Spectral Synthesis in the Heisenberg Group

Transcription

1 Intrnational Journal of Mathmatical Analysis Vol. 13, 19, no. 1, 1-5 HIKARI Ltd, Spctral Synthsis in th Hisnbrg Group Yitzhak Wit Dpartmnt of Mathmatics, Univrsity of Haifa, Isral Copyright 19 Yitzhak Wit. This articl is distributd undr th Crativ Commons Attribution Licns, which prmits unrstrictd us, distribution, and rproduction in any mdium, providd th original work is proprly citd. Abstract W introduc som sts of spctral synthsis in som commutativ Banach algbras of intgrabl functions on th Hisnbrg group. Mathmatics Subjct Classification: 43A45, 43A7 Kywords: spctral synthsis, Hisnbrg group 1. Introduction and Prliminaris Considr th Hisnbrg group H as th st {(z, t): z ϵ, t ϵ R} with th group opration (z, t) (w, s) (z + w, t + s + Im z w). Lt us dnot by L 1 (H) th intgrabl functions on H which ar radial on : L 1 (H) {f ϵ L 1 (H): f(z, t) f( z,t) }. It is wll-known that L 1 (H) is a commutativ Banach algbra undr th convolution f*g(z, t) f (z-w, t - s - Im z w )g(s,w)dsdw whr dw is Lbsgu masur on ([]). Th rgular maximal idals of L 1 (H) ar th annihilators of th sphrical functions, ϵ R,, {, 1,,...} (th Lagurr part) and B, (th Bssl part) whr (z,t) i t z L(4 z ) whr d x L ( x ) (th Lagurr! dx polynomials) and B (z) J ( z ) whr J is th Bssl function of th first kind of ordr. Hulanicki and Ricci showd that Winr Taubrian thorm holds x

2 Yitzhak Wit for L 1 (H) ([4]). That it, vry propr closd idal is containd in som rgular maximal idal. Th dual spac of L 1 (H) is L (H) {f ϵ L (H): f(z, t) f( z, t)}. By duality, Winr thorm implis that for vry f ϵ L (H), f, th subspac V(f) w*- closur of {f* h: h ϵ L 1 (H)} contains a sphrical function. Th spctrum of f is dfind as Sp (f) { }{(, )} whn B and basic "radial translat" of a function f in L (H) is T r, s (f)(z, t) blong to V( f). Th 1 f(z - r i, t - s -Im (z r i ))d and thn V(f) is th w*- closd subspac spannd by all "radial translats" of f. For f ϵ L (R) its spctrum is dfind as th st of ϵ R such that it blongs to th translation invariant w*- closd subspac gnratd by f ([5]). Th purpos of this not is to study som sts of spctral synthsis in L (H) and to giv a simpl proof to Winr thorm for L 1 (H) introducd in [1].. Main rsults Thorm 1. Each point is a st of spctral synthsis. That is, if I is a propr closd idal in L 1 (H) which is containd in xactly on rgular maximal idal M thn I M. By duality, for f ϵ L (H) if Sp (f) { } thn f(z, t) C J ( z ) and if Sp (f ) {(, )} thn f(z,t) C (z, t). Suppos first that Sp (f ) { } for som and assum that f is continuous. W claim that f (z, t) f (z) for ach t ϵ R and z ϵ. Suppos that k(t) f( z, t ) C for som zϵ. Sinc th singlton {} is a st of spctral synthsis in L (R) ([5]) thr xists in th spctrum of k. Lt ϵ L 1 (R) so that ˆ is supportd in ( -, +) and k *. Thn F (z, t) f (z, t-s) (s) ds, F ϵ V(f) and for ach z th spctrum of F(z, ) lis in ( -, + ).

