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

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 $B{V}^{\alpha ,p}({\mathbb{R}}^n)$ 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 $B{V}^{\alpha ,p}$ function.

arXiv comment32 pages

References in corpus (6)

• A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up
• A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I
• An $L^1$-type estimate for Riesz potentials
• A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II
• Regularity for a fractional p-Laplace equation
• Leibniz rules and Gauss-Green formulas in distributional fractional spaces

Cited by in corpus (3)

• Leibniz rules and Gauss-Green formulas in distributional fractional spaces
• Extending linear growth functionals to functions of bounded fractional variation
• Fractional divergence-measure fields, Leibniz rule and Gauss-Green formula