TitleLectures on harmonic analysis on Lie groups and relatedtopics( Some properties of Fourier transform on Riemanniansymmetric spaces. / M. Eguchi )

Author(s) Hirai, T.; Schiffmann, Gerard

Citation Lectures in Mathematics (1982), 14

Issue Date 1982

URL http://hdl.handle.net/2433/84919

Right

The following article has been remained unpublicized byrequest of the owner of copyright. [H.Midorikawa: "Clebsch-Gordan coefficiects for a tensor product representation Ad ⊗π of a maximal compact subgroup of a real semi-simple Liegroup." pp.149-175]

Type Book

Textversion publisher

Kyoto University

Lec. in Math.,KyotoUniv.No. 14 Lectures on Harmonic Analysis cm Lie Groups and Related Topics

pp. 9-43

Some Properties of Fourier Transform

on Riemannian Symmetric Spaces

Masaaki EGUCHI

Hiroshima University

§1. Introduction. This paper is a continuation of the

previous paper [1]. Concerning Harish-Chandra's Schwartz space

C(G) on a reductive Lie group G([6]), it is known that C(G) is a

Frechet space and one of the spaces CP(G)(p>0), that is, C(G)=

C2(G). These spaces CP(G) are also known to be Frechet spaces

and satisfy that, if 0<p<q, then

17(G) C CP(G) C CcI(G)

and

CP(G) C LP(G)

as dense subsets, here 17(M) denotes the space of C functions

with compact support on a C manifold M and, for p>0, LP(M)

denotes the set of all measurable functions on a measure space

M with a measure dx such that Of0p= (fMIf(x)1Pdx)1/P<00

respectively. As is well known, if p:›1, then LP(M) is a Banach

space.

In the previous papers [1,3], we have given a characteriza-

tion of the Fourier image of these spaces CP(G/K)(0<p<2) on

— 9 —

Riemannian symmetric spaces G/K. And in [2] we considered an

application of this result to invariant differential equations

on G/K. The purpose of this paper is to give an another applica-

tion of the result and study other properties of CP spaces and Lp

analysis. An outline of contents of this paper is as follows.

In Section 3 we give the Riemann-Lebesgue lemma and the multiplica-

tion formula for our Fourier transform. We discuss in Section 4 an

elementary L1-theory. We study in Section 5 a convolution property

and show that, if positive numbers p and q satisfy p>q and 1/P+

1/q=1, then the relation CP(G)*CcI(G)CCP(G) holds. We discuss,

in Section 6, LP derivatives of LP functions on the symmetric space

G/K and show that if an LP function f has all LP derivatives of

orders less than a certain integer, then f is almost everywhere

continuous. In Section 7 we show that a bounded linear mapping

, from Lp0/K) into Lq(G/K) (1‹,p,q) which commute with G translations

is essentially a convolution operator by a p-tempered distribution,

when it is restricted to TP(G). Some results, as the reader knows,

are analogues of Euclidean case (cf.[10]).

§2. Notation and Preliminaries. Let R and C be the fields

of real and complex numbers respectively. The set of non-negative

integers is denoted by Z.

Let G/K be a Riemannian symmetric space. Here G and K denote

a noncompact connected semisimple Lie group with finite center and

its maximal compact subgroup respectively. Let and a be the

usual fundamental spherical functions on G (cf. e.g. [12, Chap. 8]).

—10—

We identify, as usual, functions on G/K with those on G which are

invariant under right K actions. Let g be the Lie algebra of G

and G the enveloping algebra over ge. We regard the algebra G as

the algebra of left invariant differential operators on G. And

also, since the algebra of right invariant differential operators

on G is anti-isomorphic with G, we identify each right invariant

differential operator with the element in G by this anti-iso-

morphism.

For each p>0, CP(G) denotes the Frechet space consisting of

all C functions f which satisfy for each D1, D2EG and rER

sup If(D1'°x;D2)1E(x)-2/13(1-1-a(x))r<... xEG

We denote by CP(G/K) and /P(G) the subspaces of all fECP(G) Which

are invariant under right K translations and bi-invariant under K

translations from both sides respectively. We write simply C and

I for C2 and I2. Denote by I .(G) the space of functions in 0,(G)

which are K bi-invariant. Put 1c(G)=D(G)n1(G) and, for p>0,

IP(G)=CP(G)nI.(G). We denote by IP(G) for p%1 the space of all

functions fELP(G) which are K bi-invariant.

Let G=KAN be an Iwasawa decomposition of G and g=k+a+n the

corresponding decomposition of g. We denote by a* the real dual

space of a. Let M' (resp. M) be the normalizer (resp. centralizer)

of A in K. The finite group M'/M is denoted by W and called the

(little) Weyl group. The order of W is denoted by [W]. We normal-

ize the measures as in [1]. We also use the notation in [l] with-

out referring.

