Failure of the local chain rule for the fractional variation

We prove that the local version of the chain rule cannot hold for the fractional variation defined in arXiv:1809.08575. In the case $n = 1$, we prove a stronger result, exhibiting a function $f \in BV^{\alpha}(\mathbb{R})$ such that $|f| \notin BV^{\alpha}(\mathbb{R})$. 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 results of arXiv:2111.13942 and the distributional approach of the previous papers arXiv:1809.08575, arXiv:1910.13419, arXiv:2011.03928, arXiv:2109.15263. As a byproduct, we refine the fractional Hardy inequality obtained in arXiv:1611.07204, arXiv:1806.07588 and we prove a fractional version of the closely related Meyers-Ziemer trace inequality.