{"paper":{"title":"On the maximal directional Hilbert transform","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Alessandro Marinelli, Izabella Laba, Malabika Pramanik","submitted_at":"2017-07-04T16:30:03Z","abstract_excerpt":"For any dimension $n \\geq 2$, we consider the maximal directional Hilbert transform $\\mathscr{H}_U$ on $\\mathbb R^n$ associated with a direction set $U \\subseteq \\mathbb S^{n-1}$: \\[ \\mathscr{H}_Uf(x) := \\frac{1}{\\pi} \\sup_{v \\in U} \\Bigl| \\text{p.v.} \\int f(x - tv) \\, \\frac{dt}{t}\\Bigr|.\\] The main result in this article asserts that for any exponent $p \\in (1, \\infty)$, there exists a positive constant $C_{p,n}$ such that for any finite direction set $U \\subseteq \\mathbb S^{n-1}$, \\[||\\mathscr{H}_U||_{p \\rightarrow p} \\geq C_{p,n} \\sqrt{\\log \\#U}, \\] where $\\#U$ denotes the cardinality of $U"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.01061","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"}