On the regularity of maximal operators

We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps W-1,W-p(R) x W-1,W-q(R) -> W-1,W-r(R) with 1 < p, q < infinity and r >= 1, boundedly and continuously. The same result holds on R-n when r > 1. We also investigate the almost everywhere and weak convergence under the action of the classical Hardy-Littlewood maximal operator, both in its global and local versions.