{"paper":{"title":"Quasi-Banach estimates of commutators of bilinear bi-parameter singular integrals: paraproducts","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":"2018-06-25T12:18:41Z","abstract_excerpt":"We complete our boundedness theory of commutators of bilinear bi-parameter singular integrals by establishing the following result. If $T$ is a bilinear bi-parameter singular integral satisfying suitable $T1$ type assumptions, $\\|b\\|_{\\operatorname{bmo}(\\mathbb{R}^{n+m})} = 1$ and $1 < p, q \\le \\infty$ and $1/2 < r < \\infty$ satisfy $1/p+1/q = 1/r$, then we have $$ \\|[b, T]_1(f_1, f_2)\\|_{L^r(\\mathbb{R}^{n+m})} \\lesssim \\|f_1\\|_{L^p(\\mathbb{R}^{n+m})} \\|f_2\\|_{L^q(\\mathbb{R}^{n+m})}. $$ Previously the range $r \\le 1$ was proved only in the paraproduct free situation. The main novelty lies in t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.10085","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"}