-- 11—

For XEa*, the elementary spherical function (1)2, is given* by

qbx(x) = e(iX-P)(H(xk ))dk (xEG)

K

and for each function fEI(G), its spherical transform is defined

by

(1) Ff(X) = f f(x)(1)_i(x)dx (XEa*).

Using Harish-Chandra's c-function, this is inverted by

(2) f(x) = [W]lf Ff(A)cpx(x)Ic(A)1-2dA (xEG) a*

(See Harish-Chandra [5]).

For fEC(G/K) the Fourier transform is defined by

(-iA+p)(H(a)) (3) Ff(A:kM) = f(kan)e'dadn

AxN

for AEa* and kEK. This is inverted by the inversion formula:

(4) f(x)=M-11 Ff(A:kM)e-(iX")(H(x-ik))1c(A)1-2dk dA a*x(K/M)

for xEG.

— 12 —

§3.

formula. If f E L1(G/K), by the Fubini theorem, the

integral fNf(kan)dn converges for almost all (a,kM) E A x

(K/M). Put

= ep(log a)f Af(a:kM)f(kan)dn (k E K, a E A)

N

and call it the Abel transform. If f E L1(G) write

f1(x) = JJ f(kxk')Idk dk' (x E G).KxK

Lemma 3.1. If f E L1(G), then f1satisfies the

following relations.

(1) fl(k1xk2) = f1(x) (kl, k2 E K, x E G),

(2)f11 = Hf111'

(3) fl(as) = fl(a) (a E A, s E W).

Proof. The first two relations are trivial. The last

one is the L1 case of Lemma 17 of Harish-Chandra [5],

Let L1(Ax(K/M))w be the set of L1 functions F on Ax

(K/M) which satisfy the following symmetry condition with

respect to W

-isA(log a)-(isX+p)(HOCIk)) F(a:kM)e-isX( log a)-(isX+p)(H(fik)) da dkM

Ax(K/M)(5)

=F(a:kM)e-iA( log a)-(iA+p)(H(x-lk)) da dk M'

Ax(K/M)

The Riemann-Lebesgue lemma and the multiplication

— 13 —

The following result shows that the Abel transform is a

continuous mapping from L1(G/K) into L1(Ax(K/M))w.

Lemma 3.2. If f E Ll(G/K), then Af E Ll(Ax(K/M))

and the following inequality holds. W

IlAflll‹clIfil l.

Where c is a positive constant independent of f.

Proof. We have

lAf(a:kM)Idkda < el)( loga)f If(kan)Idkdnda Ax(K/M) A NxK

= J Af1(a) da = [W]i Af (a) da A A+

= [w]i e-p(log a)e2p(loga)ff If(kan)Idk dn da. A+ NK

The last expression is bounded by

[W]f If(kan)le2p(log a) dk da dn

A xKxN

The symmetry property is seen from the

f Ff(A:kM)e- (I.A+p)(H(x

K/M

relation

-1 k ) ) dk

— 14 —

= f * (I)x(x)

and the symmetry cbsA = cpx (s E W). This completes the proof.

As the function Af(.:kM) is integrable on the abelian

group A for almost all kM E K/M by the last lemma, the

following result is obtained from the Riemann-Lebesgue lemma

in the euclidean case . (See e.g. [10]).

Theorem 3.3.(The Riemann-Lebesgue Lemma). If f E L1(G/K)

then we have

lim IFf(X:kM)1 = 0 1AI-

for almost all kM E K/M.

It is known[7] that there exist positive integers a,b such

that

1 a(1 + Ixob.

Let b0 be the smallest one of such b and put q = b0 + R + 1,

2, denoting the rank of G/K. We say a function F on a*x(K/M)

is weakly decreasing if it satisfies, for a positive constant

c,

sup IF(A:kM)I(1 + IAI)q < c. (A,kM)Ea*x(K/M)

If F is a function on a*x(K/M), we write

— 15 —

F(A:kM) = F(-A:kM) (X E a*, k E K).

Proposition 3.4.(The Multiplication Formula). If f E

L1(G/K) and F is a weakly decreasing function on a*x(K/M)

which satisfies the symmetry condition (5) with respect to

the Weyl group W, the following formula holds.

