{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:YTV6JEIPU5OROSE56SNEI3TYVS","short_pith_number":"pith:YTV6JEIP","canonical_record":{"source":{"id":"1212.4018","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-12-17T15:12:33Z","cross_cats_sorted":[],"title_canon_sha256":"09930e849621cec45e17f015b007f8a6c9d902f1be1cbef3d02b759cf79914c3","abstract_canon_sha256":"207adc868022e5d3144d0daa4ba12fa50b3f4cd1569ccd1d0f5795a2097b0faf"},"schema_version":"1.0"},"canonical_sha256":"c4ebe4910fa75d17489df49a446e78acbc5ae80cdfd8a4e377c738e9058eccb2","source":{"kind":"arxiv","id":"1212.4018","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.4018","created_at":"2026-05-18T03:29:06Z"},{"alias_kind":"arxiv_version","alias_value":"1212.4018v2","created_at":"2026-05-18T03:29:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.4018","created_at":"2026-05-18T03:29:06Z"},{"alias_kind":"pith_short_12","alias_value":"YTV6JEIPU5OR","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"YTV6JEIPU5OROSE5","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"YTV6JEIP","created_at":"2026-05-18T12:27:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:YTV6JEIPU5OROSE56SNEI3TYVS","target":"record","payload":{"canonical_record":{"source":{"id":"1212.4018","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-12-17T15:12:33Z","cross_cats_sorted":[],"title_canon_sha256":"09930e849621cec45e17f015b007f8a6c9d902f1be1cbef3d02b759cf79914c3","abstract_canon_sha256":"207adc868022e5d3144d0daa4ba12fa50b3f4cd1569ccd1d0f5795a2097b0faf"},"schema_version":"1.0"},"canonical_sha256":"c4ebe4910fa75d17489df49a446e78acbc5ae80cdfd8a4e377c738e9058eccb2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:29:06.458846Z","signature_b64":"l78H+1U3UJrG6+Zo9Jeb1euEFM/xd6wyDhCYl8zjOBN+50P1Svc5PxD2ncUn5tuJ+vFJYBJbzK9TN7qKTgk0Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c4ebe4910fa75d17489df49a446e78acbc5ae80cdfd8a4e377c738e9058eccb2","last_reissued_at":"2026-05-18T03:29:06.458411Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:29:06.458411Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1212.4018","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":"2026-05-18T03:29:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YbwalXE/WniZmIjXQI3MBH4cakkfOLPeaYulBfpnikflmeAuGaemDVIh56a75Mgn77c/1vsXyCZlt2dX1JNoBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T09:08:55.327435Z"},"content_sha256":"06cfae500224d306b4fe1bc1d0ffb45e99bbb8c82f8c7b7e535e86f75b9a63f3","schema_version":"1.0","event_id":"sha256:06cfae500224d306b4fe1bc1d0ffb45e99bbb8c82f8c7b7e535e86f75b9a63f3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:YTV6JEIPU5OROSE56SNEI3TYVS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The bilinear Bochner-Riesz problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Frederic Bernicot (LMJL), Liang Song, Lixin Yan, Loukas Grafakos (MU Mathematics)","submitted_at":"2012-12-17T15:12:33Z","abstract_excerpt":"Motivated by the problem of spherical summability of products of Fourier series, we study the boundedness of the bilinear Bochner-Riesz multipliers $(1-|\\xi|^2-|\\eta|^2)^\\delta_+$ and we make some advances in this investigation. We obtain an optimal result concerning the boundedness of these means from $L^2\\times L^2 $ into $L^1$ with minimal smoothness, i.e., any $\\delta>0$, and we obtain estimates for other pairs of spaces for larger values of $\\delta$. Our study is broad enough to encompass general bilinear multipliers $m(\\xi,\\eta)$ radial in $\\xi$ and $\\eta$ with minimal smoothness, measur"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.4018","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":"2026-05-18T03:29:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"L6IS7m9gWP1Jv5IS/eC8kyurdIiXJQghzsDwh+rK7E+arUcGb0IJJ/dsFKyJY9CFq3ryTMm22o1qp0f4G/rdBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T09:08:55.327773Z"},"content_sha256":"d226073bb72727cbcb3656369c0947421de47321f89697ab9370ca05b6bd3a61","schema_version":"1.0","event_id":"sha256:d226073bb72727cbcb3656369c0947421de47321f89697ab9370ca05b6bd3a61"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YTV6JEIPU5OROSE56SNEI3TYVS/bundle.json","state_url":"https://pith