{"paper":{"title":"Solyanik estimates and local H\\\"older continuity of halo functions of geometric maximal operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ioannis Parissis, Paul A. Hagelstein","submitted_at":"2014-09-12T18:20:59Z","abstract_excerpt":"Let $\\mathcal{B}$ be a homothecy invariant basis consisting of convex sets in $\\mathbb{R}^n$, and define the associated geometric maximal operator $M_{\\mathcal{B}}$ by $$ M_{\\mathcal{B}} f(x) :=\\sup_{x \\in R \\in \\mathcal{B}}\\frac{1}{|R|}\\int_R |f| $$ and the halo function $\\phi_{\\mathcal{B}}(\\alpha)$ on $(1,\\infty)$ by $$\\phi_{\\mathcal B}(\\alpha) :=\\sup_{E \\subset \\mathbb{R}^n :\\, 0 < |E| < \\infty}\\frac{1}{|E|}|\\{x\\in \\mathbb{R}^n : M_{\\mathcal{B}} \\chi_E (x) >1/\\alpha\\}|. $$ It is shown that if $\\phi_{\\mathcal{B}}(\\alpha)$ satisfies the Solyanik estimate $\\phi_{\\mathcal B}(\\alpha) - 1 \\leq C "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.3811","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}