{"paper":{"title":"$L^{p}$ estimates for the bilinear Hilbert transform for $1/2<p\\leq2/3$: A counterexample and generalizations to non-smooth symbols","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Guozhen Lu, Wei Dai","submitted_at":"2014-09-12T21:56:39Z","abstract_excerpt":"M. Lacey and C. Thiele proved in [27] (Annals of Math. (1997)) and [28] (Annals of Math. (1999)) that the bilinear Hilbert transform maps $L^{p_1}\\times L^{p_2}\\rightarrow L^{p}$ boundedly when $\\frac{1}{p_1}+\\frac{1}{p_2}=\\frac{1}{p}$ with $1<p_{1}, \\, p_{2}\\leq\\infty$ and $\\frac{2}{3}<p<\\infty$. Whether the $L^p$ estimates hold in the range $p\\in (1/2,2/3]$ has remained an open problem since then. In this paper, we prove that the bilinear Hilbert transform does not map $\\mathcal{F}L^{p'_{1}}\\times L^{p_{2}}\\rightarrow L^{p}$ for $p_1<2$ and $L^{p_{1}}\\times \\mathcal{F}L^{p'_{2}}\\rightarrow L"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.3875","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"}