Non Euclidean Analysis

8/8/2019 Non Euclidean Analysis

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NON-EUCLIDEAN ANALYSISSigurdur HelgasonDepartment of Mathematics,Massachusetts Institute of TechnologyCambridge, MA [email protected]

1. The Non-Euclidean PlaneIn case the work of Bolyai [Bo] and Lobatschevsky [Lo] left any doubts

about the existence of non-Euclidean geometry these doubts were re-moved by the work [Be] of Beltrami. With a modification made possibleby hindsight one can state the following result.Theorem 1.1. Given a simply connected region D c R~ D # R ~ )there exis ts a Riemannian metr ic g o n D which is invariant u nde r allconformal transformations of D . Also g is unique u p t o a constant fac-tor .

Because of the Riemann mapping theorem we can assume D to be thea - zunit disk. Given a E D the mapping cp : z -+- s conformal and1- a zp(a) = 0. The invariance of g requires

for each u E D, (the tangent space to D a t a ) . Since go is invariantunder rotations around 0,

where c is a constant. Here Do is identified with C . Let t -+ z ( t ) be acurve with z(0) = a , z'(0) = u E C . Then dcp(u) is the tangent vector


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368 NON-EUCLIDEAN GEOMETRIESThus g is the Riemannian structure

and the proof shows that it is indeed invariant.We shall now take D as the unit disk lz l < 1 with g = ds2 given by(1 .3) with c = 1. In our analysis on D we are mainly interested in thegeodesics in D (the arcs orthogonal to the boundary B = {z E C : lz l =1 ) ) and the horocycles in D which are the circles inside D tangential toB. Note that a horocycle tangential to B at b is orthogonal to all thegeodesics in D which end at b.2. The Non-Euclidean Fourier Transform

We first recall some of the principal results from Fourier analysis onRn. The Fourier transform f t or Rn is defined by

where ( , ) denotes the scalar product and d x the Lebesgue measure. Inpolar coordinates u= Xw X E R, w E Sn-I we can write

It is then inverted by

say for f E D ( R n ) = C F ( R n ) ,dw denoting the surface element on thesphere S n - l . The Plancherel formula

expresses that f t f is an isometry of L ~ ( R ~ )nto L ~ ( R +sn-1 , ~ T ) - ~ X ~ - 'X dw)

The range of the mapping f ( x )t ( X w ) as f runs through D ( R n ) isexpressed in the following theorem [He7]. A vector a = ( a l , . . ,a,) E C nis said to be isotropic if ( a ,a ) = a: + . + a; = 0 .Theorem 2.1. Th e Fourier transform f ( x ) -+ f ( X w ) m aps D ( R n ) ontothe set of functions f(Xw) = y ( X , w ) E C m ( R x Sn-l) satisfying:


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Non-Euclidean Analysis 369(i) There exists a constant A > 0 suc h that for each w E Sn-' thefunction X + y(X, w) extends to a holomorphic function o n C

wit h t he property

for each N E Z . (lm X = ima gina ry part of A).(iz) For each k E Z+ and each isotropic vector a E Cn the function

is even and holomorphic in C .Condition (2.5) expresses that the function X -, y(X, w) is of uni-form exponential type: The classical Paley-Wiener theorem states thatD(Rn)" consists of entire functions of exponential type in n variableswhereas in the description above only X enters.Formula (2.2) motivates a Fourier transform definition on D. The in-

ner product ( x ,w) equals the (signed) distance from 0 to the hyperplanethrough x with normal w. A horocycle in D through b is perpendicularto the (parallel) family of geodesics ending at b so is an analog of a hy-perplane in Rn. Thus if z E D, b E B we define ( z ,b) as the (signed)distance from 0 to the horocycle through x and b. The Fourier transformf -t j on D is thus defined by

for all b E B and X E C for which the integral converges. Here dz is theinvariant surface element on D,

The +1 term in (2.7) is included for later technical convenience.The Fourier transform (2.7) is a special case of the Fourier transformon a symmetric space X = G / K of the non-compact type, introducedin [He3]. Here G is a semisimple connected Lie group with finite centerand K is a maximal compact subgroup. In discussing the propertiesof f - below we stick to the case X = D for notational simplicitybut shall indicate (with references) the appropriate generalizations toarbitrary X. Some of the results require a rank restriction on X.


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370 NON-EUCLIDEAN GEOMETRIESTheorem 2.2. The transform f -t j in (2.7) is inverted by

f(z) = &1 1 (A, b)e(""+l)(z*b)~h ( y ) Xdb, f E V ( D ) .R B (2.9)Also the map f 4 j extends to an isometry of L ~ ( D ) nto L ~ ( R + B, p )where p is the measureand db is normalized by J db = 1.

