{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:WBTBDX74UJSNLC2N2SG62UJPGO","short_pith_number":"pith:WBTBDX74","canonical_record":{"source":{"id":"1504.03862","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-15T11:24:28Z","cross_cats_sorted":[],"title_canon_sha256":"295dbc52606d3f968a141f00f9f6cadeca4f5fc8aa98d3711444860a571be80f","abstract_canon_sha256":"a49dbfa97949d37b2d75d7c3d557727b9e23a778289f33acaf09dec082b372c3"},"schema_version":"1.0"},"canonical_sha256":"b06611dffca264d58b4dd48ded512f33b66be62957cc83231597a7a8b18a131f","source":{"kind":"arxiv","id":"1504.03862","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.03862","created_at":"2026-05-17T23:58:15Z"},{"alias_kind":"arxiv_version","alias_value":"1504.03862v2","created_at":"2026-05-17T23:58:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.03862","created_at":"2026-05-17T23:58:15Z"},{"alias_kind":"pith_short_12","alias_value":"WBTBDX74UJSN","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"WBTBDX74UJSNLC2N","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"WBTBDX74","created_at":"2026-05-18T12:29:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:WBTBDX74UJSNLC2N2SG62UJPGO","target":"record","payload":{"canonical_record":{"source":{"id":"1504.03862","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-15T11:24:28Z","cross_cats_sorted":[],"title_canon_sha256":"295dbc52606d3f968a141f00f9f6cadeca4f5fc8aa98d3711444860a571be80f","abstract_canon_sha256":"a49dbfa97949d37b2d75d7c3d557727b9e23a778289f33acaf09dec082b372c3"},"schema_version":"1.0"},"canonical_sha256":"b06611dffca264d58b4dd48ded512f33b66be62957cc83231597a7a8b18a131f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:15.326749Z","signature_b64":"CPBuwofNgnxJZ5aRDr/VQ1pn9nXy44L1YtQIh6TgJdD4xBbuW4esvG2FOUcKcmEIV0yVjDyPgYFHREyk+6QUCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b06611dffca264d58b4dd48ded512f33b66be62957cc83231597a7a8b18a131f","last_reissued_at":"2026-05-17T23:58:15.326133Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:15.326133Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1504.03862","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":"zRbvOUXTrhS+25NEh/yOoNlFlVv1f2SIrL1becSuCnpY3BMK6Y43QKZY3TiakhunVMMJD6X3cSbdKmYOYkpgDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T10:32:31.870736Z"},"content_sha256":"717d613e790151707984ae1f7de5dab451aa462f50f76a399894e15f1190482f","schema_version":"1.0","event_id":"sha256:717d613e790151707984ae1f7de5dab451aa462f50f76a399894e15f1190482f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:WBTBDX74UJSNLC2N2SG62UJPGO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spectral multipliers for sub-Laplacians on solvable extensions of stratified groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alessandro Ottazzi, Alessio Martini, Maria Vallarino","submitted_at":"2015-04-15T11:24:28Z","abstract_excerpt":"Let $G = N \\rtimes A$, where $N$ is a stratified group and $A = \\mathbb{R}$ acts on $N$ via automorphic dilations. Homogeneous sub-Laplacians on $N$ and $A$ can be lifted to left-invariant operators on $G$ and their sum is a sub-Laplacian $\\Delta$ on $G$. We prove a theorem of Mihlin-H\\\"ormander type for spectral multipliers of $\\Delta$. The proof of the theorem hinges on a Calder\\'on-Zygmund theory adapted to a sub-Riemannian structure of $G$ and on $L^1$-estimates of the gradient of the heat kernel associated to the sub-Laplacian $\\Delta$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.03862","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":"/Djt7QSrFP2TdX+9jqKA6Zk0OGGUoqDzuiAOPt8ecW5cd952ltAZZc7ZRL5NNBB8irrfAlJQmpLvfQLlX4cFCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T10:32:31.871472Z"},"content_sha256":"d1928e92861eda5e35f6296ad7338a02c03369f1b7e5b3ceed74620e8297bc63","schema_version":"1.0","event_id":"sha256:d1928e92861eda5e35f6296ad7338a02c03369f1b7e5b3ceed74620e8297bc63"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WBTBDX74UJSNLC2N2SG62UJPGO/bundle.json","state_url":"https://pith.science/pith/WBTBDX74UJSNLC2N2SG62UJPGO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WBTBDX74UJSNLC2N2SG62UJPGO/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":"2026-06-06T10:32:31Z","links":{"resolver":"https://pith.science/pith/WBTBDX74UJSNLC2N2SG62UJPGO","bundle":"https://pith.science/pith/WBTBDX74UJSNLC2N2SG62UJPGO/bundle.json","state":"https://pith.science/pith/WBTBDX74UJSNLC2N2SG62UJPGO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WBTBDX74UJSNLC2N2SG62UJPGO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:WBTBDX74UJSNLC2N2SG62UJPGO","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":"a49dbfa97949d37b2d75d7c3d557727b9e23a778289f33acaf09dec082b372c3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-15T11:24:28Z","title_canon_sha256":"295dbc52606d3f968a141f00f9f6cadeca4f5fc8aa98d3711444860a571be80f"},"schema_version":"1.0","source":{"id":"1504.03862","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.03862","created_at":"2026-05-17T23:58:15Z"},{"alias_kind":"arxiv_version","alias_value":"1504.03862v2","created_at":"2026-05-17T23:58:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.03862","created_at":"2026-05-17T23:58:15Z"},{"alias_kind":"pith_short_12","alias_value":"WBTBDX74UJSN","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"WBTBDX74UJSNLC2N","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"WBTBDX74","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:d1928e92861eda5e35f6296ad7338a02c03369f1b7e5b3ceed74620e8297bc63","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":"Let $G = N \\rtimes A$, where $N$ is a stratified group and $A = \\mathbb{R}$ acts on $N$ via automorphic dilations. Homogeneous sub-Laplacians on $N$ and $A$ can be lifted to left-invariant operators on $G$ and their sum is a sub-Laplacian $\\Delta$ on $G$. We prove a theorem of Mihlin-H\\\"ormander type for spectral multipliers of $\\Delta$. The proof of the theorem hinges on a Calder\\'on-Zygmund theory adapted to a sub-Riemannian structure of $G$ and on $L^1$-estimates of the gradient of the heat kernel associated to the sub-Laplacian $\\Delta$.","authors_text":"Alessandro Ottazzi, Alessio Martini, Maria Vallarino","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-15T11:24:28Z","title":"Spectral multipliers for sub-Laplacians on solvable extensions of stratified groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.03862","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":"sha256:717d613e790151707984ae1f7de5dab451aa462f50f76a399894e15f1190482f","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":"a49dbfa97949d37b2d75d7c3d557727b9e23a778289f33acaf09dec082b372c3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-15T11:24:28Z","title_canon_sha256":"295dbc52606d3f968a141f00f9f6cadeca4f5fc8aa98d3711444860a571be80f"},"schema_version":"1.0","source":{"id":"1504.03862","kind":"arxiv","version":2}},"canonical_sha256":"b06611dffca264d58b4dd48ded512f33b66be62957cc83231597a7a8b18a131f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b06611dffca264d58b4dd48ded512f33b66be62957cc83231597a7a8b18a131f","first_computed_at":"2026-05-17T23:58:15.326133Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:58:15.326133Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CPBuwofNgnxJZ5aRDr/VQ1pn9nXy44L1YtQIh6TgJdD4xBbuW4esvG2FOUcKcmEIV0yVjDyPgYFHREyk+6QUCQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:58:15.326749Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.03862","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:717d613e790151707984ae1f7de5dab451aa462f50f76a399894e15f1190482f","sha256:d1928e92861eda5e35f6296ad7338a02c03369f1b7e5b3ceed74620e8297bc63"],"state_sha256":"57272c080c70de0ae2e279b2a822b4a09745956f270dbd0c0c7ef9a511c058d3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Am5l7YrgGO6XADyUsLHr5PfR6m308Zl+ZghmO8bGDArwWdl6CoW/XN4jLhJ3sNsbgOesOedOfS6tMzdbr/gyCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T10:32:31.875181Z","bundle_sha256":"8a2bf40487671bfe6875a805b0ddafca65076216759c4c57beb97e480238ef91"}}