{"paper":{"title":"$L^{p}$ estimates for bilinear and multi-parameter Hilbert transforms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Guozhen Lu, Wei Dai","submitted_at":"2014-03-03T22:44:57Z","abstract_excerpt":"C. Muscalu, J. Pipher, T. Tao and C. Thiele proved in \\cite{MPTT1} that the standard bilinear and bi-parameter Hilbert transform does not satisfy any $L^{p}$ estimates. They also raised a question asking if a bilinear and bi-parameter multiplier operator defined by $$\n  T_{m}(f_{1},f_{2})(x):=\\int_{\\mathbb{R}^{4}}m(\\xi,\\eta)\\hat{f_{1}}(\\xi_{1},\\eta_{1})\\hat{f_{2}}(\\xi_{2},\\eta_{2})e^{2\\pi ix\\cdot((\\xi_{1},\\eta_{1})+(\\xi_{2},\\eta_{2}))}d\\xi d\\eta $$ satisfies any $L^p$ estimates, where the symbol $m$ satisfies $$\n  |\\partial_{\\xi}^{\\alpha}\\partial_{\\eta}^{\\beta}m(\\xi,\\eta)|\\lesssim\\frac{1}{dist"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.0624","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":""},"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"}