This result is valid for arbitrary X = G I K ([He3, He4]), suitably for-mulated in terms of the fine structure of G. While this result resembles(2.3)-(2.4) closely the range theorem for D takes a rather different form.Theorem 2.3. The range V(D)" consists of the functions cp(X, b) which(in A) are holomorphic of uniform exponential type and satisfy the func-tional equation

One can prove that condition (2.11) is equivalent to the followingconditions (2.12) for the Fourier coefficients cpk(X) of cp,

c~k(-X)~k(-iX) = ~k(X)pk(iX)where pk (x) is the polynomial

Again these results are valid for arbitrary X = G/K ([He5, He71).The Paley-Wiener type theorems can be extended to the Schwartzspaces SP(D) (0 < p 5 2 ) . Roughly speaking, f belongs to SP(D) if eachinvariant derivative Df belongs to LP(D), more precisely, it is rapidlydecreasing in the distance from 0 even after multiplication by the pthroot of the volume element. Let Spdenote the strip I Im XI < : 1 in Cand S(Sp B) the space of smooth functions on S(Spx B) holomorphic(in A) in Spand rapidly decreasing (uniformly for b E B) on each lineX = [ + i n (171


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Non-Euclidean Analysis 371Theorem 2.4. The Fourier transform f + f o n D is a bijection ofSP(D) onto the set of cp E S(Sp B) satisfying (2.11).

The theorem holds for all X = G/K (Eguchi [Eg]). The proof iscomplicated. For the case of K-invariant functions (done for p = 2 byHarish-Chandra [HI and Trombi-Varadarajan [TV] for general p) a sub-stantial simplification was done by Anker [A]. A further range theoremfor the space of functions for which each invariant derivative has arbi-trary exponential decay was proved by Oshima, Saburi and Wakayama[OSW]. See also Barker [Bar, p. 271 for the operator Fourier transformof the intersection of all the Schwartz spaces on SL(2, R) .In classical Fourier analysis on Rn the Riemann-Lebesgue lemmastates that for f E L'(R), f tends to 0 at m. For D the situationis a bit different.

Theorem 2.5. Let f E L'(D). T h en there exists a null set N i n B suchthat if b E B - N, X -+ f (X , b) is holomorphic in the strip I Im XI < 1and

uniform ly for 1771 5 1.The proof [HRSS] is valid even for symmetric spaces X = G/K ofarbitrary rank. Moreover

uniformly in the strip I Im XI 5 1,and this extends to f E LJ' (1 5 p < 2)this time in the strip I Im A ( < - 1 ([SS]). In particular, if f E LP(D)then there is a null set N in B such that !(A, b) exists for b $ N and allX in the strip I Im XI < % - 1.The classical inversion formula for the Fourier transform on Rn nowextends to f E LP(D) (1 5 p < 2) as follows.Theorem 2.6. Let f E Lp(D) and assume f E L1(R x B, p) (with p asi n (2.10)). T he n the inversion formula (2.9) holds for almost all x E D(the Lebesgue set for f).

Again this holds for all X = GIK. A result of this type was provedby Stanton and Thomas [ST] without invoking f explicitly (since theexistence had not been established). The version in Theorem 2.6 is fromPSIIn Schwartz's theory of mean-periodic functions [Sc] it is proved thatany closed translation-invariant subspace of Cm(R) contains an expo-nential epx. The analogous question here would be:


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372 NO N- EU CL I D E A N G EO M E T RI E SDoes a n arbitrary closed inva rian t subspace of C m ( D ) contain an ex-ponential

e,,b(z) = e'('lb) (2.15)for some p E C and some b E B ?Here the topology of C o o ( D ) s the usual Frhchet space topology and

"invariant" refers to the action of the group G = S U ( 1 , l ) on D . Theanswer is yes.Theorem 2.7. Each closed invariant subspace E of C m ( D ) contains anexponential ep,b.

This was proved in [HS] for all symmetric X = G/K of rank 1. Hereis a sketch of the proof. By a result of Bagchi and Sitaram [BS], Econtains a spherical function

For either X or -A it is true ([Heg, Lemma 2.3, Ch. 1111) that the Poissontransform PA : F + where

maps L ~ ( B )nto the closed invariant subspace of E generated by y x .On the other hand it is proved in [He9, Ex. B1 in Ch. 1111 that eiA+l,b isa series of terms P x ( F n ) where Fn E L ~ ( B )nd the series converges inthe topology of C m ( D ) . Thus eiA+l,b E E as desired.

