{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:FELH6AGE6E5ASD6NNFJSWVPLNJ","short_pith_number":"pith:FELH6AGE","canonical_record":{"source":{"id":"1609.05278","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-09-17T03:47:59Z","cross_cats_sorted":[],"title_canon_sha256":"6108c77cb4f84f3748fd75b43fc9926aa215edfdf4f629e9d522857844f2864c","abstract_canon_sha256":"111ad561fe59de5bc9ff75d4b45faab8973dfa3249f4a29ee80e46b931d05939"},"schema_version":"1.0"},"canonical_sha256":"29167f00c4f13a090fcd69532b55eb6a40519c0160424d6f1475282008c293c3","source":{"kind":"arxiv","id":"1609.05278","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.05278","created_at":"2026-05-18T01:04:26Z"},{"alias_kind":"arxiv_version","alias_value":"1609.05278v1","created_at":"2026-05-18T01:04:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.05278","created_at":"2026-05-18T01:04:26Z"},{"alias_kind":"pith_short_12","alias_value":"FELH6AGE6E5A","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FELH6AGE6E5ASD6N","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FELH6AGE","created_at":"2026-05-18T12:30:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:FELH6AGE6E5ASD6NNFJSWVPLNJ","target":"record","payload":{"canonical_record":{"source":{"id":"1609.05278","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-09-17T03:47:59Z","cross_cats_sorted":[],"title_canon_sha256":"6108c77cb4f84f3748fd75b43fc9926aa215edfdf4f629e9d522857844f2864c","abstract_canon_sha256":"111ad561fe59de5bc9ff75d4b45faab8973dfa3249f4a29ee80e46b931d05939"},"schema_version":"1.0"},"canonical_sha256":"29167f00c4f13a090fcd69532b55eb6a40519c0160424d6f1475282008c293c3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:26.878539Z","signature_b64":"wlDenQ5+6MaHeKJlGW9XeplzIktoaXAVx6tSB96Obxcabkk5WaWdcZYy4u4swMHsIiJLxqzm1BS/nXInu7PDDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"29167f00c4f13a090fcd69532b55eb6a40519c0160424d6f1475282008c293c3","last_reissued_at":"2026-05-18T01:04:26.877797Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:26.877797Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1609.05278","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":"2026-05-18T01:04:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6SdzJayEv9ZL1myUUobV6D7OvH92inVdqNg4a1m1YS67P4URXRd9UcYPpdu4ShLSSVIOHJ1cESr7fMWG/QJCAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T04:53:54.902593Z"},"content_sha256":"bbe77c5bd76a2bdcec0ea2d05db0246345b0a605b3101555c4c121e5145ffa67","schema_version":"1.0","event_id":"sha256:bbe77c5bd76a2bdcec0ea2d05db0246345b0a605b3101555c4c121e5145ffa67"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:FELH6AGE6E5ASD6NNFJSWVPLNJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Inclusion relations between Modulation and Triebel-Lizorkin spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Guoping Zhao, Huoxiong Wu, Weichao Guo","submitted_at":"2016-09-17T03:47:59Z","abstract_excerpt":"In this paper, we obtain the sharp conditions of the inclusion relations between modulation spaces $M_{p,q}^s$ and Triebel-Lizorkin spaces $F_{p,r}$ for $p\\leq 1$, which greatly improve and extend the results for the embedding relations between local Hardy spaces and modulation spaces obtained by Kobayashi, Miyachi and Tomita in [Studia Math. 192 (2009), 79-96]."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.05278","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":"2026-05-18T01:04:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hFh2uG2RstdwV8sPQAi/ahanlntmT0XdjYOAQ9YWpvWNFftPqi9O4pHIRGvYsLMofN+9JOt6u/ZzwukjWzRaAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T04:53:54.902937Z"},"content_sha256":"40aba2a80b8a2a3e5edf752bbb5750bcba7298a357a0de19d5c97e49318dc23c","schema_version":"1.0","event_id":"sha256:40aba2a80b8a2a3e5edf752bbb5750bcba7298a357a0de19d5c97e49318dc23c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FELH6AGE6E5ASD6NNFJSWVPLNJ/bundle.json","state_url":"https