{
  "abstract": "We prove a local $Tb$ theorem with testing conditions on metric balls, for possibly non-doubling measures of polynomial growth on geometrically doubling metric spaces. It resolves the maximal-truncation version of Hofmann's local $Tb$ problem for all $1<p,q\\leq\\infty$, without assuming antisymmetry of the Calderón-Zygmund kernel. Our theorem allows accretive systems of complex testing measures satisfying a quantitative small-set condition; these measures need not be absolutely continuous with respect to the underlying measure. Testing requires only a one-level $L^0$ bound for maximal truncations, with an alternative for balls on which the underlying measure concentrates in a set of small relative diameter. We also remove antisymmetry from the known buffered and unbuffered fixed-truncation criteria. The proof combines a mixed corona decomposition adapted to the two testing systems with two-measure $L^2$ estimates for suppressed operators. As an application in the parabolic space $\\mathbb P^n$, local time-directed $A_\\infty$-type conditions on caloric and adjoint caloric measure imply $L^2(σ)$-boundedness of the spatial gradient of the heat single layer potential for boundary measures $σ$ with upper parabolic $(n+1)$-growth. This gives a parabolic counterpart of the harmonic-measure theorem of Mourgoglou and Tolsa without Ahlfors regularity, corkscrew or Harnack chain assumptions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40135",
  "author_total": 3,
  "authors": [
    {
      "name": "Andrea Merlo"
    },
    {
      "name": "Mihalis Mourgoglou"
    },
    {
      "name": "Carmelo Puliatti"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dw7t5k2t5tqmm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40135",
  "fulltext_url": null,
  "id": "dw7t5k2t5tqmm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40135",
  "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.CA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40135",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dw7t5k2t5tqmm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.40135",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40135"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Local $Tb$ theorems in non-homogeneous metric spaces, and applications to caloric measure",
  "title_text": "Local Tb theorems in non-homogeneous metric spaces, and applications to caloric measure",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/dw7t5k2t5tqmm",
  "venue": "arXiv",
  "versions": []
}