Rearrangement Inequalities on the Lattice Graph

Abstract

The Polya-Szegő inequality in ${\mathbb{R}}^n$ states that, given a non-negative function $f:{\mathbb{R}}^n\to {\mathbb{R}}_{}$, its spherically symmetric decreasing rearrangement $f^{\ast }:{\mathbb{R}}^n\to {\mathbb{R}}_{}$ is `smoother' in the sense of $\Vert \nabla f^{\ast }{\Vert }_{L^p}\le \Vert \nabla f{\Vert }_{L^p}$ for all $1\le p\le \infty$. We study analogues on the lattice grid graph ${\mathbb{Z}}^2$. The spiral rearrangement is known to satisfy the Polya-Szegő inequality for $p=1$, the Wang-Wang rearrangement satisfies it for $p=\infty$ and no rearrangement can satisfy it for $p=2$. We develop a robust approach to show that both these rearrangements satisfy the Polya-Szegő inequality up to a constant for all $1\le p\le \infty$. In particular, the Wang-Wang rearrangement satisfies $\Vert \nabla f^{\ast }{\Vert }_{L^p}\le 2^{1/p}\Vert \nabla f{\Vert }_{L^p}$ for all $1\le p\le \infty$. We also show the existence of (many) rearrangements on ${\mathbb{Z}}^d$ such that $\Vert \nabla f^{\ast }{\Vert }_{L^p}\le c_d\cdot \Vert \nabla f{\Vert }_{L^p}$ for all $1\le p\le \infty$.