Rearrangement inequalities on the lattice graph

The Polya–Szegő inequality in $R^n$ states that, given a nonnegative function $f:R^n\to R$, its spherically symmetric decreasing rearrangement $f^{\ast }:R^n\to R$ is ‘smoother’ in the sense of $\parallel \nabla f^{\ast }{\parallel }_{L^p}⩽{\parallel \nabla f\parallel }_{L^p}$ for all $1⩽p⩽\infty$. We study analogues on the lattice grid graph $Z^2$. 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 $1⩽p⩽\infty$. In particular, the Wang-Wang rearrangement satisfies $\parallel \nabla f^{\ast }{\parallel }_{L^p}⩽2^{1/p}{\parallel \nabla f\parallel }_{L^p}$ for all $1⩽p⩽\infty$. We also show the existence of (many) rearrangements on $Z^d$ such that $\parallel \nabla f^{\ast }{\parallel }_{L^p}⩽c_d·{\parallel \nabla f\parallel }_{L^p}$ for all $1⩽p⩽\infty$.