Fractional Medians and Their Maximal Functions

Abstract

In this article, we introduce the fractional medians, provide a representation for the set of all fractional medians in terms of non-increasing rearrangements, and investigate the mapping properties of the fractional maximal operators defined by these medians. Our maximal operator is a generalization of the one introduced by Strömberg (Indiana Univ Math J 28(3):511–544, 1979). It turns out that our maximal operator is smoother than the usual fractional maximal operator. Furthermore, we provide an alternative proof of the embedding from BV to \(L^{n/(n-1),1}\) due to Alvino (Boll Un Mat Ital A 14(1):148–156, 1977) by using the usual medians.