{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:5TTAFZA7JPTOYHKF6YF64QZEC7","short_pith_number":"pith:5TTAFZA7","schema_version":"1.0","canonical_sha256":"ece602e41f4be6ec1d45f60bee432417ee027e55200968f9946099355608be9d","source":{"kind":"arxiv","id":"2507.20069","version":1},"attestation_state":"computed","paper":{"title":"Fractional Trudinger-Moser type inequalities with logarithmic convolution potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Huxiao Luo, Shiying Wang","submitted_at":"2025-07-26T21:51:32Z","abstract_excerpt":"We establish the following fractional Trudinger-Moser type inequality with logarithmic convolution potential $$ \\sup_{u\\in W^{\\frac{1}{2},2}_0(I),\\|u\\|_{W_0^{\\frac{1}{2},2}}\\leq1}\\int_{I} \\int_{I} \\log \\frac{1}{|x-y|} G(u(x))G(u(y)) \\, dx \\, dy<+\\infty,$$ where $G(s)\\leq C\\frac{e^{\\pi s^{2}}}{(1 + |s|)^{\\gamma}}~ \\forall s\\in\\mathbb{R}$ with some constant $C>0,\\gamma\\geq1$, the domain $I\\subset\\mathbb{R}$ is a bounded interval. This type of inequality in the entire space $\\mathbb{R}$ is also considered. Moreover, we study the existence of corresponding extremal functions.\n  In addition, by the"},"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":"2507.20069","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-07-26T21:51:32Z","cross_cats_sorted":[],"title_canon_sha256":"1b81b42d6737288cc3b7aa9373ece949fe45718f4991b38fcfa383e7c3a6b8db","abstract_canon_sha256":"bf103be72ef5c50fe25fec303170e92e66c2b6456f5924e48e630609b6e64286"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:44:12.781744Z","signature_b64":"dpQMnIwLSFEaZDZzJYxDJx4Q9xOHkH4/UfxaedbekCCS0qJB+icYAi39nj1V0i2UICOzH/oqtMlhY2gc9cCMBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ece602e41f4be6ec1d45f60bee432417ee027e55200968f9946099355608be9d","last_reissued_at":"2026-07-05T11:44:12.781279Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:44:12.781279Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fractional Trudinger-Moser type inequalities with logarithmic convolution potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Huxiao Luo, Shiying Wang","submitted_at":"2025-07-26T21:51:32Z","abstract_excerpt":"We establish the following fractional Trudinger-Moser type inequality with logarithmic convolution potential $$ \\sup_{u\\in W^{\\frac{1}{2},2}_0(I),\\|u\\|_{W_0^{\\frac{1}{2},2}}\\leq1}\\int_{I} \\int_{I} \\log \\frac{1}{|x-y|} G(u(x))G(u(y)) \\, dx \\, dy<+\\infty,$$ where $G(s)\\leq C\\frac{e^{\\pi s^{2}}}{(1 + |s|)^{\\gamma}}~ \\forall s\\in\\mathbb{R}$ with some constant $C>0,\\gamma\\geq1$, the domain $I\\subset\\mathbb{R}$ is a bounded interval. This type of inequality in the entire space $\\mathbb{R}$ is also considered. Moreover, we study the existence of corresponding extremal functions.\n  In addition, by the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.20069","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/2507.20069/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":"2507.20069","created_at":"2026-07-05T11:44:12.781337+00:00"},{"alias_kind":"arxiv_version","alias_value":"2507.20069v1","created_at":"2026-07-05T11:44:12.781337+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.20069","created_at":"2026-07-05T11:44:12.781337+00:00"},{"alias_kind":"pith_short_12","alias_value":"5TTAFZA7JPTO","created_at":"2026-07-05T11:44:12.781337+00:00"},{"alias_kind":"pith_short_16","alias_value":"5TTAFZA7JPTOYHKF","created_at":"2026-07-05T11:44:12.781337+00:00"},{"alias_kind":"pith_short_8","alias_value":"5TTAFZA7","created_at":"2026-07-05T11:44:12.781337+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/5TTAFZA7JPTOYHKF6YF64QZEC7","json":"https://pith.science/pith/5TTAFZA7JPTOYHKF6YF64QZEC7.json","graph_json":"https://pith.science/api/pith-number/5TTAFZA7JPTOYHKF6YF64QZEC7/graph.json","events_json":"https://pith.science/api/pith-number/5TTAFZA7JPTOYHKF6YF64QZEC7/events.json","paper":"https://pith.science/paper/5TTAFZA7"},"agent_actions":{"view_html":"https://pith.science/pith/5TTAFZA7JPTOYHKF6YF64QZEC7","download_json":"https://pith.science/pith/5TTAFZA7JPTOYHKF6YF64QZEC7.json","view_paper":"https://pith.science/paper/5TTAFZA7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2507.20069&json=true","fetch_graph":"https://pith.science/api/pith-number/5TTAFZA7JPTOYHKF6YF64QZEC7/graph.json","fetch_events":"https://pith.science/api/pith-number/5TTAFZA7JPTOYHKF6YF64QZEC7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5TTAFZA7JPTOYHKF6YF64QZEC7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5TTAFZA7JPTOYHKF6YF64QZEC7/action/storage_attestation","attest_author":"https://pith.science/pith/5TTAFZA7JPTOYHKF6YF64QZEC7/action/author_attestation","sign_citation":"https://pith.science/pith/5TTAFZA7JPTOYHKF6YF64QZEC7/action/citation_signature","submit_replication":"https://pith.science/pith/5TTAFZA7JPTOYHKF6YF64QZEC7/action/replication_record"}},"created_at":"2026-07-05T11:44:12.781337+00:00","updated_at":"2026-07-05T11:44:12.781337+00:00"}