We continue the study of the fractional variation following the distributional approach developed in the previous works Bruè et al. (2021), Comi and Stefani (2019), Comi and Stefani (2019). 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.

我们继续按照之前的作品 Bruè 等人开发的分布方法研究分数变化。(2021)、科米和史蒂芬尼 (2019)、科米和史蒂芬尼 (2019)。我们提供了对 \(L^p\) 函数的分布空间\(BV^{\alpha ,p}({\mathbb {R}}^n)\) 的一般分析，其中\ (p\in [1 ,+\infty ]\)，具有阶\(\alpha \in (0,1)\)的有限分数变化。我们的两个主要结果涉及相对于 Hausdorff 测度的分数变分的绝对连续性和\(BV^{\alpha ,p}\)函数的精确表示的存在。