{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:6IAQT67MISJNO5GSWU5V5HQQDD","short_pith_number":"pith:6IAQT67M","canonical_record":{"source":{"id":"1907.03573","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-07-04T03:51:40Z","cross_cats_sorted":[],"title_canon_sha256":"4dbf69caaf96743be6cde7e200504bcbe12fe074c50447597179c1444304b75e","abstract_canon_sha256":"c3fd83438ff28fec408999ea05e59866480c3fd6013c9bdebdf06f751083ab9e"},"schema_version":"1.0"},"canonical_sha256":"f20109fbec4492d774d2b53b5e9e1018cf3995b9232f782bc4983948c64e90c2","source":{"kind":"arxiv","id":"1907.03573","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.03573","created_at":"2026-05-17T23:41:14Z"},{"alias_kind":"arxiv_version","alias_value":"1907.03573v1","created_at":"2026-05-17T23:41:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.03573","created_at":"2026-05-17T23:41:14Z"},{"alias_kind":"pith_short_12","alias_value":"6IAQT67MISJN","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_16","alias_value":"6IAQT67MISJNO5GS","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_8","alias_value":"6IAQT67M","created_at":"2026-05-18T12:33:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:6IAQT67MISJNO5GSWU5V5HQQDD","target":"record","payload":{"canonical_record":{"source":{"id":"1907.03573","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-07-04T03:51:40Z","cross_cats_sorted":[],"title_canon_sha256":"4dbf69caaf96743be6cde7e200504bcbe12fe074c50447597179c1444304b75e","abstract_canon_sha256":"c3fd83438ff28fec408999ea05e59866480c3fd6013c9bdebdf06f751083ab9e"},"schema_version":"1.0"},"canonical_sha256":"f20109fbec4492d774d2b53b5e9e1018cf3995b9232f782bc4983948c64e90c2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:14.292425Z","signature_b64":"R6ebvXsqsQuICL3x+P7xPCbMo+PlGUjcqa4GKbgFqbe5COE3c3/ViC/8JoBrxisK5fVb+p8Fa1biT5hzGNy8Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f20109fbec4492d774d2b53b5e9e1018cf3995b9232f782bc4983948c64e90c2","last_reissued_at":"2026-05-17T23:41:14.292002Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:14.292002Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1907.03573","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":"2026-05-17T23:41:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sp+YCFowABE5qq22mnZ1dR2gYdI/jy4CFuQMaXtENs46Tg3M/io9721zPe+SbM9NM2njnFKLNd5l2E2Clz7nAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T02:04:31.588691Z"},"content_sha256":"dc19dc962c7dc5c206fea83ea5b243beca1e6e17731a79ab4dc667e07e1b027d","schema_version":"1.0","event_id":"sha256:dc19dc962c7dc5c206fea83ea5b243beca1e6e17731a79ab4dc667e07e1b027d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:6IAQT67MISJNO5GSWU5V5HQQDD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Morrey spaces for Schr\\\"odinger operators with nonnegative potentials, fractional integral operators and the Adams inequality on the Heisenberg groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Hua Wang","submitted_at":"2019-07-04T03:51:40Z","abstract_excerpt":"Let $\\mathcal L=-\\Delta_{\\mathbb H^n}+V$ be a Schr\\\"odinger operator on the Heisenberg group $\\mathbb H^n$, where $\\Delta_{\\mathbb H^n}$ is the sublaplacian on $\\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse H\\\"older class $RH_s$ with $s\\in[Q/2,\\infty)$. Here $Q=2n+2$ is the homogeneous dimension of $\\mathbb H^n$. For given $\\alpha\\in(0,Q)$, the fractional integral operator associated with the Schr\\\"odinger operator $\\mathcal L$ is defined by $\\mathcal I_{\\alpha}={\\mathcal L}^{-{\\alpha}/2}$. In this article, the author introduces the Morrey space $L^{p,\\kappa}_{\\rho,\\inf"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.03573","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":"2026-05-17T23:41:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UmO5sR8xDWbq4fmIu9R6DLE1yhcfsQE2DPjsivkdpLnHsoH6ZmwW6NC/BmCNew6P9quebKLvhXVllz6XthMrDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T02:04:31.589359Z"},"content_sha256":"11b5041f0df78253a6a9b23ae732f1a52120c84bcc15a84ddff7b4ac28bafc7a","schema_version":"1.0","event_id":"sha256:11b5041f0df78253a6a9b23ae732f1a52120c84bcc15a84ddff7b4ac28bafc7a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6IAQT67MISJNO5GSWU5V5HQQDD/bundle.json","state_url":"https://