Sobolev regularity of polar fractional maximal functions

Regularity of the centered fractional maximal function on radial functions

“…It is noted that the main result in Theorem 1.2 is the endpoint Sobolev bound (1.3). The continuity statement can be deduced via a similar scheme to the one used by the authors [2] in the non-centered case, together with a new idea recently introduced by González-Riquelme [8]. Interestingly, he obtained endpoint Sobolev results for a version of M β f defined on the sphere S d−1 if f is polar; see also [5] for similar results when β = 0.…”

“…As mentioned in the Introduction, the proof follows the strategy used by the authors [2] in the noncentered case, together with a new idea recently introduced by González-Riquelme [8] that allows to obtain smallness of ∇M β f j a small ball around the origin. In what follows we put this strategy in action; see also [8,Theorem 25] for a similar approach.…”

“…We next show that there exists a small neighbourhood around the origin for which there is convergence. To this end, we use an idea recently introduced by González-Riquelme in [8]. Lemma 6.7.…”