{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:G4OCPZ2DPHENA2MQZXPWI5WYV3","short_pith_number":"pith:G4OCPZ2D","canonical_record":{"source":{"id":"1602.04585","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-15T08:04:26Z","cross_cats_sorted":[],"title_canon_sha256":"6cb81c8224a18c5ce7c8eafbd5dbbf615af360405e38ef79793d2651911ef0e6","abstract_canon_sha256":"2f35de394721fb3a0eb2c6209cd2690080f8efa7b8b1ae6bea03664183c1ea6d"},"schema_version":"1.0"},"canonical_sha256":"371c27e74379c8d06990cddf6476d8aec770907d210f6914e2ee6219642e34f6","source":{"kind":"arxiv","id":"1602.04585","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.04585","created_at":"2026-05-18T01:20:49Z"},{"alias_kind":"arxiv_version","alias_value":"1602.04585v1","created_at":"2026-05-18T01:20:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.04585","created_at":"2026-05-18T01:20:49Z"},{"alias_kind":"pith_short_12","alias_value":"G4OCPZ2DPHEN","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"G4OCPZ2DPHENA2MQ","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"G4OCPZ2D","created_at":"2026-05-18T12:30:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:G4OCPZ2DPHENA2MQZXPWI5WYV3","target":"record","payload":{"canonical_record":{"source":{"id":"1602.04585","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-15T08:04:26Z","cross_cats_sorted":[],"title_canon_sha256":"6cb81c8224a18c5ce7c8eafbd5dbbf615af360405e38ef79793d2651911ef0e6","abstract_canon_sha256":"2f35de394721fb3a0eb2c6209cd2690080f8efa7b8b1ae6bea03664183c1ea6d"},"schema_version":"1.0"},"canonical_sha256":"371c27e74379c8d06990cddf6476d8aec770907d210f6914e2ee6219642e34f6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:20:49.598595Z","signature_b64":"O0swU2k/iTAj1nCWohmraljXxzVBR0eyWDs8Cg5jhtHm/4f39U+CEJMqMwYjb+FLEzeE+/IC9XfYYFkU4mq6Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"371c27e74379c8d06990cddf6476d8aec770907d210f6914e2ee6219642e34f6","last_reissued_at":"2026-05-18T01:20:49.598136Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:20:49.598136Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1602.04585","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:20:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V1NZDr79P5deJig/3m0MqQRIze9/cWUDDIUTx6LoKyqs+G5m9a2WaMiqKrPYmHJiFwjCbTpNRW9i5gn8WpdIAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T10:12:56.378288Z"},"content_sha256":"db591b89c2de1e9184ddad7ea168ba0a17958b9bd5c0836c31f83b737e6bf314","schema_version":"1.0","event_id":"sha256:db591b89c2de1e9184ddad7ea168ba0a17958b9bd5c0836c31f83b737e6bf314"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:G4OCPZ2DPHENA2MQZXPWI5WYV3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Extremal function for Moser-Trudinger type Inequality with Logarithmic weight","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Prosenjit Roy","submitted_at":"2016-02-15T08:04:26Z","abstract_excerpt":"On the space of weighted radial Sobolev space, the following generalization of Moser-Trudinger type inequality was established by Calanchi and Ruf in dimension 2 : If $\\beta \\in [0,1)$ and $w_0(x) = |\\log |x||^\\beta $ then $$ \\sup_{\\int_B |\\grad u|^2w_0 \\leq 1 , u \\in H_{0,rad}^1(w_0,B)} \\int_B e^{\\alpha u^{\\frac{2}{1-\\beta}}} dx < \\infty,$$ if and only if $\\alpha \\leq \\alpha_\\beta = 2\\left[2\\pi (1-\\beta) \\right]^{\\frac{1}{1-\\beta}}.$ We prove the existence of an extremal function for the above inequality for the critical case when $\\alpha = \\alpha_\\beta$ thereby generalizing the result of Car"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.04585","kind":"arxiv","version":1},"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":"2026-05-18T01:20:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JkWH9/FIDsJ6sOwFwVFA5CI+un+btY0C2reEJGaSwthyceFUFQI9xLifvm2G4WzBl8psqg1ffOtZdRGgWHwqBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T10:12:56.378850Z"},"content_sha256":"1bb225757ce8ade240cfb609115e1d77bc5c8ab4f791f4438777bfdd3b328afa","schema_version":"1.0","event_id":"sha256:1bb225757ce8ade240cfb609115e1d77bc5c8ab4f791f4438777bfdd3b328afa"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/G4OCPZ2DPHENA2MQZXPWI5WYV3/bundle.json","state_url":"https