Rearrangement Inequalities on the Lattice Graph

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 $1 \leq p \leq \infty$.