Functional calculus in weighted group algebras

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the constant C 1 and ‖u(f)‖ 1 by max(‖u(f)‖ 1 , ‖u(−f)‖ 1 ) in the constant C 2 . We thenhaveand‖u(nf)‖ ω ≤ C 1(1 ′ + |n|)(1 + |n|r(|n|)) Q C2 s(|n|r(|n|)) · e ′ 3 ( s 2 (r(|n|)) ) , n ∈ Z,‖e inf ∗ F ‖ ω ≤ C 2(1 ′ + |n|)(1 + |n|r(|n|)) Q C2 s(|n|r(|n|)) · e ′ 3 ( s 2 (r(|n|)) ) , n ∈ Z,where the constants C ′ 1, C ′ 2, C ′ 3 are obtained in the following way:C ′ 1 = 2K 1 2 e‖f‖ 2+ 1≥ 2K 1 2 max(‖u(f)‖2 , ‖u(−f)‖ 2 ) + 1C ′ 2 = K 1 2 e‖f‖ 1(1 + q) Q 2 s(q)‖F ‖2 + ‖F ‖ ω≥ K 1 2 max(‖u(f)‖1 , ‖u(−f)‖ 1 )(1 + q) Q 2 s(q)‖F ‖2 + ‖F ‖ ω ,where q ∈ N ∗ is such that suppF ⊂ U q . As f has compact support, there exists p ∈ N ∗such that suppf ⊂ U p . Hence|n||n|‖f‖ ω3≤ e C′p ‖f‖ 1 ≤ e C′p ‖f‖ ωandC ′ 3 = 2e (eC′p )‖f‖ ω≥ 2e ‖f‖ ω 3 ≥ 2 max(‖u(f)‖ω3 , ‖u(−f)‖ ω3 ).So the constants depend essentially only on the original weight ω and on the functions fand F .3.11. Choice of the radius r(n): Motivated by the techniques of functional calculusexposed later on in this paper, let’s putr(n) = [ln(ln n) + 1], ∀n ≥ e e ,where [z] denotes the integer part of z. Thenln(ln |n|) ≤ r(|n|) ≤ ln(ln |n|) + 1 ≤ 2 ln(ln |n|),for |n| ≥ e eandHences 2 (r(|n|)) ≥ e C(ln(ln |n|)) = (ln |n|) C .e C′ 3|n|s 2 (r(|n|))≤ e C′ 3|n|(ln |n|) Cands(|n|r(|n|)) ≤ s(|n|) r(|n|) ≤ s(|n|) 2 ln(ln |n|) , for |n| ≥ e e .With this choice of the radius we get the following bounds:16
3.12. Theorem: Let G be a locally compact, compactly generated group with polynomialgrowth. Let ω be an arbitrary weight on G. Let f = f ∗ : G → C be continuous with compactsupport. Let F : G → C be an arbitrary continuous function with compact support. Wehave the following bounds: There exist strictly positive constants K 1 , K 2 , K 3 such that‖u(nf)‖ ω ≤ K 1 (1 + |n|)(1 + |n| 2 ) Q 2 s(|n|)2 ln(ln |n|) e K 3(‖e inf ∗ F ‖ ω ≤ K 2 (1 + |n|)(1 + |n| 2 ) Q 2 s(|n|)2 ln(ln |n|) e K 3(|n|(ln |n|) C )|n|(ln |n|) C ) ,for |n| ≥ e e . If U is a generating neighbourhood of G and p, q ∈ N ∗ such that suppf ⊂ U pand suppF ⊂ U q , thenK 1 = 2 Q 2 +1 K 1 2 e‖f‖ 2+ 1K 2 = 2 Q 2 K12 e‖f‖ 1(1 + q) Q 2 s(q)‖F ‖2 + ‖F ‖ ωK 3 = 2e (eC′p )‖f‖ ω,where the constants C, K and C ′ just depend on the growth of the weight ω and whereC > 1.Proof: By (3.10.) and (3.11.).4. Functional calculus4.1. In this section we shall first construct functions ϕ whose Fourier transform havea strong decrease and may hence operate, under certain conditions, on the self-adjointfunctions of Cc ∞ . This will be done using a result of Paley-Wiener. The use of thesefunctions in connection with the bound of ‖u(nf)‖ ω obtained in (3.), will give us a conditionon the growth of the weight ω. This condition will ensure the existence of functionalcalculus on a total part of L 1 (G, ω) and will be called the non-abelian Beurling-Domarcondition, because of its similarity with the results of Beurling and Domar.4.2. A result of Paley-Wiener: For any function f ∈ L 2 (R) the following twopropositions are equivalent:(i) There is a function g ∈ L 2 (R) such that |g| = |f| and ĝ vanishes on a half line.(ii)∫| ln |f(x)||1 + x 2 dx < +∞See ([P.W.]) or ([Rei.]).We shall apply the previous result in the following situation:R17
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: where C 1 = (K 1 2 ‖u(f)‖ 2 + 1
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
Page 34 and 35: where the constants C, K, C ′ jus

Functional calculus in weighted group algebras

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