Abstract: In the paper, we investigate the iterated non-centered Hardy-Littlewood maximal function and the iterated centered Hardy-Littlewood maximal function. We prove that the limit of the iterated maximal function is just a fixed point of maximal operator. As an application of the fixed point theory, we finally obtain that the fixed point is||f||_∞ for non-centered Hardy-Littlewood maximal operator. The same is true for the centered Hardy-Littlewood maximal operator only for n=1,2.

Key words: Hardy-Littlewood maximal function, fixed point, iterated Hardy-Littlewood maximal function