{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:CDMGG4MU4FE6KJ4YAIDF2HYCOY","short_pith_number":"pith:CDMGG4MU","canonical_record":{"source":{"id":"1710.05217","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-10-14T17:55:07Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"a59311ec2ef2f9d475ce4ddb1eeab78c7926c80b4feebb4644e9273447ba19d4","abstract_canon_sha256":"6c2749dacd56d2c37966c71c5903fa0d05354e261cc46c48cabb55fb74aa70d7"},"schema_version":"1.0"},"canonical_sha256":"10d8637194e149e5279802065d1f02763b330ad4476560f3c47ec740c1d4bedf","source":{"kind":"arxiv","id":"1710.05217","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1710.05217","created_at":"2026-05-18T00:32:25Z"},{"alias_kind":"arxiv_version","alias_value":"1710.05217v2","created_at":"2026-05-18T00:32:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.05217","created_at":"2026-05-18T00:32:25Z"},{"alias_kind":"pith_short_12","alias_value":"CDMGG4MU4FE6","created_at":"2026-05-18T12:31:10Z"},{"alias_kind":"pith_short_16","alias_value":"CDMGG4MU4FE6KJ4Y","created_at":"2026-05-18T12:31:10Z"},{"alias_kind":"pith_short_8","alias_value":"CDMGG4MU","created_at":"2026-05-18T12:31:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:CDMGG4MU4FE6KJ4YAIDF2HYCOY","target":"record","payload":{"canonical_record":{"source":{"id":"1710.05217","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-10-14T17:55:07Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"a59311ec2ef2f9d475ce4ddb1eeab78c7926c80b4feebb4644e9273447ba19d4","abstract_canon_sha256":"6c2749dacd56d2c37966c71c5903fa0d05354e261cc46c48cabb55fb74aa70d7"},"schema_version":"1.0"},"canonical_sha256":"10d8637194e149e5279802065d1f02763b330ad4476560f3c47ec740c1d4bedf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:25.374748Z","signature_b64":"pQEm6W9/WyHvJEN4q7MsboGurK8J5NbnYASm594PNvGSRDC35/0nXZ3T4W/nWMEIc4aj7FUrZYV2KKhyD6CWDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"10d8637194e149e5279802065d1f02763b330ad4476560f3c47ec740c1d4bedf","last_reissued_at":"2026-05-18T00:32:25.374094Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:25.374094Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1710.05217","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":"2026-05-18T00:32:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gt4ANyE3GyaoOzD1sJY2WJQsgSP/1aamLKafb6ojbzPZP87ZpF/Kui4D4Wwj6v/4ij8BjxXAF/c+7bmkn36BDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T05:38:36.108500Z"},"content_sha256":"93bc8e8d0bfad00a378f34d3f9555a5ce8d50ddb233c4f9ba2286d3a863ec49d","schema_version":"1.0","event_id":"sha256:93bc8e8d0bfad00a378f34d3f9555a5ce8d50ddb233c4f9ba2286d3a863ec49d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:CDMGG4MU4FE6KJ4YAIDF2HYCOY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Modular inequalities for the maximal operator in variable Lebesgue spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Alberto Fiorenza, David Cruz-Uribe, Giovanni Di Fratta","submitted_at":"2017-10-14T17:55:07Z","abstract_excerpt":"A now classical result in the theory of variable Lebesgue spaces due to Lerner [A. K. Lerner, On modular inequalities in variable $L^p$ spaces, Archiv der Math. 85 (2005), no. 6, 538-543] is that a modular inequality for the Hardy-Littlewood maximal function in $L^{p(\\cdot)}(\\mathbb{R}^n)$ holds if and only if the exponent is constant. We generalize this result and give a new and simpler proof. We then find necessary and sufficient conditions for the validity of the weaker modular inequality \\[ \\int_\\Omega Mf(x)^{p(x)}\\,dx \\ \\leq c_1 \\int_\\Omega |f(x)|^{q(x)}\\,dx + c_2, \\] where $c_1,\\,c_2$ ar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.05217","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":"2026-05-18T00:32:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BpmA/l/Pgo2BZoqu8Sfp4YQivi/GJHmCFkrReiUQhq+6cB/ZalsAEOWNBhpgP6TmUqJjo/DzKZmPmBtBRA5GAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T05:38:36.108849Z"},"content_sha256":"e84512273578e484cd63063d5a9eebdbfbf663961f055935669339ff731e6dca","schema_version":"1.0","event_id":"sha256:e84512273578e484cd63063d5a9eebdbfbf663961f055935669339ff731e6dca"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CDMGG4MU4FE6KJ4YAIDF2HYCOY/bundle.json","state_url":"https://pith.science