Harmonic analysis and integral geometry 9781584881834, 1584881836, 9781138441743

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Harmonic analysis and integral geometry

CHAPMAN & HALL/CRC Research Notes in Mathematics Series Main Editors H. Brezis, Universite de Paris R.G. Douglas, Texas A&M University A. Jeffrey, University o f Newcastle upon Tyne (Founding Editor)

Editorial Board H. Amann, University o f Zurich R. Aris, University o f Minnesota G. I. Barenblatt, University o f Cambridge H. Begehr, Freie Universitat Berlin R Bullen, University o f British Columbia R.J. Elliott, University o f Alberta R.P. Gilbert, University o f Delaware D. Jerison, Massachusetts Institute o f Technology B. Lawson, State University o f New York at Stony Brook

B. Moodie, University o f Alberta S. Mori, Kyoto University L.E. Payne, Cornell University D.B. Pearson, University o f Hull 1. Raeburn, University o f Newcastle, Australia G.F. Roach, University o f Strathclyde I. Stakgold, University o f Delaware W.A. Strauss, Brown University J. van der Hoek, University o f Adelaide

Submission of proposals for consideration Suggestions for publication, in the form of outlines and representative samples, are invited by the Editorial Board for assessment. Intending authors should approach one of the main editors or another member of the Editorial Board, citing the relevant AMS subject classifications. Alternatively, outlines may be sent directly to the publisher's offices. Refereeing is by members of the board and other mathematical authorities in the topic concerned, throughout the world.

Preparation of accepted manuscripts On acceptance of a proposal, the publisher will supply full instructions for the preparation of manuscripts in a form suitable for direct photo-lithographic reproduction. Specially printed grid sheets can be provided. Word processor output, subject to the publisher's approval, is also acceptable. Illustrations should be prepared by the authors, ready for direct reproduction without further improvement. The use of hand-drawn symbols should be avoided wherever possible, in order to obtain maximum clarity of the text. The publisher will be pleased to give guidance necessary during the preparation of a typescript and will be happy to answer any queries.

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Massimo A Picardello

Harmonic analysis and integral geometry

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

A CHAPMAN & HALL BOOK

CRCPress Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 First issued in hardback 2019 © 2001 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Infonna business No claim to original U.S. Government works ISBN-13: 978-1-58488-183-4 (pbk) ISBN-13: 978-1-138-44174-3 (hbk) This book contains infonnation obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and infonnation, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if pennission to publish in this fonn has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as pennitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any fonn by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any infonnation storage or retrieval system, without written pennission from the publishers. For pennission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Library of Congress Cataloging-in-Publication Data Harmonic analysis and integral geometry I edited by Massimo A. Picardello. p. cm.--(Chapman & HalUCRC research notes in mathematics series; 422) Includes bibliographical references. ISBN 1-58488-183-6 (alk. paper) I. Harmonic analysis--Congresses. 2. Integral geometry--Congresses. I. Picardello, Massimo A.. 1949- II. Series. QA403 .H2225 2000 515' .2433--dc21

Library of Congress Card Number 00-055484

00-055484

Preface This volume contains the proceedings of the first Summer University of Safi, Morocco. The Summer University of Safi is an annual advanced re­ search school and congress, supported by several institutions in Morocco, with a significant contribution by the University “Hassan II” of Casablanca. The beginning sessions were devoted to mathematical research. The sub­ ject of the first session, held July 15-20, 1998, was “Harmonic Analysis and Integral Geometry.” It was a lively and very successful scientific event, at­ tended by many of the research leaders in this field, who gave individual lectures and courses. The extensive participation of mathematicians from Maghreb, whose conferences gave full evidence of the advanced level of local research in mathematics, confirms the continuing tradition of mathematical excellence in North Africa that began thousands of years ago. Besides the contributors to this volume, many other participants held conferences at the school. To them, and to all the other participants, I wish to express my gratitude for the stimulating scientific environment that they helped create. Massimo Picardello

Contents

P reface F ulton B. G onzalez John’s Equation and the Plane to Line Transform on R3 ..............1 T omoyuki K akehi Radon Transforms on Compact Grassmann Manifolds and Invariant Differential Operators of Determinantal T y p e.......9 T akaaki N omura Invariant Berezin Transform s................................................................ 19 S imon G indikin Integral Geometry on Hyperbolic S p aces.........................................41 M assimo A. P icardello The Geodesic Radon Transform on T rees.........................................47 E nrico C asadio Tarabusi, J oel M. C ohen and F lavia C olonna The D istribution-V alued Horocyclic Radon Transform on T rees.........................................................................................................55 Laura Atanasi Integral Geometry on Affine B uildings...............................................67 A hmed A bouelaz Integral Geometry in the Sphere Sd .................................................... 83 A ndrea D ’A gnolo and C orrado M arastoni A Topological Obstruction for the Real Radon Transform.......127 Hacen D ib and Mohammed M esk On Laguerre Polynomials of two V ariables..................................... 135 S amira Ibenmouloud and M ohamed S bai Poisson Transform on H3 ....................................................................... 139 H assan S ami Transfert Formula in the Real Hyperbolic Space B n...................145 A bderrahman E ssadiq q-Analogue of W atanabe Unitary Transform Associated to the (/-Continuous Gegenbauer P olynom ials.........155

M eryem E l B eggar Realization of a Holomorphic Discrete Series o f the Lie Group 517(1,2) as Star-Representation

. 163

JOHN’S EQUATION AND THE PLANE-TO-LINE TRANSFORM ON R3 FULTON B. GONZALEZ

1. Introduction

In this note we provide a range characterization result for the plane-toline transform on R3, in terms of the ultrahyperbolic equations of Fritz John [2]. This equation was used to characterize the range of the X-ray transform on rapidly decreasing functions on R3, and has the feature of being invariant under the group of orientation-preserving rigid motions of R3. While this range result is likely not new, the proof below is fairly elementary and uses only basic facts from vector calculus, Riemannian geometry, and a classical integral formula involving Legendre polynomials. 2. T he P lane- to-L ine T ransform

Let M be a manifold acted on by a Lie group G. We denote the Lie group action by (g,p) »-*» g · p. Then G also acts on the functions on M by g · f(x ) = f(g ~ *l - x). By differentiating this action, we obtain an action, via vector fields, of the Lie algebra g of G on the smooth functions of M: Z · f(x ) = d /d t(f (exp(-tZ) · *))Ι*=ο, for all Z G g, / G By composing the above vector fields, we obtain a representation of the universal enveloping algebra u(g) on Each element of u(g) then acts as a differential operator on M (possibly the zero operator). If U G u(g), g G G, and / G C°°(M), then g-(U · / ) = (Ad(g)U) · (g · /). Thus if G is connected and U belongs to the center 3(g) of u(g), then U acts as a G-invariant differential operator on M. Let G (l,3) and G(2,3) denote, respectively, the manifolds of unoriented lines and planes in R3. These are vector bundles over the compact Grassmannian manifolds Gi,s and £ 2,3 of lines and planes, respectively, through the origin in R3. (Both spaces equal projective 2-space RP2.) The projec­ tion of = (D · t.p) o 1r11 • By (1) and (2), we have (6)

D · ci>(w, x)

= (E1X23- E2X13 + E3X12)xci>(w, x) ((EI)x(X23)w- (E2)x(XI3)w

+ (E3)x(XI2)w) ci>(w, x).

Here the subscript w indicates differentiation with respect to the first argument, and the subscript x differentiation with respect to the second argument. By part (a), the first term on the right vanishes, so that

D·ci>(w, x) = (- V xci>(w, x), ((X23)wci>(w, x), -(XI3)wci>(w, x), (XI2)wci>(w, x))). Now V xci>(w, x) is parallel tow, so we write V xci>(w, x) = h(w, x) w, for some hE C 00 (S2 x JR 3 . Thus,

D · ci>(w,x) = -h(w,x) [wi(X23)wci>(w,x)- w2(XI3)wci>(w,x)+ +w3(X12)wci>(w, x)]. If we fix x in the equation above, Lemma 2 below will show that the factor in brackets vanishes, which will prove that D · ci> = 0, and so D · 4> = 0 as

o

~ll.

Lemma 2.2. Let u(w) be a smooth function on 8 2 . Then

w1X23u(w)- w2X13u(w)

+ w3X12u(w) = 0.

u

Proof. Fix t > 0, and extend u to a smooth function on the spherical shell 1-t < llxll < 1 +~:on JR 3 by setting u(tw) = u(w) for all t E (1-t, 1 +~:) and wE 8 2. Lemma 2 will follow if we can show that x1X23u(x)- x2X13u(x) + x3X12u(x) = 0 on this shell. But one can deduce this immediately from equation (3). 0 We note that the plane-to-line transform R commutes with the action of E(3) on the smooth functions on G(2, 3) and G(1, 3), respectively. This can be seen directly from formula (5), for example. Differentiating inside the integral sign, we obtain R(U · t.p) = U · Rt.p for all U E u(e(3)) and t.p E C 00 (G(2,3)). Since D acts as the zero operator on 0(2,3), we obtain a necessary condition on the range of R, namely D · R( t.p) = 0 for all functions t.p E C 00 (G(2,3)). The theorem below shows that this condition is also sufficient.

Theorem 1. Let '1/J be a smooth function on G(1, 3). Then '1/J = Rt.p for some t.p E C 00 (G(2,3)) if and only if D · '1/J = 0. We note that the equation D · '1/J = 0 is the group-theoretic version of John's differential equation on the line space G(1, 3). (See [3].)

Proof. Assume that D · '1/J = 0. Let W = '1/J o 1r1 • Then D · W = 0, so for each ( 0', X) E 8 2 X JR 3 , we have as in equation (6) 0

= (E1X23- E2X13 + E3X12)x W(O', x)

(7) =

+ ((EI)x(X23)u- (E2)x(XI3)u + (E3)x(XI22)u) ((EI)x(X23)u- (E2)x(XI3)u + (E3)x(X122)u) W(O',x),

W(O', x)

JOHN'S EQUATION AND THE PLANE-TO-LINE TRANSFORM ON

rre

5

since the first expression on the right above vanishes by Lemma 1 (a). Now the great circle transform is a bijection on the space of smooth even functions on S 2. Thus, for each x E JR3 , there exists a unique even function w f--> cllx(w) on S 2 such that

w(a,x) = 21 {

(8)

1f

}A(a)

cllx(w)da(w).

We write cllx(w) = cll(w,x). By differentiating inside the integral sign and using uniqueness, or by directly using the inversion formula for the great circle transform on C~en(S2 ) it is easy to see that cll(w, x) is a smooth function on S2 x JR3 . If we can show that the gradient \7xcll(w,x) is a multiple of w, then there will exist a smooth function r.p on the plane space G(2, 3) such that r.p o 1r11 = ell and Rr.p = '1/J. We note that the great circle transform commutes with the action of S0(3) on S2 , so applying (7) inside the integral on the right-hand side of equation (8), we obtain 0

=

J

[(X23)w(El)xcll(w, x)- (X13)w(E2)xcll(w, x)

A(a)

+ (X12)w(E3)xcll(w, x)] da(w).

For fixed X E JR3 ' the integrand above is an even function of w E S2 ' so that by the injectivity of the great circle transform,

(9) 0 = (X23)w(El)xcll(w, x)- (X13)w(E2)xcll(w, x)

+ (X12)w(E3)xcll(w, x).

We now consider the even, JR3 -valued vector field

G(w)

G on S2 given by

= -\7 xcll(w, x) = ((El)xcll(w, x), (E2)xcll(w, x), (E3)xcll(w, x)).

Our objective is to prove that G(w) is a multiple of w. To simplify notation, we write G(w) = (G1(w),G2(w),G3(w)); then each Gi is a smooth even function on S 2 ' and by (9) we have

X23 · G1 - X13 · G2 + X12 · G3 =

o.

We resolve G(w) into radial and tangential components: G(w) = T(w) + F(w), where T(w) ..l w and F(w) II w. T(w) and F(w) are smooth even vector fields on S2 . It suffices for us to show that T(w) = 0 for all wE S2 . Writing f = (T1, T2, T3), F = (H, F2, F3), we have (X23T1- X13T2 + X12T3)(w) + (X23F1- X13F2 + X12F3)(w) = 0. It is easy to show that (X23Fl - x13F2 + x12F3)(w) = 0: write F(w) f(w)w, where f is a smooth (odd) function on S2. Then

(10) (X23Fl- x13F2 + x12F3)(w) = W1X23f(w)- w2X13f(w) + W3X12f(w) + f(w) (X23W1- X13W2 + X12w3) =0 by (2.2) and a trivial direct calculation.

=

FULTON B. GONZALEZ

6

Also, applying the condition (a, Y'xW(a,x)) = 0 to (8) one obtains

0= { (a,Y'x~(w,x))du(w) }A(u) = {

(a, G(w)) du(w)

f

(a, T(w)) du(w)

}A(u)

(11) =

JA(u)

= {

JA(u)

+ f

JA(u)

(a, F(w)) du(w)

(a, T(w)) du(w),

since F(w) .l a on A(a). Thus T(w) is an even vector field on (11) and the first order equation

s'2 satisfying

(12)

We will show that the only such vector field is 0. First we claim that there exists an odd 0 00 function t(w) on 8 2 such that T(w) = grads2t (w), where grads2 denotes the gradient on (the Riemannian manifold) 8 2 . For this, we extend f as usual to a smooth vector field on a spherical shell 1- E < llxll < 1 + E, with T(rw) = T(w) for all r E (1- E, 1 +E). Equation (12) still holds on the spherical shell, with w replaced by x. Using (3), (12) becomes (curlT(x),x) = 0. By Stokes' theorem, this implies that the line integral off along any smooth closed curve in 8 2 vanishes; since 8 2 is simply connected, this in turn implies that T(w) = grads2t (w), for some t E 0 00 (82 ). Resolving t into odd and even components, we note that the even component must be constant, so we can assume that tis odd. Let H (a) denote the hemisphere on the side of the great circle A (a) away from a. Since a is the outward-pointing unit normal on its boundary in 8 2 , we can apply the divergence theorem (on 8 2 ) to the integral in (11) to obtain 0= =

f

(a, gradsd(w)) du(w)

f

(Ls2t)(w) dw

JA(u) Jn(u)

for all a E 8 2 , where Ls2 denotes the Laplace-Beltrami operator on 8 2 , and dw represents the area element on 8 2 . This shows that Ls2t(w) is a smooth odd function on 8 2 , whose integral vanishes over all hemispheres. The following lemma shows that the only such function is zero, which means that t is harmonic on 8 2 . Since 8 2 is compact and t is odd, we must have t(w) = 0. Then T(w) = grads2 t(w) = 0, proving the theorem. D

Lemma 2.3. LetT denote the hemisphere transform on 8 2 , T(h)(a) = In(u) h(w) dw, for h E 0 00 (82 ), a E 8 2 . Then T(h) = 0 only if h is an even function.

JOHN'S EQUATION AND THE PLANE-TO-LINE TRANSFORM ON JR3

7

commutes with the action of 80(3) on S 2 , it can be diagonalized. Let 1tm denote the space of degree m spherical harmonics on S 2 , m = 0, 1, 2, ... ; we write h(w) = 2:~= 1 hm(w) where hm(w) E 1tm. 80(3) acts irreducibly on each 1tm, soT must equal a scalar operator Cm there. Cm can be calculated by integrating a zonal harmonic in 1tm over an appropriate hemisphere; this gives

Proof. Since

T

Cm =

=

r/2 Pm(cosO)sin(}d(} Jo

1 1

Pm(x) dx,

where Pm(x) is the Legendre polynomial of degree m. It is a classical result (see [5]) that the latter integral equals 1 if m = 0, 0 if m is even, and D .- 1/ 2 dt.

Proposition 3.2 (Dougall's formula). If m1 + m2 + l = 2s E 2 Z+ and if the triangular inequality lm1 - m2l ~ l ~ m1 + m2 holds, then D>.(m1, m2, l)

21- 2>. 1r f(s + 2>.) f(s- m 1 + >.)f(s- m2 + >.)f(s - l + >.) = f(>.) 4 f(s + >. + 1) f(s- m 1 + 1)f(s- m2 + 1)f(s - l + 1)" Otherwise one has D>.(m1, m2, l) = 0.

See the paper [24] or the book [46, 9.4.11] for a proof. Since the Gegenbauer polynomials are orthogonal over the interval (-1, 1) with respect to the weight function w(t) = (1 - t 2 )>.- 1/ 2 (cf. [29, Chapter V]), we have >.

m1 +m2

>.

"""'

Cml (t)Cm2 (t) =

D>. (

m1' m2, v>-(l, l, 0)

Li

l)

>.

Cz (t).

l=lm1-m2l l::m1+m2 (2) In particular, setting o:(m,j) = v. ( z, z) d~-t >. be the Berezin measure associated to H~ (D). By (31) and (32), we see that d~-to is a positive number multiple of the Ginvariant measure described in (30). Let K be the stabilizer at (0, e) E D in G. Then K is a maximal compact subgroup of G, and L 2 (D, d~-to) is naturally identified with L 2(G I K). Since G is a semisimple Lie group with trivial center, the theory of Helgason's Fourier transform [23] on GIK can be applied here to get a G-irreducible decomposition of L 2 (D, d~-to). To do so, it is necessary to specify some of the subgroups of G and of the Lie subalgebras of g = Lie (G). Elements of g are holomorphic polynomial vector fields p(z)oloz on D, and the bracket operation in g is the Poisson bracket

[p(z):z' q(z):z] = (p'(z)(q(z)) -q'(z)(p(z))):z. We will drop the symbol I f)z in what follows to facilitate the notation. Thus we simply think of elements of g as holomorphic polynomial mappings Z--+ Z. With the JTS frame {e1, ... ,er} we set Cl = L:~~~rR(ejDej)· Then Cl is a commutative subalgebra of g such that ad Cl consists of semisimple operators on g. To see the positive restricted root spaces, we decompose the space U as U = L:t~~r Uj, where

a

Ui

=

{u E U: (ekDek)u

= t&jkU

(1 ~ k ~ r)}.

Recalling (28), we set

g?i = { x D ei : x E Vii}

g~/ 2 ={u+2eDu:uEUj} g]k = {ia: a E Vjk}

( 1 ~ i < j ~ r), (j=1, ... ,r),

(1 ~ j ~ k ~ r),

and n = (L:f. on a'f_ such that upon identifying L 2 (D, dJ.Lo) with L 2 (G/K), we have dv [ 61 B>. ~ la• b>.(v) lc(v)l2 · +

In other words, we have (\J!B>.f)(v) = b>.(v)\J!f(v) for any f E Cgo(D). Let . is a symmetric non-negative function on D x D such that for every z E D,

(42)

{ A>.(z, z') dJ.Lo(z') =

jD

/

) {

~>. Z,Z }D

l~>.(z, z')l 2 dJ.L>.(z') =

1.

TAKAAKI NOMURA

38

Moreover AA is G-invariant by (9):

AA(g · z, g · z') = AA(z, z')

(43)

(g E G).

We set aA(gK) = AA(g · (O,e), (O,e)) forgE G. It is clear from (42) and (43) that aA E L 1 (K\G/K). Proposition 4.1. The function bA in (39) equals (aA)~, the spherical Fourier transform of aA. Theorem 6. If .A> p- 1, one has

(aA)~(ll) = fn( -ill+ p +.A- Njr) fn(ill*- p* +.A) fn(.A- N /r) rn(.A) where(*= ((r, ... , ( 1) for ( = ((1 , ... , (r) E ([;r =a(;.

(ll E a*),

Consequently the direct integral decomposition

BA ~

(:B fn( -ill+ p +.A- Njr) fn(ill*- p* +.A)

la+

fn(.A- Njr) fn(.A)

~ lc(ll)l2

can be considered as the spectral decomposition of the selfadjoint operator BA. Let us outline briefly the proof of Theorem 6. By (37) and (40), what we have to compute is

(aA)~(ll) =

2rA

fv

lr;,A(z, (0, e)) 12 ~~iv+p(z) dJLA(z).

By using (31), (32), and (41), the computation will be terminated by the following integral formula, a corrected version of the formula in [22, Proposition 2.6]. Proposition 4.2. If Repj > d(j- 1)/2, Re Qj > d(j- 1)/2 (j = 1, ... , r), then fn(P) fn(q) Llp-njr(x)Ll-p-q+2po(e + x) dx = r ( ) ' n np+q where 2p0 (H) = trad (H)Ino (HE a).

1

Remark 7. Since fn(s*) = fn(s + 2p0 ) (cf. [18, p.312]), we see by Proposition 4.2 that if both p and q are scalars (p, ... ,p) and (q, ... , q), respectively, then f A(x)p-n/r A(e + x)-p-q dx = fn(p) fn(q).

h

This is the formula given by [18, Exercise VII.4].

~~+~

REFERENCES [1] J. Arazy, S. D. Fisher, J. Peetre, Hankel operators on weighted Bergman spaces, Amer. J. Math. 110 (1988), 989-1054. [2] S. Axler, D. Zheng, The Berezin transform on the Toeplitz algebra, Studia Math. 127 (1998), 113-136. [3] D. Bar-Moshe, M. S. Marinov, Berezin quantization and unitary representations of Lie groups, Amer. Math. Soc. Trans!. 177 (1996), 1-21. [4] F. A. Berezin, Quantization, Math. USSR Izv. 8 (1974), 1109-1165.

INVARIANT BEREZIN TRANSFORMS

39

[5] F. A. Berezin, Quantization in complex symmetric spaces, Math. USSR Izv. 9 (1975), 341379. [6] F. A. Berezin, General concept o f quantization, Comm. Math. Phys. 4 0 (1975), 153-174. [7] F. A. Berezin, A connection between the co- and contravariant symbols of operators on classical complex symmetric spaces, Soviet Math. Dokl. 19 (1978), 786-789. [8] C. A. Berger, L. A. Coburn, Toeplitz operators on the Segal-Bargmann space, Trans. Amer. Math. Soc. 301 (1987), 813-829. [9] C. A. Berger, L. A. Coburn, K. H. Zhu, Function theory on Cartan domains and the BerezinToeplitz symbol calculus, Amer. J. Math. 110 (1987), 921-953. [10] M. Cahen, S. G utt, J. Rawnsley, Quantization o f Kahler manifolds I: geometric interpretation of Berezin’s quantization, J. Geom. Phys. 7 (1990), 45-62; II, Trans. Amer. Math. Soc. 3 3 7 (1993), 73-98; III, Lett. Math. Phys. 3 0 (1994), 291-305; IV, Lett. Math. Phys. 3 4 (1995), 158-168. [11] R. R. Coifman, G. Weiss, ‘Analyse harmonique non-commutative sur certains espaces ho­ mogenes’, Lecture Notes in Math. 242, Springer-Verlag, Berlin-Heidelberg-New York, 1971. [12] H. Ding, Ramanujan’s master theorem fo r Hermitian symmetric spaces, Ramanujan J. 1 (1997), 35-52. [13] M. Englis, Functions invariant under the Berezin transform, J. Functional Anal. 121 (1994), 133-154. [14] M. Englis, Toeplitz operators and the Berezin transform on H 2, Lin. Alg. Appl. 2 2 3 /2 2 4 (1995) , 171-204. [15] M. Englis, Berezin transform and the Laplace-Beltrami operator, St. Petersburg Math. J. 7 (1996) , 633-647. [16] M. Englis, Berezin quantization and reproducing kernels on complex domains, Trans. Amer. Math. Soc. 3 48 (1996), 411-479. [17] J. Faraut, Analyse harmonique etfonctions spiciales, in ‘Deux Cours d ’Analyse Harmonique’, Birkhauser, Basel-Boston, 1987, 1-151. [18] J. Faraut and A. Koranyi, ‘Analysis on symmetric cones’, Oxford M athematical Monographs, The Clarendon Press, Oxford, 1994. [19] E. Fujita, T. Nomura, Spectral decompositions of Berezin transformations on Cn related to the natural U(n)-action, J. Math. Kyoto Univ. 36 (1996), 877-888. [20] E. Fujita, T. Nomura, Berezin transforms on 2 x 2 matrix spaces related to the U( 2) x U{ 2)action, Int. Equations and Operator Theory, 32 (1998), 152-179. [21] J. E. Gilbert, M. A. M. Murray, ‘Clifford algebras and Dirac operators in harmonic analysis’, Cambridge Studies in Adv. Math. 26, Cambridge Univ. Press, Cambridge, 1991. [22] S. G. Gindikin, Analysis in homogeneous domains, Russian Math. Surveys, 19 (4) (1964), 1-89. [23] S. Helgason, Geometric Analysis on Sym metric Spaces, Amer. Math. Soc., M athematical Surveys and Monographs 39, Providence, 1994. [24] H.-Y. Hsii, Certain integrals and infinite series involving ultraspherical polynomials and Bessel functions, Duke Math. J. 4 (1938), 374-383. [25] K. Koike, On representation of the classical groups, Amer. Math. Soc. Transl. 183 (1998), 79-100. [26] K. Koike, I. Terada, Young-diagrammatic methods fo r the representation theory o f the clas­ sical groups of type B n , Cn , D n , J. Algebra, 10 7 (1987), 466-511. [27] T. H. Koornwinder, Clebsch-Gordan coefficients fo r SU (2) and Hahn polynomials, Nieuw Arch. Wisk. 29 (1981), 140-155. [28] O. Loos, ‘Bounded symmetric domains and Jordan pairs’, Lecture Notes, Univ. California at Irvine, 1977. [29] W. Magnus, F. Oberhettinger, R. P. Soni, ‘Formulas and theorems for the special functions of mathematical physics’, Grundlehren math. Wissensch. 52, Springer-Verlag, New York, 1966. [30] V. F. Molchanov, Quantization on para-Hermitian symmetric spaces, Amer. Math. Soc. Transl. 175 (1996), 81-95. [31] C. Moreno, *-products on some Kahler manifolds, Lett. Math. Phys. 11 (1986), 361-372. [32] T. Nomura, Berezin transforms and group representations, J. Lie Theory 8 (1998), 433-440. [33] B. 0 rsted , G. Zhang, Weyl quantization and tensor products of Fock and Bergman spaces, Indiana Univ. Math. J. 4 3 (1994), 551-583.