pith.science/pith/6IAQT67MISJNO5GSWU5V5HQQDD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6IAQT67MISJNO5GSWU5V5HQQDD/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":"2026-06-11T02:04:31Z","links":{"resolver":"https://pith.science/pith/6IAQT67MISJNO5GSWU5V5HQQDD","bundle":"https://pith.science/pith/6IAQT67MISJNO5GSWU5V5HQQDD/bundle.json","state":"https://pith.science/pith/6IAQT67MISJNO5GSWU5V5HQQDD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6IAQT67MISJNO5GSWU5V5HQQDD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:6IAQT67MISJNO5GSWU5V5HQQDD","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":"c3fd83438ff28fec408999ea05e59866480c3fd6013c9bdebdf06f751083ab9e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-07-04T03:51:40Z","title_canon_sha256":"4dbf69caaf96743be6cde7e200504bcbe12fe074c50447597179c1444304b75e"},"schema_version":"1.0","source":{"id":"1907.03573","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.03573","created_at":"2026-05-17T23:41:14Z"},{"alias_kind":"arxiv_version","alias_value":"1907.03573v1","created_at":"2026-05-17T23:41:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.03573","created_at":"2026-05-17T23:41:14Z"},{"alias_kind":"pith_short_12","alias_value":"6IAQT67MISJN","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_16","alias_value":"6IAQT67MISJNO5GS","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_8","alias_value":"6IAQT67M","created_at":"2026-05-18T12:33:10Z"}],"graph_snapshots":[{"event_id":"sha256:11b5041f0df78253a6a9b23ae732f1a52120c84bcc15a84ddff7b4ac28bafc7a","target":"graph","created_at":"2026-05-17T23:41:14Z","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 $\\mathcal L=-\\Delta_{\\mathbb H^n}+V$ be a Schr\\\"odinger operator on the Heisenberg group $\\mathbb H^n$, where $\\Delta_{\\mathbb H^n}$ is the sublaplacian on $\\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse H\\\"older class $RH_s$ with $s\\in[Q/2,\\infty)$. Here $Q=2n+2$ is the homogeneous dimension of $\\mathbb H^n$. For given $\\alpha\\in(0,Q)$, the fractional integral operator associated with the Schr\\\"odinger operator $\\mathcal L$ is defined by $\\mathcal I_{\\alpha}={\\mathcal L}^{-{\\alpha}/2}$. In this article, the author introduces the Morrey space $L^{p,\\kappa}_{\\rho,\\inf","authors_text":"Hua Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-07-04T03:51:40Z","title":"Morrey spaces for Schr\\\"odinger operators with nonnegative potentials, fractional integral operators and the Adams inequality on the Heisenberg groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.03573","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:dc19dc962c7dc5c206fea83ea5b243beca1e6e17731a79ab4dc667e07e1b027d","target":"record","created_at":"2026-05-17T23:41:14Z","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":"c3fd83438ff28fec408999ea05e59866480c3fd6013c9bdebdf06f751083ab9e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-07-04T03:51:40Z","title_canon_sha256":"4dbf69caaf96743be6cde7e200504bcbe12fe074c50447597179c1444304b75e"},"schema_version":"1.0","source":{"id":"1907.03573","kind":"arxiv","version":1}},"canonical_sha256":"f20109fbec4492d774d2b53b5e9e1018cf3995b9232f782bc4983948c64e90c2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f20109fbec4492d774d2b53b5e9e1018cf3995b9232f782bc4983948c64e90c2","first_computed_at":"2026-05-17T23:41:14.292002Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:41:14.292002Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"R6ebvXsqsQuICL3x+P7xPCbMo+PlGUjcqa4GKbgFqbe5COE3c3/ViC/8JoBrxisK5fVb+p8Fa1biT5hzGNy8Cg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:41:14.292425Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.03573","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dc19dc962c7dc5c206fea83ea5b243beca1e6e17731a79ab4dc667e07e1b027d","sha256:11b5041f0df78253a6a9b23ae732f1a52120c84bcc15a84ddff7b4ac28bafc7a"],"state_sha256":"ce033c9a8740714021d80de78d7c1a0b26636d0912175a6a70bebbf1952facf2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oyh1wD5qUOqV6GDUa9xQ1Ta0djMFWA++iOTTUZoOdwwlpI+rLSZY9kg0vP0PROjO67dCgC0kFR25KMk7+dkyAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T02:04:31.592894Z","bundle_sha256":"32e13c2ffa16b9d9f0c9f7fc8b23df254f0ecc0c3b413c4353c9471de40ad0ec"}}