Proposition 7.13 (Nonlinear error estimate) . [056F] 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 7.13 (Nonlinear error estimate).
There exists a constant C N > 0 C_{N}>0 0 < | t | ≪ 1 0<|t|\ll 1
(7.166) ϱ ∈ ( 0 , 1 2 ) \varrho\in(0,\frac{1}{2})
and
(7.167) − 1 ∂ ∂ ¯ ϕ 2 ∈ B ϱ ( 𝟎 ) ¯ ⊂ 𝔄 , − 1 ∂ ∂ ¯ ϕ 2 ∈ B ϱ ( 𝟎 ) ¯ ⊂ 𝔄 , \sqrt{-1}\partial\bar{\partial}\phi_{2}\in\overline{B_{\varrho}(\bm{0})}\subset\mathfrak{A},\quad\sqrt{-1}\partial\bar{\partial}\phi_{2}\in\overline{B_{\varrho}(\bm{0})}\subset\mathfrak{A},
we have the pointwise estimate
(7.168) ‖ 𝒩 ( − 1 ∂ ∂ ¯ ϕ 1 ) − 𝒩 ( − 1 ∂ ∂ ¯ ϕ 2 ) ‖ 𝔅 ≤ C N ⋅ ϱ ⋅ ‖ − 1 ∂ ∂ ¯ ( ϕ 1 − ϕ 2 ) ‖ 𝔄 . \displaystyle\|\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_{1})-\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_{2})\|_{\mathfrak{B}}\leq C_{N}\cdot\varrho\cdot\|\sqrt{-1}\partial\bar{\partial}(\phi_{1}-\phi_{2})\|_{\mathfrak{A}}.