40

TAKAAKI NOMURA

[34] J. Peetre, The Berezin transform and Ha-plitz operators, J. O perator Theory 24 (1990), 165-186. [35] J. Peetre, G. Zhang, A weighted Plancherel formula III. The case o f the hyperbolic m atrix ball, Collect. Math. 43 (1992), 273-301. [36] F. Radulescu, ‘The Γ-equivariant form of the Berezin quantization of the upper half plane’, Mem. Amer. Math. Soc. 630, Amer. Math. Soc,. Providence, Rhode Island, 1998. [37] J. Repka, Tensor products of holomorphic discrete series representations, Canad. J. Math. 31 (1979), 836-844. [38] I. Satake, ‘Algebraic structures of symmetric domains’, Iwanami Shoten and Princeton Univ. Press, Tokyo-Princeton, 1980. [39] K. StroethofF, The Berezin transform and operators on spaces o f analytic functions, Banach Center Publ. 38 (1997), 361-380. [40] A. Unterberger, H. Upmeier, The Berezin transform and invariant differential operators, Comm. Math. Phys. 164 (1994), 563-597. [41] H. Upmeier, Jordan algebras in analysis, operator theory and quantum mechanics, CBMS Reg. Conf. Ser. Math. 67, Amer. Math. Soc., Providence, Rhode Island, 1987. [42] G. van Dijk, S. C. Hille, Canonical representations related to hyperbolic spaces, J. Functional Anal. 147 (1997), 109-139. [43] G. van Dijk, S. C. Hille, Maximal degenerate representations, Berezin kernels and canoni­ cal representations, in ‘Lie groups and Lie algebras’, B. P. Komrakov et al. (eds.), Kluwer Academic Publ., Dordrecht, 1998, 285-298. [44] G. van Dijk and V. F. Molchanov, The Berezin form fo r rank one para-Hermitian sym metric spaces, J. Math. Pures et Appl. 77 (1998), 747-799. [45] M. Vergne, H. Rossi, Analytic continuation of the holomorphic discrete series o f a sem i­ simple Lie group, A cta Math. 136 (1976), 1-59. [46] N. Ja. Vilenkin, A. U. Klymik, ‘Representation of Lie Groups and Special Functions’, Vol. 3, Mathematics and its Applications 75, Kluwer Academic Publishers, Dordrecht, 1992. [47] N. R. Wallach, The analytic continuation o f the discrete series. I, Trans. Amer. Math. Soc. 251 (1979), 1-17; II, ibid., 19-37. D epartment of Mathematics, Faculty of Science, Kyoto U niversity, K itashirakawaO iwake- cho, Sakyo- ku 606-8502, Kyoto , J apan E-mail address: nomura®kusm.kyoto-u.ac.jp

INTEGRAL GEOMETRY ON HYPERBOLIC SPACES SIMON GINDIKIN ABSTRACT. We discuss a general way to write inversion formulas for different versions of spherical transforms on the hyperbolic spaces, including the geodesic and horospherical Radon transforms.

We will discuss using the example of the hyperbolic integral geometry some important general constructions, arising from the Gelfand-Graev-Shapiro method of operator K.. Basically we specialize here more general constructions of the conformal integral geometry on quadrics [2, 3]. We will obtain the inversion formulas for the geodesic and horospherical Radon transform (which is of course well known) as partial cases of more general formula. It is important that the geometrical background of this construction is elementary and does not use group considerations. We will consider the realization of n-dimensional hyperbolic sp~ce Hn as the upper sheet of the two-sheeted hyperboloid in Rn:

= xi We consider only functions f (1)

Ox

x~ - · · · - x~+ 1

= 1,

x1

> 0.

E C~(Hn). Sometimes it is convenient to consider them as even functions on the complete hyperboloid. Let us consider the bilinear form, corresponding to the quadratic form (1):

(2) Let

~·X= 6x1- 6x2- · · · - ~n+1Xn+1· L(~,p)

be the sections of the hyperboloid (1) by hyperplanes ~·x=p.

In hyperbolic geometry there are different kind of spheres, including as degenerate cases geodesic hyperplanes (p = 0) and horospheras (D~ = 0). The set of all spheres depends on (n + 1) parameters; subsets of geodesic hyperplanes and horospheras are n-parametric. The group 80(1, n) acts transitively on Hn, but we will not extensively use this action. Families of geodesic hyperplanes and horospheras are only invariant subfamilies of spheres. Let dx = dx1 1\ dx2 1\ · · · 1\ dxn+l and

(3)

J..L(x, dx)

= d(Dx)Jdx =

dx 2 1\

... 1\

2x1

dxn+l

be the invariant measure on Hn. 1991 Mathematics Subject Classification. Integral Transforms. Operational Calculus 1991 MSC: 44A12. Key words and phrases. Hyperbolic spaces, hyperbolic spheras and horospheras, operator '"41

SIMON GINDIKIN

42

Let us consider the spherical Radon transform on Hn:

Rf(~,p)

(4)

= {

}Hn

f(x)8(~ · x- p)J.L(x, dx),

pER.

We have Rf(~,p) = 0 if 0~- p 2 ;::: 0. Of course we can interpret this transform as the usual Radon transform at Rn+ 1 of a function, multiplied by the 8-function on Hn. It is also convenient to consider another version of the Radon transform: f(x) (5) f(~,p) = ~ J.L(dx), p E C\R,

1

A

Hn

·X- p

where we use the Cauchy kernel instead of 8-function. This version is more convenient in some relations; in particular, there is no difference in the inversion formulas for even or odd dimensional cases. There is a simple connection between them: 1 Rf(~,p)

= (21ri) (f(~,p- iO)- f(~,p + iO)),

}(~,p)

A

=

A

1 RJ(~, 00

-oo

q) dq, p E C\R.

q- p

So R f is the even part of j. As we already mentioned, the family of hyperplanes depends on n + 1 parameters. The basic problem is the reconstruction off through restrictions of Rf (or}) on different n-parametric subfamilies. We will denote through [a1, a2, ... , an+l] the determinant with the columns a1, ... , an+l· Some of them can be columns of 1-forms and we use the cup-product for computing the determinant. As a result a determinant with identical columns of 1-forms can be different from zero. The notation a{k} means that the column-form a repeats k times. In the projective analysis the Leray form .C(~, ~)

(6)

=

[~, ~{n}]

plays an important role (a projective version of the volume element). In our computations we find useful the following formula, which can be verified directly using the Euler formula for homogeneous functions (cf. also [2]):

d[a(~), ~' d~{n- 1 }] =

(7)

cL: ~jaai ;a~j).C(~, d~),

where a(~) is a column of homogeneous functions ai(~) of the degree -n. Our principal object is the differential form

(8)

r;,f =

(~. (~(:)w))nJ.L(X, dx) A [x + w, ~' ~{n- 1 }],

Ox= Ow= 1.

Here we write the coordinates of the point x + w as the column and a point w E Hn is fixed. The basic fact is

Proposition 1. The form r;,f is closed. The proof is a direct application of (7). On x we have the form of maximal degree, so we need only care about the differentiation on~- We include the factor (~ · (x- w))-n in the first column and after the application of (7) we obtain the factor Ox - Ow = 0.

INTEGRAL GEOMETRY ON HYPERBOLIC SPACES

43

Let us explain how we will use the form "'f for inversion formulas. We have u E Hn and for the point w we take the boundary values when w ---+ u E Hn, ~w ---+ 0,.; · ~w < 0, so that we have in (8)

(9) The form is positive homogeneous (relative to.;---+ p.;, p > 0) of degree zero, but it has nontrivial even and odd parts following the decomposition · ( l)n-1 (10) (t ± iO)-n =en =F z1r ()Cn- 1)(t). (n- 1)! As the result "'f can be moved down to the base of the fibering on.;: Rn+1\{0}---+

sn

relative to positive dilations. In "'f we can keep only the even part of (9) for odd n and only the odd part for even n. Correspondingly we will have local or nonlocal formulas. We will integrate "'f on cycles Hn x "(, where 'Y is an cycle of sections L(.;,.; · u) passing through u E Hn; we use.; =j:. 0 for parameterization of its points.

Theorem 1. We have

(11) where c( "() depends only on the cycle 'Y and is independent of f.

Depending on the parity of n we can keep in "'f only local or nonlocal parts (the integration of any other one will give zero for all cycles 'Y). It is the usual trick with the parity of Rf on (.;, p ). The sketch of the proof for an important class of cycles will be given below. Now we explain how to use (11) for inversion formulas. For a fixed .; we integrate the first part of ( 11) on Hn and express the integral through Rf (or}). In the general situation the column x+u can deliver in the result of the integration an unpleasant differential operator on Rf, but in some cases, among which are our examples, it is possible to eliminate this as an operator. It is convenient to decompose the determinant [x + u, .;, dc;{n- 1}] in the sum of two: [2u, .;, d.;{n- 1}] +[x-u,.;, dc;{n- 1 }] and correspondingly to decompose the form: "'f (12)

{

JH"X"f

K, 1

f=

2

= "'' f

+"'"f. We have

,jJ~n- 1 )(.;,.;.u-iO)[u,.;,dc;{n- 1 }J,

(n- 1).

'Y

where f~n-1) is the partial derivation on p of the order n - 1. It is the principal term of the integral and in some important examples (but not always) the second integral is equal to zero. Basic examples 1. Let us start from the reconstruction f through Rf(.;,p) where

(13)

6 = 6.

44

SIMON GINDIKIN

This subfamily is invariant relative to translations on p and we can express J(~,p) through RJ(~,p) inside the subfamily. Conversely, this subfamily is not S0(1, n)-invariant. For the computation of the determinant let us subtract the second row from the first row. Then in the first row only the first element u1 - u2 is different from zero and for the first integral we have

{

}Hnx{f;,=6}

_ 4(ul- u2) { - (n- 1)! Js'!-1

"

(14)

=

c(r)f(u),

'(n-1)

!p

K

'!

.

(~,~ u

4(27ri)n-l (n- 1)! '

c(r) =

_ .

- -

zO)b=6.C(~,~)

~ = (6, ... '~n+d·

To prove (13) is sufficient to remark that we have here the inversion formula for the usual Radon transform of the function be continuous and bounded on r. Define a dual Radon transform R* by

R*¢>(x) = 2(qx + 1)

(2: Qyq~ Y"-'X

1 (

3 Y _ __!__

1

}~~x Qx -y3x

)¢>(!) drx(!)

for all x E T. Then R*R

= ELl€

as operators on f 1 (T), where

E

is a parity.

To complete the inversion of the geodesic Radon transform we need to invert the operator .Ll. The inversion of the Laplace operator is usually achieved through its fundamental solution, the Green function (or better, kernel) G(x, y) = L:~=O pn(x, y) (composition powers). If this series converges for some (hence all) x, y E T (i.e., if P is transient), it is easy to verify that the operator -G inverts .Ll on the space of finitely supported functions. However, the Green kernel is explicitly computed only on homogeneous trees. On general trees, a computable inversion formula was obtained for .Ll: Proposition 2.2 ([2, Proposition 2.8]). An inverse of the Laplace operator .Ll onf 1 (T) is the summation operator L given by Lf(x) = L:yET L(x, y)f(y), where

n

k-1 n

_1 L(x' Y) - Qy + 1 '"' L.J Qx + 1 n=0j=1 Qxj

of ,or each x, y E T ,

and {xj}o~~k is the path from xo = x to Xk = y.

On the other hand, if Tis homogeneous the Green kernel or, equivalently, the convolution powers of J.L1 are excellent computable tools to invert the Laplace operator. In this case the convolution kernel corresponding to R*R is a multiple of J.Lo + 2 L:~ 1 J.Lk· Then we can prove a more elegant inversion formula for the Radon transform (different from the previous one). One computes first the operator R*R. The computation, performed in [2, Proposition 3.2], yields R*R=

2 (q~ 1 ) (J.Lo+ t,2J.Ln)

onf1 (T).

Then one observes that in a homogeneous tree the family of operators J.Lk satisfies the product relations 1 q for each n > 0. J.L1J.Ln = q + 1 J.Ln-1 + q + 1 J.Ln+l These two formulas together easily yield * q(q- 1) (J.Lo- J.L1)R R = 2(q + 1)2 (J.Lo

(J.Lo + J.L1)E whence (J.Lo obtain:

+ J.L1),

2(q + 1) 2 1) (J.Lo- J.L1),

= q(q _

+ J.L1)E(R*R) = J.Lo + J.L1

and (J.Lo- J.L1)(R*R)E

= J.Lo- J.L1·

We

THE GEODESIC RADON TRANSFORM ON TREES

51

Theorem 2 ([2, Corollary 3.5]; see also [1]). The geodesic Radon transform on a homogeneous tree is inverted on l 1 (T) by cSR*R = id, where c =

!~: ~ ~ ~:,

and S is the convolution operator against the radial kernel J-to

+

2 L~=l (-l)nJ-tn· 2.3. Range. In order to describe the range of R we must first agree on an appropriate choice of its domain. Absolute convergence is convenient to deal with inversion, so the choice in the previous subsection was £1 (T). Here we want to consider, instead, the largest natural domain of R, that is, the space of all functions f on T such that LxE-y f (x) converges for every 1 E but not necessarily absolutely. However, we must define in which sense the doubly infinite series should converge. For this goal, consider the space A of functions fonT such that for each ray p = {p(j)}r;.:o the series Z f (p) = L:;o f (p(j)) converges-namely, in the natural order of consecutive vertices in a ray-but not necessarily absolutely. Now let p, a be disjoint rays with neighboring initial vertices, and let the geodesic 1 = pa be their union. Then we define the geodesic sum as Rf('Y) = Zf(p) + Zf(a); that is, in the doubly infinite series each infinite sum converges separately. Obviously R/(1) is independent of the way 1 is decomposed into a pair of disjoint rays. In this way we have introduced an operator Z from functions on T to functions on the set of rays R. The range of Z consists of functions a on R that respect the obvious compatibility conditions: whenever p, jj are disjoint rays with neighboring initial vertices such that pa = pu, then a(p) +a( a)= a(p) + a(u), and a vanishes at the empty ray. Therefore R factors as R=YZ,

r,

where the operator Yon functions on R is given by Ya('Y) = a(p) + a(a) if 1 = pa. Observe that Y is well defined on the functions that satisfy the compatibility conditions above, in particular, on the range of Z. We shall invert the operators Y, Z, hence the Radon transform R. However, for this goal we need to impose some decay conditions on the ranges of these operators.

Definition 1 (combs). A comb inTis a family (Pn)~00 , where the Pn's are pairwise disjoint rays, and their respective starting vertices Pn(O) form a geodesic that we call its supporting geodesic. Such geodesic decomposes into the rays P+ = {Pn(O)}n:;,:o and P- = {Pn(O)}n~O· Observe that if Pn is nonempty then Pn(O) is not flat (i.e., it does not have exactly two neighbors). Given a ray p, for every k denote by p(k) the ray {p(j)}j:;,:k; in particular, set p' = p(l).

Definition 2 (truncated comb condition). We say that a function on satisfies the truncated comb condition if for every comb (Pn) we have k-1

lim h,k-+oo

L

j=-h

(-l)i(PiPi+I)

+ (-l)k(PkP~))- (-l)h(P-hP~-h)) =

0.

r

52

MASSIMO A. PICARDELLO

Remark 1. It is immediately seen that the truncated comb condition holds for the functions in the range of the Radon transform that have a suitable decay at infinity, say, Zf(Pn) ~ 0 as n ~ oo on every comb (Pn)· We can finally introduce the appropriate conditions of decay at infinity on the domains of the operators above:

Definition 3 (domain of Rand Z). Denote by A the space of all functions fonT such that for each ray p the series Zf(p) = L~=O f(p(n)) converges, and such that for any comb (Pn). lim Zf(Pn) = 0 n-+oo Definition 4 (domain of Y). Denote by B the space of functions a on R that satisfy the compatibility conditions stated at the beginning of this section, and such that lim a(pn) = 0 for any comb (Pn) n-+oo and for any ray p. An elementary computation now yields the inverse of Z:

Proposition 2.3 ([3, Proposition 1.3]). {1) The operator Z is a bijection of A onto B. {2) For a ray p define Na(p) = a(p)- a(p'). Then Na(p) depends only on p(O), and N: B ~A is the inverse of Z. It remains to compute the inverse of the operator Y. This involves some technical difficulties if the tree contains flat vertices. For this case the reader is referred to [3, Sections 3 and 4]; we shall limit attention here to trees without flat vertices, except for a short remark at the end.

Definition 5 (half combs). A set H = (Pn)n~o of pairwise disjoint rays is a half comb if the Pn(O)'s form a ray, called the supporting ray of the half comb. (Observe that two disjoint half combs form a comb if the union of their supporting rays forms a geodesic.) Given a function ¢> on r satisfying the truncated comb condition, and given a half comb H = (Pn)n~o, define M¢>(H) = 2:~=0 ( -1)n¢>(PnPn+I)· We shall show that M, on an appropriate domain, is the inverse of Y. First of all, the following proposition shows that a function in the range of M is defined on R and satisfies the appropriate compatibility conditions.

Proposition 2.4 ([3, Lemmas 2.1 and 2.2]). {1) If T has no flat vertices, ¢> satisfies the truncated comb condition, and H = (Pn)n~O is a half comb, then M¢>(H) depends only on Po (and so it can be regarded as a function of the initial ray of the half comb). {2) Let po, Pl be disjoint rays that begin at neighboring vertices, so that PoP! is a geodesic. Then (PoPl) = M¢>(po) + M¢>(p1). The decay condition to impose on the domain of M are the natural ones: Definition 6. We denote by C the space of functions on r that vanish on the empty geodesic and satisfy the truncated comb condition.

THE GEODESIC RADON TRANSFORM ON TREES

53

We are ready for the final step of the inversion procedure:

Theorem 3 ([3, Proposition 2.4]). On a tree with no fiat vertices, the operator Y is a bijection from B to C, and its inverse isM. To summarize, we have proved the inversion theorem for the Radon transform by factoring it as the product of two operators (on appropriate spaces) and inverting each factor, as is clear from the diagram

z

R

y

A~B~C. N

M

So we have the following characterization of the range of R, obtained in [3, Theorem 2.5], with some improvements suggested by A. Le Donne (personal communication):

Corollary 2.5. If the tree T has no fiat vertices, the range of the geodesic Radon transform on the space A of functions on T with doubly convergent geodesic sums is the space C of functions on the space of geodesics ofT that satisfy the truncated comb condition. Remark 2. If there are fiat vertices, then the inversion theorem requires some modification. We have already observed that the functions on r that are Radon transforms of functions on T satisfy the switch condition (1). If there are no fiat vertices, the truncated comb condition implies the switch condition [3, Proposition 3.2]. In the general case, the switch condition must be imposed on the range of R, and in addition one must consider combs whose supporting ray may contain fiat vertices. For details see [3, Section 3]. For a functional analytic discussion see [4]. No characterization for the range of R on £1 (T) is available yet. REFERENCES [1] G. Ahumada Bustamante, Analyse harmonique sur l'espace des chemins d'un arbre, These de Doctorat d'Etat, Universite de Paris-Sud (Orsay), 1988. [2] C. A. Berenstein, E. Casadio Tarabusi, J. M. Cohen, M. A. Picardello, Integral geometry on trees, Amer. J. Math. 113 (1991), 441-470. [3] E. Casadio Tarabusi, J. M. Cohen, M.A. Picardello, Range of the X-ray transform on trees, Adv. Math. 109 (1994), 153-167. [4] J. M. Cohen, F. Colonna, The functional analysis of the X-ray transform on trees, Adv. in Appl. Math. 14 (1993), 123-138. DIPARTIMENTO D1 MATEMATICA, UNIVERSITA D1 RoMA "TOR VERGATA", VIA DELLA RICERCA

00133 ROMA, ITALY E-mail address: picardGmat. uniroma2. it

SCIENTIFICA,

THE DISTRIBUTION-VALUED HOROCYCLIC RADON TRANSFORM ON TREES ENRICO CASADIO TARABUSI, JOEL M. COHEN AND FLAVIA COLONNA A bstract. This article contains recent results on injectivity and inversion of the Radon transform R on the set Ή of horocycles of a homogeneous tree T, as well as on the characterization of its range for different domains. The Radon transform maps the set of functions of finite support on T onto the set of func­ tions of compact support on H th a t satisfy the Radon conditions. This result holds also in the non-compact case, provided th a t the appropriate decay cri­ teria are added, but functions on Ή need to be replaced by distributions. We prove here an inversion formula valid on such space of distributions, generaliz­ ing a formula th a t was previously known (for T not necessarily homogeneous) on the image of L l (T).

