimensionalBesseloperator D²_x

+ ( ν

²

₋

1

/

4

)

x⁻² admitsa factorizationastheproductofafirst-order operatoranditsadjoint;thisfirst-orderoperator isthentreatedasanelementaryderivative.

Using a variationalapproach, similar ellipticestimates were studied by HolzegelandWarnick [27,29,49,50]. However, onlythe “classical” self-adjointboundary conditionswere handledwhen0

< ν <

1; thesearethe Dirichlet(T

= γ

₋) and Robinboundaryconditions(T

= γ

₊

+ β γ

₋ with

β

a real-valued function).The approachtakenhereaccountsforalarger classofnon-self-adjointboundaryconditions.

Usingresultsofthispaper,discretenessofquasinormalfrequenciesonKerr–AdSblackholeswitharbitraryrotationspeed is established in [23]. Thesefrequencies replace eigenvalues in scatteringproblems [14,17,23,31,34,50]. When 0

< ν <

1, arbitrary boundary conditions satisfyingthe Lopatinskiˇıcondition maybe imposed on the field (althoughof course one does not have any completeness statement).This generalizes earlier work of Warnick[50], where,as notedabove, only DirichletorRobinboundaryconditionswereconsidered.

TheresultsofthispapershouldalsobecomparedtoearlierworksofVasy[46] andHolzegel[27] onaAdSspaces,where amorerestrictivemeasureisusedtodefinethespaceofsquareintegrablefunctions.Inthoseworks,thesquareintegrability conditionisequivalenttothegeneralizedDirichletboundarycondition.Thislimitstherangeofapplications,sincedifferent boundaryconditionsareemployedthroughoutthephysicsliteratureonaAdSspaces[3,6,10,11,16,32,52].

ThereisalsoageneralmicrolocalapproachtodegenerateboundaryvalueproblemsdevelopedbyMazzeo–Melrose[39]

andMazzeo[43],culminatingintheworkofMazzeo–Vertman[41] ongeneralboundaryvalueproblems.Inparticular,the elliptictheoryin[41] couldlikelyreproducetheresultsofthispaper,andisalsoapplicabletomuchmoregeneralclasses of ellipticoperators. On theother hand,the approach developedhereis directlymotivated by thephysics literature. For instance,theSobolevspacesusedinthispaperwereoriginallydefinedin[49] togiveprecisemeaningtotheenergyrenor- malizationimplicit inthework ofBreitenlohner–Freedman [10].Furthermore,the parameter-dependent(or semiclassical) theory isnotdevelopedin[41].Thereisalsoa simplicityadvantageinusingphysicalspacemethods,ratherthana more sophisticatedmicrolocalapproach.ItshouldalsobestressedthatBesseloperatorsoftheprecisekindstudiedherearisein numerouscontextsoutsideofgeneralrelativitywithnegativecosmologicalconstant,bothmathematicalandphysical.

2. Preliminaries

2.1. Conventionsfordifferentialoperators

If P isasmoothdifferentialoperatoronamanifoldY,theninlocalcoordinates,

P

=

|α|≤^m

a_α

(

y

)

D^α_y

.

(2.1)

The orderof P issaidtonogreaterthanm,writtenord

(

P

) ≤

^m.Iford

(

P

) ≤

^{m,thenthesymbol}

σ

m

(

P

)

of P withrespect tomisthepolynomialfunctiononT^∗Y giveninlocalcoordinatesby

σ

m

(

P

)(

y

, η ) =

|α|=^m

a_α

(

y

) η

^α

, (

y

, η ) ∈

^T∗Y

.

The spaceofsmoothdifferential operatorsoforderno greaterthanm isdenoted Diff^m

(

Y

)

.Thefollowingconvention will be usefulthroughoutSection 5: iford

(

P

) ≤

^{m withm}

<

0, then P

=

^{0;converselyif P}

=

^{0,then P canbe assignedany} negativeorder.

The classofparameter-dependent differentialoperatorsonamanifold Y isdefinedasfollows: P

∈

^Diffm_(λ)

(

Y

)

ifinlocal coordinates,

P

(

y

,

D_y

; λ) =

j+|α|≤^m

a_α,j

(

x

)λ

^jD^α_y

.