(6) j f(x)(F11')(x) dx = [W]-1J (Ff)(A:kM)F(A:kM)1c(X)1-2 dA dkm . G a*x(K/M)

Proof. If F satisfies the conditions, then F_1F is

well defined and bounded on G, so that by the Riemann-Lebesgue

lemma the integrals of the both sideS can be considered. The

right hand side is equal to

iwi-lf f f(x)e(iX-P)(H(x 11°)dxF(X:kM)Ic(X)1 2dkmdX.a*x(K/M) G

Using the Fubini theorem, we see that the last expression equals

DA[W]-11 f(x)dxf F(X:kM)e (iX-p)(H(x-1k)) lc(A)I-2dkmdX G a*x(K/M)

= I f(x)(F-1F)(x) dx. G

as Ic(-X )1- = This proves our assertion.

- 16 -

§4. An elementary L1 property. As in the usual euclid-

ean Fourier transform, the Fourier transform Ff of L1 func-

tion f is not necessarily invertible. We give here an L1

approximation of an L1 function on G/K by using its Fourier

transform, modifying the euclidean case (cf. [10]).

Fix a function IP E 11(G) with nonnegative values and

f01,b(x) dx = 1. As the relation

I f(x) dx = J f(exp H)A(H) dH

a

holds for f E I1(G), here a denoting the positive Weyl

chamber with the euclidean measure dH on a and A(H) =

aEFsinha(H), we define a functionIP(c > 0) on A+ by

+

06(exp H)A(H) = s=24(exp(H/e))A(H/E).

We extend to a function in 11(G) by assuming K bi-in-

variance for it. Then we have

( 6(x) dx =ipc(exp H)A(H) dH 'G °a+

= IP(H)A(H) dH .a+

= f zp(x) dx = 1. 0

Lemma 4.1. Let 11,6 be the function as above. If f E

LP(G/K)(1 < p < then

Ilf* IPC - f Hp -> 0

— 17 —

as E + 0.

Proof. By the assumption on Ips we have

(f * Ipc)(g) - f(g) = [f(gx-1) - f(g)Dpc(x) dx,

which is equal to, according to the Cartan decomposition,

J[f(gkexp(-H)k') - f(g)]tpc(exp H)A(H) dH dk dk'Kxa xK

= Ef {f(gkexp(-cH)) - f(g)}dk]lp(exp H)A(H) dH. a K

Let q be the positive number such that 1/p + 1/q = 1. Then,

making use of the HOlder-Riesz inequality, we get

1(f * 1PE)(g) - f(g)1P

(J (exp H)A(H)I dH)P/ga

If {f(gkexp(-EH)) - f(g)} dkIP1p(exp H)A(H) dH a K

=If {f(gexp(-EAd(k)H)) - f(g)} dkIPIp(exp H)A(H) dH. a+ K

Using again the H31der-Riesz inequality, we see that the last

integral is bounded by

J{f If(gexp(-EAd(k)H)) - f(g)IP dklIp(exp'H)A(H)--dH. a+ K

If f E LP(G/K), the following inequality is obtained from

— 18 —

the above argument and the Minkovski inequality.

(J If(g exp(-EAd(k)H)) - f(g)IP dk dg)1/PGxK

< (J If(g exp(-EAd(k)H))IP dk dg)l/P +(JIf(g)IPdk dg)l/P.GxK GxK

The last expression is bounded by AfHp. Thus we know that

the integral

If(g exp(-cAd(k)H)) f(g)IPdk dgGxK

converges uniformly in E. This fact allows us to take the

limit E 0 inside the integral. Therefore we have

lim J If(g exp(-eAd(k)H)) - f(g)IPdk dg = 0. E-)-0 GxK

This Proves that

IIf * tpc - fllp-4- 0

as E 0.

Lemma 4.2. Let4)E6be as before. WriteT= Flpe

for each e > 0. Then the following equality holds for f E

Ll(G/K).

[w]-11 Ff(A:kM)e- (1A+p)(H(g-1k)) (A)Ic(A)I-2 dkMdAa*x(K/M)

f * 4,c(g) (g e G).

— 19 —

Proof. The left hand side equals

rw1-11 f(x)e (iX-p)(H(x-1k))dx a*x(K/M) G

e-(iX+p)(H(glk))-2 TE(A)Ic(A)I dkM dA.

As H(xg lk) = H(xl<(g-1k)) + H(g lk) for all x, g E G and

k E K, by a simple calculation we obtain

(iX-p)(H(x-ik))—(iX+p)(H(g-ik))=(iA-p)(H(x-igkg))-2p(H(g-ik)),

-1

here kg= K(g-1k). As is well known dk = k))dkg dk

holds. By these relations, the last integral equals

-1

Ewl-11 f(x)e(iX-P)(H(xgk))dxTc(A)1c(A)-21 dk dA.a*x(K/M) G

Applying the Fubini theorem, we see that the last expression

equals

Ewi-if f(x)q)x(x-ig)TE(A)Ic(A)I-2 dx dAa* G

= m-lf f(gx)CYx-1)T E(A)le(A)1-2 dx dx. a * xG

Using again the Fubini theorem, we know this equals

f * Ips(g)•

Proposition 4.3. Assume p 1. Let pcbe as in

Lemma 4.2. If f E L (G/K) n LP(G/K)

— 20 —

_(ix+p)(1-1(g-1k))4,6(A)Ic(A)1-2dkmdA. [W]-1J Ff(A:kM)e

a*x(K/M)

converges to f(g) in LP as a 0.

