We write \(\lesssim\) if \(\le\) holds up to a positive numerical constant.

Using the estimates given in (0x68ed5419) , \(\alpha_-< \alpha_+\) and \(R_1< R_2< 1\), we obtain

\begin{align*} |\Delta \vartheta|^2 &= |\Delta \vartheta_1\, \vartheta_2 + 2 \nabla \vartheta_1 \cdot \nabla \vartheta_2 + \vartheta_1\, \vartheta_2''|^2\\ &\lesssim ( |\Delta \vartheta_1|^2 𝟙_{\operatorname{supp} \vartheta_2} +(|\nabla \vartheta_1|^2 |\vartheta_2'|^2)𝟙_{\operatorname{supp} \vartheta_1 \cap \operatorname{supp} \vartheta_2} +|\vartheta''_2|^2 𝟙_{\operatorname{supp} \vartheta_1})\\ &\lesssim \tfrac{\beta^2(1+\alpha_-^2)^2}{\delta^4 R_2^4}\, 𝟙_{\operatorname{supp} \vartheta_2 \cap \operatorname{supp} \nabla \vartheta_1} + \\ &\quad + \tfrac{\alpha_+^2(1+\alpha_-^2)}{\delta^4 R_2^2 R_1^2}\, 𝟙_{\operatorname{supp} \nabla \vartheta_1 \cap \operatorname{supp} \vartheta_2'} + \\ &\quad + \tfrac{\alpha_+^4}{\delta^4 R_1^4}\, 𝟙_{\operatorname{supp} \vartheta_2' \cap \operatorname{supp} \vartheta_1})\\ & \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\}}. \end{align*}