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Let $T_1=(-\\Delta+V)^{-1}V,\\ T_2=(-\\Delta+V)^{-1/2}V^{1/2}$ and $T_3=(-\\Delta+V)^{-1/2}\\nabla$. We obtain that $[b,T_j] (j=1,2,3)$ are bounded operators on $L^p(\\mathbb{R}^n)$ when $p$ ranges in a interval, where $b \\in \\mathbf{BMO}(\\mathbb{R}^n)$. Note that the kernel of $T_j (j=1,2,3)$ has no smoothness."},"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":"0802.3128","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2008-02-21T15:10:44Z","cross_cats_sorted":[],"title_canon_sha256":"f531a598caa691b3e31c8ecad4d0182397ad704efdc6b8393eceaa4f7b3b617a","abstract_canon_sha256":"d3d40b5c3834bd269b20f5d7ddfa740cfb676ffd04d05b2005ab39e6e14b0757"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:16:00.952877Z","signature_b64":"UA3l/mMhjZYMq85FnOARa7pfvHVeebTw71YPXePKlV9E2WeOuH7O7YOJk+sIm83agcgwx5LkECiUYqT8cWWqCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c05dc17eedace3db0a11e1d40ec3775ab20ef08e8b45dbc724338ef2c0b87a57","last_reissued_at":"2026-05-18T02:16:00.952491Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:16:00.952491Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$L^p$ Boundedness of Commutators of Riesz Transforms associated to Schr\\\"{o}dinger Operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Lizhong Peng, Pengtao Li, Zihua Guo","submitted_at":"2008-02-21T15:10:44Z","abstract_excerpt":"In this paper we consider $L^p$ boundedness of some commutators of Riesz transforms associated to Schr\\\"{o}dinger operator $P=-\\Delta+V(x)$ on $\\mathbb{R}^n, n\\geq 3$. 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