on fundamental differential operators and the p-plane radon transform

Ann Global Anal. Geom.Vol. 8, No. 1(1990), 61-75

On fundamental differential operatorsand the p-plane Radon transform

F. RICHTER

1. Introduction

Let us consider the Euclidean space E" and the manifold E, vp of all p-dimensional planesin E" to be homogeneous spaces of the Euclidean group E(n), actually,

E" = E(n)/O(n), En,p = E(n)/E(p)x O(n - p).

Then the Lie algebra e(n) of E(n) acts on both E" and E, , by fundamental vector fields.This action defines two mappings 2, A of the complexified universal enveloping algebraof e(n) (designed by U(n)) into the algebras of differential operators on E", E,p by thefollowing equations:

A(Y)f(x)= - f(exp - tY x)= odt It=,

A(Yx ... Yl):= (Y1) ... 2(Yr),

where Y, Y1, ... , Y e (n);fE C (E") and (Y1 ... Y,) e 11(n). The mapping A is analogouslydefined.

It is the main tool of this paper to describe the kernel of 2 explicitely (Theorem A,section 2 of this paper).

Furthermore, we apply Theorem A to investigate the range of the p-plane Radontransform of rapidly decreasing functions f on E" which is defined by

(pf) () .= f(x)dr(x), (1)

where 5 e E, p,fis rapidly decreasing and d5(x) denotes the induced Euclidean measureon .

Since Mp commutes with the action of E(n), we obtain

p A(_P) = (P) t for all P U(n).Hence

A(P) (pf) = 0 for all P E Ker2, (2)

which gives the necessary condition that a function qp on E., belongs to the range of P.By the explicite description of Ker the system (2) reduces to a finite system of second

order differential equations which turns out to be also sufficient for p < n - 1 (TheoremB, Section 4 of this paper).

62 F. Richter

It has been known for a long time that the range of the p-plane Radon transform canbe characterized by second order differential equations written in local coordinates (see[2, 3, 4]). That these equations arise from fundamental differential operators seems notto have been investigated in detail by now. Furthermore, the range of the Radon transformcan be characterized as the image of certain projection operators introduced by Weinstein.On the other hand, the range of such a projection operator can always be described bya system of peudodifferential equations (see [10] for detailed information).

Note that Ker i c Ker A for the classical Radon transform (p = n - 1). Here therange of ,n- 1 is described by a so-called moment condition (Helgason [5]).

2. On the characterisation of Ker A

The Lie algebra e(n) is a semi-direct product of the abelian Lie algebra R" and theorthogonal Lie algebra o(n):

e(n) = R" E o(n). (3)

In what follows we suppose that n > 3. We choosea basis (Z1 , ..., ZX 1 2 , X...,_- ,)of e(n) by defining

a) (Zi)i=,= 1... is the standard basis of R" and

b) (Xij)i< j is a basis of o(n) (as the set of skew symmetric matrices).

In U(n) we define now the following elements:

ijl:= ZiXj - ZXIi + ZjXj, I i < j < I _ n,

U Xilm = XijXlm - XilXjm + XimXjl, 1 i < j < I < m _ n.

Let 3(n) c U(n) be the left ideal generated by the elements iji and Uijm,. To avoid adetailed, lengthy distinction of several cases we agree upon Xij = -Xi if i > j aidsubordinate the following formulas to this agreement.

From the commutator relations

[Zi, Xj] = 6ijZ - ilZ j, [Zi, Z] = 0,

[Xij, XIm] = -ilXjm + jiXim + imXjl - 6jX i

we obtain in U(n):

-ijZm = Zm_-iji,

YijlXm., = XmVijl + imjlr - irJfjlm -b jmVitr + bjrVilm + bimijr - Lryijm,

UijlmZ r = ZrUijlrn - 6iryjlm + bjrilm - birk + 6mriJl, (5)

UijlnXrs = XrsUijlm + irtUsjlm - isUrm + 65jrUislm

- 6jsUirim + 61rUijs - 61siUijrm + 6mrnUijls - msUijir

for all possible indices.

On fundamental differential operators and the p-plane Radon transform

The formulae (5) prove

Corollary 1. 3(n) is a two-sided ideal. [

Our main result is

Theorem A. Ker = 3(n) (n 3).

Note that is injective for n = 1 and n = 2.

