{"paper":{"title":"Absolute continuity and $\\alpha$-numbers on the real line","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Tuomas Orponen","submitted_at":"2017-03-08T17:46:49Z","abstract_excerpt":"Let $\\mu,\\nu$ be Radon measures on $\\mathbb{R}$, with $\\mu$ non-atomic and $\\nu$ doubling, and write $\\mu = \\mu_{a} + \\mu_{s}$ for the Lebesgue decomposition of $\\mu$ relative to $\\nu$. For an interval $I \\subset \\mathbb{R}$, define $\\alpha_{\\mu,\\nu}(I) := \\mathbb{W}_{1}(\\mu_{I},\\nu_{I})$, the Wasserstein distance of normalised blow-ups of $\\mu$ and $\\nu$ restricted to $I$. Let $\\mathcal{S}_{\\nu}$ be the square function $$\\mathcal{S}^{2}_{\\nu}(\\mu) = \\sum_{I \\in \\mathcal{D}} \\alpha_{\\mu,\\nu}^{2}(I)\\chi_{I},$$ where $\\mathcal{D}$ is the family of dyadic intervals of side-length at most one. I p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.02935","kind":"arxiv","version":3},"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"}