.science/pith/YTV6JEIPU5OROSE56SNEI3TYVS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YTV6JEIPU5OROSE56SNEI3TYVS/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":"2026-06-06T09:08:55Z","links":{"resolver":"https://pith.science/pith/YTV6JEIPU5OROSE56SNEI3TYVS","bundle":"https://pith.science/pith/YTV6JEIPU5OROSE56SNEI3TYVS/bundle.json","state":"https://pith.science/pith/YTV6JEIPU5OROSE56SNEI3TYVS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YTV6JEIPU5OROSE56SNEI3TYVS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:YTV6JEIPU5OROSE56SNEI3TYVS","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":"207adc868022e5d3144d0daa4ba12fa50b3f4cd1569ccd1d0f5795a2097b0faf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-12-17T15:12:33Z","title_canon_sha256":"09930e849621cec45e17f015b007f8a6c9d902f1be1cbef3d02b759cf79914c3"},"schema_version":"1.0","source":{"id":"1212.4018","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.4018","created_at":"2026-05-18T03:29:06Z"},{"alias_kind":"arxiv_version","alias_value":"1212.4018v2","created_at":"2026-05-18T03:29:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.4018","created_at":"2026-05-18T03:29:06Z"},{"alias_kind":"pith_short_12","alias_value":"YTV6JEIPU5OR","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"YTV6JEIPU5OROSE5","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"YTV6JEIP","created_at":"2026-05-18T12:27:30Z"}],"graph_snapshots":[{"event_id":"sha256:d226073bb72727cbcb3656369c0947421de47321f89697ab9370ca05b6bd3a61","target":"graph","created_at":"2026-05-18T03:29:06Z","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":"Motivated by the problem of spherical summability of products of Fourier series, we study the boundedness of the bilinear Bochner-Riesz multipliers $(1-|\\xi|^2-|\\eta|^2)^\\delta_+$ and we make some advances in this investigation. We obtain an optimal result concerning the boundedness of these means from $L^2\\times L^2 $ into $L^1$ with minimal smoothness, i.e., any $\\delta>0$, and we obtain estimates for other pairs of spaces for larger values of $\\delta$. Our study is broad enough to encompass general bilinear multipliers $m(\\xi,\\eta)$ radial in $\\xi$ and $\\eta$ with minimal smoothness, measur","authors_text":"Frederic Bernicot (LMJL), Liang Song, Lixin Yan, Loukas Grafakos (MU Mathematics)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-12-17T15:12:33Z","title":"The bilinear Bochner-Riesz problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.4018","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":"sha256:06cfae500224d306b4fe1bc1d0ffb45e99bbb8c82f8c7b7e535e86f75b9a63f3","target":"record","created_at":"2026-05-18T03:29:06Z","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":"207adc868022e5d3144d0daa4ba12fa50b3f4cd1569ccd1d0f5795a2097b0faf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-12-17T15:12:33Z","title_canon_sha256":"09930e849621cec45e17f015b007f8a6c9d902f1be1cbef3d02b759cf79914c3"},"schema_version":"1.0","source":{"id":"1212.4018","kind":"arxiv","version":2}},"canonical_sha256":"c4ebe4910fa75d17489df49a446e78acbc5ae80cdfd8a4e377c738e9058eccb2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c4ebe4910fa75d17489df49a446e78acbc5ae80cdfd8a4e377c738e9058eccb2","first_computed_at":"2026-05-18T03:29:06.458411Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:29:06.458411Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"l78H+1U3UJrG6+Zo9Jeb1euEFM/xd6wyDhCYl8zjOBN+50P1Svc5PxD2ncUn5tuJ+vFJYBJbzK9TN7qKTgk0Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:29:06.458846Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.4018","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:06cfae500224d306b4fe1bc1d0ffb45e99bbb8c82f8c7b7e535e86f75b9a63f3","sha256:d226073bb72727cbcb3656369c0947421de47321f89697ab9370ca05b6bd3a61"],"state_sha256":"4fd5f66aac5884aa561033dddfbf41160ec7fe0e4cdc11b88a366629a21c6be0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QAPQVZlGyANmMpTmNS+1lLfSy7j22mXJKbhYF/ikoj44eBCyD5j25zbFKEFEAZyTZ2BtA2XaICRA53wHCjRrDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T09:08:55.330029Z","bundle_sha256":"224129f73bce7f2dd36fbf4e2ed9a884c19957959e26bc5be82d2b10f5a7b175"}}