On the continuous and discontinuous maximal operators

In the first part of this paper we study the regularity properties of a wide class of maximal operators. These results are used to show that the spherical maximal operator is continuous W1,p(Rn)↦W1,p(Rn), when p>nn−1. Other given applications include fractional maximal operators and maximal singular integrals. On the other hand, we show that the restricted Hardy-Littlewood maximal operator Mλ, where the supremum is taken over the cubes with radii greater than λ>0, is bounded from Lp(Rn) to W1,p(Rn) but discontinuous.