Proof. The result follows from Lemma 4.1 and Lemma 4.2.

We remark here that

ITS(A)1 < 1 (A E a*) and lim Tc(A) = 1

holds. In fact we have

TE(A) = bE(exp H)LA(exp H)4(H) dH 'a+

ip(exp H)o -X(exp el-1)4(H) dH.

As ILA(exp sH)1 C 1 for all A E a* and a > 0, the

last integral converges absolutely and uniformly in a. and

ITE(A)I < 1.

Thus we can take the limit a 0 inside the integral and

obtain

ITE(A)1 1 for all A E a*.

as a 0, since

lim Lx(exp cH) = 1. e--0

Corollary 4.4. If f E Li(G/K) and Ff E Li(a*x(K/M))

— 21—

-2with the measure Ic(A)1'dAdkM'then

f(x) = [W]-11 Ff(A:kM)e- (iA+p)(H(x 11()), Ic(X)I 2d.kmdA a*x(K/M)

holds for almost all x E G/K.

Proof. From the assumption that Ff E L1 and the above

remark and the proposition, we see that the integral

[w]-11 Ff(A:kM)e- (iX+p)(H(x-1k)) Ts(X)Ic(A)I-2 dkmdAa*x(K/M)

converges to f(x) in L1 uniformly in e. The assertion is

obtained by taking the limit E 0 inside the integral.

Corollary 4.5. Assume that f1, f2 E L1(G/K) and

Ff1(A:kM) = Ff2(A.:kM) for almost all A E a* and k E K.

Then fl(x) f2(x) holds for almost all x E G/K.

Proof. We obtain our assertinon by applying Corollary

4.4 to the function f = fl — f2.

§5. A convolution property of functions in C. Let

0 < p < 2. Then

Chandra[7, §14]

p = 2 case and

Theorem 5.1

the following

and Rader( see

by Milin6[9]

. Let 0 < p

result

e.g.[12

for the

< 2.

is obtained

, Chap. 8,

other case.

Then CP(G)

by Harish-

§3.71) for

is a Frdchet

— 22 —

algebra under the convolution. That is, (f, g) f * g is a

continuous mapping of CP(G) x CP(G) into CP(G).

Corollary 5.2. Let 0 < q p < 2. Then (f, g)

f * g is a continuous mapping of CP(G/K) x CcI(G/K) (resp.

/P(G) x Icl(G)) into CP(G/K) (resp. IP(G)).

Remark. The theorem was proved by studing the function

and the corollary is obtained if one confine the theorem to the

subspaces CP(G/K) and IP(G). But we can also get the co-

rollary as an application of the Fourier transform of CP(G/K)

and IP(G). Because we have the relation

F(f * g) = Ff.Fg f,g E /P(G)

and, as is easily seen, Ff.Fg is contained in the image

F(IP(G)). (See Trombi-Varadarajan[10]). And for the space

C (G/K), the following proposition shows our assertion is true,

since Ff•Fg1 satisfies the growth condition and the symmetry

condition for the image F(CP(G/K)). (See Eguchi[l]),

Proposition 5.3. If f, g E C(G/K), then

F(f * g)(X:kM) = Ff(X:kM)•Fgl(X) (X E a*, k E K)

holds. Here gl is defined by

g1(x) = f g(kx) dx. K

Proof. We have, by the G-invariance of the Haar measure,

— 23—

-1, -1(iX-p)(HlYx))dx dy. F(f* g)(A:kM) =f(kx)g(Y)e

GxG

As H(y lx) = H(y-1K(x)) + H(x), this is equal to

J (iX-13)(H(y-1<(x))) + (iA-p)(H(x)) dx dy f(hx-i)g(Y)e

GxG

=Jf(kn-1a-1)Fg(A:k1M)e (iA+p)(H(a)) da dn. KxAxN

-1Substituting n for ana we see that the last integral equals

J f(ka-1n)e(iX-0(H(a)) da dn Fg(A:k'M) dkk f

K AxN

= Fg,(X)•Ff(X:kM).

We next study the case when p and q satisfy 1/p + 1/q

= 1 and p > 2 .

