{
  "abstract": "We study the heat equation in the half-space with nonhomogeneous Dirichlet boundary data. For the caloric extension $v$ of the boundary data $g$, we prove maximal regularity estimates in mixed Lebesgue norms $L^p_tL^q_x$ for any order derivative of $v$ in terms of mixed Besov and Lizorkin--Triebel type norms of $g$. We also establish the corresponding reverse inequalities, which are caloric trace estimates recovering the boundary regularity of $g$ from the mixed-norm regularity of $v$. As a model case, our results show that the natural \\[\\dotc W^{1,p}\\big(\\R;L^q(\\R^d_+)\\big)\\cap L^p\\big(\\R;\\dotc W^{2,q}(\\R^d_+)\\big)\\] regularity norm of $v$ is controlled by the \\[\\dotc {F}^{1-\\frac{1}{2q}}_{p,q}\\big(\\R;\\,L^{q}(\\R^{d-1})\\big)\\cap L^{p}\\big(\\R;\\,\\dotc{B}_{q,q}^{2-\\frac{1}{q}}(\\R^{d-1})\\big)\\] norm of $g$. The maximal regularity estimate holds for $1\\leq p,q<\\infty$, while the caloric trace estimate holds for $1<p<\\infty$ and $1\\leq q\\leq\\infty$. In particular, the endpoint cases $p=1$ or $q=1$ in the maximal regularity estimate are included and appear to be new. These endpoint estimates may be useful in the analysis of free-boundary Navier--Stokes problems with small initial data, whereas the caloric trace estimates may be relevant to the construction of Stokes or Navier--Stokes flows exhibiting strong boundary singularities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.16438",
  "author_total": 3,
  "authors": [
    {
      "name": "Hui Chen"
    },
    {
      "name": "Su Liang"
    },
    {
      "name": "Tai-Peng Tsai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a9h4vfw4kn8f0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.16438",
  "fulltext_url": null,
  "id": "a9h4vfw4kn8f0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.16438",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2606.16438",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a9h4vfw4kn8f0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2606.16438",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.16438"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Maximal regularity and caloric trace estimates in mixed Lebesgue norms for the heat equation",
  "title_text": "Maximal regularity and caloric trace estimates in mixed Lebesgue norms for the heat equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/a9h4vfw4kn8f0",
  "venue": "arXiv",
  "versions": []
}