{
  "abstract": "Let $0<α<2$ and $p=2/(2-α)$. We study critical nonlinear convolution inequalities for kernels $K:\\mathbb{R}^2\\setminus\\{0\\}\\to\\mathbb{R}^m$ and functions $Φ:\\mathbb{R}^m\\to\\mathbb{R}$ positively homogeneous of degrees $α-2$ and $p$. We assume angular Lipschitz regularity for $Φ$ and angular $C^{0,γ}$ regularity for $K$, where $0<γ\\le1$ and $γ=1$ if $p<2$. On bounded finitely cornered planar domains with a common Dini tangent modulus on the boundary branches, we characterize $|\\int_ΩΦ(K*f) dx|\\le C\\|f\\|_{L^1}^p$ for real densities $f\\in L^1(\\mathbb{R}^2)\\cap L^s_{\\mathrm{loc}}(\\mathbb{R}^2)$, $1<s\\le\\infty$, by signed angular cancellation on the plane, tangent half-planes, and full vertex tangent cones. An analogous criterion holds on infinite sectors for compactly supported mean-zero densities. For the Newton kernel and each nonzero angularly Lipschitz, positively two-homogeneous $Φ:\\mathbb{R}^2\\to\\mathbb{R}$ with vanishing half-circle moments, we construct a bounded convex $C^1$ domain $Ω_Φ$, independent of $\\varepsilon$, and smooth tests $u_\\varepsilon$ satisfying $\\|Δu_\\varepsilon\\|_{L^1(Ω_Φ)}+\\|\\partial_nu_\\varepsilon\\|_{L^1(\\partialΩ_Φ)}\\le2$, while, despite tangent cancellation, their nonlinear integrals grow in magnitude like $\\log\\log(1/\\varepsilon)$. On the Dini class, a constant-preserving linear extension with compact support modulo the domain mean and controlled measure-valued Laplacian yields the same necessary-and-sufficient criterion for the Newton-kernel PDE inequality, without exact corner charts. We also construct infinite-dimensional cancellation spaces containing nonquadratic functions and classify the quadratic locus in polygon moduli.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12755",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhouyu Long"
    },
    {
      "name": "Wenming Zou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2hxab19bqh0xe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12755",
  "fulltext_url": null,
  "id": "2hxab19bqh0xe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12755",
  "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.CA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7212928568",
  "pdf_url": "https://arxiv.org/pdf/2609.12755",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2hxab19bqh0xe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2609.12755",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12755"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Tangent-cone cancellation and Maz'ya's $Φ$-inequalities on finitely cornered planar domains",
  "title_text": "Tangent-cone cancellation and Maz'ya's Φ-inequalities on finitely cornered planar domains",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/2hxab19bqh0xe",
  "venue": "arXiv",
  "versions": []
}