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

Feng LiuSuzhen Mao — 2017

Czechoslovak Mathematical Journal

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 W 1 , p ( ) W 1 , p ( ) with 1 < p < , 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 ( M + ( f ) ) Var ( f ) and Var ( M - ( f ) ) Var ( f ) 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 ) < .

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