Failure of the local chain rule for the fractional variation

TL;DR

This paper demonstrates that the local chain rule fails for fractional variation, revealing surprising rigidity properties and refining related inequalities, with specific results in one-dimensional cases and implications for fractional Hardy inequalities.

Contribution

It proves the failure of the local chain rule for fractional variation and establishes new rigidity results and refined inequalities in the fractional setting.

Findings

Counterexample for the local chain rule in fractional variation

Stronger results for one-dimensional functions in BV^α

Refinement of fractional Hardy and Meyers-Ziemer trace inequalities

Abstract

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.