{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:MNT64GHPNJJGTR2AC5G6H3QQAA","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":"219589fc12efe638245578d61eebda34b5010a4bd727ee2349ec5930aef03585","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-08T23:53:20Z","title_canon_sha256":"68974bac12c68ff31382efb37ab57c5128c5c3306ded061e6135370f3c43ba2b"},"schema_version":"1.0","source":{"id":"1602.02821","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.02821","created_at":"2026-05-18T01:19:52Z"},{"alias_kind":"arxiv_version","alias_value":"1602.02821v2","created_at":"2026-05-18T01:19:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.02821","created_at":"2026-05-18T01:19:52Z"},{"alias_kind":"pith_short_12","alias_value":"MNT64GHPNJJG","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"MNT64GHPNJJGTR2A","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"MNT64GHP","created_at":"2026-05-18T12:30:32Z"}],"graph_snapshots":[{"event_id":"sha256:1909677ff9ea8df869d1904e240b5549e8548803917165f65818995cf3e5e947","target":"graph","created_at":"2026-05-18T01:19:52Z","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":"Fix $d\\geq 2$, and $s\\in (d-1,d)$. We characterize the non-negative locally finite non-atomic Borel measures $\\mu$ in $\\mathbb{R}^d$ for which the associated $s$-Riesz transform is bounded in $L^2(\\mu)$ in terms of the Wolff energy. This extends the range of $s$ in which the Mateu-Prat-Verdera characterization of measures with bounded $s$-Riesz transform is known.\n  As an application, we give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator $(-\\Delta)^{\\alpha/2}$, $\\alpha\\in (1,2)$, in terms of a well-known capacity","authors_text":"Benjamin Jaye, Fedor Nazarov, Maria Carmen Reguera, Xavier Tolsa","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-08T23:53:20Z","title":"The Riesz transform of codimension smaller than one and the Wolff energy"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.02821","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:e64c9f962982bcfd04993a98db500166ccd165f8db841966539f5c917717088a","target":"record","created_at":"2026-05-18T01:19:52Z","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":"219589fc12efe638245578d61eebda34b5010a4bd727ee2349ec5930aef03585","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-08T23:53:20Z","title_canon_sha256":"68974bac12c68ff31382efb37ab57c5128c5c3306ded061e6135370f3c43ba2b"},"schema_version":"1.0","source":{"id":"1602.02821","kind":"arxiv","version":2}},"canonical_sha256":"6367ee18ef6a5269c740174de3ee10000e7a47dce2e08db0579a4d182edfe8aa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6367ee18ef6a5269c740174de3ee10000e7a47dce2e08db0579a4d182edfe8aa","first_computed_at":"2026-05-18T01:19:52.960590Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:52.960590Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"60gDzsaAfY0VBEPgANYFbZ08XiaVNH80B0UJky0sgMB+74k2DJ3mDMiUmfy7XZ9WRB7ce/zTYYXNgVTbx3G4Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:52.961172Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.02821","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e64c9f962982bcfd04993a98db500166ccd165f8db841966539f5c917717088a","sha256:1909677ff9ea8df869d1904e240b5549e8548803917165f65818995cf3e5e947"],"state_sha256":"163d811e8b54d4517988f21079340ff5f7d70ee8b9feda64d05917805690f116"}