Failure of the Pólya-Szegő inequality for the fractional gradient

Abstract: We prove that the Pólya-Szegő inequality for the $L<sup>p$ norm of the Riesz fractional $s$-gradient fails for every $p\in[1,2)$ and $s\in(0,1)$, thus providing a negative answer to a question posed by Nguyen and Squassina in 2017.