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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1909.02883","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-09-06T16:44:27Z","cross_cats_sorted":[],"title_canon_sha256":"5787d32cc1664a41a094a0263f44b25967f1274211bcb040cd339e96ecbfbecb","abstract_canon_sha256":"49c1588184c5b3ddca56b2c08c912483f2eeefc67a49b85f1723485bcbacc885"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:11:41.526354Z","signature_b64":"c+WTcEC5D7fNU2L8iJSnpSWyIvmBnRrr+pMe69iCoVecBbYqsD61qo14SBnvXjiJIoj4H0pLXJ9wghzNkOKJAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ca4fed9500d472c69d1ae9f87a77c6ac85ec646684e31370e6adfeba2af75a54","last_reissued_at":"2026-07-05T01:11:41.525765Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:11:41.525765Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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