# 'Diamagnetic' inequality for negative Sobolev spaces

Let us look at the subspace of smooth complex functions of $$L^2(\mathbb{R}^n,\mathbb{C})$$, call $$H^s$$ the Sobolev spaces. By the diamagnetic inequality $$\lvert \nabla \lvert\psi\rvert\rvert (x) \le \lvert\nabla \psi\rvert(x)$$ (a proof is here), we have \begin{align*} \lVert\,\lvert\psi\rvert\,\rVert_{H^s} \le c_s \lVert \psi\rVert_{H^s} \end{align*} for $$s=1$$ and $$c_1 = 1$$, where $$c_1$$ does not depend on $$\psi$$. It is also true for $$s=0$$ with $$c_0 = 1$$. Do we have such a result for $$s=-1$$, with $$c_{-1} < + \infty$$?

This is not true in the case of $$H^{-1}(\Omega) = H_0^1(\Omega)^*$$ (real spaces), where $$\Omega \subset \mathbb R^d$$ is bounded:
• In https://math.stackexchange.com/questions/336834/decomposition-of-functionals-on-sobolev-spaces, we see that $$|\psi|$$ cannot be defined for all $$\psi \in H^{-1}(\Omega)$$.
• Even if $$|\psi|$$ can be defined (as a measure), it might not belong to $$H^{-1}(\Omega)$$, see https://math.stackexchange.com/questions/1402697/decomposition-of-measures-acting-on-sobolev-spaces
• Even if $$|\psi| \in H^{-1}(\Omega)$$, we do not get a bound. On $$\Omega = (0,1)$$, you can consider $$\psi_n(x) = \sin(n x)$$ as an element in $$H^{-1}(\Omega)$$. Then, $$\|\psi_n\|_{H^{-1}} \to 0$$, but $$\||\psi_n|\|_{H^{-1}} \not\to 0$$.