The Variation of the Fractional Maximal Function of a Radial Function

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 $q=n\mathrm{/}(n-\beta )$ and $f$ is a radial function, then $|DM_{\beta}f|{L^{q}($\mathbb{R}$^n)}\leq C(n,\beta)|Df|{L^{1}($\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}$^n)$.