{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:MGD5EFLNIYQXZIW6NHWH4D7M6F","short_pith_number":"pith:MGD5EFLN","canonical_record":{"source":{"id":"1501.02321","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-01-10T08:53:39Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"8329f64c8b1beef2e4bdefaf0dc7d38ff61fa841b3c59266dd783c2ae94af5ee","abstract_canon_sha256":"9e876cb80a99b909e8c0adc2b750b7a22a32d5857e49b64bf837b0cd83a18a41"},"schema_version":"1.0"},"canonical_sha256":"6187d2156d46217ca2de69ec7e0fecf16c8a9d7509ce5931fb9e95289eeb8fa9","source":{"kind":"arxiv","id":"1501.02321","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.02321","created_at":"2026-05-17T23:58:15Z"},{"alias_kind":"arxiv_version","alias_value":"1501.02321v2","created_at":"2026-05-17T23:58:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.02321","created_at":"2026-05-17T23:58:15Z"},{"alias_kind":"pith_short_12","alias_value":"MGD5EFLNIYQX","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"MGD5EFLNIYQXZIW6","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"MGD5EFLN","created_at":"2026-05-18T12:29:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:MGD5EFLNIYQXZIW6NHWH4D7M6F","target":"record","payload":{"canonical_record":{"source":{"id":"1501.02321","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-01-10T08:53:39Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"8329f64c8b1beef2e4bdefaf0dc7d38ff61fa841b3c59266dd783c2ae94af5ee","abstract_canon_sha256":"9e876cb80a99b909e8c0adc2b750b7a22a32d5857e49b64bf837b0cd83a18a41"},"schema_version":"1.0"},"canonical_sha256":"6187d2156d46217ca2de69ec7e0fecf16c8a9d7509ce5931fb9e95289eeb8fa9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:15.352599Z","signature_b64":"7uA4rwv5uExBuM2xU4FblUmDwQmN0QjXOwNOtUtsmt6Fv7bb+gQrmbLrqTZUyUBZoRQ1DQ0mWJWGYenktphjAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6187d2156d46217ca2de69ec7e0fecf16c8a9d7509ce5931fb9e95289eeb8fa9","last_reissued_at":"2026-05-17T23:58:15.352105Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:15.352105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.02321","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":"2026-05-17T23:58:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n5KSlEztMO3UFzQKgxKaLZr7XWDSBfdORjk3h12X57UdtxpDz6RuEMw4KZ//r9QzpnAYXIm9xSHWqIppC19ODA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T19:09:31.831141Z"},"content_sha256":"269aa88f61e16dcbed8bef93b9add0474098b7bd0080eb84d5cfa4c03f29e9ab","schema_version":"1.0","event_id":"sha256:269aa88f61e16dcbed8bef93b9add0474098b7bd0080eb84d5cfa4c03f29e9ab"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:MGD5EFLNIYQXZIW6NHWH4D7M6F","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spectral multipliers for the Kohn Laplacian on forms on the sphere in $\\mathbb{C}^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Adam Sikora, Alessio Martini, Michael G. Cowling, Valentina Casarino","submitted_at":"2015-01-10T08:53:39Z","abstract_excerpt":"The unit sphere $\\mathbb{S}$ in $\\mathbb{C}^n$ is equipped with the tangential Cauchy-Riemann complex and the associated Laplacian $\\Box_b$. We prove a H\\\"ormander spectral multiplier theorem for $\\Box_b$ with critical index $n-1/2$, that is, half the topological dimension of $\\mathbb{S}$. Our proof is mainly based on representation theory and on a detailed analysis of the spaces of differential forms on $\\mathbb{S}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.02321","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":"2026-05-17T23:58:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gj6EpoL1ws4RKPGfVdw3f6/gx0YaK3JZ+09TUzwykptFys7/x3mR+nZXVmEkVtyLrS3IiBzG+k/5PGtEwHgsCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T19:09:31.831792Z"},"content_sha256":"2eba7ab4e3b5fdd827673ee1953de7572d5870a8ceb226d7806856b1df89d547","schema_version":"1.0","event_id":"sha256:2eba7ab4e3b5fdd827673ee1953de7572d5870a8ceb226d7806856b1df89d547"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MGD5EFLNIYQXZIW6NHWH4D7M6F/bundle.json","state_url":"https://pith.science