Thestatementord^(λ)

(

P

) ≤

^mabouttheparameter-dependentorderof P isdefinedbyassigningto

λ

^jthesameweightasa derivativeoforder j.Thustheparameter-dependentprincipalsymbolof P isgivenby

σ

m^(λ)

(

A

) =

j+|α|=^m

a_α,j

(

t

)λ

^j

η

^α

, (

y

, η , λ) ∈

^T∗Y

× C.

IfY isacompactmanifoldwithoutboundary,theparameter-dependentSobolevnormsonY aredefinedfors

≥

^{0 by}

|

u

|

²_Hs(Y)

= |λ|

^2s

u

²_H0(Y)

+

u

²_Hs(Y)

,

andanoperator P

∈

^Diffm(λ)

(

Y

)

isboundedH^s

(

Y

) →

^Hs−m

(

Y

)

uniformlywithrespectto

| λ |

^{inthesenorms.}

2.2. Manifoldswithboundary

Let X

=

^X

∪ ∂

X denotean n-dimensional manifold with compact boundary

∂

X and interior X.A boundary defining functionfor

∂

X isafunctionx

∈

^C∞

(

X

)

satisfying

x⁻¹

(

0

) = ∂

X

,

x

>

0 onX

,

dx

|

∂X

=

⁰

.

Givenx,thereexistsanopen subset W

⊇ ∂

X,anumber

ε >

0,andadiffeomorphism

φ : [

⁰

, ε ) × ∂

X

→

^{W suchthatx}

◦ φ

agrees withthe projection

[

⁰

, ε ) × ∂

X

→ [

⁰

, ε )

. A collarof this type is said to be compatible with x. Unless otherwise specified,amanifold withboundary X willalwaysrefer to X equippedwitha distinguishedboundarydefiningfunction x andachoiceofcompatiblecollardiffeomorphism

φ

.

2.3. Besseloperators

Given

ν ∈

R^{,formallydefinethe}differentialoperator

∂

_ν bytheformula

∂

ν

= ∂

x

+ ( ν −

¹

/

2

)

x⁻¹

=

^x1/2−ν

∂

xx^ν−1/2

.

Furthermore,let

∂

_ν^∗

= −

^xν−1/2

∂

xx^1/2−ν,whichistheformaladjointof

∂

ν withrespecttoLebesguemeasureonR+.Similarly, let Dν

= −

ⁱ

∂

_ν andD^∗ν

=

ⁱ

∂

_ν^∗.Notethat

|

^Dν

|

²

:=

^D∗_ν^Dν

=

^D2_x

+ ( ν

²

−

¹

/

4

)

x⁻²

istheone-dimensionalBesseloperatorinSchrödingerform.

Definition2.1. Let X denote a manifold with compact boundary. A differential operator P

∈

^Diff2

(

X

)

is called a Bessel operatoroforder

ν >

0 ifthereexist

A

=

^A

(

x

,

y

,

D_y

) ∈

^Diff2

(∂

X

),

B

=

^B

(

x

,

y

,

D_y

) ∈

^Diff1

(∂

X

)

dependingsmoothlyonx

∈ [

⁰

, ε )

,suchthat B

(

0

,

y

,

D_y

) =

^{0 and}

φ

^∗P

= |

D_ν

|

²

+

B

(

x

,

y

,

Dy

)

D_ν

+

A

(

x

,

y

,

Dy

).

(2.2)

ThesetofsuchoperatorsisdenotedbyBessν

(

X

)

.

The requirement that

|

^Dν

|

^{2 appears with unit coefficient is not at all essential. If the coefficient is a} nonvanishing functionsmoothuptox

=

^{0,thenthequotientofP bythiscoefficientisaBesseloperatorasabove,andthis}normalization doesnotaffectanyoftheargumentsinSections5,6.

The classofBessel operators dependsstrongly onthe pair

(

x

, φ)

,where xisa boundarydefining function and

φ

isa collardiffeomorphismcompatiblewithxinthesenseofSection2.2.ThisdependencewillbewrittenasBessν

(

X

;

^x

, φ)

when necessary. Ontheother hand,let

(˜

x

, ˜

y

)

denotearbitrarylocalcoordinates near

∂

X,where

˜

xis alocalboundary defining function.If

(

x

,

y

)

arelocalcoordinatesinducedby

φ

,then P isstilloftheform(2.2) (uptoasmoothnonvanishingmultiple) in

(˜

^x

,

^y

˜ )

coordinatesprovidedthat

˜

^x

/

xand^y

˜

^{areevenfunctionsofxmodulo}O(^x3

)

.

