{"paper":{"title":"Endpoint bounds for the quartile operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ciprian Demeter, Francesco Di Plinio","submitted_at":"2012-06-17T21:46:48Z","abstract_excerpt":"It is a result by Lacey and Thiele that the bilinear Hilbert transform maps L^{p_1}(R) \\times L^{p_2}(R) into L^{p_3}(R) whenever (p_1,p_2,p_3) is a Holder tuple with p_1,p_2 > 1 and p_3>2/3. We study the behavior of the quartile operator, which is the Walsh model for the bilinear Hilbert transform, when p_3=2/3. We show that the quartile operator maps L^{p_1}(R) \\times L^{p_2}(R) into L^{2/3,\\infty}(R) when p_1,p_2>1 and one component is restricted to subindicator functions. As a corollary, we derive that the quartile operator maps L^{p_1}(R) \\times L^{p_2,2/3}(R) into L^{2/3,\\infty}(R). We a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.3798","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"}