{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2021:N64OKLT5MUSKCTZ25652676XA6","short_pith_number":"pith:N64OKLT5","schema_version":"1.0","canonical_sha256":"6fb8e52e7d6524a14f3aefbbaf7fd707b35144b86c528ab62f7d0dff94bdeac4","source":{"kind":"arxiv","id":"2102.01778","version":2},"attestation_state":"computed","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"},"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":"2102.01778","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2021-02-02T22:09:32Z","cross_cats_sorted":[],"title_canon_sha256":"50d8c71992b5be46dc779fb049843e320dc853fd5ca58d35954f39fbf8ebe13d","abstract_canon_sha256":"006bd818986824b32dfbf054d26e5bf83bd0c67f96052eb7a4a789a57d7bf1e3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:10:11.723760Z","signature_b64":"gN6say8Q3+XgZtgPxn3d99yZQo4fjT5FPHI3LjR4T6Zp8H7CcuwviTtvjD8kfSPhXn+npcGnZZTB1rosxl7WBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6fb8e52e7d6524a14f3aefbbaf7fd707b35144b86c528ab62f7d0dff94bdeac4","last_reissued_at":"2026-07-05T04:10:11.723278Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:10:11.723278Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2102.01778","created_at":"2026-07-05T04:10:11.723340+00:00"},{"alias_kind":"arxiv_version","alias_value":"2102.01778v2","created_at":"2026-07-05T04:10:11.723340+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2102.01778","created_at":"2026-07-05T04:10:11.723340+00:00"},{"alias_kind":"pith_short_12","alias_value":"N64OKLT5MUSK","created_at":"2026-07-05T04:10:11.723340+00:00"},{"alias_kind":"pith_short_16","alias_value":"N64OKLT5MUSKCTZ2","created_at":"2026-07-05T04:10:11.723340+00:00"},{"alias_kind":"pith_short_8","alias_value":"N64OKLT5","created_at":"2026-07-05T04:10:11.723340+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2608.08893","citing_title":"Sharp $\\ell^p$-Improving Estimates for Fixed-Radius Discrete Spherical Averages","ref_index":5,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/N64OKLT5MUSKCTZ25652676XA6","json":"https://pith.science/pith/N64OKLT5MUSKCTZ25652676XA6.json","graph_json":"https://pith.science/api/pith-number/N64OKLT5MUSKCTZ25652676XA6/graph.json","events_json":"https://pith.science/api/pith-number/N64OKLT5MUSKCTZ25652676XA6/events.json","paper":"https://pith.science/paper/N64OKLT5"},"agent_actions":{"view_html":"https://pith.science/pith/N64OKLT5MUSKCTZ25652676XA6","download_json":"https://pith.science/pith/N64OKLT5MUSKCTZ25652676XA6.json","view_paper":"https://pith.science/paper/N64OKLT5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2102.01778&json=true","fetch_graph":"https://pith.science/api/pith-number/N64OKLT5MUSKCTZ25652676XA6/graph.json","fetch_events":"https://pith.science/api/pith-number/N64OKLT5MUSKCTZ25652676XA6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/N64OKLT5MUSKCTZ25652676XA6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/N64OKLT5MUSKCTZ25652676XA6/action/storage_attestation","attest_author":"https://pith.science/pith/N64OKLT5MUSKCTZ25652676XA6/action/author_attestation","sign_citation":"https://pith.science/pith/N64OKLT5MUSKCTZ25652676XA6/action/citation_signature","submit_replication":"https://pith.science/pith/N64OKLT5MUSKCTZ25652676XA6/action/replication_record"}},"created_at":"2026-07-05T04:10:11.723340+00:00","updated_at":"2026-07-05T04:10:11.723340+00:00"}