In this note we give a simple proof of the endpoint regularity for the uncentred Hardy–Littlewood maximal function on $\mathbb{R}$ . Our proof is based on identities for the local maximum points of the corresponding maximal functions, which are of interest in their own right.

The index groups for which weighted Young inequalities hold in both the continuous case and discrete case are characterized. As applications, the index groups for the product inequalities on modulation spaces are characterized, and we also obtain the weak

This paper gives a criterion on the weighted norm estimates of the oscillatory and variation operators for the commutators of Calderón–Zygmund singular integrals in dimension 1. As applications, the weighted L^p -boundedness of the oscillation operators and the ρ-variation operators for commutators of the Hilbert transform and the Hermitian Riesz transform are obtained. In addition, the corresponding results for the λ-jump operators and the numbers of up-crossings associated with these operators are also given.