The following result for the Fourier transform on Rn is closely relatedto the Wiener Tauberian theorem.Let f E L ' ( R ~ ) be such that f ( u ) # 0 for all u E Rn. T h e n t hetranslates of f span a dense subspace of L1( R n ) ,

There has been considerable activity in establishing analogs of thistheorem for semisimple Lie groups and symmetric spaces. See e.g. [EM,Sa, Sil , Si2]. The neatest version for D seems to me to be the followingresult from [SS, MRSS] which remains valid for X = G/K of rank 1.

Let d ( z ,w) denote the distance in D and if E > 0 , let L , ( D ) denote thespace of measurable functions f on D such that If ( z ) e 'd(OJ) z < GO.Let T, denote the strip I Im X I 5 1 + E .Theorem 2.8. Let f E L, (X) and as sume f is not almost everywhereequal to a ny real analytic fun ction . Let

z= X E T , : ?(A,.) r ) .


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Non-Euclidean Analysis 373If Z = 0 then the translates o f f span a dense subspace of L 1 ( D ) .

A theorem of Hardy's on Fourier transforms on Rn asserts in a precisefashion that f and its Fourier transform cannot both vanish too fast atinfinity. More precisely ([Ha])

AssumeI f ( x ) < ~ e - " l ~ l ' ~ f ( u )< ~ - ~ ~ I ' ,

where A , B , a and ,tl are positive constants and a@> $. Then f = 0.Variations of this theorem for L p spaces have been proved by Morgan

[MI and Cowling-Price [CP]For the Fourier transform on D the following result holds.

Theorem 2.9. Let f be a measurable function on D satisfying

where C, a , ,tl are positive constants. If a@> 16 then f - 0 .This is contained in Sitaram and Sundari [SiSu, 51 where an exten-

sion to certain symmetric spaces X = G / K is also proved. The theoremfor all such X was obtained by Sengupta [Se], together with refinementsin terms of L P ( X ) .Many such completions of Hardy's theorem have been given, see [RS,CSS, NR, Shi].3. Eigenfunctions of the Laplacian

Consider first the plane R2 and the Laplacian

Given a unit vector w E R2 and X E C the function x -+ ei'("yw) is aneigenfunction

0 i X ( x , w ) = - ~ 2 ~ i A ( x , w )L& (3 .1)Because of (2.3) one might expect all eigenfunctions of L to be a "de-composition" into such eigenfunctions with fixed X but variable w .

Note that the function w -+ ei'(xlw) is the restriction to S 1 of theholomorphic function

z - xp [ $ i X ) x l z + z - ' ) + ;Xxz (z - -')I z E C - ( 0 )which satisfies a conditionsup ( l f ( z ) e - a ~ Z ~ - b ~ z ~ - l )cm

Z


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374 NON-EUCLIDEAN GEOMETRIESwith a, b > 0. Let Ea,bdenote the Banach space of holomorphic functionssatisfying (3.2), the norm being the expression in (3.2). We let E denotethe union of the spaces Ea,band give it the induced topology. We identifythe elements of E with their restrictions to S1 and call the members ofthe dual space E' entire functionals.Theorem 3.1. ([He6]) The eigenfunctions of Lo on R2 are preciselythe harmonic functions and the functions

where X E C - 0) and T is an entire functional on S'.For the non-Euclidean metric (1.3) (with c = 1) the Laplacian is given

bv

and the exponential function epIb (z) ep('tb) is an eigenfunction:

In particular, the function z -+ e2('lb) s a harmonic function and in factcoincides with the classical Poisson kernel from potential theory:

Again the eigenfunctions of L are obtained from the functions ep,bbysuperposition. To describe this precisely consider the space A(B) ofanalytic functions on B. Each F E A(B) extends to a holomorphicfunction on a belt B, : 1- < lz l < 1+ E around B. The space 'H(B,)of holomorphic functions on B, is topologized by uniform convergenceon compact subsets. We can view A(B) as the union U?'H(B~,,) andgive it the inductive limit topology. The dual space Ar(B) then consistsof the analytic functionals on B (or hyperfunctions in B).Theorem 3.2. ([Hed,IV, $11) The eigenfunctions of L are precisely thefunctions

u(z) = ep('yb)dT (b) ,JBwhere p E C and T E Ar(B).

Lewis in [L] has proved (under minor restriction on p) that T in (3.7)is a distribution if and only if u has at most an exponential growth (in


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Non-Euclidean Analysis 375d(0, x)). On the other hand, Ban and Schlichtkrull proved in [Bas] thatT E Cm(B) if and only if all the invariant derivatives of u have the sameexponential growth.