A(smooth,positive)density

μ

onX issaidtobeofproducttypenear

∂

X if

φ

^∗

μ = |

^dx

| ⊗ μ

∂X

forafixeddensity

μ

∂X on

∂

X.Itisofcoursealways possibletochooseadensityofproducttypenear

∂

X.Thisisuseful inlight ofthenext lemma.If X iscompact,then L²

(

X

)

maybedefinedasthespaceofsquare integrablefunctionswith respecttoanysmoothdensity

μ

on X,inparticularoneofproducttypenear

∂

X.

Lemma2.2.Supposethat

μ

isofproducttypenear

∂

X .IfP

∈

^Bessν

(

X

)

,thenP^∗

∈

^Bessν

(

X

)

,whereP^∗istheformaladjointofP withrespectto

μ

.

Proof. ThepullbackofP^∗ to

(

0

, ε ) × ∂

X isgivenby

|

^Dν

|

²

+

^D∗_ν^B∗

+

^A∗

,

where A^∗

,

B^∗ aretheformaladjointsof A

,

B withrespectto

μ

∂X.Ontheotherhand, D^∗_νB^∗

=

B^∗D^∗_ν

+ [

Dx

,

B^∗

].

Furthermore,since B

=

^xB1 forafirst-orderoperatorB1 on

∂

X dependingsmoothlyonx

∈ [

⁰

, ε )

,itfollowsthat B^∗D^∗_ν

=

^xB∗1D^∗_ν

=

^B∗1

(

xD_x

−

ⁱ

(

1

/

2

− ν )) =

^B∗Dν

−

ⁱ

(

1

/

2

− ν )

B^∗₁

,

whichcompletestheproofsincethemultipleofB^∗₁ aswellas

[

^Dx

,

B^∗

]

^{maybeabsorbedintoA∗.} 2

Forthelocaltheory,itisconvenienttoworkonTⁿ₊

=

R+

×

T^n−1,whereTⁿ⁻¹

= (

R

/

2πZ

)

ⁿ⁻¹.ThesetofBesseloperators onTⁿ₊^{isdefinedwithrespecttothecanonicalproduct}decompositiononTⁿ₊^{.Thus P}

∈

^Bessν

(

Tⁿ₊

)

if

P

(

x

,

y

,

D_ν

,

Dy

) = |

^Dν

|

²

+

|β|≤1

b_β

(

x

,

y

)

D^β_yD_ν

+

|α|≤2

a_α

(

x

,

y

)

D^α_y

forb_β

∈

^xC∞

(

Tⁿ₊

)

andaα

∈

^C∞

(

Tⁿ₊

)

.WhenworkingonTⁿ₊^{,thefunctionsb}β

,

aα arereferredtoasthecoefficientsof P.

FixacoordinatechartY

⊆ ∂

X andadiffeomorphism

θ :

^Y

→

^{V,whereV isanopensubsetof}T^{n−1.Setting} U

= φ ( [

⁰

, ε ) ×

^Y

),

themap

ψ :

^U

→ [

⁰

, ε ) ×

^{V givenby}

ψ = (

1

× θ ) ◦ φ

⁻¹ definesaboundarycoordinatecharton X.GivenP

∈

^Bessν

(

X

)

,there clearlyexists P_U

∈

^Bessν

(T

ⁿ₊

)

suchthat

P u

=

^PU

(

u

◦ ψ )

for each u

∈

^Cc^∞

(

U^◦

)

.Furthermore, itis always possible toarrange it sothat the coefficientsof PU (in thesense of the previousparagraph)areconstantoutsideacompactsubsetofTⁿ₊^.

2.4. Ellipticityandtheboundarysymbol

Given P

∈

^Bessν

(

X

)

whichnear

∂

X hastheform P

= |

^Dν

|

²

+

^{B D}ν

+

^A

,

let A0

(

y

,

Dy

) =

^A

(

0

,

y

,

Dy

)

.Ellipticityof P atapoint p

∈ ∂

X isdefinedviathefunction

ξ

²

+ σ

2

(

A₀

)(

p

, η ),

(2.3)

whichisahomogeneouspolynomialofdegreetwoin

(ξ, η ) ∈

R

×

^T∗p

∂

X.

