Leibniz rules and Gauss-Green formulas in distributional fractional spaces

TL;DR

This paper extends fractional Leibniz rules and Gauss-Green formulas to distributional fractional spaces, providing new tools for fractional PDE analysis and boundary-value problems.

Contribution

It introduces new fractional Leibniz rules for $BV^{eta,p}$ and $S^{eta,p}$ functions within a distributional framework, revising properties of fractional operators in Besov spaces.

Findings

Established new fractional Leibniz rules for distributional fractional spaces.

Revised elementary properties of fractional operators in Besov spaces.

Proved well-posedness for boundary-value problems of fractional elliptic operators.

Abstract

We apply the results established in arXiv:2109.15263 to prove some new fractional Leibniz rules involving $BV^{\alpha,p}$ 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.