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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 $WM_{q}^{p}$. 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