3. Proof of Theorem A

Let us denote by U(R") and U(o(n)) the complexified universal enveloping algebras ofR" and o (n). These Lie algebras are filtered by the Lie subalgebras Ul(R") and Um(o(n)):

U(R") U Um(R"), U(o(n)) = U Um(O(n)).m=0 m=O

By the Poincar6-Birkhoff-Witt Theorem the elements X22 ... X 4x,,Zi ... Zn form abasis of U(n) if the exponents i 2, ... , in- 1 ,, il, ..., i run through all non-negative integers.Hence, we have

U(n) = span (U(o(n)) U· (R")).

For each integer M > 0 we define

UU(n) = span (UM(o(n)) ·l(R')). (6)

Definition 1. An element P e lI(n) has X-degree M, iff P E UM(n) and there is no integerM' < M with P E UM (n).

The group E(n) acts on U(n) by the adjoint representation:

(Ad(g) P) (Z1,..., Z, Xl2 ... Xn,- ,n) := P (Ad(g) Z, .. , Ad(g) X, -,.),

P lU(n), g E(n).

For P l(n) we define

Zi(P):= Ad (exp Z i) P - P, i = 1, ... , n,

and Zl ... Zn(P) for all integers i. ... i, 0 by iteration. This notion is justified bythe equality

Zi(Zj(P)) = Zj(Zi(P)) for all i,j a {1 ... n}.

Corollary 2. If P E UM(n), then Zi(P)E UM- l(n).

Proof. This is an easy consequence of the identities

Ad (exp Z) Zj = Zj, Ad (exp Z) Xjl = Xi, + 5bZt - 6 iZji. (7)

63

64 F. Richter

Definition 2. Let M > 0.

a) RM(n) = {P E UM(n) I P has X-degree M and for alli..., i. > 0 with i + ... + i = Mit holds Z' ... Zn(P) = 0},

b) R(n)= U RM(n) u {0}.M=1

Proposition 3. Ker A = (n).

Proof. An easy computation yields

(z) = , (Xi)= xi - - xj a, (8)ax, ax. i

hence, the restriction of i to U°(n) = U(R") is injective. Assume, P E U(n) has X-degreegreater than zero (P e UM(n) for some M > 0) and satisfies the condition

(P) = A (Ad (exp Y) P) for all Y R. (9)

We represent P according to (6) by

P = cP, = CJ, XP= i fZil ... Z CJ E C . (10)

For arbitraryfe C'(E") we compute now

iA(PJ) f(x) = - f(xj(O) + AJ(' (11)

with

t = (tl, .... t, t2, ., t-," ),

XJ) ,= (exp -tZ,)' ... (exp -tlZ,)" E R",

AJ(O := (exp -t n -,nX_ ,n)in- . ... (exp -tl2Xl2)i l2 E O(n),

a alJi

at atl ... atil_,n

Since exp: R" - R is the identity map, we choose exp Y = x and obtain from (9), (10),(11) and the relation exp (Ad (g) Z) = g(exp Z) g- 1 for all g E E(n), Z E e(n):

A (Ad (exp Y) P) f(x) = f(x + xj( + Ajd )O t=o

= -f( + xJ0) (12)

= 0.


Since P E U UM(n), the operator - contains a derivation with respect to some tij.M=1 at

That is why the last identity in (12) holds true. Because of (9) we now have (_P) f(x) = 0for all x E E" and allf E C (E"). This implies P E Ker 2. Thus we have proved the following.

Lemma 4. If P has X-degree greater than zero and satisfies

2(P) = 1(Ad (exp Y) P) for all Y R,

then P Ker A.

Using Lemma 4 we continue to prove Proposition 3. At first we show that P hasX-degree greater than zero if

PEKerA and P$0.

Assume, that P E Ker A, P * 0 and P = P + P2 with P, E U(R") and P2 E U UM(n).Since 2(Ad (g) P) = g(2(P)) for all g E E(n) we conclude M=1

o = l(Ad (exp Y) P) = (P1 ) + (Ad (exp Y) P2) for all Y R" .

Since A(P1) + (P2) = 0 we obtain

-(P 2) = (P1 ) = -(Ad (exp Y) P2

for all YE R. This implies P2 e Ker 2 by Lemma 4. Thus, P, E Ker . But 2 lU(RA) isinjective which yields

Ker \ {0} c U l(n).M=1

Let P Ker 2 have X-degree M. Since then Ad (g) P Ker i for all g E E(n), we deducefrom Corollary 2:

Zi1... Zn(P) E (U(Rn) n Ker ) = {0} for i + ... + in = M .

