{"paper":{"title":"A bilinear Rubio de Francia inequality for arbitrary squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Cristina Benea (LMJL), Frederic Bernicot (LMJL)","submitted_at":"2016-02-05T08:01:23Z","abstract_excerpt":"We prove the boundedness of a smooth bilinear Rubio de Francia operator associated with an arbitrary collection of squares (with sides parallel to the axes) in the frequency plane\\[\\left(f, g \\right)\\mapsto \\left( \\sum\\_{\\omega \\in \\Omega}\\left| \\int\\_{\\mathbb{R}^2} \\hat{f}(\\xi) \\hat{g}(\\eta) \\Phi\\_{\\omega}(\\xi, \\eta) e^{2 \\pi i x\\left(\\xi+\\eta \\right)} d \\xi d \\eta\\right|^r \\right)^{1/r},\\] provided $r\\textgreater{}2$. More exactly, we show that the above operator maps $L^p \\times L^q \\to L^s$ whenever $p, q, s'$ are in the \"local $L^{r'}$\" range, i.e.  $\\displaystyle \\frac{1}{p}+\\frac{1}{q}+"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.01948","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"}