Mathematics

Abstract: Differently from their integer versions, the fractional Sobolev spaces $W^{\alpha,p}(\mathbb{R}^n)$ do not seem to have a clear distributional nature. By exploiting suitable notions of fractional gradient and divergence already existing in the literature, we introduce via a distributional approach the new spaces $BV^{\alpha,p}(\mathbb{R}^n)$ of $L^p$ functions with bounded $\alpha$-fractional variation in $\mathbb{R}^n$, for $\alpha \in (0,1)$ and $p \in [1, \infty]$. In addition, we define in a similar way the distributional fractional Sobolev spaces $S^{\alpha,p}(\mathbb{R}^n)$, which extend naturally $W^{\alpha,p}(\mathbb{R}^n)$.In this talk, we shall focus on the absolute continuity property of the fractional variation with respect to a suitable Hausdorff measure and on the existence of precise representatives for $BV^{\alpha,p}$ functions. Subsequently, we shall derive fractional Leibniz rules and Gauss-Green formulas involving distributional fractional Sobolev and $BV$ functions. As an application of these results, we will give an alternative proof of the fractional Hardy inequality.