Therefore, Ker c R(n).Now let P E SRM(n). The condition Z 1 ... Zn(P) = 0 for all i + ... + i, = M supplies

A(Ad (exp Y) (ZI' ... Zj"(P))) = A(Z' ... Z(P))

for all Y R and all j + ... + j, = M - 1.Lemma 4 provides Zil ... ZJn(P) e Ker A for all j, + ... + i, = M - 1. Successively,

we conclude that P e Ker A.

Remark. Proposition 3 can be obtained from [1], Corollary 3.6. Note that, by Proposition3, R(n) is a two-sided ideal and a vector space which is not obvious from the definitionof A(n).

The formulas (8) imply immediately that 3(n) c Ker A. It remains to show R(n) c 3(n).

5 Annals Bd. 8, Heft 1 (1990)

65

66 F. Richter

In order to prove this we should overcome some difficulties arising from thenon-commutativity of the multiplication in U(n).

We consider the complexified symmetric algebra 3Y(n) over the vector space e(n).According to (4) we define elements Uijlm, Vj, in 9'(n) and an ideal f(n) generated bythem. Furthermore, we introduce the subspaces YM(n) analogously to (6).

Now we define the X-degree of an element P e 5(n) and the subspaces ,YM(n), *'(n)according to the Definitions 1, 2. The main step for showing (n) c 3(n) is.

Proposition 5. It holds in 9(n) that /(n) = ~(n).

We continue to prove Theorem A and will add the proof of Proposition 5 afterwards.Let 0: 9(n) - U(n) be the so-called symmetrisation map (cf. Helgason [6]):

INY, ... .1 ), (13)r!. s,

where S, denotes the symmetric group and Y, ... , Y E e (n). This mapping d has threeimportant properties:

a) It is a linear isomorphism (n) - U(n).

b) It commutes with Ad (g) for each g E E(n).

c) It preserves the filtrations of 92 (n) and 11(n).

These properties supply

Corollary 6. (.X(n)) = R(n) is a linear isomorphism and 0(.*M(n)) = StM(n) for allM> 1.

Now Theorem A is obtained from the following

Lemma 7. R(n) c (n).

Proof. Let P e RM(n). By Corollary 6 there is a unique Q M(n) with O(Q) = P.Moreover, by Proposition 5 we have that Q e (/(n) n YM(n)). We can represent Q by

Q = QijlVil + E Qij)mUijm (14)i<j<l i<j<<m

with

Qijl M-' (n) and Qijim - 9 2 (n)

(for a proof of this assertion see Lemma 9 below).Let us introduce the following notation: If R e 59(n) has a representation

R = Y ... Y,, Y1, .. , Y, e(n),

we denote by R the corresponding element in U(n), that means R = Y ... Y, withmultiplication in (n). Note, that in general O(R) * R.


Now we conclude from (5), (13) and (14):

P = (Q) = Z QijVij + E Qij1ui milm + Ti<j<l i<j<l<m

and Thas X-degree <M. Moreover, we have TE St(n). If M = 1, then TE (SR(n) nr U°(n))and therefore T = 0, P E 3(n). Using induction on M we deduce TE 3(n) and Lemma7 is proved. [

In order to prove Proposition 5 we start with some easy conclusions. Analogously toCorollary 2 it holds:

Corollary 8. If P e 9M(n), then Zi(P) E "M-'(n). 1

An easy computation yields the following important formula:

Zi(Q1Q2 ) = Zi(Q1)Zi(Q2) + Zi(Q1) Q2 + Q1Zi(Q2) (15)

for all i = 1, ... , n; Q1, Q2 E (n).

Furthermore, we verify

Z,(Z) = 0,Z.(Xij) = iZj - Zi, (16)

z(iji) = 0,Zr(Uijtm) = irVjam - 6 jrvilm + 5IrVj - .Vj

We denote by ,,s(n) the subset in "(n) consisting of all symmetric tensors with orderr in Z, ... , Zn and order q in X 2, ... Xn-,n

Remark. 9r,s(n) can be identified with the set of all polynomials on the dual algebrae(n)* which are homogeneous of degree r in Z1 , ... , Zn and homogeneous of degree s inX12 , .. , Xn -l,n. Then

Y(n) = ( ,, (n),r,s=O

"(n) = ( (3 °J,s(n) ,k=O r+s=k

M

YSm (n) = ( ( -,.s(n).r=o qO=

The following lemma seems to be obvious, however, an analogous proposition is nottrue in U(n).