1. I n t r o d u c t i o n

The Radon transform (RT) associates to each (reasonably well-behaved) function on M2 its Lebesgue integrals along all affine straight lines [15]. This transform has found its main applications in tomography (with CATscanning and related techniques) and has led to several generalizations on other spaces, cf. [14]. In particular, in the Poincare disk H2, lines correspond to two different kinds of one-dimensional submanifolds: geodesics and horo­ cycles, giving rise to two distinct RTs (for an application of the former see [2]). Trees, widely regarded as discrete counterparts of H2, also feature two distinct kinds of RTs, the geodesic R T (or X-ray transform, reminiscent of the CAT-scan procedure), and the horocyclic RT. The geodesic RT on trees has been investigated by various authors: e.g., in [3], [1] for injectivity and inversion, [10] for range characterization, and [11] from the point of view of function spaces. In this article we give an overview of the results on the horocyclic RT on a tree T, focusing mainly on the most recent developments. For T homoge­ neous, we state those [7], [8] on injectivity and characterization of the range of the RT (for the range of the Helgason-Fourier transform see [13]). The ranges on suitable spaces of infinitely supported functions on T are explicitly described as spaces of distributions on the space Ti of horocycles of T, on which we show here the validity of an inversion formula proven in [9] (for non-homogeneous T) for the image of L l (T). Detailed proofs of the injec­ tivity and the inversion in the latter setting can be found, for homogeneous trees, in [6] for radial functions and [5] for functions in the Schwartz space, and in [9] in the general non-homogeneous L l case.*5 1991 Mathematics Subject Classification. Prim ary 44A12; Secondary 05C05, 43A85. Key words and phrases. Radon transform, trees, horocycles, distributions. 55

56

ENRICO CASADIO TARABUSI, JOEL M. COHEN AND FLAVIA COLONNA

2. PRELIMINARIES

We start by recalling that a horocycle in lHl 2 is a circle in the (closed) disk that is tangent to the boundary. For each boundary point w, the horocycles tangent at w are the level sets of the Poisson kernel (with respect to the origin) P(z, w) = (1 - lzl 2 ) /lw - zl 2 regarded as a function of z E lHl 2 . More generally, the horocycles tangent at w are the level sets of the Poisson kernel with respect to any given wE lHl 2 , namely Pw(z,w) = P(Tw(z),w), where Tw is the conformal automorphism of lHl 2 mapping w to 0 and fixing w, since

P(Tw(z),w)

= P(z,w)jP(w,w).

Let T be a tree. As is customary, a function on T will mean a function on its vertices. The boundary n ofT is the set of equivalence classes of onesided infinite paths of vertices under the relation generated by [vo, v1, ... ] "' [vl, V2, ... ]. There is a unique path [u, w) in the class w E n that starts at a given vertex u, whence n can be identified with the set of paths starting at u. An edge [u, v] of T is positively oriented with respect to w E n if v E [u, w). The horocycle index ~w ( u, v) with respect to w of an ordered pair of vertices u, v is the number of positively oriented edges minus the number of negatively oriented edges in the path from u to v. The degree of a vertex v is the number qv + 1 of its neighbors, and we assume that it is greater than 2. If the degree is q + 1 for all vertices, then T is homogeneous. In this case the Poisson kernel (with respect to the vertex u) is given by p, (v w) _ qk..,(u,v) u

'

-

'

since it satisfies the following properties, in analogy to the classical case: 1. for fixed u E T and w E 0, the function v t--+ Pu (v, w) is harmonic on T (the value at any vertex is the average of the values at its neighbors); 2. if J.Lu is the natural probability measure on n with respect to u (defined in (1) below), then, for any measurable function jon n, the function

f(v) =

J

](w)Pu(v,w) dJ.Lu(w)

for vET

is harmonic on T, and, conversely, every harmonic function f on T yields a measurable function j on the boundary for which the above integral representation holds (cf. [16]). In analogy to the Poincare disk, the horocycles on the tree are the level sets of the Poisson kernel. In fact, the horocycle of index n E Z with respect to u E T and w E 0 is h~n ={wET: ~w(u,w) = '

n}.

In particular, h~ 0 is the horocycle through u and w. Given u E T and w E n, the set of 'vertices of T may therefore be decomposed as the disjoint union llnEZ h~,n and, for u fixed, (n, w) t--+ h~,n is a one-to-one map of Z x n onto the set 1-l of horocycles. The family of horocycles through a fixed w does not depend on the choice of the reference vertex u, although indices do: h~,n = h~,n+~e..,(u,v)·

HOROCYCLIC RADON TRANSFORM ON TREES

57

Definition 1. The L 1-horocyclic Radon transform Ron Tis defined by Rf(h) =

L f(v)

for f E L 1(T) and hE 'H..

vEh

(The set of vertices of T is assumed endowed with the counting measure.) The topology generated by the sets { h~,o : w E 0, v E [u, w)}, for u, v ranging in T, makes 1i totally disconnected. Then 1i is homeomorphic to l. X 0, where f2 is endowed with the compact topology generated by I;:= {wE 0: v E [u,w)}. For each given u E T, a probability measure J.Lu on n that arises naturally (cf. [3, Section 2]), and that reduces to the unique rotation-invariant one if T is homogeneous, is given by

(1)

J.Lu(I:) = {

1

1 qVo

+1

rr~-1 ~ _

-

3 1

if u = v, i f U 1-1- V,

qVj

where [vo = u,v1, ... ,vk = v] is the finite path from u to v, and the distance d( u, v) between them is consequently k. In the remainder of this section we shall assume that T is homogeneous. Therefore J.Lu(I;:) = 1/ck, where

Ck=

{

1

ifk=O,

(q+1)qk- 1 ifk>O

is the number of vertices ofT at distance k from a fixed vertex. For simplicity of notation we fix a root e throughout, and set hw,n = h~,n• J.L = J.Le, dw = dJ.Le(w), k(v,w) = Kw(e,v), and Iv =I;. Notice that dJ.Lv(w) = qk(v,w) dw. We call lvl = d(e, v) the length of the vertex v. For w En, let Wn be the vertex of length n in [e,w); analogously, for vET and n ~ lvl, let Vn be the vertex of length n in [e, v]. The set of descendants of v of length n 2: lvl is defined as Dn(v) = {u: lui= n, ulvl = v}. The set Sv = Un:?:lvl Dn of all descendants of vis called the sector determined by v.

Definition 2. We define the Radon conditions on a function r.p on 1i as

follows:

is independent of v and w; n

ln{ r.p(h~ 'n) dJ.Lv(w) = q-n Jn{ r.p(h~ '-n) dJ.Lv(w)

for every v E T and n E Z.

Condition (R2 ) was first observed in [4] and [5] for the RT of radial functions.

Proposition 2.1. Iff the Radon conditions.

E

£ 1 (T), then Rf is a continuous function satisfying

58

ENRICO CASADIO TARABUSI, JOEL M. COHEN AND FLAVIA COLONNA

The proof is based on showing that the Radon conditions are satisfied for the function c.p = Rxu, where Xu is the characteristic function of {u}, and then using linearity to extend the result to f E L 1 (T) [8, Proposition 3.3]. There are, however, continuous functions satisfying the Radon conditions that are of the form Rf for f rJ. L 1(T) [8, Example 3.7]. Fix a vertex v of length m. For 0::::; t::::; m, let~= {wE 0: k(v,w) = 2tm}. Then~= Iv, -Iv,+ 1 fort ::j:. m, while I~= Iv, and 0 = Il~0 I~. Using the relations h~,n = hw,n+k(v,w) and df..Lv(w) = qk(v,w) dw, condition (R2) may be rewritten as

(~)

f

q2t-m /, c.p(hw,n+2t-m) dw = q-n It

t=O

f

q2t-m /, c.p(hw,-n+2t-m) dw. I!

t=O

The inversion formula for the RT on homogeneous trees first appeared in [6] for radial functions in L 1 (T); it was then found in [5] for the Schwartz space of rapidly decreasing functions. In [9] it was shown that the RT is injective on L 1 (T) for T not necessarily homogeneous, and an inversion formula was given, which we illustrate in Section 3. In the subsequent sections we restrict attention to homogeneous trees. In Section 4 we characterize the range of the RT on the set of functions of finite support on T. In Section 5, after defining the Radon transform of a function on T as a distribution on 1-l, we give decay criteria to both the domain and range of R to obtain a similar characterization for functions of infinite support; full details appear in [8]. Finally, in Section 6 we extend the inversion formula of Corollary 3.1 to the distribution-valued RT setting.

3.

INVERSION OF THE

RT

ON

L1

FUNCTIONS

In this section the tree T is not necessarily homogeneous. If u is a given vertex and v lies at even distance 2n from u, denote the finite path from u to v by [vo = u,v1, ... ,v2n = v].

Theorem 1. The Radon transform R is one-to-one on L 1 (T) and, if c.p Rf for an f E L 1 (T), then f(u)

=

L

v: d(u,v) even

where the coefficient

A~,

A~/, c.p(h~,d(u,v)) df..Lu(w)

for all u E T,

I;:

for even d( u, v), is given recursively by ifv = u, ifv ::J. u.

A closed form can be provided for the coefficients n

A~=l+ _L(-l)k k=l

L aE{O,l}k with ak=O

A~,

namely

k

(-l)laliTqv;-l+min{i:e;;a;=oJ' j=l

=

HOROCYCLIC RADON TRANSFORM ON TREES

The first few A~ = A~2 n are (setting

Avo Av2 Av4 Av6 AvM

Qj = Qvj

59

for brevity):

= 1, = 1- Ql, = 1- Ql + Q1Q3=

=

Q2Q3,

1- Ql + Q1Q3- Q2Q3- Q1Q3Q5 + Q2Q3Q5 + QlQ4Q5- Q3Q4Q5, 1 - Ql + Ql Q3 - Q2Q3 - Ql Q3Q5 + Q2Q3Q5 + Ql Q4Q5 - Q3Q4Q5 +~~~0-~~~0-~~~0+~~~0 -

Q1Q3Q6Q7

+ Q2Q3Q6Q7 + QlQ5Q6Q7- Q4Q5Q6Q7·

Although these coefficients grow fast (in general), it is proved in [9, Theorem 6.1], by sufficiently accurate estimates, that the above inversion formula holds for functions in L 1 (T). In the homogeneous case this formula is easily seen to take a much simpler expression:

Corollary 3.1. Assume T is homogeneous. If cp = Rf where f E L 1 (T), then

(2) f(v)

= {

Jn

cp(h~,o) dp,v(w) + (1- q)

f:n=lJn{cp(h~,2n)

dp,v(w)

for all vET.

4. RANGE OF THE RT ON FUNCTIONS OF FINITE SUPPORT

The Radon conditions and the compactness of the support completely characterize the range of the RT on functions of finite support:

Theorem 2. The image of Ron the space of functions on T of finite {i.e., compact) support is the space of functions on 1i of compact support satisfying the Radon conditions.

The proof is based on a generalization of radiality:

Definition 3. Let N be a non-negative integer. A function f on T is: Nradial if, for all v E T with lvl ~ N, the value f(v) depends only on VN and IvI (a 0-radial function is generally called radial); N -supported if its support is contained in {v E T: Ivi :5 N}. A function cp on 1i is: N -radial if cp(hw,n) depends only on WN and n; N -supported if, whenever lnl > N, we have cp(hw,n) = 0 for all w E 0. We actually prove a more precise version of Theorem 2, specifically that the image under R of the set of N -supported functions on T is the set of continuous N -supported functions on 1i satisfying the Radon conditions. This result is established by means of Propositions 4.1 and 4.2 below, whose proofs we outline here. For N ~ 0, let EN be the set of N-radial N-supported functions on 1i satisfying (RI) and (R2).

Proposition 4.1. EN= EN-I EB ffilvi=NCRXv·

60

ENRICO CASADIO TARABUSI, JOEL M. COHEN AND FLAVIA COLONNA

It follows by induction that EN is the image under R of the set of N-radial N-supported functions on T, since the characteristic function of a vertex v is !vi-radial and !vi-supported.

Proposition 4.2. If t.p is a function on 1t of compact support satisfying the Radon conditions, then there exists N such that t.p E EN.

Let {v 1 , ... , vcN} be an enumeration of the vertices of length N. If v E T and lvl ~ N, let At = {j: lv; ~ ~}. Thus~ = lljeA~ I vi. If jo is the index such that v = vio' then A~ = {jo}. Observe that {1, ... 'CN} = ul~o At, and recall that 0 = IJl~o~· Let t.p E EN, and set an,j = t.p(hw,n) for WN =vi. Then(~) becomes M

M

L q2t L an+2t-M,j = q-n L q2t L a-n+2t-M,j

(~)

t=O

t=O

jEA~

jEA~

for lvl = M ~ N. The proof of Propositions 4.1 and 4.2 is based on repeated applications of (~) for various values of nand M. For instance, if we set M =Nand n = 2N, the left-hand side of(~) reduces to EjeA~ aN,j, since n+2t-M > N except for t = 0. On the right-hand side, a-n+2t-M = 0 except for t = M = N, leaving just EjeA~ a-N,j, which is a-N,j 0 , where v = vio. Thus EjeA~ aN,j = a-N,j0 QN. In particular, if aN,j = 0 for all j, then a-N,j = 0 for all j. If t.p E EN, then the function tj; = t.p- Ej~ 1 aM,jR(Xvi) has the property that tj;(hw,n) = 0 for n = N as well as for ini > N. Hence aN,j = 0 for all j, and so, by what we just proved, a-N,j = 0 for all j. Thus tj; E EN-l, proving Proposition 4.1. Now let t.p be a function with compact support satisfying the Radon conditions. Since topologically 1t ~ z X n with n compact, there is some positive integer N such that the support of 'P is contained in [-N, N] x n, i.e., 'P is N-supported. Again using (~) it is possible to show that 'Pis N-radial. Thus t.p E EN, proving Proposition 4.2, hence Theorem 2. 5. INFINITE SUPPORT - THE DISTRIBUTION-VALUED RT

In this section we develop a parallel theory for distributions on 1t and define appropriate decay conditions for functions on T and distributions on H. A more elegant, although equivalent, approach is used in [8], but we chose the present formulation to simplify the calculations. For r > 0, let Ar be the space of functions f on T that satisfy the decay condition:

f

n=lvl

tnJ

L uEDn(v)

f(u)l


the Radon transform R is injective on

Ar, and

its image is Br.

The proof is based on the use of N-radial functions and N-radial distributions.

Definition 5. A distribution 'P on 1t is N -radial if cp(Hv,n) depends only on nand VN· Given a positive number r and a non-negative integer N, let A[:' be the space of N-radial functions in Ar, and let B;:' be the space of N-radial distributions in Br. Then [8, Proposition 5.4]

Proposition 5.1. For r

A[:',

and its image is

B;:'.

> 1/ ..fii,

the Radon transform R is injective on

The folwwing example shows that the use of distributions is necessary: Example 1. Let .>.1, ... , Aq be complex numbers of modulus 1 and such that 1 Aj = 2/3; also, set Aq+l = .>. 1. Label the vertices as follows: let x1, ... , Xq+l be the vertices of length 1. If v # e has already been labeled, write the immediate descendants of v as vx 1 , ... , vxq. Thus a typical vertex v of length N is labeled as Xi, · · · XiN, where the ij are between 1 and q, except for i1 which can also be q + 1. Define the function

2:J=

f(v)

Then

{

1

= .>.·

~1

I '2:

···.A·~N (i)N 3

f(u)l

if v = e, if e ..J.. r v = x·~1 · · ·x·~N •

= lf(v)l (~) n-N =

(~) n (~) N

uEDn(v)

IfO < t < 9/8, then Ln tn(~tG)N converges, so f E Ag;s· By Theorem 3, Rf is well defined and is in 8 9 ; 8 . Nevertheless, we now show that Rf cannot be evaluated at any horocycle. A horocycle hw,n is the disjoint union of Dn+2k(wn+k) \ Dn+2k(Wn+k+l) over the set of all non-negative integers k, for n ~ 0. Thus Rf(hw,n)

=

f( '2: k=O

vEDn+2k(wn+k)

f(v)-

'2:

f(v)),

vEDn+2k(wn+k+t)

where for j for j

= k, = k + 1.

63

HOROCYCLIC RADON TRANSFORM ON TREES

Since the sum corresponding to j = k + 1 has a larger absolute value, the absolute value of the difference is at least ~ ( ~) n ( ~~) k, which is greater than 1 for all sufficiently large k. Thus the series defining Rf(hw,n) does not converge for positive n. For negative n the decomposition

=II (Dn+2k(wk) \ Dn+2k(Wk+I)), 00

hw,n

k=O

similarly yields the same conclusion. Since point evaluation cannot be defined, Rf is not induced by any function. 6. THE INVERSION FORMULA In this section we give an inversion formula that extends Corollary 3.1. We present a detailed proof, since this result has not appeared before.

Theorem 4. For r > 1/ ..;q let cp E Br. Then the function f E Ar such that Rf = cp is given, for all v E T, by f(v)

=

t;q lvl

( 2t-lvl cp(H~,2t-lvl)

oo

+ (1- q) ~ cp(H~,2n+2t-lvl)

)

·

Observe that the above inversion formula agrees with (2) iff E L 1 (T). Indeed, following the same argument used to derive (Hi) from (R2), we may write (2) as follows:

f(v)

=

t; lvl

q2t-lvl

(

oo

1~ cp(hw,2t-lvl) dw + (1- q) ~ 1~ cp(hw,2n+2t-lvl) dw

)

·

J

Recalling that Rj(Hu,n) = 1, Rf(hw,n) dw for f E L 1 (T), we obtain the inversion formula of Theorem 4. The proof of Theorem 4 is based on showing that the inversion formula holds for theN-radial approximations of the given distribution for any positive integer N.

Definition 6. For an integer N ;::: 0 the N -radialization of a function f on T is the function f N on T defined by if if

f(u) fN(u) = { qN-Iul"' f(w) L....wEDiul(uN)

lui ~ N, lui > N.

The N -radialization of a distribution cp on 1i is the distribution cp N on 1i defined by

) _ {cp(Hu,n) ( 'PN Hu,n - qN-Iuicp(HuN,n)

if if

lui ~ N, lui > N.

In particular, a function f on T (respectively, a distribution cp on 1i) is N-radial if and only iff= fN (respectively, cp = 'PN). The N-radialization operator commutes with the Radon transform, cf. [8, Proposition 2.9]:

64

ENRICO CASADIO TARABUSI, JOEL M. COHEN AND FLAVIA COLONNA

Proposition 6.1. Iff is a function on T such that Rf is defined, and if N 2': 0, then RfN is defined and equals (Rf)N. In particular, iff is N -radial, then so is Rf. For

1/vfQ. If

0 and m > 0 then

= n +m

then

q m(Ax nAy)= q2 (N-l).

b) If n

> 0, m = 0

or (n

= 0, m > 0)

m(Ax nAy)=

then

(q + 1) 1 2 q2(N-l).

Lemma 4.4. 1. The kernel of R* R, defined on the space of finitely supported functions, with respect to the basis { 8x }xE'Eo is given by K(x, y)

= m(Ax nAy)

for each pair (x, y) E ~o x ~O· 2. For each x E ~o the function y -

1 < p < oo.

K(x, y) belongs to LP(~o) for each

Proof. (1) The sequence {8x}xE'Eo is a basis for the space of finitely supported functions. With respect to this basis K(x, y) = R* R8x(y) = m(Ax nAy)

By Proposition 3.2 we obtain the expression of K(x, y).

76

LAURA ATANASI

If we set

(4)

f3 = i(q + 1)(q2 + q + 1) the kernel K(x, y) can be written as f3 if X= y

K(x, y)

=

J!o = Jfn if Y E Vn,o(x) or y E Vo,n(x), n > 0 N:~m if y

EVn,m(x), n, m > 0 .

One can see that

L

K(x,y)P

= O(q(p-l)!(n+m)

yEVm,n(x)

for each m ~ 0, n ~ 0 and 1 < p < oo. Then (2) follows as :Eo= Un~O,m~oVm,n(x)

Remark 1. From Lemma 4.4 (2) it follows that R* R: L 1 (:Eo) for each E > 0, while R* R does not map L 1 into L 1 .

0 ----+

£l+t(:E0 )

For each pair (m, n) there is a natural averaging operator J.lmn defined on the space of complex valued functions on :Eo: 1 f(y). J.lmnf(x) = ~ mn yEVmn(x) The linear span of the operators J.lmn is a Banach algebra B with identity which is generated by J.liO and J.LOI, [7]. Each operator in B maps L 2(:Eo) into L 2 (:Eo) and is bounded. Let us denote by B2 the closure of Bin the space of bounded operators on L 2 and by II · 112,2 the norm of bounded operators acting on L 2 (:Eo).

L

Lemma 4.5. For each m, n

~

0

IIJ.Lm,nll2,2

~

C(m,n) qn+m

where C(n, m) is the symmetric cubic polynomial given by C(m, n) = { (q- 1) 3 nm(n + m) + (q- 1) 2 (q + 1)(n2 + 4nm + n 2 )

+ 3(q- 1)(q + 1) 2 (n + m) +2(q + 1)(l + q + 1)} 0

Proof. [7, Lemma 3.3]. Proposition 4.6. For each f E L 1 (:E0 )

(5)

R* Rf

=

(f3J.Lo,o + 3/3

L J.lm,n) f · L J.lO,n + 6/3 m>O,n>O L J.lm,O + 3/3 n>O

m>O

The series f3J.Lo,o + 3/3 l:n>O J.ln,O + 3{3 l:m>O J.Lo,m + 6/3 l:n>O,m>O J.ln,m converges absolutely in the operator norm on L 2 (:E0 ). Therefore (5) gives rise to a bounded extension of R* R to L 2 (:Eo).

77

INTEGRAL GEOMETRY ON AFFINE BUILDINGS

Proof. The action of R* R on the space of finitely supported functions is given by R*Rg(v)

=

L K(v, w)g(w) = K(v, v)g(v) + L K(v, w)g(w) L K(v, w)g(w) + L K(v, w)g(w). wEVm,o(v)

wEVo,n(v)

wEVm,n(v)

Then (5) follows from (4) and from the definition of J.Lm,n· The absolute convergence of the right-hand side is an immediate consequence of Lemma 4.5. For general f the claim follows by density of finitely supported function in £ 1 . 0

--

We conclude this section by computing the symbol R* R of R* R. We recall that the spectrum sp2(J.Lw) acting on L 2(:Eo) was computed in [7] and [11]. It is the region bounded by the hypocycloid q(q2 + q + 1)- 1(2ei8 + e- 2i 8 ), 0 ~ () ~ 27r. The region SP2(J.L 10 ) can be characterized as the set of z E C such that

4q(q2 + q + 1)3(z3 + z-3)- (q2 + q + 1)4z2:z2- 18q2(q2 + q + 1)2zz + 27q4 ;;;:: 0. With respect to the usual scalar product in L 2(:Eo), the adjoint J.Lio of /-LlO is /-LOl· Then B2 is a commutative C* -algebra. There is a bijective correspondence between sp2(J.Lw) and the Gelfand spectrum of B2 given by z ---> Az,:z, where Az,z is the linear multiplicative functional of B2 onto C defined by Az,z(J.Lw) = z and Az,:z(J.LOI) = z. We introduce the operator

T

6(q2+q+1)

2

2

= (q + 1)(q _ 1)2 [(q- 1) - (q- 1) (J.LlO + /-LOI) + (q 2 + 1)/-LIO/-LOl- q(J.LI ,o + 1-Lio)J.

By applying the composition rules in the algebra generated by /-LlO and /-LOl (see [7, proposition 3.1]), one obtains

TR* R = [(q4 + 2q3 - 3l + 2q + 1) + (2q 4 + q3 - 3q2 + q + 2)(J.LI,o + J.Lo,I) + (q 2 - 3q + 1)(l + q + 1)J.LI,oJ.Lo,I- q(q2 + q + 1)(1-LI,o + 1-LI,o)l ·

---

--

In order to compute R* R it is enough to compute T R* R( z), which is the real continuous (indeed analytic) function on sp2(J.L1 0) given by

'fli!:R(z) = [(q4 + 2q3 - 3q2 + 2q + 1) + (2q 4 + q3 - 3q2 + q + 2)(z + z) + (q 2 - 3q + 1)(q2 + q + 1)zz- q(q 2 + q + 1)(z2 + z 2)] .

--

To prove that R* R has inverse in B2 it is enough to show that T R* R( z) is nowhere vanishing on SP2(J.LIO).

