{
  "abstract": "We establish an exact trace theorem for vector-valued time-Morrey spaces and complementary continuity theorems for time-BMO evolution classes. For a Banach couple $X_1\\hookrightarrow X_0$, $1<p<\\infty$ and $0<λ<1$, we prove that the trace of the class $u\\in\\mathcal M^{p,λ}(X_1)$, $u_t\\in\\mathcal M^{p,λ}(X_0)$ is exactly $(X_0,X_1)_{θ,\\infty}$, where $θ=1-(1-λ)/p$. A constructive extension realizes every admissible trace, and counterexamples establish optimality of both the smoothness exponent and the weak fine index. Time-BMO control yields continuous finite-index interpolation traces and, under a mixed boundedness hypothesis, a logarithmic time modulus whose base-space logarithm is necessary. For the quadratic viscous Hamilton--Jacobi equation on the torus, we characterize the bounded initial data generating Morrey-regular solutions by the exact Besov condition $u_0\\in B_{2,\\infty}^{2θ}$. The proof uses Hopf--Cole linearization and a direct composition argument, and requires no smallness assumption. Lacunary nonlinear solutions realize the sharp trace threshold. A forced Hamilton--Jacobi example exhibits genuinely logarithmic time regularity, and a forward--backward mean-field-game system illustrates initial and terminal trace compatibility. \\par\\smallskip\\noindent\\textbf{Keywords:} Morrey spaces; trace spaces; real interpolation; Hamilton--Jacobi equations; bounded mean oscillation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29148",
  "author_total": 2,
  "authors": [
    {
      "name": "Rishad Shahmurov"
    },
    {
      "name": "Veli Shahmurov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4yp466k674ytm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29148",
  "fulltext_url": null,
  "id": "4yp466k674ytm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29148",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.29148",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4yp466k674ytm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2609.29148",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29148"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Exact Traces for Vector-Valued Morrey Spaces",
  "title_text": "Exact Traces for Vector-Valued Morrey Spaces",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/4yp466k674ytm",
  "venue": "arXiv",
  "versions": []
}