Lemma 9. If P E /(n) has X-degree M > 1, then there are Pi j E yM- 1(n)

and Pijim E 9M- 2(n) with

P = PijiJ + E P E PijlmUijimi<j<l i< j<l<m

5*

67

68 F. Richter

Proof. Let P e (/(n)n r M(n)) for M > 1 and

P= E QijVit + E QijtmUijimti<j<t i<j<t<m

where

Qij = Pij, + Ri,

Qijlm = Pijlm + Rijimwith

P e S p M

-l(n), Pijm E

9M

- 2(n),

N N

Rije · ( ® ,,(n), Rijim e (nr=O s=M r=O s=M-1

for some N > M. Then it follows that

P= Y PijtjU + E PijlmUijm + P2i< j<l i< j<l<m

withs N+2

P 2 i 3 ' r,s(n)r=O s=M+1

Since P e yM(n), we conclude P2 = 0. O

Analogously, we obtain for M = 1 and P e (/(n) n 9l1(n)):

P = E PijlVijl with Pjjl el °(n).i<j<

Together with Lemma 9 the formulas (15) and (16) immediately supply

Corollary 10. (n) c %'(n). 1

So we have to show that . (n) c / (n). To this end we use the following three lemmas.

Lemma 11. Let Q .E f(n). Then there are non-negative integers M1, ... , M, satisfyingZM Q e (n)for all i = 1, ... , n.

Lemma 12. Let Q 9(n). If there are non-negative integers M 1, ..., Mn such thatZiM Q e /(n)for i = 1, ... , n, then Q e aWd (n), where bad /(n) denotes the radicalof /(n).

Lemma 13. The set Jr'(n) is a prime ideal in S"(n).

Remark. Lemma 13 can also be deduced from results in [1]. We give a direct proof.Assume that Lemma 11-13 would be proved. For the following facts from ideal

theory we refer to [7] and [9].By Lemma 11 and Lemma 12 we have Vl(n) c Bad /(n). Since f(n) c .f(n) by

Corollary 10, we deduce from Lemma 13 that Aad /(n) = Y(n), where (n) is prime.That means, f(n) is a so-called quasi-primary ideal with respect to the prime ideal .k(n).

On fundamental differential operators and the p-plane Radon transform 69

Let f(n) = I n ... rn am be the decomposition into primary components. Then for thecorresponding prime ideals /i = Mad Yi it holds

~F(n) c i for all i= 1, ..., m (17)

(see [7], chapter 2.19). Since Y(n) = O°(n) ) U YM(n), we have ZMi j J.(n) for alli= l,...,n. M=1

Let Q E Y.(n) and Z'i Q E /(n) by Lemma 11. Now formula (17) and Theorem 29in [7], chapter 2.14, provide

I(n): (Zg') = J(n). (18)

But Zm' · Q E f(n) implies Q E (f(n): (')). Hence, Q E (n) by (18). Thus Proposition5 is proved.

Proof of Lemma 11. Let Q E IM(n). We insert

Z1Xij = (ZiXlj-ZjXi + Vj) for 1 < i < j n

in Z Q and obtain

M.Q = Q1 + R, (19)

where Ql E (n) and R, depends only on Z1, ... , Z,, X 12, ... , X1 ,. Moreover, R1 E cM(n).We represent R by

R, = R1,JXj122.. Xj + P1, (20)IJI =M

with P1 E SM-l(n) and R1,J E °(n). The formulas (15) and (16) supply for arbitrary0 < K < M and i > 1:

Z(R 1) = R1,JX2 ... Z(Xii) ... Xi + Z(P 1). (21)IJI =M

Using (7) we compute for 1 K Land i > 1:

m=O m

withK

aKm = Z (_l)i ()jmj=1

Note that aK m = 0 for K > m and aKK = (-1)K K!. Hence

9z(Xli) = K! (-, ) (L) ZlXi + Pi, KL, (22)

where P,K,L E L-K- 1 (n).

