The Variation of the Fractional Maximal Function of a Radial Function

Abstract

In this paper we study the regularity of the non-centered fractional maximal operator $M_β$. As the main result, we prove that there exists $C(n,β)$ such that if $q=n/(n-β)$ and $f$ is a radial function, then $\|DM_βf\|_{L^{q}(\mathbb{R}^n)}\leq C(n,β)\|Df\|_{L^{1}(\mathbb{R}^n)}$. The corresponding result was previously known only if $n=1$ or $β=0$. Our proofs are almost free from one-dimensional arguments. Therefore, we believe that the new approach may be very useful when trying to extend the result for all $f\in W^{1,1}(\mathbb{R}^n)$.

14 pages.To appear in IMRN