Leibniz rules and Gauss-Green formulas in distributional fractional spaces

Abstract

We apply the results established in arXiv:2109.15263 to prove some new fractional Leibniz rules involving $B{V}^{\alpha ,p}$ and $S^{\alpha ,p}$ functions, following the distributional approach adopted in the previous works arXiv:1809.08575, arXiv:1910.13419, arXiv:2011.03928. 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 $2\alpha$-order fractional elliptic operator in divergence form.

References in corpus (3)

• A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II
• Regularity for a fractional p-Laplace equation
• The fractional variation and the precise representative of $B{V}^{\alpha ,p}$ functions

Cited by in corpus (6)

• The fractional variation and the precise representative of $B{V}^{\alpha ,p}$ functions
• Failure of the local chain rule for the fractional variation
• The divergence theorem and nonlocal counterparts
• Extending linear growth functionals to functions of bounded fractional variation
• Fractional divergence-measure fields, Leibniz rule and Gauss-Green formula
• $H$-compactness for nonlocal linear operators in fractional divergence form