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

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

\begin{align*} |\nabla \vartheta|^2 &\leq (|\nabla \vartheta_1|^2 \vartheta_2^2 + \vartheta_1^2 |\vartheta'_2|^2) \\ &\lesssim \frac{1+\alpha_-^2}{\delta^2 R_2^2} \, 𝟙_{\operatorname{supp} \nabla \vartheta_1 \cap \operatorname{supp} \vartheta_2} + \frac{\alpha_+^2}{\delta^2 R_1^2}\, 𝟙_{\operatorname{supp} \vartheta_1 \cap \operatorname{supp} \vartheta_2'}\\ &\le \frac{1+\alpha_-^2}{\delta^2 R_2^2} \, 𝟙_{\Bigl\{ - 3R_2 \le z \leq -2R_1,\, y < \tfrac{R_1}{2a}\Bigr\}} + \frac{1+\alpha_+^2}{\delta^2 R_1^2}\, 𝟙_{\Bigl\{z \geq -3R_2,\, \tfrac{R_1}{4a} < y < \tfrac{R_1}{2a}\Bigr\}}. \end{align*}