/pith/CDMGG4MU4FE6KJ4YAIDF2HYCOY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CDMGG4MU4FE6KJ4YAIDF2HYCOY/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":"2026-05-31T05:38:36Z","links":{"resolver":"https://pith.science/pith/CDMGG4MU4FE6KJ4YAIDF2HYCOY","bundle":"https://pith.science/pith/CDMGG4MU4FE6KJ4YAIDF2HYCOY/bundle.json","state":"https://pith.science/pith/CDMGG4MU4FE6KJ4YAIDF2HYCOY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CDMGG4MU4FE6KJ4YAIDF2HYCOY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:CDMGG4MU4FE6KJ4YAIDF2HYCOY","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":"6c2749dacd56d2c37966c71c5903fa0d05354e261cc46c48cabb55fb74aa70d7","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-10-14T17:55:07Z","title_canon_sha256":"a59311ec2ef2f9d475ce4ddb1eeab78c7926c80b4feebb4644e9273447ba19d4"},"schema_version":"1.0","source":{"id":"1710.05217","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1710.05217","created_at":"2026-05-18T00:32:25Z"},{"alias_kind":"arxiv_version","alias_value":"1710.05217v2","created_at":"2026-05-18T00:32:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.05217","created_at":"2026-05-18T00:32:25Z"},{"alias_kind":"pith_short_12","alias_value":"CDMGG4MU4FE6","created_at":"2026-05-18T12:31:10Z"},{"alias_kind":"pith_short_16","alias_value":"CDMGG4MU4FE6KJ4Y","created_at":"2026-05-18T12:31:10Z"},{"alias_kind":"pith_short_8","alias_value":"CDMGG4MU","created_at":"2026-05-18T12:31:10Z"}],"graph_snapshots":[{"event_id":"sha256:e84512273578e484cd63063d5a9eebdbfbf663961f055935669339ff731e6dca","target":"graph","created_at":"2026-05-18T00:32:25Z","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":"A now classical result in the theory of variable Lebesgue spaces due to Lerner [A. K. Lerner, On modular inequalities in variable $L^p$ spaces, Archiv der Math. 85 (2005), no. 6, 538-543] is that a modular inequality for the Hardy-Littlewood maximal function in $L^{p(\\cdot)}(\\mathbb{R}^n)$ holds if and only if the exponent is constant. We generalize this result and give a new and simpler proof. We then find necessary and sufficient conditions for the validity of the weaker modular inequality \\[ \\int_\\Omega Mf(x)^{p(x)}\\,dx \\ \\leq c_1 \\int_\\Omega |f(x)|^{q(x)}\\,dx + c_2, \\] where $c_1,\\,c_2$ ar","authors_text":"Alberto Fiorenza, David Cruz-Uribe, Giovanni Di Fratta","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-10-14T17:55:07Z","title":"Modular inequalities for the maximal operator in variable Lebesgue spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.05217","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":"sha256:93bc8e8d0bfad00a378f34d3f9555a5ce8d50ddb233c4f9ba2286d3a863ec49d","target":"record","created_at":"2026-05-18T00:32:25Z","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":"6c2749dacd56d2c37966c71c5903fa0d05354e261cc46c48cabb55fb74aa70d7","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-10-14T17:55:07Z","title_canon_sha256":"a59311ec2ef2f9d475ce4ddb1eeab78c7926c80b4feebb4644e9273447ba19d4"},"schema_version":"1.0","source":{"id":"1710.05217","kind":"arxiv","version":2}},"canonical_sha256":"10d8637194e149e5279802065d1f02763b330ad4476560f3c47ec740c1d4bedf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"10d8637194e149e5279802065d1f02763b330ad4476560f3c47ec740c1d4bedf","first_computed_at":"2026-05-18T00:32:25.374094Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:32:25.374094Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pQEm6W9/WyHvJEN4q7MsboGurK8J5NbnYASm594PNvGSRDC35/0nXZ3T4W/nWMEIc4aj7FUrZYV2KKhyD6CWDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:32:25.374748Z","signed_message":"canonical_sha256_bytes"},"source_id":"1710.05217","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:93bc8e8d0bfad00a378f34d3f9555a5ce8d50ddb233c4f9ba2286d3a863ec49d","sha256:e84512273578e484cd63063d5a9eebdbfbf663961f055935669339ff731e6dca"],"state_sha256":"1aeebe6111a359343fde5cdcc27eff5bcd2ea5b23667df76e6d4bb7b2d536968"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+8lMepz5i5EMwxWn4j2nKpTa4jLP1FaetV2038U9MZMV1+eX4j54XulSM7oaM9L7iHYcQLpGpuCYIVij9lJTBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T05:38:36.110999Z","bundle_sha256":"3eb93d90d1f761f8bd0be739876f2506f8c04a28dec97a43b9b6e663d575ed87"}}