{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:QU2F4VDDCQ2CZ2LB6QHAKXUQ3L","short_pith_number":"pith:QU2F4VDD","schema_version":"1.0","canonical_sha256":"85345e546314342ce961f40e055e90daf0fe971c562020d842a0214acff53c17","source":{"kind":"arxiv","id":"2607.05928","version":1},"attestation_state":"computed","paper":{"title":"Reversed inequality of the Herbst-type and the related Euler-Lagrange system","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Tiantian Zhou, Yutian Lei","submitted_at":"2026-07-07T07:31:08Z","abstract_excerpt":"In 2008, Beckner (Proc. Amer. Math. Soc. 136(5), 1871-1885) proved two inequalities of the Herbst type, which are the critical forms of the Stein-Weiss inequality. In 2018, Chen et al. (Tran. Amer. Math. Soc. 370(12), 8429-8450) established the reversed Stein-Weiss inequality. In this paper, we are concerned about its critical case and give a reversed Herbst inequality. Namely,\n  $$\n  \\left|\\int_{\\mathbb{R}^n}\\int_{\\mathbb{R}^n}|x-y|^{\\alpha/q'-n}|y|^{\\alpha/q'}g(x)h(y)dxdy\\right|\n  \\geq C_{n,\\alpha,p,q'}\\|g\\|_{L^{q'}(\\mathbb{R}^n)}\\|h\\|_{L^p(\\mathbb{R}^n)}\n  $$ holds for any nonnegative funct"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.05928","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.AP","submitted_at":"2026-07-07T07:31:08Z","cross_cats_sorted":[],"title_canon_sha256":"257487a1a01d7fdc106fd226624103538ed66d0c9ad1c2f378180ac8b8134068","abstract_canon_sha256":"9f448f0a5f6fc46fef270c40120cc5ae3c4dca9ccf2c8aaf2d5cc2364a702c48"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-08T01:18:50.534606Z","signature_b64":"SEaNBijFxKX6amZiHXjHngwVGcclLeGYoaTO50dINoBwtBedmv88gEFNJNifBdCqKT3y24Graegw8deltlyRBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"85345e546314342ce961f40e055e90daf0fe971c562020d842a0214acff53c17","last_reissued_at":"2026-07-08T01:18:50.534116Z","signature_status":"signed_v1","first_computed_at":"2026-07-08T01:18:50.534116Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reversed inequality of the Herbst-type and the related Euler-Lagrange system","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Tiantian Zhou, Yutian Lei","submitted_at":"2026-07-07T07:31:08Z","abstract_excerpt":"In 2008, Beckner (Proc. Amer. Math. Soc. 136(5), 1871-1885) proved two inequalities of the Herbst type, which are the critical forms of the Stein-Weiss inequality. In 2018, Chen et al. (Tran. Amer. Math. Soc. 370(12), 8429-8450) established the reversed Stein-Weiss inequality. In this paper, we are concerned about its critical case and give a reversed Herbst inequality. Namely,\n  $$\n  \\left|\\int_{\\mathbb{R}^n}\\int_{\\mathbb{R}^n}|x-y|^{\\alpha/q'-n}|y|^{\\alpha/q'}g(x)h(y)dxdy\\right|\n  \\geq C_{n,\\alpha,p,q'}\\|g\\|_{L^{q'}(\\mathbb{R}^n)}\\|h\\|_{L^p(\\mathbb{R}^n)}\n  $$ holds for any nonnegative funct"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.05928","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/2607.05928/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.05928","created_at":"2026-07-08T01:18:50.534189+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.05928v1","created_at":"2026-07-08T01:18:50.534189+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.05928","created_at":"2026-07-08T01:18:50.534189+00:00"},{"alias_kind":"pith_short_12","alias_value":"QU2F4VDDCQ2C","created_at":"2026-07-08T01:18:50.534189+00:00"},{"alias_kind":"pith_short_16","alias_value":"QU2F4VDDCQ2CZ2LB","created_at":"2026-07-08T01:18:50.534189+00:00"},{"alias_kind":"pith_short_8","alias_value":"QU2F4VDD","created_at":"2026-07-08T01:18:50.534189+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/QU2F4VDDCQ2CZ2LB6QHAKXUQ3L","json":"https://pith.science/pith/QU2F4VDDCQ2CZ2LB6QHAKXUQ3L.json","graph_json":"https://pith.science/api/pith-number/QU2F4VDDCQ2CZ2LB6QHAKXUQ3L/graph.json","events_json":"https://pith.science/api/pith-number/QU2F4VDDCQ2CZ2LB6QHAKXUQ3L/events.json","paper":"https://pith.science/paper/QU2F4VDD"},"agent_actions":{"view_html":"https://pith.science/pith/QU2F4VDDCQ2CZ2LB6QHAKXUQ3L","download_json":"https://pith.science/pith/QU2F4VDDCQ2CZ2LB6QHAKXUQ3L.json","view_paper":"https://pith.science/paper/QU2F4VDD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.05928&json=true","fetch_graph":"https://pith.science/api/pith-number/QU2F4VDDCQ2CZ2LB6QHAKXUQ3L/graph.json","fetch_events":"https://pith.science/api/pith-number/QU2F4VDDCQ2CZ2LB6QHAKXUQ3L/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QU2F4VDDCQ2CZ2LB6QHAKXUQ3L/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QU2F4VDDCQ2CZ2LB6QHAKXUQ3L/action/storage_attestation","attest_author":"https://pith.science/pith/QU2F4VDDCQ2CZ2LB6QHAKXUQ3L/action/author_attestation","sign_citation":"https://pith.science/pith/QU2F4VDDCQ2CZ2LB6QHAKXUQ3L/action/citation_signature","submit_replication":"https://pith.science/pith/QU2F4VDDCQ2CZ2LB6QHAKXUQ3L/action/replication_record"}},"created_at":"2026-07-08T01:18:50.534189+00:00","updated_at":"2026-07-08T01:18:50.534189+00:00"}