Fractional divergence-measure fields, Leibniz rule and Gauss-Green formula

Given $\alpha \in (0,1]$ and $p\in [1,+\mathrm{\infty }]$, we define the space ${\mathcal{D}\mathcal{M}}^{\alpha ,p}({\mathbb{R}}^n)$ of $L^p$ vector fields whose $\alpha$-divergence is a finite Radon measure, extending the theory of divergence-measure vector fields to the distributional fractional setting. Our main results concern the absolute continuity properties of the $\alpha$-divergence-measure with respect to the Hausdorff measure and fractional analogues of the Leibniz rule and the Gauss-Green formula. The sharpness of our results is discussed via some explicit examples.