Theorem 5.4. Assume that 0 < q < p and l/p + 1/q =1.

Then

(f, g) f * g (f E CP(G), g E el(G))

is a continuous mapping from CP(G) x CcI(G) into CP(G).

Proof. If p = q = 2, the result was obtained by Harish-

-24—

Chandra[6] and Rader as we mentioned above. We now assume p

> 2 and prove the assertion by modifying the proof for the

case p = q = 2. For any function f on G we write

f(x : y) = f(xy) (x, y E G).

Let f E CP(G), g E Cci(G) and put h = f * g. For D1, D2 E

G, we have

Ih(DI;x;I2)1<flf(I)1Jx;D2:y-1)g(y) I dy. G

By the G-invariance of the Haar measure, the right hand side

equals

J If(D1ixy-1)g(y;D2)1 dy. G

From the assumption on f and g, it follows that for any posi-

tive integers r, s there exists a constant s C'rsuch that

,

2/q -1E(xy-1) 22/p If(D :xy)g(y;D < C' r,s (1 + a(xy-1))r (1 + a(y))s

for all x, y E G. As the inequality

1 1 + o(y)

1 + u(xy-1) 1 + a(x)

is deduced from the well known inequality

1 + a(uv) < (1 + a(u))(1 + a(v)) (u, v E G).

— 25 —

We see that the last integral is bounded by

_

C'(1 + o(x))-r( -(xy-1)2/p=(y)2/q(1 + a(y))r-s dy r,s

G

s

= C'r(1 + o(x))-rJE(xky-1)2/pE(y)2/q(1 + a(y)) r-s dk dy.

,

GxK

Using the H51der-Riesz inequality and the well known relation

J E(xky) dk = E(x)E(y), K

we have

,(y,,cy)2/p 6k (..„(x)_(y))2/P ,

K

Thus we obtain from the above argument that for any positive

integer u

Ih(D1;x;D2)1(1 + a(x))11E(x)-2/13

< C.1",.,s(1 + a(x))u-rf E(y)2(1 + a(y))r-s dy. G

If we take a sufficiently large r the last integral converges

(see Harish-Chandra[6, Lemma 11]). This proves that h E

CP(G) and (f, g) f * g is a continuous mapping from

CP(G) x Cci(G) into CP(G).

The following result is now obvious.

— 26—

Corollary 5.5. Assume that 0 < q < p and l/p + 1/q

= 1 . Then

(f, g) f * g

is a continuous mapping from CP(G/K) x CcI(G/K) (resp. /P(G)

x 1q(G)) into CP(G/K) (resp. 7P(G)).

§6. Derivatives of LP Functions. Throughout this sec-

tion we fix a number p such that p 7 1. Let X E g and f

E LP(G). If there exists a g E LP(G) such that

(I If(x exp ix) - f(x) g(x)IP dx)1/P -4- 0

as t 4- 0, then we say f has the right derivative g with

respect to X in LP and f is differentiable from the right in

L- with respect to X. By substituting exp tX-x for x.exp tX

in the limit we define the left derivative and the differenti-

ability of f from the left with respect to X. Obviously, if f

has a left (rasp. right) derivative with respect to X, then it

is unique in L.

Proposition 6.1. If f E CP(G), then f is differenti-

able infinitely in LP. Especially, for any D E g, f(x;D)

(resp. f(Dx)) is the right (resp. left) derivative of f in

LP with respect to D.

Proof. Let X be any element in g. We then prove

— 27 —

1{f(exp tX .x) - f(x)1 f(Xx)

in LP as t 0. By the Taylor theorem, for t there exists

a constant 0(0 < 0 < 1) such that

1 Tif(exp tX•x) - f(x)} - f(X;x)

= -tf(exp OtX•x) . 2!

Therefore we have

1,-{f(exp tX x) - f(x)1 - f(Xx)IP dx)1/P0

=Md.If(exp 0tX.x)lp dx)l/p 2!

_

2! p

which tends to 0 as t 4- 0. By iteration we obtain our

assertion.

Let 0 be the Casimir operator in G.

Lemma 6.2. Assume f E LP(G/K) has all right deri-

vatives of order < 2 in LP and let h(.) = f(.;0). Then

for all k C K

h(xk) = h(x)

in L.

- 28-

Proof. Let X1,..., Xn be an orthonormal basis of g

and g.1 < i < n) be the LP derivatives of f with respect

to X.. Then1

(j Ih(xk) - h(x)IP dx)1/P

( Ih(xk) sXj) - gi(xk)11Pdx)1/P °

G

+ ( IE7fg,(xk•exp - gi(xk)1

J

Es ff(xkexpsXiexptXi)-f(xkexpsy+f(xkexptXi)-f(xk)IPdx)1/P1

+ (J IE-st1 -{f(x.exp sAd(k)X ..exp Ad(k)Xi)-f(x•exp sAd(k)Xi) .

