Let \(\vartheta\) be defined as in the proof of (0x68c9184d) .
Supports Link to heading
- \(\supp \vartheta \subset B_{1/2} \times [-\delta r_0, \delta r_0]\) (see (0x68ca9ad1) ).
- \(\supp \vartheta \cap (\mathbb{R}^{d-1} \times \{0\}) \subset \{ x \in \mathbb{R}^{d-1} : \sqrt{4\delta R_2} < |x| < \sqrt{6\delta R_2} \}\).
- \( \supp \nabla \vartheta_1 \circ z \subset \{ (x,y)\in \mathbb{R}^d : -3R_2< z(x,y)< -2R_2 \}. \)
- \( \supp \vartheta_2' \subset \{ y\in \mathbb{R} : \frac{R_1}{4a}< y<\frac{R_1}{2a} \}. \)
Bounds on \(\mathbb{R}^{d-1}\times \{0\}\) Link to heading
We write \(\lesssim\) if \(\le\) holds up to a positive numerical constant.
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We have
\[ |\nabla \vartheta(x,0)|^2 \lesssim \tfrac{1+\alpha_-^2}{\delta^2R_2^2} \](see (0x68cb9821) ).
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We have
\[ \lvert \nabla \vartheta(x,0)-\nabla \vartheta(y,0)\rvert^2 \lesssim \frac{1+\alpha_-^2}{\delta^3} R_2^{-3}\lvert x-y\rvert^2 \](see (0x68d29180) ).
Bounds on level sets of \(z\) Link to heading
We write \(\lesssim\) if \(\le\) holds up to a positive numerical constant.
- \(\lvert \vartheta_2'(y)\rvert^2 \lesssim \frac{\alpha_+^2}{\delta^2 R_1^2}\) (see (0x68edf29b) ).
- \(\lvert \vartheta_2''(y)\rvert^2 \lesssim \frac{\alpha_+^4}{\delta^4 R^4_1}\) (see (0x68edf29b) ).
- \(\lvert \nabla \vartheta_1(z(x,y))\rvert^2 \lesssim \frac{1+\alpha_-^2}{\delta^2 R_2^2}\) (cf. (0x68cb9821) ).
- We have \[ \lvert \Delta \vartheta_1(z(x,y))\rvert^2 \lesssim \frac{\beta^2(1 + \alpha_-^2)^2}{\delta^4 R_2^4}. \] (see (0x68edf6c2) ).
- We have \[ \lvert \nabla \vartheta\rvert^2 \lesssim \tfrac{1+\alpha_-^2}{\delta^2 R_2^2}\, 𝟙_{\Bigl\{ - 3R_2 \le z \leq -2R_1,\, y < \tfrac{R_1}{2a}\Bigr\}} + \tfrac{1+\alpha_+^2}{\delta^2 R_1^2}\, 𝟙_{\Bigl\{z \geq -3R_2,\, \tfrac{R_1}{4a} < y < \tfrac{R_1}{2a}\Bigr\}} \] (see (0x68ee143b) ).
- We have \[ \lvert \Delta \vartheta\rvert^2 \lesssim \tfrac{\beta^2(1+\alpha_-^2)^2}{\delta^4 R_2^4}\, 𝟙_{\Bigl\{ - 3R_2 \le z \leq -2R_1,\, y < \tfrac{R_1}{2a}\Bigr\}} +\tfrac{(1+\alpha_+^2)^2}{\delta^4 R_1^4}\, 𝟙_{\Bigl\{z \geq -3R_2,\, \tfrac{R_1}{4a} < y < \tfrac{R_1}{2a}\Bigr\}}. \] (see (0x68ee1432) ).