{
  "abstract": "We prove global reverse Hölder estimates in \\(\\mathrm L^2\\) for the Poisson kernel of scalar divergence-form parabolic operators in the upper half-space. The coefficient matrix is real, symmetric, and periodic in every spatial variable \\(X=(λ,x)\\), while no periodicity in time is assumed. If its boundary trace \\(A^0=A^0(x,t)\\) has a quantitative uniform modulus of continuity in time and its convergence to \\(A^0\\) in the transverse variable is controlled by a square-Dini modulus, then the Poisson kernel satisfies a uniform \\(\\mathrm{RH}_2\\) estimate at scales below the period. Full spatial periodicity is then used to propagate this estimate to every scale. The main new ingredient in the large-scale argument is a positive adjoint height coordinate \\(Φ^*=λ+O(1)\\). Boundary comparison with \\(Φ^*\\) yields the required large-scale decay of the Green function and hence the global \\(\\mathrm{RH}_2\\) estimate. Consequently, the \\(\\mathrm L^2\\) Dirichlet problem is uniquely solvable among weak solutions \\(u\\) satisfying \\(N_\\ast u\\in\\mathrm L^2\\). We also establish, for fully spatially periodic coefficients, a nonsymmetric \\(A_\\infty\\) theorem under square-Dini convergence to a transverse-independent boundary trace, with no temporal regularity beyond measurability. We explain how the results transfer to time-independent Lipschitz graph domains under the stated coefficient and periodicity-compatibility hypotheses, and we derive scale-uniform boundary estimates for the periodic homogenization families considered below. The condition \\(D_t^{1/2}A^0\\in\\mathrm L^\\infty_x(\\mathrm{BMO}_t)\\) provides a natural sufficient criterion for the temporal continuity required in the symmetric theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11626",
  "author_total": 1,
  "authors": [
    {
      "name": "Kaj Nyström"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/edmg7r2j9p9za.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11626",
  "fulltext_url": null,
  "id": "edmg7r2j9p9za",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11626",
  "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|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7212225555",
  "pdf_url": "https://arxiv.org/pdf/2609.11626",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/edmg7r2j9p9za.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11626",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11626"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Solvability of the \\(\\mathrm L^2\\) Dirichlet problem for parabolic operators with spatially periodic coefficients",
  "title_text": "Solvability of the L^2 Dirichlet problem for parabolic operators with spatially periodic coefficients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/edmg7r2j9p9za",
  "venue": "arXiv",
  "versions": []
}