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Specifically, we show that if $Q$ is a cube in $\\mathbb{R}^n$ and $h:[0,\\infty)\\to[0,\\infty)$ is such that $h(t)\\underset{t\\to\\infty}{\\longrightarrow}\\infty,$ then\n  $$ \\sup_{J \\text{subcube} Q} \\frac1{|J|}\\int_J h(|\\varphi-\\frac1{|J|} \\int_J\\varphi |)<\\infty \\Longrightarrow \\varphi\\in BMO(Q). $$\n  Under some additional assumptions on $h$ we obtain estimates on $\\|\\varphi\\|_{BMO}$ in terms of the supremum above. We also show that even though the condition $h(t)\\underset{t\\to\\infty}{\\longrightarrow}\\infty$ is not ne"},"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":"1309.6780","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-09-26T10:05:42Z","cross_cats_sorted":[],"title_canon_sha256":"a7cf4c248a92b346d46aa4dc63fa644885d165df0bc3a36ba9582989074bfa4d","abstract_canon_sha256":"f5244d49f79e19ea7d680c8e80909e0e6d54fe65d61cb067de89faf8a7484f7e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:02:17.367152Z","signature_b64":"XpI3gpOQaAV5ujXueKFWaiXD3CNoGErpGuyl9sGiOpPzEbLS3NcwzSUhaEhtv6AefN2wnlQmHXPd1tOp2d7wCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"922e9b298d3ea7032fa984e35c35e67a37f61175bbacab86080ca22b823336ec","last_reissued_at":"2026-05-18T03:02:17.366433Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:02:17.366433Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weak integral conditions for BMO","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Alexander A. Logunov, Dmitriy M. Stolyarov, Leonid Slavin, Pavel B. Zatitskiy, Vasily Vasyunin","submitted_at":"2013-09-26T10:05:42Z","abstract_excerpt":"We study the question of how much one can weaken the defining condition of BMO. Specifically, we show that if $Q$ is a cube in $\\mathbb{R}^n$ and $h:[0,\\infty)\\to[0,\\infty)$ is such that $h(t)\\underset{t\\to\\infty}{\\longrightarrow}\\infty,$ then\n  $$ \\sup_{J \\text{subcube} Q} \\frac1{|J|}\\int_J h(|\\varphi-\\frac1{|J|} \\int_J\\varphi |)<\\infty \\Longrightarrow \\varphi\\in BMO(Q). $$\n  Under some additional assumptions on $h$ we obtain estimates on $\\|\\varphi\\|_{BMO}$ in terms of the supremum above. We also show that even though the condition $h(t)\\underset{t\\to\\infty}{\\longrightarrow}\\infty$ is not ne"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.6780","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":""},"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":"1309.6780","created_at":"2026-05-18T03:02:17.366544+00:00"},{"alias_kind":"arxiv_version","alias_value":"1309.6780v2","created_at":"2026-05-18T03:02:17.366544+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.6780","created_at":"2026-05-18T03:02:17.366544+00:00"},{"alias_kind":"pith_short_12","alias_value":"SIXJWKMNH2TQ","created_at":"2026-05-18T12:27:59.945178+00:00"},{"alias_kind":"pith_short_16","alias_value":"SIXJWKMNH2TQGL5J","created_at":"2026-05-18T12:27:59.945178+00:00"},{"alias_kind":"pith_short_8","alias_value":"SIXJWKMN","created_at":"2026-05-18T12:27:59.945178+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/SIXJWKMNH2TQGL5JQTRVYNPGPI","json":"https://pith.science/pith/SIXJWKMNH2TQGL5JQTRVYNPGPI.json","graph_json":"https://pith.science/api/pith-number/SIXJWKMNH2TQGL5JQTRVYNPGPI/graph.json","events_json":"https://pith.science/api/pith-number/SIXJWKMNH2TQGL5JQTRVYNPGPI/events.json","paper":"https://pith.science/paper/SIXJWKMN"},"agent_actions":{"view_html":"https://pith.science/pith/SIXJWKMNH2TQGL5JQTRVYNPGPI","download_json":"https://pith.science/pith/SIXJWKMNH2TQGL5JQTRVYNPGPI.json","view_paper":"https://pith.science/paper/SIXJWKMN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1309.6780&json=true","fetch_graph":"https://pith.science/api/pith-number/SIXJWKMNH2TQGL5JQTRVYNPGPI/graph.json","fetch_events":"https://pith.science/api/pith-number/SIXJWKMNH2TQGL5JQTRVYNPGPI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SIXJWKMNH2TQGL5JQTRVYNPGPI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SIXJWKMNH2TQGL5JQTRVYNPGPI/action/storage_attestation","attest_author":"https://pith.science/pith/SIXJWKMNH2TQGL5JQTRVYNPGPI/action/author_attestation","sign_citation":"https://pith.science/pith/SIXJWKMNH2TQGL5JQTRVYNPGPI/action/citation_signature","submit_replication":"https://pith.science/pith/SIXJWKMNH2TQGL5JQTRVYNPGPI/action/replication_record"}},"created_at":"2026-05-18T03:02:17.366544+00:00","updated_at":"2026-05-18T03:02:17.366544+00:00"}