70 F. Richter

As Rt e ~.M(n), the equations (21) and (22) yield, for i2 + ... + in = M,

2... Z(R 1) = i2 ! ... i! (-Z 1 )M R1 (i2..... ) = 0, (23)

which implies R,. = 0 for all J.Consequently, R1 = P1 e M- 1 (n).If M = 1, it follows R1 E(°(n) n .F(n)) = {0) and therefore Z1 Q E f(n). By

induction on M there is a non-negative integer N1 with

z1 R E (n).

Hence, Z + q' Q E f(n) and we put M1 ,= M + N1.Now we can show ZMI Q E (n) for i = 2, ... , n and some Mi analogously. [

Proof of Lemma 12. We identify e(n) with the dual space e(n)* and the correspondingalgebras Y(n) and 9Y(n)* (:= 5°(e(n)*)). So we look at 9(n) as the ring of polynomialfunctions in the variables Z, ..., Z,, X1 2 ..., X,, n (over the field C).

We abbreviate Z = (Z1, ... ,Zn), := (X 2 ..., Xn _,) and define:

.X(/(n)):= {(Z, J) Cn x C2" 1 ) Vjj = 0; Uijlm = 0 for all possibleindices},

Xi((n)) = {(Z, X) E A(J(n)) I Zi = 0},

Jgo(f(n)) := {(2, X) E Xr(/(n)) I Z .= }.

Let Q E 9(n) with Z '. Q E /(n) for i = 1, ..., n. Then

Q(Z, X) = 0 for all (Z, X) e Xr(,(n)) \ Xj(/(n)) and i = 1,...,n.

Thus,

Q(2, X) = 0 for all (Z, X) E X (f(n)) \ X 0o(,(n)). (24)

We also want to prove that

Q(O, X) = 0 for all (0, ) E .o(jf(n)). (25)

Then (24) and (25) would imply the assertion of Lemma 12 by the Nullstellensatz. Inorder to show (25) we prove the following proposition (*):

(*) For any (0, X(°)) E jX(J,(n)) there is a sequence TZ' ) such thata) 2(') - 0 and

b) (2r, X(O)) e X(A(n)) \ x.ro((n)) for all r.

Then (25) is obtained from (24) and the continuity of Q. If (0, X(O)) E Xo(j(n)) andX(o) = 0, then the equations VKj, = O and Ujm = 0 are fulfilled for any 2 and proposition(*) is clearly satisfied. Let (0, (O)) E XJro(/(n)) and X(o) * 0. We may assume X(1) * 0.


Then the equations V121 = 0 and U121m = 0 provide for Z('):

Zr )= (Z()X(1) - Z?()X(?)) for I > 2,(26)

XI = (X_[(O)X(O) - X()X()) for m > I > 2.

By (26) an easy calculation shows that for each such (Z( ), X(0)) the equations Vjl = 0and Uijlm = 0 are fulfilled for all possible indices. But in (26) Z(r) and Z(2

) can be choosenarbitrarily. Hence, proposition (*) is satisfied in this case, too.

Proof of Lemma 13. Let P, Q E Y(n), P has X-degree M and Q has X-degree N. Suppose,that N > 1. Then PQ has X-degree M + N.

We verify for j + ... + jn = M + N:

Z1 '"Z Qn =(P 'Q) () ZEZ ... Z (P)Zj- Zn r(Q) (27)

IRI = M0 r, < j

O r j.

where R = (rl, ... , rn) is a multi-index.If P E t(n), then Z', ..., Z"(P) = 0 for all r + + r = M. Thus, Zi' ... Z

x (P Q) = O for all j + ... + j, = M + N and (n) is an ideal.Let now P Q e A'(n) and P 0 Y (n). We have to prove that Q e ~(n). We define a

lexicographic ordering on the set of indices R = (r1, ... , r,) with r + ... + rn = M and,analogously, for S = (s., ... , s,) with s + ... + s, = N in the usual manner:

R > R' iff there is one j {1, ... ,n} with

a) r1 = r1 ,..., rj_ 1 = rj_1 ,

b) r > r'j.

Since P ¢ JF(n) there is an index (ix, ... , i,) with i + ... + in = M and

a) Z ... Z(P) O,

b) Z ... Z(P ) = 0 for all (r, ... , r) > (i1 ... in).

