{"paper":{"title":"Averaging with the Divisor Function: $\\ell^p$-improving and Sparse Bounds","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Christina Giannitsi","submitted_at":"2021-02-02T22:09:32Z","abstract_excerpt":"We study averages along the integers using the divisor function $d(n)$, and defined as $$K_N f (x) = \\frac{1}{D(N)} \\sum _{n \\leq N} d(n) \\,f(x+n) , $$ where $D(N) = \\sum _{n=1} ^N d(n) $. We shall show that these averages satisfy a uniform, scale free $\\ell^p$-improving estimate for $p \\in (1,2)$, that is $$ \\left( \\frac{1}{N} \\sum |K_Nf|^{p'} \\right)^{1/p'} \\lesssim \\left(\\frac{1}{N} \\sum |f|^p \\right)^{1/p} $$ as long as $f$ is supported on $[0,N]$.\n  We also show that the associated maximal function $K^*f = \\sup_N |K_N f|$ satisfies $(p,p)$ sparse founds for $p \\in (1,2)$, which implies th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2102.01778","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/2102.01778/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"}