Endpoint Sobolev and BV continuity for maximal operators

In this paper we investigate some questions related to the continuity of maximal operators in W-1,W-1 and BV spaces, complementing some well-known boundedness results. Letting (M) over tilde be the one-dimensional uncentered Hardy-Littlewood maximal operator, we prove that the map f -> ((M) over tilde f)' is continuous from W-1,W-1(R) to L-1(H). In the discrete setting, we prove that (M) over tilde : BV (G) -> BV(Z) is also continuous. For the one-dimensional fractional Hardy-Littlewood maximal operator, we prove by means of counterexamples that the corresponding continuity statements do not hold, both in the continuous and discrete settings, and for the centered and uncentered versions. (C) 2017 Elsevier Inc. All rights reserved.