# How to prove Kato's inequality $\nabla\vert u\vert \leq \operatorname{Re}( \operatorname{sgn} u \cdot\nabla_{H}u)$?

Let $a_{k}\;(k=1,...,n)$ be a real-valued function on $\mathbb{R}^n$, $a_{k}\in L_\text{loc}^q(\mathbb{R}^n)$, $q\geq 4$. And $\nabla(a_{1},...,a_{n})=0$ in distribution. Let $\nabla_{H}=\sum_{j=1}^n(\partial_{j}-ia_{j})^2$. Show that Kato's inequality: $$\nabla\vert u\vert \leq \operatorname{Re}( \operatorname{sgn} u \cdot \nabla_{H} u),\ (*)$$ where $\forall u\in L_\text{loc}^1(\mathbb{R}^n), \nabla_{H} u\in L_\text{loc}^2(\mathbb{R}^n).$

In Kato's paper https://link.springer.com/content/pdf/10.1007%2FBF02760233.pdf, he has proved Lemma A in P138 $$L_{0}\vert u \vert \geq \operatorname{Re}( \operatorname{sgn} u \cdot L u)$$

Is there something wrong in $(*)$? Why is it different from the Kato's conclusion?

I find a solution for special case $\Delta$: In textbooks: https://link.springer.com/content/pdf/10.1007%2F978-1-4612-0741-2.pdf Theorem 8.12