://pith.science/pith/FELH6AGE6E5ASD6NNFJSWVPLNJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FELH6AGE6E5ASD6NNFJSWVPLNJ/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":"2026-06-20T04:53:54Z","links":{"resolver":"https://pith.science/pith/FELH6AGE6E5ASD6NNFJSWVPLNJ","bundle":"https://pith.science/pith/FELH6AGE6E5ASD6NNFJSWVPLNJ/bundle.json","state":"https://pith.science/pith/FELH6AGE6E5ASD6NNFJSWVPLNJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FELH6AGE6E5ASD6NNFJSWVPLNJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:FELH6AGE6E5ASD6NNFJSWVPLNJ","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":"111ad561fe59de5bc9ff75d4b45faab8973dfa3249f4a29ee80e46b931d05939","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-09-17T03:47:59Z","title_canon_sha256":"6108c77cb4f84f3748fd75b43fc9926aa215edfdf4f629e9d522857844f2864c"},"schema_version":"1.0","source":{"id":"1609.05278","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.05278","created_at":"2026-05-18T01:04:26Z"},{"alias_kind":"arxiv_version","alias_value":"1609.05278v1","created_at":"2026-05-18T01:04:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.05278","created_at":"2026-05-18T01:04:26Z"},{"alias_kind":"pith_short_12","alias_value":"FELH6AGE6E5A","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FELH6AGE6E5ASD6N","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FELH6AGE","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:40aba2a80b8a2a3e5edf752bbb5750bcba7298a357a0de19d5c97e49318dc23c","target":"graph","created_at":"2026-05-18T01:04:26Z","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":"In this paper, we obtain the sharp conditions of the inclusion relations between modulation spaces $M_{p,q}^s$ and Triebel-Lizorkin spaces $F_{p,r}$ for $p\\leq 1$, which greatly improve and extend the results for the embedding relations between local Hardy spaces and modulation spaces obtained by Kobayashi, Miyachi and Tomita in [Studia Math. 192 (2009), 79-96].","authors_text":"Guoping Zhao, Huoxiong Wu, Weichao Guo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-09-17T03:47:59Z","title":"Inclusion relations between Modulation and Triebel-Lizorkin spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.05278","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:bbe77c5bd76a2bdcec0ea2d05db0246345b0a605b3101555c4c121e5145ffa67","target":"record","created_at":"2026-05-18T01:04:26Z","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":"111ad561fe59de5bc9ff75d4b45faab8973dfa3249f4a29ee80e46b931d05939","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-09-17T03:47:59Z","title_canon_sha256":"6108c77cb4f84f3748fd75b43fc9926aa215edfdf4f629e9d522857844f2864c"},"schema_version":"1.0","source":{"id":"1609.05278","kind":"arxiv","version":1}},"canonical_sha256":"29167f00c4f13a090fcd69532b55eb6a40519c0160424d6f1475282008c293c3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"29167f00c4f13a090fcd69532b55eb6a40519c0160424d6f1475282008c293c3","first_computed_at":"2026-05-18T01:04:26.877797Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:04:26.877797Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wlDenQ5+6MaHeKJlGW9XeplzIktoaXAVx6tSB96Obxcabkk5WaWdcZYy4u4swMHsIiJLxqzm1BS/nXInu7PDDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:04:26.878539Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.05278","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bbe77c5bd76a2bdcec0ea2d05db0246345b0a605b3101555c4c121e5145ffa67","sha256:40aba2a80b8a2a3e5edf752bbb5750bcba7298a357a0de19d5c97e49318dc23c"],"state_sha256":"b8e6b510287fe1ef0b0a9c081d61a17b1da6d4b5e29e2a44c635123e77d6d56d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YRvyy9fHtaotjZT2EffUxYtMH8FJJOGC2fxWZ1jEonCmIYGVciMtQCY499VWVNaL6UJuyXyqfyz5WDGSxcK7DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T04:53:54.904827Z","bundle_sha256":"10be53bd48bcecdd3942adcc04a858999af60abed6562b41ccf19d9909b479a2"}}