Proposition 7.15 (Global injectivity estimates) . [056H] Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.
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Proposition 7.15 (Global injectivity estimates).
For all parameters α ∈ ( 0 , 1 ) \alpha\in(0,1) δ > 0 \delta>0 μ , ν ∈ ℝ \mu,\nu\in\mathbb{R}
(7.170) 0 < δ < δ G , − 1 < ν < 0 , ν + α < 0 , μ = ( 1 − 1 n ) ( ν + 2 + α ) , \displaystyle 0<\delta<\delta_{G},\quad-1<\nu<0,\quad\nu+\alpha<0,\quad\mu=(1-\frac{1}{n})(\nu+2+\alpha),
there exists a uniform constant C > 0 C>0 t t u ∈ C 2 , α ( X ^ t ) u\in C^{2,\alpha}(\widehat{X}_{t})
(7.171) ‖ ∇ u ‖ C δ , ν + 1 , μ 0 ( X ^ t ) + ‖ ∇ 2 u ‖ C δ , ν + 2 , μ 0 ( X ^ t ) ≤ C ⋅ ‖ Δ u ‖ C δ , ν + 2 , μ 0 , α ( X ^ t ) , \displaystyle\|\nabla u\|_{C_{\delta,\nu+1,\mu}^{0}(\widehat{X}_{t})}+\|\nabla^{2}u\|_{C_{\delta,\nu+2,\mu}^{0}(\widehat{X}_{t})}\leq C\cdot\|\Delta u\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\widehat{X}_{t})},
(7.172) [ u ] C δ , ν , μ 2 , α ( X ^ t ) ≤ C ⋅ ‖ Δ u ‖ C δ , ν + 2 , μ 0 , α ( X ^ t ) . \displaystyle[u]_{C_{\delta,\nu,\mu}^{2,\alpha}(\widehat{X}_{t})}\leq C\cdot\|\Delta u\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\widehat{X}_{t})}.