78

LAURA ATANASI

5. HOROCYCLES

Let X be an A2-building. We use the same notation as in Section 3. The boundary 0 of X is defined as the set of equivalence classes of sectors: two sectors are equivalent if their intersection contains a sector. For any w E n and o E :Eo there is a unique sector Q 0 = Q 0 (w) in the class w having base vertex o [12, chapter 9, section 3], and so we can think of n as the set of sectors in X with base vertex o. There is a natural topology on n which makes 0 a topological, totally disconnected, compact Hausdorff space [11]. A basis of open and closed sets for this topology is given by the sets 0 0 ( v) which consist of the elements w E n such that the sector based at o and representing w contains v. If we fix m, n::;::: 0, then n = UvEV,,n0 0 (v), a disjoint union of Nm,n sets. Let wE 0, o E :Eo and A 0 an apartment containing Q0 . Every vertex in Q 0 has sector coordinates defined as in Section 3. We extend the coordinates to all A 0 and we denote by Xm,n the vertex in A 0 with coordinates (m, n). There is a bijective correspondence between Z 2 and the set of vertices of Ao given by (i,j)--+ Xi,j-i· Let x be any vertex in X. There exists a subsector Q of Q0 such that x and Q are contained in an apartment A [12, lemma 9.4]. We denote by 1ro,w (x) the image of x under the unique isomorphism of A onto Ao which fixes A n Ao pointwise. This image does not depend on A, and it does not depend on the sector Q as two subsectors of Q 0 intersect not trivially [11, chapter 9, section 3]. Then we can define a map 1r = 1ro,w : X--+ A 0 , which is a retraction of X onto A 0 • We refer to 1r as the retraction of X onto A 0 with center w and relative to o. For each pair (m, n) E Z2 , we set

Hm,n(w) = 7r- 1 {xm,n} . We call Hm,n(w) the horocycle of w with parameters (m, n). For each o E :Eo, w E n and (m, n) E Z 2 , each horocycle can be defined in this way. Observe that for each fixed wando the set { Hm,n(w) : (m, n) E Z 2 } is a partition of :Eo. Let 1t be the set of horocycles in X. There is a bijection between Z 2 x 0 and 1-l: this correspondence is not unique but it depends on the choice of a vertex in X. 1t inherits a natural topology via this bijection. The horocyclic Radon transform Rf of a function defined on :Eo is given by Rf(H) =

L f(x)

for every H E 1t

xEH

when the right-hand side converges absolutely for each HE 'H. The Radon transform of a compactly supported function f has compact support because the support of f intersects only a finite number of horocycles. The Radon transform of a function in L 1 (:Eo) is bounded. This transform has been studied. In the case of trees, see for example [3] for the homogeneous case and [9] for the non-homogeneous case.

INTEGRAL GEOMETRY ON AFFINE BUILDINGS

79

When f is a hi-radial function with respect to a vertex o, that is f has constant value on each Vm,n(o), then the Radon transform can be written more explicitly as

Rf(h, k,w)

L

=

f(x)

=L

IHh,k(w) n Vm,nlf(m, n)

m,n

xEHh,k(w)

and Rf does not depend on w. The following is a classical argument which allows us to connect the spherical Fourier transform, the Radon transform and the classical Fourier transform. The spherical functions for an A2-building have been studied in [7] and [11]. These functions may be indexed by pairs of complex numbers (z, w). The spherical function 'Pz,w can be characterized as the hi-radial function (with respect too), such that 'Pz,w(o) = 1, f-l1,0'Pz,w = Z'Pz,w and f-l0,1'Pz,w = W'Pz,w· The spherical functions 'Pz,w may be also indexed by the group S = {8 = (81, 82, 83) E C 3 : 818283 = 1}. These two indexing methods are connected by and

w =

q

q2 +q+ 1

(8 -1 1

+ 8 2-1 + 8 3-1) .

When 818283 = 1 then

.- 1] + 1. +cost - 1 2.4. Inversion formula for the Radon transform on the sphere. In the remainder of this paper, we shall use Theorems 3.1 and 3.2 in [2]. The statements of these theorems in [2] includes the assumption that f vanishes in a neighborhood of the south pole. However, their proofs hold provided that f(a1r) = 0, that is, that f vanishes at the south pole. For the reader's convenience, we shall rephrase these two theorems in an unified statement:

Theorem 2. (Radon inversion formula). Let >. = d2l and (>.)m be as in 1. Suppose that f E C 00 (GIIK) vanishes at the south pole. Then, for every s E (0, 1r): 1.

(5)

2. (6 )

f(as)

=

17r( 1 d).X >.! 2.x - sinsds Rf(s)

if dis odd,

and both sides of the equality have limit for s -. 0,

1

f(as)

1

7r

= ~ (>.)m 2.x

r(

}8

-

1 d) sintdt

m

1r.

sin t Rf(t) vcoss- cost dt if d is even, say d =2m.

In this statement, the only point that is not exactly as in the proof given in [2] is the existence of limits in (5). But this is trivial, because f has limits at the poles, since it is a C 00 function on the sphere. 3.

IMAGE OF THE DUAL RADON TRANSFORM

We shall now find a characterization of the range of R and R*, respectively. We let >. = d;- 1 . We give here as separate lemmas some preliminary computations based on integration by parts.

Lemma 3.1. For j, f3 EN, 0

~

j

< /3,

one has

(1 + cost).B = 0 ((t(,i3-j)) (__;_!!_)j smt dt 1r) 2

fort__.

1r.

AHMED ABOUELAZ

92

In particular, (1 + cost)f3 ( _;_~)j smtdt

(7)

for 0

~

j

=0 t=7r

< (3.

Proof. This is obvious for j = 0. Suppose it is true for j = n < (3. Then ( _;_dd

smt t

)n (1 + cost)f31 t=-rr = 0. Thus, by recurrence on n, 1 d) n+ 1 ( -.-(1 + cost)f3 = 0 ((t- 1r) 2 1

(8)

smtdt

(t-1r)2

(/3-n)) ,

whence the first part of the statement. The second part is already built in this induction argument: see (8). D Before presenting our last preliminary lemma, we introduce our candidates for the range of the dual Radon transform. Definition 4. Let cq~(GIIK) and cr;n(GIIK) denote, respectively, the sets: ' '

= {f E C00 (GIIK): f(a-rr) = f'(a-rr) = 0 and

C2~d(GIIK)

(-si~t:t)k (~i:ti)lt=O =0, O~k~A-1}, for d odd, and Ci~n(GIIK)

'

= {f E C 00 (GIIK): f(a-rr) = f'(a-rr) = 0 = f(ao) and 1-rr R- 1(s---+ (1 +cos s)A- 1)(t)f(at) dt = 0}

for d even, d

> 5.

Observe that the function (1 +cos s )A- 1) has a well defined inverse Radon transform, by Theorem 2, as this function vanishes at 1r. Observe also that the equalities f( a-rr) = f' (a-rr) = 0 are equivalent to the condition (!(at)/ sint)lt=-rr = 0. We can now state the lemmas on integration by parts that we need in this section. This is the first: Lemma 3.2. For f E c~,:t(GIIK) and A= (d- 1)/2 one has: '

r (1 + cost)A-1~dt (_;_~)A-1 (f~at)) dt smt dt smt

(9a)

(9b)

}0

=

r

Jo

((dd _;_)A-1 dd (1+cost)A-1) tsmt t

(f~at)) smt

dt.

INTEGRAL GEOMETRY IN THE SPHERE

93

§d

Proof. The statement is obviously true for A= 1, because

r.!!:... f(at) dt = }0

dt sin t

f(at)

=0

111"

sin t t=o

as f E C~~(GIIK) (see the remarks preceding the statement). For A > 1 one proce~ds by induction, or more precisely by finite descent. Here is the first step:

r (1 + cost)A-1!!:._dt (~!!:...)A-1 (f~at)) dt smt dt smt =- r !!:._(1 + cost)A-1 (~!!:...)A-1 (f~at)) dt Jo dt sm t dt sm t

Jo

- (1 + cost)A-1

(~!!:...)A-1 (!~ut)) smt dt

smt

71" ,

0

but the boundary term in the right-hand side vanishes because 1 +cos 1r

1 1 (-~!!:...)A( smt ~at)) I smtdt

0=

= 0 as

=

f E codd(GIIK). 2 ·71"

t=O

Another integration by parts transforms the right-hand side into the following expression:

r dt

}0

!!:.__1_!!:._(1 + cost)A-1 (-1_!!:._)A-1 sin t dt sin t dt

(f(ut)) dt t sin

- _1_!!:._(1 + cost)A-1 (-1_!!:._)A-1 sint dt sint dt

(!(ut)) sint

71"

0

'

but the boundary terms vanish again (we now make use of the fact that, for

f E

2 c~:(GIIK), (-~dd)A(!~at)) I =0 ). • sm t t sm t t=O

The statement is

obtained by A - 2 iterations of these steps, thanks to the conditions (

d)k (f(at))l _0

1 -sin t dt

sin t

t=O -

'

O~k~A-1.

D

Finally, here is the other lemma: Lemma 3.3. (10)

r

Jo

~)A-1 ddt (1 + cost)A-1) (f~at)) smt

((dd tsmt

dt = 0.

AHMED ABOUELAZ

94

Proof. By recurrence on

>. - 2. Then we have:

r

>., we assume that (10) is true for the exponent

~)>.- 1 ddt (1 + cost)>.- 1) (!~O-t)) smt

}0

((dd tsmt

=

kr

((dd t~t

=

(1- >.)

dt

~)>.- 1 (-(>.-1)sint(1+cost)>.- 2 ) (!~at)) ~t

r

}0

~)>.- 2 ddt (1 + cost)>.-2) (!~at)) smt

((dd tsmt

dt

dt=

= 0. D

These preliminaries allow us to characterize the range of the dual Radon transform:

Theorem 3. (dodd).

R*(C~+(A)) = C~(GIIK) Proof. We shall show that

R*(C~(A))

(11)

IfF E

C~+(A),

c

cr:(GIIK).

it follows from Definition 2 that R* F(t) lt=1r= (R* F)'(t) lt='IT= 0

since (R* F)'(t)

2>.>. = -cost 7r

1t 0

F(s)(coss- cos t)>.- 1ds

2>.>. + (>.- 1 ) - sin2 t 7r

1t 0

F(s)(coss- cost)>.- 2 ds.

By (2), one also has (R* F(t)/ sin t) lt=O = 0. For every k iterations of (2) one obtains [2, p. 380]: (

1 d)k (R*F(t)) sin t dt sint 2>. = ->.(>. -1) ... (>.- k) 7r

1 d)k(R*F(t)) ( -sintdt sint

t=O

1t

=O,

0

< >.

E N, after k

F(s)(coss- cost)>.-k- 1 ds,

k = 0, . . . ' >.

- 1.

This proves inclusion (11). Now we prove the converse inclusion:

INTEGRAL GEOMETRY IN THE SPHERE Sd

95

For f E C~(GIIK), we shall show that there exists FE C~+(A) such that f = R* F, where

F t = _!_~~ (-1 ~).>.-1 () .A! 2.>. dt sint dt

(!(at)) sint '

for every A E N .

From Theorem 2 one has R* F =f. In addition FE C~+(A) because 1 1r , 12 .>. "'·

11r (1+cost) .>.-1-dd ( -.--d 1 d ).>.- 1 (!(at)) -.t

0

sm t t

sm t

dt=O, 0

by Lemma 3.2 and Lemma 3.3.

Theorem 4. (d even, d = 2m, and d > 5). c~~n (GIIK).

One has R* (C~+(A)) =

'

Proof. Let f be a function of Ci~n(GIIK), and d =2m with mE N. We use Notation 1. We have shown i~ 2.4 that R* F = f if

rt

1 1 . 1/2 . ( 1 d )m F(t) = (.A)m 2.>..A smt Jo (coss-cost)- (sms) sinsds

(!(as)) sins ds

This implies that FE C~+(A). Indeed, J;(l+cost).>.- 1F(t) dt = f01r R- 1 (s--+ (1 +cost)>-- 1)(t).f(t) dt = o by the definition of c~~n(GIIK) (Definition 4). ' Hence c~:;;n(GIIK) c R*(C~+(A)). We now show that R*(C~+(A)) c c~:;;n(GIIK). Once again, Definition 2 shows that, ifF E C~+(A),

R* F(1r) = (R* F)'(t) lt=7r= 0 and

R* F(t) sin t But ifF E

C~+(A),

I

t=O

=

R* F(t) sin t

I

t='lr

= 0.

it follows immediately from Definition 3 that

fo'lr R- 1 (t--+ (1 + cost).>.- 1 )(t) R* F(t) dt = fo'lr (1 + cost).>.-l F(t) = because FE

C~+(A).

Theorem 5. For all j

This completes the proof of Theorem 4.

> -1, we have

2>- ..x r(..x)r( · + 1) R(t--+ (1 + cost)i)(s) = - r(.A ~ ) (1 + coss).>.+i. 7r +J+1

Proof. Recall that R is the operator defined as follows: 2.>..A 11r Rf(s) = !(at) sint (coss- cost).>.- 1dt. 7r

8

0,

0

AHMED ABOUELAZ

96

Put f(a 8 ) = (1 + coss)i in (1). Then .

2-X,X/71'

R(t -+ (1 + cost)J)(s) = 7r

=

.

(1 + cost)J sint (coss- cost).x- 1 dt

8

c 171' (1 + cost)i(coss- cost).x- 1 d(- cost)

where c = 2:.x. Let x =cost with t E [0,1r]. Then the equality above leads to (12)

R(t-+ (1 + cost)i)(s)

=-el-l

(1 + x)i(coss- x).x- 1 dx.

coss

By putting y = 1 + x, equality (12) can be transformed as follows:

R(t-+ (1 + cost)i)(s) =

rl+coss

cJo

yi(1 + coss- y).x- 1 dy.

If o: = 1 + cos s and y = o:z, we obtain

R(t-+ (1 + cost)i)(s) = c

1 1

(o:z)i(o:- o:z).X-ld(o:z)

= co:i+A

1 1

(z)i(1- z).x- 1dz 2-X,X

.

= -B(j + 1, .X)(1 +cos s)A+J, 7r

where B(., .) is Euler's (3 function. 4.

0

THE PALEY-WIENER AND PALEY-WIENER-SCHWARTZ THEOREMS IN §d

4.1. Paley-Wiener theorems in §d (dodd). Recall that the spherical Fourier transform (see [2, (8)]) is defined as follows: (13)

J(n) =

171' cos [(n + .X)t] Rf(t) dt,

for all n E N.

In this subsection we prove the Paley-Wiener theorem for the spherical Fourier transform associated to the Gelfand's pair (G, K) where G = SO(d+ 1) and K = SO(d).

Theorem 6. Let L be the Laplace-Beltrami operator in §d, and .X~ 2. For

f

in

(14)

c:r71'(GIIK), '

Lf(z) = (.X 2 - (z + .X) 2 ) J(z),

Proof. From (13), the Fourier transform (15)

J(z)

= 171' cos [(z +

for all z

E

C.

J can be extended to C by .X)t] Rf(t) dt.

INTEGRAL GEOMETRY IN THE SPHERE

The function z

§d

97

J(z) is complex analytic. We now prove (14):

--t

Lf(z)

= 11r cos [(z + >.)t] R(Lf)(t) dt.

By (3) (see Theorem 1) we have

Lf(z) = Since ftRf(t)

11r cos [(z + >.)t] ( !: + >.

2)

Rf(t) dt.

lt=o= ftRf(t) lt=7r= 0 and

d

dt cos [(z + >.)t] Rf(t)

d

lt=o= dt cos [(z + >.)t] Rf(t) lt=7r= 0,

integration by parts gives

11r cos [(z + >.)t] : 2 Rf(t) dt = 11r ~ (cos [(z + >.)t]) Rf(t) dt. Therefore

=

11r cos [(z + >.)t] ( :t22 + >. Rf(t) dt 11r (::2 + >- (cos [(z + >.)t]) Rf(t) dt

=

(>. 2

=

Lf(z)

2)

2)

-

(z

(>.2- (z

+ >.) 2 ) 17r cos [(z + >.)t] Rf(t) dt + >.)2)J(z). 0

From [2, (8), p. 356] and {14),

L.2

(16)

-

(z + >.) 2 ).(z) =

13 117r exp( -izt) (d) dt Rf(t) dt,

"2

-1r

for all /3 E N,

since supp(t--+ Rf(t)) C supp(t--+ f(at)), see [2, p. 369]). From (21) one has jz 13 h(z)

I ~ ltl~r sup iexp( -izt)i r I::Rf(t) Idt. }ltl~r t

INTEGRAL GEOMETRY IN THE SPHERE Sd

99

Therefore,

lzPh(z)l

~ ~

sup

lexp(-izt)llldd~Rf(t)ll £1([-1r,7r])

ltl~r t Cp exp(r llm(z)J),

{3 EN.

Then for all {3 E N, one has

(1 + lzl 2 ),611(z)l

~ C,aexp(r llm(z)l),

It is clear that the function z -

for all z E C.

h. (z) is even.

D

Let us now turn our attention to the converse of the Paley-Wiener theorem. For clarity of exposition we study first the case where dis odd. We need the following results. Lemma 4.1. Let f be a function ofC00 (GIIK), and let r, r' be two elements

of [0, 1r] such that r < r'. Then

(22)

~ [(1 +cos r).\ 1r ~ IRf(t)l sinx dx ~ -;;

(1 +cos r').\]

1~ r If( as) I sins ds.

Proof. Replace a (respectively, {3) by 1 (respectively, ). - 1) in [2, (52)]. We obtain: 2.\).

Rf o arccos o'l/J = -r(.X)¢.\ * f

(23)

7r

o

arccos o'l/J.

We apply Holder's inequality to (23): 2,\).

IIRf o arccoso'l/JIIu(I,dx) ~ -;:-r(.X) 114>.\IILl(J,dx) ·llfllu(I,dx), where I= [cosr' + 1, cosr + 1]. But

1

cosr+l T.\-1

(24)

=

cosr'+l

--dT r(.X)

= .xr~.X) { (1 +

cosr).\- (1 + cosr').\}.

Recall that

¢,(x) put T

= cosx +

~ { ~~;

if X;;:: 0 if

X< 0

1. Then (24) implies that:

Lemma 4.2. Let f be a function in C 00 (GIIK). Then supp(t- Rf(t)) C supp(t- f(at))

AHMED ABOUELAZ

100

Proof. If r rt supp(t -+ !(at)), there exists r' very near to r such that [r, r'] n supp(t-+ f(at)) = 0. We apply {22) and obtain:

ir'

IRJ(x)l sinxdx

= 0, 0

Hence Rf(r) = 0 (since x-+ Rf(x) is continuous)

Theorem 9. {Support theorem in §d). For every d, let f be a function in C 00 (GJJK) that vanishes to order d- 3 at the south pole. Then, for every r E

[0, 1r],

supp(t-+ Rj(t)) C [-r, r] {::} supp(t-+ f(at)) C [-r, r]. Proof. Let us first consider the case d odd. By Lemma 4.2, supp(t -+ f(at)) C [-r, r] implies that supp(t-+ Rf(t)) C [-r, r]. Let us prove that supp(t-+ Rj(t)) c [-r,r] implies that supp(t-+ f(at)) C [-r,r]. We suppose Rf(s) = 0 for lsi ~ r. Hence 0

= 11r Rj(s) r

-)A

(dd -.1 ssms

f(s)ds.

Since f vanishes to order d- 3 = 2).- 2 at the south pole a1r and Rf(r) = 0, we have

The Radon inversion theorem (see subsection 2.4) yields: 0

= 11r f(a8)f(a8) ds = 11r lf(a 8)j 2

ds.

As f is a C 00 function, we obtain then f(a 8) = 0 for Jsl ~ r. This proves the Theorem ford odd. Let us now take d even. Suppose that Rf(s) = 0 for lsi ~ r. Then we have 0 = 11r Rf(t) 8

(.!!.._1_)m dt sin t

sint dt. .Jcos s - cos t

This equality implies 0 = 11r 8

(--1_.!!_)m Rj(t) sintdt

sint dt, .Jcoss-cost

with m- 1/2 = ).. The inversion theorem {6) gives f(s) = 0 for all s such that Jsl > r. 0

-1, 1, 1, ... ,1, -1, 1).

Remark 2. Let f E C 00 (GJJK) and ko = diag( Then for all t E R we have f(koa-tko) = f(at) = J(a_t) Thus t-+ f{at) is even.

INTEGRAL GEOMETRY IN THE SPHERE Sd

101

Theorem 10. (Paley-Wiener theorem for §d, d odd, sufficient condition). Let d be odd and f be an analytic even function vanishing to order d - 3 at 1r. We assume that for every n E N there exists a constant Cn such that: 1/(z)l ~ Cn(l + lz + .XI 2 )-n exp(r IIm(z)l),

(25)

for all z E C, with r ~ 1r. Then there exists a function F of c=(GIIK) s;:pported in the compact set Kr = {katk': t E [-r, r], k, k' E K} such that F = f>...

=K

Proof. For k, k' E SO( d) (26)

F(kask')

and s E ]-1r, 1r[, put

= F(a = ~! 2;_ 1 8)

(-

si~ s :s) >. (L exp(isx)f;.(x) dx) .

It is clear that F(a7r) = 0 (see (20). The functions....,. JR. exp(isx)f(x->.) dx vanishes at order 2d- 4 at 1r iff vanishes to order d- 3 at 1r (see [2]). Then (26) yields f>. =F. It remains to show that F is supported in

Kr = {katk': ltl ~ r

with k, k' E SO( d)} .

It then suffices to shows that supp( t ....,. F (at)) c [-r, r ]. For this we make use of the classical Fourier transform. If 7J is real,

L

exp(is(x + i7J))f(x + i7]- .X) dx =

and

L

exp(isx)f>.(x) dx,

=

!) (L exp(is(x + i7J))f(x + i7]- .X) dx) ~!~ (- si~s!).>. (L exp(isx)f;.(x)dx)

=

F(s).

~! ~ (- si~

(27)

8

>.

Since f>..(x) = f(x- >.), (27) implies that

L

exp(is(x + i7]))/;.(x + i7]) dx = RF(s)

By {25) we have: IRF(s)l By setting 7J

~ Cn exp(r 1771- S7J)

(see [2, {39)]).

L

(1 ::2)n ·

= as with a > 0, we have IRF(s)l ~ Cn exp (a lsi {r -lsi)).

When a....,. oo, the above inequality yields RF(a8 ) = 0 if lsi > r. From the support theorem (Theorem 9) we obtain

F(ka 8 k1 ) = F(a 8 ) = 0

if lsi ~ r . D

AHMED ABOUELAZ

102

4.2. Paley-Wiener Theorem in

§d

(d even).

Theorem 11. (d even). Let f be an even analytic function vanishing to

order {d- 3} at the south pole an. We assume that for every n E N there exists a constant Cn such that:

(28)

+ iz + >.1 2 )-n exp(r IIm(z)i),

if(z)i:::;; Cn(1

for all z E C, with r < 1r. Then there exists a function F in C 00 (GiiK), supported in the compact set [-r, r] such that F =f.

Proof. Using Notation 1 again, we let

1( 7r = -') 2>., Am

A

in ( smt1 d ) m[l 8

--.--d t

IR

exp(itx)f>.(x) dx

]

sin t dt v'coss- cost

with d =2m and f>.(x) = f(x- >.), s E ]-1r, 1r[, k, k' E SO( d). We remark that F (an) = 0 (see (20). The function s - fiR exp( isx) f (x - >.) d:_ vanishes to order 2d - 4 at 1r if f vanishes to order d - 3. It is clear that F = />.. It remains to prove that F is supported in Kr = [-r, r]. For rJ E JR,

i

exp(it(x + iTJ))f(x + iTJ- >.) dx =

i

exp(itx)f>.(x) dx,

therefore

(>.~m 2~>.1n (- si~t!)m [i exp(it(x+iTJ))f(x+iTJ->.)dx] · sin t dt v'coss- cost = ('2) A

m

2~'1n (-_;_dd)m sm t t A

8

[ { exp(itx)f>.(x)dx] v' sint dt. }IR cos s - cost

This implies that

i

exp(it(x + iTJ))f(x + iTJ- >.) dx = RF(at),

(see (6)). The rest of this proof is exactly as in the odd case.

