have $$\begin{aligned}
\int_I |U'(y)||\nabla_k \psi(y)|^2\dd y \leq C\sum_{j\in \J}\int_{I_j} |U'(y)||\nabla_k \psi_j(y)|^2\dd y,\end{aligned}$$ for some $C\geq 1$, independent of $k$.

From Lemma \[lem:weighted\_elliptic\], we have $$\begin{aligned}
\label{eq:intpart}
\gamma_{0} \int_I |U'(y)||\nabla_k \psi(y)|^2\dd y\leq -\l \Delta \psi, |U'|\psi\r
=-\sum_{j\in\J}\l \Delta \psi_j, \chi_j|U'|\psi\r.\end{aligned}$$ Recalling and using the product formula $$\begin{aligned}
\Delta_k (|U'|\psi\chi_j)=\de_{yy}|U'|\psi\chi_j+2\de_y|U'|\de_y(\psi\chi_j)+|U'|[\Delta_k\psi_j+2\de_y\psi\chi_j'+\psi\chi_j''],\end{aligned}$$ for each fixed $j\in\J$, several integration by parts yield $$\begin{aligned}
\label{eq:estim0}
\l \Delta_k \psi_j, |U'|\psi\chi_j\r
&=\l  \psi_j, \de_{yy}|U'|\psi\chi_j+2\de_y|U'|\de_y(\psi\chi_j)+|U'|[\Delta_k\psi_j+2\de_y\psi\chi_j'+\psi\chi_j'']\r\notag\\
&=\l  \Delta_k\psi_j,|U'|\psi_j\r+\l  \psi_j, \de_{yy}|U'|\psi\chi_j\r
+2\l  \de_y|U'|,\psi_j\de_y(\psi\chi_j)\r
+\l  |U'|\psi_j,2\de_y\psi\chi_j'+\psi\chi_j''\r\notag\\
&=\l  \Delta_k\psi_j,|U'|\psi_j\r+\l  \psi_j, \de_{yy}|U'|\psi\chi_j\r
-2\l  |U'|,\de_y(\psi_j\de_y(\psi\chi_j))\r
+\l  |U'|\psi_j,2\de_y\psi\chi_j'+\psi\chi_j''\r\notag\\
&=\l  \Delta_k\psi_j,|U'|\psi_j\r-2\l \de_{yy}\psi \chi_j,|U'|\psi_j\r
+\l  \psi_j, \de_{yy}|U'|\psi\chi_j\r\notag\\
&\quad- 2\l\de_y \psi_j ,|U'|\de_y\psi \chi_j\r- 2\l  \de_y\psi_j,|U'|\psi\chi'_j\r- 2\l\psi_j  ,|U'|\de_y\psi \chi'_j\r-\l \psi_j,|U'|\psi\chi''_j\r\notag\\
&=-\l  \Delta_k\psi_j,|U'|\psi_j\r+2k^2\l\psi \chi_j,|U'|\psi_j\r
+\l  \psi_j, \de_{yy}|U'|\psi\chi_j\r\notag\\
&\quad- 2\l\de_y \psi_j ,|U'|\de_y\psi \chi_j\r- 2\l  \de_y\psi_j,|U'|\psi\chi'_j\r- 2\l\psi_j  ,|U'|\de_y\psi \chi'_j\r-\l \psi_j,|U'|\psi\chi''_j\r\notag\\
&=-\l  \Delta_k\psi_j,|U'|\psi_j\r+\l  \psi_j, \de_{yy}|U'|\psi\chi_j\r\notag\\
&\quad- 2\l\nabla_k \psi_j ,|U'|\nabla_k\psi \chi_j\r- 2\l  \de_y\psi_j,|U'|\psi\chi'_j\r- 2\l\psi_j  ,|U'|\de_y\psi \chi'_j\r-\l \psi_j,|U'|\psi\chi''_j\r\end{aligned}$$ We estimate the right-hand side above term by term. Appealing to , it is not hard to see that $$\begin{aligned}
&|\l\nabla_k \psi_j ,|U'|\nabla_k\psi \chi_j\r| +|\l  \de_y\psi_j,|U'|\psi\chi'_j\r|+ |\l\psi_j  ,|U'|\de_y\psi \chi'_j\r|+|\l \psi_j,|U'|\psi\chi''_j\r|\notag\\
&\qquad\leq C\int_{I_j} |U'(y)| \left[|\nabla_k \psi_j(y)|(|\nabla_k \psi(y)|+|\psi(y)|) + | \psi_j(y)|(|\nabla_k \psi(y)|+|\psi(y)|) \right]\dd y\notag\\
&\qquad\leq C\int_{I_j} |U'(y)| |\nabla_k \psi_j(y)||\nabla_k \psi(y)|\dd y.\end{aligned}$$ Moreover, in view of , and the mild degeneracy assumptions, we have that $$\begin{aligned}
|\l  \psi_j, \de_{yy}|U'|\psi\chi_j\r|=|\l  \psi_j, \sign(U')U'''\psi\chi_j\r|\leq 