The Abel, Fourier and Radon Transforms on Symmetric Spaces.

21 Pages·2006·0.33 MB·English

Indag.Mathem.,N.S.,16(3–4),531–551 December19,2005
TheAbel,FourierandRadontransformsonsymmetricspaces
bySigurdurHelgason
77MassachusettsAvenue,Cambridge,MA02139,USA
DedicatedtoGerritvanDijkontheoccasionofhis65thbirthday
CommunicatedbyProf.J.J.DuistermaatatthemeetingofJune20,2005
1. INTRODUCTION
In this paper we prove some recent results on the three transforms in the title and
discusstheirrelationshipstoolderresults.Thespaceswedealwitharesymmetric
spaces X = G/K of the noncompact type, G being a connected noncompact
semisimpleLiegroupwithfinitecenterandK amaximalcompactsubgroup.
For the two natural Radon transforms on X we prove a new inversion formula
andasharpeningofanoldsupporttheorem;fortheAbeltransformweprovesome
new identities with some applications and for the Fourier transform a result for
integrable functions which has a strong analog of the Riemann–Lebesgue lemma.
TheselatterresultsarefromacollaborationwithRawat,SenguptaandSitaram.
Notation. Following Schwartz we use the notation D(X) for C∞(X), E(X) for
c
C∞(X)andS(Rn)forthespaceofrapidlydecreasingfunctionsonRn.
2. DIFFERENT RADON TRANSFORMS ON THE SYMMETRIC SPACE X
Radon’s paper [40] suggested the general problem of determining a function on a
manifold on the basis of its integrals over certain submanifolds. A natural case of
E-mail:[email protected](S.Helgason).
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thisproblemisthe inversion oftheX-raytransformonaRiemannian manifold.It
isthetransformf →fˆdefinedbythearc-lengthintegral
(cid:1)
(2.1) fˆ(γ)= f(x)dm(x),
γ
f beingan“arbitrary”functionontheRiemannianmanifoldXandγ anycomplete
geodesicinX.
In general this injectivity problem seems to be unresolved. For a Cartan sym-
metric space X (cid:3)=Sn the injectivity, however, holds. For a symmetric X of the
noncompacttheinjectivityholdsinthestrongerformofthe
Supporttheorem [26]. Iffˆ(γ)=0forallgeodesicsγ disjointfromaballB⊂X
thenf(x)=0forx∈/B.
Thislastresultrequiresstrongerdecayassumptionat∞thantheinjectivityresult
does.
Here we shall prove an explicit inversion formula for the X-ray transform for
rankX>1.SeeSection5forthecontactwithRouvière’sdifferentsolution.
Funk [11] and Radon [40] inverted this transform for the sphere S2 and R2.
Denotingthesetofgeodesicsby(cid:2)wehavethecosetspacerepresentations
S2=O(3)/O(2), (cid:2)=O(3)/O(2)Z2,
R2=M(2)/O(2), (cid:2)=M(2)/M(1)Z ,
2
M(n)denotingtheisometrygroupofRn.
Thissuggeststhefollowinggeneralization.LetX=G/K and(cid:2)=G/H becoset
spaces of the same locally compact group G, K and H being closed subgroups.
Here it will be convenient to assume all these groups as well as L=K∩H to be
unimodular. We do not assume the elements ξ ∈(cid:2) to be subsets of X but instead
useChern’sconceptofincidence:
x=gK isincidenttoξ =γH
ifgK∩γH (cid:3)=∅assubsetsofG.Givenx∈X,ξ ∈(cid:2)define
xˇ={ξ ∈(cid:2): x,ξ incident},
ξˆ ={x∈X: x,ξ incident}.
Theseareorbitsofcertainsubgroupsof Gandhavenaturalmeasuresdm,dµ(up
tofactors)andwedefinetheabstractRadontransformf →fˆanditsdual ϕ→ϕˇ
by
(cid:1) (cid:1)
(2.2) fˆ(ξ)= f(x)dm(x), ϕˇ(x)= ϕ(ξ)dµ(ξ).
ξˆ xˇ
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Thenormalizationsofdmanddµareunifiedbytakingx =eK,ξ =eH and
0 0
(cid:1) (cid:1)
(2.3) fˆ(γH)= f(γh·x )dh , ϕˇ(gK)= ϕ(gk·ξ )dk
0 L 0 L
H/L K/L
theinvariantmeasuresdh ,dk beingfixedbyHaarmeasuresofH,K andL.
L L
Mainproblems.
(i) Injectivityoff →fˆ,ϕ→ϕˇ.
