Regularity of the centered fractional maximal function on radial functions

We study the regularity properties of the centered fractional maximal function Mβ. More precisely, we prove that the map f↦|∇Mβf| is bounded and continuous from W1,1(Rd) to Lq(Rd) in the endpoint case q=d/(d−β) if f is a radial function. For d=1, the radiality assumption can be removed. This corresponds to the counterparts of known results for the non-centered fractional maximal function. The main new idea consists in relating the centered and non-centered fractional maximal function at the derivative level.