{"paper":{"title":"Continuity of halo functions associated to homothecy invariant density bases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Oleksandra V. Beznosova, Paul A. Hagelstein","submitted_at":"2012-12-18T00:18:07Z","abstract_excerpt":"Let $\\mathcal{B}$ be a collection of open sets in $\\mathbb{R}^{n}$ such that, for any $x \\in \\mathbb{R}^{n}$, there exists a set $U \\in \\mathcal{B}$ of arbitrarily small diameter {containing $x$.} $\\mathcal{B}$ is said to be a \\emph{density basis} provided that, given a measurable set $A \\subset \\mathbb{R}^{n}$, for a.e. $x \\in \\mathbb{R}^{n}$ we have $$\\lim_{k \\rightarrow \\infty}\\frac{1}{|R_{k}|}\\int_{R_{k}}\\chi_{A} = \\chi_{A}(x)$$ holds for any sequence of sets $\\{R_{k}\\}$ in $\\mathcal{B}$ containing $x$ whose diameters tend to 0. The geometric maximal operator $M_{\\mathcal{B}}$ associated t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.4199","kind":"arxiv","version":1},"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"}