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 $M_{\beta }$. As the main result, we prove that there exists $C(n,\beta )$ such that if $q=n/(n-\beta )$ and $f$ is a radial function, then $\Vert D{M}_{\beta }f{\Vert }_{L^q({\mathbb{R}}^n)}\le C(n,\beta )\Vert Df{\Vert }_{L^1({\mathbb{R}}^n)}$. The corresponding result was previously known only if $n=1$ or $\beta =0$. 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