{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:FZHEYHPB25AULUSC2YMLQ2ALHR","short_pith_number":"pith:FZHEYHPB","canonical_record":{"source":{"id":"1301.1051","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-01-06T20:41:33Z","cross_cats_sorted":[],"title_canon_sha256":"3fe18f2762e145306afcffb2282a3533542816bebe9f65c89a1894297f2941f7","abstract_canon_sha256":"dbf8a089fcabe519e245d3e92bac1d36f53e03a48002bf6fa760b6457d23bb46"},"schema_version":"1.0"},"canonical_sha256":"2e4e4c1de1d74145d242d618b8680b3c5cd2146eb928fa2eda38082ab98eb4e3","source":{"kind":"arxiv","id":"1301.1051","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1301.1051","created_at":"2026-05-18T03:36:06Z"},{"alias_kind":"arxiv_version","alias_value":"1301.1051v2","created_at":"2026-05-18T03:36:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.1051","created_at":"2026-05-18T03:36:06Z"},{"alias_kind":"pith_short_12","alias_value":"FZHEYHPB25AU","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"FZHEYHPB25AULUSC","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"FZHEYHPB","created_at":"2026-05-18T12:27:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:FZHEYHPB25AULUSC2YMLQ2ALHR","target":"record","payload":{"canonical_record":{"source":{"id":"1301.1051","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-01-06T20:41:33Z","cross_cats_sorted":[],"title_canon_sha256":"3fe18f2762e145306afcffb2282a3533542816bebe9f65c89a1894297f2941f7","abstract_canon_sha256":"dbf8a089fcabe519e245d3e92bac1d36f53e03a48002bf6fa760b6457d23bb46"},"schema_version":"1.0"},"canonical_sha256":"2e4e4c1de1d74145d242d618b8680b3c5cd2146eb928fa2eda38082ab98eb4e3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:36:06.013708Z","signature_b64":"4zCXDBynukuxTgMpHVnZUiPvHzR0V8H2ebh3OXFNy08ipi/lM2Ji6dPVan1JcwqAybx5Cfu/LneQ7aElvkd0Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2e4e4c1de1d74145d242d618b8680b3c5cd2146eb928fa2eda38082ab98eb4e3","last_reissued_at":"2026-05-18T03:36:06.012923Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:36:06.012923Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1301.1051","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:36:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8DKndYTxlpPMEdoAYez3rtfsgaZvDRTqlCOujtQI1gTnoTfP6m3B3ITPmoWFhuiwR+5egQPmtD97mfnhjposAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T02:41:47.202486Z"},"content_sha256":"f7df8bd60faa2fa136f276ba0b17348e162ba7d1838e2241a8b35dd7e3af1f97","schema_version":"1.0","event_id":"sha256:f7df8bd60faa2fa136f276ba0b17348e162ba7d1838e2241a8b35dd7e3af1f97"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:FZHEYHPB25AULUSC2YMLQ2ALHR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On sharp aperture-weighted estimates for square functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Andrei K. Lerner","submitted_at":"2013-01-06T20:41:33Z","abstract_excerpt":"Let $S_{\\a,\\psi}(f)$ be the square function defined by means of the cone in ${\\mathbb R}^{n+1}_{+}$ of aperture $\\a$, and a standard kernel $\\psi$. Let $[w]_{A_p}$ denote the $A_p$ characteristic of the weight $w$. We show that for any $1<p<\\infty$ and $\\a\\ge 1$, $$\\|S_{\\a,\\psi}\\|_{L^p(w)}\\lesssim \\a^n[w]_{A_p}^{\\max(1/2,\\frac{1}{p-1})}.$$ For each fixed $\\a$ the dependence on $[w]_{A_p}$ is sharp. Also, on all class $A_p$ the result is sharp in $\\a$. Previously this estimate was proved in the case $\\a=1$ using the intrinsic square function. However, that approach does not allow to get the abo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.1051","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:36:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"prXrTiYDqZ9wsepdoPA1VG1KtvFn7D7bgki37EIqCu8Svl1GAmmNO7h3BEUJfo2mhK4+qV7IAvlPVBFO0t9SDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T02:41:47.202832Z"},"content_sha256":"df4cfaa801a87231668fe92b8f317b2ab3ce8ece05b99af85e376e0868555a56","schema_version":"1.0","event_id":"sha256:df4cfaa801a87231668fe92b8f317b2ab3ce8ece05b99af85e376e0868555a56"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FZHEYHPB25AULUSC2YMLQ2ALHR/bundle.json","state_url":"https://pith.science