{
  "abstract": "For every finite family of Euclidean balls in $\\mathbb{R}^n$, $n\\ge2$, and every $0<δ<1/2$, we select a subfamily whose $(1+δ)$-dilations cover the original union and whose undilated balls have multiplicity at most $C_nδ^{-(n-1)/2}$. This proves the covering estimate conjectured by Han and Lu \\cite{HL}. As an application, we determine the optimal Solyanik asymptotic \\[ \\mathcal C_n(α)-1\\asymp_n(1-α)^{2/(n+1)} \\qquad(α\\uparrow1) \\] for the uncentered Hardy--Littlewood maximal operator over Euclidean balls, which is Conjecture~1(b) of Hagelstein and Parissis \\cite{HP14}. For the modified uncentered maximal operators, we determine the optimal weak $(1,1)$ growth rate $(k-1)^{-(n-1)/2}$ as $k\\downarrow1$, uniformly over Radon measures. We also obtain the corresponding $L^p$ and Fefferman--Stein bounds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07880",
  "author_total": 1,
  "authors": [
    {
      "name": "Mayukh Mukherjee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/awr155a2pzdzw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07880",
  "fulltext_url": null,
  "id": "awr155a2pzdzw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07880",
  "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.CA"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7212053508",
  "pdf_url": "https://arxiv.org/pdf/2609.07880",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/awr155a2pzdzw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07880",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07880"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A sharp covering theorem and Solyanik estimates for Euclidean balls",
  "title_text": "A sharp covering theorem and Solyanik estimates for Euclidean balls",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/awr155a2pzdzw",
  "venue": "arXiv",
  "versions": []
}