I am trying to understand a special case of Diamagnetic inequality mentioned in Appendix [B] of the book Nonlinear Dispersive Equations by Tao. Here is the statement,
Let $Q\in H_x^{1}(\mathbb{R}^d).$ Then we have that,$$-|\nabla Q|\leq |\nabla |Q||\leq |\nabla Q|$$ in the sense of distributions. In particular $|Q|\in H^{1}_x(\mathbb{R}^d).$
Here is the justification provided in the book.
If $Q$ is Schwartz then one easily verifies that $$-|\nabla Q|\leq \nabla (\epsilon^2 + |Q|^2)^{1/2}\leq |\nabla Q|\quad \quad \quad (*)$$ for all $\epsilon>0.$ Taking limints in $H^1_x$ we see that the same inequality also hold in the distributional sense for all $Q\in H^{1}_x(\mathbb{R}^d).$
I do not know how to verify $(*)$ if $Q$ is Schwartz. For the second part, I am guessing that since the Schwartz functions are dense in $H^1_x(\mathbb{R}^d)$ we can generalize the inequality for all $Q\in H^1_x(\mathbb{R}^d)$ although I am not quite sure. Any hints/comments regarding these two questions will be much appreciated.