{
  "abstract": "On a volume doubling metric measure space endowed with a family of $p$-energies such that the Poincaré inequality and the capacity upper bound with $p$-walk dimension $β_p$ hold, for $p$ in an open interval $I\\subseteq(1,+\\infty)$, we prove the following dichotomy: either $β_p=p$ for all $p\\in I$, or the function $p\\mapsto \\frac{β_p}{p}$ takes values in $(1,+\\infty)$ and is strictly decreasing on $I$. Moreover, if the metric measure space is $d_h$-Ahlfors regular, we prove the following dichotomy: either $β_p=d_h+p-1$ for all $p\\in I$, or the function $p\\mapsto \\frac{β_p-d_h}{p-1}$ takes values in $(-\\infty,1)$ and is strictly increasing on $I$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02422",
  "author_total": 1,
  "authors": [
    {
      "name": "Meng Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9dcv6v97n77va.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02422",
  "fulltext_url": null,
  "id": "9dcv6v97n77va",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02422",
  "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.FA"
    ],
    [
      "arxiv",
      "math.MG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02422",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9dcv6v97n77va.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02422",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02422"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On the dichotomy of $p$-walk dimensions on metric measure spaces II",
  "title_text": "On the dichotomy of p-walk dimensions on metric measure spaces II",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/9dcv6v97n77va",
  "venue": "arXiv",
  "versions": []
}