Proposition

Lemma 10 ([Growth along phase orbits]). Suppose Q≥2 , k_j=Q^j , and the positive functions W_0=1,W_1,…,W_n , with n≥1 , satisfy c_0W_ j-1 ≤W_j≤2W_ j-1 , |TW_j|≤K_*k_jW_j (1≤j≤n). Put R=2 c_0 , β=_2R. For every p∈S , the function …

Lemma 10 ([Growth along phase orbits]). Suppose Q≥2 , kj=Qj , and the positive functions W0=1,W1,…,Wn , with n≥1 , satisfy c0W_ j-1 ≤Wj≤2W_ j-1 , |TWj|≤K*kjWj (1≤j≤n). Put R=2 c0 , β=2R. For every p∈S , the function Sp(θ)=Wn(eiθ p) satisfies Sp(θ+u) ≤eK_* R(1+kn|u|)βSp(θ), |u|≤π. In particular, the constants in this bound are independent of Q,n,p .

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Lemma 9 ([Moment preservation]). For every stage of, the phase bandwidth of W_j…Theorem 4.1 ([Uniform representing densities]). There is a constant K_*>0 ,…Proposition
Choose the largest 1≤l≤n such that k_l|u|≤1 , or put l=0 if no such index exists. The comparisons between successive densities give W_n(e^ i(θ+u) p) W_n(e^ iθ p) ≤R^ n-l W_l(e^ i(θ+u) p) W_l(e^ iθ p) . When l≥1 , logarithmic differentiation and the phase derivative bound show that the last ratio is at most (K_*k_l|u|)≤e^ K_* . For l=0 it equals one. If l=n , this already proves the assertion. Otherwise the maximality of l , including the case l=0 , gives k_n|u|>Q^ n-l-1 ≥2^ n-l-1 . It follows that R^ n-l ≤R(1+k_n|u|)^β , proving .
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