{"paper":{"title":"Bilinear generalized Radon transforms in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Alex Iosevich, Allan Greenleaf, Allen Liu, Ben Krause","submitted_at":"2017-04-04T03:07:21Z","abstract_excerpt":"Let $\\sigma$ be arc-length measure on $S^1\\subset \\mathbb R^2$ and $\\Theta$ denote rotation by an angle $\\theta \\in (0, \\pi]$. Define a model bilinear generalized Radon transform, $$B_{\\theta}(f,g)(x)=\\int_{S^1} f(x-y)g(x-\\Theta y)\\, d\\sigma(y),$$ an analogue of the linear generalized Radon transforms of Guillemin and Sternberg \\cite{GS} and Phong and Stein (e.g., \\cite{PhSt91,St93}). Operators such as $B_\\theta$ are motivated by problems in geometric measure theory and combinatorics. For $\\theta<\\pi$, we show that $B_{\\theta}: L^p({\\Bbb R}^2) \\times L^q({\\Bbb R}^2) \\to L^r({\\Bbb R}^2)$ if $\\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.00861","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"}