IJMMS 2003:37, 2345–2347 PII. S0161171203203045 http://ijmms.hindawi.com
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THE CONJUGATION OPERATOR ONAq(G)
SANJIV KUMAR GUPTA Received 5 March 2002
Letq >2. We prove that the conjugation operatorHdoes not extend to a bounded operator on the space of integrable functions defined on any compact abelian group with the Fourier transform inlq.
2000 Mathematics Subject Classification: 42A50, 43A17.
LetGbe a compact abelian group with dualΓ. For 1≤q <∞, the spaceAq
is defined as
Aq(G)=f:f∈L1(G),fˆ∈lq(Γ)
(1) with the normfAq= fL1+fˆlq. ThenAq(G)is a commutative semisim- ple Banach algebra with maximal ideal spaceΓ, in which the set of trigonomet- ric polynomials is dense [4]. TheAp-spaces have been studied in [1,6].
IfGis, in addition, a connected group, then its dual can be ordered; there exists a semigroupP⊂Γsuch thatP∩−P= {0},P∪−P=Γ (see [5]), and we say thatγ∈Γis positive ifγ∈P. Iff=
γ∈Ff (γ)γˆ is a trigonometric polynomial, the conjugation operatorHis defined as
Hf=
γ∈F
sgn(γ)f (γ)γ,ˆ (2)
where sgn(γ)= +1 ifγ∈P,−1 ifγ∈ −P, and 0 ifγ=0.
If 1≤q≤2, thenAq(G)⊂L2(G), and it is easy to see thatHextends to a bounded operator onAq(G). It was mentioned in [5] that the corresponding result forq >2 is not known. Note thatHextends to a bounded operator on Aq(G)if and only ifHextends to a bounded operator fromAq(G)toL1(G). In [5], the following theorem was proved.
Theorem1. LetGbe a compact, connected abelian group andPany fixed order onΓ. Ifq >2andφis a Young’s function, then the conjugation operator Hdoes not extend to a bounded operator fromAq(G)toLφ(G).
We prove inTheorem 2thatH does not extend to a bounded operator on Aq(G),q >2, thus answering the problem mentioned in [5]. Also,Theorem 1 follows from our theorem (Theorem 2).Theorem 2was proved for the circle group in [2] but for the completeness sake, we give it below.
2346 SANJIV KUMAR GUPTA
Theorem2. LetGbe a compact, connected abelian group andPany fixed order on Γ. If q >2, then the conjugation operator H does not extend to a bounded operator onAq(G).
Proof. We will show thatHdoes not extend to a bounded operator from Aq(G)toL1(G). The proof is divided into three steps.
Step1. LetG=T, the circle group. Suppose thatHextends as a bounded operator fromAq toL1. Chooseµ0∈M(T), ˆµ0∈lq such thatµ0is not abso- lutely continuous. DefineT:L1→L1by
T f=H f∗µ0
, (3)
whereTis well defined asf∗µ0∈AqandHmapsAqintoL1by our assump- tion onH. Hence,T is a multiplier fromL1toL1, and therefore is given by a measureµ∈M(T)(see [3]). Hence
sgn(n)ˆµ0(n)=µ(n).ˆ (4)
Observe that
µˆ0=µˆ0+µˆ
2 +µˆ0−µˆ
2 . (5)
Now,(ˆµ0+µ)/2 andˆ (ˆµ0−µ)/2 are absolutely continuous by F. and M. Rieszˆ theorem. Hence, ˆµ0is absolutely continuous, which contradicts the choice of µ0. Hence,His unbounded onAq,q >2.
Step2. LetI be a closed subgroup ofGsuch thatHdoes not extend as a bounded operator onAq(G/I). ThenHdoes not extend as a bounded operator onAq(G).
Proof. Let(fn)be a sequence of trigonometric polynomials onG/I such that
HfnL1(G/I) → ∞, fnA
q(G/I) →0, asn → ∞. (6) Letgn=fn◦q, whereq:G→G/Iis the quotient map. Then it is easily seen that Hgn=(Hfn)◦q,HgnL1(G)= HfnL1(G/I), andfn◦qAq(G)= fnAq(G/I). Hence
HgnL1(G) → ∞, gnA
q(G) →0, asn → ∞, (7)
andStep 2follows.
Step3. SinceG is connected,Γ contains an element of infinite order, say γ0(see [3]). Let S denote the subgroup generated byγ0and letI=S⊥. Then G/His isomorphic to the circle groupT. Now, the proof of the theorem follows from Steps1and2.
THE CONJUGATION OPERATOR ONAq(G) 2347 Acknowledgment. This research was supported by Sultan Qaboos Uni- versity, Oman.
References
[1] L. M. Bloom and W. R. Bloom,Multipliers on spaces of functions withp-summable Fourier transforms, Harmonic Analysis (Luxembourg, 1987), Lecture Notes in Math., vol. 1359, Springer, Berlin, 1988, pp. 100–112.
[2] J. T. Burnham, H. E. Krogstad, and R. Larsen,Multipliers and the Hilbert distribution, Nanta Math.8(1975), no. 2, 95–103.
[3] E. Hewitt and K. A. Ross,Abstract Harmonic Analysis. Vol. I: Structure of Topolog- ical Groups. Integration Theory, Group Representations, Die Grundlehren der Mathematischen Wissenschaften, vol. 115, Academic Press, New York, 1963.
[4] ,Abstract Harmonic Analysis. Vol. II: Structure and Analysis for Compact Groups. Analysis on Locally Compact Abelian Groups, Die Grundlehren der Mathematischen Wissenschaften, vol. 152, Springer-Verlag, New York, 1970.
[5] W. Rudin,Fourier Analysis on Groups, Interscience Tracts in Pure and Applied Mathematics, no. 12, Interscience Publishers, New York, 1962.
[6] U. B. Tewari and A. K. Gupta,The algebra of functions with Fourier transforms in a given function space, Bull. Austral. Math. Soc.9(1973), 73–82.
Sanjiv Kumar Gupta: Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khaud 123, Sultanate of Oman
E-mail address:[email protected]
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