Failure of the local chain rule for the fractional variation

Abstract

We prove that the local version of the chain rule cannot hold for the fractional variation defined in in our previous article (2019). In the case $n=1$, we prove a stronger result, exhibiting a function $f\in B{V}^{\alpha }(\mathbb{R})$ such that $|f|\notin B{V}^{\alpha }(\mathbb{R})$. The failure of the local chain rule is a consequence of some surprising rigidity properties for non-negative functions with bounded fractional variation which, in turn, are derived from a fractional Hardy inequality localized to half-spaces. Our approach exploits the distributional techniques developed in our previous works (2019–2022). As a byproduct, we refine the fractional Hardy inequality obtained in works of Shieh and Spector (2018) and Spector (2020) and we prove a fractional version of the closely related Meyers–Ziemer trace inequality.