Functional calculus in weighted group algebras

(1 + (n + m) 2 ) Q 2 ≤((1 + n 2 )(1 + m 2 ) + nm ) Q 2e K 3(n+mand, if n ≥ m ≥ e e ,(ln(n+m)) C ) ≤ e K 3(≤≤ 2 Q 2 (1 + n 2 ) Q 2 (1 + m 2 ) Q 2( ) k ( ) k≤ 2 Q 2 (1 + n 2 ) Q 2 (1 + m 2 ) Q 2 ∀k ≥ 1, ∀n, m ∈ N,n + m(ln(n + m)) C ≤n(ln(n)) C ) e K 3((e K 3(n(ln(n)) C )) k(n(ln n) C +m(ln(m)) C )e K 3(ln(ln(n + m)) = ln(ln(n(1 + m n )))m≤ ln(ln n + ln 2)(= ln (ln n)(1 + ln 2 )ln n )≤ ln(ln n) + ln 2Similarly if m ≥ n ≥ e e . Hence, if n ≥ m ≥ e e ,m(ln m) C ,(ln(m)) C )) k, ∀k ≥ 1, ∀n, m ≥ e,≤ ln(ln n) + ln(ln m) + ln 2()≤ 2 ln(ln n) + ln(ln m) , n, m ≥ e e .s(n + m) 2 ln(ln(n+m)) ≤ s(n) 4 ln(ln(n+m))16 ln(ln n)≤ s(n)≤ s(n) 16 ln(ln n) 16 ln(ln m)s(m)=(s(n) 2 ln(ln n)) 8(s(m) 2 ln(ln m)) 8.Similarly for m ≥ n. Hence the third condition of (4.4.) is satisfied for w and for k = 8.As w(n) ≥ C(1 + n), the fifth and sixth conditions of (4.4.) are satisfied for the functionw. The fourth condition of (4.4.) admits the following equivalent formulations:∑ ln(w(n))1 + n 2 < +∞n∈N,n≥e e⇔∑ ln C 11 + n 2 + ∑ ln(1 + n)1 + n 2 + ∑n∈N,n≥e e n∈N,n≥e e⇔+ 2Q2n∈N,n≥e e∑ (ln(ln n)) ln(s(n)) ∑1 + n 2 + K 2n∈N,n≥e e∑ (ln(ln n)) ln(s(n))1 + n 2 < +∞n∈N,n≥e e23ln(1 + n 2 )1 + n 2n∈N,n≥e en(ln n) C (1 + n 2 ) < +∞