{"paper":{"title":"Endpoint $ \\ell ^{r}$ improving estimates for Prime averages","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"Hamed Mousavi, Michael T. Lacey, Yaghoub Rahimi","submitted_at":"2021-01-25T20:36:23Z","abstract_excerpt":"Let $ \\Lambda $ denote von Mangoldt's function, and consider the averages \\begin{align*} A_N f (x) &=\\frac{1}{N}\\sum_{1\\leq n \\leq N}f(x-n)\\Lambda(n) . \\end{align*} We prove sharp $ \\ell ^{p}$-improving for these averages, and sparse bounds for the maximal function. The simplest inequality is that for sets $ F, G\\subset [0,N]$ there holds \\begin{equation*} N ^{-1} \\langle A_N \\mathbf 1_{F} , \\mathbf 1_{G} \\rangle \\ll \\frac{\\lvert F\\rvert \\cdot \\lvert G\\rvert} { N ^2 } \\Bigl( \\operatorname {Log} \\frac{\\lvert F\\rvert \\cdot \\lvert G\\rvert} { N ^2 } \\Bigr) ^{t}, \\end{equation*} where $ t=2$, or as"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2101.10401","kind":"arxiv","version":2},"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/2101.10401/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"}