{"paper":{"title":"Improving estimates for discrete polynomial averages","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Fan Yang, Jos\\'e Madrid, Michael Lacey, Rui Han, Vjekoslav Kova\\v{c}","submitted_at":"2019-10-31T17:16:05Z","abstract_excerpt":"For a polynomial $P$ mapping the integers into the integers, define an averaging operator $A_{N} f(x):=\\frac{1}{N}\\sum_{k=1}^N f(x+P(k))$ acting on functions on the integers. We prove sufficient conditions for the $\\ell^{p}$-improving inequality \\begin{equation*} \\|A_N f\\|_{\\ell^q(\\mathbb{Z})} \\lesssim_{P,p,q} N^{-d(\\frac{1}{p}-\\frac{1}{q})} \\|f\\|_{\\ell^p(\\mathbb{Z})}, \\qquad N \\in\\mathbb{N}, \\end{equation*} where $1\\leq p \\leq q \\leq \\infty$. For a range of quadratic polynomials, the inequalities established are sharp, up to the boundary of the allowed pairs of $(p,q)$. For degree three and h"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1910.14630","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1910.14630/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}