{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:SXQPEPCS46AZOXJCYHD3ICXV4N","short_pith_number":"pith:SXQPEPCS","canonical_record":{"source":{"id":"1704.01490","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-04-05T15:32:24Z","cross_cats_sorted":["math-ph","math.FA","math.MP","math.SP"],"title_canon_sha256":"b34ba874d516b1c87ed134f2af747f3b3257285ab727280d5671b1bfcbad5699","abstract_canon_sha256":"3d6e469fe2f0e5d3f43b0146e23459a9ad8d889e6f2ed6b96717e95aa1139bf2"},"schema_version":"1.0"},"canonical_sha256":"95e0f23c52e781975d22c1c7b40af5e37adcb2a5003194195e67558e4c62f0d3","source":{"kind":"arxiv","id":"1704.01490","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.01490","created_at":"2026-05-18T00:46:56Z"},{"alias_kind":"arxiv_version","alias_value":"1704.01490v1","created_at":"2026-05-18T00:46:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.01490","created_at":"2026-05-18T00:46:56Z"},{"alias_kind":"pith_short_12","alias_value":"SXQPEPCS46AZ","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_16","alias_value":"SXQPEPCS46AZOXJC","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_8","alias_value":"SXQPEPCS","created_at":"2026-05-18T12:31:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:SXQPEPCS46AZOXJCYHD3ICXV4N","target":"record","payload":{"canonical_record":{"source":{"id":"1704.01490","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-04-05T15:32:24Z","cross_cats_sorted":["math-ph","math.FA","math.MP","math.SP"],"title_canon_sha256":"b34ba874d516b1c87ed134f2af747f3b3257285ab727280d5671b1bfcbad5699","abstract_canon_sha256":"3d6e469fe2f0e5d3f43b0146e23459a9ad8d889e6f2ed6b96717e95aa1139bf2"},"schema_version":"1.0"},"canonical_sha256":"95e0f23c52e781975d22c1c7b40af5e37adcb2a5003194195e67558e4c62f0d3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:56.336644Z","signature_b64":"FpMorhoqpYla6BxbUnnW02EiOGi/DJ42uourxaMlOO/v8Cb9o8qVmyrBnIz2u1kM0h48vBLDr+iSYLsOe/B0Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95e0f23c52e781975d22c1c7b40af5e37adcb2a5003194195e67558e4c62f0d3","last_reissued_at":"2026-05-18T00:46:56.336134Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:56.336134Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1704.01490","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":"2026-05-18T00:46:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"luX3CX3SN3g5qP9izWB8zfH8sBOBagaLhTj5luDVP4IaYUiDLAxFy/7oKY61EN6Uuj1EPsqothxI94YLZD0RAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T03:11:10.611714Z"},"content_sha256":"334b1ea590988bd0c3729ed0652ad6b3a41d5305f7cac97ccfd1baed6496dacf","schema_version":"1.0","event_id":"sha256:334b1ea590988bd0c3729ed0652ad6b3a41d5305f7cac97ccfd1baed6496dacf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:SXQPEPCS46AZOXJCYHD3ICXV4N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Best constants in Sobolev and Gagliardo-Nirenberg inequalities on graded groups and ground states for higher order nonlinear subelliptic equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP","math.SP"],"primary_cat":"math.AP","authors_text":"Michael Ruzhansky, Niyaz Tokmagambetov, Nurgissa Yessirkegenov","submitted_at":"2017-04-05T15:32:24Z","abstract_excerpt":"In this paper the dependence of the best constants in Sobolev and Gagliardo-Nirenberg inequalities on the precise form of the Sobolev space norm is investigated. The analysis is carried out on general graded Lie groups, thus including the cases of $\\mathbb R^n$, Heisenberg, and more general stratified Lie groups. The Sobolev norms may be defined in terms of Rockland operators, i.e. the hypoelliptic homogeneous left-invariant differential operators on the group. The best constants are expressed in the variational form as well as in terms of the ground state solutions of the corresponding nonlin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.01490","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":"2026-05-18T00:46:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PouuXPnERKT5QnLvyRHfBZDMoQDV9hT99l4MavXRelsdwLQldG3WGHB+SOMzuiWfpz7jlIMhWkJvvHljsnJADg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T03:11:10.612370Z"},"content_sha256":"1f65047f902335230982a485435635986f235c7a8af8f501e55392b8a4c53279","schema_version":"1.0","event_id":"sha256:1f65047f902335230982a485435635986f235c7a8af8f501e55392b8a4c53279"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SXQPEPCS46AZOXJCYHD3ICXV4N/bundle.json","state_url":"https://pith.science/pith