Definition2.3. The Bessel operator P

∈

^Bessν

(

X

)

issaid to be (properly) ellipticat p

∈ ∂

X if foreach

η ∈

^T∗p

∂

X

\

^{0 the} polynomial

ξ → ξ

²

+ σ

2

(

A₀

)(

p

, η )

(2.4)

hasnorealroots.

Thus, ellipticityimpliestheexistence ofnonrealroots

± ξ(

p

, η )

,whereIm

ξ(

p

, η ) <

0 byconvention. Foreach

(

p

, η ) ∈

T^∗

∂

X

\

^0,thesymbol

σ

2

(

A₀

)(

p

, η )

determinesafamilyofone-dimensionalBesseloperatorsgivenby

P(p,η)

= |

D_ν

|

²

+ σ

2

(

A0

)(

p

, η ).

(2.5)

TheoperatorP_(p,η)iscalledtheboundarysymboloperatorofP.LetM+

(

p

, η )

denotethespaceofsolutionstotheequation

P₍p,η)u

=

⁰

whichareboundedasx

→ ∞

^.Ellipticityatp

∈ ∂

X impliesthatdimM₊

(

p

, η ) =

^{1 foreach}

η _∈

T^∗_p

∂

X

\

^{0.Indeed,thespace} ofsolutionstoP₍p,η)u

=

^{0 isspannedbythemodifiedBesselfunctions}

x^1/2K_ν

(

i

ξ(

p

, η )

x

),

x^1/2I_ν

(

i

ξ(

p

, η )

x

)

.

SinceRe i

ξ(

p

, η ) >

0,itfollowsthat x^1/2K_ν

(

i

ξ(

p

, η )

x

) =

O

e^−x/C

,

x

→ ∞ ,

whilethesecondsolutiongrowsexponentially[42,Chapter7.8].ThusonlythefirstsolutioncanpossiblylieinM+

(

p

, η )

. 2.5. Parameter-dependentBesseloperators

Definition2.4.Let X denotea compactmanifoldwithboundaryasinSection2.2.Adifferentialoperator P

(λ) ∈

^Diff2_(λ)

(

X

)

iscalledaparameter-dependentBesseloperatoroforder

ν >

0 ifthereexist

A

(λ) =

A

(

x

,

y

,

Dy

; λ) ∈

Diff²_(λ)

(∂

X

),

B

(λ) =

B

(

x

,

y

,

Dy

;λ) ∈

Diff¹_(λ)

(∂

X

)

dependingsmoothlyonx

∈ [

⁰

, ε )

,suchthat B

(

0

,

y

,

D_y

; λ) =

^{0 and}

φ

^∗P

(λ) = |

D_ν

|

²

+

B

(

x

,

y

,

Dy

; λ)

D_ν

+

A

(

x

,

y

,

Dy

; λ).

(2.6) ThesetofsuchoperatorsisdenotedbyBess^(λ)ν

(

X

)

.

Ellipticity with parameter is defined by replacing the standard principal symbol of A with its parameter-dependent version.Beginbyfixinganangularsector

⊆

C^.

Definition2.5.Aparameter-dependentBesseloperator P

(λ)

issaidtobe(properly)parameter-ellipticwithrespectto

at p

∈ ∂

X ifforeach

( η , λ) ∈

^T∗p

∂

X

× \

^{0,thepolynomial}

ξ → ξ

²

+ σ

₂^(λ)

(

A0

)(

p

, η _; λ)

(2.7)

hasnorealroots.

Similarly,for

(

p

, η , λ) ∈

T^∗

∂

X

× \

0,define

P₍p,η;λ)

= |

D_ν

|

²

+ σ

₂^(λ)

(

A0

)(

p

, η ; λ),

andthenlet M+

(

p

, η ; λ)

denotethespaceofsolutionstoP_(p,η_;λ)u

=

^{0 whichare boundedasx}

→ ∞

^{.AsinSection 2.4,} thisspaceisone-dimensional.

3. Motivation:asymptoticallyanti-deSittermanifolds

Thissection recalls thenotionofan asymptotically anti-deSitter(aAdS)metric. Then,a convenientexpressionforthe Klein–Gordon equation isgivenwithrespectto acertain product decompositionneartheconformal boundary.Bymeans ofa Fouriertransform, the initial boundaryvalue problemforthe Klein–Gordon equation isreduced tothe studyofthe boundary value problem fora stationary partialdifferential equation depending polynomially on the spectral parameter.

