Functional calculus in weighted group algebras

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3.4. For n ∈ N, let us make the following estimations (using the methods of [Hu.1]):‖(δ e + u(f)) ∗n ‖ ω = ‖ ( (δ e + g) + k ) ∗n‖ωn∑ ∑≤ ‖(δ e + g) a 1∗ k b 1∗ · · · ∗ (δ e + g) a n∗ k b n‖ ω ,m=0 a,bwhere a = (a 1 , . . . , a n ) ∈ {0, 1} n , b = (b 1 , . . . , b n ) ∈ {0, 1} n , |a| = m, |b| = n − m,a i + b i = 1, (δ e + g) a j= δ e if a j = 0 and k b j= δ e if b j = 0. Fix a and b as above. Assumethat a ≠ 0. Then‖(δ e + g) a 1∗ k b 1∗ · · · ∗ (δ e + g) a n∗ k b n‖ ω∫ ∫≤ · · · ‖(δ e + g) a 1∗ δ b 1s 1∗ · · · ∗ (δ e + g) a n∗ δ b nsn‖ ωGG· |k(s 1 ) b1 · · · k(s n ) b n|ds 1 · · · ds n ,where the convention is such that there is no integral with respect to ds l if b l = 0. Onemay check that‖(δ e + g) a 1∗ δ b 1s 1∗ · · · ∗ (δ e + g) a n∗ δ b nsn‖ ω≤ ∑ ‖(δ e + g) a 1∗ δ b 1s 1∗ · · · ∗ (δ e + g) a j−1∗ δ b j−1s j−1∗ g ∗ δ b js j∗ · · · ∗ δ b nsn‖ ω + ∏a j =1Similarly,b i =1ω(s i ).‖e inf ∗ F ‖ ω = ‖(δ e + u(f)) ∗n ∗ F ‖ ωn∑ ∑≤ ‖(δ e + g) a 1∗ k b 1∗ · · · ∗ (δ e + g) a n∗ k b n∗ F ‖ ωm=0 a,bandFinally,‖(δ e + g) a 1∗ k b 1∗ · · · ∗ (δ e + g) a n∗ k b n∗ F ‖ ω∫ ∫≤ · · · ‖(δ e + g) a 1∗ δ b 1s 1∗ · · · ∗ (δ e + g) a n∗ δ b nsn∗ F ‖ ωGG· |k(s 1 ) b1 · · · k(s n ) b n|ds 1 · · · ds n .‖(δ e + g) a 1∗ δ b 1s 1∗ · · · ∗ (δ e + g) a n∗ δ b nsn∗ F ‖ ω≤ ∑ a j =1‖(δ e + g) a 1∗ δ b 1s 1∗ · · · ∗ (δ e + g) a j−1∗ δ b j−1s j−1∗ g ∗ δ b js j∗ · · · ∗ δ b nsn∗ F ‖ ω+ ∏ω(s i )‖F ‖ ω .b i =112
3.5. Let us denote B(n) = U n . Then, for a ≠ 0, the function(δ e + g) a 1∗ δ b 1s 1∗ · · · ∗ (δ e + g) a j−1∗ δ b j−1s j−1∗ g ∗ δ b js j∗ · · · ∗ δ b nsnis supported by B(nr(n) + |s 1 | b 1+ · · · + |s n | b n) with the convention that |s l | b l= 0 if b l = 0.Moreover, by (1.2.),∣ B(nr(n) + |s1 | b 1+ · · · + |s n | b n) ∣ ∏≤ K · (1 + nr(n))Q(1 + |s i |) Q .b i =1Similarly, the function (δ e + g) a 1∗ δ b 1s 1∗ · · · ∗ (δ e + g) a j−1∗ δ b j−1s j−1∗ g ∗ δ b js jsupported by B(q + nr(n) + |s 1 | b 1+ · · · + |s n | b n), if suppF ⊂ U q , and∣ B(q + nr(n) + |s1 | b 1+ · · · + |s n | b n) ∣ ≤ K · (1 + q) Q (1 + nr(n)) ∏ Q3.6. If h is any function with compact support in B(r), then∫‖h‖ ω = |h(x)|ω(x)dx ≤ s(r)‖h‖ 2 |B(r)| 1 2 .U rObviously ‖g‖ 2 ≤ ‖u(f)‖ 2 . Hence, by (3.3.)-(3.5.), we obtainb i =1∗ · · · ∗ δ b nsn∗ F is(1 + |s i |) Q .‖(δ e + g) a 1∗ δ b 1s 1∗ · · · ∗ (δ e + g) a n∗ δ b nsn‖ ω≤ ∑ (1 + ‖u(f)‖ ) a1 +···aω j−1 3· ‖u(f)‖2 · ∣∣ B(nr(n) + |s1 | b 1+ · · · + |s n | b n) ∣ 1 2sa j =1 2 (r(n))· s(nr(n) + |s 1 | b 1+ · · · + |s n | b n) + ∏ω(s i )b i =1∑ (≤ K 1 2 ‖u(f)‖2 1 + ‖u(f)‖ ) a1 +···+aω 3j−1( ) Q1 + nr(n)2sa j =1 2 (r(n))· s(nr(n) + |s 1 | b 1+ · · · + |s n | b n) ∏∏(1 + |s i |) Q 2 +b i =1b i =1ω(s i )(≤ K 1 2 ‖u(f)‖2 · n(1 + nr(n)) Q 2 · s(nr(n)) · 1 + ‖u(f)‖ ) n ∏ω 3(· (1 + |si |) Q 2 · ω1 (s i ) )s 2 (r(n))b i =1+ ∏ω(s i )b i =1((≤ K 1 2 ‖u(f)‖2 · n(1 + nr(n)) Q 2 · s(nr(n)) · 1 + ‖u(f)‖ ω 3s 2 (r(n))· ∏ ((1 + |si |) Q 2 · ω1 (s i ) ) + ∏ω(s i )b i =1b i =1) s2 (r(n))) ns 2 (r(n))≤ K 1 2 ‖u(f)‖2 · n(1 + nr(n)) Q n‖u(f)‖2 · s(nr(n)) · eω3 ·(s 2 (r(n)) ) · ∏ ((1 + |si |) Q 2 · ω1 (s i ) )+ ∏ω(s i ).b i =113b i =1
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: For any complex valued function f d
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
Page 34 and 35: where the constants C, K, C ′ jus

Functional calculus in weighted group algebras

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