{"paper":{"title":"Restriction of the Fourier transform to some oscillating curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Dashan Fan, Lifeng Wang, Xianghong Chen","submitted_at":"2017-01-02T18:49:02Z","abstract_excerpt":"Let $\\phi$ be a smooth function on a compact interval $I$. Let $$\\gamma(t)=\\left (t,t^2,\\cdots,t^{n-1},\\phi(t)\\right).$$ In this paper, we show that $$\\left(\\int_I \\big|\\hat f(\\gamma(t))\\big|^q \\big|\\phi^{(n)}(t)\\big|^{\\frac{2}{n(n+1)}} dt\\right)^{1/q}\\le C\\|f\\|_{L^p(\\mathbb R^n)}$$ holds in the range $$1\\le p<\\frac{n^2+n+2}{n^2+n},\\quad 1\\le q<\\frac{2}{n^2+n}p'.$$ This generalizes an affine restriction theorem of Sj\\\"olin (1974) for $n=2$. Our proof relies on ideas of Sj\\\"olin (1974) and Drury (1985), and more recently Bak-Oberlin-Seeger (2008) and Stovall (2016), as well as a variation bound"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.00477","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"}