,

G

+f(xexptAd(k)Xi)-f(x)} - E-s-4-ff (xexpsXiexpty-f(xexpsXi)

+f(x.exp tXi) - f(x)/IP dx)1/'0

+(lIE.-1-f(x*exPsIXi-exiDtX. i).--f(x.exiDsX.,)+1"(x.exPtxi)-f(x)) j-st _ 1

U

+ Ei{gi(x•exp sXi) - gi(xk)11Pdx)1/P1

+(f lis gi(x•exp sxi) gi(x)} - h(x)(P dx)1/P.

Here the five terms except the third one tend to 0 as t, s

.4- 0 by definition of gi's and h. The third term also

- 29-

tends to 0 when t, s 0 since C is G-invariant.

Corollary 6.3. Assume f E IP(G) has all right deri-

vatives of order < 2 inLP and let h(.) = f(.;Q). Then

for all kl, k2 E K

h(kixk2) = h(x)

in L.

If f E Li(G) and f satisfies

R f = f

in L1, we can define the Fourier transform Ff of f by (3).

Proposition 6.4. Assume f E Li(G/K) has all left deri-

vatives of order < 2 and let h(.) = f(.;Q). Then

Fh(A:kM) = -(<A, A> + <0, p>)Ff(A:kM)

holds for almost all k E K.

Proof. For A E a* and k E K, put

(x) =e(iX-P)(H(x -1k)) A,k

Then first of all we see that, if g is the left L1 derivative

of f with respect to X E g, then

if (g(x) f(exp sx.) - f(x). )1)A,k(x) dkl 0 G

— 30 —

for almost all k E K as s O.

f(exp (x) = g(x) -

In fact

sX•x) -

, if we

f(x)

put

we have

X, s

fIf uxs(x)xk(x) dxl dk K G

luX,s(x)le- p(H(xlk)) dx dk

KxG

HuX,s(x)10 as s 0.

Thus we have the assertion.

We now assume that g is the left L1-derivative of f

respect to a Y E g and that h is the left L'-derivative

g with respect to an X E g. Then

J h(x4A,k(x) dx G

-g(exp sX•x) - g(x)

'

= lim f (h(x)Vk(x) dx

s

s-->-0 G,

+ lim.1-, [I g(exp sX•x)11)Ak(x) dx - f g(x)1Xk(x) dx] ,, s--0 G G

The first limit equals 0 and the second one equals

lim1 g(x)[4)X,k(exp (-sX).x) -A,k(x)] dx

G

with

of

— 31 —

1for almost all k E K. Since g E L_, blpxk is bounded

for each b E G (see [3]), we can take the limit inside the

integral and we see that .the last expression equals, for almost

all k E K,

f g(x)tpx,k(-X;x) dx. G

By a similar argument, we see that this is equal to

f(x)tpx,k((-Y)(-X)ix) dx.

Therefore we have the desired relation from the following

equality.

4)),31,(Q;.x) = (-<X, X> - <P5 P>)1PA,k(x)

This proves the proposition.

Corollary 6.5. If f E I1(G) has all left L1-deri-

vatives of order < 2, then

F(S2f)(X) = -(<A, A> + <p, p>)Ff(A).

Proposition 6.6. Choose a positive integer n so that

the integral

(Ix/2 ip12)-nic(x)1 -2 dx < co. a*

Assume that f E LP(G/K) has all LP-derivatives of order

2n. Then there exist a unique function g E C(G/K), finite

— 32 —

differential operators D (a E I) of order < n and a a

constant C > 0 depending only on p and n such that almost

everywhere

f = g

and

Ig(1)1 < C PDap aEI

Proof. Assume first p = 1. As

IFf(A:kM)1 = (IX12 1P12)-n1F(S2nf)(X:kM)13

we obtain, by using the Riemann-Lebesgue lemma,

HFfli1[w]-1'IFf(A:kM)11c(A)1-2 dkH dAa*x(K/M)

< C sup I IF(Orif)(A:kM)1 dk < Cilefil 2Ea* K/M

here C is a constant given by

C= [14]-11 (1Al2 102)-nic001-2 dA. a*

Hence Ff E L'(a* x (K/M), lc(A)1-2dkmdA)w. We then put

g(x) = D-1 (iX+p)(H(xknic(x)I-2[W]-1Ff(A:kM)e-dl dka*x(K/M)

M'

— 33 —

Then clearly g(x) is a continuous function on G/K and

f(x) = g(x) holds for almost all x E G/K. Moreover we

then have

Ig(1)1 < IIFflll < CHQnfill.

