Endpoint Sobolev Regularity of Multilinear Maximal Operators

In this paper, we establish some new endpoint regularity properties of multilinear maximal operators and their fractional variants, both in the centered and uncentered cases. The main results we obtain not only answer a question of Liu, Wu, Xue and Yabuta, but also provide some new endpoint Sobolev boundedness and continuity results for multilinear fractional maximal operators. More importantly, we present a new approach to deal with the endpoint Sobolev continuity for the above operators.