Boundedness and continuity of phase-corrected gradients on Sobolev spaces

For a function $\psi : \mathbb{R}^d \to \mathbb{C}$, we define the "phase-corrected gradient" $\nabla_\circ \psi = |\psi|^{-1}\overline{\psi}\,\nabla\psi$ and study the regularity of the map $\psi \mapsto \nabla_\circ\psi$ on Sobolev spaces. Since $\operatorname{Re}[\nabla_\circ\psi] = \nabla|\psi|$, the phase-corrected gradient fits naturally into the framework of nonlinear composition operators. Its imaginary part $\Lambda = \operatorname{Im}[\nabla_\circ\psi]$, on the other hand, plays a key role in quantum hydrodynamic models. We prove that $\psi \mapsto \nabla_\circ\psi$ is bounded and continuous from $W^{s,2}(\mathbb{R}^d)$ to $W^{s-1,2}(\mathbb{R}^d)$ for $1 \leq s < \frac{3}{2}$, and we show that the threshold $s = \frac{3}{2}$ is sharp. In particular, this answers an open question concerning the continuity of the map $\psi \mapsto |\psi|$ on Sobolev spaces. We also show that our results remain valid for functions that do not necessarily decay at infinity, a relevant setting in the context of Ginzburg--Landau theory. Finally, we extend our analysis to more general phase-corrected gradients of the form $f(|\psi|^{-1}\psi)\nabla\psi$, where $f : \mathbb{S}^1 \to \mathbb{C}$ is Lipschitz, and discuss implications for certain parabolic PDEs.