{
  "abstract": "We establish a non-Archimedean Caffarelli-Silvestre extension theory for the Vladimirov-Taibleson operator $D^s$ for $s>0$ and, more generally, for hierarchical Laplacians $L_C$. For every $f\\in\\mathcal{S}(\\mathbb{Q}_p^n)$, we equip the Bruhat-tits tree $\\mathcal{T}_{p^n}$ with the weight $w_{v_{k-1}v_k}=p^{k(s-n)}$ under the horocyclic coordinates. We prove that the resulting Dirichlet problem admits a unique bounded weighted-harmonic solution that is continuous on the end compactification. Its boundary traces converge to $f$ uniformly and in $L^2$, while the associated deformed normal derivative converges pointwise and in $L^2$ to $D^s f$. We derive explicit Fourier and Poisson representations of the extension and establish an energy identity between its weighted tree energy and the quadratic form of $D^s$. Motivated by $p$-adic AdS/CFT, we also give an equivalent massive formulation on the unweighted tree, based on a renormalized trace and a scale-corrected normal derivative. Finally, under natural local-finiteness and scaling assumptions, we extend the construction to hierarchical Laplacians $L_C$ on ultrametric spaces. We characterize the canonical edge conductances by requiring the cancellation Green operator on the ultrametric tree $\\mathcal{T}_X$ to coincide with $L_C^{-1}$ on $\\mathcal{S}_0(X)$. In the corresponding canonical flux class, the extension is unique, its Dirichlet-to-Neumann map is $L_C$, and it satisfies the associated energy identity. The Vladimirov-Taibleson construction is recovered as the homogeneous special case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03647",
  "author_total": 1,
  "authors": [
    {
      "name": "Yaojia Sun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7snsmjkmdprkv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03647",
  "fulltext_url": null,
  "id": "7snsmjkmdprkv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03647",
  "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"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.03647",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7snsmjkmdprkv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03647",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03647"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Extension Problems for the Vladimirov--Taibleson Operator and the Hierarchical Laplacian",
  "title_text": "Extension Problems for the Vladimirov--Taibleson Operator and the Hierarchical Laplacian",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/7snsmjkmdprkv",
  "venue": "arXiv",
  "versions": []
}