{
  "abstract": "We study weighted endpoint estimates for the centered fractional maximal operator on the infinite rooted $k$-ary tree, $T$. Our first result is a Fefferman--Stein type estimate \\[ w\\bigl(\\{M_αf>λ\\}\\bigr) \\lesssim_{α,s} \\frac1λ\\sum_{x\\in T}|f(x)|M_{α,s}w(x), \\qquad s>1, \\] with a constant independent of the branching parameter $k$. Although the proof is inspired by the non-fractional argument of Ombrosi, Rivera-Ríos and Safe in \\cite{ORRS}, the fractional normalization introduces a genuinely new scale interaction. At radius $r$, one must renormalize the function by $k^{rα}$ while keeping the dyadic cutoff at $k^r$; this leads to a new summation mechanism involving simultaneously $k^{rα}$ and $k^{r(1-α)}$. We then study sharpness and the geometry of admissible weights. We construct counterexamples showing that the estimate cannot in general be reduced to $s=1$ for any $0<α<1$. In the subcritical range $0<α<1/2$, we characterize power-type radial weights satisfying the natural fractional $A_1$ condition and prove that the failure persists after any fixed finite number of iterations of $M_α$. We also prove that $M_αw\\lesssim w$ is necessary for the corresponding weighted weak type estimate but is not sufficient in the subcritical range. Finally, we identify a phase transition: for $1/2\\leα<1$ no nonzero global weight can satisfy $M_αw\\lesssim w$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08653",
  "author_total": 2,
  "authors": [
    {
      "name": "Gonzalo Ibañez-Firnkorn"
    },
    {
      "name": "Jorgelina Recchi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/axea7m9689smp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08653",
  "fulltext_url": null,
  "id": "axea7m9689smp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08653",
  "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.CA"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.MG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08653",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/axea7m9689smp.ris",
  "sources": [
    {
      "first_seen": "2026-11-10T06:21:07Z",
      "native_id": "2611.08653",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08653"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Fractional Fefferman--Stein inequalities on infinite rooted $k$-ary trees",
  "title_text": "Fractional Fefferman--Stein inequalities on infinite rooted k-ary trees",
  "type": "preprint",
  "updated_at": "2026-11-10T06:21:07Z",
  "url": "https://research.draconitic.com/paper/axea7m9689smp",
  "venue": "arXiv",
  "versions": []
}