The corresponding operator is a Bessel operator whose order

ν

dependson the Klein–Gordon parameter; the condition

ν >

0 translatesintothewell-knownBreitenlohner–Freedmanbound.

3.1. aAdSmetrics

Let X denoteann-dimensionalcompactmanifoldwithboundaryasinSection 2.2,andset M

=

R

×

^X.Heret

∈

R^will denoteaglobaltimecoordinate.Aboundarydefiningfunction

ρ

onMsatisfying

∂

t

ρ ₌

0 issaidtobestationary.Thereisan obviousone-to-onecorrespondencebetweenstationaryboundarydefiningfunctionsonM andboundarydefiningfunctions xforX.

Definition3.1. A smooth Lorentzian metric g on M is said to asymptotically simple ifthere exists a boundary defining function

ρ ∈

^C∞

(

M

)

withthefollowingproperties.

(1) theLorentzianmetric g

¯ = ρ

²g admitsasmoothextensionto M, (2) therestriction ^g

¯ |

T∂M isagainLorentzian.

The map g

→ ¯

^g

|

∂M dependson g anda choice of boundarydefining function. However, the conformal class

[¯

^g

|

T∂M

]

dependsonlyon g,sinceanytwoboundarydefiningfunctionsdifferbyapositivemultiple.Alsonotethatifg isasymptot- icallysimple,thend

ρ

isspacelikeforg

¯ |

∂M.

Definition3.2.Anasymptoticallysimplemanifold

(

M

,

g

)

issaidtobeaAdSifthereexistsaboundarydefiningfunction

ρ

suchthat

|

^d

ρ |

²_g¯

= −

^{1 on}

∂

M.

TheaAdSpropertydoesnotdependonthechoiceofboundarydefiningfunction.InadditiontobeingaAdS,supposethat g isstationaryinthesense that

∂

t isaKillingvectorfieldforg.Fortheremainderofthissection,allmetricsareassumed tobestationary.

Thefollowingisawell-knownobservationofGraham–Lee[24],adaptedtotheLorentziansetting.

Lemma3.3.Supposethat

(

M

,

g

)

isanaAdSspacetime.Ifg and

γ

0

∈ [¯

^g

|

T∂M

]

^arestationary,thenthereexistsauniquestationary boundarydefiningfunctionx withthefollowingproperties.

(1) x²g

|

T∂M

= γ

0,

(2)

|

^dx

|

²_x2g

= −

^1inacollarneighborhoodof

∂

M.

Proof. Theproofof[24,Lemma5.2] goesthroughunchanged. 2

IfxsatisfiestheconditiondescribedinLemma3.3,thenxissaidtobeageodesicboundarydefiningfunction.Notethe integralcurvesof

∇

x²gxaregeodesicsofx²gnear

∂

M.ThustheGausslemmaimpliesthat

φ

^∗

(

g

) = −

ds²

+ γ (

s

)

s² (3.1)

on

[

⁰

, ε ) × ∂

M,where

φ : [

⁰

, ε ) × ∂

M

→

C ^isthecollardiffeomorphismobtainedbytheflow-outof

−∇

x²gx.Heres

→ γ (

s

)

is asmooth familyofstationaryLorentzian metrics on

∂

M suchthat

γ (

0

) = γ

0.Furthermore,

φ

^∗x

=

^{s,soby an abuseof} notationitisconvenienttowrite g

=

^x−2

( −

^dx2

+ γ (

x

))

onthecollarneighborhoodC^.

Definition3.4.Themetricg asinLemma3.3issaidtoevenmoduloO(x³

)

(inthesenseofGuillarmou[26])ifthereexists atwo-tensor

γ

1 on

∂

Msuchthat

γ (

s

) = γ

0

+

^s2

γ

1

+

O

(

s³

).

Asin[26,Proposition2.1],thisevennesspropertyisinstrinsictotheconformalclass

[¯

^g

|

T∂M

]

^.ThefundamentalclassofaAdS metricswhichareevenmoduloO

(

x³

)

aretheEinsteinaAdSmetrics;see[2,Section2] forexample.Thisincludesallofthe physicallymotivatedaAdSspaces.TheEinsteincondition alsoenforcesadditionalconditionsontheexpansionof

γ (

s

)

that arenotexploitedhere(intheasymptoticallyhyperbolicsetting,seeMazzeo–Pacard[40,Section2]).

