By L. Auslander, R. Tolimieri (auth.), A. Figà Talamanca (eds.)

Lectures: A. Auslander, R. Tolimeri: Nilpotent teams and abelian varieties.- M Cowling: Unitary and uniformly bounded representations of a few uncomplicated Lie groups.- M. Duflo: building de representations unitaires d’un groupe de Lie.- R. Howe: On a suggestion of rank for unitary representations of the classical groups.- V.S. Varadarajan: Eigenfunction expansions of semisimple Lie groups.- R. Zimmer: Ergodic concept, workforce representations and rigidity.- Seminars: A. Koranyi: a few functions of Gelfand pairs in classical research.

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C0 j J n+1 We shall be interested in the group O(q) of all linear transformations of F ... ( 1 - 5- . which preserve this form. One of our principal aims here is to describe sane aspects of the harmonic analysis of O(q). Along the way, we shall meet other groups, on which we shalldescribe some aspects of harmonic analysis, some of which will be directly pertinent to our study of O(q) and some of which will be complementary. These groups are compact, abelian, and nilpotent. Harmonic analysis on compact groups has been under intensive investigation during the last half-century, and on abelian groups for much longer, but harmonic analysis on nilpotent groups is less well known.

S. 3) i s n o t t r i v i a l . M. which i s based on work of J . G . 1 ~ y g ]on p. 197. Perhaps t h e e a s i e s t method S t e i n and S. Wainger [ s w ~ ] , van d e r Corput, d e s c r i b e d i n A. Zygmund's book I n o u r c a s e , t h e p r o o f s s i m p l i f y , and we reproduce t h e s e f o r completeness. u *f i s t h e p r o d u c t of t h e 5 0 t r a n s f o r m s of t h e f a c t o r s , we may r e s o l v e t h e problem by e s t i m a t i n g (T I-. a q) = a o~(P,Q) - Since and 2 exp(i[sp + s q] s - a s 2) .

S t e i n [KnS] ( s e e p. The f i r s t i s due 494; t h e h y p o t h e s i s of t h e i r theorem may be r e l a x e d s u b s t a n t i a l l y u s i n g t h e same method of p r o o f , b u t n o t t o R i n C(C). tly p r e s e n t a second p r o o f , which u s e s complex i n t e r - p o l a t i o n ; t h e philosophy of t h i s second proof i s t h a t i f a n o p e r a t o r e x i s t s i n a weak s e n s e (K ) and i f t h e ambient o p e r a t o r s (K w i t h 5 # 0 ) a r e bounded i n 0 5' some s t r o n g e r s e n s e , t h e n s o i s t h e o r i g i n a l o p e r a t o r .