This proves the assertion for the case when p = 1.

We next assume p > 1. Fix a function ¢ E D(G/K)

such that

1 if a(x) < 1, ¢(x) =

`, 0 if o(x) > 2 .

Then by a simple calculation it is shown that ¢f satisfies

the assumption of the lemma for p = 1. Therefore, by the

above argument, there exists a function h on G/K which is

continuous and almost everywhere

¢f = h

and for a conxtant > 0 and an n E Z +

h ( ) < stn ( (pr )

holds. On the other hand, as SP(¢f) can be written in the

form

Qn(¢f) = Cy(D)(Df), 110,11,14

— 34 —

where D's are elements in G of degree < 2n and C's P3v

are certain constants, we have

42114f)ll1< IC-p,vIf IDPflIDvfl dx P3v a(x)<2

< C' ysupID P IDvfl dx

n

p,V a(x)<2 a(x)52

< CI, X• IDfl dx, 'F'n deg D<2nia(x)<2here Cn and C' are positive constants. Making use of

(1),n

the HOlder-Riesz inequality, we see that the last expression

is bounded by

CHDO' deg D.-C:2n P ,

where Ccp,nis a positive constant independent of f. Therefore

there exists a constant C > 0 such that

Ih(1)I < C HDfil deg D<2nP

As 4(x) = 1 on the ball B1 consiting of all x E G such

that a(x) < 1, f is equal almost everywhere to a continuous

function on B1. Moreover

Ig(1)1 = Ih(l)I < C IIDfll . deg D<2n

By choosing q appropriately, the argument clearly shows that f

— 35 —

equals almost everywhere a continuous function of any sphere

about the origin. The uniqueness is trivial. This proves the

proposition.

§7. Convolutions of Distributions with Functions and

Their Fourier Transform. We shall show that the space CP(G/K)

for each p (1 p < 2) plays a role of the Schwartz space

S in LP analysis on euclidean space (cf. [10]). We first

define the convolution u E IP(G)' with cp E IP(G).

If u, ^15, E IP(G), then

(u * 14)(ip) = (u * (p)(x)Ip(x) dx

= J u(xwxy-lw(y) dy dx GxG

= u(4) * (Pt) = u(cPt *

ftwherefis defined for a function f on G by

ft(x) = f(x-1) (x E G).

So, for u E IP(G)' and cb E IP(G), we define the convolu-

tion u * 41 by

.t(u * = u(cP *

Then u * (1) clearly belongs to IP(G)'.

According to Trombi-Varadarajan[11], let IP(a*) be the

Fourier image . F(IP(G)) and we denote the dual of it by

—36—

IP(a*)'. The Fourier transform for p-tempered distributions

is defined by

U (F*)-lu (u E IP(G)').

We denote this mapping simply by d.

If v E 1P(a*)' and (;) E /P(a*), we define the product

vq) E 713(0*)' by

(v0(0 = v(clip)

for all ip E 7P(a*). Then we have for ip E F(Ic(G))

(u *(p)^(4)) = (u * ¢)(F-1V) = u(q)t* F-11p)

= u(F-1(Fcpt = d(F4)t .11))

Ca•F(4)t)7(4)).

Thus we have the following result.

Proposition 7.1. Assume that 1 < p < 2. For u E

/P(G)' and cp E 1P(G), the relation

(u * (P)" = a•F(¢fi)

holds.

For u E CP(G/K)' and cp E 1P(G) define a function

fcb,u by

f it,,u(x) = u(Lx(pt).

— 37 —

G acts on CP(G/K)' by, for x C G and f E CP(G/K)3

(L u)f = u(L -1f). x

Proposition 7.2. Assume that 1 < p < 2. Fix 4) E

IP(G), u E CP(G/K)° and define the function f as abo as above. cp,u

Then it is a Cc° function on G. Moreover, if u is left K-

invariant, then fcpu is K bi-invariant.

Proof. Let X E g. Then

,f,t(1- - t(x-1y)} IL=Ticpt(exp(-tX).xly) t(x.exptX)Y\Y/Lx('t(Y)1

converges to

t ( -X;x ly) = ((-X)01))(Lx(1)t)(y)

in the topology of CP(G/K) as t 0. Hence

1 r ff(1),u(x.exp tX) f(1),u(x)}

1 =LIET IL(x.exp tx)tLxg't11'

This tends to u[(X 0 1)(Lx$ )] by the continuity of u as t

0. Taking account of that (X 0 1)(Lx(pt) E CT(G/K), by

iteration we see that fcp,u is C . Ascis K bi-invariant,

obviously

(k E K, x, yG) Lx0(Y) = Lx(I'(Y)

— 38—

holds. On the other hand, if u is left K-invariant, we then

have

f (1),u(kx) = 1-1[Lkxq4tI = u[LkL0] = (L _111)[1,0t

= u[LOt] = f(1),u(x).

