On a generalization of the Hörmander condition

We consider a natural generalization of the classical Hörmander condition in the Calderón–Zygmund theory. Recently the author [J. Fourier Anal. Appl. 27 (2021)] proved the L p L^p boundedness of singular integral operators under the L 1 L^1 mean Hörmander condition, which was originally introduced by Grafakos and Stockdale [Bull. Hellenic Math. Soc. 63 (2019), pp. 54–63]. In this paper, we show that the L 1 L^1 mean condition actually coincides with the classical one. On the other hand, we introduce a new variant of the Hörmander condition, which is strictly weaker than the classical one but still enough for the L p L^p boundedness. Moreover, it still works in the non-doubling setting with a little modification.