Characterizations of the BMO and Lipschitz Spaces via Commutators on Weak Lebesgue and Morrey Spaces

We prove that the weak Morrey space WMqp is contained in the Morrey space Mq1p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{{q_1}}^p$$\end{document} for 1 ≤ q1 < q ≤ p < ∞. As applications, we show that if the commutator [b, T] is bounded from Lp to Lp,∞ for some p ∈ (1, ∞), then b ∈ BMO, where T is a Calderón-Zygmund operator. Also, for 1 < p ≤ q < ∞, b ∈ BMO if and only if [6, T] is bounded from Mqp to WMqp. For b belonging to Lipschitz class, we obtain similar results.