/SXQPEPCS46AZOXJCYHD3ICXV4N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SXQPEPCS46AZOXJCYHD3ICXV4N/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":"2026-06-10T03:11:10Z","links":{"resolver":"https://pith.science/pith/SXQPEPCS46AZOXJCYHD3ICXV4N","bundle":"https://pith.science/pith/SXQPEPCS46AZOXJCYHD3ICXV4N/bundle.json","state":"https://pith.science/pith/SXQPEPCS46AZOXJCYHD3ICXV4N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SXQPEPCS46AZOXJCYHD3ICXV4N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:SXQPEPCS46AZOXJCYHD3ICXV4N","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":"3d6e469fe2f0e5d3f43b0146e23459a9ad8d889e6f2ed6b96717e95aa1139bf2","cross_cats_sorted":["math-ph","math.FA","math.MP","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-04-05T15:32:24Z","title_canon_sha256":"b34ba874d516b1c87ed134f2af747f3b3257285ab727280d5671b1bfcbad5699"},"schema_version":"1.0","source":{"id":"1704.01490","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.01490","created_at":"2026-05-18T00:46:56Z"},{"alias_kind":"arxiv_version","alias_value":"1704.01490v1","created_at":"2026-05-18T00:46:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.01490","created_at":"2026-05-18T00:46:56Z"},{"alias_kind":"pith_short_12","alias_value":"SXQPEPCS46AZ","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_16","alias_value":"SXQPEPCS46AZOXJC","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_8","alias_value":"SXQPEPCS","created_at":"2026-05-18T12:31:43Z"}],"graph_snapshots":[{"event_id":"sha256:1f65047f902335230982a485435635986f235c7a8af8f501e55392b8a4c53279","target":"graph","created_at":"2026-05-18T00:46:56Z","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 the dependence of the best constants in Sobolev and Gagliardo-Nirenberg inequalities on the precise form of the Sobolev space norm is investigated. The analysis is carried out on general graded Lie groups, thus including the cases of $\\mathbb R^n$, Heisenberg, and more general stratified Lie groups. The Sobolev norms may be defined in terms of Rockland operators, i.e. the hypoelliptic homogeneous left-invariant differential operators on the group. The best constants are expressed in the variational form as well as in terms of the ground state solutions of the corresponding nonlin","authors_text":"Michael Ruzhansky, Niyaz Tokmagambetov, Nurgissa Yessirkegenov","cross_cats":["math-ph","math.FA","math.MP","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-04-05T15:32:24Z","title":"Best constants in Sobolev and Gagliardo-Nirenberg inequalities on graded groups and ground states for higher order nonlinear subelliptic equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.01490","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":"sha256:334b1ea590988bd0c3729ed0652ad6b3a41d5305f7cac97ccfd1baed6496dacf","target":"record","created_at":"2026-05-18T00:46:56Z","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":"3d6e469fe2f0e5d3f43b0146e23459a9ad8d889e6f2ed6b96717e95aa1139bf2","cross_cats_sorted":["math-ph","math.FA","math.MP","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-04-05T15:32:24Z","title_canon_sha256":"b34ba874d516b1c87ed134f2af747f3b3257285ab727280d5671b1bfcbad5699"},"schema_version":"1.0","source":{"id":"1704.01490","kind":"arxiv","version":1}},"canonical_sha256":"95e0f23c52e781975d22c1c7b40af5e37adcb2a5003194195e67558e4c62f0d3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"95e0f23c52e781975d22c1c7b40af5e37adcb2a5003194195e67558e4c62f0d3","first_computed_at":"2026-05-18T00:46:56.336134Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:46:56.336134Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FpMorhoqpYla6BxbUnnW02EiOGi/DJ42uourxaMlOO/v8Cb9o8qVmyrBnIz2u1kM0h48vBLDr+iSYLsOe/B0Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:46:56.336644Z","signed_message":"canonical_sha256_bytes"},"source_id":"1704.01490","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:334b1ea590988bd0c3729ed0652ad6b3a41d5305f7cac97ccfd1baed6496dacf","sha256:1f65047f902335230982a485435635986f235c7a8af8f501e55392b8a4c53279"],"state_sha256":"128018b14430a3fba7ad7479aca2628f97cdeadc20a81f82d36820d437b435bb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n2zhPpcdx9lM0EztFLvL8L1zdNoSPpDixHMLjns6ot+HE6jANJbj1BPiSmj3TwM6VkCKdhl9C+tKJq2X9dcYCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T03:11:10.616148Z","bundle_sha256":"e52ecddc3c3d759b29209eccce9b63abf152c71f3111b056cd74b0efa415b310"}}