Boundedness and continuity of phase-corrected gradients on Sobolev spaces

Abstract

For a function ψ: ℝ^d → ℂ, we define the phase-corrected gradient ∇_∘ ψ= |ψ|⁻¹ ψ̅ ∇ψ and study the regularity of the map ψ↦ ∇_∘ ψ on Sobolev spaces. Since Re[∇_∘ ψ] = ∇ |ψ|, the phase-corrected gradient fits naturally into the framework of nonlinear composition operators. Its imaginary part Λ= Im[∇_∘ ψ], on the other hand, plays a key role in quantum hydrodynamic models. We prove that ψ↦ ∇_∘ ψ is bounded and continuous from W^s,2(ℝ^d) to W^s−1,2(ℝ^d) for 1 ≤ s < 3/2, and we show that the threshold s = 3/2 is sharp. In particular, this answers an open question concerning the continuity of the map ψ↦ |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) ∇ ψ, where f: 𝕊¹ → ℂ is Lipschitz, and discuss implications for certain parabolic PDEs.

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BibTeX RIS

BibTeX

@misc{hientzsch2026boundedness,
  title = {{Boundedness and continuity of phase-corrected gradients on Sobolev spaces}},
  author = {Lars Eric Hientzsch and Raffaele Scandone and Robert Wegner},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35584},
  eprint = {2610.35584},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35584},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Boundedness and continuity of phase-corrected gradients on Sobolev spaces
AU  - Lars Eric Hientzsch
AU  - Raffaele Scandone
AU  - Robert Wegner
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35584
UR  - https://arxiv.org/abs/2610.35584
AB  - For a function ψ: ℝ^d → ℂ, we define the phase-corrected gradient ∇_ ∘ ψ= |ψ|^−1 ψ̅ ∇ψ and study the regularity of the map ψ↦ ∇_ ∘ ψ on Sobolev spaces. Since Re[∇_ ∘ ψ] = ∇ |ψ|, the phase-corrected gradient fits naturally into the framework of nonlinear composition operators. Its imaginary part Λ= Im[∇_ ∘ ψ], on the other hand, plays a key role in quantum hydrodynamic models. We prove that ψ↦ ∇_ ∘ ψ is bounded and continuous from W^s,2(ℝ^d) to W^s−1,2(ℝ^d) for 1 ≤ s < 3/2, and we show that the threshold s = 3/2 is sharp. In particular, this answers an open question concerning the continuity of the map ψ↦ |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) ∇ ψ, where f: 𝕊^1 → ℂ is Lipschitz, and discuss implications for certain parabolic PDEs.
ER  -

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