{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:C4VIMXWHAT43CWRZJSCPOBLX27","short_pith_number":"pith:C4VIMXWH","schema_version":"1.0","canonical_sha256":"172a865ec704f9b15a394c84f70577d7c260f207697177c73eb511bb6f3e61b4","source":{"kind":"arxiv","id":"1408.1213","version":3},"attestation_state":"computed","paper":{"title":"The Maximal Function and Square Function Control the Variation: An Elementary Proof","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CA","authors_text":"Bartosz Trojan, Ben Krause, Kevin Hughes","submitted_at":"2014-08-06T08:45:48Z","abstract_excerpt":"In this note we prove the following good-$\\lambda$ inequality, for $r>2$, all $\\lambda > 0$, $\\delta \\in \\big(0, \\frac{1}{2} \\big)$ \\[ \\nu\\big\\{ V_r(f) > 3 \\lambda ; \\mathcal{M}(f) \\leq \\delta \\lambda\\big\\} \\leq 4 \\nu\\{s(f) > \\delta \\lambda\\} + {\\delta^2 \\left(1+\\frac{16}{r-2}\\right)^2} \\cdot \\nu\\big\\{ V_r(f) > \\lambda\\big\\}, \\] where $\\mathcal{M}(f)$ is the martingale maximal function, $s(f)$ is the conditional martingale square function. This immediately proves that $V_r(f)$ is bounded on $L^p$, $1 < p <\\infty$ and moreover is integrable when the maximal function is."},"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":"1408.1213","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-08-06T08:45:48Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"1482528c16db5f1a1650255acac204fb17ede29c52fca6e477d2bbc3f65037f0","abstract_canon_sha256":"ab98a62aa35e753db45594d6dd4f160a29a92944fce105ddb5d2ad441ea8c53d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:41.812617Z","signature_b64":"eOnG1r9GrcgQbrmO3Z180LmKfQH5/+HbdB1WpQ/9pmokk9yoVWeDJP3H6ASyCbvM5VvU3wZLCghdJjGiHpZNDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"172a865ec704f9b15a394c84f70577d7c260f207697177c73eb511bb6f3e61b4","last_reissued_at":"2026-05-18T01:32:41.811892Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:41.811892Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Maximal Function and Square Function Control the Variation: An Elementary Proof","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CA","authors_text":"Bartosz Trojan, Ben Krause, Kevin Hughes","submitted_at":"2014-08-06T08:45:48Z","abstract_excerpt":"In this note we prove the following good-$\\lambda$ inequality, for $r>2$, all $\\lambda > 0$, $\\delta \\in \\big(0, \\frac{1}{2} \\big)$ \\[ \\nu\\big\\{ V_r(f) > 3 \\lambda ; \\mathcal{M}(f) \\leq \\delta \\lambda\\big\\} \\leq 4 \\nu\\{s(f) > \\delta \\lambda\\} + {\\delta^2 \\left(1+\\frac{16}{r-2}\\right)^2} \\cdot \\nu\\big\\{ V_r(f) > \\lambda\\big\\}, \\] where $\\mathcal{M}(f)$ is the martingale maximal function, $s(f)$ is the conditional martingale square function. This immediately proves that $V_r(f)$ is bounded on $L^p$, $1 < p <\\infty$ and moreover is integrable when the maximal function is."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.1213","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1408.1213","created_at":"2026-05-18T01:32:41.812000+00:00"},{"alias_kind":"arxiv_version","alias_value":"1408.1213v3","created_at":"2026-05-18T01:32:41.812000+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.1213","created_at":"2026-05-18T01:32:41.812000+00:00"},{"alias_kind":"pith_short_12","alias_value":"C4VIMXWHAT43","created_at":"2026-05-18T12:28:22.404517+00:00"},{"alias_kind":"pith_short_16","alias_value":"C4VIMXWHAT43CWRZ","created_at":"2026-05-18T12:28:22.404517+00:00"},{"alias_kind":"pith_short_8","alias_value":"C4VIMXWH","created_at":"2026-05-18T12:28:22.404517+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/C4VIMXWHAT43CWRZJSCPOBLX27","json":"https://pith.science/pith/C4VIMXWHAT43CWRZJSCPOBLX27.json","graph_json":"https://pith.science/api/pith-number/C4VIMXWHAT43CWRZJSCPOBLX27/graph.json","events_json":"https://pith.science/api/pith-number/C4VIMXWHAT43CWRZJSCPOBLX27/events.json","paper":"https://pith.science/paper/C4VIMXWH"},"agent_actions":{"view_html":"https://pith.science/pith/C4VIMXWHAT43CWRZJSCPOBLX27","download_json":"https://pith.science/pith/C4VIMXWHAT43CWRZJSCPOBLX27.json","view_paper":"https://pith.science/paper/C4VIMXWH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1408.1213&json=true","fetch_graph":"https://pith.science/api/pith-number/C4VIMXWHAT43CWRZJSCPOBLX27/graph.json","fetch_events":"https://pith.science/api/pith-number/C4VIMXWHAT43CWRZJSCPOBLX27/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/C4VIMXWHAT43CWRZJSCPOBLX27/action/timestamp_anchor","attest_storage":"https://pith.science/pith/C4VIMXWHAT43CWRZJSCPOBLX27/action/storage_attestation","attest_author":"https://pith.science/pith/C4VIMXWHAT43CWRZJSCPOBLX27/action/author_attestation","sign_citation":"https://pith.science/pith/C4VIMXWHAT43CWRZJSCPOBLX27/action/citation_signature","submit_replication":"https://pith.science/pith/C4VIMXWHAT43CWRZJSCPOBLX27/action/replication_record"}},"created_at":"2026-05-18T01:32:41.812000+00:00","updated_at":"2026-05-18T01:32:41.812000+00:00"}