{"paper":{"title":"Weak type commutator and Lipschitz estimates: resolution of the Nazarov-Peller conjecture","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Denis Potapov, Dmitriy Zanin, Fedor Sukochev, Martijn Caspers","submitted_at":"2015-06-02T07:32:13Z","abstract_excerpt":"Let $\\mathcal{M}$ be a semi-finite von Neumann algebra and let $f: \\mathbb{R} \\rightarrow \\mathbb{C}$ be a Lipschitz function. If $A,B\\in\\mathcal{M}$ are self-adjoint operators such that $[A,B]\\in L_1(\\mathcal{M}),$ then $$\\|[f(A),B]\\|_{1,\\infty}\\leq c_{abs}\\|f'\\|_{\\infty}\\|[A,B]\\|_1,$$ where $c_{abs}$ is an absolute constant independent of $f$, $\\mathcal{M}$ and $A,B$ and $\\|\\cdot\\|_{1,\\infty}$ denotes the weak $L_1$-norm. If $X,Y\\in\\mathcal{M}$ are self-adjoint operators such that $X-Y\\in L_1(\\mathcal{M}),$ then $$\\|f(X)-f(Y)\\|_{1,\\infty}\\leq c_{abs}\\|f'\\|_{\\infty}\\|X-Y\\|_1.$$ This result re"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.00778","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"}