Leibniz rules and Gauss–Green formulas in distributional fractional spaces

Abstract

We apply the results established in [12] to prove some new fractional Leibniz rules involving BVα,p and Sα,p functions, following the distributional approach adopted in the previous works [8,13,14]. In order to achieve our main results, we revise the elementary properties of the fractional operators involved in the framework of Besov spaces and we rephraze the Kenig–Ponce–Vega Leibniz-type rule in our fractional context. We apply our results to prove the well-posedness of the boundary-value problem for a general 2α-order fractional elliptic operator in divergence form.