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

Abstract

We prove that the weak Morrey space $W{M}_q^p$ is contained in the Morrey space $M_{q_1}^p$ for $1\le q_1<q\le p<\infty$. 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 Calderón-Zygmund operator. Also, for $1<p\le q<\infty$, $b\in \mathrm{BMO}$ 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.

9 pages