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

Given α∈(0,1] and p∈[1,+∞], we define the space DMα,p(Rn) of Lp 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.