{
  "abstract": "In this paper, we prove that we can subtract the full spectral gap $\\left(\\frac{N-1}{p}\\right)^{p}$ of the $p$-Laplacian and keep the sharp Sobolev constant $S(N,p)$ of $\\mathbb{R}^N$ at the same time in a large family of $p$. For $p=2$, this was proved by Benguria, Frank and Loss in dimension three, and it was shown by Hebey to be false in dimensions $N\\geq4$. For $p\\neq2$, it was proved by Nguyen for $N\\geq4$ and $\\frac{2N}{N-1}\\leq p<N$. By employing, among other tools, a suitable conjugation that maps the radial $p$-Dirichlet integral on $\\mathbb{H}^N$ onto its Euclidean counterpart on $\\mathbb{R}^N$ while preserving the $L^p$ norm, together with a one-dimensional tangent line inequality, we establish such sharp Poincaré--Sobolev inequalities for $N=3$ and $2\\leq p<3$, for $N=4$ and $\\frac{1+\\sqrt{13}}{2}\\leq p<4$, and for $N\\geq5$ and $\\min\\{3,\\frac{1+\\sqrt{4N-3}}{2}\\}\\leq p<N$. In particular, in all cases we also obtain an explicit remainder term. Moreover, since the change of function keeps the $L^p$ norm, we prove the sharp logarithmic Sobolev inequalities with the full spectral gap and the sharp Euclidean constant for $N=3$ and $2\\leq p\\leq3$, and for $N\\geq4$ on an explicit interval of $p$. For $p\\geq3$, we also use the change of variable $t=\\sinhρ$ and the Hardy inequality, which give a second remainder term. At the endpoint $p=N$, this remainder gives a Hardy--Trudinger--Moser inequality on $\\mathbb{H}^N$ under a weaker constraint than the one of Liang, Lu, Wang and Yang, in every dimension $N\\geq3$. Finally, we show that for $1<p<\\frac{1+\\sqrt{4N-3}}{2}$, which includes all $1<p<2$, the full spectral gap can still be subtracted with some positive constant in front of the Sobolev term, and that this constant is smaller than $S(N,p)^{p}$ when $p<\\min\\{2,\\sqrt N\\}$. Together with our other results, this shows that the full spectral gap can be subtracted for every $1<p<N$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01161",
  "author_total": 4,
  "authors": [
    {
      "name": "Anh Do"
    },
    {
      "name": "Debdip Ganguly"
    },
    {
      "name": "Nguyen Lam"
    },
    {
      "name": "Guozhen Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a1f238hxjnv5t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01161",
  "fulltext_url": null,
  "id": "a1f238hxjnv5t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01161",
  "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.DG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01161",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a1f238hxjnv5t.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01161",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01161"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Optimal AB-program for Poincaré--Sobolev and logarithmic Sobolev inequalities on hyperbolic spaces",
  "title_text": "Optimal AB-program for Poincaré--Sobolev and logarithmic Sobolev inequalities on hyperbolic spaces",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/a1f238hxjnv5t",
  "venue": "arXiv",
  "versions": []
}