{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:FZQJV52FCLWVUQM7UFMN76AJQH","short_pith_number":"pith:FZQJV52F","canonical_record":{"source":{"id":"1204.1159","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-04-05T09:27:49Z","cross_cats_sorted":[],"title_canon_sha256":"2027b0918e2a31ddb367025a34b589df4fd2d88c6e24fd7a71bf382f52eec3ba","abstract_canon_sha256":"bb9b0e3d178eb837a9115a6465ef9b8bea49f1bef197a232e269ab13c49281d7"},"schema_version":"1.0"},"canonical_sha256":"2e609af74512ed5a419fa158dff80981f2b6f0f6c3a7b5cf406062b2c651cb57","source":{"kind":"arxiv","id":"1204.1159","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.1159","created_at":"2026-05-18T03:30:53Z"},{"alias_kind":"arxiv_version","alias_value":"1204.1159v1","created_at":"2026-05-18T03:30:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.1159","created_at":"2026-05-18T03:30:53Z"},{"alias_kind":"pith_short_12","alias_value":"FZQJV52FCLWV","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"FZQJV52FCLWVUQM7","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"FZQJV52F","created_at":"2026-05-18T12:27:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:FZQJV52FCLWVUQM7UFMN76AJQH","target":"record","payload":{"canonical_record":{"source":{"id":"1204.1159","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-04-05T09:27:49Z","cross_cats_sorted":[],"title_canon_sha256":"2027b0918e2a31ddb367025a34b589df4fd2d88c6e24fd7a71bf382f52eec3ba","abstract_canon_sha256":"bb9b0e3d178eb837a9115a6465ef9b8bea49f1bef197a232e269ab13c49281d7"},"schema_version":"1.0"},"canonical_sha256":"2e609af74512ed5a419fa158dff80981f2b6f0f6c3a7b5cf406062b2c651cb57","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:30:53.837241Z","signature_b64":"H80XAFkBBHfvEBwqt2zT7BrZsHyWIALaYnnzvE0d2wIYnSrr3o49L/cXp8g7jio5mWRd4i0/DDPT4xoLYvkYBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2e609af74512ed5a419fa158dff80981f2b6f0f6c3a7b5cf406062b2c651cb57","last_reissued_at":"2026-05-18T03:30:53.836402Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:30:53.836402Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1204.1159","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":"2026-05-18T03:30:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fDlwUcERVG6bYH+jBQxWV0L881qOBBeMCCrmgErMT2JJxbGo2el+PrdW5peuoSPXzb5J/encxlSsstpQpTd7Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T13:22:18.328544Z"},"content_sha256":"54dddc39cb5fb5ee0ef8fa27af4ad2fb6f5786d8ccd8e610951c6a571366a16b","schema_version":"1.0","event_id":"sha256:54dddc39cb5fb5ee0ef8fa27af4ad2fb6f5786d8ccd8e610951c6a571366a16b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:FZQJV52FCLWVUQM7UFMN76AJQH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weighted Plancherel estimates and sharp spectral multipliers for the Grushin operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Adam Sikora, Alessio Martini","submitted_at":"2012-04-05T09:27:49Z","abstract_excerpt":"We study the Grushin operators acting on $\\R^{d_1}_{x'}\\times \\R^{d_2}_{x\"}$ and defined by the formula \\[ L=-\\sum_{\\jone=1}^{d_1}\\partial_{x'_\\jone}^2 - (\\sum_{\\jone=1}^{d_1}|x'_\\jone|^2) \\sum_{\\jtwo=1}^{d_2}\\partial_{x\"_\\jtwo}^2. \\] We obtain weighted Plancherel estimates for the considered operators. As a consequence we prove $L^p$ spectral multiplier results and Bochner-Riesz summability for the Grushin operators. These multiplier results are sharp if $d_1 \\ge d_2$. We discuss also an interesting phenomenon for weighted Plancherel estimates for $d_1 <d_2$. The described spectral multiplier"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.1159","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":"2026-05-18T03:30:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gu/vIASXlqcFrz4Wdxg1T8ERWBQufFDfl6EwL6iqSQneTd/3uoxT+7ZJBOT8Pe+MzWDOuzYyxbWGV2/AlS+HCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T13:22:18.328875Z"},"content_sha256":"5bd5eaee54433208ae018cab82a3dc81f61554b7677ba56cbd1e92bfa73abe79","schema_version":"1.0","event_id":"sha256:5bd5eaee54433208ae018cab82a3dc81f61554b7677ba56cbd1e92bfa73abe79"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FZQJV52FCLWVUQM7UFMN76AJQH/bundle.json","state_url":"https://