{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:6MIIPY3DEOPEA24PX6O2M5WFKA","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":"e8f3b1282def5766a2a2dac4e9723eafaa9332e23bb28de5d7ff09560b93a2dd","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-08-16T21:16:59Z","title_canon_sha256":"7511b248105be7ef372def5db73d918cfb561a36e0439ee7293509befc827336"},"schema_version":"1.0","source":{"id":"1908.06179","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.06179","created_at":"2026-07-04T23:58:13Z"},{"alias_kind":"arxiv_version","alias_value":"1908.06179v1","created_at":"2026-07-04T23:58:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.06179","created_at":"2026-07-04T23:58:13Z"},{"alias_kind":"pith_short_12","alias_value":"6MIIPY3DEOPE","created_at":"2026-07-04T23:58:13Z"},{"alias_kind":"pith_short_16","alias_value":"6MIIPY3DEOPEA24P","created_at":"2026-07-04T23:58:13Z"},{"alias_kind":"pith_short_8","alias_value":"6MIIPY3D","created_at":"2026-07-04T23:58:13Z"}],"graph_snapshots":[{"event_id":"sha256:e2c84b94fcde8e9676f3d0c64bf84e6454cb58d1995cdf440ffb4d82b7a94b34","target":"graph","created_at":"2026-07-04T23:58:13Z","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/1908.06179/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $d \\ge 1$, $p \\ge d$, and let $\\Omega$ be a smooth bounded open subset of $\\mathbb{R}^d$. We prove some exponential integrability in the spirit of Moser-Trudinger's inequalities for measurable functions $u$ defined in $\\Omega$ such that $$ \\mathop{\\int_{\\Omega} \\int_{\\Omega}}_{|u(x) - u(y)| > \\delta} \\frac{1}{|x-y|^{d+p}} \\, dx \\, dy < + \\infty, $$ for some $\\delta > 0$. This double integral appeared in characterizations of Sobolev spaces and involved in improvements of the Sobolev inequaliies, Poincar\\'e inequalities, and Hardy inequalities.","authors_text":"Arka Mallick, Hoai-Minh Nguyen","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-08-16T21:16:59Z","title":"Exponential integrability in the spirit of Moser-Trudinger's inequalities of functions with finite non-local, non-convex energy"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.06179","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":"sha256:034fed7c8f61cfb1c734793c7431be144ca249ca029e393b1cd7b6967fddc919","target":"record","created_at":"2026-07-04T23:58:13Z","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":"e8f3b1282def5766a2a2dac4e9723eafaa9332e23bb28de5d7ff09560b93a2dd","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-08-16T21:16:59Z","title_canon_sha256":"7511b248105be7ef372def5db73d918cfb561a36e0439ee7293509befc827336"},"schema_version":"1.0","source":{"id":"1908.06179","kind":"arxiv","version":1}},"canonical_sha256":"f31087e363239e406b8fbf9da676c5500f5be822e6ae1df16d8da05bab42cbd9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f31087e363239e406b8fbf9da676c5500f5be822e6ae1df16d8da05bab42cbd9","first_computed_at":"2026-07-04T23:58:13.330031Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T23:58:13.330031Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IrCZpYjgAKrDRJDZNSyrgrQbCM8zNrA+LQjGpIIXvL5SgzKkDg1dFGE+IXHiiBigOKCU9yV38gAAzCNf2dH7Dw==","signature_status":"signed_v1","signed_at":"2026-07-04T23:58:13.330414Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.06179","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:034fed7c8f61cfb1c734793c7431be144ca249ca029e393b1cd7b6967fddc919","sha256:e2c84b94fcde8e9676f3d0c64bf84e6454cb58d1995cdf440ffb4d82b7a94b34"],"state_sha256":"c0e96a98b47d79cd41edde16d67bbf0229300c97afba4e0aa62881b60b379fd0"}