A note on Hardy-Littlewood maximal operators

In this paper, we will prove that, for $1< p<\infty$ , the $L^{p}$ norm of the truncated centered Hardy-Littlewood maximal operator $M^{c}_{\gamma}$ equals the norm of the centered Hardy-Littlewood maximal operator for all $0<\gamma<\infty$ . When $p=1$ , we also find that the weak $(1,1)$ norm of the truncated centered Hardy-Littlewood maximal operator $M^{c}_{\gamma}$ equals the weak $(1,1)$ norm of the centered Hardy-Littlewood maximal operator for $0<\gamma<\infty$ . 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.