Rearrangement Inequalities on the Lattice Graph

TL;DR

This paper investigates rearrangement inequalities on the lattice graph ^2, extending classical Polya-Szeg
53 inequalities to discrete settings and establishing bounds for various rearrangements across all p-norms.

Contribution

It develops a robust method to show that certain lattice rearrangements satisfy Polya-Szeg
53 inequalities up to a constant for all p, generalizing previous results.

Findings

Wang-Wang rearrangement satisfies  abla f^*_{L^p} q; 2^{1/p} \u00a0 abla f_{L^p} for all p.

Existence of multiple rearrangements on ^d with bounded  abla f^*_{L^p} norms.

Extension of inequalities to all p in [1, ] with explicit bounds.

Abstract

The Polya-Szeg\H{o} inequality in $\mathbb{R}^n$ states that, given a non-negative function $f:\mathbb{R}^{n} \rightarrow \mathbb{R}_{}$, its spherically symmetric decreasing rearrangement $f^*:\mathbb{R}^{n} \rightarrow \mathbb{R}_{}$ is `smoother' in the sense of $\| \nabla f^*\|_{L^p} \leq \| \nabla f\|_{L^p}$ for all $1 \leq p \leq \infty$. We study analogues on the lattice grid graph $\mathbb{Z}^2$. The spiral rearrangement is known to satisfy the Polya-Szeg\H{o} 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-Szeg\H{o} inequality up to a constant for all $1 \leq p \leq \infty$. In particular, the Wang-Wang rearrangement satisfies $\| \nabla f^*\|_{L^p} \leq 2^{1/p} \| \nabla f\|_{L^p}$ for all $1 \leq p \leq \infty$. We also show the existence of (many) rearrangements on $\mathbb{Z}^d$ such that $\| \nabla f^*\|_{L^p} \leq c_d \cdot \| \nabla f\|_{L^p}$ for all $1 \leq p \leq \infty$.