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Consider the set $H_M$ of all functions $u$ harmonic in $P:=\\{(x,y)\\in \\mathbb{R}^n: x\\in \\mathbb{R}^{n-1}, y\\in \\mathbb{R}, |x|<1, |y|<1 \\}$ and satisfying $|u(x,y)| \\leq M(|y|)$. We prove that $H_M$ is a normal family in $P$."},"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":"1407.4934","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-07-18T10:21:51Z","cross_cats_sorted":["math.CA","math.CV"],"title_canon_sha256":"d14b0069f7b4a3305be1fbc06970052ecd4bcecfb0e573724c47072c06d89e2e","abstract_canon_sha256":"1f9fed58872290d8c0f37868c4d737fcaf7bee64e61e00e7f7cba1ffe7e1aeff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:45:55.536109Z","signature_b64":"LI9oLRJXH072NbFoRO/9CUuZJcnxC0rY572Fc8gfvkvhUD5FLHDs70xkbXtCBdb+nUKPCOGfw0eeiWeFydYWDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95bee854cd8db12aeec0beb3c60ac510bf6b360d6044785cacfcecd9568ce48a","last_reissued_at":"2026-05-18T02:45:55.535520Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:45:55.535520Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the higher-dimensional harmonic analog of the Levinson log log theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.CV"],"primary_cat":"math.AP","authors_text":"Alexander Logunov","submitted_at":"2014-07-18T10:21:51Z","abstract_excerpt":"Let $M\\colon (0,1) \\to [e,+\\infty)$ be a decreasing function such that $\\int\\limits_{0}^{1}\\log\\log M(y)dy<+\\infty$. Consider the set $H_M$ of all functions $u$ harmonic in $P:=\\{(x,y)\\in \\mathbb{R}^n: x\\in \\mathbb{R}^{n-1}, y\\in \\mathbb{R}, |x|<1, |y|<1 \\}$ and satisfying $|u(x,y)| \\leq M(|y|)$. We prove that $H_M$ is a normal family in $P$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.4934","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":"1407.4934","created_at":"2026-05-18T02:45:55.535619+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.4934v2","created_at":"2026-05-18T02:45:55.535619+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.4934","created_at":"2026-05-18T02:45:55.535619+00:00"},{"alias_kind":"pith_short_12","alias_value":"SW7OQVGNRWYS","created_at":"2026-05-18T12:28:49.207871+00:00"},{"alias_kind":"pith_short_16","alias_value":"SW7OQVGNRWYSV3WA","created_at":"2026-05-18T12:28:49.207871+00:00"},{"alias_kind":"pith_short_8","alias_value":"SW7OQVGN","created_at":"2026-05-18T12:28:49.207871+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/SW7OQVGNRWYSV3WAX2Z4MCWFCC","json":"https://pith.science/pith/SW7OQVGNRWYSV3WAX2Z4MCWFCC.json","graph_json":"https://pith.science/api/pith-number/SW7OQVGNRWYSV3WAX2Z4MCWFCC/graph.json","events_json":"https://pith.science/api/pith-number/SW7OQVGNRWYSV3WAX2Z4MCWFCC/events.json","paper":"https://pith.science/paper/SW7OQVGN"},"agent_actions":{"view_html":"https://pith.science/pith/SW7OQVGNRWYSV3WAX2Z4MCWFCC","download_json":"https://pith.science/pith/SW7OQVGNRWYSV3WAX2Z4MCWFCC.json","view_paper":"https://pith.science/paper/SW7OQVGN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.4934&json=true","fetch_graph":"https://pith.science/api/pith-number/SW7OQVGNRWYSV3WAX2Z4MCWFCC/graph.json","fetch_events":"https://pith.science/api/pith-number/SW7OQVGNRWYSV3WAX2Z4MCWFCC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SW7OQVGNRWYSV3WAX2Z4MCWFCC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SW7OQVGNRWYSV3WAX2Z4MCWFCC/action/storage_attestation","attest_author":"https://pith.science/pith/SW7OQVGNRWYSV3WAX2Z4MCWFCC/action/author_attestation","sign_citation":"https://pith.science/pith/SW7OQVGNRWYSV3WAX2Z4MCWFCC/action/citation_signature","submit_replication":"https://pith.science/pith/SW7OQVGNRWYSV3WAX2Z4MCWFCC/action/replication_record"}},"created_at":"2026-05-18T02:45:55.535619+00:00","updated_at":"2026-05-18T02:45:55.535619+00:00"}