The Hardy-Littlewood Maximal Function on Sobolev Space

In this paper, we discussed the Hardy-Littlewood maximal function on Sobolev space $W^{1,p}(R^n)$. We proved that the maximal function belongs to $W^{1,p}(R^n)$ and is bounded for $p>1$. Also, we proved that the derivative of maximal function is bounded for $p=1$.