Let

φ : [

⁰

, ε ) × ∂

M

→

C ^{denotethecollar} diffeomorphismassociatedwitha geodesic boundary definingfunction x as above, andlett

:

^M

→

R^{be thetime functionassociated withthe}identification M

=

R

×

^X.Ift0 denotesthe restriction oft to

∂

M,thent₀ givesa timecoordinateon

[

⁰

, ε ) × ∂

M.Ingeneralit isnottruethat

φ

^∗

(

t

) =

^t0 unlessx⁻²g⁻¹

(

dx

,

dt

)

vanishes identically. Ontheother hand,by stationaritythere isat0-invariant function h suchthat

φ

^∗

(

t

) =

^t0

+

^{h,so the} map

φ

0

(

x

,

y

,

t0

) = φ (

x

,

y

,

t0

−

^h

)

(3.2)

doessatisfy

φ

₀^∗

(

t

) =

^t0,and

φ

0

: [

⁰

, ε ) × ∂

M

→

C ^{isstillcompatiblewithx.}Furthermore,if

{

^t

=

⁰

}

^meets

∂

M orthogonally withrespecttox²g,thendh

|

∂M

=

^{0 aswell.}

3.2. TheKlein–Gordonequation

Fix a stationary aAdS spacetime

(

M

,

g

)

anda geodesic boundary defining function x as in Lemma 3.3. Furthermore, suppose that g isevenmodulo O(^x3

)

inthesense ofDefinition 3.4.Inlightoftheproductdecomposition(3.1) itfollows thatnear

∂

M,

2

g

=

^x2D2x

+

ⁱ

(

n

−

¹

+

^e

(

x

))

xD_x

+

^x2

2

γ(x)

,

wherex

→

^e

(

x

)

isasmoothfamilyoffunctionson

∂

M suchthat e

(

x

) =

^x2e0

+

O

(

x³

)

forsomee₀

∈

^C∞

(∂

M

)

and

∂

te

(

x

) =

^0.Indeed,e

(

x

) = (

1

/

2

)

x

∂

xlog

(

det

γ (

x

))

,anddet

γ (

x

) =

^det

γ

0

+

O(^x2

)

.Given

ν >

0,let P_g

=

^x−(n+3)/2

(2

g

+ ν

²

−

ⁿ²

/

4

)

x^(n−1)/2

,

which corresponds to conjugatingthe Klein–Gordon operator withmass

ν

²

−

ⁿ²

/

4 by x^(n−1)/2 andthen dividingby x². Explicitly,

P_g

=

^D2x

+ ( ν

²

−

¹

/

4

)

x⁻²

+

^ix−1e

(

x

)

D_x

+

ⁿ⁻₂¹

x⁻²e

(

x

) + 2

γ(x)

.

(3.3)

Inwhatfollows,X willbeidentifiedwiththetimeslice

{

^t

=

⁰

}

^{withinM,andfunctionsonX areidentifiedwith}t-invariant functionson M.Usingthat g isstationary,itispossibletodefineadifferentialoperatoron X,dependingon

λ ∈

C^,by

P

(λ)

u

=

^eiλtPg

(

e^−iλtu

),

(3.4)

whereu

∈

^C∞

(

X

)

.

Lemma3.5.Let

(

M

,

g

)

denoteastationaryaAdSspacetime.Supposethatx isageodesicboundarydefiningfunction,andg iseven moduloO

(

x³

)

.Let

φ

0begivenby(3.2).IfX meets

∂

M orthogonallywithrespecttox²g andP

(λ)

isgivenby(3.4),then

P

(λ) ∈

^Bess(λ)ν

(

X

;

^x

, φ

₀

).

Proof. Note that

φ

0

= φ ◦

^{H, where H}

(

x

,

y

,

t0

) = (

x

,

y

,

t0

−

^h

(

x

,

y

))

inthenotation of (3.2). Asnotedfollowing (3.2), the differentialofhvanishesalong

∂

M.Itremainstocombinethisobservationwiththeexpression(3.3) for Pgwithrespectto theproductstructureinducedby

φ

. 2

Lemma3.6.Thefollowingellipticitypropertieshold.

(1) If

∂

tistimelikefor

γ

0,thenP

(λ)

isellipticon

∂

X inthesenseofSection2.4.