D

4.3. Paley-Wiener-Schwartz Theorem for §d. Let T E C~(GIIK)' (respectively, T E c~::n(GIIK)') where C~(GIIK)' (respectiv~ly, c~::n (GIIK)') is the topological dual of C~(GIIK) (respectively, c~~;n(GIIK)). Define the Radon transform of T as: (RT, ¢) = (T, R*¢),

with¢ E C~+(A).

Observe that R*¢ E C~(GIIK) if T E C~(GIIK)', and that R*¢ E c~v;n(GiiK) if T E C~V:n(GIIK)' (see Theore'ms 3 and 4). Let A be the operator '

INTEGRAL GEOMETRY IN THE SPHERE Sd

Af(s) =

_!_~ ( -1- .!!:._)>. f(s) if dis odd .X! 2>. sins ds ' 1 1 1-rr(- 1 d)m(f(t)) -;====::;: sint dt,

{

(.X)m 2>..x 8 sin t dt The dual operator A* of A is given by

v'cos s- cost

103

. d =2m. If

(29) A*F(t) =

1 .:£)>.- 1 (F(t)_\ _!_~_:£ ( -

{ .X!2>.dt sintdt

sint~'

_1__1-sint { (-1_.!!:._) (.X)m 2>..x } 0 sins ds

if dis odd

(F(s)) sins ds sins v'cos s- cost ' with d =2m.

Theorem 12. LetT be a distribution ofC~(GIIK)'uc~::n(GIIK)'. Then ART=T Proof. Let¢ be an element of C~(GIIK) or c~::n(GIIK). We have

{ART,¢) = = Since AR = I, the statement follows. Theorem 13. ForTE

C~+(A)',

{RT,A*¢) {T, R* A*¢). 0

one has

A*R*T = T. Proof. Let¢ be an element of C~(GIIK) U CiV:n(GIIK). Then ' ' {A* R*T, ¢) = {R*T, A¢) = (T,RA¢) ,

but AR =I, because A= R- 1 .

0

LetT be a distribution of C~(GIIK)' U Ci::n(GIIK)', and let

T(z) = (RT, cos(z + .X)t),

with z E C.

Since

1-r (cos(n + .X)t) (cost- cosr)>.- dt = T>. B(.X, 1/2)cpn(cos r)(sin r) >.-1, 1

then

lor

(

COS(n

2

+ .X)t) (COSt+ 1)>.- 1 dt = clim 'Pn( COST) (sin r) 2>.- 1 = 0, -r--rr

(see [2, (5),(6)], and also [6, p, 1030]). In order to show that

1-rr (cos(z + .X)t) (1 + cost)>.- 1 dt = 0,

for all z E C,

AHMED ABOUELAZ

104

we use the Phragmen-Lindelof theorem as in ([2, p. 361]). We need the following lemma. Lemma 4.3. For n EN, the dual Radon transform (Definition 2) satisfies the identity R* (s

~ cos(n + .\)s) (t) = ( 2~)!B(.\ + 1/2, 1/2)- 1 C~(cost)(sint) 2 A,

where C~(cost) is the Gegenbauer polynomial. Proof. Let I be an element of C 00 (GIIK) and n EN. The properties of the spherical Fourier transform ([11], [12]) yield f(n) = ( 2:!)n B(.\ + 1/2, 1/2)- 1

17r f(at)C~(cos t)(sin t) Adt 2

(notation as in Notation 1). But f(n) =

11r cos ((n + .\)t) Rf(t) dt,

(see [2, (12)]). Therefore the above equality can be written as J(n) =

11r R* (s ~ cos(n + .\)s) (t)f(at) dt.

[2, (6)] gives f(n) =

11r B(.\ + 1/2, 1/2)- cpn(cost)(sint) 1

2 A /(at)

dt,

where B(., .) is Euler's (3 function, and t ~ 'Pn(t) is a spherical function for the Gelfand pair (G, K) (G = SO(d + 1), K =SO( d)). Now,

11r R* (s ~ cos(n + .\)s) (t)f(at) dt = 11r B(.\ + 1/2, 1/2)- cpn(cost)(sint) 1

This equality is true for all functions

2 A f(at)

dt.

IE G 00 (GIIK). The lemma is proved.

D

Lemma 4.4. For all z E C, R* (s ~ cos(z + .\)s) (t) = B(.\ + 1/2, 1/2)- 1 cpz(cost)(sint) 2 \ where cpz(cost) is an extension of the spherical function 'Pn(cost) := 'Pn(t). Proof. Let f be an element of G 00 (GIIK), vanishing in a neighborhood of the south pole. Let

(30)

WJ(Z) =

= 11r [R* (s ~ cos(z + .\)s) (t) -

B(.\ + 1/2, 1/2)- 1 cpz(t)(sin t) 2 A J f(t) dt.

INTEGRAL GEOMETRY IN THE SPHERE

105

§d

From the last lemma, we have WJ(n) = 0 for all n EN. Since f = 0 in a neighborhood of the south pole 1r, this implies that there exist constants Co and c1 such that:

c1 < 1r.

with Now z----+ WJ(z) is an analytic function inC , since 'Pz( cosT)

= 2>-(B(>.., 1/2)) - 1 (sin T ) 1- 2>. x

loT (cos(z + >..)t) (cost- cosr)>.-

1

dt.

The last expression is Mehler's integral, see [5, p. 177], and also [2, (5)]. By applying the Phragmen-Lindelof theorem as before, one has wz (cosT) = 0 for all z E C and f E C 00 (GIIK) which vanishes in a neighborhood of the south pole a1r· So the lemma is proved. D Let o:o be an element of ]0, 1r[. Let Ka 0 [o:o, 1r- o:o]}.

= {katk' /t

E

[o:o- 1r, -o:o] U

Theorem 14. (Paley-Wiener-Schwartz). Assume that T E C 00 (GIIK)' is supported in the compact set Ka 0 • Then conditions {1) and {2) are equivalent to, respectively:

1. T is the image of the Fourier transform of the distribution T E C 00 (911K)') with suppT C Ka 0 ; 2. T extends to an analytic function on C which satisfies the following properties: there exist mE N and o:0 > 0 such that, for all z E C, (31)

lf(z)l

~ c(1 + lzl 2 )m exp(rollmzl),

ro = 1r- o:o

{Recall that T(z) = B(>..+ 1/2, 1/2)- 1 < T,'f?z(cost)(sint) 2>. >.)

In order to prove this theorem, we need the following lemma. Lemma 4.5. For r, mEN, we have:

(32) dm ----,-----,--(sin rr d(cosr)m

(33)

d(c::r)m

= (-l)m2mm!(l- cos2 rr/ 2 -m p(r/2 -m,r/ 2 -ml(cosr), m

(loT (cos(z + >..)t) (cost- cosr)>.- dt) = C>.,m loT (cos(z + >..)t) (cost- cosr)>.- -m dt. 1

1

Here C>.,m = (>.. -1)(>..- 2) ... (>..+1-m) and pj,;/ 2 -m,r/ 2 -m)(cosr) is the Jacobi polynomial. Proof. We know that p(a,(3)(x) = ( -1)m 1 dm [(1- x)m+a(l + x)m+f3]. m 2mm! (1- x)a(1 + x)f3 dxm

AHMED ABOUELAZ

106

For o: = f3 =

~ -

m and x = cosT, the above equality leads to:

Then

am

--:-:--~(sin Tr =

a(cosT)m

2mm!( -l)m(l- cos2 Tr/2-mp(r/2-m,r/2-m)(cosT). m

For (33), we apply the same reasoning and recurrence on m.

0

Proof. {Theorem 14). LetT be a distribution of C 00 (GIIK)'. We know that, for n EN,

= (RT, cos(n +

T(n)

>-.)t).

By Lemma 4.3 the mapping n __. T(n) can be extended to an analytic function z __. T(z) with:

T(z)

=

B().. + 1/2, 1/2) -l \ T, 'Pz( cos t)(sin t) 2.X)

=

(RT, cos(z + >-.)t).

But, by Mehler's integral ([2, (5), (6)]),

'Pz(cosT) = 2-XB().. + 1/2, 1/2)- 1 (sint) 1 - 2.x

1 7

(cos(z + >-.)t) (cost- cosT).x- 1 at.

Then Lemma 4.5 yields: lr(z)l = I(T,R*cos(z+>-.)t)l

~c

sup Ia(

m~mo TEKa 0

am )m 'Pz(cosT)(sinT) 2.xl

COST

sup Ia( am )m ~ c m~mo COST -rEKa: 0

[(sinT)

r (cos(z+>-.)t)(cost-cosT).X-latJI

Jo

where C is an absolute constant. Let ro = 1r - o:o and observe that 22.x- 1 1rB(>-. + 1/2, 1/2)- 1 sup lcos(z + >-.)tl tEK,. 0

~

Cexp(rollmzl).

By the Leibnitz formula, Lemma 4.5 and (34), this implies that there exists mo EN such that (31) holds.

INTEGRAL GEOMETRY IN THE SPHERE Sd

107

Sufficiency: Let T/t(t) E C27r(GIIK) be a spherical approximate identity (i.e. limt---.0 'flt = 8o, the delt~ measure at the origin of X = G / K = §d).

Then

supp 'flt C BE = { katk; k, k' E K and It I ~

E} .

The Fourier transform of "h depends only on z E C, and ift(z) --+ 1 when E--+ .---......-0 uniformly on compact set. ForTE £~,7r(GIIK) we also have T * 'flt(z) =

T(z)ik(z).

.---......-

By (32), (25) and (28), we see that (25) holds for a function z--+ T Then we apply Theorems 8 and 10 to finish the proof.

* rtt(z). 0

Remark 3. By making use of [2, (68), p. 376], we have

J2

1t

27r

0

Ri 1; 2(s--+ coszs)(t) = - sint '

coszs

J cos s -

cos t

ds,

or

J21t 1r o

cos zs - Ri,l/2(cos zs)(t) ds-2 . -Pz-lj2(cost), J cos s - cost sm t

where t--+ Pz- 1; 2(t) is the Legendre function (see the definition of Ra.,f3 and R~,f3 in [2]). Fix, now, a point Zo E hemisphere

§d,

{x E

and let §d :

f be a function supported in the

x · Zo >

0} .

Define the characters or the multiplicative operators on

Mzo+zu(x) = [x · (Zo

+ w)r).-z,

z E C,

§d

as follows

u E Z~,

d-1

>.=-2 .

The conjugates characters are defined by

z E C. (see [16, p. 415]). The Fourier transform in

§d

can be defined as:

Fzof(z, u) = { f(x)Mzo+zu(x) dx.

J§d

This transform is inverted as follows:

f(x)

= { {

lrlzt

(Fzof) (z, u)Mzoz+zu(x)h(z) dudz,

where h(y) dy is the Plancherel measure defined in [16, (23), (24)]. The study of Fzof is very difficult because Mzoz+zu(x) has many singularities. For Zo = (0, 0, ... , 0, 1), the north pole, the reader is referred to [14].

AHMED ABOUELAZ

108

Remark 4. We know that

>') ,

T(z) = B(.A + 1/2, 1/2) - 1 ( T, 'Pz( cos t)(sin t) 2

where Tis a distribution of C~7r(GIJK)' = £27r(GIIK) and 'Pz(cost) is an ' extension of a spherical functio~. 1

.

Since smt = - (exp(zt)- exp( -zt)) we have: 2z 1 2>. C2). ( -1)m->. exp (2zt(.A- m)) . (sin t) 2>. = 22 >.

L

m=O

Put

Tm(z) = (am'Pz(cost) ·T,exp(zt(2.A-2m))), with 1

am= B(.A + 1/2, 1/2)- 122>. C2).( -1)m->.. By combining the above equalities, we obtain 2>. z E C. Tm(z), T(z) =

L

m=O

If we write

T(2.A- 2m, z) =Fe( 'Pz( cost) · T)(2,\- 2m), it follows 2>.

T(z) =

L

amT(2.A- 2m, z).

m=O

Here Fe (cp z (cost) · T) ( 2.-\ - 2m) is the classical Fourier transform of the distribution 'Pz(cost) ·Tat the point 2.-\- 2m. Proposition 4.6. LetT E C)()(GIIK)' = £'(GIIK). If the singular support ofT is contained in Ka 0 , then there are a constant No and a sequence of constants Cm (m = 1, 2, ... ) such that: lf(z) I ~ Cm ( 1 + lz1 2)

No

exp (ro lim zl),

provided that

IImzl

~

mlog ( 1 + lz1 2),

with ro

= 1r - ao.

Proof. We can write T = T1 + T2 where suppT1 C Kao+l/m and T2 E C 00 (GIIK). Denote by No- 1 the order ofT, which is equal to the order of T1 Then it follows from the proof of the Paley-Wiener-Schwartz theorem (Theorem 14) that

IT1(z)l

~ Cm

(1

+ lz1 2)No-l exp((1r- ao -1/m) llmzl),

INTEGRAL GEOMETRY IN THE SPHERE Sd

109

for every z E C. Therefore IT1(z)l if IImzl

~ Cm (1 + izi 2)No exp((1r- ao -1/m) llmzl),

~ mlog ( 1 + lz1 2) .

It also follows from Theorems 8 and 10 that T2(z)--+ 0 when lzl--+ oo. Then the inequality in the statement holds for sufficiently large constants Cm. 0 5. APPLICATION OF PALEY-WIENER'S THEOREM AND GENERALIZED RADON'S TRANSFORM

Lemma 5.1. [10] Let f an entire function of a complex variable, and W be a non negative integrable function in lR with compact support. Let P be a polynomial on C [z] of degree m. Then (35)

lf(O) · p(r)(O)

where Cm,r

=

f

lzlr 'lf(izl) dzi

~ Cm,r

f

lf(z)l!P(z)l w(lzi) dz,

(m~!r)! and p(r)(O) is the derivative at 0 of P(z).

Proof. See [10, pp. 65-66].

0

Theorem 15. Let L be the Laplace-Beltrami operator in §d, lL be the operator L- >..2, and P be a polynomial on C [z] of degree m. Let v E £'(GIIK) be such that suppv c Kao and v(z)j P( -(z + >..) 2) is entire. Then there exists a distribution J.L E £'(GIIK) with suppv contained in a compact set Ka 0 such that v(z) J.L(z) = P( -(z + >..)2)' that is, P(lL)J.L

= v.

Proof. Let F(z) = P(-C~~,\)2). By replacing F(z) by F(z + ~) (~ E q and P( -(z + >..) 2) by P( -(z + >.. + ~) 2 ) in (35), with w(~) = 1 if 1~1 ~ 1 and W(~) = 0 otherwise, we obtain

IF(~)p(r)( -(~ + >..) 2)1 ~ C sup lv(z +~)I. lzl~l

We can chooser such that constant C such that (36)

p(r)

is a constant# 0. Then there exists another

IF(~)I ~ C sup lv(z +~)I, lzl~l

Because of Theorem 14 we now have

AHMED ABOUELAZ

110

Recall that ro = rr- o:o. From Theorem 14 we obtain F = ji, that is, ji(z) = P(-(~~A)2) and suppJL C Kao· The last equality implies ~(z) = v(z) for all z E C. Since

~(z- A)= P(-z 2 )ji(z- A), 0

for all z E C. Then P(lL)JL = v with supp JL C Ka 0 •

Corollary 5.2. The equation P(lL)JL = T has a solution JL E C 00 (GIIK)' = £'(GIIK) if and only ifT E £'(GIIK) andsuppT C Ka 0 , T(z)/P(-(z+A) 2 ) entire. 5.1. Generalized Radon transform. Let f be an even C 00 function with compact support defined on [-rr, rr]. In [12] Koornwinder has defined the spherical Fourier transform as follows.

(37)

- n) = f(

1~ f( t )r(a,,6) (cost) (sin t/2) 2a+l (cos t/2) 2,6+! dt 1 ' n f(o:+1) 0

where n = 0,1,2, ... , ~~a,,6)(cost) = PAa,,6)(cost)/PAa,,6)(1) and, as before, PJa,,6)(cost) denotes the Jacobi polynomial. We assume that o: > f3 > -1/2; foro:= f3 = A-1/2 > 0 (37) coincides with [2, (9)]. From [11, (5.10), p. 158] we have:

(38)

f(n)

=

1~ cos [(n + 1/2(o: + f3 + 1))t] ( Ca,,61~ Aa,,a(t, r)(R1,,6-lj2f)(r) dr)

dt,

where

ca, ,6 --

2a-3,6-1/2

--,-----,------- -

f(o:- j3)f(j3 + 1/2)(/3 + 1/2)'

Aa,,a(t,r) = (cost/2-cosr/2)a-,6-lsin( r/2) and

(39) (R1,,6-l/2f)(r) =

2,6+1/2(/3 + 1/2) 1~

(see [2, (49), p. 366]).

7r

7

f(a 8 ) sins(cosr-coss),6-l/2ds

INTEGRAL GEOMETRY IN THE SPHERE Sd

111

Let

=

(R(a,,B)) j(T)

c

11r Aa,,B(T,t)(Rl,,B-1/2f)(t)dt 2a-2,8-3/2

=

2y0iT(a- (3)f((3 + 112) x

11r sinsl2(cosTI2-cossl2)a-,8-l/2

x

(17r f(at)sint(coss-cost).B-l/2dt) ds

with c = c(a, (3). The radial part of the Laplace-Beltrami operator on compact symmetric spaces of rank one on GI K is ~

t:l.a,,B = dt 2

d

+ (2(a- (3) cot(t) + (4(3 + 2) cot(2t)) dt

[13, pg. 1]. The eigenfunctions of this operator are P~a,,B)(cos(2t). Then we can define the Radon transform on compact symmetric spaces G I K of rank one as follows: (40)

R(a,,B) f(T) = C(a,,B)

(41) -2-XA R(a,,B) f( T) = 7r

11r Aa,,B(T, t)Rl,,B-lj2(t)f(t) dt 11r f( at) (cosT -

if a> (3 > -112,

cos t).X-l sin t dt if a

= (3 = A -

112.

T

The Laplace-Beltrami operator on and [8, p. 172]):

a2

t:l.r = Br2 + ('y(pcot('yr)

§d

can be written as follows (see [9]

a

+ 2qcot(2-yr)) Br,

0 < r < L.

Here t:l.r is a multiple of t:l.a,,B· The following properties hold: (1) X= §d,

a= (3 =A -112

(2) X = pd(~) ,

a= (3 =A -112

{3) X= pd(C),

a = A - 112 , (3 = 0

(4) X = pd(JHI) ,

a = A- 112 , (3 = 0

(5) X =

JP>16 (K)

,

a=11,(3=3

where K denotes the space of Cayley numbers. We shall now study the properties of R(a,,B). Proposition 5.3. Let

f be a function on ~- Then

R(a,,B) f ( -t) = R(a,,B) f (t) for all t E R

112

AHMED ABOUELAZ

(42)

(R(a,f3)J) (t+27r) = (-1)a-{3 (R(a,f3)J) (t)

Proof. In [2, pp. 361, 366], for all t E lR we have Rl,f3-112f (t + 21r)

For a> f3 > -1/2, replace R(a,f3)J(-T)

T

by

-T

=

(Rl,f3-1/2f) ( -t)

=

Rl,f3-1/2f(t).

in (41) to obtain

=

Ca,f31: Aa,{3(-T,t) (Rl,f3-1/2f) (t)dt

=

Ca,f31: Aa,f3 (T, t) (Rl,f3-1/2f) (t) dt.

The equality above can be written: R(a,f3)J(-T)

=

Ca,f31: Aa,f3(T,t) (Rl,f3-1/2f) (t)dt +Ca,f311r Aa,f3 (T, t) (Rl,f3-1/2f) (t) dt.

Since t

---t

Aa,f3 (T, t) (R1,f3-l/ 2f) (t) is odd, we obtain

(R(a,f3)J)(-T)

=

Ca,f311r Aa,(3(T,t)(Rl,f3-1/2f)(t)dt

=

(R(a,f3)J)(T).

Now we prove (42). Observe that R(a,f3) f (T + 27r) = Ca,f311r

Putting T

= t- 21r,

Aa,f3 (T + 271', t) (Rl,f3-1/2f) (t) dt.

T+27r

we obtain

R(a,f3) f (T + 21r) = Ca,f311r Aa,f3 (T + 27r, T

But (43)

+ 27r) (Rl,f3-1/2f) (T) dT.

Aa,(3(T+27r,T+27r) = (-1)a-f3 Aa,(3(T,T)

and 1-1r Aa,f3 (T + 21r, T + 211') (Rl,f3-1/2f) (T) dT

(44) =

11r Aa,f3 (T + 21r, T

+ 21r) (Rl,f3-1/2f) (T) dT +

+ 1-1r Aa,f3 (T + 21r, T By (43) and the fact t reduces to

(45)

---t

=

+ 211') (Rl,f3-1/2f) (T) dT.

Aa,f3 (T, t) (Rl,f3-l/ 2f) (t) is a odd function, (44)

R(a,f3) f (T + 27r) = ( -1)a-f3 Ca,f311r Aa,f3 (T, t) (Rl,f3-1/2f) (t) dt.

INTEGRAL GEOMETRY IN THE SPHERE Sd

113

D Remark 5. From [12, p. 158] one has f(n) = r(a + 1)- 1

1-rr cos [ ( n +a+~+ l) t]

(R(a,f1) f) (t) dt.

For a= {3 =A- 1/2, an equivalent of (45) was proved in [2]. 5.2. Inversion formula for the Radon transform. The inversion formula for R(a,f1) is given in the following theorem.

Theorem 16. Let f be a function on [-1r, 1r] which vanishes at two real numbers such that {3 - a rf. - N. Then: 1. if {3 + 1/2 E N,

d)

_ 1r1l 2r(f3 + 1/2) ( 1 !1+1/2 f(t)- ({3- 1/2)!2a-2{1-3/2 - sint dt ( ~{1-a where

tr-1

~r(t) = r(r), ~r(t) =

2. if {3

= m- 1,

0,

* R(

a

1r

and a, {3

,{1) f) (t)

0

ift

~

ift

< 0;

m E N,

11r (-sins 1 d ) !1+1( ) ds ~{1-a * R(a,f1) f (s)

r(f3 + 1/2)7!"- 112 f(t) = 2a-2{1-3/2((3 + 1/2)({1+1) t

x

3. if {3 + 1/2 f(s)

=

rf. N

and {3

rf. N,

7!"1/2 2a-2f1-3/2r( -{3- 1/2)

sins d s· y'cost- coss '

1-rr (~{1-a * R(a,{1) f) (t) s

. (cos s- cos t)f1+1/2 sm t dt.

Remark 6. Let f be an even C 00 function, and let

(O(a,{1)f) (r) = Ca,{1

1-rr Aa,{1 (r,t) (Ra,{1f) (t) dt,

where Ra,f1 is the extended Radon transform in §d (39). The operator O(a,,B) is called the extended Radon transform on a compact symmetric space of rank one. Theorem 16 gives an inversion formula for the operator O(a,,a); see also [2, (50), (51), (54)]. Proof. (of Theorem 16). For a> {3 > -1/2, the Radon transform in a compact symmetric space is defined by:

(46) ( R(a,,B) f) (t)

= Ca,{1 1-rr sin s/2 (cos t/2- cos s/2)a-,8- 1 (R1,,8-1/2f) (s) ds.