We show Q E it(n) by induction on the sequence S1 > S2 > ... of all indices withS1 + ... + s, = N. Let us start with S1 = (N, 0, ..., 0). Then (27) yields

= i +N i2 . Z( p Q)=( i l+N ) Z... Z" (P) Z(Q),

hence, ZN(Q) = O.

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72 F. Richter

For shortness we write

ZSK(Q) := Z1 ... ZS"(Q)

for SK = (s1, ..., sI). Let us suppose that

(*) ZSk(Q)= O for all k<K

and assert that ZS"(Q)= 0. We put SK-1 = (s', ..., S). Let m {1, ..., n} such thats', ..., s > and s,+ = ... = S = O. Then

SK = (SI, ... ,sn- 1, 0., 0) for m nand

SK = (S, ... , Sn- 1 - 1, Sn + 1) for m=n.

Formula (27) and our assumption (*) yield

= Z z... Z ZSK(P · Q),

o = zi... Z(P) ZSK(Q),

hence ZS"K(Q) = 0; Q E oV(n) by induction. [

4. On a range characterization of the Radon transform

In this section we want to discuss the range of 9(E") (the set of rapidly decreasingfunctions on E") with respect to the p-plane Radon transform (1).

We consider the mapping A from U(n) into the algebra of differential operators onE,, explained in the introduction and the elements jr E 1(n) defined by (4). Furthermore,the space En.p is a vector bundle over the Grassmann manifold Gn, of p-dimensionalsubspaces of E". The fibre in each point a e G,p is identified with the vector space al.Then a smooth function p on Enp is said to be rapidly decreasing, roughly speaking,if it is so in each fibre (for precise definition of the space Y(E,.) of rapidly decreasingfunction on E, p see [81).

Now we can formulate our.

Theorem B. Let n > 3 and 1 p < n - 1. Then the p-plane Radon transform p, is alinear bijection from 9(E") onto the set of allfunctions qp e Y9(Ep.,) satisfying

A(Vij) p = O for all 1 < i < j < I < n. (28)

Proof of Theorem B. We fix the standard basis (o, ..., e") in E and define for eachmulti-index J = (, ... ,j) an open neighborhood in G, p by:

U | G | the orthogonal projectio n of a onto}

span(ej, ... ej) is a bijection

Then U Uj = G.,p and the corresponding local coordinates are denoted byJ


If a E U, we identify the fibre a' with span(ejip + ,, ... ej.). Let yj. be the correspondingcoordinates in a'. Then (aj.j., yj,) is a local coordinate system in E,,p. For a fixed J letH, = (hip) be the n x p-matrix with entries

a) hj..= ajxj. for x = p + 1, ... , n;a = ... ,p,

b) the remaining p rows form the p x p-unit matrix.

If ir: En p -* G,,p denotes the natural projection, for a function qi on E,p and eachJ = (j, ... ,j) we define a function oj by

Qpj := (det HH,)- (pl.- uj). (29)

In [8] we proved the following proposition (see also [2, 3, 4]):

Proposition 14. For n > 3 and 1 p < n - 1 the p-plane Radon transform is a linearbijection from Y(E") onto the space of all functions p E 9Y(E,p) which satisfy in eachcoordinate neighborhood Uj x R" -P the system

a2 a2

y aj. ~ - ayj jd ¢j = 0 (30)

for all A, x = p + 1, ... , n and = 1,..., p.

In order to prove Theorem B it remains to show the equivalence of (28) and (30).We compute the operators A(Vij) in local coordinates, but only in the neighborhood

Uo x R"-P, where UO = U1 ... ) (J = (1, 2, ... , n)).First we obtain

A (Zx) = - aayxA(Z~) = yj

a aA(Xp) = aI -- - ap ,

Xa = p+1 OXi aa, X aa

A (XX) = a a a- a + YX YAx~l aa."1 aaxa, ay' ay.

