Fractional medians and their maximal functions

TL;DR

This paper introduces fractional medians, explores their properties, and studies the associated maximal operators, revealing their smoother nature and providing new proofs for known embeddings.

Contribution

It defines fractional medians, analyzes their maximal functions, and demonstrates their smoother behavior compared to traditional fractional maximal operators.

Findings

Fractional medians are characterized via non-increasing rearrangements.

The fractional maximal operator based on medians is smoother than the classical one.

A new proof of the embedding from BV to Lorentz spaces is provided.

Abstract

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 $L^{n/(n-1),1}$ due to Alvino by using the usual medians.