AHMED ABOUELAZ

114

Recall that

c- c

(47)

-

2o:-3(3-7 /2 y'1i

(3 - _ _ _ _ _..,.....:.,.-_ _____,._ o:, - r(o:- f3)f(/3 + 1/2)(/3 + 1/2)

Putting T = cos s /2 we have ( R(o:,(3)

f) (t) = 2C Jo

rcoss/2

(cos t/2- T)o:-f3-l(R 1 ,(3-l/ 2f) (2 arccos T dT) = 2Cf(o:-

/3) (.

"'m"'

111" hm(x,t) s

sint dt, y'coss- cost

with hm(x, t)

=-

-1 d)m ( -. --d expixt. smt t

In the general case, {38) can be extended to C: we use the same techniques as in the case of §d. Finally we have r(a 1+ 1) ( R(a,f3)!) (t) =

21

e(x+l/2(a+f3+1)(t)f (x) dx,

where e(x+l/2(a+f3+1)(t) = exp (it(x + 1/2{a + (3 + 1)).

By Theorem 16, we obtain for (3 + 1/2 E N:

INTEGRAL GEOMETRY IN THE SPHERE Sd

119

Part (a) of Theorem 16 leads to: (56)

-1 d ){3+1/2 /m f(t) = C ( -.--d (q,f3-o: smt t lR

* e(x+l/2(o:+f3+1)(t)) f- (x) dx,

with

c _ y0rr(a + 1)r(,8 + 1)

- (,6- 1/2)!20:-2{3-1/2 .

From (56), we can make the Plancherel measure explicit. The other case can be treated the same way. We have then the following theorem. Theorem 18. (Plancherel's measure in §d). Let f be a function ofC 00 (G II K), where G = SO(d + 1) and K = SO(d). We assume that f is vanishing at the south pole a1r. Then we have

f(t) =

1.

J(x)h(t, x) dx,

where h(t,x) =

~!; (s~n1t!)>. exp(i(x+.X)t),

2 1 h (t,x ) =Km (x+.X,t ) = -(') , 2>. 1\ m 1\ -1 d with hm(x, t) =- ( -.--d smt t

17r 8

)m exp(ixt).

ifd is odd

hm(X +.X, s) . sms ds, ..;'cost- coss if d is even, d = 2m

6.2. Poisson's equation in §d. In this subsection, we shall give the explicit solution for the Poisson equation in §d. We solve also the generalized Poisson equation. The Poisson equation on §d given by

(57)

Lu = uo,

where u is the function of C 00 (GIIK) vanishing to second order at is supposed vanish at 1r. We need some intermediate results.

1r

and uo

uo is the

Lemma 6.1. Let uo be as in (57). We assume that u 0 (0) = 0 ( spherical Fourier transform (13) ) and that uo vanishes at 1r. Then

t

~ j_t'Tr sin (.Xt- .Xs) Ruo(s) ds

is even. Proof. Putting f (t) =

j_t'Tr sin .X (t- s) Ruo (s) ds,

AHMED ABOUELAZ

120

we have f ( -t)

=

1:

1:t sin ( ->..t- >..s) Ruo (s) ds

1:

sin ( ->..t- >..s) Ruo (s) ds

- sin(>..t)

+ 1-t sin ( ->..t- >..s) Ruo(s) ds

cos(>..s)Ruo(s) ds

+ 1-t sin ( ->..t- >..s) Ruo(s) ds.

Since s---> Ruo(s) is even, it follows from (13) that f( -t)

-2 sin(>..t):Uo(O) =

+ ltrr sin (>..t- >..s) Ruo(s) ds

[rr sin (>..(t- s)) Ruo(s) ds = f(t).

Since r::rrsin(>..s)Ruo(s)ds = 0, and f:_rrcos(>..s)Ruo(s)ds = 2uo(O) by (13). D Lemma 6.2. Let uo be a function satisfying (57) and vanishing at n, u E C2rr(G II K) and)..~ 2. Then '

Ru(t) = a1 cos(>..t)

+ b1 sin(>..t) +

±1:

sin (>..t- >..s) Ruo(s) ds,

with t E [-n, n] and a1, b1 absolute constants. Proof. We apply the transmutation formula (Theorem 1) in transformed as follows ( : :2

+ ).. 2 )

§d.

(57) can be

Ru = Ruo .

The above equation has the solution Ru(t) = a1 cos(>..t)

+ b1 sin(>..t) + ~ jrr )..

sin (>..t- >..s) Ruo(s) ds,

-'Tr

with t E [-n, n] and a1, b1 absolute constants.

D

Theorem 19. Let d be odd, d ~ 5. Let u and UQ belong to C 00 (G II K) {G = SO(d + 1), K =SO( d)). We assume that u vanishes to second order at n, and u 0 vanishes at n. The Poisson equation {57) has a solution if and only if uo(O) = 0. If this condition is satisfied, the unique solution of Lu = uo is given by 1 1

7r

u(t)=-:\>..! 2).

(

1

d ) ). (jt ) -rrsin(>..t->..s)Ruo(s)ds .

-sintdt

INTEGRAL GEOMETRY IN THE SPHERE

121

§d

Proof. The condition uo(O) = 0 is necessary. In fact, suppose that the equation (57) has a solution u. Then Lu = uo implies ~(n) = uo(n) for all n EN, where uo is the spherical Fourier transform. But

Lu(n)

= -n(n + 2.\)u(n),

therefore

~(n) Consequently, uo(O) have

= 0.

= -n(n + 2.\)u(n) = uo(n). Suppose now that uo(O) = 0.

Ru(t) = a1 cos(>.t)

By Lemma 6.2 we

+b1 sin(>.t) +} j_t1f sin (>.t- >.s) Ruo(s) ds.

Replacing t by 1r in the equality above we obtain: Ru(1r) = a1 cos(>.1r)

+ (-l).H1

j_:

sin (>.s) Ruo(s) ds.

Since s ---t Ruo(s) is an even function and Ru(1r) = 0, we have then a1 cos(>.1r) = 0 (>. E N*), hence Ru(t)

= b1 sin(>.t)

+} [1f sin (>.t- >.s) Ruo(s) ds.

Let now w(t)

=} j_t1f sin (>.t- >.s) Ruo(s) ds.

Since Ru(t) is even, we have Ru(t) = b1 sin(>.t) Ru( -t)

As t

---t

\[1 ( t)

+} [1f sin (>.t- >.s) Ruo(s) ds

= Ru(t) = -b1 sin(>.t) + w( -t).

is even we have { Ru(t) = b1 sin(>.t) + w(t) Ru( -t) = -b1 sin(>.t) + w(t),

(w(t) = w( -t), see Lemma 6.1.) Then b1 sin(>.t) = 0 ,

hence b1 = 0. Finally Ru(t) =

lit

:X

-1r

for all t E [-1r, 1r],

sin (>.t- >.s) Ruo(s) ds. D

122

AHMED ABOUELAZ

Remark 9. For all t E [-1r, 1r], Ru (t)

11-t

-1r

=

~

sin (- >-.t - >-.s) Ruo ( s) ds

=

-1~-t T -1r sin (>-.t +

>-.s) Ru 0 (s) ds.

Replaces by-sin the above equality. We have then: Ru ( -t) =

~

lt

sin (>-.t- .Xs) Ruo (s) ds.

This can be written Ru ( -t) =

~ l-1r sin (>-.t- >-.s) Ruo (s) ds + ~ [1!" sin (>-.t- >-.s) Ruo (s) ds

=

~2 sin (>-.t) uo (0) + ~ [1!" sin (>-.t- >-.s) Ruo (s) ds

=

~2 sin (.Xt) uo (0) + Ru (t).

Since t - Ru (t) is even, we obtain then

Uo (0) =

0.

Now we treat the even case. Assume that dis even ().. = m- ~, m E N* ) . Theorem 20. (d;;::: 6 even). Let u and uo be functions in C 00 (GIIK) (G = SO (d + 1), K = SO (d)). We assume that u vanishes to second orderat 1r, and uo vanishes at 1r. The Poisson equation (57) has a solution if and only if Uo (0) = 0. If this condition is satisfied, the unique solution to Luo = uo is given by

(58)

u(s)

=

(>-.~m 2 ~)..11!" [(- si~ t!) m (a

jt

1

cos(>-.t) +

+~ sin (>-.t- >-.r) Ruo(r) dr)] "' -1r

sint dt v'cos s- cost

with a1 an absolute constant and).. = m- 1/2. Here again, (>-.)m is as in Notation 1. Proof. From Lemma 6.2 we have: Ru(t) = a1 cos(.Xt) + b1 sin(>-.t) +

~ j_t1l" sin (>-.t- >-.s) Ruo(s) ds.

Fort= 1r, we obtain 0 = a1 cos(>-.1r) + b1 sin(>-.1r)

+~/_:sin (>-.1r- >-.s) Ruo(s) ds.

Since s - Ruo(s) is even and)..= m- 1/2, the above equation becomes 0 = b1( -1)m + ~( -l)m+ 1uo(O).

INTEGRAL GEOMETRY IN THE SPHERE

123

§d

As uo(O) = 0 we obtain

Finally Ru(t) =

a1

cos(>.t)

+

l j_t7r

sin (>.t- >.s) Ruo(s) ds,

by the Radon inversion formula (2). This yields (58).

0

Now, we state the generalized Poisson's equation Lnu = uo, n E N*. We begin with the case n = 2. Consider the equation L 2u = uo can be transformed as follows: (59)

d2 ( dt2

+ >.

2) (dt2d + >.2) Ru = Ruo, 2

uo(O)

=0

Assume that dis odd. (59) leads to

l j_:

2 ( :t2 + >. 2) Ru(t)

sin (>.t- >.s) Ruo(s) ds

'Yl (t).

=

By Theorem 20 we obtain

lit-1r

Ru(t) = -:x:

The above equation becomes

11

7r u (t) = \ , 1 --.>:

lis-1r

A A.

'YI(s) = -:x:

2

(

sin (>.t- >.s) 'YI(s) ds = 'Y2(t).

1

d) - -.- -d Sill t

t

A

(it-7r .

Sill ( >.t

- >.s) 'Yl (s) ds

)

sin (>.s- >.r) Ruo(r) dr.

Then, for d odd, the solution of the Poisson equation Lnu = uo is

d)A 'Yn(t)

ll1r( 1 u(t) = -:x: >.! 2A -sin t dt

'Yj(t) =

l 1:1r

sin (>.t- >.s) 'Yj-I(s) ds, 1

~j ~n

'Yo(t) = Ruo(t).

Theorem 21. Let d be odd, d ;;:;: 5. Let u and uo belong to C 00 (GIIK) (G =SO (d + 1), K =SO (d)). We assume that u vanishes to second order at 1r, and uo vanishes at 1r. The Poisson equation Lnu = uo has a solution if and only if uo(O) = ::Y1(0) = ... = ::Yn(O) = 0. If this condition is satisfied,

124

AHMED ABOUELAZ

the unique solution of Lnu = uo is given by u(t)

=

!n(t) =

l ~!; l [7r

( _

s~nt :t) A ([7r sin {At- As) ln-l(s) ds)

sin (At- As) ln-l(s) ds, 1::;; j::;; n

lo(s) = Ruo(s). Theorem 22. Let d be even, d ~ 6. Let u and uo belong to C 00 ( GIIK) (G =SO (d + 1), K =SO (d)). We assume that u vanishes to second order at 1r and uo vanishes at 1r. The Poisson equation Lnu = uo has a solution if and only if uo(O) = 1h{O) = ... = vn(O) = 0 (where vi is the spherical Fourier transform of the function Vj). If this condition is satisfied the unique solution to Lnu = uo is given by:

r

u(s) = _1__1_ {A)m 2AA } 8 with Vj(t)

(--1_!!_)m Vn-l(t) sint dt' sin t dt v'cos s- cost

11t

= a1j cos(At) + ~

-1r

sin (At- As) Vj-l{s) ds,

{1 ::;; j ::;; n)

and vo(t) = Ruo(t), where

a1j

(j = 1, 2, ... , n) are absolute constants.

Theorem 23. Let P be a polynomial with complex coefficients. If P( -x(x+ 2A)) # 0 for all x E R, then

P(L) : coo (GIIK) -

coo (GIIK)

is bijective, where L is the Laplace-Beltrami operator in §d. Proof. Let u be an element of C 00 (GIIK) such that P(L)u = 0. Taking the spherical Fourier transform, we obtain P( -n(n + 2A))u(n)

=0

,

for all n EN.

Since P( -n(n + 2A)) # 0, this implies that u = 0. Let now v E C 00 {GIIK). We shall find u E C 00 (GIIK) such that P(L)u v. This implies that (60)

P( -x(x + 2A))u(x)

= v(x) ,

=

for all x E R

where -denotes the spherical Fourier transform. The equality (60) implies

v(x) u(x) = P( -x(x + 2A)) By Theorem 7 we have

2 1r u(s)= A!2A

(

d)

1 -sinsds

A (

{

.

v(x- A)

)

J~.exp(zsx)P(>,2-x2)dx'

INTEGRAL GEOMETRY IN THE SPHERE Sd

125

if >. E N* and d odd, or 2

1

u(s) = (>.)m 2>.

11r JJR{ (- sint 1 d) dt s

m

. v(x - >.) (exp(ztx)) P(>.2- x2) dx

sint d t, ..;cos s- cost if d =2m, with (>.)mas in Notation 1. It is obvious that u E C 00 (GIIK). 0 X

ACKNOWLEDGMENT

The author is grateful to Massimo Picardello for several suggestions and to the referee for his comments. This paper was partially supported by the exchange agreement between the University of Rome "Tor Vergata" and the University "Hassan II" at Casablanca. REFERENCES [1] A. Abouelaz, Transformation de Radon sur le groupe special orthogonal SO(d + 1), Africa Math. 6 (1996), 101-111. [2] A. Abouelaz, R. Daher, Sur la tronsformation de Radon de la sphere §d, Bull. Soc. Math. France 121 (1993), 353-382. [3] J. L. Clerc, Les representations des groupes compacts, in 'Analyse Harmonique', CIMPA, Nancy, 1980. [4] J. Dadok, Paley- Wiener theorem for singular supports of K -finite distribution on symmetric spaces, J. Functional Anal. 31 (1979), 341-354. [5] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, 'Tables of Integral 'fransforms', Vol. I and II, McGraw-Hill, New York-Toronto-London, 1954. [6] I. S. Gradshteyn, I.M. Ryzhik, 'Table of Integrals, Series and Products', Academic Press, New York-London-Toronto, 1980. [7] S. Helgason, 'Groups and Geometric Analysis', Academic Press, New York, 1984. [8] S. Helgason, Duality for symmetric spaces with applications to group representations, Advances Math. 22 (1976), 187-219. [9] S. Helgason, The Radon tronsform on Euclidean spaces, compact two-point homogeneous spaces and Grossmann manifolds, Acta Math. 113 (1965), 153-180. [10] L. Hormander, 'Linear Partial Differential Operators', Grundlehren math. Wissensch. 116, Springer-Verlag, New York, 1969. [11] T. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie groups, in 'Spherical functions: Group Theoretical Aspects and Applications', R. A. Askey et al. eds, Reidel, Dordrecht-Boston, 1984, 1-85. [12] T. Koornwinder, A new proof of a Paley- Wiener type theorem for the Jacobi Transform, Ark. Math. 13 (1975). [13] T. Koornwinder, Explicit formulas for special functions related to symmetric spaces, in 'Harmonic analysis on homogeneous spaces', Proc. Sympos. Pure Math., Vol XXVI, Williams College, Williamstown, Mass. 1972, Amer. Math. Soc., Providence, Rhode Island, 1973, 351354. [14] T. Sherman, Fourier analysis on the sphere, 'frans. Amer. Math. Soc. 209 (1975), 1-31. [15] R. Strichartz, Harmonic analysis as spectrol theory of Laplacians, J. Functional Anal. 87 (1989), 51-148. [16] R. Strichartz, Local harmonic analysis on spheres, J. Functional Anal. 27 (1988), 403-433. FACULTY OF SCIENCES AiN CHOCK, UNIVERSITY "HASSAN II", B.P. 5366, MAARIF, CASABLANCA, MOROCCO E-mail address: abouelaziDfacsc-achok. ac .ma

A TOPOLOGICAL OBSTRUCTION FOR THE REAL RADON TRANSFORM ANDREA D’AGNOLO AND CORRADO MARASTONI A bstract. The recently developed theory of integral transforms for sheaves and ^-m odules provides a general framework to deal with problems of integral geometry. Our aim here is to illustrate such techniques by reconsidering the most classical example of integral transforms, namely, the Radon hyperplane transform for C °°-functions.

I n t r o d u c t io n

The theory of integral transforms for sheaves and D-modules provides a natural framework to deal with the problems of integral geometry. In particular, a general adjunction formula allows one to separate the analytical aspects of the problem from the topological ones. In [3, 1] such theory was applied to the study of the Radon hyperplane transform (see [2] for an exposition). The Radon transform associates to a homogeneous C°° function on a real projective space P its integrals along the family of hyperplanes, thus yielding a homogeneous C°° function on the dual projective space P*. It is a well-known fact that such transform is invertible in a suitable range of the degree of homogeneity. A classical approach (see, e.g., [5]) is to deduce this result from the inversion formula for the Fourier transform. Another approach, closer to ours, is that of [8, Proposition 4.1.3], where the real Radon transform is considered as the “boundary value” of the complex one (see also [7] for a similar point of view). Here we will use the methods of [3, 1] to discuss the case of arbitrary homogeneity. In particular, we will calculate exactly the finite dimensional obstruction to invertibility, and we will show how such obstruction is purely topological. 1. N o t a t io n s

Let us recall some notations and results on the hyperplane Radon trans­ form, referring the reader, e.g., to [5]. Affine hyperplanes in R n 3 y are described by equations and are thus parameterized by (η, r) G ((R71)* \ {0}) x R. The classical Radon transform associates to a rapidly decreasing C°° function / on R n 1991 Mathematics Subject Classification. Primary: 44A12; Secondary: 32L25. Key words and phrases. Radon transform, algebraic analysis. We present here the results of a paper th a t will appear elsewhere. 127

128

ANDREA D' AGNOLO AND CORRADO MARASTONI

its integrals along affine hyperplanes, and reads f(y)

(1)

~ g(1J, r) =

J(y) dJ.t11 ,r,

{ }(y,7J)+r=O

where dj.t71 ,7 is the measure on the hyperplane (y, 17) + T = 0 induced by the volume element dy1 1\ ···I\ dYn· Let P be a real projective space of dimension n, and [x] = [xo, ... , Xn] a system of homogeneous coordinates on P. FormE Z and c: E Z/2Z, let us denote by Cpc'(m!e) the C 00 line bundle on P whose sections r.p satisfy for ,\ E R x.

r.p(,\x) = (sgn ,\)e ,xmr.p(x)

Let Vp = C~,(n) ®orp be the sheaf of densities on P, where orp denotes the orientation sheaf. Recall the isomorphism Vp ~ Cpc'(-n-11-n-1). The Leray form n

w(x) = ~( -1)ixjdxo 1\ ···I\ dxj 1\ ···I\ dxn j=O

is the global section of Vp(n+lln+l) corresponding to 1 E cpo. (Here, for a Cpc'(m!e).) C 00-module F, we set F(mle) = F ®c"" p Let P* be the dual projective space, and let [~] be the system of homogeneous coordinates dual to [x]. The manifold P* parameterizes projective hyperplanes in P by the correspondence~ -o = {x: (x,~) = 0}. The projective Radon transform is defined by

€

(2)

r.p(x)

~ '1/J(~) =

i r.p(x)8((x,~))w(x),

where, in order for the integral to make sense, r.p E r(P;Cpc'(-n!-n)). It is clear that '1/J E r(P*;C~(-11-1)).

Remark 1. Let us identify Rn with the affine chart xo =j; 0 of P, so that Yi = Xj/Xo for j = 1, ... , n. If f(y) is rapidly decreasing, r.p(x) = lxol-n f(y) is a well-defined global section of Cpc'(-nl-n), vanishing up to infinite order on the hyperplane at infinity xo = 0. In this sense, (2) is the natural projective extension of (1). Form< 0, consider the family k(mle) of distributions in R defined by (3) form< 0:

d )-m-1

k(ml1)(t) = ( dt

8(t),

k(miO)(t)

1

= p.v. t-m'

where p.v. stands for principal value. Note that k(mle) is (mJc:)-homogeneous. A natural generalization of (2) is then given by the following.

Definition 1. The generalized projective Radon transform is defined by (4) 'R..(m•!e•): r(P;Cpc'(m•je•)) ---+ r(P*;C~(m!e)) r.p(x)

~ '1/J(~) =

where m* = -n - 1 - m,

c:*

i r.p(x)k(mle)((x,~))w(x),

= -n -

1 - c: mod 2.

A TOPOLOGICAL OBSTRUCTION FOR THE REAL RADON TRANSFORM

129

Let R(mlc:) be the transform obtained by interchanging the roles of P and P*, i.e., (5)

R(mlc:): f(P*;Cp.~) )w(>.~) =(sgn >.)n+l'lj;(~)k(m•JO) ( (x, ~) )w(~)

+ (sgn >.)n+llog 1>-l'l/J(~) (x, ~)m• w(~).

Denote by C[x](m•) the space of homogeneous polynomials of degree m*, and consider the transform

(8)

Cm: f(P*;Cp Dqi, and ® denote the (derived) functors of proper direct image, inverse image, and tensor product for 1J-modules. Similarly, for sheaves G on Y and K on X x Y (better, objects of the bounded derived categories of sheaves of Cvector spaces), one considers

K o ·: G ~ K o G = Rq1,(K ®q2 1 G). Assume that G is R-constructible, JC is regular holonomic, and K = Sol(JC, OxxY) is its associated perverse sheaf. Under a natural transversality hypothesis-satisfied in the case of the Radon transform-one has the following adjunction formula between global solution complexes (11)

Sol(N,C 00 (K o G[dimXJ)) ~ Sol(M g /C,C 00 (G)),

where G ~ G[l] is the shift functor in the derived category, and C00 (G) = G® Oy is the formal cohomology functor of [9]. (Recall that C00 (G) = Cfl if G = C N is the constant sheaf along a totally real submanifold N C Y, of which Y is a complexification.) Finally, recall that in the transversality hypothesis, one has the following isomorphisms, asserting that the morphisms N --+ M g JC and M g JC --+ N are described by an integral kernel. (12) (13)

a: Sol(Mv ~ N, JC) ...=::. Homvy (N, M {3: Sol(M ~Nv,JCv) ...=::. Homv)M

g /C),

g JC,N),

where Mv = R1iomvx (M, 1Jx) ®ox O)c is the dual of M as a left 1Jymodule. Let lP 3 [z] be a complex projective space of dimension n, JP* 3 [(] the dual space, § C lP x JP* the incidence relation (z, () = 0, and §c = (JP x JP*) \§. One denotes by Bsc the regular holonomic 'Dpxr·-module of meromorphic functions on lP x JP*, with poles along §. For m E Z, let OrCm) be the holomorphic line bundle whose sections

•( -m) ~ Vp( -m*) a(tm•): Vp( -m*) ~ B¥c

g Bsc,

g Vp. ( -m).

(iii) By (11), a(sm) and a(tm•) induce mutually inverse isomorphisms a(s,)

(16)

RI'(JP>;C 00 (Csc o C~}(nJ)(m•)) - - - f(P*;Cp.(e•lm)). a(t,.)

(To explain the notations in (15), recall that on C 3 r one has Bcx = Vc/(Vc · 8rr), Btx = Vc/(Vc · r8r)· One denotes by Y(r) the canonical generator of Btx, and sets 8(k)(r) = a:Y(r).) 4.

