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.
Categories: math.FA