(ii) Inversionformulas.
(iii) Rangeandkernelquestionforthesetransforms.
(iv) Applicationselsewhere.
An easy general result relevant to problem (iii) is the following. For a suitable
normalizationofthemeasuresdx=dg ,dξ =dγ wehave
K H
(cid:1) (cid:1)
(2.4) f(x)ϕˇ(x)dx= fˆ(ξ)ϕ(ξ)dξ,
X (cid:2)
aresultwhichsuggeststheextensionof(2.3)todistributions.
3. d-PLANES IN Rn
Here we consider the space X =Rn and (cid:2)=G(d,n) the set of d-dimensional
planesinRn.ThesearebothhomogeneousunderthegroupG=M(n).Fixx ∈Rn,
0
ξ ∈G(d,n)atdistanced(x ,ξ )=p.Thenwehave
0 0 0
(3.1) X=Rn=M(n)/K , (cid:2)=G(d,n)=M(n)/H ,
p p
whereK andH ,respectively,arethestabilitygroupsofx andξ .Sincevarious
p p 0 0
p will be considered the transforms (2.2) will be denoted by fˆ and ϕˇ . Since
p p
the action of M(n) on X and (cid:2) is quite rich it turns out that for the coset space
representation(3.1)
z∈Xisincidenttoη∈(cid:2) ⇔ d(z,η)=p.
Thusthetransformfˆ andϕˇ canbewritten
p p
(cid:1) (cid:1)
(3.2) fˆ (ξ)= f(x)dm(x), ϕˇ (x)= ϕ(ξ)dµ(ξ).
p p
d(x,ξ)=p d(x,ξ)=p
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ˆ ˆ
Inparticular, f istheusuald-planetransformf,butinordertoinvertitweneed
0
ϕˇ forvariablep.Oneofseveralversionsoftheinversionformulaisthefollowing
p
(see[25,27]):
(cid:2)(cid:3) (cid:4) (cid:1)∞ (cid:7)
(3.3) f(x)=c(d) d d p(cid:5)p2−r2(cid:6)d/2−1(cid:5)fˆ(cid:6)∨(x)dp
d(r2) p
r r=0
ˆ ∨
withc(d)andconstant.Notethat(f) (x)istheaverageoftheintegralsoff over
p
alld-planesatdistancepfromx.
Ford=1thisformulareducesto
(cid:1)∞
(cid:5)(cid:5) (cid:6) (cid:6)
(3.4) f(x)=−1 d fˆ ∨(x) dp,
π dp p p
0
which for n=2 coincides with Radon’s original formula. Radon’s proof is very
elegant and is based on an exhaustion of the exterior |x|>r by lines. As far as I
know this proof has not been extended to higher dimensions n. Formula (3.4) for
n>2iscrucialfortheinversionof(2.1)giveninTheorem5.1.
4. d-DIMENSIONAL TOTALLY GEODESIC SUBMANIFOLDS IN HYPERBOLIC SPACE Hn
Asimilarmethodworkshereandtheanalogof(3.3)istheformula
(cid:2) (cid:7)
(cid:3) (cid:4) (cid:1)∞
(4.1) f(x)=C(d) d d (cid:5)t2−r2(cid:6)d/2−1td(cid:5)fˆ(cid:6)∨ (x)dt ,
d(r2) s(t)
r r=1
whereC(d)isaconstantands(p)=cosh−1(p)(see[25,27]).Otherversionsofthe
inversionexist(e.g.,[28]and[6]).Ford=1thisreducesto
(cid:1)∞
(cid:5)(cid:5) (cid:6) (cid:6)
(4.2) f(x)=−1 d fˆ ∨(x) dp
π dp p sinhp
0
aformulawhichforn=2isstatedwithoutproofinRadon[40].
5. X-RAY INVERSION ON THE SYMMETRIC SPACE X=G/K
In communication from 2003, Rouvière proved an extension of formula (4.2) to
symmetricspacesX ofrank(cid:7)=1.Inspiredbyhismethods,Iprovedtheinversion
formula (5.2) for the X-ray transform for X of rank (cid:7)>1. Then Rouvière [43]
extendedhisformulatoXofarbitraryrank(cid:7).Actuallyhehasseveralsuchformulas
buttheyarealldifferentfromtheformula(5.2)below.
