{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35584",
  "author_total": 3,
  "authors": [
    {
      "name": "Lars Eric Hientzsch"
    },
    {
      "name": "Raffaele Scandone"
    },
    {
      "name": "Robert Wegner"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bcrtkw6wdsnb7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35584",
  "fulltext_url": null,
  "id": "bcrtkw6wdsnb7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35584",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7224020893",
  "pdf_url": "https://arxiv.org/pdf/2610.35584",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bcrtkw6wdsnb7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.35584",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35584"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Boundedness and continuity of phase-corrected gradients on Sobolev spaces",
  "title_text": "Boundedness and continuity of phase-corrected gradients on Sobolev spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/bcrtkw6wdsnb7",
  "venue": "arXiv",
  "versions": []
}