The fractional variation and the precise representative of $BV^{\alpha,p}$ functions

TL;DR

This paper advances the understanding of fractional variation in $BV^{eta,p}$ functions, establishing their absolute continuity and the existence of a precise representative, thus deepening the mathematical theory of fractional BV spaces.

Contribution

It provides a comprehensive analysis of the distributional space $BV^{eta,p}$, including absolute continuity and the precise representative, extending previous fractional variation studies.

Findings

Fractional variation is absolutely continuous with respect to Hausdorff measure.

Existence of a precise representative for $BV^{eta,p}$ functions.

General analysis of $BV^{eta,p}$ functions in $ ^n$.

Abstract

We continue the study of the fractional variation following the distributional approach developed in the previous works arXiv:1809.08575, arXiv:1910.13419 and arXiv:2011.03928. We provide a general analysis of the distributional space $BV^{\alpha,p}(\mathbb{R}^n)$ of $L^p$ functions, with $p\in[1,+\infty]$, possessing finite fractional variation of order $\alpha\in(0,1)$. Our two main results deal with the absolute continuity property of the fractional variation with respect to the Hausdorff measure and the existence of the precise representative of a $BV^{\alpha,p}$ function.