We consider now the natural group representations on the eigenspaces.The group M(2) of isometries of R2 acts transitively on R~ and leavesthe Laplacian Lo invariant: LO(fo T) = (Lo ) o T for each T E M(2) . IfX E C the eigenspace

is invariant under the action f + f o T - ~ O we have a representation TAof M(2) on Ex given by TA(7)(f)= f OT-', the eigenspace representation.Theorem 3.3. ( [He6]) The representation Tx is im-educible if and onlyif x # 0.

Similarly the group G = SU(1 , l ) of conformal transformations

leaves (1.3) and the operator L in (3.4) invariant. Thus we get again aneigenspace representation TA of G on each eigenspace

Theorem 3.4. ( [Hed]) The representation TA is irreducible if and onlyif i X + 1 $2 22.

Again all these results extend to Euclidean spaces of higher dimensionsand suitably formulated, to all symmetric spaces GIK of the noncompacttype.4. The Radon TransformA. The Euclidean Case

Let d be a fixed integer, 0 < d < n and let G(d,n) denote the spaceof d-dimensional planes in Rn. To a function f on Rn we associate afunction f^ on G(d, n) by

dm being the Euclidean measure on


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376 NO N- EU CL I D E A N G EO M E T RI E SThe inversion of the transform f --t f^ is well known (case d = n - 1

in [R, J, GS], general d in [F, Hel, He21). We shall give another group-theoretic method here, resulting in alternative inversion formulas.The group G = M(n) acts transitively both on Rn and on G(d,n).In particular, Rn = G / K where K = O(n). Let p > 0. Consider apair x E Rn, J E G(d, n) at distance p = d(x,E). Let g E G be suchthat g 0 = x. Then the family kg-' J constitutes the set of elementsin G(d , n) at distance p from 0. Along with the transform f + f^ weconsider the dual transform cp -t $ given by

the average of cp over the set of d-planes passing through x. More gen-erally we put

the average of cp over the set of d-planes at distance p from x. Since Kacts transitively on the set of d-planes through 0 we see by the abovethat

(4.4)dk being the normalized Haar measure on K . Let (Mr )( x) denote themean-value of f over the sphere S,(x) of radius r with center x. Ifx E Rn has distance r from 0 we then have

We thus see that since d(0,g-l . y) = d(x, y),

Let xo be the point in E at minimum distance p from x. The integrand( M ~ ( ~ ~ Y )(x)) is constant in y on each sphere in J with center xo. I tfollows that

(4.6)where r = d(xo,y), q = d(x, y), f ld denoting the area of the unit sphere inR ~ . e have q2 = p2 + r2 so putting F(q ) = (Mq f)(x) , F ( ~ ) (f^):(x)


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Non-Euclidean Analysis 377we have

2 d / 2 - 1F ( P ) = a d lm!9 ) (q2- ) dq - (4.7)This Abel-type integral equation has an inversion

where c ( d ) is a constant, depending only on d. Putting r = 0 we obtainthe inversion formula

Note that in (4.8)

so in (4 .8) we can use integration by parts and the integral becomes

d 1 dA P P ~ Y ~ ~ E(r2)= 5 i ~o this integral reduces the exponent d / 2 by\ ,1. For d odd we continue the differentiation times until the expo-

nent is -+. For d even we continue until the exponent is 0 and thenreplace J ~ ' ( p )p by - F ( r ) . This ~ ( r )s an even function so taking( d / d ( ~ ~ ) ) ~ / ~t r = 0 amounts to taking a constant multiple of ( d / d ~ ) ~at r = 0. We thus get the following refinement of (4 .9) where we recallthat (f): ( x ) s the average of the integrals of f over the d-planes tangentto S , ( x ) .Theorem 4.1. Th e d-plane tran sform is inverted as follows:

(2) If d is even then

(ii) If d is odd then


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378 NON-EUCLIDEAN GEOMETRIES(zzz) If d = 1 then

f (x) = --JW.!(!),v(x) dp.7i - 0 P ~ PFor n = 2 formula (4.12) is proved in Radon's original paper [R]. Notethat the constant - l /n is the same for all n. In the case d = n - 1 theformula in coincides with formula (21) in Rouvikre [Ro].Another inversion formula ([Hel, He21) valid for all d and n is

f = c(-L)~'~((~")")where

Here the fractional power of L is defined in the usual way by the Fouriertransform. The parity of d shows up in the same way as in Theorem 4.1.For range questions for the transform f + f^ see an account in [HelO]and references there.B. The Hyperbolic Case