pith.science/pith/FZQJV52FCLWVUQM7UFMN76AJQH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FZQJV52FCLWVUQM7UFMN76AJQH/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":"2026-06-05T13:22:18Z","links":{"resolver":"https://pith.science/pith/FZQJV52FCLWVUQM7UFMN76AJQH","bundle":"https://pith.science/pith/FZQJV52FCLWVUQM7UFMN76AJQH/bundle.json","state":"https://pith.science/pith/FZQJV52FCLWVUQM7UFMN76AJQH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FZQJV52FCLWVUQM7UFMN76AJQH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:FZQJV52FCLWVUQM7UFMN76AJQH","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":"bb9b0e3d178eb837a9115a6465ef9b8bea49f1bef197a232e269ab13c49281d7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-04-05T09:27:49Z","title_canon_sha256":"2027b0918e2a31ddb367025a34b589df4fd2d88c6e24fd7a71bf382f52eec3ba"},"schema_version":"1.0","source":{"id":"1204.1159","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.1159","created_at":"2026-05-18T03:30:53Z"},{"alias_kind":"arxiv_version","alias_value":"1204.1159v1","created_at":"2026-05-18T03:30:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.1159","created_at":"2026-05-18T03:30:53Z"},{"alias_kind":"pith_short_12","alias_value":"FZQJV52FCLWV","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"FZQJV52FCLWVUQM7","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"FZQJV52F","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:5bd5eaee54433208ae018cab82a3dc81f61554b7677ba56cbd1e92bfa73abe79","target":"graph","created_at":"2026-05-18T03:30:53Z","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":"We study the Grushin operators acting on $\\R^{d_1}_{x'}\\times \\R^{d_2}_{x\"}$ and defined by the formula \\[ L=-\\sum_{\\jone=1}^{d_1}\\partial_{x'_\\jone}^2 - (\\sum_{\\jone=1}^{d_1}|x'_\\jone|^2) \\sum_{\\jtwo=1}^{d_2}\\partial_{x\"_\\jtwo}^2. \\] We obtain weighted Plancherel estimates for the considered operators. As a consequence we prove $L^p$ spectral multiplier results and Bochner-Riesz summability for the Grushin operators. These multiplier results are sharp if $d_1 \\ge d_2$. We discuss also an interesting phenomenon for weighted Plancherel estimates for $d_1 <d_2$. The described spectral multiplier","authors_text":"Adam Sikora, Alessio Martini","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-04-05T09:27:49Z","title":"Weighted Plancherel estimates and sharp spectral multipliers for the Grushin operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.1159","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":"sha256:54dddc39cb5fb5ee0ef8fa27af4ad2fb6f5786d8ccd8e610951c6a571366a16b","target":"record","created_at":"2026-05-18T03:30:53Z","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":"bb9b0e3d178eb837a9115a6465ef9b8bea49f1bef197a232e269ab13c49281d7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-04-05T09:27:49Z","title_canon_sha256":"2027b0918e2a31ddb367025a34b589df4fd2d88c6e24fd7a71bf382f52eec3ba"},"schema_version":"1.0","source":{"id":"1204.1159","kind":"arxiv","version":1}},"canonical_sha256":"2e609af74512ed5a419fa158dff80981f2b6f0f6c3a7b5cf406062b2c651cb57","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2e609af74512ed5a419fa158dff80981f2b6f0f6c3a7b5cf406062b2c651cb57","first_computed_at":"2026-05-18T03:30:53.836402Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:30:53.836402Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"H80XAFkBBHfvEBwqt2zT7BrZsHyWIALaYnnzvE0d2wIYnSrr3o49L/cXp8g7jio5mWRd4i0/DDPT4xoLYvkYBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:30:53.837241Z","signed_message":"canonical_sha256_bytes"},"source_id":"1204.1159","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:54dddc39cb5fb5ee0ef8fa27af4ad2fb6f5786d8ccd8e610951c6a571366a16b","sha256:5bd5eaee54433208ae018cab82a3dc81f61554b7677ba56cbd1e92bfa73abe79"],"state_sha256":"24591517c70fcf09fe497ff4eb3ef21e350e7997280041ea42ca1cce95d6f05f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1VteURpEobhnGaLv5qpBN3+ZUDUSpcihTawtE8nHtUE27SdryKzrzOcXnEU8O+15dudCjrmmWRxMAojnY0DkAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T13:22:18.330693Z","bundle_sha256":"96c62e3d566ba5bc2bc1da7a7e7b9d725e4d4998ea3b9884124ca2781ea4876d"}}