(2) If

∂

tanddt arebothtimelikefor

γ

0,thenP

(λ)

isparameter-ellipticon

∂

X inthesenseofSection2.5withrespecttoanyangular sector

⊆

C^disjointfromR

\

^0.

Proof. See[47,Section3.2] and[23,Lemma2.3]. 2 4. Functionspacesandmappingproperties

ThepurposeofthissectionistodefineSobolev-typespacesH^{s basedontheelementary}derivativesDν and

|

^Dν

|

^2,both onTⁿ₊^{andonamanifoldwithboundary.Finally,itisshownthatBesseloperatorsact}continuouslybetweenthesespaces.

The expositionis closestto that of[49],where these“twisted”Sobolevspaceswere firstintroduced inthe contextof aAdSgeometry.The relationshipbetweenH¹ andcertainweighted Sobolevspaceswas exploitedbothin[49] andalsoin thecloselyrelatedstudyofasymptoticallyhyperbolicspacesin[15].

Throughoutthissection,thespaces L²

(

Tⁿ₊

)

andL²

(

Tⁿ⁻¹

)

are definedwithrespecttoordinary Lebesguemeasure,and H^m

(T

ⁿ⁻¹

)

denotesthestandardSobolevspaceofordermon T^n−1.Here T^{n−1 willalways denotetheboundary}

∂T

ⁿ₊^.The notationH⁰

(

Tⁿ₊

) :=

^L2

(

Tⁿ₊

)

isalsofrequentlyused.

4.1. TheweightedspaceH¹μ

(T

ⁿ₊

)

Given

μ ∈

R^,let

H¹_μ

(T

ⁿ₊

) = {

^u

∈

D

(T

ⁿ₊

) :

^xμ2Dαu

∈

^L2

(T

ⁿ₊

)

for

| α | ≤

¹

},

whichisaHilbertspaceunderthenorm

^u

²_H1

μ(Tⁿ₊)

=

|α|≤¹

^xμ2Dαu

²_L2(Tⁿ₊)

.

Furthermore,let H

˙

¹μ

(T

ⁿ₊

)

denotetheclosureofC^∞_c

(T

ⁿ₊

)

in H¹μ.Thesespacesare wellstudied; seeLions[36] or Grisvard [25] forexample.

Lemma4.1.Thefollowingholdfor

μ ∈

R^. (1) If

| μ | <

1,thenC_c^∞

(

Tⁿ₊

)

isdenseinH¹μ

(

Tⁿ₊

)

. (2) If

| μ _{| ≥}

1,thenH¹μ

(

Tⁿ₊

) = ˙

^H1μ

(

Tⁿ₊

)

.

Proof. Proofsofthesefactsmaybefoundin[25,36]. 2

GivenaHilbertspaceE,let H¹μ

(

R+

;

^E

)

denotetheHilbertspaceofE-valueddistributionsu

∈

D

(

R+

;

^E

)

suchthat x^μ2u

∈

^L2

( R

+

;

^E

),

x^μ2u

∈

^L2

( R

+

;

^E

),

equippedwithobviousnorm.TheSobolevembeddingtheoreminthissetting,[25,Proposition1.1’],is H¹_μ

( R

+

;

^E

) →

^C0

( R

+

;

^E

), μ <

1

,

so u

→

^u

(

0

)

iscontinuous H¹μ

(

R+

;

^E

) →

^{E.Taking E}

=

^L2

(

Tⁿ⁻¹

)

,it followsthat when

μ <

1,any u

∈

^H1μ

(

Tⁿ₊

)

admitsa trace

u

→

^u

|

_Tⁿ⁻¹

∈

^L2

( T

ⁿ⁻¹

).

(4.1)

Furthermore,thekernelofu

→

^u

|

_Tn−1 is H

˙

¹μ

(T

ⁿ₊

)

(see[25,Proposition1.2]).Thenextlemmaimprovesupontheregularity ofthisrestriction.

Lemma4.2.If

| μ | <

1,thentherestriction u

→

^u

|

_Tⁿ−1

,

u

∈

^C_c^∞

(T

ⁿ₊

)

extendsuniquelytocontinuousmap

γ :

^H1μ

(

Tⁿ₊

) →

^H(1−μ)/2

(

Tⁿ⁻¹

)

.Furthermore,

γ

admitsacontinuousrightinverse.