The hyperbolic space Hn s the higher-dimensional version of (1.3)and its Riemannian structure is given by

in the unit ball 1x1 < 1. The constant 4 is chosen such tha t the curvatureis now -1. The d-dimensional totally geodesic submanifolds are spher-ical caps perpendicular to the boundary B : 1x1 = 1. They are naturalanalogs of the d-planes in Rn. e have accordingly a Radon transformf + f , where!(F) = J f (x ) d m b ) F E E, (4.15)

Ewhere E s the space of d-dimensional totally geodesic submanifolds ofHn.The group G of isometries of Hn cts transitively on E s well. As in(4.2)-(4.3) we consider the dual transform


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Non-Euclidean Analysis 379and more generally for p > 0 ,

the mean value of cp over the set of J E E t distance p from x. Theformula (f1 (4= ( ~ ~ ( ~ ~ ~ ) f(x) dm(y)h (4.18)is proved just as before. Let xo be the point in J at minimum distancep from x and put r = d(xo,y), q = d(x, y). Since the geodesic triangle(xxoy) is right angled at xo we have by the cosine rule

cosh q = coshp cosh r . (4.19)Also note that since ,$ is totally geodesic, distances between two pointsin J are the same as in Hn. In particular (Md("J) f) (x) is constant as yvaries on a sphere in J with center xo. Therefore (4.18) implies

(f): (x) = ~d L m ( M q ) x) sinhd-I r drFor x fixed we put

F cosh q) = (Mq ) x) , F (cosh p) = (f (x) ,substitute in (4.20) and use (4.19). Writing t = coshp, s = coshr weobtain the integral equation

Putting here u = ts, ds = t-I du we get the Abel-type integral equa-tiont d - ~ ( t ) R~lm- 1 ~ ( u ) ( u 2 - 2)d/2IU du ,

which by (4.8) is inverted by

Here we put r = 1 and s(p) = cosh-lp. We then obtain the followingvariation of Theorem 3.12, Ch. I in [Heg]:


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380 NON-EUCLIDEAN GEOMETRIESTheorem 4.2. The transform f + f" is inverted by

f (x) = C [(A)i " ( t 2 - 2ldI2 Itd (4.23)r= l .As in the proof of Theorem 4.1 we can obtain the following improve-ment.

Theorem 4.3.(i) If d is even the inversion can be written

(ii) If d = 1 then

Proof. Par t (i) is proved as (4.10) except that we no longer can equate( d / d ( ~ ~ ) ) ~ / ~ith ( d / d ~ ) ~t r = 1.For (ii) we deduce from (4.22) since t(t2- 2)-lI2 = $(t2 - r2)lI2that

Putting again t = coshp, dt = sinhp dp our expression becomesw l d- ((!)pV) (4 p.sinhp dp

Remark. For n = 2, d = 1 formula (4.25) is stated in Radon [R, Part C].The proof (which is only indicated) is very elegant but would not workfor n > 2.For d even (4.24) can be written in a simpler form ([Hell) namely


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Non-Euclidean Analysis 381r(q)where c = (-ln)d,2r(q) and Q d is the polynomial

The case d = 1, n = 2 is that of the X-ray transform on the non-Euclidean disk ((4.15) for n = 2). Here are two further alternatives tothe inversion formula (4.25). Let S denote the integral operator

Then L S ( ~ ) "= -4r2 f , f E D ( X ) (4.28)This is proved by Berenstein-Casadio [BC]; see [HelO] for a minor sim-plification. By invariance it suffices to prove (4.28) for f radial and thenit is verified by taking the spherical transform on both sides. Less ex-plicit versions of (4.28) are obtained in [BC] for any dimension n andd.

One more inversion formula was obtained by Lissianoi and Ponomarev[LP] using (4.23) for d = 1,n = 2 as a starting point. By parameterizingthe geodesics y by the two points of intersection of y with B they provea hyperbolic analog of the Euclidean formula:

which is an alternative to (4.12). Here Y p s a normalized Hilbert trans-form in the variable p and f (w,p) is f ( ~ )or the line (x, w) = p, whereIwI = 1.In the theorems in this section we have not discussed smoothness anddecay at infinity of the functions. Here we refer to [Je, Rul, Ru2, BeR1,BeR21 as examples.Additional inversion formulas for the transform f -t f can be foundin [Sem, Ru3, K]. The range problem for the transform f -t f is treatedin [BCK, I].Bibliography[A] J. Anker, The spherical Fourier transform of rapidly decreasing functions-a simple proof of a characterization due to Harish-Chandra, Helgason,

Trombi and Varadarajan. J. F'unct. Anal. 96 (1991), 331-349.[BS] S.C. Bagchi and A. Sitaram, Spherical mean-periodic functions on semisim-ple Lie groups. Pac. J. Math. 84 (1979), 241-250.


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