{"paper":{"title":"Cancellation for the multilinear Hilbert transform","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CA","authors_text":"Terence Tao","submitted_at":"2015-05-24T20:49:18Z","abstract_excerpt":"For any natural number $k$, consider the $k$-linear Hilbert transform $$ H_k( f_1,\\dots,f_k )(x) := \\operatorname{p.v.} \\int_{\\bf R} f_1(x+t) \\dots f_k(x+kt)\\ \\frac{dt}{t}$$ for test functions $f_1,\\dots,f_k: {\\bf R} \\to {\\bf C}$. It is conjectured that $H_k$ maps $L^{p_1}({\\bf R}) \\times \\dots \\times L^{p_k}({\\bf R}) \\to L^p({\\bf R})$ whenever $1 < p_1,\\dots,p_k,p < \\infty$ and $\\frac{1}{p} = \\frac{1}{p_1} + \\dots + \\frac{1}{p_k}$. This is proven for $k=1,2$, but remains open for larger $k$.\n  In this paper, we consider the truncated operators $$ H_{k,r,R}( f_1,\\dots,f_k )(x) := \\int_{r \\leq "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.06479","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"}