{
  "abstract": "Let $m\\geq1$, $k\\geq0$, $n=m+k$, and write $x=(v,z)\\in\\mathbb R^m\\times \\mathbb R^k$, where $v$ is the active velocity variable and $z$ is passive with respect to $Y=v\\cdot\\nabla_y+\\partial_t$. We consider \\[ \\operatorname{div}_x(A\\nabla_xu)-Y(wu)=0, \\] where $w=w(x)\\in A_2(\\mathbb R^n)$ and the measurable matrix $A$ has ellipticity and size controlled by $w$. For arbitrary such weights, including weights depending on the active variables, we prove a weighted hypoelliptic transfer estimate, a kinetic Sobolev inequality, local boundedness, the weak Harnack and Harnack inequalities, local Hölder continuity, and the strong minimum principle. The Harnack argument uses a full-variable weighted measure-data compactness theorem. An $\\mathrm L^p$ velocity-averaging estimate gives compactness of smooth moments of $wu$, and weighted Poincaré inequalities in all diffusive variables reconstruct $u$. At a two-phase limit, testing with non-negative profiles produces transport directions given by weighted mean velocities. A $\\mathrm{BV}$ argument applied to the limiting subsolution inequality yields the no-jump principle needed for expansion of positivity and a weighted kinetic ink-spots lemma. The theory includes the weight $|λ|^{1-2s}$ in the extension of Garofalo and Tralli for fractional powers of the Kolmogorov operator.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21737",
  "author_total": 1,
  "authors": [
    {
      "name": "Kaj Nyström"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/70f2tvtecd9ks.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21737",
  "fulltext_url": null,
  "id": "70f2tvtecd9ks",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21737",
  "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|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7213866672",
  "pdf_url": "https://arxiv.org/pdf/2609.21737",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/70f2tvtecd9ks.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.21737",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21737"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Weighted Kolmogorov equations: transfer estimates, local boundedness, and Harnack inequalities",
  "title_text": "Weighted Kolmogorov equations: transfer estimates, local boundedness, and Harnack inequalities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/70f2tvtecd9ks",
  "venue": "arXiv",
  "versions": []
}