{"paper":{"title":"Bloom type inequality for bi-parameter singular integrals: efficient proof and iterated commutators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Emil Vuorinen, Henri Martikainen, Kangwei Li","submitted_at":"2018-06-07T15:57:39Z","abstract_excerpt":"Utilising some recent ideas from our bilinear bi-parameter theory, we give an efficient proof of a two-weight Bloom type inequality for iterated commutators of linear bi-parameter singular integrals. We prove that if $T$ is a bi-parameter singular integral satisfying the assumptions of the bi-parameter representation theorem, then $$ \\| [b_k,\\cdots[b_2, [b_1, T]]\\cdots]\\|_{L^p(\\mu) \\to L^p(\\lambda)} \\lesssim_{[\\mu]_{A_p}, [\\lambda]_{A_p}} \\prod_{i=1}^k\\|b_i\\|_{\\operatorname{bmo}(\\nu^{\\theta_i})} , $$ where $p \\in (1,\\infty)$, $\\theta_i \\in [0,1]$, $\\sum_{i=1}^k\\theta_i=1$, $\\mu, \\lambda \\in A_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.02742","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"}