On the L p conjecture for locally compact groups


 Mervin Henry
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1 Arch. Math. 89 (2007), c 2007 Birkhäuser Verlag Basel/Switzerlad 0003/889X/ , ublished olie DOI 0.007/s x Archiv der Mathematik O the L cojecture for locally comact grous F. Abtahi, R. NasrIsfahai ad A. Rejali Abstract. Let be a locally comact grou. For <<, it is wellkow that f g exists ad belogs to L () for all f, g L () if ad oly if is comact. Here, for 2 <<, we show that f g exists for all f, g L () if ad oly if is comact. We also show that this result does ot remai true for < 2. Mathematics Subject Classificatio (2000). Primary: 43A5; Secodary: 43A20. eywords. Covolutio, locally comact grou, L sace, L cojecture.. Itroductio. Throughout the aer, let be a locally comact grou with a fixed left Haar measure λ. For <<, the Lebesgue sace L () with resect to λ is as defied i [5]; i.e. the equivalece classes of measurable fuctios o with ( / f := f(x) dλ(x)) <. For measurable fuctios f ad g o, the covolutio multilicatio (f g)(x) = f(y) g(y x) dλ(y) is defied at each oit x for which this makes sese; i.e. the fuctio y f(y) g(y x) is λitegrable. The f g is said to exist if (f g)(x) exists for almost all x. The L cojecture asserts that if f g exists ad belogs to L () for all f,g L (), the is comact. This cojecture was first formulated by Rajagoala i his Ph.D. thesis i 963. However, the first result related to this cojecture is due to Zelazko [8] ad Urbaik [7] i 96; they roved that the cojecture is true for all locally
2 238 F. Abtahi, R. NasrIsfahai ad A. Rejali Arch. Math. comact abelia grous. The truth of the cojecture has bee established for >2 by Zelazko [9] ad Rajagoala [] ideedetly; see also Rajagoala s works [0] for the case where 2 ad is discrete, [] for the case where = 2 ad is totally discoected, ad [2] for the case where > ad is either ilotet or a semidirect roduct of two locally comact grous. I the joit work [3], they showed that the cojecture is true for > ad ameable grous; this result ca be also foud i reeleaf s book [4]. Rickert [5] cofirmed the cojecture for = 2. For related results o the subject see also Crombez [] ad [2], audet ad amle [3], Johso [6], uze ad Stei [7], Lohoue [8], Miles [9], Rickert [4], ad Zelazko [20]. Fially, i 990, Saeki [6] gave a affirmative aswer to the cojecture by a comletely selfcotaied roof. Motivated by the L cojecture, we cosider oly the roerty that f g exists for all f,g L (), ad rove the followig result which is ideed the urose of this work. Theorem.. Let be a locally comact grou ad >2. Iff g exists for all f,g L (), the is comact. The followig examle shows that Theorem. is, i geeral, ot true for < 2. Examle.2. Let be a locally comact grou. (a) If is uimodular, it follows from Hölder s iequality that f g exists for all f,g L 2 (). (b) If < 2 ad is discrete, the f g exists for all f,g L (); this follows from (a) together with the fact that L () L 2 (). As a cosequece of these observatios together with the solutio of the L cojecture ad Theorem., we have the followig corollary. Corollary.3. Let be a locally comact grou ad <<. Cosider the followig coditios. (a) is comact. (b) f g exists ad belogs to L () for all f,g L (). (c) f g exists for all f,g L (). The (a) (b) (c) ad also the followig assertios hold. (i) If <<2 ad is discrete, the (c) always holds. (ii) If =2ad is uimodular, the (c) always holds. (iii) If >2, the (a) (b) (c). This corollary leads us to the followig atural questio. Questio Let be a locally comact grou ad < 2. Doesf g exist for all f,g L ()?
3 Vol. 89 (2007) O the L cojecture for locally comact grous The roof. Proof of Theorem.. Let be a fixed comact symmetric eighbourhood of the idetity elemet of. The 0 <λ() λ( 2 ) < ad there exists a costat C>0 such that (x) <C (x ), where deotes the modular fuctio of. Suose o the cotrary that is ot comact. The \ is a oemty symmetric subset of. Thus, there is a elemet a of \ with (a ). Iductively, we may fid a sequece (a )i with (a ) such that a \ m= The for every m, with m<, a m 4 ( 2). a m 2 a 2 = ad a m a =. For each A, let χ A deote the characteristic fuctio of A o, ad defie the fuctios f ad g o by ad f(x) = (x ) / g(x) = χ a (x) χ a 2(x) for all x. The f,g L (); ideed, for each wehave (x ) χ a (x) dλ(x) = (x ) χ (xa ) dλ(x) = (a ) (a x ) χ (x) dλ(x) = (a ) (a ) (x ) dλ(x) = (x ) dλ(x) Cλ()
4 240 F. Abtahi, R. NasrIsfahai ad A. Rejali Arch. Math. from which it follows that f(x) dλ(x) = Moreover, <. g(x) dλ(x) = Cλ() = = (x ) χ a (x) dλ(x) χ a2(x) dλ(x) = λ( 2 ) <. χ 2(a x) dλ(x) χ 2(x) dλ(x) We ext show that (f g)(x) = for all x. To that ed, recall that (a ) ad / > 0 ad hece (a ) / ( ). We thus have (y ) / χ a (y) dλ(y) = (y ) / χ (ya ) dλ(y) = (a ) (a y ) / χ (y) dλ(y) = (a ) (a ) / (y ) / χ (y) dλ(y) = (a ) / (y ) / dλ(y) (y ) / dλ(y) C / λ().