/pith/MGD5EFLNIYQXZIW6NHWH4D7M6F/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MGD5EFLNIYQXZIW6NHWH4D7M6F/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":"2026-06-05T19:09:31Z","links":{"resolver":"https://pith.science/pith/MGD5EFLNIYQXZIW6NHWH4D7M6F","bundle":"https://pith.science/pith/MGD5EFLNIYQXZIW6NHWH4D7M6F/bundle.json","state":"https://pith.science/pith/MGD5EFLNIYQXZIW6NHWH4D7M6F/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MGD5EFLNIYQXZIW6NHWH4D7M6F/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:MGD5EFLNIYQXZIW6NHWH4D7M6F","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":"9e876cb80a99b909e8c0adc2b750b7a22a32d5857e49b64bf837b0cd83a18a41","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-01-10T08:53:39Z","title_canon_sha256":"8329f64c8b1beef2e4bdefaf0dc7d38ff61fa841b3c59266dd783c2ae94af5ee"},"schema_version":"1.0","source":{"id":"1501.02321","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.02321","created_at":"2026-05-17T23:58:15Z"},{"alias_kind":"arxiv_version","alias_value":"1501.02321v2","created_at":"2026-05-17T23:58:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.02321","created_at":"2026-05-17T23:58:15Z"},{"alias_kind":"pith_short_12","alias_value":"MGD5EFLNIYQX","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"MGD5EFLNIYQXZIW6","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"MGD5EFLN","created_at":"2026-05-18T12:29:32Z"}],"graph_snapshots":[{"event_id":"sha256:2eba7ab4e3b5fdd827673ee1953de7572d5870a8ceb226d7806856b1df89d547","target":"graph","created_at":"2026-05-17T23:58:15Z","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":"The unit sphere $\\mathbb{S}$ in $\\mathbb{C}^n$ is equipped with the tangential Cauchy-Riemann complex and the associated Laplacian $\\Box_b$. We prove a H\\\"ormander spectral multiplier theorem for $\\Box_b$ with critical index $n-1/2$, that is, half the topological dimension of $\\mathbb{S}$. Our proof is mainly based on representation theory and on a detailed analysis of the spaces of differential forms on $\\mathbb{S}$.","authors_text":"Adam Sikora, Alessio Martini, Michael G. Cowling, Valentina Casarino","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-01-10T08:53:39Z","title":"Spectral multipliers for the Kohn Laplacian on forms on the sphere in $\\mathbb{C}^n$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.02321","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":"sha256:269aa88f61e16dcbed8bef93b9add0474098b7bd0080eb84d5cfa4c03f29e9ab","target":"record","created_at":"2026-05-17T23:58:15Z","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":"9e876cb80a99b909e8c0adc2b750b7a22a32d5857e49b64bf837b0cd83a18a41","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-01-10T08:53:39Z","title_canon_sha256":"8329f64c8b1beef2e4bdefaf0dc7d38ff61fa841b3c59266dd783c2ae94af5ee"},"schema_version":"1.0","source":{"id":"1501.02321","kind":"arxiv","version":2}},"canonical_sha256":"6187d2156d46217ca2de69ec7e0fecf16c8a9d7509ce5931fb9e95289eeb8fa9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6187d2156d46217ca2de69ec7e0fecf16c8a9d7509ce5931fb9e95289eeb8fa9","first_computed_at":"2026-05-17T23:58:15.352105Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:58:15.352105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7uA4rwv5uExBuM2xU4FblUmDwQmN0QjXOwNOtUtsmt6Fv7bb+gQrmbLrqTZUyUBZoRQ1DQ0mWJWGYenktphjAg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:58:15.352599Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.02321","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:269aa88f61e16dcbed8bef93b9add0474098b7bd0080eb84d5cfa4c03f29e9ab","sha256:2eba7ab4e3b5fdd827673ee1953de7572d5870a8ceb226d7806856b1df89d547"],"state_sha256":"d77024ed8f996f82d76a1b4e7a08e3a5dd20d9bd8d15995657b6bdb27dd7ccd8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8lLFn0NjZq8KVUIcvM+JX5FgUIkoM+xSblc7NOdihh8v3eYqnh5wb7mIlGgCxXrPbmPuyO2oYY/goAErRBVFDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T19:09:31.835008Z","bundle_sha256":"24e234bbe61a7fcd25a3762d7b8d82566436974fa7b0747c101b2084a87bbeb1"}}