The Variation of the Fractional Maximal Function of a Radial Function

TL;DR

This paper establishes a new regularity estimate for the fractional maximal operator applied to radial functions in higher dimensions, extending previous one-dimensional results and introducing a potentially broadly applicable proof technique.

Contribution

The paper proves a higher-dimensional regularity estimate for the fractional maximal operator on radial functions, generalizing known one-dimensional results and offering a new proof approach.

Findings

Established a bound for the fractional maximal operator on radial functions in 9 dimensions.

Extended the regularity results from one-dimensional to higher-dimensional cases.

Provided a proof technique that minimizes reliance on one-dimensional arguments.

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 $\|DM_{\beta}f\|_{L^{q}(\mathbb{R}^n)}\leq C(n,\beta)\|Df\|_{L^{1}(\mathbb{R}^n)}$. 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 $f\in W^{1,1}(\mathbb{R}^n)$.