the following, we need a more localized version of the above estimate, adapted to the partition $\chi_j$, where we morally would want to replace $I$ by $I_{j}=\text{supp}(\chi_{j})$. As the Biot Savart law is *non-local*, such an estimate can not hold exactly. However, we show that in summed sense and in suitably modified and weighted spaces such a localization can indeed be established.

We begin with a preliminary key lemma establishing weighted elliptic estimates.

\[lem:weighted\_elliptic\] Let $J\subset I$ be an interval. For any complex-valued $g\in H^1_0(J)$ and any $k\geq 1$ there holds $$\begin{aligned}
- \l  |U'| g, \Delta_k g\r \geq \gamma_{0}\int_J |U'(y)||\nabla_k g(y)|^2\dd y.\end{aligned}$$ In particular, the left-hand-side is a positive definite bilinear form in $\nabla_k g$.

We integrate by parts twice and use that $g$ vanishes at the boundary to obtain $$\begin{aligned}
- \l g |U'|, \Delta_k g\r  
&= \int_J \left[|U'(y)| |\nabla_k g(y)|^{2} +\frac12 \de_{y}|U'(y)| \de_{y}|g(y)|^{2} \right]\dd y \notag\\
&= \int_J\left[ |U'(y)| |\nabla_k g(y)|^{2}- \frac{1}{2}|g(y)|^{2} \de_{yy}|U'(y)|\right]\dd y.\end{aligned}$$ Since $U$ and $B$ are mildly degenerate, in the sense of weak derivatives we have $$\begin{aligned}
\label{eq:weakder}
\de_{yy}|U'|=\sign(U')U''' +2 \sum_{\bar{y}: U'(\bar{y})=0} \delta_{\bar{y}} U''(\bar{y})=\sign(U')U''',\end{aligned}$$ and therefore we can exploit to deduce that $$\begin{aligned}
- \l g |U'|, \Delta_k g\r   
 &= \int_J\left[ |U'(y)| |\nabla_k g(y)|^{2}- 