{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:OEKLGPZKTF7K4IB7IRBKL372TM","short_pith_number":"pith:OEKLGPZK","schema_version":"1.0","canonical_sha256":"7114b33f2a997eae203f4442a5effa9b31ce0ac72e7ca5897c74762027cce7a6","source":{"kind":"arxiv","id":"1703.03089","version":1},"attestation_state":"computed","paper":{"title":"Weak type operator Lipschitz and commutator estimates for commuting tuples","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Dmitriy Zanin, Fedor Sukochev, Martijn Caspers","submitted_at":"2017-03-09T00:59:23Z","abstract_excerpt":"Let $f: \\mathbb{R}^d \\to\\mathbb{R}$ be a Lipschitz function. If $B$ is a bounded self-adjoint operator and if $\\{A_k\\}_{k=1}^d$ are commuting bounded self-adjoint operators such that $[A_k,B]\\in L_1(H),$ then $$\\|[f(A_1,\\cdots,A_d),B]\\|_{1,\\infty}\\leq c(d)\\|\\nabla(f)\\|_{\\infty}\\max_{1\\leq k\\leq d}\\|[A_k,B]\\|_1,$$ where $c(d)$ is a constant independent of $f$, $\\mathcal{M}$ and $A,B$ and $\\|\\cdot\\|_{1,\\infty}$ denotes the weak $L_1$-norm. If $\\{X_k\\}_{k=1}^d$ (respectively, $\\{Y_k\\}_{k=1}^d$) are commuting bounded self-adjoint operators such that $X_k-Y_k\\in L_1(H),$ then $$\\|f(X_1,\\cdots,X_d)-"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1703.03089","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2017-03-09T00:59:23Z","cross_cats_sorted":[],"title_canon_sha256":"c60d2cfb5ca14271aeaf3dbc60eddf5b77289d39f965849d619c811b4f5c8b2d","abstract_canon_sha256":"6ce6e48faafc93a075a409b6d4eb63cc28bc31f4b6a9302487a13e6baceab43a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:01.609102Z","signature_b64":"PHCb2aFkZKUoexOIX+3UK8Ze4Wb0TH24aW+QrFz/imf0ihJzatOAUIoY1wUKWpgo0KzascV0KQ1lIpc57WcUDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7114b33f2a997eae203f4442a5effa9b31ce0ac72e7ca5897c74762027cce7a6","last_reissued_at":"2026-05-18T00:49:01.608307Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:01.608307Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weak type operator Lipschitz and commutator estimates for commuting tuples","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Dmitriy Zanin, Fedor Sukochev, Martijn Caspers","submitted_at":"2017-03-09T00:59:23Z","abstract_excerpt":"Let $f: \\mathbb{R}^d \\to\\mathbb{R}$ be a Lipschitz function. If $B$ is a bounded self-adjoint operator and if $\\{A_k\\}_{k=1}^d$ are commuting bounded self-adjoint operators such that $[A_k,B]\\in L_1(H),$ then $$\\|[f(A_1,\\cdots,A_d),B]\\|_{1,\\infty}\\leq c(d)\\|\\nabla(f)\\|_{\\infty}\\max_{1\\leq k\\leq d}\\|[A_k,B]\\|_1,$$ where $c(d)$ is a constant independent of $f$, $\\mathcal{M}$ and $A,B$ and $\\|\\cdot\\|_{1,\\infty}$ denotes the weak $L_1$-norm. If $\\{X_k\\}_{k=1}^d$ (respectively, $\\{Y_k\\}_{k=1}^d$) are commuting bounded self-adjoint operators such that $X_k-Y_k\\in L_1(H),$ then $$\\|f(X_1,\\cdots,X_d)-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.03089","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":"1703.03089","created_at":"2026-05-18T00:49:01.608448+00:00"},{"alias_kind":"arxiv_version","alias_value":"1703.03089v1","created_at":"2026-05-18T00:49:01.608448+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.03089","created_at":"2026-05-18T00:49:01.608448+00:00"},{"alias_kind":"pith_short_12","alias_value":"OEKLGPZKTF7K","created_at":"2026-05-18T12:31:34.259226+00:00"},{"alias_kind":"pith_short_16","alias_value":"OEKLGPZKTF7K4IB7","created_at":"2026-05-18T12:31:34.259226+00:00"},{"alias_kind":"pith_short_8","alias_value":"OEKLGPZK","created_at":"2026-05-18T12:31:34.259226+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/OEKLGPZKTF7K4IB7IRBKL372TM","json":"https://pith.science/pith/OEKLGPZKTF7K4IB7IRBKL372TM.json","graph_json":"https://pith.science/api/pith-number/OEKLGPZKTF7K4IB7IRBKL372TM/graph.json","events_json":"https://pith.science/api/pith-number/OEKLGPZKTF7K4IB7IRBKL372TM/events.json","paper":"https://pith.science/paper/OEKLGPZK"},"agent_actions":{"view_html":"https://pith.science/pith/OEKLGPZKTF7K4IB7IRBKL372TM","download_json":"https://pith.science/pith/OEKLGPZKTF7K4IB7IRBKL372TM.json","view_paper":"https://pith.science/paper/OEKLGPZK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1703.03089&json=true","fetch_graph":"https://pith.science/api/pith-number/OEKLGPZKTF7K4IB7IRBKL372TM/graph.json","fetch_events":"https://pith.science/api/pith-number/OEKLGPZKTF7K4IB7IRBKL372TM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OEKLGPZKTF7K4IB7IRBKL372TM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OEKLGPZKTF7K4IB7IRBKL372TM/action/storage_attestation","attest_author":"https://pith.science/pith/OEKLGPZKTF7K4IB7IRBKL372TM/action/author_attestation","sign_citation":"https://pith.science/pith/OEKLGPZKTF7K4IB7IRBKL372TM/action/citation_signature","submit_replication":"https://pith.science/pith/OEKLGPZKTF7K4IB7IRBKL372TM/action/replication_record"}},"created_at":"2026-05-18T00:49:01.608448+00:00","updated_at":"2026-05-18T00:49:01.608448+00:00"}