3 Spctral synthsis in th Hisnbrg group 3 For ach fixd lt P (z, t) F(z r i, t s Im(zr i )). Th function P(z,) is a translat in t of F(z - r i, ) implying that th spctrum of P(z,) lis in ( -, + ) for ach z. Taking avrag ovr prsrv this proprty implying that ach " radial translat" T r, s F and hnc ach function in V(F) shars this proprty. So V(F) cannot contain a sphrical Bssl function and by Winr thorm thr xists (, ) in Sp (F) Sp (f) contradicting that Sp (f) { } proving that f (z, t) f (z) t ϵ R and z ϵ. In this cas V (f) is th w*- closur of {f * h: h radial in L 1 (R )}. Sinc, as provd by Hrtz ([H]), th circl C (z ϵ : z } is a st of spctral synthsis, ach radial boundd function on R with spctrum C is of th form C ( z ) J which provs th first part of th thorm. Suppos now that Sp (f) {(, )}. W claim that for ach z th spctrum of f(z, ) is { }. Suppos that for som z th spctrum of f ( z,) contains. Lt ϵ L 1 (R) such that ˆ is supportd in U [ -, + ] so that U and g (z, t) f(z, t -s) (s)ds. Th function g ϵ V(f) and for ach z th spctrum of g(z, ) is containd in U. This proprty is shard by all functions in V(g) V(f). By Winr thorm Sp (g) contains som (, ) with, contradicting our assumption. Sinc th point { } is of spctral synthsis w obtain f(z, t) som. It rmains to show that (z) W W considr W E(z, t) ϵ V(f ) whr E(z, t) ( i t (z) z L ( 4 ) z it. (z), for as a finit singular masur on H supportd on. Hnc (z)) * W i (tim z w) (z-w) W (w)dw i t ( * i ( t W ) * W )(z) i t ( W * i t W <, W > sinc i t W )(z) is a charactr.

4 4 Yitzhak Wit Hr f * h dnots th twistd convolution of functions on. But Sp (f) {(, )} implying that <, W > for all and it follows that C W. Corollary. Th st B { } is a st of spctral synthsis. That is, if Sp (f) B thn f is containd in th w* - closur of th subspac spannd by { ( z ) J : }. As in th first part of th proof of Thorm 1 it follows that f (z, t) f ( z ) implying that f is containd in th w* - closur of th subspac spannd by { J ( z ) : }. Lt us dnot by L 1 (H) th closd sub-algbra of L 1 (H) dfind by L 1 (H ) {f ϵ L 1 (H): f(z, t)dt, z ϵ }. In th following w giv a simpl proof to Thorm 4.7 in [1]. Thorm. Winr Taubrian thorm holds for L 1 (H). That is, ach propr closd idal is containd in a maximal rgular idal which is in th Lagurr part of th maximal idal spac of L 1 (H). Lt f (z, t) b a continuous function in L (H) which is not a function of z only. L 1 Lt Q(f) th w* - closur of {f * h: h ϵ (H) that Q(f) contains a function it W Suppos that f(z, ) C. }. By duality, w hav to show for som ϵ R\{} and intgr. Lt b in th spctrum of f(z,) and ϵ L 1 (R) with ˆ ( ), ˆ ( ) in U [ - ϵ, ϵ ], U. Lt F(z, t ) f(z, t-s) (s) ds and F ϵ V(f). Thn for ach z th spctrum of F(z, ) R \ U and all functions in V(F) shar this proprty as shown in th proof of Thorm 1. By Winr Taubrian thorm for L 1 (H) th subspac V(F) and hnc V(f) contains a sphrical function which cannot b in th Bssl part

5 Spctral synthsis in th Hisnbrg group 5 implying that it show that it W W ϵ V(f) for som ϵ R \ {} and intgr. It rmains to ϵ Q(f ). W know that thr xists a nt g ϵ L 1 (H) such that f* g it W in w*. Lt ϵ L 1 (R), ˆ () and ˆ ( ) 1. Thn (f* g )(z, t-s) (s)ds it W in w*. By Fubini, it follows that f* E it W in w* whr E (w, ) (w, s) ( + s)ds satisfy E (w, )d. Hnc E L 1 (H), and it ϵ Q(f) which complts th proof of th thorm. g W Rfrncs [1] M. Agranovsky, C. Brnstin and D.C. Chang, Morra thorm for holomorphic Hp spacs in th Hisnbrg group, J. Rin Angw. Math., 1993 (1993), no. 443, [] S. Hlgason, Groups and Gomtric Analysis, Amrican Marhmatical Socitty, Nw York-London-Toronto, [3] C.S. Hrz, Spctral synthsis for th circl, Ann. of Math., 68 (1958), [4] A. Hulanicki and F. Ricci, A Taubrian thorm and tangntial convrgnc for boundary harmonic functions on balls in n, Invnt. Math., 6 (198), [5] Y. Katznlson, An Introduction to Harmonic Analysis, John Wily and Sons, Inc, Rcivd: Dcmbr 9, 18; Publishd: January 7, 19