{"paper":{"title":"Weighted Estimates for Multilinear Commutators of Marcinkiewicz Integrals with Bounded Kernel","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Jianglong Wu, Qingguo Liu","submitted_at":"2013-04-16T13:28:32Z","abstract_excerpt":"Let $\\mu_{\\Omega,\\vec{b}}$ be the multilinear commutator generalized by $\\mu_{\\Omega}$, the $n$-dimensional Marcinkiewicz integral with the bounded kernel, and $b_{j}\\in \\Osc_{\\exp L^{r_{j}}}(1\\le j\\le m)$. In this paper, the following weighted inequalities are proved for $\\omega\\in A_{\\infty}$ and $0<p<\\infty$,\n  $$\\|\\mu_{\\Omega}(f)\\|_{L^{p}(\\omega)}\\leq C\\|M(f)\\|_{L^{p}(\\omega)}, \\ \\ \\|\\mu_{\\Omega,\\vec{b}}(f)\\|_{L^{p}(\\omega)}\\leq C\\|M_{L(\\log L)^{1/r}}(f)\\|_{L^{p}(\\omega)}.$$\n  The weighted weak $L(\\log L)^{1/r}$ -type estimate is also established when $p=1$ and $\\omega\\in A_{1}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.4434","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"}