Proof. BytheSobolevembedding,any

ϕ _∈

C_c^∞

(

R+

) ⊆

^H1μ

(

R+

)

admitsanestimateoftheform

| ϕ (

0

)|

²

≤

C

R+

x^μ

| ϕ |

²

+ | ϕ

|

²

dx

.

Applythisinequalitytothefunction

ϕ (

sx

)

,andthenchooses(dependingon

ϕ

)satisfying

R

x^μ

| ϕ |

^2dx

=

^s2

R

x^μ

| ϕ

|

^2dx

.

Thisyieldstheestimate

| ϕ (

0

)|

²

≤

^2C

⎛

⎜ ⎝

R+

x^μ

| ϕ |

^2dx

⎞

⎟ ⎠

(1−μ)/2

⎛

⎜ ⎝

R+

x^μ

| ϕ

|

^2dx

⎞

⎟ ⎠

(1+μ)/2

.

(4.2)

Now consider u

∈

^H1_μ

(T

ⁿ₊

)

andlet ^u

ˆ (

q

)

denoteits Fouriercoefficients, whereq

∈

Z^{n−1. Itsuffices toapply theinequality} (4.2) to^u

ˆ (

q

)

,whichliesinH¹μ

(R

+

)

foreachq

∈

Z^n−1.Multiplying(4.2) by

^q

^{1−μandsummingoverallq,itfollowsthat}

γ

u

H⁽¹⁻μ)/2

≤

^C

^u

_H¹_μ(R+)

.

When

| μ _| <

1,theuniquecontinuationof

γ

followsfromthedensityofC^∞_c

(

Tⁿ₊

)

in H¹μ

(

Tⁿ₊

)

.That

γ

admitsarightinverse isalsostraightforward;seeLemmaA.3foracloselyrelatedresult. 2

The trace u

→ γ

u definedinLemma4.2agreeswiththerestrictiongivenby(4.1) sincethey bothagreeonthe dense setC^∞_c

(T

ⁿ₊

)

.

4.2. ThespaceH¹

(T

ⁿ₊

)

Given

ν ∈

R^,define

H¹

( T

ⁿ₊

) = {

^u

∈

D

( T

ⁿ₊

) :

^Dν^jD^α_yu

∈

^L2

( T

ⁿ₊

)

forj

+ | α _{| ≤}

1

} ,

where Dν^jDα

yuistakeninthesenseofdistributionsonTⁿ₊^;thenH¹

(

Tⁿ₊

)

isaHilbertspacewhenequippedwiththenorm

^u

²_H1(Tⁿ₊)

=

j+|α|≤¹

^Dν^jD^α_yu

²_L2(Tⁿ₊)

.

The spaceH¹_∗

(

Tⁿ₊

)

isdefinedanalogously by replacing Dν withits formal adjoint D^∗ν.Let H

˙

¹

(

Tⁿ₊

)

denote theclosureof C_c^∞

(T

ⁿ₊

)

inH¹

(T

ⁿ₊

)

,andsimilarlyforH

˙

_∗¹

(T

ⁿ₊

)

.

Lemma4.3.If

ν =

^0,thenH

˙

¹

(T

ⁿ₊

) = ˙

^H1

(T

ⁿ₊

)

withanequivalenceofnorms.

Proof. ThiscanbededucedfromHardy’sinequality

(

1

/

4

)

^x−1u

²_L2(Tⁿ₊)

≤

^Dxu

²_L2(Tⁿ₊)

,

validforu

∈

^Cc^∞

(

Tⁿ₊

)

,andthedensityofC_c^∞

(

Tⁿ₊

)

inbothspaces. 2

The basic observation concerning H¹

(

Tⁿ₊

)

is that the map D

(

Tⁿ₊

) →

D

(

Tⁿ₊

)

given by u

→

^{xν−1/2u restricts to an} isometricisomorphism

H¹

(T

ⁿ₊

) →

^H1_1−2ν

(T

ⁿ₊

).

It follows fromLemma 4.1that x^1/2−νC^∞

(

Tⁿ₊

)

isdense in H¹

(

Tⁿ₊

)

if 0

< ν <

1,and H¹

(

Tⁿ₊

) = ˙

H¹

(

Tⁿ₊

)

if

ν ≥

^{1. Using} Lemma4.2,itisalsopossibletodefineweightedtracesofH¹

(

Tⁿ₊

)

functions,aswillbeexplainedinSection4.4.