Pólya–Szegő inequality for the Riesz fractional gradient when p>2

Determine whether the Pólya–Szegő inequality for the L^p norm of the Riesz fractional s-gradient, namely whether every non-negative compactly supported smooth function u satisfies \(\int_{\mathbb{R}^N}|\nabla^s u^\star|^p\,dx\leq\int_{\mathbb{R}^N}|\nabla^s u|^p\,dx\), holds for \(p>2\) and \(s\in(0,1)\).

Background

The paper studies the analogue of the classical Pólya–Szegő rearrangement inequality for the Riesz fractional gradient. It establishes that the inequality fails for every $p\in[1,2)$ and every $s\in(0,1)$, while the case $p=2$ is known to hold because the squared fractional-gradient norm is equivalent to the fractional $W^{s,2}$ seminorm and the classical fractional Pólya–Szegő inequality applies.

Consequently, the unresolved range identified by the paper is the range $p>2$. Determining whether symmetric decreasing rearrangement decreases the Riesz fractional-gradient energy in that range remains an open problem.