Fix a flat totally geodesic submanifold E of X with dimE = (cid:7) > 1
((cid:7)therankofX)passingthroughtheorigino=eK ofX.Letp>0andS=S (o)
p
be the sphere in E with radius p and center o. The geodesics γ in E tangent to
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S are permuted transitively by the orthogonal group O(E). Let du and dk denote
thenormalizedHaarmeasuresonU andK.Thespacesk·E ask runsthroughK
constitute all flat totally geodesic subspaces of X through o of dimension (cid:7). Thus
theimagesk·γ (k∈K,γ tangenttoS)constitutetheset(cid:8) ofallgeodesicsγ inX
p
lyingin someflat (cid:7)-dimensionaltotally geodesic submanifoldof X through o and
d(o,γ)=p.Theset(cid:8) hasanaturalmeasureω givenbythefunctional
p p
(cid:8) (cid:9)
(cid:1) (cid:1)
(cid:5) (cid:6)
(5.1) ω :ϕ→ ϕ k(u·γ) du dk.
p
K O(E)
Theorem5.1. TheX-raytransform(2.1)onasymmetricspaceX=G/K ofrank
(cid:7)>1isinvertedbytheformula
(cid:8) (cid:9)
(cid:1)∞ (cid:1)
(5.2) f(o)=−1 d fˆ(γ)dω (γ) dp, f ∈D(X).
p
π dp p
0 (cid:8)p
Since(cid:8) anddω areK-invarianttheformulaholdsateachpointx byreplacing
p p
f byf ◦g whereg∈Gissuchthatg·o=x.
Proof. First assume f to be K-invariant and consider the restriction f|E. Fix
an orthonormal frame H ,H ∈E , the tangent space to E at o, consider the one
0 0
parameter subgroups exptH , exptH and the geodesic γ (t)=exptH ·o. Then
0 0 0
the geodesic γ(t)=exppH ·γ (t) lies in E and is tangent to S (o). Because of
o p
(3.4)wehave
(cid:1)∞
(cid:5) (cid:6)
(5.3) f(o)=−1 d fˆ E(o)dp,
π dp p p
0
wherethesuperscriptE standsforthedualtransformongeodesicsinthespaceE.
Thus
(cid:1) (cid:1)
(cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6)
(5.4) fˆ E(o)= fˆ (γ)dν(γ)= fˆ (u·γ)du,
p
γ⊂E O(E)
d(o,γ)=p
whereν standsfortheaverageoverthesetofgeodesicstangenttoS (o).
p
Forf ∈D(X)arbitraryweuse(5.2)onthefunction
(cid:1)
f(cid:11)(x)= f(k·x)dk.
K
Takingintoaccountthedefinition(5.1)theinversionformula(5.2)followsimme-
diately. (cid:1)
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Remark. Note that the measure ω in (5.1) is a kind of convolution of the Haar
p
measuresdk and du.Howeveritisnotastrictconvolutionsincetheproduct ku is
notdefined.
6. THE HOROCYCLE TRANSFORM IN X=G/K
Consider the usual Iwasawa decomposition of G, G=NAK where N and A are
nilpotent and abelian, respectively. A horocycle is by definition [12] an orbit in X
ofaconjugategNg−1 ofN.ThegroupGpermutesthehorocyclestransitivelyand
thespace(cid:2)ofhorocyclescanbewritten(cid:2)=G/MN whereM isthecentralizerof
AinK.Inthedoublefibration
G/M
(cid:1)(cid:1) (cid:2)
(cid:1) (cid:2)(cid:2)
X=G/K G/MN =(cid:2)
itturnsoutthatx=gK isincidenttoξ =γMN ifandonlyifx∈ξ.Thetransforms
(2.2)become
(cid:1) (cid:1)
(6.1) fˆ(γMN)= f(γn·o)dn, ϕˇ(gK)= ϕ(gk·ξ )dk,
0
N K
whereξ =N ·o.Whilethemap ϕ→ϕˇ hasabigkernel,thehorocycletransform
o
f →fˆisinjective(see[17]or[13]).Thefollowingresultfrom[21]isconsiderably
stronger.
Theorem6.1 (Supporttheorem). LetB beaclosedballinX.Then
fˆ(ξ)=0 forξ ∩B=∅ implies
f(x)=0 forx∈/B.
Hereonerequiresstrongerdecayconditionsonf thanfortheinjectivity.Adif-
ferentproofwasgivenin[14].WehavealsothefollowinginversionandPlancherel
formula for the Radon transform [18,19]. The pseudodifferential operator (cid:12) and
thedifferentialoperator (cid:1) belowareconstructedbymeansoftheHarish-Chandra
c-function, and w denotes the order of the Weyl group. For G complex a result
similarto(6.2)appearsin[12].
