ON BOUNDEDNESS OF MAXIMAL FUNCTIONS IN SOBOLEV SPACES

Kinnunen and Lindqvist proved in (9) that the local Hardy{Littlewood maximal function, dened in an open set in the Euclidean space, is a bounded operator from W 1;p () to W 1;p () , 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 () , p > n=(n 1). We also raise many questions concerning boundedness of maximal operators in Sobolev spaces.