{"paper":{"title":"Propagation of smallness near codimension two for gradients of harmonic functions","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Benjamin Foster, Josep Gallegos","submitted_at":"2025-08-28T21:09:06Z","abstract_excerpt":"Let $u$ be a harmonic function in the unit ball $B_1 \\subset \\mathbb R^n$, normalized so that its gradient has magnitude at most 1 on the unit ball. We show that if the gradient of $u$ is $\\epsilon$-small in size on a set $E\\subset B_{1/2}$ with positive $(n-2+\\delta)$-dimensional Hausdorff content for some $\\delta>0$, then $\\sup_{B_{1/2}} |\\nabla u| \\leq C \\epsilon^\\alpha$ with $C,\\alpha>0$ depending only on $n,\\delta$ and the $(n-2+\\delta)$-Hausdorff content of $E$. This is an improvement over a similar result of Logunov and Malinnikova that required $\\delta>1-c_n$ for a small dimensional co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.21214","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2508.21214/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}