On the regularity of the one-sided Hardy-Littlewood maximal functions

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 (?) → W ^1,p (?) with 1 < p < 1, boundedly and continuously. In addition, we show that the discrete versions M ⁺ and M ^? map BV(?) → BV(?) boundedly and map l ¹(?) → BV(?) 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(?), where Var(f) is the total variation of f on ? and BV(?) is the set of all functions f: ? → ? satisfying Var(f) < 1.