SKETCH OF PROOF

As it was done in [3, 1] for the case -n- 1 < m < 0, we will show here how Theorem 1 can be obtained as a corollary of Theorem 2. A purely topological computation (see [3, Proposition 5.16]) gives W) (n] C p•

( 17)

C §c

(18)

Cll'\P ~ Csc o Cp. (n] ~ Cll' ~ [ + 1].

0

~

c(l) p , (o•)

The isomorphism (9) is obtained by plugging (17) in (16). In order to get (10), we need to describe the complex F = Rf(JP>;C 00 (Csc o C~~·)[nJ)(m•)). By (18), we get a distinguished triangle 0 ~ Rf(JP>;C 00 (CI!'\P)(m•)) ~ F ~ C[x](m•) ~ 0,

ANDREA D' AGNOLO AND CORRADO MARASTONI

132

where we used Serre's isomorphism Rr(IP; C00 (CJP)(m•)) ~ Rr(IP; OJP(m•)) ~ C[x]cm·)· Moreover, the short exact sequence 0---+ CIP\P---+ CIP---+

Cp---+

0 gives

0---+ C[x](m•)---+ r(P;Cpc>cm•]o))---+ Rr(IP;C 00 (CIP\P)(m•))---+ 0. Combining the two short exact sequences above, we get the following commutative diagram, whose first row is exact F

0---+ C[xJcm•)---+ r(P;Cpocm•]o))

-------

C[x]cm•)---+ 0.

l!a(s,.)c. __ /

n..(w, 0). Lemma 2.1. Let). E JR.*. There exists a positive constant c(.X) such that Vr E [0, 1[:

ISr,>.(w, 0)1 ~ c(.X)Jw- OJ- 2 w' (} E 8 2 c IR.3 ISr,>.(W, 0)- Sr,>.(w', 0)1 ~ c(.X)Iw- w'llw- oj- 3 when lw - OJ

~

2Jw - w'J.

These inequalities follow from Taylor expansion at order 1 of the function ~8 E 0 00 (IR.3 \ { 0}) defined, for (} fixed in 8 2 , by

POISSON TRANSFORM ON H:3

IP9(x)=

Lemma 2.2. that

(1-r< ,0>): 1 1

1

141

-iA

{1) Let A E JR.*. There exists a positive constant c(A) such {

Jo~d2~1w-9l~dl

Sr,A(w, 8)d8

~ c(A)

'l:/d1, d2.

{2) Let a> 0. There exists a positive constant C(a) such that {

Jlw-9l~d

ISr,A(w, B)llw-

Blade~ C(a)da.

{3) Let (3 > 0. There exists a positive constant C((3) such that

f

Jlw-9l~d

ISr,A(w, O)llw- 81-.Bde

~ C((3)d-.B.

{4) For all a E [0, 1[, there exists a positive constant C(a, .X) such that {

Jlw-91 ~2lw-w'l

ISr,A(w, 8)- Sr,A(w', B)llw-

Blade~ C(a, .X)Iw- w'la.

Point 1 follows from a direct computation in spherical coordinates, the other points follow from Lemma 2.1.

Step 2: a-molecular resolution. To introduce the notion of a-molecules on 8 2 , where a E]O, 1] is fixed, one considers the weight functions on 8 2 X 8 2, Od(O, w) = 4~da(d+ 18 -wl)- 2-a, ford E]O, 1].

Definition 1. A function m : S 2 - - - t C is called a a-molecule of width d (a E]O, 1], d E]O, 1]) centered at the point wE S 2 , if the following 3 conditions are satisfied: 1) fs2 m(O)dO = 0 2) lm{B)I ~ nd(O,w) 3) lm{O)- m{O')I ~ d-a10- B'la(nd(O,w) + nd(O',w)) We denote by M(a,d,w) the space of such functions. The sequence {~j; j E N} of kernels on 8 2 : ~i(O,w)

= P{{1- 2- 1 -i)O,w)- P{{1- 2-i)O,w)

where P(x, w)( is the Poisson kernel {1), defines by convolution a resolution of the identity in the space

{ f E L2 {S2 );

fs

2

f(O)dO = 0}.

This resolution is a-molecular. More precisely we have:

142

SAMIRA IBENMOULOUD AND MOHAMED SBAI

Proposition 2.3. Let a E]O, 1] and let w be in 8 2 ; for any j EN the kernel Llj(O,w) is in M(a,2-i,w). This follows from estimates on the Poisson kernel P((1- d)x, 0) obtained from Taylor expansion at order 1, with integral remainder, of the function wd E coo (JR3 \ { 0}), defined for 0 E 8 2 fixed, by

1:1' 0 >) + d2 )

wd(x) = d2 (2- d) 2 ( 2(1- d)(1-
. on a-molecules. Proposition 2.4. Let 0 < (3 < a < 1. There exists a positive constant J.t( a, (3) such that J.t(a, (3)Sr,>. (M(a, d, w)) C M((3, d, w) '.(0, v)Llj(v, w)dv

and the operator s;,>. defined by this kernel, one has the decomposition:

8r,>.

=L j

8r,>. . Llj

=L

8;,>..

j

To apply the "concrete version" of Cotlar-Stein's lemma to the family (8r,>.)r, one must check a result analogous to lemma 6 in [2]:

Lemma 2.5. Let a E]0.1],j EN and set d = 2-i. Then (1) 8;,>.(0,w) E M(a,d,w),

(2) fs2 8;,>.(0,w)di..J = 0, (3) 3 positive constant c(r,j, >..) l8;,>.(0,w)- 8;,>.(0, v)l ~ c(r,j>.)lw- vlada(nd(B,w)

+ nd(B, v)).

The first point follows from propositions 2.3 and 2.4. The next two points follow from symmetry properties of the kernel 8r,>. (8, w) and of the weight function nd(e, w). 3.

PROOF OF THEOREM

1

The theorem is proved in two steps: first the continuity, then the bijectivity of P>..

Proposition 3.1. For any >.. E JR.*, there exists a positive constant A(>.) such that

POISSON TRANSFORM ON IHI 3

143

D

Proposition 3.2. For any uA E ,BA(JHI3 ) there exists f E £ 2(82) such that uA = PA(f). Furthermore, there exists a positive constant CcA) such that

IIPA(f)ll* ~ C(A)Iifli£2(82)· Proof. Helgason proves in [5] that given any uA E ,BA(H3 ) there exists a hyperfunction f on the sphere 8 2 such that PAJ = uA. To show that f E £ 2(82), one proceeds as in [3]. The spherical functions 'Pk,A on JHI3 are solutions (smooth in the radial variable r) of the differential equation:

J2

d

(dr 2 + 4coth2r dr

+ 4((sh2r)- 2k(k + 1)) + A2 + 4)cpk,A(r) =

0

and can be expressed as:

'Pk,A(r)

=

3

iA

r( 2)r(k + 1 +!) (th r)k(1- (th r) 2)-k r(k + ~)r(k + ~;) F2(k

+ 1- iA k + 1 + iA k + ~

2' 2' 2' where F2 is the hypergeometric function. One defines 2 2 1 {R Ik,A(R) = R Jo i'Pk,A(r)i (sh2r) dr.

-(sh r) 2 )

The proofrelies on the following two points (see [3] or [1]): ----;-;::-;-

1 (R) =I (R) _ (sh2R) 2 'Pk+1,A(R)'Pk+~,A(R) k+2,A k,A R 2(k + 2- ~;) limR--++oo lk,A(R) = c(A)

>0

where c is the Harish-Chandra function of JHI3 .

D

REFERENCES [1] S. Benmouloud-Sbai, Caracterisation de !'image de L 2 (S 2 ) par Ia racine carree complexe du noyau de Poisson sur l'espace hyperbolique lHI 3 , these de troisieme cycle, Universite Mohamed V, Rabat, Maroc, fevrier 1991. [2] Y. Meyer, 'Les nouveaux operateurs de Calderon-Zygmund', Asterique 132, 2 {1985). [3] R. Strichartz, Harmonic analysis as spectral theory of Laplacian, J. Functional Anal. 87 (1989), 51-148.

144

SAMIRA AND MOHAMED SAMIRA IBENMOULOUD IBENMOULOUD AND MOHAMED SBAI SBAI

(4) Harmonic Laplacian, Functional [4] R. R. Strichartz, Strichartz, corrigendum corrigendum to to H a rm o n ic analysis an a lysis as spectral theory th eory of of L aplacian , J. J. Functional Anal. Anal. 102 1 0 2 (1992), (1992), 457-460. 457-460. [5] S. S. Helgason, Helgason, Eigenspaces E igen spaces of o f the Laplacian, L aplacian , integral re p re se n ta tio n s and a n d in'educibility, irre d u c ib ility , J. J. Func­ (5) representations Functional Anal. Anal. 17 1 7 (1974), (1974), 569-645. 569-645. tional D epartment OF of MATHEMATICS, Mathematics, UNIVERSITY U niversity "IBN “Ibn TOFAIL", T ofail”, B.P. enitra, MOROCCO Morocco DEPARTMENT B.P. 1018, K KENITRA,

TRANSFERT FORMULA IN THE REAL HYPERBOLIC SPACE

an

HASSAN SAMI ABSTRACT. Let s E IC such that -1 < Re s < n _: 1 . Then we show that every eigenfunction f of the Lobatchevsky Laplacian D in the real unit ball Bn, with (s 2 - 1)(n - 1) 2 as eigenvalue can be written as an integral transform (transfert formula) of a unique Euclidean harmonic function Fin Bn: n-2

f(x) =

r(~)(1-lxl 2 ),-

1

1[

r((1+s)(n21))re-•1,j Yi,j (u) j

l):O

is nothing but the series expansion of the harmonic function F(tru) with respect to Yz,j, one has:

] SCT+~ dt t [ r(!!)(1- r2)(1+s)CT [1 d-u . (BsF)(ru) = r((12+ s)o-)rO- so-) lo F(tru) (1- t)(1- tr2) This completes the proof of Theorem 1. 4.

APPLICATION OF THE TRANSFERT FORMULA

We begin this section with an LP-inequality on the spherical Poisson transformation associated to P8 •

Proposition 4.1. Lets be a real number such that -1 < s < n~ 1 and let p;;::: 1 be a real number. For every eigenfunction f in 1t 8 (Bn) the following estimate holds:

(fsif(ru)IP du)* :::;: Cf!s(r)(fs!F(ru)IP du)* for every 0 :::;: r < 1. Here F is the Euclidean harmonic function such that 08 F =f.

Proof. First, we consider the case of p = 1. For every f in 1ts(Bn), the transfert formula of Theorem 1 yields a harmonic function F in 1te(Bn) such that: f(ru) =

r2)(1+s)u r(!!)(12 r((1 + s)a-)r(~- so-)

11 0

F(tru)

[

] su+~ dt t - 3 -. 2 t2-u (1- t)(1- tr )

HASSAN SAMI

150

Hence, for each fixed r E [0, 1[,

1

f(!!)(1- r2)(1+s)u 2 s If (ru) Idu ~ r ((1 + s ) u ) r - su

e2

)

(fs IF(tru)l du) fo [(1- t)/1 _ tr2)ru+~ t~u · = f(!!)(1-r2)(1+s)u 11[ t ]su+~ dt 1

Then the result for p 2

1 follows by observing that

r((1 + s)u)r( 2 - su) 0 1

-3-

(1- t)(1- tr 2)

t2-u

= ~s(r)

and by the fact that the application

r--"

Is

IF(ru)l du

is increasing in r E [0, 1[ (since F is harmonic). Let us now turn our attention to the case p > 1. Let h be in Lq(S) with ~ + = 1 and llhllq ~ 1. Then

i

lis f(ru)h(u) dul

~

r(!!)(1- r2)(1+s)u rl f((12+ s)u)r(!- su) } 0

Ir

Js F(tru)h(u) du

~

r( ~) ( - 2)u(l+s) fl .._, r((1 + s)u)r(!- su) 1 r }0

II

I[

t Jsu+~ dt (1- t)(1- tr 2) d-u

II [ Ftr P

t ] su+~__!!:!__ (1- t)(1- tr 2) d-u ·

The result follows by using again the fact that the LP-slice norm (fsiF(rtu)IP 1

0

du)"P is increasing in r E [0, 1[.

As a consequence of Proposition 4.1, we prove an inequality for the spherical function ~ 8 • More precisely: Corollary 4.2. Let s be a real number and let p ;;;:;: 1. Then:

~(s+l)p-l(r) ~ (1- r 2 )Pe;n)(~ 8 (r))P~~-l(r) for every r E [0, 1[. Proof. Recall that (BsPe)(ru) = P8 (reo, u) where eo= (1, ... , 0) denotes the unit vector of S. By Proposition 4.1:

(Is

1

IPs(reo, u)IPdu);;

~ ~s(r)

(Is

1

IPe(reo, u)IPdu);;

The inequality follows by using on the right-hand side the identity (see [6])

[Pe(reo, u)] = (1- r 2 ) 1 -~ P_1 (reo, u) n-1

TRANSFER FORMULA IN THE REAL HYPERBOLIC SPACE B"

151

and on the left-hand side the fact that I

( [ IPs(reo,u)IPdu);;

= q,p(s+l)-l(r). D

We shall now use the transfert formula and Proposition 4.1 to show that, for some s E C, the Poisson transformation P8 is a continuous application from LP(S)(p ;;::: 2) into Lq(Bn, d~t) for every q > p, where d~t(x) = (1lxl 2 )-n. More precisely, we establish the following result:

Theorem 2. Let p ;;::: 2 be a real number and let s E C such that Res = ~ 1. Then the Poisson transform P 8 is continuous from LP(S) into Lq(Bn, d~t) for every q > p. That is: Ps(LP (S)) c n Lq (En, d~t). q>p

Remark 1. Theorem 2 gives another proof of the result of Eymard and Lohow~ on the square root of the Poisson transform [3]. Our approach, which makes use of the transfert formula, is much simpler than the argument of Eymard and Lohoue.

Before giving the proof of Theorem 2, we first recall the following result on Euclidean harmonic functions [6]:

Lemma 4.3. Let p > 1 be a real number and let F be a harmonic function in Bn which is the Poisson transform by Pe of some gin LP(S) (i.e. F =Peg). Then, for each q > p there exists a positive constant c = c(p, q) such that the following estimate holds:

1IIFrll~ 1

In this inequality

IIFrllq

(1- r 2 )-kdr:::; c

llgll~ ·

denotes the Lq-norm I

IIFrllq = ([ lf(ruW du) 7i and k

= 1 + (n-

1)(~-

1).

Proof of Theorem 2. We first observe that, for each gin V(S), (lgl) · IPs(g)l :::; Pa_l p

Hence, to prove that P8 (LP(S)) C Lq(Bn,d~t) for every q show that Pa_l(LP(S)) p

for every q

> p.

c

Lq(Bn,d~t)

> p, it suffices to

HASSAN SAMI

152

Now, let f = P:L 1 (9) with 9 E £P(S), and let F be the Euclidean harp monic function in Bn such that

Then, by Proposition 4.1,

Is

lf(ru)IQ du

~ ( ;_ 1(r) r

(is IF(ru)IQ du) .

By the expression in polar coordinates of the volume measure dJ.L of the real hyperbolic space, one has:

Ln Let k

lf(x)!QdJ.L(X)

= 1 + (n-

1)(~-

1 (;-1(r)fi1Frllg(l-r )-nrn-1dr.

~

1

1). The right-hand side of this inequality is

1 ;-1(r)f IIFrllrp 1IIFrll~ 1

2

(

(1- r 2 )-n+k

IIFrll~ (1- r 2 )-krn- 1dr.

By making use of the estimate given in Lemma 4.3 1

(1- r 2 )-k dr

~ c(p, q) 11911~

one has

Ln

lf(x)lq dj.t(x)

~ c(p,q) 11911~ sup { ( ;-1(r)r IIFrllrP(l- r2 )k-n}.

Now, recall that

Fr(u)

=Is

Pe(ru, v)9(v) dv.

But S = SO(n)/SO(n- 1), and the functions inS can be identified with the functions defined on SO(n) which are SO(n- I)-invariant. Therefore (1)

Fr(u) = {

lso(n)

Pe(rk, h)9(h) dh

(k E SO(n)) .

Now write

(Pe)r(u) = Pe(ru, eo). One can regard 1 as a convolution over the compact group SO(n):

Fr

= (Pe)r * 9 ·

Hence, by Jensen-Young's inequality, 1. with 1q = 1p + 1v

TRANSFER FORMULA IN THE REAL HYPERBOLIC SPACE Bn

153

On the other hand, 1

II(Pe)rllv =

(1- r 2 ) 1 -~ ( { IP-1 (reo, uWdu);

ls

= (1-

n-1

r2)1-~ ( 0, Watanabe considered in [6] the following Fock-Hilbert space H 2;>.(D) consisting of analytic functions of one complex variable z E C on the unit disk D equipped with the Hermitian scalar product:

< f; 9 >:=

l

f(z)g(z)p>.(lzl 2 ) dx dy

where the density is given by 1

(t) _ P>. -

{

f(2A- 1)

t>.- 1

(z

= x + iy ED)

1 1

r

s->.(1- s) 2>.- 2 ds

if

t

[f(l -A) 1 ] >.-1 ->. 2>.-2 I'f f(A) - f(2A- 1) t Jo s (1- s) ds

The Gegenbauer polynomials C~(x)n~o form a complete orthogonal system of the Hilbert space G>.(] -1, 1[)=L2 (] -1, 1[, (1-x2 )>.-~ dx). In [6], Watanabe has given an explicit unitary integral transform between the above space G>.(]- 1; 1[) and the Hilbert space H 2 ;>.(D) of Fock type. More explicitly, for E G>.(]- 1; 1[) Watanabe's unitary integral transform is (W>.)(z)

=

j_ A>.(z; 1 1

x)¢(x)(1-

x 2 )>.-~ dx,

where the kernel A>.(z; x) is given by:

A >. (z· x ) - 2>.-~r(A + 1)(1- z2) ' - 11'(1 - 2xz + z 2 ):A+ 1 The purpose of this paper is to show that Watanabe's procedure for the Date: March 30, 1999. 1991 Mathematics Subject Classification. Primary: 44A12; Secondary: 43A85. Key words and phrases. q-analysis. 155

ABDERRAHMAN ESSADIQ

156

classical Gegenbauer polynomials works also for the q-continuous Gegenbauer polynomials C~(x; q) introduced by Rogers in [4], and to construct a q-analogue of Watanabe's unitary integral transform. The plan of the paper is as follows. In section 2, we define the q-continuous Gegenbauer polynomials C~(x; q) of Rogers [4] and consider some of their properties. Section 3 introduces the q-analogue of the Fock-Watanabe space H 2;>.(D) as well as q-analogue of G>.(]-1; 1[), and gives an explicit formula for the reproducing kernel of H:;>.(D). In section 4 we construct a unitary integral transform between H:;>.(D) and Gq,>.(] - 1; 1[). We also give some remarks on the possible extension of this procedure to q-continuous Laguerre polynomials on ]0; +oo[.

2. q-CONTINUOUS GEGENBAUER POLYNOMIALS AND BASIC PROPERTIES

Let 0 < q < 1 and A> 0. For fixed x in]- 1; 1[ we consider the function z----+

II

k~O

[1-

2xzqk+>.

1 - 2xzqk

+ z2q2(k+>.)l + z2q2k

which is analytic on D = { z E C : Iz I < 1}. The coefficients of the associated generating function are the q-continuous Gegenbauer polynomials C~(x; q):

+ z2q2(k+>.)]- "'""c>-(x· q)zn II [1-12xzqk+>- 2xzqk + z2q2k - ~ n '

(1)

k~O

n~O

for all lzl < 1. See [5] for more details. These polynomials satisfy the following orthogonality relations:

1 C~(x; q)C~(x; 1

(2)

-1

q)aq,>.(x)dx = hnt5n,m

where

(3)

aq,>.(x)

1- 2(x2 - 1)qk + q2k

]

1

= II [ 1- 2(x2- 1)qk+>. + q2(k+>.) Jf=X2' k~O

hn > 0 is a positive constant and t5n,m is the usual Kronecker symbol. The orthogonality relations yield the following recursion relation:

2x(1- qn+>.)C~(x; q) = (1- qn+ 1 )C~+l(x; q)

with

c: (x; q) = 0 and CB(x; q) = 1.

+ (1- qn+ 2 >.- 1 )C~_ 1 (x; q)

1

The q-continuous Gegenbauer polynomials form a complete orthogonal system in the Hilbert space L 2(] - 1, 1[, aq,>.(x)dx), henceforth denoted by Gq,>.(] -1; 1[). Note that aq,>.(x) - t (1- x 2 )>.-~ as q - t 1-.

q-ANALOGUE OF WATANABE UNITARY TRANSFORM ASSOCIATED...

157

30 THE HILBERT SPACE H~;>.(D) OF ANALYTIC FUNCTIONS ON THE UNIT DISK OF THE COMPLEX PLANE

Let .A > 0, 0 < q < 1 and let D be the unit disk of the complex plane defined by D = {z E C: lzl < 1}0 For f and g analytic functions on D we put:

< !; g >>.,q:= 1 : q

(4)

1[1 1

2

7r

f(rei 0 )g(rei 0 ) dO] P>.,q(qr 2 ) dq(r 2 )

where dO is the usual Lebesgue measure on [0, 27r] and the integral with respect to dq(r 2 ) is related to the q-Jackson integral on [0, 1] defined by Jackson in [3] as:

To define the q-analogue P>.(t) of the density function given by Watanabe in [6] we need to introduce the following ingredients which are very common in the q-literature (see for example [5])0 Let 0 < q < 1, a E C and set: 1- qa [a]q = 1- q

(a,q)o = 1

(a, q)k = (1- q)(1- qa)(1 -la)

(a,

0

q)oo = IT (1- qka)

0

0

(1- qk- 1a)

o

k~O

Also, for

~(a)

> 0 we define a q-extension of the r function by

/~q\oo

(1- q)l-a q ,q 00 A q-extension of the (3 function is provided by a q-analogue integral representation: for (a, b) E C 2 and ~(a; b) > 0,

fq(a)

:=

(Jq(a, b)= with (t, q)z

Remark 1.

=

q~~q) ,q 00

1 1

ta-l(qt, q)b-1 dqt

for z E C and ~(a; b)

> Oo

1. (3 ( q

a,

b)= fq(a)fq(b) fq(a +b)

2 For q __... 1-, 0

fq(a) __... r(a) and

r(a)r(b) (Jq(a, b) __... (J(a, b) = r(a +b)

0

ABDERRAHMAN ESSADIQ

158

With notation as above we define P>.,q(t) as t>.- 1 { 1 s->.(1- s) 2>.- 2 ds

1

P>.

(t) _ { r(2.X- 1) lt [r(1- .X) 1 ] t>.-1 {t s->.(1- s)2>.-2 ds r(.X) r(2.X- 1) lo

Then the Hilbert space Hg;>.(D) is defined as

H;;>.(D) := {! : D-+ C, f analytic and llfll>.,q < +oo} where llfll~,q :=< f;f >>.,q· Note that P>.,q(t)-+ P>.(t) as q-+ 1-. Next, we shall study in detail some Hilbert properties of the above q-FockHilbert space of analytic functions on D.

Proposition 3.1. 1. The set {1, z;, z 2 ; ... } forms a complete orthogonal system in Hg;>.(D). More precisely,

(a ) < z m ; z n > >.,q=

27rq>-r~(m+1)

~

(l+q)[>.+m .r.( 2>.+m) un,m

(b) For f E Hg;>.(D) written in the form

f(z) =

L

amzm

m~O

one has

I

2

lfll>.,q =

21rq>.rq(m + 1)

""'

2

~ (1 + q) [.X+ m]qrq(2.X + m) lam I .

