G is dual (a maximal set of

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1 194 TRANSFERRING FOURIER MULTIPLIERS A.H. Dooley 1. FOURIER MULTIPLIERS OF LP(Gl Let G be a comact Lie grou, and G is dual (a maximal set of irreducible reresentations of G). The Fourier transform of f f(x) o(x G a acts). associates to a E G, the d xd matrix dx (where d a is the dimension of the sace in which The Fourier multiliers of L(G) are sequences (A ) of matrices so that if (f (a) ) is the Fourier series of an L function, so is (Aa f(crll. Examle. If G su (2) A ' G - {,~,1,... } and if!/, A E G ' cr!/, dimension 2!/,+1, and we look for sequences,a,,, where A is '2 )(, a (2l,+l) x (2!/,+1) matrix. has 2. EXAMPLES OF MULTIPLIERS (i) Central mult.iliers. We restrict to A = c I for c E ~ a o o This is the case which has b9en most studied. Clere [1] and Clere [2] have shown that the For examle, Bonami and Poisson kernel Gauss kernel Riesz kernel (o > ll are bounded summability kernels in L(SU(2)). These results also

2 195 hold for general G, where 5I, is relaced by ].!, the eigenvalue of x under the biinvariant Lalacian on G. Coifman and Weiss [4] unified these results; m I is a!v 5I, multilier of if 5I, ~ (2t+l) mt - (2t-l) mt-l is a multilier of Fourier series. This result also holds in some g enerali ty. This criterion can be alied to the above kernels. The idea is to use the Weyl integration formula to reduce the roblem to the maximal torus. (ii) Noncentral multiliers. Very little has been roved in generality. Coifman and Weiss [3] gave an extremely detailed study of SU ( 2), and showed tha t the Riesz kernels are bow1ded. One defines - - oerators B 1 '. B 1, B 2 q B 2, as certain differential oerators and comutes their Fourier transfonns to be m o 1 2I ht(it-1) 121 n 251,-1 -n " B2(crt) 1 2Q. A i3 2 <crr.l 251, -n shows that, for a given harmonic olynomial which is zero at the 3 origin, F f+ k s.f. is a generalized analytic function, where j=l are a basis for the quaternions and f. are the Riesz transfonns f l f2 = i (Bl + Bl) f

3 196 This generalizes the,classical Riesz transform. The critical oint is to rove that the B. attemt an HP theory. The roof that the are bounded on L (SU ( 2) ) for then one may B. are bounded given by Coifman and Weiss deends on a detailed structural analysis of SU(2), together with some theory of seudo differential oerators. It is hard to estimate the multilier,norm of the B. I wish to roose a new aroach, based uon some of my recent work on contractions of Lie grous which gives a simler roof lus better control over the constants. 3. CONTRACTION OF g ONTO G We consider, as have many other authors, the family of mas X TIA : ~ + G : X~ ex I (A> ). We use some results of c.s. Herz [6], on eriodification of multiliers to "transfer" multiliers of L(g) onto multiliers of L(G). Secifically, we can rove Theorem. There.is a canonical norm non-increasing ma A (G) + B (g) = Proof. For X locally comact Hausdorff, Herz defines V (X) to be the set of functions of two variables which are ointwise multiliers of RP R,' (X) B may be identified as the elements of V invariant under right translation in both variables. According to a theorem of Herz, F t-+ F o TIA x TIA is a norm non-increasing ma V (G) + V (g) The ma which takes F E V (g) to the invariant mean = = F of Z t-+ F (X+ Z, Y + Z) gives a rojection V (g) + B (g) = = Combining these mas with the injection A (G) C-r V (G) gives the ma D

4 197 It remains to comute ia The following theorem is roved in [5]. Theorem. Let f E A (G) (*) where T is any maximal torus for G, t is its Lie algebra, and )t denotes rojection onto t (*) is a natural generalization of the usual eriodification ma A ('] ) + B (lr) - We may dualize ia, obtaining ia* :B*(~) any f E L 1 (~), ia*f E M(G) and + M(G) Thus, for (where Ill Ill denotes the multilier norm). Using (*),we may comute the Fourier transform of ia*f For simlicity, we restrict to the case G = SU(2), although the formula holds in generality ~when suitably modified) Theorem. Let {~}~=-~ be the usual orthono~al basis for H a~ (**) (i, *c> "(a ) A ;v n,m ~. ~ (.<I...:.!.) k=-~ G/T A 4. APPLICATIONS (i) Suose c is a radial multilier of LP(m 3 ) (alias an Ad(SU(2)) invariant multilier of L(SU(2)) ). Then :h ~ [g A ~ k) is indeendent of g, so we may use the orthogonality relations on (**) to obtain

5 198 Thus ia* is a central multilier. Knowing the L boundedness of certain radial multiliers on R 3 now allows us to deduce the boundedness of certain other multiliers on SU(2) In fact, we may solve the equation for, obtaining which is effectively the theorem of Coifman and Weiss alluded to above. (ii) More interes ting are the noncentral multiliers. Take (x) f ( jxj) where Y is a sherical hannonic of degree s s,q In fact, identifying Y s,q with SU (2), we comute the Fourier trcmsform of and -s s q s s, by using the Bochner-Heeke formula, as Q, s-~ (i/ A(O{,) I: 2Tii -s ( j ~<ll hll A r s - f,s+r,, 1 f t tktk(g) dg m,n k=-(,, A ) A. SU(2) qo m n where f( ' ) is the Bessel transform of f. 1'he integral over SU(2) is jus t a Clebsch-Gordan coefficient which may be evaluated. Taking in articular and Yl,O, one comutes easily that 2-3/2 -~. 1:"' d, '1 1 jxj-3 'IT la "'~ I +B 2, an SlmL.ar y becomes I +B 2 One may further check that has for image yl,l[g) 1~ /2 k 'IT 2(-i) Bl ' and that

6 199 It follows at once from our aroach that and :s; Ill 1 Ill.. 3 /8iT,IR 1 :s; Ill Ill /8iT 2,IR iTP+l 'I'hese estimates are extremely recise, and avoid comletely any mention of seudo differential oerators. REFERENCES [1] A. Bonami and.l. Clerk, 'Sommes de Cesaro et multilicateurs des develoements en harmonique sheriques', T~ans. Ame~. Math. Soa. 183 (1973) [2].L. Clerk, 'Sommes des Riesz et multilicateurs sur un groue de Lie comact', Ann. Inst. Fourier (Grenoble)! (1974) [3] R. Coifman and G. Weiss, Analyse harmonique non-commutative au!' certains eaaces homogenes, Sringer Lectures Notes in Mathematics 242 (1971). [4] R. Coifman and G. Weiss, 'Central multilier theorems for comact Lie grous', BuZZ. A.mer. Math. Soa. 8 (1974) [5] A. H. Dooley, 'Transferring L multiliers' (submitted). [6] c.s. Herz, 'Asymmetry of convolution norms II'., Pr-oc. Sym, Pu1 e Math. (Williamstown 1978). School of Mathematics University of New South Wales Kensington NSW 233 AUSTRALIA