Characterizations of the BMO and Lipschitz spaces via commutators on weak Lebesgue and Morrey spaces

We prove that the weak Morrey space $W{M}_q^p$ is contained in the Morrey space $M_{q_1}^p$ for $1\le q_1<q\le p<\mathrm{\infty }$. As applications, we show that if the commutator $[b,T]$ is bounded from $L^p$ to $L^{p,\mathrm{\infty }}$ for some $p\in (1,\mathrm{\infty })$, then $b\in \mathrm{B}\mathrm{M}\mathrm{O}$, where $T$ is a Calder'on-Zygmund operator. Also, for $1<p\le q<\mathrm{\infty }$, $b\in \mathrm{B}\mathrm{M}\mathrm{O}$ if and only if $[b,T]$ is bounded from $M_q^p$ to $W{M}_q^p$. For $b$ belonging to Lipschitz class, we obtain similar results.