Kinnunen and Lindqvist proved in  that the local Hardy-Littlewood maximal function, defined in an open set Ω in the Euclidean space, is a bounded operator from W1,p(Ω) to W1,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.
|Original language||English (US)|
|Number of pages||10|
|Journal||Annales Academiae Scientiarum Fennicae Mathematica|
|State||Published - May 10 2004|
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