Fractional medians and their maximal functions

In this article, we introduce the fractional medians, give an expression of the set of all fractional medians in terms of non-increasing rearrangements and then investigate mapping properties of the fractional maximal operators defined by such medians. The maximal operator is a generalization of that in Stromberg. It turns out that our maximal operator is a more smooth operator than the usual fractional maximal operator. Further, we give another proof of the embedding from $BV$ to $L^{n\mathrm{/}(n-1),1}$ due to Alvino by using the usual medians.