{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:HRNPENA7BIQXNZJCT2CFIOB6QX","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":"0366d4cbe2b028d7703d18aeb411e45d7d5b33d00b502bb10725e073f805faec","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-20T07:30:33Z","title_canon_sha256":"28e62e1038d32de081972f353d6ba724214f39d18237b2a754a1366b61d296e2"},"schema_version":"1.0","source":{"id":"1806.07588","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.07588","created_at":"2026-05-18T00:06:24Z"},{"alias_kind":"arxiv_version","alias_value":"1806.07588v2","created_at":"2026-05-18T00:06:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.07588","created_at":"2026-05-18T00:06:24Z"},{"alias_kind":"pith_short_12","alias_value":"HRNPENA7BIQX","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_16","alias_value":"HRNPENA7BIQXNZJC","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_8","alias_value":"HRNPENA7","created_at":"2026-05-18T12:32:28Z"}],"graph_snapshots":[{"event_id":"sha256:32fe743e744d8444dea836208b4bf97468b95bf71ddb4125d4707b34d0355160","target":"graph","created_at":"2026-05-18T00:06:24Z","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 establish an optimal Lorentz space estimate for the Riesz potential acting on curl-free vectors: There is a constant $C=C(\\alpha,d)>0$ such that \\[ \\|I_\\alpha F \\|_{L^{d/(d-\\alpha),1}(\\mathbb{R}^d;\\mathbb{R}^d)} \\leq C \\|F\\|_{L^1(\\mathbb{R}^d;\\mathbb{R}^d)} \\] for all fields $F \\in L^1(\\mathbb{R}^d;\\mathbb{R}^d)$ such that $\\operatorname*{curl} F=0$ in the sense of distributions. This is the best possible estimate on this scale of spaces and completes the picture in the regime $p=1$ of the well-established results for $p>1$.","authors_text":"Daniel Spector","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-20T07:30:33Z","title":"An Optimal Sobolev Embedding for $L^1$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.07588","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":"sha256:7af1f55bc235f5ec75d3fcfff5b6a4f78460fc71b5ab559d45cca318abca14a1","target":"record","created_at":"2026-05-18T00:06:24Z","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":"0366d4cbe2b028d7703d18aeb411e45d7d5b33d00b502bb10725e073f805faec","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-20T07:30:33Z","title_canon_sha256":"28e62e1038d32de081972f353d6ba724214f39d18237b2a754a1366b61d296e2"},"schema_version":"1.0","source":{"id":"1806.07588","kind":"arxiv","version":2}},"canonical_sha256":"3c5af2341f0a2176e5229e8454383e85fd9c10150eed3729f6edcbbb9c779470","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3c5af2341f0a2176e5229e8454383e85fd9c10150eed3729f6edcbbb9c779470","first_computed_at":"2026-05-18T00:06:24.697851Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:06:24.697851Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BhbNlDLehJSIwfczRAaHBZJBXdmhhNvXSEXOZqampgxjr7aJoPTmPdserog2cMnVu952DhCk8uGg8Fj5prljBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:06:24.698520Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.07588","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7af1f55bc235f5ec75d3fcfff5b6a4f78460fc71b5ab559d45cca318abca14a1","sha256:32fe743e744d8444dea836208b4bf97468b95bf71ddb4125d4707b34d0355160"],"state_sha256":"6dc7a0fffec26c769afe0bef5f3dae90fdc0786d4fec7ab4a826bff53854dd29"}