://pith.science/pith/G4OCPZ2DPHENA2MQZXPWI5WYV3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/G4OCPZ2DPHENA2MQZXPWI5WYV3/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":"2026-06-07T10:12:56Z","links":{"resolver":"https://pith.science/pith/G4OCPZ2DPHENA2MQZXPWI5WYV3","bundle":"https://pith.science/pith/G4OCPZ2DPHENA2MQZXPWI5WYV3/bundle.json","state":"https://pith.science/pith/G4OCPZ2DPHENA2MQZXPWI5WYV3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/G4OCPZ2DPHENA2MQZXPWI5WYV3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:G4OCPZ2DPHENA2MQZXPWI5WYV3","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":"2f35de394721fb3a0eb2c6209cd2690080f8efa7b8b1ae6bea03664183c1ea6d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-15T08:04:26Z","title_canon_sha256":"6cb81c8224a18c5ce7c8eafbd5dbbf615af360405e38ef79793d2651911ef0e6"},"schema_version":"1.0","source":{"id":"1602.04585","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.04585","created_at":"2026-05-18T01:20:49Z"},{"alias_kind":"arxiv_version","alias_value":"1602.04585v1","created_at":"2026-05-18T01:20:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.04585","created_at":"2026-05-18T01:20:49Z"},{"alias_kind":"pith_short_12","alias_value":"G4OCPZ2DPHEN","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"G4OCPZ2DPHENA2MQ","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"G4OCPZ2D","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:1bb225757ce8ade240cfb609115e1d77bc5c8ab4f791f4438777bfdd3b328afa","target":"graph","created_at":"2026-05-18T01:20:49Z","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":"On the space of weighted radial Sobolev space, the following generalization of Moser-Trudinger type inequality was established by Calanchi and Ruf in dimension 2 : If $\\beta \\in [0,1)$ and $w_0(x) = |\\log |x||^\\beta $ then $$ \\sup_{\\int_B |\\grad u|^2w_0 \\leq 1 , u \\in H_{0,rad}^1(w_0,B)} \\int_B e^{\\alpha u^{\\frac{2}{1-\\beta}}} dx < \\infty,$$ if and only if $\\alpha \\leq \\alpha_\\beta = 2\\left[2\\pi (1-\\beta) \\right]^{\\frac{1}{1-\\beta}}.$ We prove the existence of an extremal function for the above inequality for the critical case when $\\alpha = \\alpha_\\beta$ thereby generalizing the result of Car","authors_text":"Prosenjit Roy","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-15T08:04:26Z","title":"Extremal function for Moser-Trudinger type Inequality with Logarithmic weight"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.04585","kind":"arxiv","version":1},"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:db591b89c2de1e9184ddad7ea168ba0a17958b9bd5c0836c31f83b737e6bf314","target":"record","created_at":"2026-05-18T01:20:49Z","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":"2f35de394721fb3a0eb2c6209cd2690080f8efa7b8b1ae6bea03664183c1ea6d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-15T08:04:26Z","title_canon_sha256":"6cb81c8224a18c5ce7c8eafbd5dbbf615af360405e38ef79793d2651911ef0e6"},"schema_version":"1.0","source":{"id":"1602.04585","kind":"arxiv","version":1}},"canonical_sha256":"371c27e74379c8d06990cddf6476d8aec770907d210f6914e2ee6219642e34f6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"371c27e74379c8d06990cddf6476d8aec770907d210f6914e2ee6219642e34f6","first_computed_at":"2026-05-18T01:20:49.598136Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:20:49.598136Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"O0swU2k/iTAj1nCWohmraljXxzVBR0eyWDs8Cg5jhtHm/4f39U+CEJMqMwYjb+FLEzeE+/IC9XfYYFkU4mq6Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:20:49.598595Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.04585","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:db591b89c2de1e9184ddad7ea168ba0a17958b9bd5c0836c31f83b737e6bf314","sha256:1bb225757ce8ade240cfb609115e1d77bc5c8ab4f791f4438777bfdd3b328afa"],"state_sha256":"d28b85e488b74928d714f4021aa6eabac8c0f236d13610aceb1e52418eb0d2ed"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UCLDx1uN0gfHKx+zwPhJJy5nebp7//LeLkk93USkJmXJ+tppJsgV8ICuraR7I8vU33QU2dI9g5dPBksVkl17AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T10:12:56.381785Z","bundle_sha256":"47144b2e90d0b125868a241109353d1e214204fd05a19a52c87f74ab1a2f3a00"}}