Functional calculus in weighted group algebras

More documents

Recommendations

Info

where the constants C, K, C ′ just depend on the growth of the weight and where q ∈ N ∗is such that suppF ⊂ U q . Moreover, as the support of the functions f s is contained ina fixed compact set, there exists p ∈ N ∗ such that suppf s ⊂ U p and hence ‖f s ‖ ω ≤sup x∈U p ω(x)‖f s ‖ 1 = sup x∈U p ω(x) = s(p), for all s. Hence the constants K 2 and K 3 mayin fact be chosen independently of s. Moreover, up to a constant, the bound is exactly thesame as the one obtained for ‖u(nf s )‖ ω . Hence, if the weight ω satisfies (BDna), thereexist functions ϕ ∈ Φ satisfying the required conditions such that (∗) holds. This provesthe following result:7.4. Theorem: Let G be a compactly generated, locally compact group with polynomialgrowth. Let ω be a weight on G that satisfies condition (BDna). Then L 1 (G, ω) has theWiener property.Proof: By (7.3.), j(∅) contains all of C c (G) and hence equals L 1 (G, ω), as all the ϕ{f s }are in m(∅) ⊂ j(∅). Finally, if I is a closed two-sided ideal of L 1 (G, ω) such that h(I) = ∅,then L 1 (G, ω) = j(∅) ⊂ I by (6.3.).7.5. Examples: a) In (4.10.a), c)) we have examples of weights that satisfy (BDna) andhence give a group algebra L 1 (G, ω) that has the Wiener property.b) Examples (4.10. d), e)) give certain growth conditions on the coefficients c i for thecorresponding weight ω to satisfy (BDna). Under these conditions L 1 (G, ω) has the Wienerproperty.c) Let ω(x) = e C |x|(ln(|x|+1)) γ . Then ω satisfies (BDna) if and only if γ > 1. Hence, for γ > 1,L 1 (G, ω) has the Wiener property. This is the best possible result. In fact, if G = R, theresult of Vretblad ([Vr.]) shows that if L 1 (R, ω) has the Wiener property, then γ > 1. Soeven if G = R, we get the same condition on γ that we had already in this paper for thenon abelian case.References[ Beu.] Beurling, A., Sur les intégrales de Fourier absolument convergentes et leur applicationà une transformation fonctionnelle, 9ème congrès des mathématiciens scandinaves,août 1938, Helsinki, 345-366, (1939).[ Beu.1] Beurling, A., Sur les spectres des fonctions, Colloques internationaux duC.N.R.S., Analyse harmonique, Nancy, 9-29, (1947).[ Boi.] Boidol, J., Leptin, H., Schürman, J., Vahle, D., Räume primitiver Ideale vonGruppenalgebren, Math. Ann., 236, 1-13, (1978).[ Di.] Dixmier, J., Opérateurs de rang fini dans les représentations unitaires, Publ.Math. IHES 6, 305-317, (1960).[ Do.] Domar, Y., Harmonic analysis based on certain commutative Banach algebras,Acta Mathematica, 96, 1-66, (1956).[ Fe.Gr.Lei.Lu.Mo.] Fendler, G., Gröchenig, K., Leinert, M., Ludwig, J., Molitor-Braun,C., Weighted Group Algebras on Groups of polynomial Growth, accepted by Math.Zeitschrift.34
[ Hu.] Hulanicki, A., Subalgebras of L 1 (G) associated with the Laplacian on a Liegroup, Colloq. Math. 31, 259-287, (1974).[ Hu.1] Hulanicki, A., A functional calculus for Rockland operators on nilpotent Liegroups, Studia Math. 78, 253-266, (1984).[ Ka.] Kahane, J.-P., Séries de Fourier absolument convergentes, Springer, Berlin,(1970).[ Katz.] Katznelson, Y., An introduction to harmonic analysis, J. Wiley and Sons, NewYork, (1968).[ Lu.] Ludwig, J., Polynomial growth and ideals in group algebras, manuscripta math.30, 215-221, (1980).[ Lu.1] Ludwig, J., Minimal C ∗ -Dense Ideals and Algebraically Irreducible Representationsof the Schwartz-Algebra of a Nilpotent Lie group, Harmonic Analysis, LectureNotes in Math. 1359, Springer Verlag, 209-217, (1987).[ Ma.] Mandelbrojt, S., Séries de Fourier et classes quasi-analytiques de fonctions,Gauthier-Villars, Paris, (1935).[ Ma.1] Mandelbrojt, S., Séries adhérentes, régularisation des suites, applications,Gauthier-Villars, Paris, (1952).[ Mi.Mo.] Mint Elhacen, S., Molitor-Braun, C., Etude de l’algèbre L 1 ω(G) avec G groupede Lie nilpotent et ω poids polynomial, Publications du Centre Universitaire de Luxembourg,Travaux mathématiques X, 77-94, (1998).[ P.W.] Paley, R.E.A.C., Wiener, N., Fourier transforms in the complex domain, ColloquiumPublications 19, AMS, New York, (1934).[ Py.] Pytlik, T., On the spectral radius of elements in group algebras, Bull. Acad.Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21, 899-902, (1973).[ Py.1] Pytlik, T., Symbolic calculus on weighted group algebras, Studia Math. LXIII,169-176, (1982).[ Rei.] Reiter, H., Classical Harmonic Analysis and Locally Compact Groups, OxfordUniv. Press, (1967).[ Vr.] Vretblad, A., Spectral analysis in weighted L 1 spaces on R, Ark. Mat. 11,109-138, (1973).Institute of MathematicsDépartement de MathématiquesWroclaw UniversityUniversité de MetzPl. Grunwaldzki 2/4Ile de Saulcy50-384 Wroclaw F-57045 Metz cedex 1PolandFrancejdziuban@math.uni.wroc.plludwig@poncelet.sciences.univ-metz.frSéminaire de mathématiqueCentre Universitaire de Luxembourg162A, Avenue de la FaïencerieL-1511 LuxembourgLuxembourgmolitor@cu.lu35
Page 2 and 3: For the group G = R, Vretblad even
Page 4 and 5: Condition (∗) is satisfied for th
Page 6: is a weight on G. The same is true
Page 11 and 12: For any complex valued function f d
Page 13 and 14: 3.5. Let us denote B(n) = U n . The
Page 15 and 16: where C 1 = (K 1 2 ‖u(f)‖ 2 + 1
Page 17 and 18: 3.12. Theorem: Let G be a locally c
Page 19: Let now be two intervals [q, r] ⊂
Page 30 and 31: 5.6. We then have Domar’s propert
Page 32 and 33: 7. Wiener propertyLet us recall the

Functional calculus in weighted group algebras

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?