  We provide a characterization of $\mathrm{BMO}$ in terms of endpoint
boundedness of commutators of singular integrals. In particular, in one
dimension, we show that $\|b\|_{\mathrm{BMO}}\eqsim B$, where $B$ is the best
constant in the endpoint $L\log L$ modular estimate for the commutator $[H,b]$.
We provide a similar characterization of the space $\mathrm{BMO}$ in terms of
endpoint boundedness of higher order commutators of the Hilbert transform. In
higher dimension we give the corresponding characterization of $\mathrm{BMO}$
in terms of the first order commutators of the Riesz transforms. We also show
that these characterizations can be given in terms of commutators of more
general singular integral operators of convolution type.