These prove that fcbu is K bi-invariant.

Theorem 7.3.• Let 1 < p, q < Assume that B is a

bounded linear mapping from LP(G/K) into Lq(G/K) and com-

mutative with G translations. Then there exists a unique

uECP(K\G)' which is right K-invariant and

Bci) = u *

for all (1) E IP(G).

Proof. First of all we prove that, if cb E CP(G/K),

then Bci) has all left Lq-derivatives and DB = BD holds

for each D E G. In fact, for X E g, by the assumption on

B, we have

,T((Lexp tX - = B( -f(Lexp (P))•

Since

T (Lexp tX(P q4) ((-K) 0 1"

in LP by Proposition 6.1, from the continuity of B it follws

— 39 —

that

(Lexp tX(B0) - B(((-X) 0 1)0)

in This This proves that 130 has all left Lg-derivatives

of first order. By iteration we obtain our assertion. There-

fore, by Proposition 6.6 and the above argument, there exist

a continuous function g0on G/K and a constant C > 0 such

that

1) 130= gq) almost everywhere,

2) ly1)I < C HDBO degD<2n

< CIIBH Y HDOH_. degD<2n

The last inequality shows that the mapping u1 : g(1)

is continuous linear functional on CP(G/K) in LP topology.

Moreover, since

g(x) = B(L_10)(x) = L_1(B0)(x) = B0(kx) = go(kx)-1L(k

holds almost everywhere, here L(x-1) denoting L _1 for

x e G, we have

u,(L _10) = yk.1) = gq)(1) = u1(0).

k

This shows the invariance of u1 under K left-translations.

We next prove that u = ul has the property in the

— 40 —

theorem. For (1) E CP(G),

u * O(x) = u(R _1(1)t) = ut[I, _10 = ul(L _10 x x x

= g -1(1) = B(L_10(1) = (L_,Bep)(1) L(x)(1)

= Bcp,(x)

holds almost everywhere. The uniqueness of such u E CP(K\G)'

is proved as follows. If u' E CP(K\G)' is right K-invariant

and

u' * (1) =

for all yb E IP(G), it follows from the above argument that

u't(L _10 = u/(L x x

Denoting u't = u' we have from this that

u1(L -1(1b) = ui,(L-1cb) x x

holds for almost all x E G and all (1) E IP(G). As ui and

u1 -1are continuous and L E CP(G/K), the last equality x

holds for all x E G. Especially, we have for all qh E IP(G)

L11(0 = ul(q)).

Now if f E CP(G/K), by making use of the left K-invariance

—41—

of u' we have

ut1(f) = (Lk u'1)(f) = ui1(L f)

k -1(k E K).

Integrating this on K,

and the correspondence

as u' is continuous linear mapping

f f, f(x) f(kx) dk,

K

is continuous from CP(G/K) into (G), we obtain

J K

u'1(L -1f) dk = u'(f). 1

Thus

ul(f) = (T)ut= u 11 (f) = g

f(1) = gf(l) = u1(f).

This proves the theorem.

References

[1] M. Eguchi, Asymptotic expansions of Eisenstein integrals

and Fourier transform on symmetric spaces, J. Funct.

Anal., 31 (1979).

[2] M. Eguchi, An application of topological Paley-Wiener

theorems to invariant differential equations on sym-

metric spaces. Lecture Notes in Mathematics, 739

(1979), 192-206, Springer-Verlag, Berlin, Heidelberg,

New York.

—42—

[3]

[4]

[57

[6]

[7]

[8]

[9]

[10]

[11]

[12]

M. Eguchi, M. Hashizume and K. Okamoto, The Paley-Wiener

theorem for distributions on symmetric spaces.

Hiroshima Math. J., 3(1973), 109-120.

M. Eguchi and K. Okamoto, The Fourier transform of the

Schwartz space on a symmetric space. Proc. Japan

Acad., 53(1977), 237-241.

Harish-Chandra, Spherical functions on a semisimple Lie

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553-613.

Harish-Chandra, Discrete series for semisimple Lie

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Harish-Chandra, Harmonic analysis on real reductive Lie

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Anal., 19(1975), 104-204.

S. Helgason, Differential Geometry and Symmetric Spaces,

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E.M. Stein and G. Weiss, Introduction to Fourier AnaZy-

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(1980, January)

— 43—