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

We continue the study of the fractional variation following the distributional approach developed in the previous works Brue et al. (2021), Comi and Stefani (2019), Comi and Stefani (2019). We provide a general analysis of the distributional space BV alpha,p(R-n) of L-p functions, with p is an element of [1, +infinity], possessing finite fractional variation of order alpha is an element of (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.