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 $| \nabla |\psi|| (x) \le |\nabla \psi|(x)$ (a proof is here), we have \begin{align*} \left| \! \left| \,\, |\psi| \,\, \right| \! \right|_{H^s} \le c_s \left| \! \left| \psi\right| \! \right|_{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 a such a result for $s=-1$, with $c_{-1} < + \infty$  ?

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 $c_{-1} < + \infty$?

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 $| \nabla |\psi|| (x) \le |\nabla \psi|(x)$ (a proof is here), we have \begin{align*} \left| \! \left| \,\, |\psi| \,\, \right| \! \right|_{H^s} \le c_s \left| \! \left| \psi\right| \! \right|_{H^s} \end{align*} for $s=1$ and $c_1 = 1$, where $c_1$ does not depend on $\psi$. Do we have a such a result for $s=-1$, with $c_{-1} < + \infty$ ?

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 $| \nabla |\psi|| (x) \le |\nabla \psi|(x)$ (a proof is here), we have \begin{align*} \left| \! \left| \,\, |\psi| \,\, \right| \! \right|_{H^s} \le c_s \left| \! \left| \psi\right| \! \right|_{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 a such a result for $s=-1$, with $c_{-1} < + \infty$ ?