5 Vol. 89 (2007) O the L cojecture for locally comact grous 24 Now, let x ad ote that y x a 2 for all y a ad. Therefore (f g)(x) = f(y) g(y x) dλ(y) = (y ) / χ a (y) dλ(y) C / λ() =. It follows that f g does ot exist whereas f,g L (). This cotradictio comletes the roof. Remark. It follows from Theorem. that if is discrete, >2 ad f g exists for all f,g L (), the is fiite. Let us coclude this work with a direct roof of this fact. Suose o the cotrary that is ifiite. The there is a elemet a with {a,a } \{e}. Iductively, we may fid a sequece (a )i such that {a +,a + } \{e, a,a,..., a,a } for all. Defie the fuctio f o by f(a )=f(a )=/ for all ad f(x) = 0 for all x {a,a,a 2,a 2,... }. The f L (), but (f f)(e) does ot exist; ideed, (f f)(e) = f(a )f(a )= /. Sice is discrete, this shows that f f does ot exist, a cotradictio. Ackowledgemet. The authors would like to thak the referee of the aer for ivaluable commets. This research was artially suorted by the Ceters of Excellece for Mathematics at the Isfaha Uiversity of Techology ad the Uiversity of Isfaha. Refereces []. Crombez, A characterizatio of comact grous. Quart. J. Pure Al. Math. 53, 9 2 (979). [2]. Crombez, A elemetary roof about the order of the elemets i a discrete grou. Proc. Amer. Math. Soc. 85, (983). [3] R. J. audet ad J. L. amle, A elemetary roof of art of a classical cojecture. Bull. Austral. Math. Soc. 3, (970).
6 242 F. Abtahi, R. NasrIsfahai ad A. Rejali Arch. Math. [4] F. P. reeleaf, Ivariat meas o locally comact grous ad their alicatios. Math. Studies 6, Va Nostrad, New York, 969. [5] E. Hewitt ad. Ross, Abstract harmoic aalysis I. SrigerVerlag, New York, 970. [6] D. L. Johso, A ew roof of the L cojecture for locally comact grous. Colloq. Math. 47, 0 02 (982). [7] R. uze ad E. Stei, Uiformly bouded reresetatios ad harmoic aalysis of the 2 2 real uimodualr grou. Amer. J. Math. 82, 62 (960). [8] N. Lohoue, Estimatios L des coefficiets de reresetatio et oerateurs de covolutio. Adv. Math. 38, (980). [9] P. Miles, Covolutio of L fuctios o ocommutative grous. Caad. Math. Bull. 4, (97). [0] M. Rajagoala, Othel saces of a discrete grou. Colloq. Math. 0, (963). [] M. Rajagoala, L cojecture for locally comact grous I. Tras. Amer. Math. Soc. 25, (966). [2] M. Rajagoala, L cojecture for locally comact grous II. Math. A. 69, (967). [3] M. Rajagoala ad W. Zelazko, L cojecture for solvable locally comact grous. J. Idia Math. Soc. 29, (965). [4] N. W. Rickert, Covolutio of L fuctios. Proc. Amer. Math. Soc. 8, (967). [5] N. W. Rickert, Covolutio of L 2 fuctios. Colloq. Math. 9, (968). [6] S. Saeki, The L cojecture ad Youg s iequality. Illiois. J. Math. 34, (990). [7]. Urbaik, A roof of a theorem of Zelazko o L algebras. Colloq. Math 8, 2 23 (96). [8] W. Zelazko, O the algebras L of a locally comact grou. Colloq. Math. 8, 2 20 (96). [9] W. Zelazko, A ote o L algebras. Colloq. Math. 0, (963). [20] W. Zelazko, O the Burside roblem for locally comact grous. Sym. Math. 6, (975). F. Abtahi, Deartmet of Mathematics, Uiversity of Isfaha, Isfaha, Ira R. NasrIsfahai, Deartmet of Mathematical Scieces, Isfaha Uiversity of Techology, Isfaha , Ira A. Rejali, Deartmet of Mathematics, Uiversity of Isfaha, Isfaha, Ira Received: 23 Aril 2006
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