On boundedness of maximal functions in Sobolev spaces

Kinnunen and Lindqvist proved in [9] that the local Hardy-Littlewood maximal function, defined in an open set Ω in the Euclidean space, is a bounded operator from W^1,p(Ω) to W^1,p (Ω), provided p > 1. In this note we give a simpler argument which leads to a slightly more general result. Furthermore, the argument also allows us to prove boundedness of the spherical maximal function in the Sobolev space W ^1,p(Ω), p > n/(n - 1). We also raise many questions concerning boundedness of maximal operators in Sobolev spaces.