{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:ELACHIQFIXBBSYPF5HTHXATIWC","short_pith_number":"pith:ELACHIQF","canonical_record":{"source":{"id":"2607.02084","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-07-02T12:25:47Z","cross_cats_sorted":[],"title_canon_sha256":"82ef39dbb4197764f549b487a2c5907a709b0b5c35681301d7a176f0e4c7d3a9","abstract_canon_sha256":"f836ca1124b057d38b9d897ceeedbbcaf7101b5d7425f6289786e5ec1552cc2b"},"schema_version":"1.0"},"canonical_sha256":"22c023a20545c21961e5e9e67b8268b0a1e99cbd9c743c756436ed217ac0441e","source":{"kind":"arxiv","id":"2607.02084","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.02084","created_at":"2026-07-03T01:17:41Z"},{"alias_kind":"arxiv_version","alias_value":"2607.02084v1","created_at":"2026-07-03T01:17:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.02084","created_at":"2026-07-03T01:17:41Z"},{"alias_kind":"pith_short_12","alias_value":"ELACHIQFIXBB","created_at":"2026-07-03T01:17:41Z"},{"alias_kind":"pith_short_16","alias_value":"ELACHIQFIXBBSYPF","created_at":"2026-07-03T01:17:41Z"},{"alias_kind":"pith_short_8","alias_value":"ELACHIQF","created_at":"2026-07-03T01:17:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:ELACHIQFIXBBSYPF5HTHXATIWC","target":"record","payload":{"canonical_record":{"source":{"id":"2607.02084","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-07-02T12:25:47Z","cross_cats_sorted":[],"title_canon_sha256":"82ef39dbb4197764f549b487a2c5907a709b0b5c35681301d7a176f0e4c7d3a9","abstract_canon_sha256":"f836ca1124b057d38b9d897ceeedbbcaf7101b5d7425f6289786e5ec1552cc2b"},"schema_version":"1.0"},"canonical_sha256":"22c023a20545c21961e5e9e67b8268b0a1e99cbd9c743c756436ed217ac0441e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-03T01:17:41.746931Z","signature_b64":"Um5KvFMQ7ANfevWlCTFyM98JHMAcbgHGIjKUWJkrcaySkiiWE1YXLEQ4WlHwOBeL2+PNHCQDYocJFbS+wwzLCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"22c023a20545c21961e5e9e67b8268b0a1e99cbd9c743c756436ed217ac0441e","last_reissued_at":"2026-07-03T01:17:41.746489Z","signature_status":"signed_v1","first_computed_at":"2026-07-03T01:17:41.746489Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.02084","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":"2026-07-03T01:17:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"19aWOKkR+LI84tfXZRIyqd+zOo1d5hAGsvPuqw9NvhM5gLoqrw0qojwi+CMe5wTJ4jXu6hcj1VoP0vB2zutqBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-10T08:55:07.047128Z"},"content_sha256":"4f62b8c28a3fcb05347ec6962fb971d694e1a24a7d34b59fb0c67b0f62ebe102","schema_version":"1.0","event_id":"sha256:4f62b8c28a3fcb05347ec6962fb971d694e1a24a7d34b59fb0c67b0f62ebe102"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:ELACHIQFIXBBSYPF5HTHXATIWC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weighted Extensions of Stein's Theorem for Linear and Multilinear Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Bae Jun Park, Mar\\'ia Jes\\'us Carro","submitted_at":"2026-07-02T12:25:47Z","abstract_excerpt":"We study weighted estimates for linear and multilinear integral operators whose kernels satisfy only size conditions. Extending a theorem of E. Stein and its refinement by Soria and Weiss, we prove weighted estimates on Herz and Ces\\`aro type spaces, together with multilinear strong-type and weak-type analogues. As applications, we derive consequences for a range of rough singular integral operators and related variants, including linear, oscillatory, and multilinear settings."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02084","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.02084/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-03T01:17:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U2CEBBvJOdWpRZv53e3WfJnBblw5eKUa8CwA2PzkBwY9h7l5pCz1RZNADGEAfE0KZxnY5QjHgk/2YRb25MzfDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-10T08:55:07.047514Z"},"content_sha256":"e68d1b7dedcd11936f90763841465817342a37acaedd8285c14cd67d2a001fd3","schema_version":"1.0","event_id":"sha256:e68d1b7dedcd11936f90763841465817342a37acaedd8285c14cd67d2a001fd3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ELACHIQFIXBBSYPF5HTHXATIWC/bundle.json","state_url":"https://pith.