2. For f E Hg;>.(D) and zED one has

lf(z)l

~

llfll>.,qVh>.,q(lzl 2 )

with (5)

Proof. 1a) It is clear that form=!= n < zm; zn >>.,q= 0. Thus, we need only to compute < zm; zn >>.,q for m = n. For this, let m E N and write z = rei 9 with 0 < r < 1 and 0 < () ~ 27r. Then < zm; zm > >.,q can be written: 1 < z m ; z m > >.,q= (127r + q) Jo{ r 2m P>.,q (qr 2) dq(r 2) .

So it suffices to compute the integral

(6)

159

q-ANALOGUE OF WATANABE UNITARY TRANSFORM ASSOCIATED...

For this, we shall use q-integration by parts, that is:

loa (Dqf)(x)g(qx) dq(x)

= [f(x)g(x)]g -loa (Dqg)(x)f(x) dq(x)

g a smooth functions on [0, a]. (See [5].) This provides another way to write the integral 6 for >. > ~:

with: (Dqf)(x) = f(q:~~f~x and

A-1

{1

Jo

J,

umpA,q(qu) dq(u)

= fq(2>.! 1)[m +

>.]q [um+Ag(u)J6-

lo (Dqg)(qu)um+A1

J:

s-A(qs, qhA-2 dq(s). where g(u) = u-A(qu, qhA-2 it follows:

dq(u)

From g(1) = 0 and (Dqg)(u) =

-

{1m

Jo

1

qAfq(m+1)

u PA,q(qu)dq(u)- fq(2>. + m)[m + >.]q

By the same token, for all>.> 0 we have:

A,q = . ,. .1-+---:q

for every mEN. 1b) Write f E H~;A(D) in the form f(z) = Lm~O O:mZm. That is:

zm zm;zm >A ,q ..j< zm· zm >A f(z) = "'o:mJ< L..J m~

'

~

since < zm; zm > A,q# 0 for all m E N. Then the orthogonality relations 1a) easily yield

27rqAfq(m+1) "' 2 IIJIIA,q = ~ ( 1 + q)[>. + m]qfq( 2>. + m) 2) First of all, we normalize the orthogonal system

2

lo:ml ·

(zm)m~O·

The normalized

functions are

Um,A(z, q) = Expand j, g in H~;A(D) with respect to this orthonormal basis:

f(z) =

L

O:mUm,A(z, q)

m~O

g(z) =

L m~O

f3mUm,A(z, q)

ABDERRAHMAN ESSADIQ

160

and m~O

Let f E Hg;>.(D), f(z) = Em~o amzm. For fixed zo in D, f(zo) is the Hermitian product < g; t >>.,q of the following two functions g and t in

Hg;>.(D):

g(~)

=I: m~O

t(~)

=I: m~O

Therefore, by the Cauchy-Schwartz inequality,

lf(zo)l ~ IIYII>.,qlltll>.,q On the other hand, it is clear that

IIYII>.,q = llfll>.,q and

lltll2 = "' (1 + q)[m + A]qfq(2A + m) lzol2m. >.,q LJ 21rq>-r (m + 1) m~O

q

By replacing this in the above inequality, we get:

lf(z)l

~

llfll>.,qV(h>.,q(lzl 2)

(1

+ q)[m + A]qfq(2A + m) ~m 21rq>.fq(m + 1) ·

for all fixed z E D, where

0

Corollary 3.2. For every z in D, the Dirac functional t5z defined by

f

~

f(z)

is a continuous linear form on the Hilbert space Hg;>.(D). It follows from Corollary 3.2 and the Riesz representation theorem that the Hilbert space H~;>.(D) has a reproducing kernel. The following proposition gives the explicit formula for the reproducing kernel. Proposition 3.3. The reproducing kernel formula for Hg;>.(D). Let K>.,q the reproducing kernel of Hg;"(D). Then:

K>.,q(z, w)

= (1 + q~A]~rq(2A) 1rq

1 + q"zw (zw, qh>.+I

'v'(z, w) E D 2 .

q-ANALOGUE OF WATANABE UNITARY TRANSFORM ASSOCIATED...

161

Proof. Following [1], we have: K>.,q(z,w) =

L Um,>.(Z, q)Um,>.(W, q). m~O

Let us use here the explicit expression of Um,>.(z, q). One obtains: K

(

(1 + q)[m + .X]qfq(2,\ + m) ( -)m ) _'"' ~ zw . 21rq>.rq(m + 1) m~O

>.,q z, w -

Since

and

1- qm+>. [m+-X]q=-1--

-q

the reproducing kernel K>.,q(z,w) becomes K

(

2 ) _(1+q)fq(2,\)'"'(q \q)m(1-qm+>.)( -)m 21r >. ~ ( ) zw q m~O q,qm

>.,q z' w -

'v'(z, w) E D 2 .

This amounts to (1+q)fq(2.\)'"' (q2\q)m( -)m (1+q)fq(2,\) >.'"' (q 2>.,q)m( -)m >. ~ ( ) zw >. q ~ ( ) qzw . 21rq m~O q, q m 21rq m~O q, q m

Now the q-analogue of the binomial formula (see [5]) yields - ) (q2>. ,q)m(zw)m = (q2>. zw,q oo m~O (q, q)m (zw, q)oo

L

whence we obtain the following explicit formula for the reproducing kernel: K

(1 + q)[-X]qfq(2-X) 1 + q>.zw ( ) >.,q z, w = 2-..q' ( ) . zw, q 2>.+1 " "' D

4.

THE q-ANALOGUE OF THE WATANABE FORMULA FOR THE INTEGRAL UNITARY TRANSFORM BETWEEN

Gq,>.(]- 1; 1[) AND H:;>.(D)

In this section we shall construct an isometry given by an integral operator whose kernel is given explicitly. For z E D and x E] - 1, 1[ we set

(1 + q) 1 q>.(1 _ q) [ (zeiB, q)>. (ze-iB, q)>. with x = cos (J and 0 < consider

(J

q>. (zqeiB, q)>. (zqe-iB, q)>.]

< 1r. For every function 4> in G q,>. (] - 1; 1[) we

ABDERRAHMAN ESSADIQ

162

Main Theorem. For every fixed q E]O, 1[ and .X> 0, W>.,q is an isometry from Gq,>.(]- 1; 1[) onto ng;>.(D).

Proof. It is easy to show that, for zED and x E]- 1, 1[, A>.,q(Z, x) =

L

Um,>.(Z, q).(X, q)

m~O

where

"' ( ) 'f'm,>. x, q =

fq(.X)[m+.X]q(q,q)mc>.(. ) 27r{ q2A, q)mf q{2.X) m X, q .

Moreover, the family .(x, q)m~o forms a complete orthonormal system in Gq,>.(]- 1; 1[). On the other hand, IUm,>.(Z, q)l 2 < 00

L

m~O

for all z E D. Hence:

A>.,q(z, .) E Gq,>.(]- 1; 1[) for all zED. Furthermore, W>.,q(.)(z) = Um,>.(Z, q) so that W>.,q is an isometry from Gq,>.(] - 1; 1[ onto n;;>.(D).

0

Remark 2. Some remarks on a similar construction. Proceeding as above, we can establish an isometry between the subspace of L2 (]0, +oo[; Wa,q(x)) generated by the q-continuous Laguerre polynomials and the Hilbert space of analytic functions on the unit disk D equipped with the following Hermitian product: foro:> -1, [o:] {1 {27r . < f; g >a,q:= 27rq [}0 Jo f(rei 8 )g(ret8 )](qr2, q)a-1 dq(r 2 ) where

fq{-o:) x0 Wa,q(X) = f{ -o:)f{o: + 1) ( -{1- q), q)oo . The details are carried out in [2]. REFERENCES

[1] N. Aronszajn, Theory of Reproducing Kernels, Trans. Amer. Math. Soc. 68 (1950), 337-404. [2] A. Essadiq, A. Intissar, q-analogue of Bargmann unitary transfonn, Workshop de !'ouest mediterraneen en Physique Theorique, Rabat, March 1997. [3] F.H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math. 41 (1910), 193-203. [4] L.J. Rogers, Second Memoir on the expansions of certain infinite products, Proc. London Math. Soc. 25 (1894), 318-343. [5] N. Ja. Vilenkin, A. U. Klymik, 'Representation of Lie Groups and Special Functions', Vol. 3, Mathematics and its Applications 75, Kluwer Academic Publishers, Dordrecht, 1992. [6] Hilbert spaces of analytic functions and Gegenbauer polynomials, Tokyo J. Math. 13, n. 2 (1990), 421-427. DEPARTMENT OF MATHEMATICS, UNIVERSITY "IBN TOFAIL",

B.P. 1018,

KENITRA, MOROCCO

REALIZATION OF A HOLOMORPHIC DISCRETE-SERIES OF THE LIE GROUP SU(l, 2) AS STAR-REPRESENTATION MERYEM EL BEGGAR

ABSTRACT. A new method of orbits, the deformation program, was introducted by M. Flato and C. Fronsdal [3], [4]. It is based on the use of (formal) deformations of an algebra of functions on the coadjoint orbits, called star products. We present here a deformation program for the semi-simple Lie group G = SU(1,2). For an orbit Ot; f E g• associated to discrete series, we use Berezin's symbolic calculus to define our star product [7]. By combining these two constructions we have the star exponential on g• and the adapted Fourier transform.

1. NOTATION AND DEFINITIONS

Let G = SU(1, 2) be the Lie group of linear transformations in C 3 with determinant one and which preserve the Hermitian form:

= Z'oZo- Z'1Z1- Z'2Z2 = Z'oZo- ((Z1, Z2), (Z~, Z~)) = (Z0 , Z 1, Z2) and Z' = (Zb, ZL Z~) and (., .) is the usual scalar

[Z, Z']

where Z product on C2 . Then tG is the group of all the 3 x 3 complex matrices satisfying the relation g*jg = j and detg = 1, where

. (1 -h0)

(1)

J=

0

Here !2 is the 2 x 2 identity matrix and g* = t 9 is the conjugate transpose of the matrix g. Let us write g E G = SU(1, 2) in block form as

g=(~ ~), where a E M1x1(C) = C, bE M1x2(C), c E M2x1(C) and dE M2x2(C). Condition (1) is equivalent to the following system ofrelations: aa-c*c = 1, b*a- d*c = 0, ab- c*d = 0, b*b- d*d = -12 and detg = 1. The Lie algebra

g of G

is given by:

g = {X = (

it ~ );

>.

E R, ( E

C 2 , BE u(2) and i>. + trB = 0}. Now let B2 = {Z E 2 , II Z II< 1} be the complex unit ball of the Hermitian space (C2 , (., .)). 163

MERYEM EL BEGGAR

164

Then the group G mations: (2)

= SU(1, 2)

acts on B 2 by the fractional linear transfor-

GxB2

-->

B2

(g, Z)

f---+

g · Z = ( < b, Z > +a)- 1 (dZ +c)

where Z is considered here as a column vector in C 2 [6]. It can be shown that the isotropy group of the origin 0 E B2 is given by:

~ ~)

K = {k E SU(1, 2); k = (

, a E U(1) and dE U(2)}.

and that K is a maximal compact subgroup of G = SU(1, 2). Therefore, the complex unit ball B 2 can be realized as a homogeneous space, i.e. B2 = G I K where G I K denotes the space of left cosets. 2. DESCRIPTION OF COADJOINT ORBITS

G acts on the algebraic dual 9* of 9 by the coadjoint representation defined by: G

-->

GL(Q*)

g

f---+

e(g)

e

The action is given by (e(g)JJ.) = -(!, Adg- 1f;,); here Ad is the adjoint representation of G. As 9 is semi-simple, we identify 9 to its dual 9* via the Killing form q, given by q(X, Y) = C'Ir XY, where C is a constant. Then e coincides with Ad and the coadjoint orbit of f in 9* is the subset of 9 given by Of= {gfg- 1 ,g E G}, where f E 9*.

Proposition 2.1. Let f

= J.l (

-

~i

s

2 ) , J.l E

R*.

Then 0 f is identified to B2. Indeed, this follows from the fact that the isotropy group of the

f in G is

K. Remark 1. By the lwasawa decomposition of G the group N A parameterizes OJ: every g E OJ is in one-to-one correspondence with a pair (n(W, Z), at) E cosht 0 sinht ) N x A, where A = {at = ( 0 1 0 ; t E R} is a maximal sinht 0 cosht abelian subgroup of G and

N

= { n(W, Z) = (

for all WE C, W

+W =

1+ w- liZI 2 Z

2

w- ~IZI2

0, Z E C. N is a nilpotent subgroup of G.

REALIZATION OF A HOLOMORPHIC DISCRETE SERIES FOR THE LIE GROUP...

3.

165

REALIZATION OF A DISCRETE SERIES OF REPRESENTATIONS OF

G = SU(1, 2)

An easy computation which makes use of the action 2 of G on B2 yields:

1. 1-

II g · z 11 2 = (1- II

z

2. (b, Z) +a= ((b, g · Z)

11 2 ) I (b, Z) +a l- 2

+ a)- 1 , g- 1 =

(

.

~ ~) .

3. The absolute value of the Jacobian of the real analytic transformation, Z ~---+ g · Z(Z E B2), is equal to I (b, Z) +a 1- 6 .

4. If dZ is the Euclidean measure on C 2 , then dJ.L(Z) = (1- II Z 11 2 )- 3 dZ is a G-invariant measure on B2.

L1 (

Definition 1. For A E R, we consider the Hilbert space B 2 , dJ.L) of functions on B2 satisfying II F 111= fs 2 I F(Z) 12 (1- II Z 11 2 )-'dJ.L(Z) < +oo. Remark 2. Observe that

L1

n

(B2, dJ.L) O(B2) is non-trivial if and only if A> 2. (Here O(B2) is the space of holomorphic function on B 2 .) If A is an integer,

L1 (B2, dJ.L)

is a G-module. The action is given by

(T"(g)F)

(Z) = F( ff-ztc _)(< +a

b' Z >+a)--' ' where g- 1 = ( ac d~).

T" is a discrete-series representation of G. Furthermore, by applying to G the Borel-Weil-Bott theorem [4], we can realize T" as an action of G on a space of sections. Indeed, for any fixed integer A E R, there is an associated character X-\ on K given by X-\ ( (

e~o ~

) )

= ei-'O, where(} E R, dE U(2)

and det d = e-iO. Now we form the quotient space G XK = G X I rv, where rv denotes the equivalence relation (g, Z) '"" (g', Z') ~ g' = gk- 1 and Z' = n(k)Z for some k E K. Then the space G x K C fibers holomorphically over G I K with projection map 1r: 1r: G XK C - Gl K given by 1r((g, Z)) = g · K. The bundle G x K C G I K is called the associated line bundle for X-\. We consider the section S on G I K, defined by

c

c

S(g · K) = (g, f(g)) where

f is a function on G.

Remark 3. There is a one-to-one correspondence between the sections Son GIK; S(g · K) = (g,f(g)) and the maps f: G - C such that f(gk) =

x). 1 (k)f(g).

MERYEM EL BEGGAR

166

Let us use the following notation: L~(G)

= {!: G--+ C

such that f(gk)

= x). 1(k)f(g) and

fa

I

f(g)

12

dg < +oo}

where dg is the left G-invariant Haar measure on G, and

r

= {S: GjK--+ G XK C;S(gK) = (g,f(g)),f E L~(G)}.

The group G acts on r by (g · S)(Z) = gS(g- 1 · Z), where g(g1, f(g1)) = (gg 1,J(gg1)) and g- 1 · Z is the action of G on B2. Moreover, denote by J>. the operator L~(G) --+ L~(B2, dJ.L) defined by J>.(f) = F if F(Z) = ( (b,Z)+~ )->..(1- II Z 11 2)->..12f(g), where Z = g · K and l(b,Z)+al

g- 1 = (

~ ~).

Then J>. gives an isometry isomorphic from

L~(G)

onto

L~ (B2, dJ.L).

Then one has:

Proposition 3.1. 1.

r

satisfies the following properties:

r = {S: GjK--+ GxKC: S(gK) = (g,F(Z)( 1 ~::~~!! 1 )>..(1-II Z

11 2)>.1 2,

J>.(f) = F}

2. The elements of r are generated by the section

So:GjK

--+

GxKC

g. K

1---+

(g, ( < ~· z > +~ )>..(1- II I< b,Z >+a I

z 112)>..12)

.

Proposition 3.2. (g · S)(Z) = (T>.(g)F)(Z)So(Z).

Proof.

=

gS(g- 1 · Z) gS(g- 1g1 · K) if Z g(g- 1 91. f(Y- 191))

=

(91, F(g-1 . Z)(l-

(g · S)(Z) =

=

where (g- 1g1)- 1 = (

~ ~

) .

= 91 · K II g-1 . z 112)>../2 ( < ~', g-1 . z > +~' ) >.. I< b',g-1. z >+a' I 0

Notice that (T>.(g- 1g1)F)(Z) = T>.(g- 1)(T>.(91)F)(Z) is equivalent to the following relation:

REALIZATION OF A HOLOMORPHIC DISCRETE SERIES FOR THE LIE GROUP...

where 91-1 = ( a1 c1

b1 ) J1

g

-1 = (

ac db )

167

and g = ( ac db ) .

By Lemma 3.1, one has

(< b,g- 1 · Z >+a}"'=(< b, Z > +ii)->. and

1-11 g- 1· z If= (1-ll z 11 2) I< b,z >+a l- 2 .

Therefore:

(

F(g-1 .

< b1,z > +ii1 ) I< b1,z > +ii1 I

< b,z >+a ) I< b,z >+a I

>. (

->.

Z)(1 - IIZII2).Af2( < b, z > +ii)-A ( < ~1, z > +~1 ) I< b1,Z > +a1 I

A

For a realization of the cross sections as a space of functions on the complex unit ball B2, we need to equip G xKC with the G-invariant Hermitian structure H defined by

H( (g, TJ ), (g', r,')) = ryr]'. Now we consider the Hilbert space H>. of holomorphic sections of G which are square integrable:

H(S(Z), S(Z)) dtt(Z)
.f 2) 1

Observe that So(Z) =

"'(gk,

(1-IIZII 2)>.1 2),

where k= (

di,Z>+ii l+iil 0

0

0 I+iil +ii

0

0 ) 0

.

1

Therefore So can be identified to (1-IIZII 2).A/ 2 and H(So, So) = (1-IIZII 2)>.. By Proposition 3.2, for every S E H.>., one has S(Z) = F(Z)So(Z), where F : B2 - - C is a holomorphic function for >.. > 2. Hence, if>.. > 2, < FSo, FSo >=II FSo 11 2= f 82 I F(Z) 12 ((1- II Z 11 2 )>. dtt(Z).

MERYEM EL BEGGAR

168

Then the subrepresentation T>. ofT>. acting on L~ (B2, df.l) a holomorphic discrete-series of G [6]. Therefore, one has:

n O(B2) = .C>. is

Proposition 3.3. The action of G on .C>. is the same as the action of G on 1-i>.. 4.

STAR PRODUCT AND BEREZIN'S SYMBOLIC CALCULUS

We define a star product by means of Berezin's symbolic calculus. If q a point of G XKC, the evaluation S(1r(q)) = l(S)q defines a continuous form l on 1-i>., thus a section eq (a coherent state [5]) such that l(S) = (eq, S). The coherent states are:

Proposition 4.1. eso(Z)(Z') = (1- ZZ')->.So(Z'). Proof. One has eso(Z)(Z) =< eso(Z)' eso(Z) > So(Z) because ecq = c E C*, and thus

< eso(z), eso(Z) > (1- II Z

11 2 )>.

for

=< eso(Z)' eso(Z) > (1- II Z 11 2 )>. 12 (1- II Z =< e1, €1 >< eso(O), eso(O) >

Thus, < eso(z), eso(Z) >=< eso(O)' eso(O) > (1- II Z can take< eso(z),eso(Z) >= (1- ZZ')->._ Then, I eso(Z)(Z)

c- 1 eq

def

=

11 2 )->..

11 2 )>. 12

Therefore one

< eso(Z)' eso(Z) >So ( z ') (1- ZZ')->.S0 (Z') .

Notice that c(Z) =II eso(Z) 11 2 H((So(Z), So(Z)) is a constant (by the Ginvariance of H). From the above computation we obtain the value of c: c(Z) =II eso(Z)

11 2

H(So(O), So(O)) = 1.

0 Remark 4. Every S E 1-i>. defines a function S on G XK C by S(q)q = S(1r(q)). Then K(q',q) = eq(q') = (eq,eq') is a reproducing kernel for the space {S/S E 1-i>.}. Furthermore, with the help of Proposition 4.1, one can write Kin explicit form: K(So(Z'), So(Z')) = (1- ZZ')->._

The space of Berezin symbols is by definition the space of functions on B2 such that there exists a bounded operator A = Tu on 1-i>. with: u(Z) = A(Z) = (Aeq, eq) , (eq, eq)

where 1r(q) = Z; q-=/; 0. The correspondence between operator and symbol being one to one, we can define the star product of two symbols u and v by Tu*v = Tu o Tv ([7], [1]). Let A be a symbol, and A( Z, W) its unique analytic extension, holomorphic in Z and W. A(z, W) = (Aeso(Z), eso(W)). (eso(z), eso(W)) Then the star product has the following form:

REALIZATION OF A HOLOMORPHIC DISCRETE SERIES FOR THE LIE GROUP...

Proposition 4.2.

(A* B)(Z) =

L2 A(W,

1- ~11~)1\~ II~ w -II

Z)B(Z, W) ( (

169

11 2 )) ->. dJL(W).

Now that we have a star product, we can introduce a star exponential E>.(g), g E G by the rule E>.(g) = T>.(g). In other words, E>.(g) is the Berezin symbol of the operator T>.(g). E>. is a *-representation of G, in the sense that E>.(g) * E>.(g') = E>.(gg') (see [2]). The value of the function E>.(g) at a point~ of OJ is given by:

Proposition 4.3. E>.(g)~ = (1- zz')->.

where g- 1 = (

~ ~)

1(b, z) +a 1\

and Z is the value of the parameter corresponding

to~ E OJ.

--

Proof. Since g · eq = e9 .q, one has

T>.(g)(Z)

E>.(g)~

< T>.eso(Z)• eso(Z) > < eso(Z)• eso(Z) > < g . eso(Z)' eso(Z) > < eso(Z)• eso(Z) > < e(g·So)(Z)• eso(Z) > < eso(Z)• eso(Z) > < eg·So(g- ·Z)• eso(Z) > < es0 (Z)• eso(Z) > 1

Notice that gSo(g- 1 · Z)

g(g1, (1- llg-1 . Zll2)>.f2(

-

-1

-

< ~1, g . Z > +~1 )>.) I< b1,g- 1 · z > +a1 1

~1,z > +~1 )->.) I< b1,Z > +a1 I

(gg1,(1-llg-1· Zll2)>.f2( < (gg1k', (1-

where

k'~

II

g- 1 · z

l+ciil +a-,

(

0 0

11 2 )>.12 )

0 +a,

l+a,l

0

Therefore gSo(g- 1 · Z) can be identified to (1By lemma 3.1, 1- II g- 1 · Z 11 2= (1- II Z 11 2) I
./2 . b, Z >+a l- 2. g- 1 · Z

MERYEM EL BEGGAR

170

Thus

because ecq =

c- 1 eq for c E C*.

0

The star exponentials yield a natural definition of the adapted Fourier transform c>. of .. Then, Tr(T>.('P)) =

L

n;;.:o

< T;>.(

1 = L1 =L

n;;.:o

B2

n;;.:o

B2

< i\(h(So(Z), So(Z)) dJ.L(Z) < T;>.(< eso(Z)' en> 1

< eso(Z), eso(Z) >

=

=

= =

L

n;;.:o

1

1

dJ.L(Z)

< T>.('P) < eso(z), en> en, eso(Z) >

B2