/pith/FZHEYHPB25AULUSC2YMLQ2ALHR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FZHEYHPB25AULUSC2YMLQ2ALHR/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-20T02:41:47Z","links":{"resolver":"https://pith.science/pith/FZHEYHPB25AULUSC2YMLQ2ALHR","bundle":"https://pith.science/pith/FZHEYHPB25AULUSC2YMLQ2ALHR/bundle.json","state":"https://pith.science/pith/FZHEYHPB25AULUSC2YMLQ2ALHR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FZHEYHPB25AULUSC2YMLQ2ALHR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:FZHEYHPB25AULUSC2YMLQ2ALHR","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"dbf8a089fcabe519e245d3e92bac1d36f53e03a48002bf6fa760b6457d23bb46","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-01-06T20:41:33Z","title_canon_sha256":"3fe18f2762e145306afcffb2282a3533542816bebe9f65c89a1894297f2941f7"},"schema_version":"1.0","source":{"id":"1301.1051","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1301.1051","created_at":"2026-05-18T03:36:06Z"},{"alias_kind":"arxiv_version","alias_value":"1301.1051v2","created_at":"2026-05-18T03:36:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.1051","created_at":"2026-05-18T03:36:06Z"},{"alias_kind":"pith_short_12","alias_value":"FZHEYHPB25AU","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"FZHEYHPB25AULUSC","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"FZHEYHPB","created_at":"2026-05-18T12:27:45Z"}],"graph_snapshots":[{"event_id":"sha256:df4cfaa801a87231668fe92b8f317b2ab3ce8ece05b99af85e376e0868555a56","target":"graph","created_at":"2026-05-18T03:36:06Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $S_{\\a,\\psi}(f)$ be the square function defined by means of the cone in ${\\mathbb R}^{n+1}_{+}$ of aperture $\\a$, and a standard kernel $\\psi$. Let $[w]_{A_p}$ denote the $A_p$ characteristic of the weight $w$. We show that for any $1<p<\\infty$ and $\\a\\ge 1$, $$\\|S_{\\a,\\psi}\\|_{L^p(w)}\\lesssim \\a^n[w]_{A_p}^{\\max(1/2,\\frac{1}{p-1})}.$$ For each fixed $\\a$ the dependence on $[w]_{A_p}$ is sharp. Also, on all class $A_p$ the result is sharp in $\\a$. Previously this estimate was proved in the case $\\a=1$ using the intrinsic square function. However, that approach does not allow to get the abo","authors_text":"Andrei K. Lerner","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-01-06T20:41:33Z","title":"On sharp aperture-weighted estimates for square functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.1051","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f7df8bd60faa2fa136f276ba0b17348e162ba7d1838e2241a8b35dd7e3af1f97","target":"record","created_at":"2026-05-18T03:36:06Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"dbf8a089fcabe519e245d3e92bac1d36f53e03a48002bf6fa760b6457d23bb46","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-01-06T20:41:33Z","title_canon_sha256":"3fe18f2762e145306afcffb2282a3533542816bebe9f65c89a1894297f2941f7"},"schema_version":"1.0","source":{"id":"1301.1051","kind":"arxiv","version":2}},"canonical_sha256":"2e4e4c1de1d74145d242d618b8680b3c5cd2146eb928fa2eda38082ab98eb4e3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2e4e4c1de1d74145d242d618b8680b3c5cd2146eb928fa2eda38082ab98eb4e3","first_computed_at":"2026-05-18T03:36:06.012923Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:36:06.012923Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4zCXDBynukuxTgMpHVnZUiPvHzR0V8H2ebh3OXFNy08ipi/lM2Ji6dPVan1JcwqAybx5Cfu/LneQ7aElvkd0Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:36:06.013708Z","signed_message":"canonical_sha256_bytes"},"source_id":"1301.1051","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f7df8bd60faa2fa136f276ba0b17348e162ba7d1838e2241a8b35dd7e3af1f97","sha256:df4cfaa801a87231668fe92b8f317b2ab3ce8ece05b99af85e376e0868555a56"],"state_sha256":"afe1dcafad1eeb7391f45408bb12966d741cadcd46aa5f7180e3390c65c13a27"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I75Ux20kFo+Q+7qn/x0MdM6658UCzR7p+xLVBXMZgjl8K5qSgdC1JaRGJMAIrAL2r22daE8bNp03YK3pfy9hAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T02:41:47.204768Z","bundle_sha256":"bc6c3a6e0b103c9d8ff53c03c4d0a6dc176c46b73be27fca12f4eaa5bcdae526"}}