{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:KOA7OOIDJJWNABF7J3JAVNNSZK","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":"0cf941ca4d355b5fdc763ed75a8744230fcd4e54a9a07eb50cf941bbe0beee7f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-03-06T20:20:44Z","title_canon_sha256":"611df49130c87a0162a60c0704df8e35eb20d48a7d04ed620b5f8e459deae385"},"schema_version":"1.0","source":{"id":"1803.02413","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.02413","created_at":"2026-05-18T00:21:50Z"},{"alias_kind":"arxiv_version","alias_value":"1803.02413v1","created_at":"2026-05-18T00:21:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.02413","created_at":"2026-05-18T00:21:50Z"},{"alias_kind":"pith_short_12","alias_value":"KOA7OOIDJJWN","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_16","alias_value":"KOA7OOIDJJWNABF7","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_8","alias_value":"KOA7OOID","created_at":"2026-05-18T12:32:33Z"}],"graph_snapshots":[{"event_id":"sha256:2d265e0752711f4e1b3f4dc05e6c6f42b0f4cc9c67e3957c5525503e47799661","target":"graph","created_at":"2026-05-18T00:21:50Z","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":"In this paper we provide necessary and sufficient conditions for the $\\textnormal{weak}(1,p)$ boundedness, $1< p<\\infty,$ of convolution operators on locally compact (Hausdorff) topological groups. So, we generalize a classical result due to Sobolev-Hardy-Littlewood and Stepanov. Applications to Fourier multipliers on Lie groups also are given.","authors_text":"Duv\\'an Cardona","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-03-06T20:20:44Z","title":"The weak type $(1,p)$ for convolution operators on locally compact groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.02413","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":"sha256:34b10dcf7386805ca624a7b5a2b0a0c3d4e497bb9ade990cd733b5f04fe3bffa","target":"record","created_at":"2026-05-18T00:21:50Z","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":"0cf941ca4d355b5fdc763ed75a8744230fcd4e54a9a07eb50cf941bbe0beee7f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-03-06T20:20:44Z","title_canon_sha256":"611df49130c87a0162a60c0704df8e35eb20d48a7d04ed620b5f8e459deae385"},"schema_version":"1.0","source":{"id":"1803.02413","kind":"arxiv","version":1}},"canonical_sha256":"5381f739034a6cd004bf4ed20ab5b2caa3c9189f428eaa91eccc85df1419fd71","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5381f739034a6cd004bf4ed20ab5b2caa3c9189f428eaa91eccc85df1419fd71","first_computed_at":"2026-05-18T00:21:50.365887Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:21:50.365887Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"g4+Bwakzxt0nUhkZqGmOrbAYb7pVY8eZ6WgBbZtk141oALZ20UHpSPeIX+XUBYNJSWXuEmALD8GYDW8TIok7Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T00:21:50.366495Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.02413","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:34b10dcf7386805ca624a7b5a2b0a0c3d4e497bb9ade990cd733b5f04fe3bffa","sha256:2d265e0752711f4e1b3f4dc05e6c6f42b0f4cc9c67e3957c5525503e47799661"],"state_sha256":"0af973a071dcfeea029af8372fc287f63fc9a4086e682088d5887584ed33aafb"}