{"paper":{"title":"Commutators in the Two-Weight Setting","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Brett D. Wick, Irina Holmes, Michael T. Lacey","submitted_at":"2015-06-18T16:46:24Z","abstract_excerpt":"Let $R$ be the vector of Riesz transforms on $\\mathbb{R}^n$, and let $\\mu,\\lambda \\in A_p$ be two weights on $\\mathbb{R}^n$, $1 < p < \\infty$. The two-weight norm inequality for the commutator $[b, R] : L^p(\\mathbb{R}^n;\\mu) \\to L^p(\\mathbb{R}^n;\\lambda)$ is shown to be equivalent to the function $b$ being in a BMO space adapted to $\\mu$ and $\\lambda$. This is a common extension of a result of Coifman-Rochberg-Weiss in the case of both $\\lambda$ and $\\mu$ being Lebesgue measure, and Bloom in the case of dimension one."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.05747","kind":"arxiv","version":4},"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"}