{"paper":{"title":"Restricted convolution inequalities, multilinear operators and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Alex Iosevich, Allan Greenleaf, Dan-Andrei Geba, Eric Sawyer, Eyvindur Palsson","submitted_at":"2012-09-28T17:17:03Z","abstract_excerpt":"For $ 1\\le k <n$, we prove that for functions $F,G$ on $ {\\Bbb R}^{n}$, any $k$-dimensional affine subspace $H \\subset {\\Bbb R}^{n}$, and $p,q,r \\ge 2$ with $\\frac{1}{p}+\\frac{1}{q}+\\frac{1}{r}=1$, one has the estimate\n  $$ {||(F*G)|_H||}_{L^{r}(H)} \\leq {||F||}_{\\Lambda^H_{2, p}({\\Bbb R}^{n})} \\cdot {||G||}_{\\Lambda^H_{2, q}({\\Bbb R}^{n})},$$ where the mixed norms on the right are defined by\n  $$ {||F||}_{\\Lambda^H_{2,p}({\\Bbb R}^{n})}={(\\int_{H^*} {(\\int {|\\hat{F}|}^2 dH_{\\xi}^{\\perp})}^{\\frac{p}{2}} d\\xi)}^{\\frac{1}{p}},$$ with $dH_{\\xi}^{\\perp}$ the $(n-k)$-dimensional Lebesgue measure on "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.6574","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"}