{"paper":{"title":"Characterizations of the BMO and Lipschitz spaces via commutators on weak Lebesgue and Morrey spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Dinghuai Wang, Jiang Zhou, Wenyi Chen","submitted_at":"2016-12-28T07:43:54Z","abstract_excerpt":"We prove that the weak Morrey space $WM^{p}_{q}$ is contained in the Morrey space $M^{p}_{q_{1}}$ for $1\\leq q_{1}< q\\leq p<\\infty$. As applications, we show that if the commutator $[b,T]$ is bounded from $L^p$ to $L^{p,\\infty}$ for some $p\\in (1,\\infty)$, then $b\\in \\mathrm{BMO}$, where $T$ is a Calder\\'on-Zygmund operator. Also, for $1<p\\leq q<\\infty$, $b\\in \\mathrm{BMO}$ if and only if $[b,T]$ is bounded from $M^{p}_{q}$ to $WM_{q}^{p}$. For $b$ belonging to Lipschitz class, we obtain similar results."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.08819","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"}