science/pith/ELACHIQFIXBBSYPF5HTHXATIWC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ELACHIQFIXBBSYPF5HTHXATIWC/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":"2026-07-10T08:55:07Z","links":{"resolver":"https://pith.science/pith/ELACHIQFIXBBSYPF5HTHXATIWC","bundle":"https://pith.science/pith/ELACHIQFIXBBSYPF5HTHXATIWC/bundle.json","state":"https://pith.science/pith/ELACHIQFIXBBSYPF5HTHXATIWC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ELACHIQFIXBBSYPF5HTHXATIWC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:ELACHIQFIXBBSYPF5HTHXATIWC","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":"f836ca1124b057d38b9d897ceeedbbcaf7101b5d7425f6289786e5ec1552cc2b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-07-02T12:25:47Z","title_canon_sha256":"82ef39dbb4197764f549b487a2c5907a709b0b5c35681301d7a176f0e4c7d3a9"},"schema_version":"1.0","source":{"id":"2607.02084","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.02084","created_at":"2026-07-03T01:17:41Z"},{"alias_kind":"arxiv_version","alias_value":"2607.02084v1","created_at":"2026-07-03T01:17:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.02084","created_at":"2026-07-03T01:17:41Z"},{"alias_kind":"pith_short_12","alias_value":"ELACHIQFIXBB","created_at":"2026-07-03T01:17:41Z"},{"alias_kind":"pith_short_16","alias_value":"ELACHIQFIXBBSYPF","created_at":"2026-07-03T01:17:41Z"},{"alias_kind":"pith_short_8","alias_value":"ELACHIQF","created_at":"2026-07-03T01:17:41Z"}],"graph_snapshots":[{"event_id":"sha256:e68d1b7dedcd11936f90763841465817342a37acaedd8285c14cd67d2a001fd3","target":"graph","created_at":"2026-07-03T01:17:41Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.02084/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study weighted estimates for linear and multilinear integral operators whose kernels satisfy only size conditions. Extending a theorem of E. Stein and its refinement by Soria and Weiss, we prove weighted estimates on Herz and Ces\\`aro type spaces, together with multilinear strong-type and weak-type analogues. As applications, we derive consequences for a range of rough singular integral operators and related variants, including linear, oscillatory, and multilinear settings.","authors_text":"Bae Jun Park, Mar\\'ia Jes\\'us Carro","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-07-02T12:25:47Z","title":"Weighted Extensions of Stein's Theorem for Linear and Multilinear Operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02084","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":"sha256:4f62b8c28a3fcb05347ec6962fb971d694e1a24a7d34b59fb0c67b0f62ebe102","target":"record","created_at":"2026-07-03T01:17:41Z","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":"f836ca1124b057d38b9d897ceeedbbcaf7101b5d7425f6289786e5ec1552cc2b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-07-02T12:25:47Z","title_canon_sha256":"82ef39dbb4197764f549b487a2c5907a709b0b5c35681301d7a176f0e4c7d3a9"},"schema_version":"1.0","source":{"id":"2607.02084","kind":"arxiv","version":1}},"canonical_sha256":"22c023a20545c21961e5e9e67b8268b0a1e99cbd9c743c756436ed217ac0441e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"22c023a20545c21961e5e9e67b8268b0a1e99cbd9c743c756436ed217ac0441e","first_computed_at":"2026-07-03T01:17:41.746489Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-03T01:17:41.746489Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Um5KvFMQ7ANfevWlCTFyM98JHMAcbgHGIjKUWJkrcaySkiiWE1YXLEQ4WlHwOBeL2+PNHCQDYocJFbS+wwzLCg==","signature_status":"signed_v1","signed_at":"2026-07-03T01:17:41.746931Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.02084","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4f62b8c28a3fcb05347ec6962fb971d694e1a24a7d34b59fb0c67b0f62ebe102","sha256:e68d1b7dedcd11936f90763841465817342a37acaedd8285c14cd67d2a001fd3"],"state_sha256":"81e168f882af6e0cf4900a9a17e7f926ba4b8381db01fc1bea2bc613a2bf0500"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2TRvL4Ovc1U2OhQh/NXZoq/EqtUryNxJIeUMpA8fWv+KEZadHYvAANHJl2K6DDauRiq9rqmQK6iDZAiXvZApBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-10T08:55:07.049737Z","bundle_sha256":"de1dc23d12a61c54b693cbfb007d96f679b0168f76681b11c8264df69c2514ad"}}