A (X,) = - + Y Awy - + - aAYaxflaa,X A=p+1 ayz A aaA

for all x, = p + 1, ... , n and , / = 1, ..., p. For p E C (E,p) ( o = nP(1,2,...,n)) theseformulas lead to:

a) K A( _VA) p =[ (aa~ ~-aa ) + a-A] "'oL M.# (

73

74 F. Richter

b) K .A(_, ) q [= (ix + aa a + aX7 aAp pA,)] Po,

c) K A(V.A,) (p =[E (a ,,raa + aa-X,, + aaSax,A)] (Po,

d) K A(V.p.x) = [ + + (aaOr + aaa + (a))] 'o

for all c, ,y = 1,...,p and x, A, = p + 1, ... , n. Here K =(detHTHo)- 2(Ho := H 2 .... )) and

a2 a2

ay. aa a a a

Hence, the system (28) follows from (30). If A (Vl) qp = 0, for all i, j, I then the equationsa) and c) yield

aMA/PO = (E ,IJPar i6PXA) 90 (31)

Putting XzA "= (,XA.., . px) T, the system (31) can be rewritten to

(HTHo) (.,A qpO) = O.

Since det HH o * O, we conclude 3X-.Aq O = 0 for all a, x, A. Hence, (28) is equivalentto (30).

This finishes the proof of Theorem B. [1

By Theorem A the necessary condition (2) (see Introduction) that a function on E,P,belongs to the range of Ap, is equivalent to

A(Cjl) p = 0 for all 1 i <j < 1 n

and (32)A(Uijm) (p = 0 for all 1 i <j < I < m n.

The question arises why the equations A(Vijlm) p= 0 can be omitted in Theorem B. Theanswer is given by the following.

Proposition 15. If (p E 59(E,p) and A(Vij ) = 0 for all possible indices, thenA(Uijlm) qp = O for all 1 i < j < I < m < n.

Proof. In U(n) we have

ZjUij, = Xj.Vjl + XijVjlm - Xj1iVj,

which implies

A(Zj) (A(Uijlm) (p) = 0;


and in the same way

A(Z,) (A(Uijim) q0) = 0 for r {i,j, , m}).

Furthermore, it holds for 1 i < j < I < m < r n:

ZrUijim + Z iUijmr = Xij_imr - Xjmil,r + XimVjlr + XrVijm,

ZrJijm - ZmUijlr = Xij-mr - XiIjmr + Xjiimr - Xmrviiji,

Zm Uijlr - ZlUijmr = XirVjlm - Xiji-mr - XjrVilm + XlmVijr.

These equations yield

A(Zr) (A(Uijljm) ) = 0,

which can analogously be shown for each index r {i,j, , m}. Hence

A(Zr) (A (Uij) qp) = 0 for all indices. (33)

As So E (Ep), so is A(Uijim) p. But (33) implies that A(_Uijm) qp is a constant on eachfibre. A rapidly decreasing function that is a constant must vanish. [

References

[1] BORHO, W., BRYLINSKI, J.: Differential operators on homogeneous spaces I. Invent. Math. 69(1982), 437-476.

[2] GEL'FAND, I. M., GRAEV, M. I., SAPIRO, Z. JA.: Integralnaja geometrija na k-mernich ploskost-jach. Funk. analiz 1 (1967) 1, 15-31.

[3] GEL'FAND, I. M., GRAEV, M. I., GINDIKIN, S. G.: Integralnaja geometrija v affinich i v proektiv-nich prostranstvach. Itogi Nauki, Serija: Sovr. probl. mat. 16 (1980).

[4] GRINBERG, E. L.: Euclidean Radon Transforms: Ranges and Restrictions. Preprint (1984).[5] HELGASON, S.: The Radon Transform. Boston- Basel- Stuttgart: Birkhiuser 1980.[6] HELGASON, S.: Groups and Geometric Analysis. New York: Academic Press 1984.[7] RENSCHUCH, B.: Elementare und praktische Idealtheorie. Berlin: VEB Deutscher Verlag der

Wissenschaften 1976.[8] RICHTER, F.: On the k-dimensional Radon transform of rapidly decreasing functions. Proc.

2"' Int. Symp. in Diff. Geom. at Pefiiscola 1985, Lecture Notes in Mathematics 1209,Berlin-Heidelberg-New York: Springer-Verlag 1986, pp. 24 3 - 2 58 .

[9] ZARISKI, O., SAMUEL, P.: Commutative Algebra. Princeton: van Nostrand 1958.[10] GUILLEMIN, V., STERNBERG, S.: Some problems in integral geometry and some related problems

in microlocal analysis. Amer. J. Math. 101 (1979), 915-955.

FRANK RICHTER

Humboldt-Universitit zu BerlinSektion MathematikPSF 1297DDR-1086 Berlin

(Received March 6, 1989)

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on fundamental differential operators and the p-plane radon transform

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