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.

We establish the bounds of Marcinkiewicz integrals associated to surfaces of revolution generated by two polynomial mappings on Triebel–Lizorkin spaces and Besov spaces when their integral kernels are given by functions $\unicode[STIX]{x1D6FA}\in H^{1}(\text{S}^{n-1})\cup L(\log ^{+}L)^{1/2}(\text{S}^{n-1})$ . Our main results represent improvements as well as natural extensions of many previously known results.

In this paper we study the regularity properties of the one-dimensional one-sided Hardy-Littlewood maximal operators \(\mathcal{M}^+\) and \(\mathcal{M}^-\). More precisely, we prove that \(\mathcal{M}^+\) and \(\mathcal{M}^-\) map W^1,p(ℝ) → W^1,p(ℝ) with 1 < p < 1, boundedly and continuously. In addition, we show that the discrete versions M⁺ and M⁻ map BV(ℤ) → BV(ℤ) boundedly and map l¹(ℤ) → BV(ℤ) continuously. Specially, we obtain the sharp variation inequalities of M⁺ and M⁻, that is $$Var\left( {{M^ + }\left( f \right)} \right) \leqslant Var\left( f \right)andVar\left( {{M^ - }\left( f \right)} \right) \leqslant Var\left( f \right)$$ if f ∈ BV(ℤ), where Var(f) is the total variation of f on ℤ and BV(ℤ) is the set of all functions f: ℤ → ℝ satisfying Var(f) < 1.