{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:OEKLGPZKTF7K4IB7IRBKL372TM","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":"6ce6e48faafc93a075a409b6d4eb63cc28bc31f4b6a9302487a13e6baceab43a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2017-03-09T00:59:23Z","title_canon_sha256":"c60d2cfb5ca14271aeaf3dbc60eddf5b77289d39f965849d619c811b4f5c8b2d"},"schema_version":"1.0","source":{"id":"1703.03089","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.03089","created_at":"2026-05-18T00:49:01Z"},{"alias_kind":"arxiv_version","alias_value":"1703.03089v1","created_at":"2026-05-18T00:49:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.03089","created_at":"2026-05-18T00:49:01Z"},{"alias_kind":"pith_short_12","alias_value":"OEKLGPZKTF7K","created_at":"2026-05-18T12:31:34Z"},{"alias_kind":"pith_short_16","alias_value":"OEKLGPZKTF7K4IB7","created_at":"2026-05-18T12:31:34Z"},{"alias_kind":"pith_short_8","alias_value":"OEKLGPZK","created_at":"2026-05-18T12:31:34Z"}],"graph_snapshots":[{"event_id":"sha256:5641cc7cec31a4d530f5ab9d0c373a9e52fb0dfaa8603e6159f579983a346cc0","target":"graph","created_at":"2026-05-18T00:49:01Z","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":"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)-","authors_text":"Dmitriy Zanin, Fedor Sukochev, Martijn Caspers","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2017-03-09T00:59:23Z","title":"Weak type operator Lipschitz and commutator estimates for commuting tuples"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.03089","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":"sha256:50de2940fec4afc033c78a2e81f45b96e4d0861512f3a4d957209db9d49694ad","target":"record","created_at":"2026-05-18T00:49:01Z","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":"6ce6e48faafc93a075a409b6d4eb63cc28bc31f4b6a9302487a13e6baceab43a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2017-03-09T00:59:23Z","title_canon_sha256":"c60d2cfb5ca14271aeaf3dbc60eddf5b77289d39f965849d619c811b4f5c8b2d"},"schema_version":"1.0","source":{"id":"1703.03089","kind":"arxiv","version":1}},"canonical_sha256":"7114b33f2a997eae203f4442a5effa9b31ce0ac72e7ca5897c74762027cce7a6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7114b33f2a997eae203f4442a5effa9b31ce0ac72e7ca5897c74762027cce7a6","first_computed_at":"2026-05-18T00:49:01.608307Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:49:01.608307Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PHCb2aFkZKUoexOIX+3UK8Ze4Wb0TH24aW+QrFz/imf0ihJzatOAUIoY1wUKWpgo0KzascV0KQ1lIpc57WcUDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:49:01.609102Z","signed_message":"canonical_sha256_bytes"},"source_id":"1703.03089","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:50de2940fec4afc033c78a2e81f45b96e4d0861512f3a4d957209db9d49694ad","sha256:5641cc7cec31a4d530f5ab9d0c373a9e52fb0dfaa8603e6159f579983a346cc0"],"state_sha256":"117c46e31de379a44f76ff194c010110ec422581c95d813613286f8bdb697e07"}