{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:RLEGRIR42XQP7Z47LWKL5ZV7VF","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":"b9e5fa47bf25de16402fb62d47b9a00387ad08c898f6a0da4233dc8ec133b33c","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-08-09T15:40:07Z","title_canon_sha256":"a9f7ff27dcffc929fc62a0c646e267d2c2190b6c9c69a1bc0db43f0b638489fc"},"schema_version":"1.0","source":{"id":"1508.02041","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.02041","created_at":"2026-05-18T00:06:51Z"},{"alias_kind":"arxiv_version","alias_value":"1508.02041v2","created_at":"2026-05-18T00:06:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.02041","created_at":"2026-05-18T00:06:51Z"},{"alias_kind":"pith_short_12","alias_value":"RLEGRIR42XQP","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"RLEGRIR42XQP7Z47","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"RLEGRIR4","created_at":"2026-05-18T12:29:39Z"}],"graph_snapshots":[{"event_id":"sha256:429b773cac8080490531765904b76ca1841d444998c562073c1bb96d17e1e921","target":"graph","created_at":"2026-05-18T00:06:51Z","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":"This is the first in our series of papers concerning some Hardy-Littlewood-Sobolev type inequalities. In the present paper, the main objective is to establish the following sharp reversed HLS inequality in the whole space $\\mathbb R^n$ \\[\\int_{\\mathbb R^n} \\int_{\\mathbb R^n} f(x) |x-y|^\\lambda g(y) dx dy \\geqslant \\mathscr C_{n,p,r} \\|f\\|_{L^p (\\mathbb R^n)}\\, \\|g\\|_{L^r (\\mathbb R^n)}\\] for any nonnegative functions $f\\in L^p(\\mathbb R^n)$, $g\\in L^r(\\mathbb R^n)$, and $p,r\\in (0,1)$, $\\lambda > 0$ such that $1/p + 1/r -\\lambda /n =2$. We will also explore some estimates for $\\mathscr C_{n,p,","authors_text":"Qu\\^oc-Anh Ng\\^o, Van Hoang Nguyen","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-08-09T15:40:07Z","title":"Sharp reversed Hardy-Littlewood-Sobolev inequality on $\\mathbb R^n$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.02041","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":"sha256:485dd9e64ded8b536f732f80b64c4afeb0bd23d18bacf8c7ccad1d4dd5d2cf38","target":"record","created_at":"2026-05-18T00:06:51Z","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":"b9e5fa47bf25de16402fb62d47b9a00387ad08c898f6a0da4233dc8ec133b33c","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-08-09T15:40:07Z","title_canon_sha256":"a9f7ff27dcffc929fc62a0c646e267d2c2190b6c9c69a1bc0db43f0b638489fc"},"schema_version":"1.0","source":{"id":"1508.02041","kind":"arxiv","version":2}},"canonical_sha256":"8ac868a23cd5e0ffe79f5d94bee6bfa965ddb98f07d3e4edd41409469070ae07","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8ac868a23cd5e0ffe79f5d94bee6bfa965ddb98f07d3e4edd41409469070ae07","first_computed_at":"2026-05-18T00:06:51.685407Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:06:51.685407Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"F2tc6mNhwWiGt9h3uzPBizKTe9pxtLr12qH8sdwDwn8ZVrwPfAlzX8u7LSHNfVIKRMF0hCAsAFy6bNvXVqCKDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:06:51.685970Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.02041","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:485dd9e64ded8b536f732f80b64c4afeb0bd23d18bacf8c7ccad1d4dd5d2cf38","sha256:429b773cac8080490531765904b76ca1841d444998c562073c1bb96d17e1e921"],"state_sha256":"bbfcaae17435b5578aaf497090ad5fe51d6acbf4060f8b142e331799c04ea694"}