Boundedness and continuity of phase-corrected gradients on Sobolev spaces

Abstract

For a function $ψ: \mathbb{R}^d \rightarrow \mathbb{C}$, we define the phase-corrected gradient $\nabla_{\!\circ} ψ= |ψ|^{-1} \overlineψ\,\nablaψ$ and study the regularity of the map $ψ\mapsto \nabla_{\!\circ} ψ$ on Sobolev spaces. Since $\mathrm{Re}[\nabla_{\!\circ} ψ] = \nabla |ψ|$, the phase-corrected gradient fits naturally into the framework of nonlinear composition operators. Its imaginary part $Λ= \mathrm{Im}[\nabla_{\!\circ} ψ]$, on the other hand, plays a key role in quantum hydrodynamic models. We prove that $ψ\mapsto \nabla_{\!\circ} ψ$ is bounded and continuous from $W^{s,2}(\mathbb{R}^d)$ to $W^{s-1,2}(\mathbb{R}^d)$ for $1 \leq s < \frac32$, and we show that the threshold $s = \frac32$ is sharp. In particular, this answers an open question concerning the continuity of the map $ψ\mapsto |ψ|$ 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 $f(|ψ|^{-1} ψ) \nabla ψ$, where $f: \mathbb{S}^1 \rightarrow \mathbb{C}$ is Lipschitz, and discuss implications for certain parabolic PDEs.