Improvement to characterizations of BMO in terms of endpoint bounds for commutators and some counterexamples

In this paper, we extend the technique of Lerner, Ombrosi and Rivera-Ríos (Bull. Lond. Math. Soc. 51(1): 107--119, 2019) for the necessity of the strong-type boundedness of commutators to the endpoint case, and give characterizations of BMO via the endpoint bounds for commutators of singular integrals with rough kernels and related operators, which improve the conditions of Accomazzo (Israel J. Math. 228(2): 787--800, 2018), Guo et al. (J. Geom. Anal. 30(4): 3995-4035, 2020). We also give a characterization of BMO via the endpoint bounds for commutators of non-degenerate Calderón-Zygmund operators by removing the real-valued restriction on pointwise multiplier, which extend the results of Hytönen (J. Math. Pures Appl. (9) 156: 351--391, 2021) to the endpoint case. Finally, we find counterexamples to illustrate that weighted endpoint estimate of commutator of fractional integral operator does not necessarily imply b∈ BMO, and two-weighted endpoint estimate of Bloom type for commutator of Hilbert transform does not hold, which are missing for a long time.