Abstract
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 l 1(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.
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