Functional calculus in weighted group algebras

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