A note on Hardy-Littlewood maximal operators

Abstract

Abstract In this paper, we will prove that, for 1 < p < ∞ $1< p<\infty$ , the L p $L^{p}$ norm of the truncated centered Hardy-Littlewood maximal operator M γ c $M^{c}_{\gamma}$ equals the norm of the centered Hardy-Littlewood maximal operator for all 0 < γ < ∞ $0<\gamma<\infty$ . When p = 1 $p=1$ , we also find that the weak ( 1 , 1 ) $(1,1)$ norm of the truncated centered Hardy-Littlewood maximal operator M γ c $M^{c}_{\gamma}$ equals the weak ( 1 , 1 ) $(1,1)$ norm of the centered Hardy-Littlewood maximal operator for 0 < γ < ∞ $0<\gamma<\infty$ . Moreover, the same is true for the truncated uncentered Hardy-Littlewood maximal operator. Finally, we investigate the properties of the iterated Hardy-Littlewood maximal function.

Keywords

• Hardy-Littlewood maximal function
• truncated operator
• L p $L^{p}$ norm
• weak ( 1 , 1 ) $(1,1)$ norm
• iterated Hardy-Littlewood maximal function