{
  "abstract": "Let $B=B(p, 1)\\subset\\mathbb{R}^n$ be a unit ball and $n\\ge 3$. We prove that there are positive constants $c$ and $C$, depending only on $n$, such that every non-zero real-valued harmonic function $u: 4B\\to \\mathbb{R}$ with $u(p)=0$ satisfies $$ \\mathcal{H}^{n-1}\\bigl(\\{u=0\\}\\cap 2B\\bigr)\\ge C\\mathcal{N}, $$ and $$ \\mathcal{H}^n\\bigl(\\{u>0\\}\\cap {\\frac 12}B\\bigr) \\ge c\\bigl(\\log(1+\\mathcal{N})\\bigr)^{1-n}, $$ where $\\mathcal{N}$ is the doubling index defined by $$\\mathcal{N}=\\log_2\\frac{\\sup_{B}|u|}{\\sup_{\\frac{1}{2}B}|u|}.$$ The first estimate confirms a folklore conjecture on the nodal volume of harmonic functions, and its linear dependence on $\\mathcal{N}$ is optimal. The second estimate extends the planar result of Nazarov, Polterovich, and Sodin to higher dimensions, and the logarithmic order is optimal. As a further consequence of the ideas developed in the proof, we obtain an alternative proof of Nadirashvili's conjecture that does not rely on the multiscale analysis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11444",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhehui Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9chgp794p7s7w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11444",
  "fulltext_url": null,
  "id": "9chgp794p7s7w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11444",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7212268871",
  "pdf_url": "https://arxiv.org/pdf/2609.11444",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9chgp794p7s7w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.11444",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11444"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp lower bounds for the volumes of nodal and positivity sets of harmonic functions",
  "title_text": "Sharp lower bounds for the volumes of nodal and positivity sets of harmonic functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/9chgp794p7s7w",
  "venue": "arXiv",
  "versions": []
}