{"paper":{"title":"Averages Along the Primes: Improving and Sparse Bounds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ben Krause, Fan Yang, Michael Lacey, Rui Han","submitted_at":"2019-09-06T16:44:27Z","abstract_excerpt":"Consider averages along the prime integers $ \\mathbb P $ given by \\begin{equation*} \\mathcal{A}_N f (x) = N ^{-1} \\sum_{ p \\in \\mathbb P \\;:\\; p\\leq N} (\\log p) f (x-p). \\end{equation*} These averages satisfy a uniform scale-free $ \\ell ^{p}$-improving estimate. For all $ 1< p < 2$, there is a constant $ C_p$ so that for all integer $ N$ and functions $ f$ supported on $ [0,N]$, there holds \\begin{equation*} N ^{-1/p' }\\lVert \\mathcal{A}_N f\\rVert_{\\ell^{p'}} \\leq C_p N ^{- 1/p} \\lVert f\\rVert_{\\ell^p}. \\end{equation*} The maximal function $ \\mathcal{A}^{\\ast} f =\\sup_{N} \\lvert \\mathcal{A}_N "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.02883","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/1909.02883/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"}