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

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 $p\in [1,+\mathrm{\infty }]$, 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.