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

Abstract

Given $α\in(0,1]$ and $p\in[1,+\infty]$, we define the space $\mathscr{DM}^{α,p}(\mathbb R^n)$ of $L^p$ vector fields whose $α$-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 $α$-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.