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 and . More precisely, we prove that and map with , boundedly and continuously. In addition, we show that the discrete versions and map boundedly and map continuously. Specially, we obtain the sharp variation inequalities of and , that is, if , where is the total variation of on and is the set of all functions satisfying .