A characterization of BMO in terms of endpoint bounds for commutators of singular integrals

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.