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Lemma 4.9 .
On | z | ≥ 1 |z|\geq 1
(4.124) C − 1 T ( n − 2 ) ( 1 − n ) n ( T + k ± z ) 1 − n ω ± , c y l ≤ ω ≤ C T 2 − n n ( T + k ± z ) ⋅ ω ± , c y l C^{-1}T^{\frac{(n-2)(1-n)}{n}}(T+k_{\pm}z)^{1-n}\omega_{\pm,cyl}\leq\omega\leq CT^{\frac{2-n}{n}}(T+k_{\pm}z)\cdot\omega_{\pm,cyl}
and for all k ≥ 1 k\geq 1 m k , C k m_{k},C_{k}
(4.125) | ∇ ω ± , c y l k ω | ω ± , c y l ≤ C k ( T 2 − n n ( T + k ± z ) ) m k . |\nabla^{k}_{\omega_{\pm,cyl}}\omega|_{\omega_{\pm,cyl}}\leq C_{k}(T^{\frac{2-n}{n}}(T+k_{\pm}z))^{m_{k}}.