{"paper":{"title":"Endpoint estimates for commutators of singular integrals related to Schr\\\"odinger operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Luong Dang Ky","submitted_at":"2012-03-28T18:51:10Z","abstract_excerpt":"Let $L= -\\Delta+ V$ be a Schr\\\"odinger operator on $\\mathbb R^d$, $d\\geq 3$, where $V$ is a nonnegative potential, $V\\ne 0$, and belongs to the reverse H\\\"older class $RH_{d/2}$. In this paper, we study the commutators $[b,T]$ for $T$ in a class $\\mathcal K_L$ of sublinear operators containing the fundamental operators in harmonic analysis related to $L$. More precisely, when $T\\in \\mathcal K_L$, we prove that there exists a bounded subbilinear operator $\\mathfrak R= \\mathfrak R_T: H^1_L(\\mathbb R^d)\\times BMO(\\mathbb R^d)\\to L^1(\\mathbb R^d)$ such that $|T(\\mathfrak S(f,b))|- \\mathfrak R(f,b)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.6335","kind":"arxiv","version":3},"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"}