Theorem 6.2. For f ∈ D(X) or sufficiently rapidly decreasing we have the
inversionformula
(cid:5) (cid:6)
(6.2) f = 1 (cid:12)(cid:12)¯fˆ ∨.
w
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IfallCartansubgroupsofGareconjugatetheformulahastheimprovedversion
(cid:5)(cid:5) (cid:6) (cid:6)
f = 1(cid:1) fˆ ∨ .
w
ForGarbitrary
(cid:1) (cid:1)
(cid:10) (cid:10) (cid:10) (cid:10)
(6.3) w (cid:10)f(x)(cid:10)2dx= (cid:10)(cid:12)fˆ(cid:10)2(ξ)dξ,
X (cid:2)
withasuitablenormalizationoftheinvariantmeasuresdx anddξ.
The range question (iii) for f → fˆ is more complicated. Consider first the
hyperbolic plane H2 in the Poincaré unit disk model D. Here the horocycles are
the circles in the disk tangential to the boundary {eiθ: θ ∈R}. Let ξ denote the
t,θ
horocyclethrougheiθ withdistancet (withsign)fromtheorigin.Thenwehavethe
followingresultfrom[23].
Theorem6.3. TherangeD(D)(cid:11)consistsofthefunctionsψ∈D((cid:2))
(cid:12)
ψ(ξ )= ψ (t)einθ
t,θ n
n
where
(cid:3) (cid:4) (cid:3) (cid:4)
d d
(6.4) ψ (t)=e−t −1 ··· −2|n|+1 ϕ (t)
n n
dt dt
whereϕ ∈D(D)iseven.
n
This implies a relationship between ψ(ξt,θ) and ψ(ξ−t,θ). More specifically, if
f(cid:11)(t)=f(−t),(cid:15) =etψ then∗denotingconvolutiononR
n n
(cid:15)(cid:11) =S ∗(cid:15)
n n n
wherethedistributionS onRhasFouriertransform
n
Sˆ = (iλ+1)···(iλ+2|n|−1), λ∈R.
n (iλ−1)···(iλ−2|n|+1)
Thisrelationshipbetweenψ(ξ−t,θ)andψ(ξt,θ)impliesthatin(6.3)f →(cid:12)fˆdoes
notmapL2(X)ontoL2((cid:2)).
For the generalization of (6.4) to X=G/K we need some additional notation.
Let Kˆ betheunitarydualof K and d(δ) thedegreeofa δ∈Kˆ.Given δ actingon
V let
δ
(cid:13) (cid:14)
VM = v∈V : δ(m)v=vform∈M
δ δ
537
andput(cid:7)(δ)=dimVM.Let
δ
(cid:13) (cid:14)
Kˆ = δ∈Kˆ: (cid:7)(δ)>0 .
M
Inthefollowingtheoremfrom[25]theexpansion(6.5)isageneralizationof(6.4).
Putρ(H)= 1Trace(adH|n).
2
Theorem6.4. TherangeD(X)(cid:11)consistsofthefunctionsψ∈D((cid:2))
(cid:12) (cid:5) (cid:6)
(6.5) ψ(ka·ξ )= d(δ)Tr δ(k)(cid:15) (a)
0 δ
δ∈KˆM
(Tr = Trace) where (cid:15) is a function on A with values in Hom(V ,VM), i.e.,
δ δ δ
(cid:15) ∈D(A,Hom(V ,VM)),givenby
δ δ δ
(cid:5) (cid:6)
(6.6) (cid:15) (a)=e−ρ(loga)Qδ(D) (cid:19) (a) , a∈A,
δ a δ
where
(cid:5) (cid:5) (cid:6)(cid:6)
(6.7) (cid:19) ∈D A,Hom V ,VM
δ δ δ
is W-invariant and Qδ(D) is a certain (cid:7)(δ)×(cid:7)(δ) matrix of constant coefficient
differentialoperatorsonA.
From this result we can derive the following (unpublished) refinement of the
+
support theorem above. Let A be the Weyl chamber corresponding to the choice
ofthegroupN.
Theorem6.5. Supposef ∈D(X)satisfies
fˆ(ka·ξ )=0 fork∈K, a∈A+, |loga|>R.
0
Then
fˆ(ka·ξ )=0 fork∈K, |loga|>R, a∈A
0
sobyTheorem6.1
f(x)=0 ford(0,x)>R.
