Continuity of the gradient of the fractional maximal operator on W^1,1(R^d)

We establish that the map f -> | del M(alpha)f | is continuous from W-1,W-1(R-d) to L-q(R-d), where alpha is an element of (0, d), q =( d) (d-alpha) and M-alpha denotes either the centered or non-centered fractional Hardy-Littlewood maximal operator. In particular, we cover the cases d > 1 and alpha is an element of (0, 1) in full generality, for which results were only known for radial functions.