{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:N7NNS7HKRBBU6OQLDSDYBNAEH5","short_pith_number":"pith:N7NNS7HK","schema_version":"1.0","canonical_sha256":"6fdad97cea88434f3a0b1c8780b4043f770344f13046f21a8d147325175c75c6","source":{"kind":"arxiv","id":"1808.05159","version":1},"attestation_state":"computed","paper":{"title":"User's guide to the fractional Laplacian and the method of semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.FA"],"primary_cat":"math.AP","authors_text":"P. R. Stinga","submitted_at":"2018-08-15T15:53:28Z","abstract_excerpt":"The \\textit{method of semigroups} is a unifying, widely applicable, general technique to formulate and analyze fundamental aspects of fractional powers of operators $L$ and their regularity properties in related functional spaces. The approach was introduced by the author and Jos\\'e L.~Torrea in 2009 (arXiv:0910.2569v1). The aim of this chapter is to show how the method works in the particular case of the fractional Laplacian $L^s=(-\\Delta)^s$, $0<s<1$. The starting point is the semigroup formula for the fractional Laplacian. From here, the classical heat kernel permits us to obtain the pointw"},"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":"1808.05159","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-08-15T15:53:28Z","cross_cats_sorted":["math.CA","math.FA"],"title_canon_sha256":"2c10125812b90d71d083f3ec2afa4b6c29261af0d83e3ede78008dba991fa0eb","abstract_canon_sha256":"b5a9e000d3b9e883831fa103b52e413b92a91691ad1765e8d654fea72b7f6de4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:01.447843Z","signature_b64":"HhksUH52TNUk3u1PfvCHV1kXKrwtq8KmsUJWd3wQvk4trn7y6Lo4TXgRAE7WJASg1hAPAX0ThmgPC1d9PbaVBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6fdad97cea88434f3a0b1c8780b4043f770344f13046f21a8d147325175c75c6","last_reissued_at":"2026-05-18T00:08:01.447282Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:01.447282Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"User's guide to the fractional Laplacian and the method of semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.FA"],"primary_cat":"math.AP","authors_text":"P. R. Stinga","submitted_at":"2018-08-15T15:53:28Z","abstract_excerpt":"The \\textit{method of semigroups} is a unifying, widely applicable, general technique to formulate and analyze fundamental aspects of fractional powers of operators $L$ and their regularity properties in related functional spaces. The approach was introduced by the author and Jos\\'e L.~Torrea in 2009 (arXiv:0910.2569v1). The aim of this chapter is to show how the method works in the particular case of the fractional Laplacian $L^s=(-\\Delta)^s$, $0<s<1$. The starting point is the semigroup formula for the fractional Laplacian. From here, the classical heat kernel permits us to obtain the pointw"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.05159","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":""},"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":"1808.05159","created_at":"2026-05-18T00:08:01.447365+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.05159v1","created_at":"2026-05-18T00:08:01.447365+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.05159","created_at":"2026-05-18T00:08:01.447365+00:00"},{"alias_kind":"pith_short_12","alias_value":"N7NNS7HKRBBU","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_16","alias_value":"N7NNS7HKRBBU6OQL","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_8","alias_value":"N7NNS7HK","created_at":"2026-05-18T12:32:40.477152+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/N7NNS7HKRBBU6OQLDSDYBNAEH5","json":"https://pith.science/pith/N7NNS7HKRBBU6OQLDSDYBNAEH5.json","graph_json":"https://pith.science/api/pith-number/N7NNS7HKRBBU6OQLDSDYBNAEH5/graph.json","events_json":"https://pith.science/api/pith-number/N7NNS7HKRBBU6OQLDSDYBNAEH5/events.json","paper":"https://pith.science/paper/N7NNS7HK"},"agent_actions":{"view_html":"https://pith.science/pith/N7NNS7HKRBBU6OQLDSDYBNAEH5","download_json":"https://pith.science/pith/N7NNS7HKRBBU6OQLDSDYBNAEH5.json","view_paper":"https://pith.science/paper/N7NNS7HK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.05159&json=true","fetch_graph":"https://pith.science/api/pith-number/N7NNS7HKRBBU6OQLDSDYBNAEH5/graph.json","fetch_events":"https://pith.science/api/pith-number/N7NNS7HKRBBU6OQLDSDYBNAEH5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/N7NNS7HKRBBU6OQLDSDYBNAEH5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/N7NNS7HKRBBU6OQLDSDYBNAEH5/action/storage_attestation","attest_author":"https://pith.science/pith/N7NNS7HKRBBU6OQLDSDYBNAEH5/action/author_attestation","sign_citation":"https://pith.science/pith/N7NNS7HKRBBU6OQLDSDYBNAEH5/action/citation_signature","submit_replication":"https://pith.science/pith/N7NNS7HKRBBU6OQLDSDYBNAEH5/action/replication_record"}},"created_at":"2026-05-18T00:08:01.447365+00:00","updated_at":"2026-05-18T00:08:01.447365+00:00"}