Proof. LetQ (D)bethematrixofcofactorsofQδ(D)sothat
c
(6.8) Q (D)Qδ(D)=detQδ(D)I.
c
Then(6.6)implies
(6.9) Q (D)(eρ(cid:15) )=detQδ(D)(cid:19) .
c δ δ
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Nowitisknown[34],[25,pp.267,348]thatdetQδ(D)isaproductoflinearfactors
δ(H )+cwhereH ∈aand∂(H )thecorrespondingdirectionalderivative.
i i i
Supposethefunctionψ=fˆsatisfies
ψ(ka·ξ )=0 fork∈K, a∈A+, |loga|>R.
0
Since
(cid:1)
(cid:15) (a)= ψ(ka·ξ )δ(k−1)dk
δ 0
K
wededucefrom(6.8)and(6.9)that
detQδ(D)(cid:19) (a)=0 fora∈A+, |loga|>R.
δ
+
Consider this equation on a ray in A starting at e. Because of the mentioned
factorization of detQδ(D) we deduce that on this ray (cid:19) satisfies an ordinary
δ
differential equation on the interval (R,∞). Having compact support we deduce
that(cid:19) (a)=0fora∈A+,|loga|>R.ByitsWeylgroupinvarianceitvanishesfor
δ
alla∈A,|loga|>R whichby(6.5)provesthetheorem. (cid:1)
ConsiderthecaserankX=1.LetB (o)beaballinXwithradiusR andcenter
R
0.FixaunitvectorH intheLiealgebraofAsuchthatexpH ∈A+.Puta =exptH.
(cid:15) t
TheinteriorofthehorocyclekNa ·oistheunion kNa ·o.Ahorocycleξ is
t τ>t τ
saidtobeexternaltoB (o)ifitsinteriorisdisjointfromB (o);ξ issaidtoenclose
R R
B (o)ifitsinteriorcontainsB (o).
R R
Corollary6.6. LetXhaverankoneandB (o)asabove.Letf ∈D(X).Thenthe
R
followingareequivalent:
(i) fˆ(ξ)=0wheneverξ isexternaltoB (o).
R
(ii) fˆ(ξ)=0wheneverξ enclosesB (o).
R
(iii) f ≡0outsideB (o).
R
ForhyperbolicspacethisisclearfromTheorem6.3andwasprovedinadifferent
waybyLaxandPhillips[35].
Problem(iii)forthedualtransformϕ→ϕˇ hasasatisfactoryanswer(see[25,IV
§§2 and 4]). The kernel can be described in the spirit of Theorem 6.5 and for the
rangeonehasthesurjectivity
(6.10) E((cid:2))∨=E(X).
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7. THE ABEL TRANSFORM
LetD (X)denotethespaceofK-invariantfunctionsinD(X).TheAbeltransform
K
f →Af isdefinedby
(cid:1)
(7.1) (Af)(a)=eρ(loga) f(an·o)dn, a∈A, f ∈D (X).
K
N
Except for the factor eρ it is the restriction of the Radon transform to K-invariant
functions
(7.2) Af =eρfˆ.
Someofitspropertiesarebestanalyzedbymeansofthesphericalfunctions
(cid:1)
(7.3) ϕ (g)= e(iλ−ρ)(H(gk))dk, g∈G, λ∈a∗,
λ c
K
whereH(g)∈aisdeterminedbyg∈kexpH(g)N anda∗ isthecomplexdualofa.
c
Thesphericaltransform
(cid:1)
(7.4) f˜(λ)= f(x)ϕ−λ(x)dx, f ∈DK(X)
X
(whereϕ (g·o)=ϕ (g))isahomomorphismrelativetoconvolution×onX:
λ λ
(7.5) (f ×f )∼(λ)=f˜(λ)f˜(λ).
1 2 1 2
Asprovedin[15],AintertwinesthesphericaltransformandtheEuclideanFourier
transformF →F∗ onAso
(cid:1) (cid:1)
(7.6) (Af)(a)e−iλ(loga)da= ϕ−λ(x)f(x)dx, (Af)∗=f˜.
A X
ThusAf isW-invariantandby(7.5)
(7.7) A(f ×f )=Af ∗Af ,
1 2 1 2
where∗isconvolutiononA.LetD(X)denotethealgebraofG-invariantdifferential
operators on X and (cid:8):D(X) → D (A) the isomorphism onto the W-invariant
W
constantcoefficientdifferentialoperatorsonA.
TheAbeltransformisasimultaneoustransmutationoperatorbetweenD(X)and
D (A),i.e.,
W
(7.8) ADf =(cid:8)(D)Af, D∈D(X), f ∈D (X)
K
as shown in [29] which for example can be used to prove that each D has
a fundamental solution. By the Paley–Wiener theorem for (7.3) one has that
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